mmdefinitions - Metamath Proof Explorer

Metamath Proof Explorer

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**List of Syntax, Axioms (ax-) and Definitions (df-)**
Ref	Expression (see link for any distinct variable requirements)
wn 3	wff ¬ 𝜑
wi 4	wff (𝜑 → 𝜓)
ax-mp 5	⊢ 𝜑 & ⊢ (𝜑 → 𝜓) ⇒ ⊢ 𝜓
ax-1 6	⊢ (𝜑 → (𝜓 → 𝜑))
ax-2 7	⊢ ((𝜑 → (𝜓 → 𝜒)) → ((𝜑 → 𝜓) → (𝜑 → 𝜒)))
ax-3 8	⊢ ((¬ 𝜑 → ¬ 𝜓) → (𝜓 → 𝜑))
wb 206	wff (𝜑 ↔ 𝜓)
df-bi 207	⊢ ¬ (((𝜑 ↔ 𝜓) → ¬ ((𝜑 → 𝜓) → ¬ (𝜓 → 𝜑))) → ¬ (¬ ((𝜑 → 𝜓) → ¬ (𝜓 → 𝜑)) → (𝜑 ↔ 𝜓)))
wa 395	wff (𝜑 ∧ 𝜓)
df-an 396	⊢ ((𝜑 ∧ 𝜓) ↔ ¬ (𝜑 → ¬ 𝜓))
wo 847	wff (𝜑 ∨ 𝜓)
df-or 848	⊢ ((𝜑 ∨ 𝜓) ↔ (¬ 𝜑 → 𝜓))
wif 1062	wff if-(𝜑, 𝜓, 𝜒)
df-ifp 1063	⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒)))
w3o 1085	wff (𝜑 ∨ 𝜓 ∨ 𝜒)
w3a 1086	wff (𝜑 ∧ 𝜓 ∧ 𝜒)
df-3or 1087	⊢ ((𝜑 ∨ 𝜓 ∨ 𝜒) ↔ ((𝜑 ∨ 𝜓) ∨ 𝜒))
df-3an 1088	⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ((𝜑 ∧ 𝜓) ∧ 𝜒))
wnan 1492	wff (𝜑 ⊼ 𝜓)
df-nan 1493	⊢ ((𝜑 ⊼ 𝜓) ↔ ¬ (𝜑 ∧ 𝜓))
wxo 1512	wff (𝜑 ⊻ 𝜓)
df-xor 1513	⊢ ((𝜑 ⊻ 𝜓) ↔ ¬ (𝜑 ↔ 𝜓))
wnor 1529	wff (𝜑 ⊽ 𝜓)
df-nor 1530	⊢ ((𝜑 ⊽ 𝜓) ↔ ¬ (𝜑 ∨ 𝜓))
wal 1539	wff ∀𝑥𝜑
cv 1540	class 𝑥
wceq 1541	wff 𝐴 = 𝐵
wtru 1542	wff ⊤
df-tru 1544	⊢ (⊤ ↔ (∀𝑥 𝑥 = 𝑥 → ∀𝑥 𝑥 = 𝑥))
wfal 1553	wff ⊥
df-fal 1554	⊢ (⊥ ↔ ¬ ⊤)
whad 1594	wff hadd(𝜑, 𝜓, 𝜒)
df-had 1595	⊢ (hadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ⊻ 𝜓) ⊻ 𝜒))
wcad 1607	wff cadd(𝜑, 𝜓, 𝜒)
df-cad 1608	⊢ (cadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ∧ 𝜓) ∨ (𝜒 ∧ (𝜑 ⊻ 𝜓))))
wex 1780	wff ∃𝑥𝜑
df-ex 1781	⊢ (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑)
wnf 1784	wff Ⅎ𝑥𝜑
df-nf 1785	⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
ax-gen 1796	⊢ 𝜑 ⇒ ⊢ ∀𝑥𝜑
ax-4 1810	⊢ (∀𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))
ax-5 1911	⊢ (𝜑 → ∀𝑥𝜑)
ax-6 1968	⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦
ax-7 2009	⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 → 𝑦 = 𝑧))
wsb 2067	wff [𝑦 / 𝑥]𝜑
df-sb 2068	⊢ (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) ↔ ∀𝑧(𝑧 = 𝑡 → ∀𝑥(𝑥 = 𝑧 → 𝜑))) ⇒ ⊢ ([𝑡 / 𝑥]𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
wcel 2113	wff 𝐴 ∈ 𝐵
ax-8 2115	⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧))
ax-9 2123	⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦))
ax-10 2146	⊢ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
ax-11 2162	⊢ (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑)
ax-12 2184	⊢ (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
ax-13 2376	⊢ (¬ 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
wmo 2537	wff ∃*𝑥𝜑
df-mo 2539	⊢ (∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦) ↔ ∃𝑧∀𝑥(𝜑 → 𝑥 = 𝑧)) ⇒ ⊢ (∃*𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))
weu 2568	wff ∃!𝑥𝜑
df-eu 2569	⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃*𝑥𝜑))
ax-ext 2708	⊢ (∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → 𝑥 = 𝑦)
cab 2714	class {𝑥 ∣ 𝜑}
df-clab 2715	⊢ (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ [𝑥 / 𝑦]𝜑)
df-cleq 2728	⊢ (𝑦 = 𝑧 ↔ ∀𝑢(𝑢 ∈ 𝑦 ↔ 𝑢 ∈ 𝑧)) & ⊢ (𝑡 = 𝑡 ↔ ∀𝑣(𝑣 ∈ 𝑡 ↔ 𝑣 ∈ 𝑡)) ⇒ ⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
df-clel 2811	⊢ (𝑦 ∈ 𝑧 ↔ ∃𝑢(𝑢 = 𝑦 ∧ 𝑢 ∈ 𝑧)) & ⊢ (𝑡 ∈ 𝑡 ↔ ∃𝑣(𝑣 = 𝑡 ∧ 𝑣 ∈ 𝑡)) ⇒ ⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵))
wnfc 2883	wff Ⅎ𝑥𝐴
df-nfc 2885	⊢ (Ⅎ𝑥𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴)
wne 2932	wff 𝐴 ≠ 𝐵
df-ne 2933	⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵)
wnel 3036	wff 𝐴 ∉ 𝐵
df-nel 3037	⊢ (𝐴 ∉ 𝐵 ↔ ¬ 𝐴 ∈ 𝐵)
wral 3051	wff ∀𝑥 ∈ 𝐴 𝜑
df-ral 3052	⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
wrex 3060	wff ∃𝑥 ∈ 𝐴 𝜑
df-rex 3061	⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
wreu 3348	wff ∃!𝑥 ∈ 𝐴 𝜑
wrmo 3349	wff ∃*𝑥 ∈ 𝐴 𝜑
df-rmo 3350	⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
df-reu 3351	⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
crab 3399	class {𝑥 ∈ 𝐴 ∣ 𝜑}
df-rab 3400	⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
cvv 3440	class V
df-v 3442	⊢ V = {𝑥 ∣ 𝑥 = 𝑥}
wcdeq 3721	wff CondEq(𝑥 = 𝑦 → 𝜑)
df-cdeq 3722	⊢ (CondEq(𝑥 = 𝑦 → 𝜑) ↔ (𝑥 = 𝑦 → 𝜑))
wsbc 3740	wff [𝐴 / 𝑥]𝜑
df-sbc 3741	⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑})
csb 3849	class ⦋𝐴 / 𝑥⦌𝐵
df-csb 3850	⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵}
cdif 3898	class (𝐴 ∖ 𝐵)
cun 3899	class (𝐴 ∪ 𝐵)
cin 3900	class (𝐴 ∩ 𝐵)
wss 3901	wff 𝐴 ⊆ 𝐵
wpss 3902	wff 𝐴 ⊊ 𝐵
df-dif 3904	⊢ (𝐴 ∖ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)}
df-un 3906	⊢ (𝐴 ∪ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)}
df-in 3908	⊢ (𝐴 ∩ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)}
df-ss 3918	⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
df-pss 3921	⊢ (𝐴 ⊊ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ 𝐵))
csymdif 4204	class (𝐴 △ 𝐵)
df-symdif 4205	⊢ (𝐴 △ 𝐵) = ((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐴))
c0 4285	class ∅
df-nul 4286	⊢ ∅ = (V ∖ V)
cif 4479	class if(𝜑, 𝐴, 𝐵)
df-if 4480	⊢ if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑))}
cpw 4554	class 𝒫 𝐴
df-pw 4556	⊢ 𝒫 𝐴 = {𝑥 ∣ 𝑥 ⊆ 𝐴}
csn 4580	class {𝐴}
df-sn 4581	⊢ {𝐴} = {𝑥 ∣ 𝑥 = 𝐴}
cpr 4582	class {𝐴, 𝐵}
df-pr 4583	⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
ctp 4584	class {𝐴, 𝐵, 𝐶}
df-tp 4585	⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
cop 4586	class ⟨𝐴, 𝐵⟩
df-op 4587	⊢ ⟨𝐴, 𝐵⟩ = {𝑥 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})}
cotp 4588	class ⟨𝐴, 𝐵, 𝐶⟩
df-ot 4589	⊢ ⟨𝐴, 𝐵, 𝐶⟩ = ⟨⟨𝐴, 𝐵⟩, 𝐶⟩
cuni 4863	class ∪ 𝐴
df-uni 4864	⊢ ∪ 𝐴 = {𝑥 ∣ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)}
cint 4902	class ∩ 𝐴
df-int 4903	⊢ ∩ 𝐴 = {𝑥 ∣ ∀𝑦(𝑦 ∈ 𝐴 → 𝑥 ∈ 𝑦)}
ciun 4946	class ∪ 𝑥 ∈ 𝐴 𝐵
ciin 4947	class ∩ 𝑥 ∈ 𝐴 𝐵
df-iun 4948	⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵}
df-iin 4949	⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵}
wdisj 5065	wff Disj 𝑥 ∈ 𝐴 𝐵
df-disj 5066	⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
wbr 5098	wff 𝐴𝑅𝐵
df-br 5099	⊢ (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
copab 5160	class {⟨𝑥, 𝑦⟩ ∣ 𝜑}
df-opab 5161	⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
cmpt 5179	class (𝑥 ∈ 𝐴 ↦ 𝐵)
df-mpt 5180	⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
wtr 5205	wff Tr 𝐴
df-tr 5206	⊢ (Tr 𝐴 ↔ ∪ 𝐴 ⊆ 𝐴)
ax-rep 5224	⊢ (∀𝑤∃𝑦∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦) → ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)))
ax-sep 5241	⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))
ax-nul 5251	⊢ ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥
ax-pow 5310	⊢ ∃𝑦∀𝑧(∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦)
ax-pr 5377	⊢ ∃𝑧∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧)
cid 5518	class I
df-id 5519	⊢ I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
cep 5523	class E
df-eprel 5524	⊢ E = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ 𝑦}
wpo 5530	wff 𝑅 Po 𝐴
wor 5531	wff 𝑅 Or 𝐴
df-po 5532	⊢ (𝑅 Po 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
df-so 5533	⊢ (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)))
wfr 5574	wff 𝑅 Fr 𝐴
wse 5575	wff 𝑅 Se 𝐴
wwe 5576	wff 𝑅 We 𝐴
df-fr 5577	⊢ (𝑅 Fr 𝐴 ↔ ∀𝑥((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦))
df-se 5578	⊢ (𝑅 Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V)
df-we 5579	⊢ (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴 ∧ 𝑅 Or 𝐴))
cxp 5622	class (𝐴 × 𝐵)
ccnv 5623	class ^◡𝐴
cdm 5624	class dom 𝐴
crn 5625	class ran 𝐴
cres 5626	class (𝐴 ↾ 𝐵)
cima 5627	class (𝐴 “ 𝐵)
ccom 5628	class (𝐴 ∘ 𝐵)
wrel 5629	wff Rel 𝐴
df-xp 5630	⊢ (𝐴 × 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)}
df-rel 5631	⊢ (Rel 𝐴 ↔ 𝐴 ⊆ (V × V))
df-cnv 5632	⊢ ^◡𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥}
df-co 5633	⊢ (𝐴 ∘ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)}
df-dm 5634	⊢ dom 𝐴 = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦}
df-rn 5635	⊢ ran 𝐴 = dom ^◡𝐴
df-res 5636	⊢ (𝐴 ↾ 𝐵) = (𝐴 ∩ (𝐵 × V))
df-ima 5637	⊢ (𝐴 “ 𝐵) = ran (𝐴 ↾ 𝐵)
cpred 6258	class Pred(𝑅, 𝐴, 𝑋)
df-pred 6259	⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (^◡𝑅 “ {𝑋}))
word 6316	wff Ord 𝐴
con0 6317	class On
wlim 6318	wff Lim 𝐴
csuc 6319	class suc 𝐴
df-ord 6320	⊢ (Ord 𝐴 ↔ (Tr 𝐴 ∧ E We 𝐴))
df-on 6321	⊢ On = {𝑥 ∣ Ord 𝑥}
df-lim 6322	⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴))
df-suc 6323	⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
cio 6446	class (℩𝑥𝜑)
df-iota 6448	⊢ (℩𝑥𝜑) = ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}
wfun 6486	wff Fun 𝐴
wfn 6487	wff 𝐴 Fn 𝐵
wf 6488	wff 𝐹:𝐴⟶𝐵
wf1 6489	wff 𝐹:𝐴–1-1→𝐵
wfo 6490	wff 𝐹:𝐴–onto→𝐵
wf1o 6491	wff 𝐹:𝐴–1-1-onto→𝐵
cfv 6492	class (𝐹‘𝐴)
wiso 6493	wff 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵)
df-fun 6494	⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ (𝐴 ∘ ^◡𝐴) ⊆ I ))
df-fn 6495	⊢ (𝐴 Fn 𝐵 ↔ (Fun 𝐴 ∧ dom 𝐴 = 𝐵))
df-f 6496	⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵))
df-f1 6497	⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ Fun ^◡𝐹))
df-fo 6498	⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵))
df-f1o 6499	⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹:𝐴–1-1→𝐵 ∧ 𝐹:𝐴–onto→𝐵))
df-fv 6500	⊢ (𝐹‘𝐴) = (℩𝑥𝐴𝐹𝑥)
df-isom 6501	⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐻:𝐴–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦))))
crio 7314	class (℩𝑥 ∈ 𝐴 𝜑)
df-riota 7315	⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
co 7358	class (𝐴𝐹𝐵)
coprab 7359	class {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
cmpo 7360	class (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
df-ov 7361	⊢ (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
df-oprab 7362	⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)}
df-mpo 7363	⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)}
cof 7620	class ∘_f 𝑅
cofr 7621	class ∘_r 𝑅
df-of 7622	⊢ ∘_f 𝑅 = (𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥))))
df-ofr 7623	⊢ ∘_r 𝑅 = {⟨𝑓, 𝑔⟩ ∣ ∀𝑥 ∈ (dom 𝑓 ∩ dom 𝑔)(𝑓‘𝑥)𝑅(𝑔‘𝑥)}
crpss 7667	class [⊊]
df-rpss 7668	⊢ [⊊] = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ⊊ 𝑦}
ax-un 7680	⊢ ∃𝑦∀𝑧(∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦)
com 7808	class ω
df-om 7809	⊢ ω = {𝑥 ∈ On ∣ ∀𝑦(Lim 𝑦 → 𝑥 ∈ 𝑦)}
c1st 7931	class 1^st
c2nd 7932	class 2^nd
df-1st 7933	⊢ 1^st = (𝑥 ∈ V ↦ ∪ dom {𝑥})
df-2nd 7934	⊢ 2^nd = (𝑥 ∈ V ↦ ∪ ran {𝑥})
csupp 8102	class supp
df-supp 8103	⊢ supp = (𝑥 ∈ V, 𝑧 ∈ V ↦ {𝑖 ∈ dom 𝑥 ∣ (𝑥 “ {𝑖}) ≠ {𝑧}})
ctpos 8167	class tpos 𝐹
df-tpos 8168	⊢ tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ (^◡dom 𝐹 ∪ {∅}) ↦ ∪ ^◡{𝑥}))
ccur 8207	class curry 𝐴
cunc 8208	class uncurry 𝐴
df-cur 8209	⊢ curry 𝐹 = (𝑥 ∈ dom dom 𝐹 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩𝐹𝑧})
df-unc 8210	⊢ uncurry 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(𝐹‘𝑥)𝑧}
cund 8214	class Undef
df-undef 8215	⊢ Undef = (𝑠 ∈ V ↦ 𝒫 ∪ 𝑠)
cfrecs 8222	class frecs(𝑅, 𝐴, 𝐹)
df-frecs 8223	⊢ frecs(𝑅, 𝐴, 𝐹) = ∪ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐹(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
cwrecs 8253	class wrecs(𝑅, 𝐴, 𝐹)
df-wrecs 8254	⊢ wrecs(𝑅, 𝐴, 𝐹) = frecs(𝑅, 𝐴, (𝐹 ∘ 2^nd ))
wsmo 8277	wff Smo 𝐴
df-smo 8278	⊢ (Smo 𝐴 ↔ (𝐴:dom 𝐴⟶On ∧ Ord dom 𝐴 ∧ ∀𝑥 ∈ dom 𝐴∀𝑦 ∈ dom 𝐴(𝑥 ∈ 𝑦 → (𝐴‘𝑥) ∈ (𝐴‘𝑦))))
crecs 8302	class recs(𝐹)
df-recs 8303	⊢ recs(𝐹) = wrecs( E , On, 𝐹)
crdg 8340	class rec(𝐹, 𝐼)
df-rdg 8341	⊢ rec(𝐹, 𝐼) = recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔))))))
cseqom 8378	class seq_ω(𝐹, 𝐼)
df-seqom 8379	⊢ seq_ω(𝐹, 𝐼) = (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) “ ω)
c1o 8390	class 1_o
c2o 8391	class 2_o
c3o 8392	class 3_o
c4o 8393	class 4_o
coa 8394	class +_o
comu 8395	class ·_o
coe 8396	class ↑_o
df-1o 8397	⊢ 1_o = suc ∅
df-2o 8398	⊢ 2_o = suc 1_o
df-3o 8399	⊢ 3_o = suc 2_o
df-4o 8400	⊢ 4_o = suc 3_o
df-oadd 8401	⊢ +_o = (𝑥 ∈ On, 𝑦 ∈ On ↦ (rec((𝑧 ∈ V ↦ suc 𝑧), 𝑥)‘𝑦))
df-omul 8402	⊢ ·_o = (𝑥 ∈ On, 𝑦 ∈ On ↦ (rec((𝑧 ∈ V ↦ (𝑧 +_o 𝑥)), ∅)‘𝑦))
df-oexp 8403	⊢ ↑_o = (𝑥 ∈ On, 𝑦 ∈ On ↦ if(𝑥 = ∅, (1_o ∖ 𝑦), (rec((𝑧 ∈ V ↦ (𝑧 ·_o 𝑥)), 1_o)‘𝑦)))
cnadd 8593	class +no
df-nadd 8594	⊢ +no = frecs({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On) ∧ (((1^st ‘𝑥) E (1^st ‘𝑦) ∨ (1^st ‘𝑥) = (1^st ‘𝑦)) ∧ ((2^nd ‘𝑥) E (2^nd ‘𝑦) ∨ (2^nd ‘𝑥) = (2^nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))}, (On × On), (𝑧 ∈ V, 𝑎 ∈ V ↦ ∩ {𝑤 ∈ On ∣ ((𝑎 “ ({(1^st ‘𝑧)} × (2^nd ‘𝑧))) ⊆ 𝑤 ∧ (𝑎 “ ((1^st ‘𝑧) × {(2^nd ‘𝑧)})) ⊆ 𝑤)}))
wer 8632	wff 𝑅 Er 𝐴
cec 8633	class [𝐴]𝑅
cqs 8634	class (𝐴 / 𝑅)
df-er 8635	⊢ (𝑅 Er 𝐴 ↔ (Rel 𝑅 ∧ dom 𝑅 = 𝐴 ∧ (^◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅))
df-ec 8637	⊢ [𝐴]𝑅 = (𝑅 “ {𝐴})
df-qs 8641	⊢ (𝐴 / 𝑅) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅}
cmap 8763	class ↑_m
cpm 8764	class ↑_pm
df-map 8765	⊢ ↑_m = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓 ∣ 𝑓:𝑦⟶𝑥})
df-pm 8766	⊢ ↑_pm = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓 ∈ 𝒫 (𝑦 × 𝑥) ∣ Fun 𝑓})
cixp 8835	class X𝑥 ∈ 𝐴 𝐵
df-ixp 8836	⊢ X𝑥 ∈ 𝐴 𝐵 = {𝑓 ∣ (𝑓 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵)}
cen 8880	class ≈
cdom 8881	class ≼
csdm 8882	class ≺
cfn 8883	class Fin
df-en 8884	⊢ ≈ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥–1-1-onto→𝑦}
df-dom 8885	⊢ ≼ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥–1-1→𝑦}
df-sdom 8886	⊢ ≺ = ( ≼ ∖ ≈ )
df-fin 8887	⊢ Fin = {𝑥 ∣ ∃𝑦 ∈ ω 𝑥 ≈ 𝑦}
cfsupp 9264	class finSupp
df-fsupp 9265	⊢ finSupp = {⟨𝑟, 𝑧⟩ ∣ (Fun 𝑟 ∧ (𝑟 supp 𝑧) ∈ Fin)}
cfi 9313	class fi
df-fi 9314	⊢ fi = (𝑥 ∈ V ↦ {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦})
csup 9343	class sup(𝐴, 𝐵, 𝑅)
cinf 9344	class inf(𝐴, 𝐵, 𝑅)
df-sup 9345	⊢ sup(𝐴, 𝐵, 𝑅) = ∪ {𝑥 ∈ 𝐵 ∣ (∀𝑦 ∈ 𝐴 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐵 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐴 𝑦𝑅𝑧))}
df-inf 9346	⊢ inf(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, ^◡𝑅)
coi 9414	class OrdIso(𝑅, 𝐴)
df-oi 9415	⊢ OrdIso(𝑅, 𝐴) = if((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴), (recs((ℎ ∈ V ↦ (℩𝑣 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤}∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣))) ↾ {𝑥 ∈ On ∣ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (recs((ℎ ∈ V ↦ (℩𝑣 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤}∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣))) “ 𝑥)𝑧𝑅𝑡}), ∅)
char 9461	class har
df-har 9462	⊢ har = (𝑥 ∈ V ↦ {𝑦 ∈ On ∣ 𝑦 ≼ 𝑥})
cwdom 9469	class ≼^*
df-wdom 9470	⊢ ≼^* = {⟨𝑥, 𝑦⟩ ∣ (𝑥 = ∅ ∨ ∃𝑧 𝑧:𝑦–onto→𝑥)}
ax-reg 9497	⊢ (∃𝑦 𝑦 ∈ 𝑥 → ∃𝑦(𝑦 ∈ 𝑥 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑥)))
ax-inf 9547	⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦)))
ax-inf2 9550	⊢ ∃𝑥(∃𝑦(𝑦 ∈ 𝑥 ∧ ∀𝑧 ¬ 𝑧 ∈ 𝑦) ∧ ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑧 ∈ 𝑥 ∧ ∀𝑤(𝑤 ∈ 𝑧 ↔ (𝑤 ∈ 𝑦 ∨ 𝑤 = 𝑦)))))
ccnf 9570	class CNF
df-cnf 9571	⊢ CNF = (𝑥 ∈ On, 𝑦 ∈ On ↦ (𝑓 ∈ {𝑔 ∈ (𝑥 ↑_m 𝑦) ∣ 𝑔 finSupp ∅} ↦ ⦋OrdIso( E , (𝑓 supp ∅)) / ℎ⦌(seq_ω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝑥 ↑_o (ℎ‘𝑘)) ·_o (𝑓‘(ℎ‘𝑘))) +_o 𝑧)), ∅)‘dom ℎ)))
cttrcl 9616	class t++𝑅
df-ttrcl 9617	⊢ t++𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑛 ∈ (ω ∖ 1_o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓‘𝑛) = 𝑦) ∧ ∀𝑚 ∈ 𝑛 (𝑓‘𝑚)𝑅(𝑓‘suc 𝑚))}
ctc 9643	class TC
df-tc 9644	⊢ TC = (𝑥 ∈ V ↦ ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ Tr 𝑦)})
cr1 9674	class 𝑅₁
crnk 9675	class rank
df-r1 9676	⊢ 𝑅₁ = rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅)
df-rank 9677	⊢ rank = (𝑥 ∈ V ↦ ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅₁‘suc 𝑦)})
cdju 9810	class (𝐴 ⊔ 𝐵)
cinl 9811	class inl
cinr 9812	class inr
df-dju 9813	⊢ (𝐴 ⊔ 𝐵) = (({∅} × 𝐴) ∪ ({1_o} × 𝐵))
df-inl 9814	⊢ inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩)
df-inr 9815	⊢ inr = (𝑥 ∈ V ↦ ⟨1_o, 𝑥⟩)
ccrd 9847	class card
cale 9848	class ℵ
ccf 9849	class cf
wacn 9850	class AC 𝐴
df-card 9851	⊢ card = (𝑥 ∈ V ↦ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥})
df-aleph 9852	⊢ ℵ = rec(har, ω)
df-cf 9853	⊢ cf = (𝑥 ∈ On ↦ ∩ {𝑦 ∣ ∃𝑧(𝑦 = (card‘𝑧) ∧ (𝑧 ⊆ 𝑥 ∧ ∀𝑣 ∈ 𝑥 ∃𝑢 ∈ 𝑧 𝑣 ⊆ 𝑢))})
df-acn 9854	⊢ AC 𝐴 = {𝑥 ∣ (𝐴 ∈ V ∧ ∀𝑓 ∈ ((𝒫 𝑥 ∖ {∅}) ↑_m 𝐴)∃𝑔∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝑓‘𝑦))}
wac 10025	wff CHOICE
df-ac 10026	⊢ (CHOICE ↔ ∀𝑥∃𝑓(𝑓 ⊆ 𝑥 ∧ 𝑓 Fn dom 𝑥))
cfin1a 10188	class Fin^Ia
cfin2 10189	class Fin^II
cfin4 10190	class Fin^IV
cfin3 10191	class Fin^III
cfin5 10192	class Fin^V
cfin6 10193	class Fin^VI
cfin7 10194	class Fin^VII
df-fin1a 10195	⊢ Fin^Ia = {𝑥 ∣ ∀𝑦 ∈ 𝒫 𝑥(𝑦 ∈ Fin ∨ (𝑥 ∖ 𝑦) ∈ Fin)}
df-fin2 10196	⊢ Fin^II = {𝑥 ∣ ∀𝑦 ∈ 𝒫 𝒫 𝑥((𝑦 ≠ ∅ ∧ [⊊] Or 𝑦) → ∪ 𝑦 ∈ 𝑦)}
df-fin4 10197	⊢ Fin^IV = {𝑥 ∣ ¬ ∃𝑦(𝑦 ⊊ 𝑥 ∧ 𝑦 ≈ 𝑥)}
df-fin3 10198	⊢ Fin^III = {𝑥 ∣ 𝒫 𝑥 ∈ Fin^IV}
df-fin5 10199	⊢ Fin^V = {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 ≺ (𝑥 ⊔ 𝑥))}
df-fin6 10200	⊢ Fin^VI = {𝑥 ∣ (𝑥 ≺ 2_o ∨ 𝑥 ≺ (𝑥 × 𝑥))}
df-fin7 10201	⊢ Fin^VII = {𝑥 ∣ ¬ ∃𝑦 ∈ (On ∖ ω)𝑥 ≈ 𝑦}
ax-cc 10345	⊢ (𝑥 ≈ ω → ∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧))
ax-dc 10356	⊢ ((∃𝑦∃𝑧 𝑦𝑥𝑧 ∧ ran 𝑥 ⊆ dom 𝑥) → ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛))
ax-ac 10369	⊢ ∃𝑦∀𝑧∀𝑤((𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → ∃𝑣∀𝑢(∃𝑡((𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑡) ∧ (𝑢 ∈ 𝑡 ∧ 𝑡 ∈ 𝑦)) ↔ 𝑢 = 𝑣))
ax-ac2 10373	⊢ ∃𝑦∀𝑧∃𝑣∀𝑢((𝑦 ∈ 𝑥 ∧ (𝑧 ∈ 𝑦 → ((𝑣 ∈ 𝑥 ∧ ¬ 𝑦 = 𝑣) ∧ 𝑧 ∈ 𝑣))) ∨ (¬ 𝑦 ∈ 𝑥 ∧ (𝑧 ∈ 𝑥 → ((𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦) ∧ ((𝑢 ∈ 𝑧 ∧ 𝑢 ∈ 𝑦) → 𝑢 = 𝑣)))))
cgch 10531	class GCH
df-gch 10532	⊢ GCH = (Fin ∪ {𝑥 ∣ ∀𝑦 ¬ (𝑥 ≺ 𝑦 ∧ 𝑦 ≺ 𝒫 𝑥)})
cwina 10593	class Inacc_w
cina 10594	class Inacc
df-wina 10595	⊢ Inacc_w = {𝑥 ∣ (𝑥 ≠ ∅ ∧ (cf‘𝑥) = 𝑥 ∧ ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝑥 𝑦 ≺ 𝑧)}
df-ina 10596	⊢ Inacc = {𝑥 ∣ (𝑥 ≠ ∅ ∧ (cf‘𝑥) = 𝑥 ∧ ∀𝑦 ∈ 𝑥 𝒫 𝑦 ≺ 𝑥)}
cwun 10611	class WUni
cwunm 10612	class wUniCl
df-wun 10613	⊢ WUni = {𝑢 ∣ (Tr 𝑢 ∧ 𝑢 ≠ ∅ ∧ ∀𝑥 ∈ 𝑢 (∪ 𝑥 ∈ 𝑢 ∧ 𝒫 𝑥 ∈ 𝑢 ∧ ∀𝑦 ∈ 𝑢 {𝑥, 𝑦} ∈ 𝑢))}
df-wunc 10614	⊢ wUniCl = (𝑥 ∈ V ↦ ∩ {𝑢 ∈ WUni ∣ 𝑥 ⊆ 𝑢})
ctsk 10659	class Tarski
df-tsk 10660	⊢ Tarski = {𝑦 ∣ (∀𝑧 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑦 ∧ ∃𝑤 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧 ≈ 𝑦 ∨ 𝑧 ∈ 𝑦))}
cgru 10701	class Univ
df-gru 10702	⊢ Univ = {𝑢 ∣ (Tr 𝑢 ∧ ∀𝑥 ∈ 𝑢 (𝒫 𝑥 ∈ 𝑢 ∧ ∀𝑦 ∈ 𝑢 {𝑥, 𝑦} ∈ 𝑢 ∧ ∀𝑦 ∈ (𝑢 ↑_m 𝑥)∪ ran 𝑦 ∈ 𝑢))}
ax-groth 10734	⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (∀𝑤(𝑤 ⊆ 𝑧 → 𝑤 ∈ 𝑦) ∧ ∃𝑤 ∈ 𝑦 ∀𝑣(𝑣 ⊆ 𝑧 → 𝑣 ∈ 𝑤)) ∧ ∀𝑧(𝑧 ⊆ 𝑦 → (𝑧 ≈ 𝑦 ∨ 𝑧 ∈ 𝑦)))
ctskm 10748	class tarskiMap
df-tskm 10749	⊢ tarskiMap = (𝑥 ∈ V ↦ ∩ {𝑦 ∈ Tarski ∣ 𝑥 ∈ 𝑦})
cnpi 10755	class N
cpli 10756	class +_N
cmi 10757	class ·_N
clti 10758	class <_N
cplpq 10759	class +_pQ
cmpq 10760	class ·_pQ
cltpq 10761	class <_pQ
ceq 10762	class ~_Q
cnq 10763	class Q
c1q 10764	class 1_Q
cerq 10765	class [Q]
cplq 10766	class +_Q
cmq 10767	class ·_Q
crq 10768	class *_Q
cltq 10769	class <_Q
cnp 10770	class P
c1p 10771	class 1_P
cpp 10772	class +_P
cmp 10773	class ·_P
cltp 10774	class <_P
cer 10775	class ~_R
cnr 10776	class R
c0r 10777	class 0_R
c1r 10778	class 1_R
cm1r 10779	class -1_R
cplr 10780	class +_R
cmr 10781	class ·_R
cltr 10782	class <_R
df-ni 10783	⊢ N = (ω ∖ {∅})
df-pli 10784	⊢ +_N = ( +_o ↾ (N × N))
df-mi 10785	⊢ ·_N = ( ·_o ↾ (N × N))
df-lti 10786	⊢ <_N = ( E ∩ (N × N))
df-plpq 10819	⊢ +_pQ = (𝑥 ∈ (N × N), 𝑦 ∈ (N × N) ↦ ⟨(((1^st ‘𝑥) ·_N (2^nd ‘𝑦)) +_N ((1^st ‘𝑦) ·_N (2^nd ‘𝑥))), ((2^nd ‘𝑥) ·_N (2^nd ‘𝑦))⟩)
df-mpq 10820	⊢ ·_pQ = (𝑥 ∈ (N × N), 𝑦 ∈ (N × N) ↦ ⟨((1^st ‘𝑥) ·_N (1^st ‘𝑦)), ((2^nd ‘𝑥) ·_N (2^nd ‘𝑦))⟩)
df-ltpq 10821	⊢ <_pQ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ((1^st ‘𝑥) ·_N (2^nd ‘𝑦)) <_N ((1^st ‘𝑦) ·_N (2^nd ‘𝑥)))}
df-enq 10822	⊢ ~_Q = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·_N 𝑢) = (𝑤 ·_N 𝑣)))}
df-nq 10823	⊢ Q = {𝑥 ∈ (N × N) ∣ ∀𝑦 ∈ (N × N)(𝑥 ~_Q 𝑦 → ¬ (2^nd ‘𝑦) <_N (2^nd ‘𝑥))}
df-erq 10824	⊢ [Q] = ( ~_Q ∩ ((N × N) × Q))
df-plq 10825	⊢ +_Q = (([Q] ∘ +_pQ ) ↾ (Q × Q))
df-mq 10826	⊢ ·_Q = (([Q] ∘ ·_pQ ) ↾ (Q × Q))
df-1nq 10827	⊢ 1_Q = ⟨1_o, 1_o⟩
df-rq 10828	⊢ *_Q = (^◡ ·_Q “ {1_Q})
df-ltnq 10829	⊢ <_Q = ( <_pQ ∩ (Q × Q))
df-np 10892	⊢ P = {𝑥 ∣ ((∅ ⊊ 𝑥 ∧ 𝑥 ⊊ Q) ∧ ∀𝑦 ∈ 𝑥 (∀𝑧(𝑧 <_Q 𝑦 → 𝑧 ∈ 𝑥) ∧ ∃𝑧 ∈ 𝑥 𝑦 <_Q 𝑧))}
df-1p 10893	⊢ 1_P = {𝑥 ∣ 𝑥 <_Q 1_Q}
df-plp 10894	⊢ +_P = (𝑥 ∈ P, 𝑦 ∈ P ↦ {𝑤 ∣ ∃𝑣 ∈ 𝑥 ∃𝑢 ∈ 𝑦 𝑤 = (𝑣 +_Q 𝑢)})
df-mp 10895	⊢ ·_P = (𝑥 ∈ P, 𝑦 ∈ P ↦ {𝑤 ∣ ∃𝑣 ∈ 𝑥 ∃𝑢 ∈ 𝑦 𝑤 = (𝑣 ·_Q 𝑢)})
df-ltp 10896	⊢ <_P = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ 𝑥 ⊊ 𝑦)}
df-enr 10966	⊢ ~_R = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (P × P) ∧ 𝑦 ∈ (P × P)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 +_P 𝑢) = (𝑤 +_P 𝑣)))}
df-nr 10967	⊢ R = ((P × P) / ~_R )
df-plr 10968	⊢ +_R = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ R ∧ 𝑦 ∈ R) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~_R ∧ 𝑦 = [⟨𝑢, 𝑓⟩] ~_R ) ∧ 𝑧 = [⟨(𝑤 +_P 𝑢), (𝑣 +_P 𝑓)⟩] ~_R ))}
df-mr 10969	⊢ ·_R = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ R ∧ 𝑦 ∈ R) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~_R ∧ 𝑦 = [⟨𝑢, 𝑓⟩] ~_R ) ∧ 𝑧 = [⟨((𝑤 ·_P 𝑢) +_P (𝑣 ·_P 𝑓)), ((𝑤 ·_P 𝑓) +_P (𝑣 ·_P 𝑢))⟩] ~_R ))}
df-ltr 10970	⊢ <_R = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ R ∧ 𝑦 ∈ R) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = [⟨𝑧, 𝑤⟩] ~_R ∧ 𝑦 = [⟨𝑣, 𝑢⟩] ~_R ) ∧ (𝑧 +_P 𝑢)<_P (𝑤 +_P 𝑣)))}
df-0r 10971	⊢ 0_R = [⟨1_P, 1_P⟩] ~_R
df-1r 10972	⊢ 1_R = [⟨(1_P +_P 1_P), 1_P⟩] ~_R
df-m1r 10973	⊢ -1_R = [⟨1_P, (1_P +_P 1_P)⟩] ~_R
cc 11024	class ℂ
cr 11025	class ℝ
cc0 11026	class 0
c1 11027	class 1
ci 11028	class i
caddc 11029	class +
cltrr 11030	class <_ℝ
cmul 11031	class ·
df-c 11032	⊢ ℂ = (R × R)
df-0 11033	⊢ 0 = ⟨0_R, 0_R⟩
df-1 11034	⊢ 1 = ⟨1_R, 0_R⟩
df-i 11035	⊢ i = ⟨0_R, 1_R⟩
df-r 11036	⊢ ℝ = (R × {0_R})
df-add 11037	⊢ + = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨(𝑤 +_R 𝑢), (𝑣 +_R 𝑓)⟩))}
df-mul 11038	⊢ · = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨((𝑤 ·_R 𝑢) +_R (-1_R ·_R (𝑣 ·_R 𝑓))), ((𝑣 ·_R 𝑢) +_R (𝑤 ·_R 𝑓))⟩))}
df-lt 11039	⊢ <_ℝ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ∃𝑧∃𝑤((𝑥 = ⟨𝑧, 0_R⟩ ∧ 𝑦 = ⟨𝑤, 0_R⟩) ∧ 𝑧 <_R 𝑤))}
ax-cnex 11082	⊢ ℂ ∈ V
ax-resscn 11083	⊢ ℝ ⊆ ℂ
ax-1cn 11084	⊢ 1 ∈ ℂ
ax-icn 11085	⊢ i ∈ ℂ
ax-addcl 11086	⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) ∈ ℂ)
ax-addrcl 11087	⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + 𝐵) ∈ ℝ)
ax-mulcl 11088	⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) ∈ ℂ)
ax-mulrcl 11089	⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 · 𝐵) ∈ ℝ)
ax-mulcom 11090	⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) = (𝐵 · 𝐴))
ax-addass 11091	⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶)))
ax-mulass 11092	⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)))
ax-distr 11093	⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶)))
ax-i2m1 11094	⊢ ((i · i) + 1) = 0
ax-1ne0 11095	⊢ 1 ≠ 0
ax-1rid 11096	⊢ (𝐴 ∈ ℝ → (𝐴 · 1) = 𝐴)
ax-rnegex 11097	⊢ (𝐴 ∈ ℝ → ∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0)
ax-rrecex 11098	⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → ∃𝑥 ∈ ℝ (𝐴 · 𝑥) = 1)
ax-cnre 11099	⊢ (𝐴 ∈ ℂ → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝐴 = (𝑥 + (i · 𝑦)))
ax-pre-lttri 11100	⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 <_ℝ 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 <_ℝ 𝐴)))
ax-pre-lttrn 11101	⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 <_ℝ 𝐵 ∧ 𝐵 <_ℝ 𝐶) → 𝐴 <_ℝ 𝐶))
ax-pre-ltadd 11102	⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 <_ℝ 𝐵 → (𝐶 + 𝐴) <_ℝ (𝐶 + 𝐵)))
ax-pre-mulgt0 11103	⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((0 <_ℝ 𝐴 ∧ 0 <_ℝ 𝐵) → 0 <_ℝ (𝐴 · 𝐵)))
ax-pre-sup 11104	⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 <_ℝ 𝑥) → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑥 <_ℝ 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 <_ℝ 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 <_ℝ 𝑧)))
ax-addf 11105	⊢ + :(ℂ × ℂ)⟶ℂ
ax-mulf 11106	⊢ · :(ℂ × ℂ)⟶ℂ
cpnf 11163	class +∞
cmnf 11164	class -∞
cxr 11165	class ℝ^*
clt 11166	class <
cle 11167	class ≤
df-pnf 11168	⊢ +∞ = 𝒫 ∪ ℂ
df-mnf 11169	⊢ -∞ = 𝒫 +∞
df-xr 11170	⊢ ℝ^* = (ℝ ∪ {+∞, -∞})
df-ltxr 11171	⊢ < = ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <_ℝ 𝑦)} ∪ (((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ)))
df-le 11172	⊢ ≤ = ((ℝ^* × ℝ^*) ∖ ^◡ < )
cmin 11364	class −
cneg 11365	class -𝐴
df-sub 11366	⊢ − = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥))
df-neg 11367	⊢ -𝐴 = (0 − 𝐴)
cdiv 11794	class /
df-div 11795	⊢ / = (𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥))
cn 12145	class ℕ
df-nn 12146	⊢ ℕ = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) “ ω)
c2 12200	class 2
c3 12201	class 3
c4 12202	class 4
c5 12203	class 5
c6 12204	class 6
c7 12205	class 7
c8 12206	class 8
c9 12207	class 9
df-2 12208	⊢ 2 = (1 + 1)
df-3 12209	⊢ 3 = (2 + 1)
df-4 12210	⊢ 4 = (3 + 1)
df-5 12211	⊢ 5 = (4 + 1)
df-6 12212	⊢ 6 = (5 + 1)
df-7 12213	⊢ 7 = (6 + 1)
df-8 12214	⊢ 8 = (7 + 1)
df-9 12215	⊢ 9 = (8 + 1)
cn0 12401	class ℕ₀
df-n0 12402	⊢ ℕ₀ = (ℕ ∪ {0})
cxnn0 12474	class ℕ₀^*
df-xnn0 12475	⊢ ℕ₀^* = (ℕ₀ ∪ {+∞})
cz 12488	class ℤ
df-z 12489	⊢ ℤ = {𝑛 ∈ ℝ ∣ (𝑛 = 0 ∨ 𝑛 ∈ ℕ ∨ -𝑛 ∈ ℕ)}
cdc 12607	class ;𝐴𝐵
df-dec 12608	⊢ ;𝐴𝐵 = (((9 + 1) · 𝐴) + 𝐵)
cuz 12751	class ℤ_≥
df-uz 12752	⊢ ℤ_≥ = (𝑗 ∈ ℤ ↦ {𝑘 ∈ ℤ ∣ 𝑗 ≤ 𝑘})
cq 12861	class ℚ
df-q 12862	⊢ ℚ = ( / “ (ℤ × ℕ))
crp 12905	class ℝ⁺
df-rp 12906	⊢ ℝ⁺ = {𝑥 ∈ ℝ ∣ 0 < 𝑥}
cxne 13023	class -_𝑒𝐴
cxad 13024	class +_𝑒
cxmu 13025	class ·_e
df-xneg 13026	⊢ -_𝑒𝐴 = if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴))
df-xadd 13027	⊢ +_𝑒 = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ if(𝑥 = +∞, if(𝑦 = -∞, 0, +∞), if(𝑥 = -∞, if(𝑦 = +∞, 0, -∞), if(𝑦 = +∞, +∞, if(𝑦 = -∞, -∞, (𝑥 + 𝑦))))))
df-xmul 13028	⊢ ·_e = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ if((𝑥 = 0 ∨ 𝑦 = 0), 0, if((((0 < 𝑦 ∧ 𝑥 = +∞) ∨ (𝑦 < 0 ∧ 𝑥 = -∞)) ∨ ((0 < 𝑥 ∧ 𝑦 = +∞) ∨ (𝑥 < 0 ∧ 𝑦 = -∞))), +∞, if((((0 < 𝑦 ∧ 𝑥 = -∞) ∨ (𝑦 < 0 ∧ 𝑥 = +∞)) ∨ ((0 < 𝑥 ∧ 𝑦 = -∞) ∨ (𝑥 < 0 ∧ 𝑦 = +∞))), -∞, (𝑥 · 𝑦)))))
cioo 13261	class (,)
cioc 13262	class (,]
cico 13263	class [,)
cicc 13264	class [,]
df-ioo 13265	⊢ (,) = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)})
df-ioc 13266	⊢ (,] = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 < 𝑧 ∧ 𝑧 ≤ 𝑦)})
df-ico 13267	⊢ [,) = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)})
df-icc 13268	⊢ [,] = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)})
cfz 13423	class ...
df-fz 13424	⊢ ... = (𝑚 ∈ ℤ, 𝑛 ∈ ℤ ↦ {𝑘 ∈ ℤ ∣ (𝑚 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛)})
cfzo 13570	class ..^
df-fzo 13571	⊢ ..^ = (𝑚 ∈ ℤ, 𝑛 ∈ ℤ ↦ (𝑚...(𝑛 − 1)))
cfl 13710	class ⌊
cceil 13711	class ⌈
df-fl 13712	⊢ ⌊ = (𝑥 ∈ ℝ ↦ (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))))
df-ceil 13713	⊢ ⌈ = (𝑥 ∈ ℝ ↦ -(⌊‘-𝑥))
cmo 13789	class mod
df-mod 13790	⊢ mod = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ⁺ ↦ (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦)))))
cseq 13924	class seq𝑀( + , 𝐹)
df-seq 13925	⊢ seq𝑀( + , 𝐹) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) “ ω)
cexp 13984	class ↑
df-exp 13985	⊢ ↑ = (𝑥 ∈ ℂ, 𝑦 ∈ ℤ ↦ if(𝑦 = 0, 1, if(0 < 𝑦, (seq1( · , (ℕ × {𝑥}))‘𝑦), (1 / (seq1( · , (ℕ × {𝑥}))‘-𝑦)))))
cfa 14196	class !
df-fac 14197	⊢ ! = ({⟨0, 1⟩} ∪ seq1( · , I ))
cbc 14225	class C
df-bc 14226	⊢ C = (𝑛 ∈ ℕ₀, 𝑘 ∈ ℤ ↦ if(𝑘 ∈ (0...𝑛), ((!‘𝑛) / ((!‘(𝑛 − 𝑘)) · (!‘𝑘))), 0))
chash 14253	class ♯
df-hash 14254	⊢ ♯ = (((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω) ∘ card) ∪ ((V ∖ Fin) × {+∞}))
cword 14436	class Word 𝑆
df-word 14437	⊢ Word 𝑆 = {𝑤 ∣ ∃𝑙 ∈ ℕ₀ 𝑤:(0..^𝑙)⟶𝑆}
clsw 14485	class lastS
df-lsw 14486	⊢ lastS = (𝑤 ∈ V ↦ (𝑤‘((♯‘𝑤) − 1)))
cconcat 14493	class ++
df-concat 14494	⊢ ++ = (𝑠 ∈ V, 𝑡 ∈ V ↦ (𝑥 ∈ (0..^((♯‘𝑠) + (♯‘𝑡))) ↦ if(𝑥 ∈ (0..^(♯‘𝑠)), (𝑠‘𝑥), (𝑡‘(𝑥 − (♯‘𝑠))))))
cs1 14519	class ⟨“𝐴”⟩
df-s1 14520	⊢ ⟨“𝐴”⟩ = {⟨0, ( I ‘𝐴)⟩}
csubstr 14564	class substr
df-substr 14565	⊢ substr = (𝑠 ∈ V, 𝑏 ∈ (ℤ × ℤ) ↦ if(((1^st ‘𝑏)..^(2^nd ‘𝑏)) ⊆ dom 𝑠, (𝑥 ∈ (0..^((2^nd ‘𝑏) − (1^st ‘𝑏))) ↦ (𝑠‘(𝑥 + (1^st ‘𝑏)))), ∅))
cpfx 14594	class prefix
df-pfx 14595	⊢ prefix = (𝑠 ∈ V, 𝑙 ∈ ℕ₀ ↦ (𝑠 substr ⟨0, 𝑙⟩))
csplice 14672	class splice
df-splice 14673	⊢ splice = (𝑠 ∈ V, 𝑏 ∈ V ↦ (((𝑠 prefix (1^st ‘(1^st ‘𝑏))) ++ (2^nd ‘𝑏)) ++ (𝑠 substr ⟨(2^nd ‘(1^st ‘𝑏)), (♯‘𝑠)⟩)))
creverse 14681	class reverse
df-reverse 14682	⊢ reverse = (𝑠 ∈ V ↦ (𝑥 ∈ (0..^(♯‘𝑠)) ↦ (𝑠‘(((♯‘𝑠) − 1) − 𝑥))))
creps 14691	class repeatS
df-reps 14692	⊢ repeatS = (𝑠 ∈ V, 𝑛 ∈ ℕ₀ ↦ (𝑥 ∈ (0..^𝑛) ↦ 𝑠))
ccsh 14711	class cyclShift
df-csh 14712	⊢ cyclShift = (𝑤 ∈ {𝑓 ∣ ∃𝑙 ∈ ℕ₀ 𝑓 Fn (0..^𝑙)}, 𝑛 ∈ ℤ ↦ if(𝑤 = ∅, ∅, ((𝑤 substr ⟨(𝑛 mod (♯‘𝑤)), (♯‘𝑤)⟩) ++ (𝑤 prefix (𝑛 mod (♯‘𝑤))))))
cs2 14764	class ⟨“𝐴𝐵”⟩
cs3 14765	class ⟨“𝐴𝐵𝐶”⟩
cs4 14766	class ⟨“𝐴𝐵𝐶𝐷”⟩
cs5 14767	class ⟨“𝐴𝐵𝐶𝐷𝐸”⟩
cs6 14768	class ⟨“𝐴𝐵𝐶𝐷𝐸𝐹”⟩
cs7 14769	class ⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”⟩
cs8 14770	class ⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺𝐻”⟩
df-s2 14771	⊢ ⟨“𝐴𝐵”⟩ = (⟨“𝐴”⟩ ++ ⟨“𝐵”⟩)
df-s3 14772	⊢ ⟨“𝐴𝐵𝐶”⟩ = (⟨“𝐴𝐵”⟩ ++ ⟨“𝐶”⟩)
df-s4 14773	⊢ ⟨“𝐴𝐵𝐶𝐷”⟩ = (⟨“𝐴𝐵𝐶”⟩ ++ ⟨“𝐷”⟩)
df-s5 14774	⊢ ⟨“𝐴𝐵𝐶𝐷𝐸”⟩ = (⟨“𝐴𝐵𝐶𝐷”⟩ ++ ⟨“𝐸”⟩)
df-s6 14775	⊢ ⟨“𝐴𝐵𝐶𝐷𝐸𝐹”⟩ = (⟨“𝐴𝐵𝐶𝐷𝐸”⟩ ++ ⟨“𝐹”⟩)
df-s7 14776	⊢ ⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”⟩ = (⟨“𝐴𝐵𝐶𝐷𝐸𝐹”⟩ ++ ⟨“𝐺”⟩)
df-s8 14777	⊢ ⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺𝐻”⟩ = (⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”⟩ ++ ⟨“𝐻”⟩)
ctcl 14908	class t+
crtcl 14909	class t*
df-trcl 14910	⊢ t+ = (𝑥 ∈ V ↦ ∩ {𝑧 ∣ (𝑥 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)})
df-rtrcl 14911	⊢ t* = (𝑥 ∈ V ↦ ∩ {𝑧 ∣ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧 ∧ 𝑥 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)})
crelexp 14942	class ↑_𝑟
df-relexp 14943	⊢ ↑_𝑟 = (𝑟 ∈ V, 𝑛 ∈ ℕ₀ ↦ if(𝑛 = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛)))
crtrcl 14978	class t*rec
df-rtrclrec 14979	⊢ t*rec = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ₀ (𝑟↑_𝑟𝑛))
cshi 14989	class shift
df-shft 14990	⊢ shift = (𝑓 ∈ V, 𝑥 ∈ ℂ ↦ {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ ℂ ∧ (𝑦 − 𝑥)𝑓𝑧)})
csgn 15009	class sgn
df-sgn 15010	⊢ sgn = (𝑥 ∈ ℝ^* ↦ if(𝑥 = 0, 0, if(𝑥 < 0, -1, 1)))
ccj 15019	class ∗
cre 15020	class ℜ
cim 15021	class ℑ
df-cj 15022	⊢ ∗ = (𝑥 ∈ ℂ ↦ (℩𝑦 ∈ ℂ ((𝑥 + 𝑦) ∈ ℝ ∧ (i · (𝑥 − 𝑦)) ∈ ℝ)))
df-re 15023	⊢ ℜ = (𝑥 ∈ ℂ ↦ ((𝑥 + (∗‘𝑥)) / 2))
df-im 15024	⊢ ℑ = (𝑥 ∈ ℂ ↦ (ℜ‘(𝑥 / i)))
csqrt 15156	class √
cabs 15157	class abs
df-sqrt 15158	⊢ √ = (𝑥 ∈ ℂ ↦ (℩𝑦 ∈ ℂ ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ⁺)))
df-abs 15159	⊢ abs = (𝑥 ∈ ℂ ↦ (√‘(𝑥 · (∗‘𝑥))))
clsp 15393	class lim sup
df-limsup 15394	⊢ lim sup = (𝑥 ∈ V ↦ inf(ran (𝑘 ∈ ℝ ↦ sup(((𝑥 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )), ℝ^*, < ))
cli 15407	class ⇝
crli 15408	class ⇝_𝑟
co1 15409	class 𝑂(1)
clo1 15410	class ≤𝑂(1)
df-clim 15411	⊢ ⇝ = {⟨𝑓, 𝑦⟩ ∣ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝑓‘𝑘) ∈ ℂ ∧ (abs‘((𝑓‘𝑘) − 𝑦)) < 𝑥))}
df-rlim 15412	⊢ ⇝_𝑟 = {⟨𝑓, 𝑥⟩ ∣ ((𝑓 ∈ (ℂ ↑_pm ℝ) ∧ 𝑥 ∈ ℂ) ∧ ∀𝑦 ∈ ℝ⁺ ∃𝑧 ∈ ℝ ∀𝑤 ∈ dom 𝑓(𝑧 ≤ 𝑤 → (abs‘((𝑓‘𝑤) − 𝑥)) < 𝑦))}
df-o1 15413	⊢ 𝑂(1) = {𝑓 ∈ (ℂ ↑_pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(abs‘(𝑓‘𝑦)) ≤ 𝑚}
df-lo1 15414	⊢ ≤𝑂(1) = {𝑓 ∈ (ℝ ↑_pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(𝑓‘𝑦) ≤ 𝑚}
csu 15609	class Σ𝑘 ∈ 𝐴 𝐵
df-sum 15610	⊢ Σ𝑘 ∈ 𝐴 𝐵 = (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ_≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚))))
cprod 15826	class ∏𝑘 ∈ 𝐴 𝐵
df-prod 15827	⊢ ∏𝑘 ∈ 𝐴 𝐵 = (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ_≥‘𝑚) ∧ ∃𝑛 ∈ (ℤ_≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚))))
cfallfac 15927	class FallFac
crisefac 15928	class RiseFac
df-risefac 15929	⊢ RiseFac = (𝑥 ∈ ℂ, 𝑛 ∈ ℕ₀ ↦ ∏𝑘 ∈ (0...(𝑛 − 1))(𝑥 + 𝑘))
df-fallfac 15930	⊢ FallFac = (𝑥 ∈ ℂ, 𝑛 ∈ ℕ₀ ↦ ∏𝑘 ∈ (0...(𝑛 − 1))(𝑥 − 𝑘))
cbp 15969	class BernPoly
df-bpoly 15970	⊢ BernPoly = (𝑚 ∈ ℕ₀, 𝑥 ∈ ℂ ↦ (wrecs( < , ℕ₀, (𝑔 ∈ V ↦ ⦋(♯‘dom 𝑔) / 𝑛⦌((𝑥↑𝑛) − Σ𝑘 ∈ dom 𝑔((𝑛C𝑘) · ((𝑔‘𝑘) / ((𝑛 − 𝑘) + 1))))))‘𝑚))
ce 15984	class exp
ceu 15985	class e
csin 15986	class sin
ccos 15987	class cos
ctan 15988	class tan
cpi 15989	class π
df-ef 15990	⊢ exp = (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ ℕ₀ ((𝑥↑𝑘) / (!‘𝑘)))
df-e 15991	⊢ e = (exp‘1)
df-sin 15992	⊢ sin = (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) − (exp‘(-i · 𝑥))) / (2 · i)))
df-cos 15993	⊢ cos = (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥))) / 2))
df-tan 15994	⊢ tan = (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) ↦ ((sin‘𝑥) / (cos‘𝑥)))
df-pi 15995	⊢ π = inf((ℝ⁺ ∩ (^◡sin “ {0})), ℝ, < )
ctau 16127	class τ
df-tau 16128	⊢ τ = inf((ℝ⁺ ∩ (^◡cos “ {1})), ℝ, < )
cdvds 16179	class ∥
df-dvds 16180	⊢ ∥ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ ∃𝑛 ∈ ℤ (𝑛 · 𝑥) = 𝑦)}
cbits 16346	class bits
csad 16347	class sadd
csmu 16348	class smul
df-bits 16349	⊢ bits = (𝑛 ∈ ℤ ↦ {𝑚 ∈ ℕ₀ ∣ ¬ 2 ∥ (⌊‘(𝑛 / (2↑𝑚)))})
df-sad 16378	⊢ sadd = (𝑥 ∈ 𝒫 ℕ₀, 𝑦 ∈ 𝒫 ℕ₀ ↦ {𝑘 ∈ ℕ₀ ∣ hadd(𝑘 ∈ 𝑥, 𝑘 ∈ 𝑦, ∅ ∈ (seq0((𝑐 ∈ 2_o, 𝑚 ∈ ℕ₀ ↦ if(cadd(𝑚 ∈ 𝑥, 𝑚 ∈ 𝑦, ∅ ∈ 𝑐), 1_o, ∅)), (𝑛 ∈ ℕ₀ ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑘))})
df-smu 16403	⊢ smul = (𝑥 ∈ 𝒫 ℕ₀, 𝑦 ∈ 𝒫 ℕ₀ ↦ {𝑘 ∈ ℕ₀ ∣ 𝑘 ∈ (seq0((𝑝 ∈ 𝒫 ℕ₀, 𝑚 ∈ ℕ₀ ↦ (𝑝 sadd {𝑛 ∈ ℕ₀ ∣ (𝑚 ∈ 𝑥 ∧ (𝑛 − 𝑚) ∈ 𝑦)})), (𝑛 ∈ ℕ₀ ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘(𝑘 + 1))})
cgcd 16421	class gcd
df-gcd 16422	⊢ gcd = (𝑥 ∈ ℤ, 𝑦 ∈ ℤ ↦ if((𝑥 = 0 ∧ 𝑦 = 0), 0, sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑥 ∧ 𝑛 ∥ 𝑦)}, ℝ, < )))
clcm 16515	class lcm
clcmf 16516	class lcm
df-lcm 16517	⊢ lcm = (𝑥 ∈ ℤ, 𝑦 ∈ ℤ ↦ if((𝑥 = 0 ∨ 𝑦 = 0), 0, inf({𝑛 ∈ ℕ ∣ (𝑥 ∥ 𝑛 ∧ 𝑦 ∥ 𝑛)}, ℝ, < )))
df-lcmf 16518	⊢ lcm = (𝑧 ∈ 𝒫 ℤ ↦ if(0 ∈ 𝑧, 0, inf({𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑧 𝑚 ∥ 𝑛}, ℝ, < )))
cprime 16598	class ℙ
df-prm 16599	⊢ ℙ = {𝑝 ∈ ℕ ∣ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑝} ≈ 2_o}
cnumer 16660	class numer
cdenom 16661	class denom
df-numer 16662	⊢ numer = (𝑦 ∈ ℚ ↦ (1^st ‘(℩𝑥 ∈ (ℤ × ℕ)(((1^st ‘𝑥) gcd (2^nd ‘𝑥)) = 1 ∧ 𝑦 = ((1^st ‘𝑥) / (2^nd ‘𝑥))))))
df-denom 16663	⊢ denom = (𝑦 ∈ ℚ ↦ (2^nd ‘(℩𝑥 ∈ (ℤ × ℕ)(((1^st ‘𝑥) gcd (2^nd ‘𝑥)) = 1 ∧ 𝑦 = ((1^st ‘𝑥) / (2^nd ‘𝑥))))))
codz 16690	class od_ℤ
cphi 16691	class ϕ
df-odz 16692	⊢ od_ℤ = (𝑛 ∈ ℕ ↦ (𝑥 ∈ {𝑥 ∈ ℤ ∣ (𝑥 gcd 𝑛) = 1} ↦ inf({𝑚 ∈ ℕ ∣ 𝑛 ∥ ((𝑥↑𝑚) − 1)}, ℝ, < )))
df-phi 16693	⊢ ϕ = (𝑛 ∈ ℕ ↦ (♯‘{𝑥 ∈ (1...𝑛) ∣ (𝑥 gcd 𝑛) = 1}))
cpc 16764	class pCnt
df-pc 16765	⊢ pCnt = (𝑝 ∈ ℙ, 𝑟 ∈ ℚ ↦ if(𝑟 = 0, +∞, (℩𝑧∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑟 = (𝑥 / 𝑦) ∧ 𝑧 = (sup({𝑛 ∈ ℕ₀ ∣ (𝑝↑𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ₀ ∣ (𝑝↑𝑛) ∥ 𝑦}, ℝ, < ))))))
cgz 16857	class ℤ[i]
df-gz 16858	⊢ ℤ[i] = {𝑥 ∈ ℂ ∣ ((ℜ‘𝑥) ∈ ℤ ∧ (ℑ‘𝑥) ∈ ℤ)}
cvdwa 16893	class AP
cvdwm 16894	class MonoAP
cvdwp 16895	class PolyAP
df-vdwap 16896	⊢ AP = (𝑘 ∈ ℕ₀ ↦ (𝑎 ∈ ℕ, 𝑑 ∈ ℕ ↦ ran (𝑚 ∈ (0...(𝑘 − 1)) ↦ (𝑎 + (𝑚 · 𝑑)))))
df-vdwmc 16897	⊢ MonoAP = {⟨𝑘, 𝑓⟩ ∣ ∃𝑐(ran (AP‘𝑘) ∩ 𝒫 (^◡𝑓 “ {𝑐})) ≠ ∅}
df-vdwpc 16898	⊢ PolyAP = {⟨⟨𝑚, 𝑘⟩, 𝑓⟩ ∣ ∃𝑎 ∈ ℕ ∃𝑑 ∈ (ℕ ↑_m (1...𝑚))(∀𝑖 ∈ (1...𝑚)((𝑎 + (𝑑‘𝑖))(AP‘𝑘)(𝑑‘𝑖)) ⊆ (^◡𝑓 “ {(𝑓‘(𝑎 + (𝑑‘𝑖)))}) ∧ (♯‘ran (𝑖 ∈ (1...𝑚) ↦ (𝑓‘(𝑎 + (𝑑‘𝑖))))) = 𝑚)}
cram 16927	class Ramsey
df-ram 16929	⊢ Ramsey = (𝑚 ∈ ℕ₀, 𝑟 ∈ V ↦ inf({𝑛 ∈ ℕ₀ ∣ ∀𝑠(𝑛 ≤ (♯‘𝑠) → ∀𝑓 ∈ (dom 𝑟 ↑_m {𝑦 ∈ 𝒫 𝑠 ∣ (♯‘𝑦) = 𝑚})∃𝑐 ∈ dom 𝑟∃𝑥 ∈ 𝒫 𝑠((𝑟‘𝑐) ≤ (♯‘𝑥) ∧ ∀𝑦 ∈ 𝒫 𝑥((♯‘𝑦) = 𝑚 → (𝑓‘𝑦) = 𝑐)))}, ℝ^*, < ))
cprmo 16959	class #_p
df-prmo 16960	⊢ #_p = (𝑛 ∈ ℕ₀ ↦ ∏𝑘 ∈ (1...𝑛)if(𝑘 ∈ ℙ, 𝑘, 1))
cstr 17073	class Struct
df-struct 17074	⊢ Struct = {⟨𝑓, 𝑥⟩ ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))}
csts 17090	class sSet
df-sets 17091	⊢ sSet = (𝑠 ∈ V, 𝑒 ∈ V ↦ ((𝑠 ↾ (V ∖ dom {𝑒})) ∪ {𝑒}))
cslot 17108	class Slot 𝐴
df-slot 17109	⊢ Slot 𝐴 = (𝑥 ∈ V ↦ (𝑥‘𝐴))
cnx 17120	class ndx
df-ndx 17121	⊢ ndx = ( I ↾ ℕ)
cbs 17136	class Base
df-base 17137	⊢ Base = Slot 1
cress 17157	class ↾_s
df-ress 17158	⊢ ↾_s = (𝑤 ∈ V, 𝑥 ∈ V ↦ if((Base‘𝑤) ⊆ 𝑥, 𝑤, (𝑤 sSet ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩)))
cplusg 17177	class +_g
cmulr 17178	class ._r
cstv 17179	class *_𝑟
csca 17180	class Scalar
cvsca 17181	class ·_𝑠
cip 17182	class ·_𝑖
cts 17183	class TopSet
cple 17184	class le
coc 17185	class oc
cds 17186	class dist
cunif 17187	class UnifSet
chom 17188	class Hom
cco 17189	class comp
df-plusg 17190	⊢ +_g = Slot 2
df-mulr 17191	⊢ ._r = Slot 3
df-starv 17192	⊢ *_𝑟 = Slot 4
df-sca 17193	⊢ Scalar = Slot 5
df-vsca 17194	⊢ ·_𝑠 = Slot 6
df-ip 17195	⊢ ·_𝑖 = Slot 8
df-tset 17196	⊢ TopSet = Slot 9
df-ple 17197	⊢ le = Slot ;10
df-ocomp 17198	⊢ oc = Slot ;11
df-ds 17199	⊢ dist = Slot ;12
df-unif 17200	⊢ UnifSet = Slot ;13
df-hom 17201	⊢ Hom = Slot ;14
df-cco 17202	⊢ comp = Slot ;15
crest 17340	class ↾_t
ctopn 17341	class TopOpen
df-rest 17342	⊢ ↾_t = (𝑗 ∈ V, 𝑥 ∈ V ↦ ran (𝑦 ∈ 𝑗 ↦ (𝑦 ∩ 𝑥)))
df-topn 17343	⊢ TopOpen = (𝑤 ∈ V ↦ ((TopSet‘𝑤) ↾_t (Base‘𝑤)))
ctg 17357	class topGen
cpt 17358	class ∏_t
c0g 17359	class 0_g
cgsu 17360	class Σ_g
df-0g 17361	⊢ 0_g = (𝑔 ∈ V ↦ (℩𝑒(𝑒 ∈ (Base‘𝑔) ∧ ∀𝑥 ∈ (Base‘𝑔)((𝑒(+_g‘𝑔)𝑥) = 𝑥 ∧ (𝑥(+_g‘𝑔)𝑒) = 𝑥))))
df-gsum 17362	⊢ Σ_g = (𝑤 ∈ V, 𝑓 ∈ V ↦ ⦋{𝑥 ∈ (Base‘𝑤) ∣ ∀𝑦 ∈ (Base‘𝑤)((𝑥(+_g‘𝑤)𝑦) = 𝑦 ∧ (𝑦(+_g‘𝑤)𝑥) = 𝑦)} / 𝑜⦌if(ran 𝑓 ⊆ 𝑜, (0_g‘𝑤), if(dom 𝑓 ∈ ran ..., (℩𝑥∃𝑚∃𝑛 ∈ (ℤ_≥‘𝑚)(dom 𝑓 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+_g‘𝑤), 𝑓)‘𝑛))), (℩𝑥∃𝑔[(^◡𝑓 “ (V ∖ 𝑜)) / 𝑦](𝑔:(1...(♯‘𝑦))–1-1-onto→𝑦 ∧ 𝑥 = (seq1((+_g‘𝑤), (𝑓 ∘ 𝑔))‘(♯‘𝑦)))))))
df-topgen 17363	⊢ topGen = (𝑥 ∈ V ↦ {𝑦 ∣ 𝑦 ⊆ ∪ (𝑥 ∩ 𝒫 𝑦)})
df-pt 17364	⊢ ∏_t = (𝑓 ∈ V ↦ (topGen‘{𝑥 ∣ ∃𝑔((𝑔 Fn dom 𝑓 ∧ ∀𝑦 ∈ dom 𝑓(𝑔‘𝑦) ∈ (𝑓‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (dom 𝑓 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝑓‘𝑦)) ∧ 𝑥 = X𝑦 ∈ dom 𝑓(𝑔‘𝑦))}))
cprds 17365	class X_s
cpws 17366	class ↑_s
df-prds 17367	⊢ X_s = (𝑠 ∈ V, 𝑟 ∈ V ↦ ⦋X𝑥 ∈ dom 𝑟(Base‘(𝑟‘𝑥)) / 𝑣⦌⦋(𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ X𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥))) / ℎ⦌(({⟨(Base‘ndx), 𝑣⟩, ⟨(+_g‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(+_g‘(𝑟‘𝑥))(𝑔‘𝑥))))⟩, ⟨(._r‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(._r‘(𝑟‘𝑥))(𝑔‘𝑥))))⟩} ∪ {⟨(Scalar‘ndx), 𝑠⟩, ⟨( ·_𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑠), 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ (𝑓( ·_𝑠 ‘(𝑟‘𝑥))(𝑔‘𝑥))))⟩, ⟨(·_𝑖‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑠 Σ_g (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(·_𝑖‘(𝑟‘𝑥))(𝑔‘𝑥)))))⟩}) ∪ ({⟨(TopSet‘ndx), (∏_t‘(TopOpen ∘ 𝑟))⟩, ⟨(le‘ndx), {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝑣 ∧ ∀𝑥 ∈ dom 𝑟(𝑓‘𝑥)(le‘(𝑟‘𝑥))(𝑔‘𝑥))}⟩, ⟨(dist‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ sup((ran (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(dist‘(𝑟‘𝑥))(𝑔‘𝑥))) ∪ {0}), ℝ^*, < ))⟩} ∪ {⟨(Hom ‘ndx), ℎ⟩, ⟨(comp‘ndx), (𝑎 ∈ (𝑣 × 𝑣), 𝑐 ∈ 𝑣 ↦ (𝑑 ∈ ((2^nd ‘𝑎)ℎ𝑐), 𝑒 ∈ (ℎ‘𝑎) ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑑‘𝑥)(⟨((1^st ‘𝑎)‘𝑥), ((2^nd ‘𝑎)‘𝑥)⟩(comp‘(𝑟‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥)))))⟩})))
df-pws 17369	⊢ ↑_s = (𝑟 ∈ V, 𝑖 ∈ V ↦ ((Scalar‘𝑟)X_s(𝑖 × {𝑟})))
cordt 17420	class ordTop
cxrs 17421	class ℝ^*_𝑠
df-ordt 17422	⊢ ordTop = (𝑟 ∈ V ↦ (topGen‘(fi‘({dom 𝑟} ∪ ran ((𝑥 ∈ dom 𝑟 ↦ {𝑦 ∈ dom 𝑟 ∣ ¬ 𝑦𝑟𝑥}) ∪ (𝑥 ∈ dom 𝑟 ↦ {𝑦 ∈ dom 𝑟 ∣ ¬ 𝑥𝑟𝑦}))))))
df-xrs 17423	⊢ ℝ^_𝑠 = ({⟨(Base‘ndx), ℝ^⟩, ⟨(+_g‘ndx), +_𝑒 ⟩, ⟨(._r‘ndx), ·_e ⟩} ∪ {⟨(TopSet‘ndx), (ordTop‘ ≤ )⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ if(𝑥 ≤ 𝑦, (𝑦 +_𝑒 -_𝑒𝑥), (𝑥 +_𝑒 -_𝑒𝑦)))⟩})
cqtop 17424	class qTop
cimas 17425	class “_s
cqus 17426	class /_s
cxps 17427	class ×_s
df-qtop 17428	⊢ qTop = (𝑗 ∈ V, 𝑓 ∈ V ↦ {𝑠 ∈ 𝒫 (𝑓 “ ∪ 𝑗) ∣ ((^◡𝑓 “ 𝑠) ∩ ∪ 𝑗) ∈ 𝑗})
df-imas 17429	⊢ “_s = (𝑓 ∈ V, 𝑟 ∈ V ↦ ⦋(Base‘𝑟) / 𝑣⦌(({⟨(Base‘ndx), ran 𝑓⟩, ⟨(+_g‘ndx), ∪ 𝑝 ∈ 𝑣 ∪ 𝑞 ∈ 𝑣 {⟨⟨(𝑓‘𝑝), (𝑓‘𝑞)⟩, (𝑓‘(𝑝(+_g‘𝑟)𝑞))⟩}⟩, ⟨(._r‘ndx), ∪ 𝑝 ∈ 𝑣 ∪ 𝑞 ∈ 𝑣 {⟨⟨(𝑓‘𝑝), (𝑓‘𝑞)⟩, (𝑓‘(𝑝(._r‘𝑟)𝑞))⟩}⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑟)⟩, ⟨( ·_𝑠 ‘ndx), ∪ 𝑞 ∈ 𝑣 (𝑝 ∈ (Base‘(Scalar‘𝑟)), 𝑥 ∈ {(𝑓‘𝑞)} ↦ (𝑓‘(𝑝( ·_𝑠 ‘𝑟)𝑞)))⟩, ⟨(·_𝑖‘ndx), ∪ 𝑝 ∈ 𝑣 ∪ 𝑞 ∈ 𝑣 {⟨⟨(𝑓‘𝑝), (𝑓‘𝑞)⟩, (𝑝(·_𝑖‘𝑟)𝑞)⟩}⟩}) ∪ {⟨(TopSet‘ndx), ((TopOpen‘𝑟) qTop 𝑓)⟩, ⟨(le‘ndx), ((𝑓 ∘ (le‘𝑟)) ∘ ^◡𝑓)⟩, ⟨(dist‘ndx), (𝑥 ∈ ran 𝑓, 𝑦 ∈ ran 𝑓 ↦ inf(∪ 𝑛 ∈ ℕ ran (𝑔 ∈ {ℎ ∈ ((𝑣 × 𝑣) ↑_m (1...𝑛)) ∣ ((𝑓‘(1^st ‘(ℎ‘1))) = 𝑥 ∧ (𝑓‘(2^nd ‘(ℎ‘𝑛))) = 𝑦 ∧ ∀𝑖 ∈ (1...(𝑛 − 1))(𝑓‘(2^nd ‘(ℎ‘𝑖))) = (𝑓‘(1^st ‘(ℎ‘(𝑖 + 1)))))} ↦ (ℝ^_𝑠 Σ_g ((dist‘𝑟) ∘ 𝑔))), ℝ^, < ))⟩}))
df-qus 17430	⊢ /_s = (𝑟 ∈ V, 𝑒 ∈ V ↦ ((𝑥 ∈ (Base‘𝑟) ↦ [𝑥]𝑒) “_s 𝑟))
df-xps 17431	⊢ ×_s = (𝑟 ∈ V, 𝑠 ∈ V ↦ (^◡(𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Base‘𝑠) ↦ {⟨∅, 𝑥⟩, ⟨1_o, 𝑦⟩}) “_s ((Scalar‘𝑟)X_s{⟨∅, 𝑟⟩, ⟨1_o, 𝑠⟩})))
cmre 17501	class Moore
cmrc 17502	class mrCls
cmri 17503	class mrInd
cacs 17504	class ACS
df-mre 17505	⊢ Moore = (𝑥 ∈ V ↦ {𝑐 ∈ 𝒫 𝒫 𝑥 ∣ (𝑥 ∈ 𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝑐))})
df-mrc 17506	⊢ mrCls = (𝑐 ∈ ∪ ran Moore ↦ (𝑥 ∈ 𝒫 ∪ 𝑐 ↦ ∩ {𝑠 ∈ 𝑐 ∣ 𝑥 ⊆ 𝑠}))
df-mri 17507	⊢ mrInd = (𝑐 ∈ ∪ ran Moore ↦ {𝑠 ∈ 𝒫 ∪ 𝑐 ∣ ∀𝑥 ∈ 𝑠 ¬ 𝑥 ∈ ((mrCls‘𝑐)‘(𝑠 ∖ {𝑥}))})
df-acs 17508	⊢ ACS = (𝑥 ∈ V ↦ {𝑐 ∈ (Moore‘𝑥) ∣ ∃𝑓(𝑓:𝒫 𝑥⟶𝒫 𝑥 ∧ ∀𝑠 ∈ 𝒫 𝑥(𝑠 ∈ 𝑐 ↔ ∪ (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠))})
ccat 17587	class Cat
ccid 17588	class Id
chomf 17589	class Hom_f
ccomf 17590	class comp_f
df-cat 17591	⊢ Cat = {𝑐 ∣ [(Base‘𝑐) / 𝑏][(Hom ‘𝑐) / ℎ][(comp‘𝑐) / 𝑜]∀𝑥 ∈ 𝑏 (∃𝑔 ∈ (𝑥ℎ𝑥)∀𝑦 ∈ 𝑏 (∀𝑓 ∈ (𝑦ℎ𝑥)(𝑔(⟨𝑦, 𝑥⟩𝑜𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥ℎ𝑦)(𝑓(⟨𝑥, 𝑥⟩𝑜𝑦)𝑔) = 𝑓) ∧ ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ∀𝑓 ∈ (𝑥ℎ𝑦)∀𝑔 ∈ (𝑦ℎ𝑧)((𝑔(⟨𝑥, 𝑦⟩𝑜𝑧)𝑓) ∈ (𝑥ℎ𝑧) ∧ ∀𝑤 ∈ 𝑏 ∀𝑘 ∈ (𝑧ℎ𝑤)((𝑘(⟨𝑦, 𝑧⟩𝑜𝑤)𝑔)(⟨𝑥, 𝑦⟩𝑜𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧⟩𝑜𝑤)(𝑔(⟨𝑥, 𝑦⟩𝑜𝑧)𝑓))))}
df-cid 17592	⊢ Id = (𝑐 ∈ Cat ↦ ⦋(Base‘𝑐) / 𝑏⦌⦋(Hom ‘𝑐) / ℎ⦌⦋(comp‘𝑐) / 𝑜⦌(𝑥 ∈ 𝑏 ↦ (℩𝑔 ∈ (𝑥ℎ𝑥)∀𝑦 ∈ 𝑏 (∀𝑓 ∈ (𝑦ℎ𝑥)(𝑔(⟨𝑦, 𝑥⟩𝑜𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥ℎ𝑦)(𝑓(⟨𝑥, 𝑥⟩𝑜𝑦)𝑔) = 𝑓))))
df-homf 17593	⊢ Hom_f = (𝑐 ∈ V ↦ (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ (𝑥(Hom ‘𝑐)𝑦)))
df-comf 17594	⊢ comp_f = (𝑐 ∈ V ↦ (𝑥 ∈ ((Base‘𝑐) × (Base‘𝑐)), 𝑦 ∈ (Base‘𝑐) ↦ (𝑔 ∈ ((2^nd ‘𝑥)(Hom ‘𝑐)𝑦), 𝑓 ∈ ((Hom ‘𝑐)‘𝑥) ↦ (𝑔(𝑥(comp‘𝑐)𝑦)𝑓))))
coppc 17634	class oppCat
df-oppc 17635	⊢ oppCat = (𝑓 ∈ V ↦ ((𝑓 sSet ⟨(Hom ‘ndx), tpos (Hom ‘𝑓)⟩) sSet ⟨(comp‘ndx), (𝑢 ∈ ((Base‘𝑓) × (Base‘𝑓)), 𝑧 ∈ (Base‘𝑓) ↦ tpos (⟨𝑧, (2^nd ‘𝑢)⟩(comp‘𝑓)(1^st ‘𝑢)))⟩))
cmon 17652	class Mono
cepi 17653	class Epi
df-mon 17654	⊢ Mono = (𝑐 ∈ Cat ↦ ⦋(Base‘𝑐) / 𝑏⦌⦋(Hom ‘𝑐) / ℎ⦌(𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ {𝑓 ∈ (𝑥ℎ𝑦) ∣ ∀𝑧 ∈ 𝑏 Fun ^◡(𝑔 ∈ (𝑧ℎ𝑥) ↦ (𝑓(⟨𝑧, 𝑥⟩(comp‘𝑐)𝑦)𝑔))}))
df-epi 17655	⊢ Epi = (𝑐 ∈ Cat ↦ tpos (Mono‘(oppCat‘𝑐)))
csect 17668	class Sect
cinv 17669	class Inv
ciso 17670	class Iso
df-sect 17671	⊢ Sect = (𝑐 ∈ Cat ↦ (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ {⟨𝑓, 𝑔⟩ ∣ [(Hom ‘𝑐) / ℎ]((𝑓 ∈ (𝑥ℎ𝑦) ∧ 𝑔 ∈ (𝑦ℎ𝑥)) ∧ (𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑥)𝑓) = ((Id‘𝑐)‘𝑥))}))
df-inv 17672	⊢ Inv = (𝑐 ∈ Cat ↦ (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ ((𝑥(Sect‘𝑐)𝑦) ∩ ^◡(𝑦(Sect‘𝑐)𝑥))))
df-iso 17673	⊢ Iso = (𝑐 ∈ Cat ↦ ((𝑥 ∈ V ↦ dom 𝑥) ∘ (Inv‘𝑐)))
ccic 17719	class ≃_𝑐
df-cic 17720	⊢ ≃_𝑐 = (𝑐 ∈ Cat ↦ ((Iso‘𝑐) supp ∅))
cssc 17731	class ⊆_cat
cresc 17732	class ↾_cat
csubc 17733	class Subcat
df-ssc 17734	⊢ ⊆_cat = {⟨ℎ, 𝑗⟩ ∣ ∃𝑡(𝑗 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝑗‘𝑥))}
df-resc 17735	⊢ ↾_cat = (𝑐 ∈ V, ℎ ∈ V ↦ ((𝑐 ↾_s dom dom ℎ) sSet ⟨(Hom ‘ndx), ℎ⟩))
df-subc 17736	⊢ Subcat = (𝑐 ∈ Cat ↦ {ℎ ∣ (ℎ ⊆_cat (Hom_f ‘𝑐) ∧ [dom dom ℎ / 𝑠]∀𝑥 ∈ 𝑠 (((Id‘𝑐)‘𝑥) ∈ (𝑥ℎ𝑥) ∧ ∀𝑦 ∈ 𝑠 ∀𝑧 ∈ 𝑠 ∀𝑓 ∈ (𝑥ℎ𝑦)∀𝑔 ∈ (𝑦ℎ𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑧)𝑓) ∈ (𝑥ℎ𝑧)))})
cfunc 17778	class Func
cidfu 17779	class id_func
ccofu 17780	class ∘_func
cresf 17781	class ↾_f
df-func 17782	⊢ Func = (𝑡 ∈ Cat, 𝑢 ∈ Cat ↦ {⟨𝑓, 𝑔⟩ ∣ [(Base‘𝑡) / 𝑏](𝑓:𝑏⟶(Base‘𝑢) ∧ 𝑔 ∈ X𝑧 ∈ (𝑏 × 𝑏)(((𝑓‘(1^st ‘𝑧))(Hom ‘𝑢)(𝑓‘(2^nd ‘𝑧))) ↑_m ((Hom ‘𝑡)‘𝑧)) ∧ ∀𝑥 ∈ 𝑏 (((𝑥𝑔𝑥)‘((Id‘𝑡)‘𝑥)) = ((Id‘𝑢)‘(𝑓‘𝑥)) ∧ ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ∀𝑚 ∈ (𝑥(Hom ‘𝑡)𝑦)∀𝑛 ∈ (𝑦(Hom ‘𝑡)𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦⟩(comp‘𝑡)𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓‘𝑥), (𝑓‘𝑦)⟩(comp‘𝑢)(𝑓‘𝑧))((𝑥𝑔𝑦)‘𝑚))))})
df-idfu 17783	⊢ id_func = (𝑡 ∈ Cat ↦ ⦋(Base‘𝑡) / 𝑏⦌⟨( I ↾ 𝑏), (𝑧 ∈ (𝑏 × 𝑏) ↦ ( I ↾ ((Hom ‘𝑡)‘𝑧)))⟩)
df-cofu 17784	⊢ ∘_func = (𝑔 ∈ V, 𝑓 ∈ V ↦ ⟨((1^st ‘𝑔) ∘ (1^st ‘𝑓)), (𝑥 ∈ dom dom (2^nd ‘𝑓), 𝑦 ∈ dom dom (2^nd ‘𝑓) ↦ ((((1^st ‘𝑓)‘𝑥)(2^nd ‘𝑔)((1^st ‘𝑓)‘𝑦)) ∘ (𝑥(2^nd ‘𝑓)𝑦)))⟩)
df-resf 17785	⊢ ↾_f = (𝑓 ∈ V, ℎ ∈ V ↦ ⟨((1^st ‘𝑓) ↾ dom dom ℎ), (𝑥 ∈ dom ℎ ↦ (((2^nd ‘𝑓)‘𝑥) ↾ (ℎ‘𝑥)))⟩)
cful 17828	class Full
cfth 17829	class Faith
df-full 17830	⊢ Full = (𝑐 ∈ Cat, 𝑑 ∈ Cat ↦ {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝑐 Func 𝑑)𝑔 ∧ ∀𝑥 ∈ (Base‘𝑐)∀𝑦 ∈ (Base‘𝑐)ran (𝑥𝑔𝑦) = ((𝑓‘𝑥)(Hom ‘𝑑)(𝑓‘𝑦)))})
df-fth 17831	⊢ Faith = (𝑐 ∈ Cat, 𝑑 ∈ Cat ↦ {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝑐 Func 𝑑)𝑔 ∧ ∀𝑥 ∈ (Base‘𝑐)∀𝑦 ∈ (Base‘𝑐)Fun ^◡(𝑥𝑔𝑦))})
cnat 17868	class Nat
cfuc 17869	class FuncCat
df-nat 17870	⊢ Nat = (𝑡 ∈ Cat, 𝑢 ∈ Cat ↦ (𝑓 ∈ (𝑡 Func 𝑢), 𝑔 ∈ (𝑡 Func 𝑢) ↦ ⦋(1^st ‘𝑓) / 𝑟⦌⦋(1^st ‘𝑔) / 𝑠⦌{𝑎 ∈ X𝑥 ∈ (Base‘𝑡)((𝑟‘𝑥)(Hom ‘𝑢)(𝑠‘𝑥)) ∣ ∀𝑥 ∈ (Base‘𝑡)∀𝑦 ∈ (Base‘𝑡)∀ℎ ∈ (𝑥(Hom ‘𝑡)𝑦)((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩(comp‘𝑢)(𝑠‘𝑦))((𝑥(2^nd ‘𝑓)𝑦)‘ℎ)) = (((𝑥(2^nd ‘𝑔)𝑦)‘ℎ)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝑢)(𝑠‘𝑦))(𝑎‘𝑥))}))
df-fuc 17871	⊢ FuncCat = (𝑡 ∈ Cat, 𝑢 ∈ Cat ↦ {⟨(Base‘ndx), (𝑡 Func 𝑢)⟩, ⟨(Hom ‘ndx), (𝑡 Nat 𝑢)⟩, ⟨(comp‘ndx), (𝑣 ∈ ((𝑡 Func 𝑢) × (𝑡 Func 𝑢)), ℎ ∈ (𝑡 Func 𝑢) ↦ ⦋(1^st ‘𝑣) / 𝑓⦌⦋(2^nd ‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔(𝑡 Nat 𝑢)ℎ), 𝑎 ∈ (𝑓(𝑡 Nat 𝑢)𝑔) ↦ (𝑥 ∈ (Base‘𝑡) ↦ ((𝑏‘𝑥)(⟨((1^st ‘𝑓)‘𝑥), ((1^st ‘𝑔)‘𝑥)⟩(comp‘𝑢)((1^st ‘ℎ)‘𝑥))(𝑎‘𝑥)))))⟩})
cinito 17905	class InitO
ctermo 17906	class TermO
czeroo 17907	class ZeroO
df-inito 17908	⊢ InitO = (𝑐 ∈ Cat ↦ {𝑎 ∈ (Base‘𝑐) ∣ ∀𝑏 ∈ (Base‘𝑐)∃!ℎ ℎ ∈ (𝑎(Hom ‘𝑐)𝑏)})
df-termo 17909	⊢ TermO = (𝑐 ∈ Cat ↦ {𝑎 ∈ (Base‘𝑐) ∣ ∀𝑏 ∈ (Base‘𝑐)∃!ℎ ℎ ∈ (𝑏(Hom ‘𝑐)𝑎)})
df-zeroo 17910	⊢ ZeroO = (𝑐 ∈ Cat ↦ ((InitO‘𝑐) ∩ (TermO‘𝑐)))
cdoma 17944	class dom_a
ccoda 17945	class cod_a
carw 17946	class Arrow
choma 17947	class Hom_a
df-doma 17948	⊢ dom_a = (1^st ∘ 1^st )
df-coda 17949	⊢ cod_a = (2^nd ∘ 1^st )
df-homa 17950	⊢ Hom_a = (𝑐 ∈ Cat ↦ (𝑥 ∈ ((Base‘𝑐) × (Base‘𝑐)) ↦ ({𝑥} × ((Hom ‘𝑐)‘𝑥))))
df-arw 17951	⊢ Arrow = (𝑐 ∈ Cat ↦ ∪ ran (Hom_a‘𝑐))
cida 17977	class Id_a
ccoa 17978	class comp_a
df-ida 17979	⊢ Id_a = (𝑐 ∈ Cat ↦ (𝑥 ∈ (Base‘𝑐) ↦ ⟨𝑥, 𝑥, ((Id‘𝑐)‘𝑥)⟩))
df-coa 17980	⊢ comp_a = (𝑐 ∈ Cat ↦ (𝑔 ∈ (Arrow‘𝑐), 𝑓 ∈ {ℎ ∈ (Arrow‘𝑐) ∣ (cod_a‘ℎ) = (dom_a‘𝑔)} ↦ ⟨(dom_a‘𝑓), (cod_a‘𝑔), ((2^nd ‘𝑔)(⟨(dom_a‘𝑓), (dom_a‘𝑔)⟩(comp‘𝑐)(cod_a‘𝑔))(2^nd ‘𝑓))⟩))
csetc 17999	class SetCat
df-setc 18000	⊢ SetCat = (𝑢 ∈ V ↦ {⟨(Base‘ndx), 𝑢⟩, ⟨(Hom ‘ndx), (𝑥 ∈ 𝑢, 𝑦 ∈ 𝑢 ↦ (𝑦 ↑_m 𝑥))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝑢 × 𝑢), 𝑧 ∈ 𝑢 ↦ (𝑔 ∈ (𝑧 ↑_m (2^nd ‘𝑣)), 𝑓 ∈ ((2^nd ‘𝑣) ↑_m (1^st ‘𝑣)) ↦ (𝑔 ∘ 𝑓)))⟩})
ccatc 18022	class CatCat
df-catc 18023	⊢ CatCat = (𝑢 ∈ V ↦ ⦋(𝑢 ∩ Cat) / 𝑏⦌{⟨(Base‘ndx), 𝑏⟩, ⟨(Hom ‘ndx), (𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ (𝑥 Func 𝑦))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝑏 × 𝑏), 𝑧 ∈ 𝑏 ↦ (𝑔 ∈ ((2^nd ‘𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔 ∘_func 𝑓)))⟩})
cestrc 18045	class ExtStrCat
df-estrc 18046	⊢ ExtStrCat = (𝑢 ∈ V ↦ {⟨(Base‘ndx), 𝑢⟩, ⟨(Hom ‘ndx), (𝑥 ∈ 𝑢, 𝑦 ∈ 𝑢 ↦ ((Base‘𝑦) ↑_m (Base‘𝑥)))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝑢 × 𝑢), 𝑧 ∈ 𝑢 ↦ (𝑔 ∈ ((Base‘𝑧) ↑_m (Base‘(2^nd ‘𝑣))), 𝑓 ∈ ((Base‘(2^nd ‘𝑣)) ↑_m (Base‘(1^st ‘𝑣))) ↦ (𝑔 ∘ 𝑓)))⟩})
cxpc 18091	class ×_c
c1stf 18092	class 1^st_F
c2ndf 18093	class 2^nd_F
cprf 18094	class ⟨,⟩_F
df-xpc 18095	⊢ ×_c = (𝑟 ∈ V, 𝑠 ∈ V ↦ ⦋((Base‘𝑟) × (Base‘𝑠)) / 𝑏⦌⦋(𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (((1^st ‘𝑢)(Hom ‘𝑟)(1^st ‘𝑣)) × ((2^nd ‘𝑢)(Hom ‘𝑠)(2^nd ‘𝑣)))) / ℎ⦌{⟨(Base‘ndx), 𝑏⟩, ⟨(Hom ‘ndx), ℎ⟩, ⟨(comp‘ndx), (𝑥 ∈ (𝑏 × 𝑏), 𝑦 ∈ 𝑏 ↦ (𝑔 ∈ ((2^nd ‘𝑥)ℎ𝑦), 𝑓 ∈ (ℎ‘𝑥) ↦ ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩(comp‘𝑟)(1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩(comp‘𝑠)(2^nd ‘𝑦))(2^nd ‘𝑓))⟩))⟩})
df-1stf 18096	⊢ 1^st_F = (𝑟 ∈ Cat, 𝑠 ∈ Cat ↦ ⦋((Base‘𝑟) × (Base‘𝑠)) / 𝑏⦌⟨(1^st ↾ 𝑏), (𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ (1^st ↾ (𝑥(Hom ‘(𝑟 ×_c 𝑠))𝑦)))⟩)
df-2ndf 18097	⊢ 2^nd_F = (𝑟 ∈ Cat, 𝑠 ∈ Cat ↦ ⦋((Base‘𝑟) × (Base‘𝑠)) / 𝑏⦌⟨(2^nd ↾ 𝑏), (𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ (2^nd ↾ (𝑥(Hom ‘(𝑟 ×_c 𝑠))𝑦)))⟩)
df-prf 18098	⊢ ⟨,⟩_F = (𝑓 ∈ V, 𝑔 ∈ V ↦ ⦋dom (1^st ‘𝑓) / 𝑏⦌⟨(𝑥 ∈ 𝑏 ↦ ⟨((1^st ‘𝑓)‘𝑥), ((1^st ‘𝑔)‘𝑥)⟩), (𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ (ℎ ∈ dom (𝑥(2^nd ‘𝑓)𝑦) ↦ ⟨((𝑥(2^nd ‘𝑓)𝑦)‘ℎ), ((𝑥(2^nd ‘𝑔)𝑦)‘ℎ)⟩))⟩)
cevlf 18132	class eval_F
ccurf 18133	class curry_F
cuncf 18134	class uncurry_F
cdiag 18135	class Δ_func
df-evlf 18136	⊢ eval_F = (𝑐 ∈ Cat, 𝑑 ∈ Cat ↦ ⟨(𝑓 ∈ (𝑐 Func 𝑑), 𝑥 ∈ (Base‘𝑐) ↦ ((1^st ‘𝑓)‘𝑥)), (𝑥 ∈ ((𝑐 Func 𝑑) × (Base‘𝑐)), 𝑦 ∈ ((𝑐 Func 𝑑) × (Base‘𝑐)) ↦ ⦋(1^st ‘𝑥) / 𝑚⦌⦋(1^st ‘𝑦) / 𝑛⦌(𝑎 ∈ (𝑚(𝑐 Nat 𝑑)𝑛), 𝑔 ∈ ((2^nd ‘𝑥)(Hom ‘𝑐)(2^nd ‘𝑦)) ↦ ((𝑎‘(2^nd ‘𝑦))(⟨((1^st ‘𝑚)‘(2^nd ‘𝑥)), ((1^st ‘𝑚)‘(2^nd ‘𝑦))⟩(comp‘𝑑)((1^st ‘𝑛)‘(2^nd ‘𝑦)))(((2^nd ‘𝑥)(2^nd ‘𝑚)(2^nd ‘𝑦))‘𝑔))))⟩)
df-curf 18137	⊢ curry_F = (𝑒 ∈ V, 𝑓 ∈ V ↦ ⦋(1^st ‘𝑒) / 𝑐⦌⦋(2^nd ‘𝑒) / 𝑑⦌⟨(𝑥 ∈ (Base‘𝑐) ↦ ⟨(𝑦 ∈ (Base‘𝑑) ↦ (𝑥(1^st ‘𝑓)𝑦)), (𝑦 ∈ (Base‘𝑑), 𝑧 ∈ (Base‘𝑑) ↦ (𝑔 ∈ (𝑦(Hom ‘𝑑)𝑧) ↦ (((Id‘𝑐)‘𝑥)(⟨𝑥, 𝑦⟩(2^nd ‘𝑓)⟨𝑥, 𝑧⟩)𝑔)))⟩), (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ (𝑔 ∈ (𝑥(Hom ‘𝑐)𝑦) ↦ (𝑧 ∈ (Base‘𝑑) ↦ (𝑔(⟨𝑥, 𝑧⟩(2^nd ‘𝑓)⟨𝑦, 𝑧⟩)((Id‘𝑑)‘𝑧)))))⟩)
df-uncf 18138	⊢ uncurry_F = (𝑐 ∈ V, 𝑓 ∈ V ↦ (((𝑐‘1) eval_F (𝑐‘2)) ∘_func ((𝑓 ∘_func ((𝑐‘0) 1^st_F (𝑐‘1))) ⟨,⟩_F ((𝑐‘0) 2^nd_F (𝑐‘1)))))
df-diag 18139	⊢ Δ_func = (𝑐 ∈ Cat, 𝑑 ∈ Cat ↦ (⟨𝑐, 𝑑⟩ curry_F (𝑐 1^st_F 𝑑)))
chof 18171	class Hom_F
cyon 18172	class Yon
df-hof 18173	⊢ Hom_F = (𝑐 ∈ Cat ↦ ⟨(Hom_f ‘𝑐), ⦋(Base‘𝑐) / 𝑏⦌(𝑥 ∈ (𝑏 × 𝑏), 𝑦 ∈ (𝑏 × 𝑏) ↦ (𝑓 ∈ ((1^st ‘𝑦)(Hom ‘𝑐)(1^st ‘𝑥)), 𝑔 ∈ ((2^nd ‘𝑥)(Hom ‘𝑐)(2^nd ‘𝑦)) ↦ (ℎ ∈ ((Hom ‘𝑐)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝑐)(2^nd ‘𝑦))ℎ)(⟨(1^st ‘𝑦), (1^st ‘𝑥)⟩(comp‘𝑐)(2^nd ‘𝑦))𝑓))))⟩)
df-yon 18174	⊢ Yon = (𝑐 ∈ Cat ↦ (⟨𝑐, (oppCat‘𝑐)⟩ curry_F (Hom_F‘(oppCat‘𝑐))))
codu 18209	class ODual
df-odu 18210	⊢ ODual = (𝑤 ∈ V ↦ (𝑤 sSet ⟨(le‘ndx), ^◡(le‘𝑤)⟩))
cproset 18215	class Proset
cdrs 18216	class Dirset
df-proset 18217	⊢ Proset = {𝑓 ∣ [(Base‘𝑓) / 𝑏][(le‘𝑓) / 𝑟]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) → 𝑥𝑟𝑧))}
df-drs 18218	⊢ Dirset = {𝑓 ∈ Proset ∣ [(Base‘𝑓) / 𝑏][(le‘𝑓) / 𝑟](𝑏 ≠ ∅ ∧ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∃𝑧 ∈ 𝑏 (𝑥𝑟𝑧 ∧ 𝑦𝑟𝑧))}
cpo 18230	class Poset
cplt 18231	class lt
club 18232	class lub
cglb 18233	class glb
cjn 18234	class join
cmee 18235	class meet
df-poset 18236	⊢ Poset = {𝑓 ∣ ∃𝑏∃𝑟(𝑏 = (Base‘𝑓) ∧ 𝑟 = (le‘𝑓) ∧ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑥) → 𝑥 = 𝑦) ∧ ((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) → 𝑥𝑟𝑧)))}
df-plt 18251	⊢ lt = (𝑝 ∈ V ↦ ((le‘𝑝) ∖ I ))
df-lub 18267	⊢ lub = (𝑝 ∈ V ↦ ((𝑠 ∈ 𝒫 (Base‘𝑝) ↦ (℩𝑥 ∈ (Base‘𝑝)(∀𝑦 ∈ 𝑠 𝑦(le‘𝑝)𝑥 ∧ ∀𝑧 ∈ (Base‘𝑝)(∀𝑦 ∈ 𝑠 𝑦(le‘𝑝)𝑧 → 𝑥(le‘𝑝)𝑧)))) ↾ {𝑠 ∣ ∃!𝑥 ∈ (Base‘𝑝)(∀𝑦 ∈ 𝑠 𝑦(le‘𝑝)𝑥 ∧ ∀𝑧 ∈ (Base‘𝑝)(∀𝑦 ∈ 𝑠 𝑦(le‘𝑝)𝑧 → 𝑥(le‘𝑝)𝑧))}))
df-glb 18268	⊢ glb = (𝑝 ∈ V ↦ ((𝑠 ∈ 𝒫 (Base‘𝑝) ↦ (℩𝑥 ∈ (Base‘𝑝)(∀𝑦 ∈ 𝑠 𝑥(le‘𝑝)𝑦 ∧ ∀𝑧 ∈ (Base‘𝑝)(∀𝑦 ∈ 𝑠 𝑧(le‘𝑝)𝑦 → 𝑧(le‘𝑝)𝑥)))) ↾ {𝑠 ∣ ∃!𝑥 ∈ (Base‘𝑝)(∀𝑦 ∈ 𝑠 𝑥(le‘𝑝)𝑦 ∧ ∀𝑧 ∈ (Base‘𝑝)(∀𝑦 ∈ 𝑠 𝑧(le‘𝑝)𝑦 → 𝑧(le‘𝑝)𝑥))}))
df-join 18269	⊢ join = (𝑝 ∈ V ↦ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦} (lub‘𝑝)𝑧})
df-meet 18270	⊢ meet = (𝑝 ∈ V ↦ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦} (glb‘𝑝)𝑧})
ctos 18337	class Toset
df-toset 18338	⊢ Toset = {𝑓 ∈ Poset ∣ [(Base‘𝑓) / 𝑏][(le‘𝑓) / 𝑟]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥𝑟𝑦 ∨ 𝑦𝑟𝑥)}
cp0 18344	class 0.
cp1 18345	class 1.
df-p0 18346	⊢ 0. = (𝑝 ∈ V ↦ ((glb‘𝑝)‘(Base‘𝑝)))
df-p1 18347	⊢ 1. = (𝑝 ∈ V ↦ ((lub‘𝑝)‘(Base‘𝑝)))
clat 18354	class Lat
df-lat 18355	⊢ Lat = {𝑝 ∈ Poset ∣ (dom (join‘𝑝) = ((Base‘𝑝) × (Base‘𝑝)) ∧ dom (meet‘𝑝) = ((Base‘𝑝) × (Base‘𝑝)))}
ccla 18421	class CLat
df-clat 18422	⊢ CLat = {𝑝 ∈ Poset ∣ (dom (lub‘𝑝) = 𝒫 (Base‘𝑝) ∧ dom (glb‘𝑝) = 𝒫 (Base‘𝑝))}
cdlat 18443	class DLat
df-dlat 18444	⊢ DLat = {𝑘 ∈ Lat ∣ [(Base‘𝑘) / 𝑏][(join‘𝑘) / 𝑗][(meet‘𝑘) / 𝑚]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 (𝑥𝑚(𝑦𝑗𝑧)) = ((𝑥𝑚𝑦)𝑗(𝑥𝑚𝑧))}
cipo 18450	class toInc
df-ipo 18451	⊢ toInc = (𝑓 ∈ V ↦ ⦋{⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑓 ∧ 𝑥 ⊆ 𝑦)} / 𝑜⦌({⟨(Base‘ndx), 𝑓⟩, ⟨(TopSet‘ndx), (ordTop‘𝑜)⟩} ∪ {⟨(le‘ndx), 𝑜⟩, ⟨(oc‘ndx), (𝑥 ∈ 𝑓 ↦ ∪ {𝑦 ∈ 𝑓 ∣ (𝑦 ∩ 𝑥) = ∅})⟩}))
cps 18487	class PosetRel
ctsr 18488	class TosetRel
df-ps 18489	⊢ PosetRel = {𝑟 ∣ (Rel 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟 ∧ (𝑟 ∩ ^◡𝑟) = ( I ↾ ∪ ∪ 𝑟))}
df-tsr 18490	⊢ TosetRel = {𝑟 ∈ PosetRel ∣ (dom 𝑟 × dom 𝑟) ⊆ (𝑟 ∪ ^◡𝑟)}
cdir 18517	class DirRel
ctail 18518	class tail
df-dir 18519	⊢ DirRel = {𝑟 ∣ ((Rel 𝑟 ∧ ( I ↾ ∪ ∪ 𝑟) ⊆ 𝑟) ∧ ((𝑟 ∘ 𝑟) ⊆ 𝑟 ∧ (∪ ∪ 𝑟 × ∪ ∪ 𝑟) ⊆ (^◡𝑟 ∘ 𝑟)))}
df-tail 18520	⊢ tail = (𝑟 ∈ DirRel ↦ (𝑥 ∈ ∪ ∪ 𝑟 ↦ (𝑟 “ {𝑥})))
cchn 18528	class ( < Chain 𝐴)
df-chn 18529	⊢ ( < Chain 𝐴) = {𝑐 ∈ Word 𝐴 ∣ ∀𝑛 ∈ (dom 𝑐 ∖ {0})(𝑐‘(𝑛 − 1)) < (𝑐‘𝑛)}
cplusf 18562	class +_𝑓
cmgm 18563	class Mgm
df-plusf 18564	⊢ +_𝑓 = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(+_g‘𝑔)𝑦)))
df-mgm 18565	⊢ Mgm = {𝑔 ∣ [(Base‘𝑔) / 𝑏][(+_g‘𝑔) / 𝑜]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥𝑜𝑦) ∈ 𝑏}
cmgmhm 18615	class MgmHom
csubmgm 18616	class SubMgm
df-mgmhm 18617	⊢ MgmHom = (𝑠 ∈ Mgm, 𝑡 ∈ Mgm ↦ {𝑓 ∈ ((Base‘𝑡) ↑_m (Base‘𝑠)) ∣ ∀𝑥 ∈ (Base‘𝑠)∀𝑦 ∈ (Base‘𝑠)(𝑓‘(𝑥(+_g‘𝑠)𝑦)) = ((𝑓‘𝑥)(+_g‘𝑡)(𝑓‘𝑦))})
df-submgm 18618	⊢ SubMgm = (𝑠 ∈ Mgm ↦ {𝑡 ∈ 𝒫 (Base‘𝑠) ∣ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (𝑥(+_g‘𝑠)𝑦) ∈ 𝑡})
csgrp 18643	class Smgrp
df-sgrp 18644	⊢ Smgrp = {𝑔 ∈ Mgm ∣ [(Base‘𝑔) / 𝑏][(+_g‘𝑔) / 𝑜]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧))}
cmnd 18659	class Mnd
df-mnd 18660	⊢ Mnd = {𝑔 ∈ Smgrp ∣ [(Base‘𝑔) / 𝑏][(+_g‘𝑔) / 𝑝]∃𝑒 ∈ 𝑏 ∀𝑥 ∈ 𝑏 ((𝑒𝑝𝑥) = 𝑥 ∧ (𝑥𝑝𝑒) = 𝑥)}
cmhm 18706	class MndHom
csubmnd 18707	class SubMnd
df-mhm 18708	⊢ MndHom = (𝑠 ∈ Mnd, 𝑡 ∈ Mnd ↦ {𝑓 ∈ ((Base‘𝑡) ↑_m (Base‘𝑠)) ∣ (∀𝑥 ∈ (Base‘𝑠)∀𝑦 ∈ (Base‘𝑠)(𝑓‘(𝑥(+_g‘𝑠)𝑦)) = ((𝑓‘𝑥)(+_g‘𝑡)(𝑓‘𝑦)) ∧ (𝑓‘(0_g‘𝑠)) = (0_g‘𝑡))})
df-submnd 18709	⊢ SubMnd = (𝑠 ∈ Mnd ↦ {𝑡 ∈ 𝒫 (Base‘𝑠) ∣ ((0_g‘𝑠) ∈ 𝑡 ∧ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (𝑥(+_g‘𝑠)𝑦) ∈ 𝑡)})
cfrmd 18772	class freeMnd
cvrmd 18773	class var_FMnd
df-frmd 18774	⊢ freeMnd = (𝑖 ∈ V ↦ {⟨(Base‘ndx), Word 𝑖⟩, ⟨(+_g‘ndx), ( ++ ↾ (Word 𝑖 × Word 𝑖))⟩})
df-vrmd 18775	⊢ var_FMnd = (𝑖 ∈ V ↦ (𝑗 ∈ 𝑖 ↦ ⟨“𝑗”⟩))
cefmnd 18793	class EndoFMnd
df-efmnd 18794	⊢ EndoFMnd = (𝑥 ∈ V ↦ ⦋(𝑥 ↑_m 𝑥) / 𝑏⦌{⟨(Base‘ndx), 𝑏⟩, ⟨(+_g‘ndx), (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (𝑓 ∘ 𝑔))⟩, ⟨(TopSet‘ndx), (∏_t‘(𝑥 × {𝒫 𝑥}))⟩})
cgrp 18863	class Grp
cminusg 18864	class inv_g
csg 18865	class -_g
df-grp 18866	⊢ Grp = {𝑔 ∈ Mnd ∣ ∀𝑎 ∈ (Base‘𝑔)∃𝑚 ∈ (Base‘𝑔)(𝑚(+_g‘𝑔)𝑎) = (0_g‘𝑔)}
df-minusg 18867	⊢ inv_g = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔) ↦ (℩𝑤 ∈ (Base‘𝑔)(𝑤(+_g‘𝑔)𝑥) = (0_g‘𝑔))))
df-sbg 18868	⊢ -_g = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(+_g‘𝑔)((inv_g‘𝑔)‘𝑦))))
cmg 18997	class ._g
df-mulg 18998	⊢ ._g = (𝑔 ∈ V ↦ (𝑛 ∈ ℤ, 𝑥 ∈ (Base‘𝑔) ↦ if(𝑛 = 0, (0_g‘𝑔), ⦋seq1((+_g‘𝑔), (ℕ × {𝑥})) / 𝑠⦌if(0 < 𝑛, (𝑠‘𝑛), ((inv_g‘𝑔)‘(𝑠‘-𝑛))))))
csubg 19050	class SubGrp
cnsg 19051	class NrmSGrp
cqg 19052	class ~_QG
df-subg 19053	⊢ SubGrp = (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾_s 𝑠) ∈ Grp})
df-nsg 19054	⊢ NrmSGrp = (𝑤 ∈ Grp ↦ {𝑠 ∈ (SubGrp‘𝑤) ∣ [(Base‘𝑤) / 𝑏][(+_g‘𝑤) / 𝑝]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ((𝑥𝑝𝑦) ∈ 𝑠 ↔ (𝑦𝑝𝑥) ∈ 𝑠)})
df-eqg 19055	⊢ ~_QG = (𝑟 ∈ V, 𝑖 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (Base‘𝑟) ∧ (((inv_g‘𝑟)‘𝑥)(+_g‘𝑟)𝑦) ∈ 𝑖)})
cghm 19141	class GrpHom
df-ghm 19142	⊢ GrpHom = (𝑠 ∈ Grp, 𝑡 ∈ Grp ↦ {𝑔 ∣ [(Base‘𝑠) / 𝑤](𝑔:𝑤⟶(Base‘𝑡) ∧ ∀𝑥 ∈ 𝑤 ∀𝑦 ∈ 𝑤 (𝑔‘(𝑥(+_g‘𝑠)𝑦)) = ((𝑔‘𝑥)(+_g‘𝑡)(𝑔‘𝑦)))})
cgim 19186	class GrpIso
cgic 19187	class ≃_𝑔
df-gim 19188	⊢ GrpIso = (𝑠 ∈ Grp, 𝑡 ∈ Grp ↦ {𝑔 ∈ (𝑠 GrpHom 𝑡) ∣ 𝑔:(Base‘𝑠)–1-1-onto→(Base‘𝑡)})
df-gic 19189	⊢ ≃_𝑔 = (^◡ GrpIso “ (V ∖ 1_o))
cga 19218	class GrpAct
df-ga 19219	⊢ GrpAct = (𝑔 ∈ Grp, 𝑠 ∈ V ↦ ⦋(Base‘𝑔) / 𝑏⦌{𝑚 ∈ (𝑠 ↑_m (𝑏 × 𝑠)) ∣ ∀𝑥 ∈ 𝑠 (((0_g‘𝑔)𝑚𝑥) = 𝑥 ∧ ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ((𝑦(+_g‘𝑔)𝑧)𝑚𝑥) = (𝑦𝑚(𝑧𝑚𝑥)))})
ccntz 19244	class Cntz
ccntr 19245	class Cntr
df-cntz 19246	⊢ Cntz = (𝑚 ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘𝑚) ↦ {𝑥 ∈ (Base‘𝑚) ∣ ∀𝑦 ∈ 𝑠 (𝑥(+_g‘𝑚)𝑦) = (𝑦(+_g‘𝑚)𝑥)}))
df-cntr 19247	⊢ Cntr = (𝑚 ∈ V ↦ ((Cntz‘𝑚)‘(Base‘𝑚)))
coppg 19274	class opp_g
df-oppg 19275	⊢ opp_g = (𝑤 ∈ V ↦ (𝑤 sSet ⟨(+_g‘ndx), tpos (+_g‘𝑤)⟩))
csymg 19298	class SymGrp
df-symg 19299	⊢ SymGrp = (𝑥 ∈ V ↦ ((EndoFMnd‘𝑥) ↾_s {ℎ ∣ ℎ:𝑥–1-1-onto→𝑥}))
cpmtr 19370	class pmTrsp
df-pmtr 19371	⊢ pmTrsp = (𝑑 ∈ V ↦ (𝑝 ∈ {𝑦 ∈ 𝒫 𝑑 ∣ 𝑦 ≈ 2_o} ↦ (𝑧 ∈ 𝑑 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))))
cpsgn 19418	class pmSgn
cevpm 19419	class pmEven
df-psgn 19420	⊢ pmSgn = (𝑑 ∈ V ↦ (𝑥 ∈ {𝑝 ∈ (Base‘(SymGrp‘𝑑)) ∣ dom (𝑝 ∖ I ) ∈ Fin} ↦ (℩𝑠∃𝑤 ∈ Word ran (pmTrsp‘𝑑)(𝑥 = ((SymGrp‘𝑑) Σ_g 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤))))))
df-evpm 19421	⊢ pmEven = (𝑑 ∈ V ↦ (^◡(pmSgn‘𝑑) “ {1}))
cod 19453	class od
cgex 19454	class gEx
cpgp 19455	class pGrp
cslw 19456	class pSyl
df-od 19457	⊢ od = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔) ↦ ⦋{𝑛 ∈ ℕ ∣ (𝑛(._g‘𝑔)𝑥) = (0_g‘𝑔)} / 𝑖⦌if(𝑖 = ∅, 0, inf(𝑖, ℝ, < ))))
df-gex 19458	⊢ gEx = (𝑔 ∈ V ↦ ⦋{𝑛 ∈ ℕ ∣ ∀𝑥 ∈ (Base‘𝑔)(𝑛(._g‘𝑔)𝑥) = (0_g‘𝑔)} / 𝑖⦌if(𝑖 = ∅, 0, inf(𝑖, ℝ, < )))
df-pgp 19459	⊢ pGrp = {⟨𝑝, 𝑔⟩ ∣ ((𝑝 ∈ ℙ ∧ 𝑔 ∈ Grp) ∧ ∀𝑥 ∈ (Base‘𝑔)∃𝑛 ∈ ℕ₀ ((od‘𝑔)‘𝑥) = (𝑝↑𝑛))}
df-slw 19460	⊢ pSyl = (𝑝 ∈ ℙ, 𝑔 ∈ Grp ↦ {ℎ ∈ (SubGrp‘𝑔) ∣ ∀𝑘 ∈ (SubGrp‘𝑔)((ℎ ⊆ 𝑘 ∧ 𝑝 pGrp (𝑔 ↾_s 𝑘)) ↔ ℎ = 𝑘)})
clsm 19563	class LSSum
cpj1 19564	class proj₁
df-lsm 19565	⊢ LSSum = (𝑤 ∈ V ↦ (𝑡 ∈ 𝒫 (Base‘𝑤), 𝑢 ∈ 𝒫 (Base‘𝑤) ↦ ran (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥(+_g‘𝑤)𝑦))))
df-pj1 19566	⊢ proj₁ = (𝑤 ∈ V ↦ (𝑡 ∈ 𝒫 (Base‘𝑤), 𝑢 ∈ 𝒫 (Base‘𝑤) ↦ (𝑧 ∈ (𝑡(LSSum‘𝑤)𝑢) ↦ (℩𝑥 ∈ 𝑡 ∃𝑦 ∈ 𝑢 𝑧 = (𝑥(+_g‘𝑤)𝑦)))))
cefg 19635	class ~_FG
cfrgp 19636	class freeGrp
cvrgp 19637	class var_FGrp
df-efg 19638	⊢ ~_FG = (𝑖 ∈ V ↦ ∩ {𝑟 ∣ (𝑟 Er Word (𝑖 × 2_o) ∧ ∀𝑥 ∈ Word (𝑖 × 2_o)∀𝑛 ∈ (0...(♯‘𝑥))∀𝑦 ∈ 𝑖 ∀𝑧 ∈ 2_o 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1_o ∖ 𝑧)⟩”⟩⟩))})
df-frgp 19639	⊢ freeGrp = (𝑖 ∈ V ↦ ((freeMnd‘(𝑖 × 2_o)) /_s ( ~_FG ‘𝑖)))
df-vrgp 19640	⊢ var_FGrp = (𝑖 ∈ V ↦ (𝑗 ∈ 𝑖 ↦ [⟨“⟨𝑗, ∅⟩”⟩]( ~_FG ‘𝑖)))
ccmn 19709	class CMnd
cabl 19710	class Abel
df-cmn 19711	⊢ CMnd = {𝑔 ∈ Mnd ∣ ∀𝑎 ∈ (Base‘𝑔)∀𝑏 ∈ (Base‘𝑔)(𝑎(+_g‘𝑔)𝑏) = (𝑏(+_g‘𝑔)𝑎)}
df-abl 19712	⊢ Abel = (Grp ∩ CMnd)
ccyg 19806	class CycGrp
df-cyg 19807	⊢ CycGrp = {𝑔 ∈ Grp ∣ ∃𝑥 ∈ (Base‘𝑔)ran (𝑛 ∈ ℤ ↦ (𝑛(._g‘𝑔)𝑥)) = (Base‘𝑔)}
cdprd 19924	class DProd
cdpj 19925	class dProj
df-dprd 19926	⊢ DProd = (𝑔 ∈ Grp, 𝑠 ∈ {ℎ ∣ (ℎ:dom ℎ⟶(SubGrp‘𝑔) ∧ ∀𝑥 ∈ dom ℎ(∀𝑦 ∈ (dom ℎ ∖ {𝑥})(ℎ‘𝑥) ⊆ ((Cntz‘𝑔)‘(ℎ‘𝑦)) ∧ ((ℎ‘𝑥) ∩ ((mrCls‘(SubGrp‘𝑔))‘∪ (ℎ “ (dom ℎ ∖ {𝑥})))) = {(0_g‘𝑔)}))} ↦ ran (𝑓 ∈ {ℎ ∈ X𝑥 ∈ dom 𝑠(𝑠‘𝑥) ∣ ℎ finSupp (0_g‘𝑔)} ↦ (𝑔 Σ_g 𝑓)))
df-dpj 19927	⊢ dProj = (𝑔 ∈ Grp, 𝑠 ∈ (dom DProd “ {𝑔}) ↦ (𝑖 ∈ dom 𝑠 ↦ ((𝑠‘𝑖)(proj₁‘𝑔)(𝑔 DProd (𝑠 ↾ (dom 𝑠 ∖ {𝑖}))))))
csimpg 20021	class SimpGrp
df-simpg 20022	⊢ SimpGrp = {𝑔 ∈ Grp ∣ (NrmSGrp‘𝑔) ≈ 2_o}
comnd 20048	class oMnd
cogrp 20049	class oGrp
df-omnd 20050	⊢ oMnd = {𝑔 ∈ Mnd ∣ [(Base‘𝑔) / 𝑣][(+_g‘𝑔) / 𝑝][(le‘𝑔) / 𝑙](𝑔 ∈ Toset ∧ ∀𝑎 ∈ 𝑣 ∀𝑏 ∈ 𝑣 ∀𝑐 ∈ 𝑣 (𝑎𝑙𝑏 → (𝑎𝑝𝑐)𝑙(𝑏𝑝𝑐)))}
df-ogrp 20051	⊢ oGrp = (Grp ∩ oMnd)
cmgp 20075	class mulGrp
df-mgp 20076	⊢ mulGrp = (𝑤 ∈ V ↦ (𝑤 sSet ⟨(+_g‘ndx), (._r‘𝑤)⟩))
crng 20087	class Rng
df-rng 20088	⊢ Rng = {𝑓 ∈ Abel ∣ ((mulGrp‘𝑓) ∈ Smgrp ∧ [(Base‘𝑓) / 𝑏][(+_g‘𝑓) / 𝑝][(._r‘𝑓) / 𝑡]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))))}
cur 20116	class 1_r
df-ur 20117	⊢ 1_r = (0_g ∘ mulGrp)
csrg 20121	class SRing
df-srg 20122	⊢ SRing = {𝑓 ∈ CMnd ∣ ((mulGrp‘𝑓) ∈ Mnd ∧ [(Base‘𝑓) / 𝑟][(+_g‘𝑓) / 𝑝][(._r‘𝑓) / 𝑡][(0_g‘𝑓) / 𝑛]∀𝑥 ∈ 𝑟 (∀𝑦 ∈ 𝑟 ∀𝑧 ∈ 𝑟 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))) ∧ ((𝑛𝑡𝑥) = 𝑛 ∧ (𝑥𝑡𝑛) = 𝑛)))}
crg 20168	class Ring
ccrg 20169	class CRing
df-ring 20170	⊢ Ring = {𝑓 ∈ Grp ∣ ((mulGrp‘𝑓) ∈ Mnd ∧ [(Base‘𝑓) / 𝑟][(+_g‘𝑓) / 𝑝][(._r‘𝑓) / 𝑡]∀𝑥 ∈ 𝑟 ∀𝑦 ∈ 𝑟 ∀𝑧 ∈ 𝑟 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))))}
df-cring 20171	⊢ CRing = {𝑓 ∈ Ring ∣ (mulGrp‘𝑓) ∈ CMnd}
coppr 20272	class opp_r
df-oppr 20273	⊢ opp_r = (𝑓 ∈ V ↦ (𝑓 sSet ⟨(._r‘ndx), tpos (._r‘𝑓)⟩))
cdsr 20290	class ∥_r
cui 20291	class Unit
cir 20292	class Irred
df-dvdsr 20293	⊢ ∥_r = (𝑤 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑤) ∧ ∃𝑧 ∈ (Base‘𝑤)(𝑧(._r‘𝑤)𝑥) = 𝑦)})
df-unit 20294	⊢ Unit = (𝑤 ∈ V ↦ (^◡((∥_r‘𝑤) ∩ (∥_r‘(opp_r‘𝑤))) “ {(1_r‘𝑤)}))
df-irred 20295	⊢ Irred = (𝑤 ∈ V ↦ ⦋((Base‘𝑤) ∖ (Unit‘𝑤)) / 𝑏⦌{𝑧 ∈ 𝑏 ∣ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥(._r‘𝑤)𝑦) ≠ 𝑧})
cinvr 20323	class inv_r
df-invr 20324	⊢ inv_r = (𝑟 ∈ V ↦ (inv_g‘((mulGrp‘𝑟) ↾_s (Unit‘𝑟))))
cdvr 20336	class /_r
df-dvr 20337	⊢ /_r = (𝑟 ∈ V ↦ (𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Unit‘𝑟) ↦ (𝑥(._r‘𝑟)((inv_r‘𝑟)‘𝑦))))
crpm 20368	class RPrime
df-rprm 20369	⊢ RPrime = (𝑤 ∈ V ↦ ⦋(Base‘𝑤) / 𝑏⦌{𝑝 ∈ (𝑏 ∖ ((Unit‘𝑤) ∪ {(0_g‘𝑤)})) ∣ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 [(∥_r‘𝑤) / 𝑑](𝑝𝑑(𝑥(._r‘𝑤)𝑦) → (𝑝𝑑𝑥 ∨ 𝑝𝑑𝑦))})
crnghm 20370	class RngHom
crngim 20371	class RngIso
df-rnghm 20372	⊢ RngHom = (𝑟 ∈ Rng, 𝑠 ∈ Rng ↦ ⦋(Base‘𝑟) / 𝑣⦌⦋(Base‘𝑠) / 𝑤⦌{𝑓 ∈ (𝑤 ↑_m 𝑣) ∣ ∀𝑥 ∈ 𝑣 ∀𝑦 ∈ 𝑣 ((𝑓‘(𝑥(+_g‘𝑟)𝑦)) = ((𝑓‘𝑥)(+_g‘𝑠)(𝑓‘𝑦)) ∧ (𝑓‘(𝑥(._r‘𝑟)𝑦)) = ((𝑓‘𝑥)(._r‘𝑠)(𝑓‘𝑦)))})
df-rngim 20373	⊢ RngIso = (𝑟 ∈ V, 𝑠 ∈ V ↦ {𝑓 ∈ (𝑟 RngHom 𝑠) ∣ ^◡𝑓 ∈ (𝑠 RngHom 𝑟)})
crh 20405	class RingHom
crs 20406	class RingIso
cric 20407	class ≃_𝑟
df-rhm 20408	⊢ RingHom = (𝑟 ∈ Ring, 𝑠 ∈ Ring ↦ ⦋(Base‘𝑟) / 𝑣⦌⦋(Base‘𝑠) / 𝑤⦌{𝑓 ∈ (𝑤 ↑_m 𝑣) ∣ ((𝑓‘(1_r‘𝑟)) = (1_r‘𝑠) ∧ ∀𝑥 ∈ 𝑣 ∀𝑦 ∈ 𝑣 ((𝑓‘(𝑥(+_g‘𝑟)𝑦)) = ((𝑓‘𝑥)(+_g‘𝑠)(𝑓‘𝑦)) ∧ (𝑓‘(𝑥(._r‘𝑟)𝑦)) = ((𝑓‘𝑥)(._r‘𝑠)(𝑓‘𝑦))))})
df-rim 20409	⊢ RingIso = (𝑟 ∈ V, 𝑠 ∈ V ↦ {𝑓 ∈ (𝑟 RingHom 𝑠) ∣ ^◡𝑓 ∈ (𝑠 RingHom 𝑟)})
df-ric 20411	⊢ ≃_𝑟 = (^◡ RingIso “ (V ∖ 1_o))
cnzr 20445	class NzRing
df-nzr 20446	⊢ NzRing = {𝑟 ∈ Ring ∣ (1_r‘𝑟) ≠ (0_g‘𝑟)}
clring 20471	class LRing
df-lring 20472	⊢ LRing = {𝑟 ∈ NzRing ∣ ∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)((𝑥(+_g‘𝑟)𝑦) = (1_r‘𝑟) → (𝑥 ∈ (Unit‘𝑟) ∨ 𝑦 ∈ (Unit‘𝑟)))}
csubrng 20478	class SubRng
df-subrng 20479	⊢ SubRng = (𝑤 ∈ Rng ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾_s 𝑠) ∈ Rng})
csubrg 20502	class SubRing
df-subrg 20503	⊢ SubRing = (𝑤 ∈ Ring ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ ((𝑤 ↾_s 𝑠) ∈ Ring ∧ (1_r‘𝑤) ∈ 𝑠)})
crgspn 20543	class RingSpan
df-rgspn 20544	⊢ RingSpan = (𝑤 ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘𝑤) ↦ ∩ {𝑡 ∈ (SubRing‘𝑤) ∣ 𝑠 ⊆ 𝑡}))
crngc 20549	class RngCat
df-rngc 20550	⊢ RngCat = (𝑢 ∈ V ↦ ((ExtStrCat‘𝑢) ↾_cat ( RngHom ↾ ((𝑢 ∩ Rng) × (𝑢 ∩ Rng)))))
cringc 20578	class RingCat
df-ringc 20579	⊢ RingCat = (𝑢 ∈ V ↦ ((ExtStrCat‘𝑢) ↾_cat ( RingHom ↾ ((𝑢 ∩ Ring) × (𝑢 ∩ Ring)))))
crlreg 20624	class RLReg
cdomn 20625	class Domn
cidom 20626	class IDomn
df-rlreg 20627	⊢ RLReg = (𝑟 ∈ V ↦ {𝑥 ∈ (Base‘𝑟) ∣ ∀𝑦 ∈ (Base‘𝑟)((𝑥(._r‘𝑟)𝑦) = (0_g‘𝑟) → 𝑦 = (0_g‘𝑟))})
df-domn 20628	⊢ Domn = {𝑟 ∈ NzRing ∣ [(Base‘𝑟) / 𝑏][(0_g‘𝑟) / 𝑧]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ((𝑥(._r‘𝑟)𝑦) = 𝑧 → (𝑥 = 𝑧 ∨ 𝑦 = 𝑧))}
df-idom 20629	⊢ IDomn = (CRing ∩ Domn)
cdr 20662	class DivRing
cfield 20663	class Field
df-drng 20664	⊢ DivRing = {𝑟 ∈ Ring ∣ (Unit‘𝑟) = ((Base‘𝑟) ∖ {(0_g‘𝑟)})}
df-field 20665	⊢ Field = (DivRing ∩ CRing)
csdrg 20719	class SubDRing
df-sdrg 20720	⊢ SubDRing = (𝑤 ∈ DivRing ↦ {𝑠 ∈ (SubRing‘𝑤) ∣ (𝑤 ↾_s 𝑠) ∈ DivRing})
cabv 20741	class AbsVal
df-abv 20742	⊢ AbsVal = (𝑟 ∈ Ring ↦ {𝑓 ∈ ((0[,)+∞) ↑_m (Base‘𝑟)) ∣ ∀𝑥 ∈ (Base‘𝑟)(((𝑓‘𝑥) = 0 ↔ 𝑥 = (0_g‘𝑟)) ∧ ∀𝑦 ∈ (Base‘𝑟)((𝑓‘(𝑥(._r‘𝑟)𝑦)) = ((𝑓‘𝑥) · (𝑓‘𝑦)) ∧ (𝑓‘(𝑥(+_g‘𝑟)𝑦)) ≤ ((𝑓‘𝑥) + (𝑓‘𝑦))))})
cstf 20770	class *_rf
csr 20771	class *-Ring
df-staf 20772	⊢ _rf = (𝑓 ∈ V ↦ (𝑥 ∈ (Base‘𝑓) ↦ ((_𝑟‘𝑓)‘𝑥)))
df-srng 20773	⊢ -Ring = {𝑓 ∣ [(_rf‘𝑓) / 𝑖](𝑖 ∈ (𝑓 RingHom (opp_r‘𝑓)) ∧ 𝑖 = ^◡𝑖)}
corng 20790	class oRing
cofld 20791	class oField
df-orng 20792	⊢ oRing = {𝑟 ∈ (Ring ∩ oGrp) ∣ [(Base‘𝑟) / 𝑣][(0_g‘𝑟) / 𝑧][(._r‘𝑟) / 𝑡][(le‘𝑟) / 𝑙]∀𝑎 ∈ 𝑣 ∀𝑏 ∈ 𝑣 ((𝑧𝑙𝑎 ∧ 𝑧𝑙𝑏) → 𝑧𝑙(𝑎𝑡𝑏))}
df-ofld 20793	⊢ oField = (Field ∩ oRing)
clmod 20811	class LMod
cscaf 20812	class ·_sf
df-lmod 20813	⊢ LMod = {𝑔 ∈ Grp ∣ [(Base‘𝑔) / 𝑣][(+_g‘𝑔) / 𝑎][(Scalar‘𝑔) / 𝑓][( ·_𝑠 ‘𝑔) / 𝑠][(Base‘𝑓) / 𝑘][(+_g‘𝑓) / 𝑝][(._r‘𝑓) / 𝑡](𝑓 ∈ Ring ∧ ∀𝑞 ∈ 𝑘 ∀𝑟 ∈ 𝑘 ∀𝑥 ∈ 𝑣 ∀𝑤 ∈ 𝑣 (((𝑟𝑠𝑤) ∈ 𝑣 ∧ (𝑟𝑠(𝑤𝑎𝑥)) = ((𝑟𝑠𝑤)𝑎(𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤)𝑎(𝑟𝑠𝑤))) ∧ (((𝑞𝑡𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ((1_r‘𝑓)𝑠𝑤) = 𝑤)))}
df-scaf 20814	⊢ ·_sf = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘(Scalar‘𝑔)), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥( ·_𝑠 ‘𝑔)𝑦)))
clss 20882	class LSubSp
df-lss 20883	⊢ LSubSp = (𝑤 ∈ V ↦ {𝑠 ∈ (𝒫 (Base‘𝑤) ∖ {∅}) ∣ ∀𝑥 ∈ (Base‘(Scalar‘𝑤))∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑥( ·_𝑠 ‘𝑤)𝑎)(+_g‘𝑤)𝑏) ∈ 𝑠})
clspn 20922	class LSpan
df-lsp 20923	⊢ LSpan = (𝑤 ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘𝑤) ↦ ∩ {𝑡 ∈ (LSubSp‘𝑤) ∣ 𝑠 ⊆ 𝑡}))
clmhm 20971	class LMHom
clmim 20972	class LMIso
clmic 20973	class ≃_𝑚
df-lmhm 20974	⊢ LMHom = (𝑠 ∈ LMod, 𝑡 ∈ LMod ↦ {𝑓 ∈ (𝑠 GrpHom 𝑡) ∣ [(Scalar‘𝑠) / 𝑤]((Scalar‘𝑡) = 𝑤 ∧ ∀𝑥 ∈ (Base‘𝑤)∀𝑦 ∈ (Base‘𝑠)(𝑓‘(𝑥( ·_𝑠 ‘𝑠)𝑦)) = (𝑥( ·_𝑠 ‘𝑡)(𝑓‘𝑦)))})
df-lmim 20975	⊢ LMIso = (𝑠 ∈ LMod, 𝑡 ∈ LMod ↦ {𝑔 ∈ (𝑠 LMHom 𝑡) ∣ 𝑔:(Base‘𝑠)–1-1-onto→(Base‘𝑡)})
df-lmic 20976	⊢ ≃_𝑚 = (^◡ LMIso “ (V ∖ 1_o))
clbs 21026	class LBasis
df-lbs 21027	⊢ LBasis = (𝑤 ∈ V ↦ {𝑏 ∈ 𝒫 (Base‘𝑤) ∣ [(LSpan‘𝑤) / 𝑛][(Scalar‘𝑤) / 𝑠]((𝑛‘𝑏) = (Base‘𝑤) ∧ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ ((Base‘𝑠) ∖ {(0_g‘𝑠)}) ¬ (𝑦( ·_𝑠 ‘𝑤)𝑥) ∈ (𝑛‘(𝑏 ∖ {𝑥})))})
clvec 21054	class LVec
df-lvec 21055	⊢ LVec = {𝑓 ∈ LMod ∣ (Scalar‘𝑓) ∈ DivRing}
csra 21123	class subringAlg
crglmod 21124	class ringLMod
df-sra 21125	⊢ subringAlg = (𝑤 ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘𝑤) ↦ (((𝑤 sSet ⟨(Scalar‘ndx), (𝑤 ↾_s 𝑠)⟩) sSet ⟨( ·_𝑠 ‘ndx), (._r‘𝑤)⟩) sSet ⟨(·_𝑖‘ndx), (._r‘𝑤)⟩)))
df-rgmod 21126	⊢ ringLMod = (𝑤 ∈ V ↦ ((subringAlg ‘𝑤)‘(Base‘𝑤)))
clidl 21161	class LIdeal
crsp 21162	class RSpan
df-lidl 21163	⊢ LIdeal = (LSubSp ∘ ringLMod)
df-rsp 21164	⊢ RSpan = (LSpan ∘ ringLMod)
c2idl 21204	class 2Ideal
df-2idl 21205	⊢ 2Ideal = (𝑟 ∈ V ↦ ((LIdeal‘𝑟) ∩ (LIdeal‘(opp_r‘𝑟))))
clpidl 21275	class LPIdeal
clpir 21276	class LPIR
df-lpidl 21277	⊢ LPIdeal = (𝑤 ∈ Ring ↦ ∪ 𝑔 ∈ (Base‘𝑤){((RSpan‘𝑤)‘{𝑔})})
df-lpir 21278	⊢ LPIR = {𝑤 ∈ Ring ∣ (LIdeal‘𝑤) = (LPIdeal‘𝑤)}
cpid 21291	class PID
df-pid 21292	⊢ PID = (IDomn ∩ LPIR)
cpsmet 21293	class PsMet
cxmet 21294	class ∞Met
cmet 21295	class Met
cbl 21296	class ball
cfbas 21297	class fBas
cfg 21298	class filGen
cmopn 21299	class MetOpen
cmetu 21300	class metUnif
df-psmet 21301	⊢ PsMet = (𝑥 ∈ V ↦ {𝑑 ∈ (ℝ^* ↑_m (𝑥 × 𝑥)) ∣ ∀𝑦 ∈ 𝑥 ((𝑦𝑑𝑦) = 0 ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) +_𝑒 (𝑤𝑑𝑧)))})
df-xmet 21302	⊢ ∞Met = (𝑥 ∈ V ↦ {𝑑 ∈ (ℝ^* ↑_m (𝑥 × 𝑥)) ∣ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (((𝑦𝑑𝑧) = 0 ↔ 𝑦 = 𝑧) ∧ ∀𝑤 ∈ 𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) +_𝑒 (𝑤𝑑𝑧)))})
df-met 21303	⊢ Met = (𝑥 ∈ V ↦ {𝑑 ∈ (ℝ ↑_m (𝑥 × 𝑥)) ∣ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (((𝑦𝑑𝑧) = 0 ↔ 𝑦 = 𝑧) ∧ ∀𝑤 ∈ 𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) + (𝑤𝑑𝑧)))})
df-bl 21304	⊢ ball = (𝑑 ∈ V ↦ (𝑥 ∈ dom dom 𝑑, 𝑧 ∈ ℝ^* ↦ {𝑦 ∈ dom dom 𝑑 ∣ (𝑥𝑑𝑦) < 𝑧}))
df-mopn 21305	⊢ MetOpen = (𝑑 ∈ ∪ ran ∞Met ↦ (topGen‘ran (ball‘𝑑)))
df-fbas 21306	⊢ fBas = (𝑤 ∈ V ↦ {𝑥 ∈ 𝒫 𝒫 𝑤 ∣ (𝑥 ≠ ∅ ∧ ∅ ∉ 𝑥 ∧ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑥 ∩ 𝒫 (𝑦 ∩ 𝑧)) ≠ ∅)})
df-fg 21307	⊢ filGen = (𝑤 ∈ V, 𝑥 ∈ (fBas‘𝑤) ↦ {𝑦 ∈ 𝒫 𝑤 ∣ (𝑥 ∩ 𝒫 𝑦) ≠ ∅})
df-metu 21308	⊢ metUnif = (𝑑 ∈ ∪ ran PsMet ↦ ((dom dom 𝑑 × dom dom 𝑑)filGenran (𝑎 ∈ ℝ⁺ ↦ (^◡𝑑 “ (0[,)𝑎)))))
ccnfld 21309	class ℂ_fld
df-cnfld 21310	⊢ ℂ_fld = (({⟨(Base‘ndx), ℂ⟩, ⟨(+_g‘ndx), (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 + 𝑦))⟩, ⟨(._r‘ndx), (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))⟩} ∪ {⟨(*_𝑟‘ndx), ∗⟩}) ∪ ({⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∪ {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩}))
czring 21401	class ℤ_ring
df-zring 21402	⊢ ℤ_ring = (ℂ_fld ↾_s ℤ)
czrh 21454	class ℤRHom
czlm 21455	class ℤMod
cchr 21456	class chr
czn 21457	class ℤ/nℤ
df-zrh 21458	⊢ ℤRHom = (𝑟 ∈ V ↦ ∪ (ℤ_ring RingHom 𝑟))
df-zlm 21459	⊢ ℤMod = (𝑔 ∈ V ↦ ((𝑔 sSet ⟨(Scalar‘ndx), ℤ_ring⟩) sSet ⟨( ·_𝑠 ‘ndx), (._g‘𝑔)⟩))
df-chr 21460	⊢ chr = (𝑔 ∈ V ↦ ((od‘𝑔)‘(1_r‘𝑔)))
df-zn 21461	⊢ ℤ/nℤ = (𝑛 ∈ ℕ₀ ↦ ⦋ℤ_ring / 𝑧⦌⦋(𝑧 /_s (𝑧 ~_QG ((RSpan‘𝑧)‘{𝑛}))) / 𝑠⦌(𝑠 sSet ⟨(le‘ndx), ⦋((ℤRHom‘𝑠) ↾ if(𝑛 = 0, ℤ, (0..^𝑛))) / 𝑓⦌((𝑓 ∘ ≤ ) ∘ ^◡𝑓)⟩))
crefld 21559	class ℝ_fld
df-refld 21560	⊢ ℝ_fld = (ℂ_fld ↾_s ℝ)
cphl 21579	class PreHil
cipf 21580	class ·_if
df-phl 21581	⊢ PreHil = {𝑔 ∈ LVec ∣ [(Base‘𝑔) / 𝑣][(·_𝑖‘𝑔) / ℎ][(Scalar‘𝑔) / 𝑓](𝑓 ∈ -Ring ∧ ∀𝑥 ∈ 𝑣 ((𝑦 ∈ 𝑣 ↦ (𝑦ℎ𝑥)) ∈ (𝑔 LMHom (ringLMod‘𝑓)) ∧ ((𝑥ℎ𝑥) = (0_g‘𝑓) → 𝑥 = (0_g‘𝑔)) ∧ ∀𝑦 ∈ 𝑣 ((_𝑟‘𝑓)‘(𝑥ℎ𝑦)) = (𝑦ℎ𝑥)))}
df-ipf 21582	⊢ ·_if = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(·_𝑖‘𝑔)𝑦)))
cocv 21615	class ocv
ccss 21616	class ClSubSp
cthl 21617	class toHL
df-ocv 21618	⊢ ocv = (ℎ ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘ℎ) ↦ {𝑥 ∈ (Base‘ℎ) ∣ ∀𝑦 ∈ 𝑠 (𝑥(·_𝑖‘ℎ)𝑦) = (0_g‘(Scalar‘ℎ))}))
df-css 21619	⊢ ClSubSp = (ℎ ∈ V ↦ {𝑠 ∣ 𝑠 = ((ocv‘ℎ)‘((ocv‘ℎ)‘𝑠))})
df-thl 21620	⊢ toHL = (ℎ ∈ V ↦ ((toInc‘(ClSubSp‘ℎ)) sSet ⟨(oc‘ndx), (ocv‘ℎ)⟩))
cpj 21655	class proj
chil 21656	class Hil
cobs 21657	class OBasis
df-pj 21658	⊢ proj = (ℎ ∈ V ↦ ((𝑥 ∈ (LSubSp‘ℎ) ↦ (𝑥(proj₁‘ℎ)((ocv‘ℎ)‘𝑥))) ∩ (V × ((Base‘ℎ) ↑_m (Base‘ℎ)))))
df-hil 21659	⊢ Hil = {ℎ ∈ PreHil ∣ dom (proj‘ℎ) = (ClSubSp‘ℎ)}
df-obs 21660	⊢ OBasis = (ℎ ∈ PreHil ↦ {𝑏 ∈ 𝒫 (Base‘ℎ) ∣ (∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥(·_𝑖‘ℎ)𝑦) = if(𝑥 = 𝑦, (1_r‘(Scalar‘ℎ)), (0_g‘(Scalar‘ℎ))) ∧ ((ocv‘ℎ)‘𝑏) = {(0_g‘ℎ)})})
cdsmm 21686	class ⊕_m
df-dsmm 21687	⊢ ⊕_m = (𝑠 ∈ V, 𝑟 ∈ V ↦ ((𝑠X_s𝑟) ↾_s {𝑓 ∈ X𝑥 ∈ dom 𝑟(Base‘(𝑟‘𝑥)) ∣ {𝑥 ∈ dom 𝑟 ∣ (𝑓‘𝑥) ≠ (0_g‘(𝑟‘𝑥))} ∈ Fin}))
cfrlm 21701	class freeLMod
df-frlm 21702	⊢ freeLMod = (𝑟 ∈ V, 𝑖 ∈ V ↦ (𝑟 ⊕_m (𝑖 × {(ringLMod‘𝑟)})))
cuvc 21737	class unitVec
df-uvc 21738	⊢ unitVec = (𝑟 ∈ V, 𝑖 ∈ V ↦ (𝑗 ∈ 𝑖 ↦ (𝑘 ∈ 𝑖 ↦ if(𝑘 = 𝑗, (1_r‘𝑟), (0_g‘𝑟)))))
clindf 21759	class LIndF
clinds 21760	class LIndS
df-lindf 21761	⊢ LIndF = {⟨𝑓, 𝑤⟩ ∣ (𝑓:dom 𝑓⟶(Base‘𝑤) ∧ [(Scalar‘𝑤) / 𝑠]∀𝑥 ∈ dom 𝑓∀𝑘 ∈ ((Base‘𝑠) ∖ {(0_g‘𝑠)}) ¬ (𝑘( ·_𝑠 ‘𝑤)(𝑓‘𝑥)) ∈ ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥}))))}
df-linds 21762	⊢ LIndS = (𝑤 ∈ V ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ ( I ↾ 𝑠) LIndF 𝑤})
casa 21805	class AssAlg
casp 21806	class AlgSpan
cascl 21807	class algSc
df-assa 21808	⊢ AssAlg = {𝑤 ∈ (LMod ∩ Ring) ∣ [(Scalar‘𝑤) / 𝑓]∀𝑟 ∈ (Base‘𝑓)∀𝑥 ∈ (Base‘𝑤)∀𝑦 ∈ (Base‘𝑤)[( ·_𝑠 ‘𝑤) / 𝑠][(._r‘𝑤) / 𝑡](((𝑟𝑠𝑥)𝑡𝑦) = (𝑟𝑠(𝑥𝑡𝑦)) ∧ (𝑥𝑡(𝑟𝑠𝑦)) = (𝑟𝑠(𝑥𝑡𝑦)))}
df-asp 21809	⊢ AlgSpan = (𝑤 ∈ AssAlg ↦ (𝑠 ∈ 𝒫 (Base‘𝑤) ↦ ∩ {𝑡 ∈ ((SubRing‘𝑤) ∩ (LSubSp‘𝑤)) ∣ 𝑠 ⊆ 𝑡}))
df-ascl 21810	⊢ algSc = (𝑤 ∈ V ↦ (𝑥 ∈ (Base‘(Scalar‘𝑤)) ↦ (𝑥( ·_𝑠 ‘𝑤)(1_r‘𝑤))))
cmps 21860	class mPwSer
cmvr 21861	class mVar
cmpl 21862	class mPoly
cltb 21863	class <_bag
copws 21864	class ordPwSer
df-psr 21865	⊢ mPwSer = (𝑖 ∈ V, 𝑟 ∈ V ↦ ⦋{ℎ ∈ (ℕ₀ ↑_m 𝑖) ∣ (^◡ℎ “ ℕ) ∈ Fin} / 𝑑⦌⦋((Base‘𝑟) ↑_m 𝑑) / 𝑏⦌({⟨(Base‘ndx), 𝑏⟩, ⟨(+_g‘ndx), ( ∘_f (+_g‘𝑟) ↾ (𝑏 × 𝑏))⟩, ⟨(._r‘ndx), (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (𝑘 ∈ 𝑑 ↦ (𝑟 Σ_g (𝑥 ∈ {𝑦 ∈ 𝑑 ∣ 𝑦 ∘_r ≤ 𝑘} ↦ ((𝑓‘𝑥)(._r‘𝑟)(𝑔‘(𝑘 ∘_f − 𝑥)))))))⟩} ∪ {⟨(Scalar‘ndx), 𝑟⟩, ⟨( ·_𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑟), 𝑓 ∈ 𝑏 ↦ ((𝑑 × {𝑥}) ∘_f (._r‘𝑟)𝑓))⟩, ⟨(TopSet‘ndx), (∏_t‘(𝑑 × {(TopOpen‘𝑟)}))⟩}))
df-mvr 21866	⊢ mVar = (𝑖 ∈ V, 𝑟 ∈ V ↦ (𝑥 ∈ 𝑖 ↦ (𝑓 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝑖) ∣ (^◡ℎ “ ℕ) ∈ Fin} ↦ if(𝑓 = (𝑦 ∈ 𝑖 ↦ if(𝑦 = 𝑥, 1, 0)), (1_r‘𝑟), (0_g‘𝑟)))))
df-mpl 21867	⊢ mPoly = (𝑖 ∈ V, 𝑟 ∈ V ↦ ⦋(𝑖 mPwSer 𝑟) / 𝑤⦌(𝑤 ↾_s {𝑓 ∈ (Base‘𝑤) ∣ 𝑓 finSupp (0_g‘𝑟)}))
df-ltbag 21868	⊢ <_bag = (𝑟 ∈ V, 𝑖 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ {ℎ ∈ (ℕ₀ ↑_m 𝑖) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∧ ∃𝑧 ∈ 𝑖 ((𝑥‘𝑧) < (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝑖 (𝑧𝑟𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤))))})
df-opsr 21869	⊢ ordPwSer = (𝑖 ∈ V, 𝑠 ∈ V ↦ (𝑟 ∈ 𝒫 (𝑖 × 𝑖) ↦ ⦋(𝑖 mPwSer 𝑠) / 𝑝⦌(𝑝 sSet ⟨(le‘ndx), {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (Base‘𝑝) ∧ ([{ℎ ∈ (ℕ₀ ↑_m 𝑖) ∣ (^◡ℎ “ ℕ) ∈ Fin} / 𝑑]∃𝑧 ∈ 𝑑 ((𝑥‘𝑧)(lt‘𝑠)(𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝑑 (𝑤(𝑟 <_bag 𝑖)𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤))) ∨ 𝑥 = 𝑦))}⟩)))
ces 22027	class evalSub
cevl 22028	class eval
df-evls 22029	⊢ evalSub = (𝑖 ∈ V, 𝑠 ∈ CRing ↦ ⦋(Base‘𝑠) / 𝑏⦌(𝑟 ∈ (SubRing‘𝑠) ↦ ⦋(𝑖 mPoly (𝑠 ↾_s 𝑟)) / 𝑤⦌(℩𝑓 ∈ (𝑤 RingHom (𝑠 ↑_s (𝑏 ↑_m 𝑖)))((𝑓 ∘ (algSc‘𝑤)) = (𝑥 ∈ 𝑟 ↦ ((𝑏 ↑_m 𝑖) × {𝑥})) ∧ (𝑓 ∘ (𝑖 mVar (𝑠 ↾_s 𝑟))) = (𝑥 ∈ 𝑖 ↦ (𝑔 ∈ (𝑏 ↑_m 𝑖) ↦ (𝑔‘𝑥)))))))
df-evl 22030	⊢ eval = (𝑖 ∈ V, 𝑟 ∈ V ↦ ((𝑖 evalSub 𝑟)‘(Base‘𝑟)))
cslv 22071	class selectVars
cmhp 22072	class mHomP
cpsd 22073	class mPSDer
cai 22074	class AlgInd
df-selv 22075	⊢ selectVars = (𝑖 ∈ V, 𝑟 ∈ V ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑓 ∈ (Base‘(𝑖 mPoly 𝑟)) ↦ ⦋((𝑖 ∖ 𝑗) mPoly 𝑟) / 𝑢⦌⦋(𝑗 mPoly 𝑢) / 𝑡⦌⦋(algSc‘𝑡) / 𝑐⦌⦋(𝑐 ∘ (algSc‘𝑢)) / 𝑑⦌((((𝑖 evalSub 𝑡)‘ran 𝑑)‘(𝑑 ∘ 𝑓))‘(𝑥 ∈ 𝑖 ↦ if(𝑥 ∈ 𝑗, ((𝑗 mVar 𝑢)‘𝑥), (𝑐‘(((𝑖 ∖ 𝑗) mVar 𝑟)‘𝑥))))))))
df-mhp 22079	⊢ mHomP = (𝑖 ∈ V, 𝑟 ∈ V ↦ (𝑛 ∈ ℕ₀ ↦ {𝑓 ∈ (Base‘(𝑖 mPoly 𝑟)) ∣ (𝑓 supp (0_g‘𝑟)) ⊆ {𝑔 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝑖) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑔) = 𝑛}}))
df-psd 22099	⊢ mPSDer = (𝑖 ∈ V, 𝑟 ∈ V ↦ (𝑥 ∈ 𝑖 ↦ (𝑓 ∈ (Base‘(𝑖 mPwSer 𝑟)) ↦ (𝑘 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝑖) ∣ (^◡ℎ “ ℕ) ∈ Fin} ↦ (((𝑘‘𝑥) + 1)(._g‘𝑟)(𝑓‘(𝑘 ∘_f + (𝑦 ∈ 𝑖 ↦ if(𝑦 = 𝑥, 1, 0)))))))))
df-algind 22114	⊢ AlgInd = (𝑤 ∈ V, 𝑘 ∈ 𝒫 (Base‘𝑤) ↦ {𝑣 ∈ 𝒫 (Base‘𝑤) ∣ Fun ^◡(𝑓 ∈ (Base‘(𝑣 mPoly (𝑤 ↾_s 𝑘))) ↦ ((((𝑣 evalSub 𝑤)‘𝑘)‘𝑓)‘( I ↾ 𝑣)))})
cps1 22115	class PwSer₁
cv1 22116	class var₁
cpl1 22117	class Poly₁
cco1 22118	class coe₁
ctp1 22119	class toPoly₁
df-psr1 22120	⊢ PwSer₁ = (𝑟 ∈ V ↦ ((1_o ordPwSer 𝑟)‘∅))
df-vr1 22121	⊢ var₁ = (𝑟 ∈ V ↦ ((1_o mVar 𝑟)‘∅))
df-ply1 22122	⊢ Poly₁ = (𝑟 ∈ V ↦ ((PwSer₁‘𝑟) ↾_s (Base‘(1_o mPoly 𝑟))))
df-coe1 22123	⊢ coe₁ = (𝑓 ∈ V ↦ (𝑛 ∈ ℕ₀ ↦ (𝑓‘(1_o × {𝑛}))))
df-toply1 22124	⊢ toPoly₁ = (𝑓 ∈ V ↦ (𝑛 ∈ (ℕ₀ ↑_m 1_o) ↦ (𝑓‘(𝑛‘∅))))
ces1 22257	class evalSub₁
ce1 22258	class eval₁
df-evls1 22259	⊢ evalSub₁ = (𝑠 ∈ V, 𝑟 ∈ 𝒫 (Base‘𝑠) ↦ ⦋(Base‘𝑠) / 𝑏⦌((𝑥 ∈ (𝑏 ↑_m (𝑏 ↑_m 1_o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝑏 ↦ (1_o × {𝑦})))) ∘ ((1_o evalSub 𝑠)‘𝑟)))
df-evl1 22260	⊢ eval₁ = (𝑟 ∈ V ↦ ⦋(Base‘𝑟) / 𝑏⦌((𝑥 ∈ (𝑏 ↑_m (𝑏 ↑_m 1_o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝑏 ↦ (1_o × {𝑦})))) ∘ (1_o eval 𝑟)))
cmmul 22334	class maMul
df-mamu 22335	⊢ maMul = (𝑟 ∈ V, 𝑜 ∈ V ↦ ⦋(1^st ‘(1^st ‘𝑜)) / 𝑚⦌⦋(2^nd ‘(1^st ‘𝑜)) / 𝑛⦌⦋(2^nd ‘𝑜) / 𝑝⦌(𝑥 ∈ ((Base‘𝑟) ↑_m (𝑚 × 𝑛)), 𝑦 ∈ ((Base‘𝑟) ↑_m (𝑛 × 𝑝)) ↦ (𝑖 ∈ 𝑚, 𝑘 ∈ 𝑝 ↦ (𝑟 Σ_g (𝑗 ∈ 𝑛 ↦ ((𝑖𝑥𝑗)(._r‘𝑟)(𝑗𝑦𝑘)))))))
cmat 22351	class Mat
df-mat 22352	⊢ Mat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ ((𝑟 freeLMod (𝑛 × 𝑛)) sSet ⟨(._r‘ndx), (𝑟 maMul ⟨𝑛, 𝑛, 𝑛⟩)⟩))
cdmat 22432	class DMat
cscmat 22433	class ScMat
df-dmat 22434	⊢ DMat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ {𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ∣ ∀𝑖 ∈ 𝑛 ∀𝑗 ∈ 𝑛 (𝑖 ≠ 𝑗 → (𝑖𝑚𝑗) = (0_g‘𝑟))})
df-scmat 22435	⊢ ScMat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ ⦋(𝑛 Mat 𝑟) / 𝑎⦌{𝑚 ∈ (Base‘𝑎) ∣ ∃𝑐 ∈ (Base‘𝑟)𝑚 = (𝑐( ·_𝑠 ‘𝑎)(1_r‘𝑎))})
cmvmul 22484	class maVecMul
df-mvmul 22485	⊢ maVecMul = (𝑟 ∈ V, 𝑜 ∈ V ↦ ⦋(1^st ‘𝑜) / 𝑚⦌⦋(2^nd ‘𝑜) / 𝑛⦌(𝑥 ∈ ((Base‘𝑟) ↑_m (𝑚 × 𝑛)), 𝑦 ∈ ((Base‘𝑟) ↑_m 𝑛) ↦ (𝑖 ∈ 𝑚 ↦ (𝑟 Σ_g (𝑗 ∈ 𝑛 ↦ ((𝑖𝑥𝑗)(._r‘𝑟)(𝑦‘𝑗)))))))
cmarrep 22500	class matRRep
cmatrepV 22501	class matRepV
df-marrep 22502	⊢ matRRep = (𝑛 ∈ V, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)), 𝑠 ∈ (Base‘𝑟) ↦ (𝑘 ∈ 𝑛, 𝑙 ∈ 𝑛 ↦ (𝑖 ∈ 𝑛, 𝑗 ∈ 𝑛 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑠, (0_g‘𝑟)), (𝑖𝑚𝑗))))))
df-marepv 22503	⊢ matRepV = (𝑛 ∈ V, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)), 𝑣 ∈ ((Base‘𝑟) ↑_m 𝑛) ↦ (𝑘 ∈ 𝑛 ↦ (𝑖 ∈ 𝑛, 𝑗 ∈ 𝑛 ↦ if(𝑗 = 𝑘, (𝑣‘𝑖), (𝑖𝑚𝑗))))))
csubma 22520	class subMat
df-subma 22521	⊢ subMat = (𝑛 ∈ V, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑘 ∈ 𝑛, 𝑙 ∈ 𝑛 ↦ (𝑖 ∈ (𝑛 ∖ {𝑘}), 𝑗 ∈ (𝑛 ∖ {𝑙}) ↦ (𝑖𝑚𝑗)))))
cmdat 22528	class maDet
df-mdet 22529	⊢ maDet = (𝑛 ∈ V, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑟 Σ_g (𝑝 ∈ (Base‘(SymGrp‘𝑛)) ↦ ((((ℤRHom‘𝑟) ∘ (pmSgn‘𝑛))‘𝑝)(._r‘𝑟)((mulGrp‘𝑟) Σ_g (𝑥 ∈ 𝑛 ↦ ((𝑝‘𝑥)𝑚𝑥))))))))
cmadu 22576	class maAdju
cminmar1 22577	class minMatR1
df-madu 22578	⊢ maAdju = (𝑛 ∈ V, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑖 ∈ 𝑛, 𝑗 ∈ 𝑛 ↦ ((𝑛 maDet 𝑟)‘(𝑘 ∈ 𝑛, 𝑙 ∈ 𝑛 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, (1_r‘𝑟), (0_g‘𝑟)), (𝑘𝑚𝑙)))))))
df-minmar1 22579	⊢ minMatR1 = (𝑛 ∈ V, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑘 ∈ 𝑛, 𝑙 ∈ 𝑛 ↦ (𝑖 ∈ 𝑛, 𝑗 ∈ 𝑛 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, (1_r‘𝑟), (0_g‘𝑟)), (𝑖𝑚𝑗))))))
ccpmat 22647	class ConstPolyMat
cmat2pmat 22648	class matToPolyMat
ccpmat2mat 22649	class cPolyMatToMat
df-cpmat 22650	⊢ ConstPolyMat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ {𝑚 ∈ (Base‘(𝑛 Mat (Poly₁‘𝑟))) ∣ ∀𝑖 ∈ 𝑛 ∀𝑗 ∈ 𝑛 ∀𝑘 ∈ ℕ ((coe₁‘(𝑖𝑚𝑗))‘𝑘) = (0_g‘𝑟)})
df-mat2pmat 22651	⊢ matToPolyMat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑥 ∈ 𝑛, 𝑦 ∈ 𝑛 ↦ ((algSc‘(Poly₁‘𝑟))‘(𝑥𝑚𝑦)))))
df-cpmat2mat 22652	⊢ cPolyMatToMat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ (𝑚 ∈ (𝑛 ConstPolyMat 𝑟) ↦ (𝑥 ∈ 𝑛, 𝑦 ∈ 𝑛 ↦ ((coe₁‘(𝑥𝑚𝑦))‘0))))
cdecpmat 22706	class decompPMat
df-decpmat 22707	⊢ decompPMat = (𝑚 ∈ V, 𝑘 ∈ ℕ₀ ↦ (𝑖 ∈ dom dom 𝑚, 𝑗 ∈ dom dom 𝑚 ↦ ((coe₁‘(𝑖𝑚𝑗))‘𝑘)))
cpm2mp 22736	class pMatToMatPoly
df-pm2mp 22737	⊢ pMatToMatPoly = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat (Poly₁‘𝑟))) ↦ ⦋(𝑛 Mat 𝑟) / 𝑎⦌⦋(Poly₁‘𝑎) / 𝑞⦌(𝑞 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑚 decompPMat 𝑘)( ·_𝑠 ‘𝑞)(𝑘(._g‘(mulGrp‘𝑞))(var₁‘𝑎)))))))
cchpmat 22770	class CharPlyMat
df-chpmat 22771	⊢ CharPlyMat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ ((𝑛 maDet (Poly₁‘𝑟))‘(((var₁‘𝑟)( ·_𝑠 ‘(𝑛 Mat (Poly₁‘𝑟)))(1_r‘(𝑛 Mat (Poly₁‘𝑟))))(-_g‘(𝑛 Mat (Poly₁‘𝑟)))((𝑛 matToPolyMat 𝑟)‘𝑚)))))
ctop 22837	class Top
df-top 22838	⊢ Top = {𝑥 ∣ (∀𝑦 ∈ 𝒫 𝑥∪ 𝑦 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑦 ∩ 𝑧) ∈ 𝑥)}
ctopon 22854	class TopOn
df-topon 22855	⊢ TopOn = (𝑏 ∈ V ↦ {𝑗 ∈ Top ∣ 𝑏 = ∪ 𝑗})
ctps 22876	class TopSp
df-topsp 22877	⊢ TopSp = {𝑓 ∣ (TopOpen‘𝑓) ∈ (TopOn‘(Base‘𝑓))}
ctb 22889	class TopBases
df-bases 22890	⊢ TopBases = {𝑥 ∣ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑦 ∩ 𝑧) ⊆ ∪ (𝑥 ∩ 𝒫 (𝑦 ∩ 𝑧))}
ccld 22960	class Clsd
cnt 22961	class int
ccl 22962	class cls
df-cld 22963	⊢ Clsd = (𝑗 ∈ Top ↦ {𝑥 ∈ 𝒫 ∪ 𝑗 ∣ (∪ 𝑗 ∖ 𝑥) ∈ 𝑗})
df-ntr 22964	⊢ int = (𝑗 ∈ Top ↦ (𝑥 ∈ 𝒫 ∪ 𝑗 ↦ ∪ (𝑗 ∩ 𝒫 𝑥)))
df-cls 22965	⊢ cls = (𝑗 ∈ Top ↦ (𝑥 ∈ 𝒫 ∪ 𝑗 ↦ ∩ {𝑦 ∈ (Clsd‘𝑗) ∣ 𝑥 ⊆ 𝑦}))
cnei 23041	class nei
df-nei 23042	⊢ nei = (𝑗 ∈ Top ↦ (𝑥 ∈ 𝒫 ∪ 𝑗 ↦ {𝑦 ∈ 𝒫 ∪ 𝑗 ∣ ∃𝑔 ∈ 𝑗 (𝑥 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑦)}))
clp 23078	class limPt
cperf 23079	class Perf
df-lp 23080	⊢ limPt = (𝑗 ∈ Top ↦ (𝑥 ∈ 𝒫 ∪ 𝑗 ↦ {𝑦 ∣ 𝑦 ∈ ((cls‘𝑗)‘(𝑥 ∖ {𝑦}))}))
df-perf 23081	⊢ Perf = {𝑗 ∈ Top ∣ ((limPt‘𝑗)‘∪ 𝑗) = ∪ 𝑗}
ccn 23168	class Cn
ccnp 23169	class CnP
clm 23170	class ⇝_𝑡
df-cn 23171	⊢ Cn = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ {𝑓 ∈ (∪ 𝑘 ↑_m ∪ 𝑗) ∣ ∀𝑦 ∈ 𝑘 (^◡𝑓 “ 𝑦) ∈ 𝑗})
df-cnp 23172	⊢ CnP = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ (𝑥 ∈ ∪ 𝑗 ↦ {𝑓 ∈ (∪ 𝑘 ↑_m ∪ 𝑗) ∣ ∀𝑦 ∈ 𝑘 ((𝑓‘𝑥) ∈ 𝑦 → ∃𝑔 ∈ 𝑗 (𝑥 ∈ 𝑔 ∧ (𝑓 “ 𝑔) ⊆ 𝑦))}))
df-lm 23173	⊢ ⇝_𝑡 = (𝑗 ∈ Top ↦ {⟨𝑓, 𝑥⟩ ∣ (𝑓 ∈ (∪ 𝑗 ↑_pm ℂ) ∧ 𝑥 ∈ ∪ 𝑗 ∧ ∀𝑢 ∈ 𝑗 (𝑥 ∈ 𝑢 → ∃𝑦 ∈ ran ℤ_≥(𝑓 ↾ 𝑦):𝑦⟶𝑢))})
ct0 23250	class Kol2
ct1 23251	class Fre
cha 23252	class Haus
creg 23253	class Reg
cnrm 23254	class Nrm
ccnrm 23255	class CNrm
cpnrm 23256	class PNrm
df-t0 23257	⊢ Kol2 = {𝑗 ∈ Top ∣ ∀𝑥 ∈ ∪ 𝑗∀𝑦 ∈ ∪ 𝑗(∀𝑜 ∈ 𝑗 (𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜) → 𝑥 = 𝑦)}
df-t1 23258	⊢ Fre = {𝑥 ∈ Top ∣ ∀𝑎 ∈ ∪ 𝑥{𝑎} ∈ (Clsd‘𝑥)}
df-haus 23259	⊢ Haus = {𝑗 ∈ Top ∣ ∀𝑥 ∈ ∪ 𝑗∀𝑦 ∈ ∪ 𝑗(𝑥 ≠ 𝑦 → ∃𝑛 ∈ 𝑗 ∃𝑚 ∈ 𝑗 (𝑥 ∈ 𝑛 ∧ 𝑦 ∈ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅))}
df-reg 23260	⊢ Reg = {𝑗 ∈ Top ∣ ∀𝑥 ∈ 𝑗 ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝑗 (𝑦 ∈ 𝑧 ∧ ((cls‘𝑗)‘𝑧) ⊆ 𝑥)}
df-nrm 23261	⊢ Nrm = {𝑗 ∈ Top ∣ ∀𝑥 ∈ 𝑗 ∀𝑦 ∈ ((Clsd‘𝑗) ∩ 𝒫 𝑥)∃𝑧 ∈ 𝑗 (𝑦 ⊆ 𝑧 ∧ ((cls‘𝑗)‘𝑧) ⊆ 𝑥)}
df-cnrm 23262	⊢ CNrm = {𝑗 ∈ Top ∣ ∀𝑥 ∈ 𝒫 ∪ 𝑗(𝑗 ↾_t 𝑥) ∈ Nrm}
df-pnrm 23263	⊢ PNrm = {𝑗 ∈ Nrm ∣ (Clsd‘𝑗) ⊆ ran (𝑓 ∈ (𝑗 ↑_m ℕ) ↦ ∩ ran 𝑓)}
ccmp 23330	class Comp
df-cmp 23331	⊢ Comp = {𝑥 ∈ Top ∣ ∀𝑦 ∈ 𝒫 𝑥(∪ 𝑥 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝑥 = ∪ 𝑧)}
cconn 23355	class Conn
df-conn 23356	⊢ Conn = {𝑗 ∈ Top ∣ (𝑗 ∩ (Clsd‘𝑗)) = {∅, ∪ 𝑗}}
c1stc 23381	class 1^stω
c2ndc 23382	class 2^ndω
df-1stc 23383	⊢ 1^stω = {𝑗 ∈ Top ∣ ∀𝑥 ∈ ∪ 𝑗∃𝑦 ∈ 𝒫 𝑗(𝑦 ≼ ω ∧ ∀𝑧 ∈ 𝑗 (𝑥 ∈ 𝑧 → 𝑥 ∈ ∪ (𝑦 ∩ 𝒫 𝑧)))}
df-2ndc 23384	⊢ 2^ndω = {𝑗 ∣ ∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝑗)}
clly 23408	class Locally 𝐴
cnlly 23409	class 𝑛-Locally 𝐴
df-lly 23410	⊢ Locally 𝐴 = {𝑗 ∈ Top ∣ ∀𝑥 ∈ 𝑗 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝑗 ↾_t 𝑢) ∈ 𝐴)}
df-nlly 23411	⊢ 𝑛-Locally 𝐴 = {𝑗 ∈ Top ∣ ∀𝑥 ∈ 𝑗 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗 ↾_t 𝑢) ∈ 𝐴}
cref 23446	class Ref
cptfin 23447	class PtFin
clocfin 23448	class LocFin
df-ref 23449	⊢ Ref = {⟨𝑥, 𝑦⟩ ∣ (∪ 𝑦 = ∪ 𝑥 ∧ ∀𝑧 ∈ 𝑥 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)}
df-ptfin 23450	⊢ PtFin = {𝑥 ∣ ∀𝑦 ∈ ∪ 𝑥{𝑧 ∈ 𝑥 ∣ 𝑦 ∈ 𝑧} ∈ Fin}
df-locfin 23451	⊢ LocFin = (𝑥 ∈ Top ↦ {𝑦 ∣ (∪ 𝑥 = ∪ 𝑦 ∧ ∀𝑝 ∈ ∪ 𝑥∃𝑛 ∈ 𝑥 (𝑝 ∈ 𝑛 ∧ {𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))})
ckgen 23477	class 𝑘Gen
df-kgen 23478	⊢ 𝑘Gen = (𝑗 ∈ Top ↦ {𝑥 ∈ 𝒫 ∪ 𝑗 ∣ ∀𝑘 ∈ 𝒫 ∪ 𝑗((𝑗 ↾_t 𝑘) ∈ Comp → (𝑥 ∩ 𝑘) ∈ (𝑗 ↾_t 𝑘))})
ctx 23504	class ×_t
cxko 23505	class ↑_ko
df-tx 23506	⊢ ×_t = (𝑟 ∈ V, 𝑠 ∈ V ↦ (topGen‘ran (𝑥 ∈ 𝑟, 𝑦 ∈ 𝑠 ↦ (𝑥 × 𝑦))))
df-xko 23507	⊢ ↑_ko = (𝑠 ∈ Top, 𝑟 ∈ Top ↦ (topGen‘(fi‘ran (𝑘 ∈ {𝑥 ∈ 𝒫 ∪ 𝑟 ∣ (𝑟 ↾_t 𝑥) ∈ Comp}, 𝑣 ∈ 𝑠 ↦ {𝑓 ∈ (𝑟 Cn 𝑠) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}))))
ckq 23637	class KQ
df-kq 23638	⊢ KQ = (𝑗 ∈ Top ↦ (𝑗 qTop (𝑥 ∈ ∪ 𝑗 ↦ {𝑦 ∈ 𝑗 ∣ 𝑥 ∈ 𝑦})))
chmeo 23697	class Homeo
chmph 23698	class ≃
df-hmeo 23699	⊢ Homeo = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ {𝑓 ∈ (𝑗 Cn 𝑘) ∣ ^◡𝑓 ∈ (𝑘 Cn 𝑗)})
df-hmph 23700	⊢ ≃ = (^◡Homeo “ (V ∖ 1_o))
cfil 23789	class Fil
df-fil 23790	⊢ Fil = (𝑧 ∈ V ↦ {𝑓 ∈ (fBas‘𝑧) ∣ ∀𝑥 ∈ 𝒫 𝑧((𝑓 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥 ∈ 𝑓)})
cufil 23843	class UFil
cufl 23844	class UFL
df-ufil 23845	⊢ UFil = (𝑔 ∈ V ↦ {𝑓 ∈ (Fil‘𝑔) ∣ ∀𝑥 ∈ 𝒫 𝑔(𝑥 ∈ 𝑓 ∨ (𝑔 ∖ 𝑥) ∈ 𝑓)})
df-ufl 23846	⊢ UFL = {𝑥 ∣ ∀𝑓 ∈ (Fil‘𝑥)∃𝑔 ∈ (UFil‘𝑥)𝑓 ⊆ 𝑔}
cfm 23877	class FilMap
cflim 23878	class fLim
cflf 23879	class fLimf
cfcls 23880	class fClus
cfcf 23881	class fClusf
df-fm 23882	⊢ FilMap = (𝑥 ∈ V, 𝑓 ∈ V ↦ (𝑦 ∈ (fBas‘dom 𝑓) ↦ (𝑥filGenran (𝑡 ∈ 𝑦 ↦ (𝑓 “ 𝑡)))))
df-flim 23883	⊢ fLim = (𝑗 ∈ Top, 𝑓 ∈ ∪ ran Fil ↦ {𝑥 ∈ ∪ 𝑗 ∣ (((nei‘𝑗)‘{𝑥}) ⊆ 𝑓 ∧ 𝑓 ⊆ 𝒫 ∪ 𝑗)})
df-flf 23884	⊢ fLimf = (𝑥 ∈ Top, 𝑦 ∈ ∪ ran Fil ↦ (𝑓 ∈ (∪ 𝑥 ↑_m ∪ 𝑦) ↦ (𝑥 fLim ((∪ 𝑥 FilMap 𝑓)‘𝑦))))
df-fcls 23885	⊢ fClus = (𝑗 ∈ Top, 𝑓 ∈ ∪ ran Fil ↦ if(∪ 𝑗 = ∪ 𝑓, ∩ 𝑥 ∈ 𝑓 ((cls‘𝑗)‘𝑥), ∅))
df-fcf 23886	⊢ fClusf = (𝑗 ∈ Top, 𝑓 ∈ ∪ ran Fil ↦ (𝑔 ∈ (∪ 𝑗 ↑_m ∪ 𝑓) ↦ (𝑗 fClus ((∪ 𝑗 FilMap 𝑔)‘𝑓))))
ccnext 24003	class CnExt
df-cnext 24004	⊢ CnExt = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ (𝑓 ∈ (∪ 𝑘 ↑_pm ∪ 𝑗) ↦ ∪ 𝑥 ∈ ((cls‘𝑗)‘dom 𝑓)({𝑥} × ((𝑘 fLimf (((nei‘𝑗)‘{𝑥}) ↾_t dom 𝑓))‘𝑓))))
ctmd 24014	class TopMnd
ctgp 24015	class TopGrp
df-tmd 24016	⊢ TopMnd = {𝑓 ∈ (Mnd ∩ TopSp) ∣ [(TopOpen‘𝑓) / 𝑗](+_𝑓‘𝑓) ∈ ((𝑗 ×_t 𝑗) Cn 𝑗)}
df-tgp 24017	⊢ TopGrp = {𝑓 ∈ (Grp ∩ TopMnd) ∣ [(TopOpen‘𝑓) / 𝑗](inv_g‘𝑓) ∈ (𝑗 Cn 𝑗)}
ctsu 24070	class tsums
df-tsms 24071	⊢ tsums = (𝑤 ∈ V, 𝑓 ∈ V ↦ ⦋(𝒫 dom 𝑓 ∩ Fin) / 𝑠⦌(((TopOpen‘𝑤) fLimf (𝑠filGenran (𝑧 ∈ 𝑠 ↦ {𝑦 ∈ 𝑠 ∣ 𝑧 ⊆ 𝑦})))‘(𝑦 ∈ 𝑠 ↦ (𝑤 Σ_g (𝑓 ↾ 𝑦)))))
ctrg 24100	class TopRing
ctdrg 24101	class TopDRing
ctlm 24102	class TopMod
ctvc 24103	class TopVec
df-trg 24104	⊢ TopRing = {𝑟 ∈ (TopGrp ∩ Ring) ∣ (mulGrp‘𝑟) ∈ TopMnd}
df-tdrg 24105	⊢ TopDRing = {𝑟 ∈ (TopRing ∩ DivRing) ∣ ((mulGrp‘𝑟) ↾_s (Unit‘𝑟)) ∈ TopGrp}
df-tlm 24106	⊢ TopMod = {𝑤 ∈ (TopMnd ∩ LMod) ∣ ((Scalar‘𝑤) ∈ TopRing ∧ ( ·_sf ‘𝑤) ∈ (((TopOpen‘(Scalar‘𝑤)) ×_t (TopOpen‘𝑤)) Cn (TopOpen‘𝑤)))}
df-tvc 24107	⊢ TopVec = {𝑤 ∈ TopMod ∣ (Scalar‘𝑤) ∈ TopDRing}
cust 24144	class UnifOn
df-ust 24145	⊢ UnifOn = (𝑥 ∈ V ↦ {𝑢 ∣ (𝑢 ⊆ 𝒫 (𝑥 × 𝑥) ∧ (𝑥 × 𝑥) ∈ 𝑢 ∧ ∀𝑣 ∈ 𝑢 (∀𝑤 ∈ 𝒫 (𝑥 × 𝑥)(𝑣 ⊆ 𝑤 → 𝑤 ∈ 𝑢) ∧ ∀𝑤 ∈ 𝑢 (𝑣 ∩ 𝑤) ∈ 𝑢 ∧ (( I ↾ 𝑥) ⊆ 𝑣 ∧ ^◡𝑣 ∈ 𝑢 ∧ ∃𝑤 ∈ 𝑢 (𝑤 ∘ 𝑤) ⊆ 𝑣)))})
cutop 24174	class unifTop
df-utop 24175	⊢ unifTop = (𝑢 ∈ ∪ ran UnifOn ↦ {𝑎 ∈ 𝒫 dom ∪ 𝑢 ∣ ∀𝑥 ∈ 𝑎 ∃𝑣 ∈ 𝑢 (𝑣 “ {𝑥}) ⊆ 𝑎})
cuss 24197	class UnifSt
cusp 24198	class UnifSp
ctus 24199	class toUnifSp
df-uss 24200	⊢ UnifSt = (𝑓 ∈ V ↦ ((UnifSet‘𝑓) ↾_t ((Base‘𝑓) × (Base‘𝑓))))
df-usp 24201	⊢ UnifSp = {𝑓 ∣ ((UnifSt‘𝑓) ∈ (UnifOn‘(Base‘𝑓)) ∧ (TopOpen‘𝑓) = (unifTop‘(UnifSt‘𝑓)))}
df-tus 24202	⊢ toUnifSp = (𝑢 ∈ ∪ ran UnifOn ↦ ({⟨(Base‘ndx), dom ∪ 𝑢⟩, ⟨(UnifSet‘ndx), 𝑢⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑢)⟩))
cucn 24218	class Cn_u
df-ucn 24219	⊢ Cn_u = (𝑢 ∈ ∪ ran UnifOn, 𝑣 ∈ ∪ ran UnifOn ↦ {𝑓 ∈ (dom ∪ 𝑣 ↑_m dom ∪ 𝑢) ∣ ∀𝑠 ∈ 𝑣 ∃𝑟 ∈ 𝑢 ∀𝑥 ∈ dom ∪ 𝑢∀𝑦 ∈ dom ∪ 𝑢(𝑥𝑟𝑦 → (𝑓‘𝑥)𝑠(𝑓‘𝑦))})
ccfilu 24229	class CauFil_u
df-cfilu 24230	⊢ CauFil_u = (𝑢 ∈ ∪ ran UnifOn ↦ {𝑓 ∈ (fBas‘dom ∪ 𝑢) ∣ ∀𝑣 ∈ 𝑢 ∃𝑎 ∈ 𝑓 (𝑎 × 𝑎) ⊆ 𝑣})
ccusp 24240	class CUnifSp
df-cusp 24241	⊢ CUnifSp = {𝑤 ∈ UnifSp ∣ ∀𝑐 ∈ (Fil‘(Base‘𝑤))(𝑐 ∈ (CauFil_u‘(UnifSt‘𝑤)) → ((TopOpen‘𝑤) fLim 𝑐) ≠ ∅)}
cxms 24261	class ∞MetSp
cms 24262	class MetSp
ctms 24263	class toMetSp
df-xms 24264	⊢ ∞MetSp = {𝑓 ∈ TopSp ∣ (TopOpen‘𝑓) = (MetOpen‘((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))))}
df-ms 24265	⊢ MetSp = {𝑓 ∈ ∞MetSp ∣ ((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))) ∈ (Met‘(Base‘𝑓))}
df-tms 24266	⊢ toMetSp = (𝑑 ∈ ∪ ran ∞Met ↦ ({⟨(Base‘ndx), dom dom 𝑑⟩, ⟨(dist‘ndx), 𝑑⟩} sSet ⟨(TopSet‘ndx), (MetOpen‘𝑑)⟩))
cnm 24520	class norm
cngp 24521	class NrmGrp
ctng 24522	class toNrmGrp
cnrg 24523	class NrmRing
cnlm 24524	class NrmMod
cnvc 24525	class NrmVec
df-nm 24526	⊢ norm = (𝑤 ∈ V ↦ (𝑥 ∈ (Base‘𝑤) ↦ (𝑥(dist‘𝑤)(0_g‘𝑤))))
df-ngp 24527	⊢ NrmGrp = {𝑔 ∈ (Grp ∩ MetSp) ∣ ((norm‘𝑔) ∘ (-_g‘𝑔)) ⊆ (dist‘𝑔)}
df-tng 24528	⊢ toNrmGrp = (𝑔 ∈ V, 𝑓 ∈ V ↦ ((𝑔 sSet ⟨(dist‘ndx), (𝑓 ∘ (-_g‘𝑔))⟩) sSet ⟨(TopSet‘ndx), (MetOpen‘(𝑓 ∘ (-_g‘𝑔)))⟩))
df-nrg 24529	⊢ NrmRing = {𝑤 ∈ NrmGrp ∣ (norm‘𝑤) ∈ (AbsVal‘𝑤)}
df-nlm 24530	⊢ NrmMod = {𝑤 ∈ (NrmGrp ∩ LMod) ∣ [(Scalar‘𝑤) / 𝑓](𝑓 ∈ NrmRing ∧ ∀𝑥 ∈ (Base‘𝑓)∀𝑦 ∈ (Base‘𝑤)((norm‘𝑤)‘(𝑥( ·_𝑠 ‘𝑤)𝑦)) = (((norm‘𝑓)‘𝑥) · ((norm‘𝑤)‘𝑦)))}
df-nvc 24531	⊢ NrmVec = (NrmMod ∩ LVec)
cnmo 24649	class normOp
cnghm 24650	class NGHom
cnmhm 24651	class NMHom
df-nmo 24652	⊢ normOp = (𝑠 ∈ NrmGrp, 𝑡 ∈ NrmGrp ↦ (𝑓 ∈ (𝑠 GrpHom 𝑡) ↦ inf({𝑟 ∈ (0[,)+∞) ∣ ∀𝑥 ∈ (Base‘𝑠)((norm‘𝑡)‘(𝑓‘𝑥)) ≤ (𝑟 · ((norm‘𝑠)‘𝑥))}, ℝ^*, < )))
df-nghm 24653	⊢ NGHom = (𝑠 ∈ NrmGrp, 𝑡 ∈ NrmGrp ↦ (^◡(𝑠 normOp 𝑡) “ ℝ))
df-nmhm 24654	⊢ NMHom = (𝑠 ∈ NrmMod, 𝑡 ∈ NrmMod ↦ ((𝑠 LMHom 𝑡) ∩ (𝑠 NGHom 𝑡)))
cii 24824	class II
ccncf 24825	class –cn→
df-ii 24826	⊢ II = (MetOpen‘((abs ∘ − ) ↾ ((0[,]1) × (0[,]1))))
df-cncf 24827	⊢ –cn→ = (𝑎 ∈ 𝒫 ℂ, 𝑏 ∈ 𝒫 ℂ ↦ {𝑓 ∈ (𝑏 ↑_m 𝑎) ∣ ∀𝑥 ∈ 𝑎 ∀𝑒 ∈ ℝ⁺ ∃𝑑 ∈ ℝ⁺ ∀𝑦 ∈ 𝑎 ((abs‘(𝑥 − 𝑦)) < 𝑑 → (abs‘((𝑓‘𝑥) − (𝑓‘𝑦))) < 𝑒)})
chtpy 24922	class Htpy
cphtpy 24923	class PHtpy
cphtpc 24924	class ≃_ph
df-htpy 24925	⊢ Htpy = (𝑥 ∈ Top, 𝑦 ∈ Top ↦ (𝑓 ∈ (𝑥 Cn 𝑦), 𝑔 ∈ (𝑥 Cn 𝑦) ↦ {ℎ ∈ ((𝑥 ×_t II) Cn 𝑦) ∣ ∀𝑠 ∈ ∪ 𝑥((𝑠ℎ0) = (𝑓‘𝑠) ∧ (𝑠ℎ1) = (𝑔‘𝑠))}))
df-phtpy 24926	⊢ PHtpy = (𝑥 ∈ Top ↦ (𝑓 ∈ (II Cn 𝑥), 𝑔 ∈ (II Cn 𝑥) ↦ {ℎ ∈ (𝑓(II Htpy 𝑥)𝑔) ∣ ∀𝑠 ∈ (0[,]1)((0ℎ𝑠) = (𝑓‘0) ∧ (1ℎ𝑠) = (𝑓‘1))}))
df-phtpc 24947	⊢ ≃_ph = (𝑥 ∈ Top ↦ {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ (II Cn 𝑥) ∧ (𝑓(PHtpy‘𝑥)𝑔) ≠ ∅)})
cpco 24956	class *_𝑝
comi 24957	class Ω₁
comn 24958	class Ω_𝑛
cpi1 24959	class π₁
cpin 24960	class π_n
df-pco 24961	⊢ *_𝑝 = (𝑗 ∈ Top ↦ (𝑓 ∈ (II Cn 𝑗), 𝑔 ∈ (II Cn 𝑗) ↦ (𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (𝑓‘(2 · 𝑥)), (𝑔‘((2 · 𝑥) − 1))))))
df-om1 24962	⊢ Ω₁ = (𝑗 ∈ Top, 𝑦 ∈ ∪ 𝑗 ↦ {⟨(Base‘ndx), {𝑓 ∈ (II Cn 𝑗) ∣ ((𝑓‘0) = 𝑦 ∧ (𝑓‘1) = 𝑦)}⟩, ⟨(+_g‘ndx), (*_𝑝‘𝑗)⟩, ⟨(TopSet‘ndx), (𝑗 ↑_ko II)⟩})
df-omn 24963	⊢ Ω_𝑛 = (𝑗 ∈ Top, 𝑦 ∈ ∪ 𝑗 ↦ seq0(((𝑥 ∈ V, 𝑝 ∈ V ↦ ⟨((TopOpen‘(1^st ‘𝑥)) Ω₁ (2^nd ‘𝑥)), ((0[,]1) × {(2^nd ‘𝑥)})⟩) ∘ 1^st ), ⟨{⟨(Base‘ndx), ∪ 𝑗⟩, ⟨(TopSet‘ndx), 𝑗⟩}, 𝑦⟩))
df-pi1 24964	⊢ π₁ = (𝑗 ∈ Top, 𝑦 ∈ ∪ 𝑗 ↦ ((𝑗 Ω₁ 𝑦) /_s ( ≃_ph‘𝑗)))
df-pin 24965	⊢ π_n = (𝑗 ∈ Top, 𝑝 ∈ ∪ 𝑗 ↦ (𝑛 ∈ ℕ₀ ↦ ((1^st ‘((𝑗 Ω_𝑛 𝑝)‘𝑛)) /_s if(𝑛 = 0, {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 ∈ (II Cn 𝑗)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)}, ( ≃_ph‘(TopOpen‘(1^st ‘((𝑗 Ω_𝑛 𝑝)‘(𝑛 − 1)))))))))
cclm 25018	class ℂMod
df-clm 25019	⊢ ℂMod = {𝑤 ∈ LMod ∣ [(Scalar‘𝑤) / 𝑓][(Base‘𝑓) / 𝑘](𝑓 = (ℂ_fld ↾_s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂ_fld))}
ccvs 25079	class ℂVec
df-cvs 25080	⊢ ℂVec = (ℂMod ∩ LVec)
ccph 25122	class ℂPreHil
ctcph 25123	class toℂPreHil
df-cph 25124	⊢ ℂPreHil = {𝑤 ∈ (PreHil ∩ NrmMod) ∣ [(Scalar‘𝑤) / 𝑓][(Base‘𝑓) / 𝑘](𝑓 = (ℂ_fld ↾_s 𝑘) ∧ (√ “ (𝑘 ∩ (0[,)+∞))) ⊆ 𝑘 ∧ (norm‘𝑤) = (𝑥 ∈ (Base‘𝑤) ↦ (√‘(𝑥(·_𝑖‘𝑤)𝑥))))}
df-tcph 25125	⊢ toℂPreHil = (𝑤 ∈ V ↦ (𝑤 toNrmGrp (𝑥 ∈ (Base‘𝑤) ↦ (√‘(𝑥(·_𝑖‘𝑤)𝑥)))))
ccfil 25208	class CauFil
ccau 25209	class Cau
ccmet 25210	class CMet
df-cfil 25211	⊢ CauFil = (𝑑 ∈ ∪ ran ∞Met ↦ {𝑓 ∈ (Fil‘dom dom 𝑑) ∣ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ 𝑓 (𝑑 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)})
df-cau 25212	⊢ Cau = (𝑑 ∈ ∪ ran ∞Met ↦ {𝑓 ∈ (dom dom 𝑑 ↑_pm ℂ) ∣ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ ℤ (𝑓 ↾ (ℤ_≥‘𝑗)):(ℤ_≥‘𝑗)⟶((𝑓‘𝑗)(ball‘𝑑)𝑥)})
df-cmet 25213	⊢ CMet = (𝑥 ∈ V ↦ {𝑑 ∈ (Met‘𝑥) ∣ ∀𝑓 ∈ (CauFil‘𝑑)((MetOpen‘𝑑) fLim 𝑓) ≠ ∅})
ccms 25288	class CMetSp
cbn 25289	class Ban
chl 25290	class ℂHil
df-cms 25291	⊢ CMetSp = {𝑤 ∈ MetSp ∣ [(Base‘𝑤) / 𝑏]((dist‘𝑤) ↾ (𝑏 × 𝑏)) ∈ (CMet‘𝑏)}
df-bn 25292	⊢ Ban = {𝑤 ∈ (NrmVec ∩ CMetSp) ∣ (Scalar‘𝑤) ∈ CMetSp}
df-hl 25293	⊢ ℂHil = (Ban ∩ ℂPreHil)
crrx 25339	class ℝ^
cehl 25340	class 𝔼_hil
df-rrx 25341	⊢ ℝ^ = (𝑖 ∈ V ↦ (toℂPreHil‘(ℝ_fld freeLMod 𝑖)))
df-ehl 25342	⊢ 𝔼_hil = (𝑛 ∈ ℕ₀ ↦ (ℝ^‘(1...𝑛)))
covol 25419	class vol*
cvol 25420	class vol
df-ovol 25421	⊢ vol* = (𝑥 ∈ 𝒫 ℝ ↦ inf({𝑦 ∈ ℝ^* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝑥 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^, < ))}, ℝ^, < ))
df-vol 25422	⊢ vol = (vol* ↾ {𝑥 ∣ ∀𝑦 ∈ (^◡vol* “ ℝ)(vol‘𝑦) = ((vol‘(𝑦 ∩ 𝑥)) + (vol*‘(𝑦 ∖ 𝑥)))})
cmbf 25571	class MblFn
citg1 25572	class ∫₁
citg2 25573	class ∫₂
cibl 25574	class 𝐿¹
citg 25575	class ∫𝐴𝐵 d𝑥
df-mbf 25576	⊢ MblFn = {𝑓 ∈ (ℂ ↑_pm ℝ) ∣ ∀𝑥 ∈ ran (,)((^◡(ℜ ∘ 𝑓) “ 𝑥) ∈ dom vol ∧ (^◡(ℑ ∘ 𝑓) “ 𝑥) ∈ dom vol)}
df-itg1 25577	⊢ ∫₁ = (𝑓 ∈ {𝑔 ∈ MblFn ∣ (𝑔:ℝ⟶ℝ ∧ ran 𝑔 ∈ Fin ∧ (vol‘(^◡𝑔 “ (ℝ ∖ {0}))) ∈ ℝ)} ↦ Σ𝑥 ∈ (ran 𝑓 ∖ {0})(𝑥 · (vol‘(^◡𝑓 “ {𝑥}))))
df-itg2 25578	⊢ ∫₂ = (𝑓 ∈ ((0[,]+∞) ↑_m ℝ) ↦ sup({𝑥 ∣ ∃𝑔 ∈ dom ∫₁(𝑔 ∘_r ≤ 𝑓 ∧ 𝑥 = (∫₁‘𝑔))}, ℝ^*, < ))
df-ibl 25579	⊢ 𝐿¹ = {𝑓 ∈ MblFn ∣ ∀𝑘 ∈ (0...3)(∫₂‘(𝑥 ∈ ℝ ↦ ⦋(ℜ‘((𝑓‘𝑥) / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ dom 𝑓 ∧ 0 ≤ 𝑦), 𝑦, 0))) ∈ ℝ}
df-itg 25580	⊢ ∫𝐴𝐵 d𝑥 = Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫₂‘(𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐵 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0))))
c0p 25626	class 0_𝑝
df-0p 25627	⊢ 0_𝑝 = (ℂ × {0})
cdit 25803	class ⨜[𝐴 → 𝐵]𝐶 d𝑥
df-ditg 25804	⊢ ⨜[𝐴 → 𝐵]𝐶 d𝑥 = if(𝐴 ≤ 𝐵, ∫(𝐴(,)𝐵)𝐶 d𝑥, -∫(𝐵(,)𝐴)𝐶 d𝑥)
climc 25819	class lim_ℂ
cdv 25820	class D
cdvn 25821	class D^𝑛
ccpn 25822	class 𝓑C^𝑛
df-limc 25823	⊢ lim_ℂ = (𝑓 ∈ (ℂ ↑_pm ℂ), 𝑥 ∈ ℂ ↦ {𝑦 ∣ [(TopOpen‘ℂ_fld) / 𝑗](𝑧 ∈ (dom 𝑓 ∪ {𝑥}) ↦ if(𝑧 = 𝑥, 𝑦, (𝑓‘𝑧))) ∈ (((𝑗 ↾_t (dom 𝑓 ∪ {𝑥})) CnP 𝑗)‘𝑥)})
df-dv 25824	⊢ D = (𝑠 ∈ 𝒫 ℂ, 𝑓 ∈ (ℂ ↑_pm 𝑠) ↦ ∪ 𝑥 ∈ ((int‘((TopOpen‘ℂ_fld) ↾_t 𝑠))‘dom 𝑓)({𝑥} × ((𝑧 ∈ (dom 𝑓 ∖ {𝑥}) ↦ (((𝑓‘𝑧) − (𝑓‘𝑥)) / (𝑧 − 𝑥))) lim_ℂ 𝑥)))
df-dvn 25825	⊢ D^𝑛 = (𝑠 ∈ 𝒫 ℂ, 𝑓 ∈ (ℂ ↑_pm 𝑠) ↦ seq0(((𝑥 ∈ V ↦ (𝑠 D 𝑥)) ∘ 1^st ), (ℕ₀ × {𝑓})))
df-cpn 25826	⊢ 𝓑C^𝑛 = (𝑠 ∈ 𝒫 ℂ ↦ (𝑥 ∈ ℕ₀ ↦ {𝑓 ∈ (ℂ ↑_pm 𝑠) ∣ ((𝑠 D^𝑛 𝑓)‘𝑥) ∈ (dom 𝑓–cn→ℂ)}))
cmdg 26014	class mDeg
cdg1 26015	class deg₁
df-mdeg 26016	⊢ mDeg = (𝑖 ∈ V, 𝑟 ∈ V ↦ (𝑓 ∈ (Base‘(𝑖 mPoly 𝑟)) ↦ sup(ran (ℎ ∈ (𝑓 supp (0_g‘𝑟)) ↦ (ℂ_fld Σ_g ℎ)), ℝ^*, < )))
df-deg1 26017	⊢ deg₁ = (𝑟 ∈ V ↦ (1_o mDeg 𝑟))
cmn1 26087	class Monic_1p
cuc1p 26088	class Unic_1p
cq1p 26089	class quot_1p
cr1p 26090	class rem_1p
cig1p 26091	class idlGen_1p
df-mon1 26092	⊢ Monic_1p = (𝑟 ∈ V ↦ {𝑓 ∈ (Base‘(Poly₁‘𝑟)) ∣ (𝑓 ≠ (0_g‘(Poly₁‘𝑟)) ∧ ((coe₁‘𝑓)‘((deg₁‘𝑟)‘𝑓)) = (1_r‘𝑟))})
df-uc1p 26093	⊢ Unic_1p = (𝑟 ∈ V ↦ {𝑓 ∈ (Base‘(Poly₁‘𝑟)) ∣ (𝑓 ≠ (0_g‘(Poly₁‘𝑟)) ∧ ((coe₁‘𝑓)‘((deg₁‘𝑟)‘𝑓)) ∈ (Unit‘𝑟))})
df-q1p 26094	⊢ quot_1p = (𝑟 ∈ V ↦ ⦋(Poly₁‘𝑟) / 𝑝⦌⦋(Base‘𝑝) / 𝑏⦌(𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (℩𝑞 ∈ 𝑏 ((deg₁‘𝑟)‘(𝑓(-_g‘𝑝)(𝑞(._r‘𝑝)𝑔))) < ((deg₁‘𝑟)‘𝑔))))
df-r1p 26095	⊢ rem_1p = (𝑟 ∈ V ↦ ⦋(Base‘(Poly₁‘𝑟)) / 𝑏⦌(𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (𝑓(-_g‘(Poly₁‘𝑟))((𝑓(quot_1p‘𝑟)𝑔)(._r‘(Poly₁‘𝑟))𝑔))))
df-ig1p 26096	⊢ idlGen_1p = (𝑟 ∈ V ↦ (𝑖 ∈ (LIdeal‘(Poly₁‘𝑟)) ↦ if(𝑖 = {(0_g‘(Poly₁‘𝑟))}, (0_g‘(Poly₁‘𝑟)), (℩𝑔 ∈ (𝑖 ∩ (Monic_1p‘𝑟))((deg₁‘𝑟)‘𝑔) = inf(((deg₁‘𝑟) “ (𝑖 ∖ {(0_g‘(Poly₁‘𝑟))})), ℝ, < )))))
cply 26145	class Poly
cidp 26146	class X_p
ccoe 26147	class coeff
cdgr 26148	class deg
df-ply 26149	⊢ Poly = (𝑥 ∈ 𝒫 ℂ ↦ {𝑓 ∣ ∃𝑛 ∈ ℕ₀ ∃𝑎 ∈ ((𝑥 ∪ {0}) ↑_m ℕ₀)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))})
df-idp 26150	⊢ X_p = ( I ↾ ℂ)
df-coe 26151	⊢ coeff = (𝑓 ∈ (Poly‘ℂ) ↦ (℩𝑎 ∈ (ℂ ↑_m ℕ₀)∃𝑛 ∈ ℕ₀ ((𝑎 “ (ℤ_≥‘(𝑛 + 1))) = {0} ∧ 𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))))))
df-dgr 26152	⊢ deg = (𝑓 ∈ (Poly‘ℂ) ↦ sup((^◡(coeff‘𝑓) “ (ℂ ∖ {0})), ℕ₀, < ))
cquot 26254	class quot
df-quot 26255	⊢ quot = (𝑓 ∈ (Poly‘ℂ), 𝑔 ∈ ((Poly‘ℂ) ∖ {0_𝑝}) ↦ (℩𝑞 ∈ (Poly‘ℂ)[(𝑓 ∘_f − (𝑔 ∘_f · 𝑞)) / 𝑟](𝑟 = 0_𝑝 ∨ (deg‘𝑟) < (deg‘𝑔))))
caa 26278	class 𝔸
df-aa 26279	⊢ 𝔸 = ∪ 𝑓 ∈ ((Poly‘ℤ) ∖ {0_𝑝})(^◡𝑓 “ {0})
ctayl 26316	class Tayl
cana 26317	class Ana
df-tayl 26318	⊢ Tayl = (𝑠 ∈ {ℝ, ℂ}, 𝑓 ∈ (ℂ ↑_pm 𝑠) ↦ (𝑛 ∈ (ℕ₀ ∪ {+∞}), 𝑎 ∈ ∩ 𝑘 ∈ ((0[,]𝑛) ∩ ℤ)dom ((𝑠 D^𝑛 𝑓)‘𝑘) ↦ ∪ 𝑥 ∈ ℂ ({𝑥} × (ℂ_fld tsums (𝑘 ∈ ((0[,]𝑛) ∩ ℤ) ↦ (((((𝑠 D^𝑛 𝑓)‘𝑘)‘𝑎) / (!‘𝑘)) · ((𝑥 − 𝑎)↑𝑘)))))))
df-ana 26319	⊢ Ana = (𝑠 ∈ {ℝ, ℂ} ↦ {𝑓 ∈ (ℂ ↑_pm 𝑠) ∣ ∀𝑥 ∈ dom 𝑓 𝑥 ∈ ((int‘((TopOpen‘ℂ_fld) ↾_t 𝑠))‘dom (𝑓 ∩ (+∞(𝑠 Tayl 𝑓)𝑥)))})
culm 26341	class ⇝_𝑢
df-ulm 26342	⊢ ⇝_𝑢 = (𝑠 ∈ V ↦ {⟨𝑓, 𝑦⟩ ∣ ∃𝑛 ∈ ℤ (𝑓:(ℤ_≥‘𝑛)⟶(ℂ ↑_m 𝑠) ∧ 𝑦:𝑠⟶ℂ ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ (ℤ_≥‘𝑛)∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑠 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥)})
clog 26519	class log
ccxp 26520	class ↑_𝑐
df-log 26521	⊢ log = ^◡(exp ↾ (^◡ℑ “ (-π(,]π)))
df-cxp 26522	⊢ ↑_𝑐 = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ if(𝑥 = 0, if(𝑦 = 0, 1, 0), (exp‘(𝑦 · (log‘𝑥)))))
clogb 26730	class log_b
df-logb 26731	⊢ log_b = (𝑥 ∈ (ℂ ∖ {0, 1}), 𝑦 ∈ (ℂ ∖ {0}) ↦ ((log‘𝑦) / (log‘𝑥)))
casin 26828	class arcsin
cacos 26829	class arccos
catan 26830	class arctan
df-asin 26831	⊢ arcsin = (𝑥 ∈ ℂ ↦ (-i · (log‘((i · 𝑥) + (√‘(1 − (𝑥↑2)))))))
df-acos 26832	⊢ arccos = (𝑥 ∈ ℂ ↦ ((π / 2) − (arcsin‘𝑥)))
df-atan 26833	⊢ arctan = (𝑥 ∈ (ℂ ∖ {-i, i}) ↦ ((i / 2) · ((log‘(1 − (i · 𝑥))) − (log‘(1 + (i · 𝑥))))))
carea 26921	class area
df-area 26922	⊢ area = (𝑠 ∈ {𝑡 ∈ 𝒫 (ℝ × ℝ) ∣ (∀𝑥 ∈ ℝ (𝑡 “ {𝑥}) ∈ (^◡vol “ ℝ) ∧ (𝑥 ∈ ℝ ↦ (vol‘(𝑡 “ {𝑥}))) ∈ 𝐿¹)} ↦ ∫ℝ(vol‘(𝑠 “ {𝑥})) d𝑥)
cem 26958	class γ
df-em 26959	⊢ γ = Σ𝑘 ∈ ℕ ((1 / 𝑘) − (log‘(1 + (1 / 𝑘))))
czeta 26979	class ζ
df-zeta 26980	⊢ ζ = (℩𝑓 ∈ ((ℂ ∖ {1})–cn→ℂ)∀𝑠 ∈ (ℂ ∖ {1})((1 − (2↑_𝑐(1 − 𝑠))) · (𝑓‘𝑠)) = Σ𝑛 ∈ ℕ₀ (Σ𝑘 ∈ (0...𝑛)(((-1↑𝑘) · (𝑛C𝑘)) · ((𝑘 + 1)↑_𝑐𝑠)) / (2↑(𝑛 + 1))))
clgam 26982	class log Γ
cgam 26983	class Γ
cigam 26984	class 1/Γ
df-lgam 26985	⊢ log Γ = (𝑧 ∈ (ℂ ∖ (ℤ ∖ ℕ)) ↦ (Σ𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))) − (log‘𝑧)))
df-gam 26986	⊢ Γ = (exp ∘ log Γ)
df-igam 26987	⊢ 1/Γ = (𝑥 ∈ ℂ ↦ if(𝑥 ∈ (ℤ ∖ ℕ), 0, (1 / (Γ‘𝑥))))
ccht 27057	class θ
cvma 27058	class Λ
cchp 27059	class ψ
cppi 27060	class π
cmu 27061	class μ
csgm 27062	class σ
df-cht 27063	⊢ θ = (𝑥 ∈ ℝ ↦ Σ𝑝 ∈ ((0[,]𝑥) ∩ ℙ)(log‘𝑝))
df-vma 27064	⊢ Λ = (𝑥 ∈ ℕ ↦ ⦋{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥} / 𝑠⦌if((♯‘𝑠) = 1, (log‘∪ 𝑠), 0))
df-chp 27065	⊢ ψ = (𝑥 ∈ ℝ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))(Λ‘𝑛))
df-ppi 27066	⊢ π = (𝑥 ∈ ℝ ↦ (♯‘((0[,]𝑥) ∩ ℙ)))
df-mu 27067	⊢ μ = (𝑥 ∈ ℕ ↦ if(∃𝑝 ∈ ℙ (𝑝↑2) ∥ 𝑥, 0, (-1↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥}))))
df-sgm 27068	⊢ σ = (𝑥 ∈ ℂ, 𝑛 ∈ ℕ ↦ Σ𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝑛} (𝑘↑_𝑐𝑥))
cdchr 27199	class DChr
df-dchr 27200	⊢ DChr = (𝑛 ∈ ℕ ↦ ⦋(ℤ/nℤ‘𝑛) / 𝑧⦌⦋{𝑥 ∈ ((mulGrp‘𝑧) MndHom (mulGrp‘ℂ_fld)) ∣ (((Base‘𝑧) ∖ (Unit‘𝑧)) × {0}) ⊆ 𝑥} / 𝑏⦌{⟨(Base‘ndx), 𝑏⟩, ⟨(+_g‘ndx), ( ∘_f · ↾ (𝑏 × 𝑏))⟩})
clgs 27261	class /_L
df-lgs 27262	⊢ /_L = (𝑎 ∈ ℤ, 𝑛 ∈ ℤ ↦ if(𝑛 = 0, if((𝑎↑2) = 1, 1, 0), (if((𝑛 < 0 ∧ 𝑎 < 0), -1, 1) · (seq1( · , (𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, (if(𝑚 = 2, if(2 ∥ 𝑎, 0, if((𝑎 mod 8) ∈ {1, 7}, 1, -1)), ((((𝑎↑((𝑚 − 1) / 2)) + 1) mod 𝑚) − 1))↑(𝑚 pCnt 𝑛)), 1)))‘(abs‘𝑛)))))
csur 27607	class No
clts 27608	class <s
cbday 27609	class bday
df-no 27610	⊢ No = {𝑓 ∣ ∃𝑎 ∈ On 𝑓:𝑎⟶{1_o, 2_o}}
df-lts 27611	⊢ <s = {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ No ∧ 𝑔 ∈ No ) ∧ ∃𝑥 ∈ On (∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑔‘𝑦) ∧ (𝑓‘𝑥){⟨1_o, ∅⟩, ⟨1_o, 2_o⟩, ⟨∅, 2_o⟩} (𝑔‘𝑥)))}
df-bday 27612	⊢ bday = (𝑥 ∈ No ↦ dom 𝑥)
cles 27712	class ≤s
df-les 27713	⊢ ≤s = (( No × No ) ∖ ^◡ <s )
cslts 27753	class <<s
df-slts 27754	⊢ <<s = {⟨𝑎, 𝑏⟩ ∣ (𝑎 ⊆ No ∧ 𝑏 ⊆ No ∧ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 𝑥 <s 𝑦)}
ccuts 27755	class \|s
df-cuts 27756	⊢ \|s = (𝑎 ∈ 𝒫 No , 𝑏 ∈ ( <<s “ {𝑎}) ↦ (℩𝑥 ∈ {𝑦 ∈ No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)} ( bday ‘𝑥) = ∩ ( bday “ {𝑦 ∈ No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)})))
c0s 27801	class 0_s
c1s 27802	class 1_s
df-0s 27803	⊢ 0_s = (∅ \|s ∅)
df-1s 27804	⊢ 1_s = ({ 0_s } \|s ∅)
cmade 27818	class M
cold 27819	class O
cnew 27820	class N
cleft 27821	class L
cright 27822	class R
df-made 27823	⊢ M = recs((𝑓 ∈ V ↦ ( \|s “ (𝒫 ∪ ran 𝑓 × 𝒫 ∪ ran 𝑓))))
df-old 27824	⊢ O = (𝑥 ∈ On ↦ ∪ ( M “ 𝑥))
df-new 27825	⊢ N = (𝑥 ∈ On ↦ (( M ‘𝑥) ∖ ( O ‘𝑥)))
df-left 27826	⊢ L = (𝑥 ∈ No ↦ {𝑦 ∈ ( O ‘( bday ‘𝑥)) ∣ 𝑦 <s 𝑥})
df-right 27827	⊢ R = (𝑥 ∈ No ↦ {𝑦 ∈ ( O ‘( bday ‘𝑥)) ∣ 𝑥 <s 𝑦})
cnorec 27933	class norec (𝐹)
df-norec 27934	⊢ norec (𝐹) = frecs({⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}, No , 𝐹)
cnorec2 27944	class norec2 (𝐹)
df-norec2 27945	⊢ norec2 (𝐹) = frecs({⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ ( No × No ) ∧ 𝑏 ∈ ( No × No ) ∧ (((1^st ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (1^st ‘𝑏) ∨ (1^st ‘𝑎) = (1^st ‘𝑏)) ∧ ((2^nd ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (2^nd ‘𝑏) ∨ (2^nd ‘𝑎) = (2^nd ‘𝑏)) ∧ 𝑎 ≠ 𝑏))}, ( No × No ), 𝐹)
cadds 27955	class +_s
df-adds 27956	⊢ +_s = norec2 ((𝑥 ∈ V, 𝑎 ∈ V ↦ (({𝑦 ∣ ∃𝑙 ∈ ( L ‘(1^st ‘𝑥))𝑦 = (𝑙𝑎(2^nd ‘𝑥))} ∪ {𝑧 ∣ ∃𝑙 ∈ ( L ‘(2^nd ‘𝑥))𝑧 = ((1^st ‘𝑥)𝑎𝑙)}) \|s ({𝑦 ∣ ∃𝑟 ∈ ( R ‘(1^st ‘𝑥))𝑦 = (𝑟𝑎(2^nd ‘𝑥))} ∪ {𝑧 ∣ ∃𝑟 ∈ ( R ‘(2^nd ‘𝑥))𝑧 = ((1^st ‘𝑥)𝑎𝑟)}))))
cnegs 28015	class -u_s
csubs 28016	class -_s
df-negs 28017	⊢ -u_s = norec ((𝑥 ∈ V, 𝑛 ∈ V ↦ ((𝑛 “ ( R ‘𝑥)) \|s (𝑛 “ ( L ‘𝑥)))))
df-subs 28018	⊢ -_s = (𝑥 ∈ No , 𝑦 ∈ No ↦ (𝑥 +_s ( -u_s ‘𝑦)))
cmuls 28102	class ·_s
df-muls 28103	⊢ ·_s = norec2 ((𝑧 ∈ V, 𝑚 ∈ V ↦ ⦋(1^st ‘𝑧) / 𝑥⦌⦋(2^nd ‘𝑧) / 𝑦⦌(({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝑥)∃𝑞 ∈ ( L ‘𝑦)𝑎 = (((𝑝𝑚𝑦) +_s (𝑥𝑚𝑞)) -_s (𝑝𝑚𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝑥)∃𝑠 ∈ ( R ‘𝑦)𝑏 = (((𝑟𝑚𝑦) +_s (𝑥𝑚𝑠)) -_s (𝑟𝑚𝑠))}) \|s ({𝑐 ∣ ∃𝑡 ∈ ( L ‘𝑥)∃𝑢 ∈ ( R ‘𝑦)𝑐 = (((𝑡𝑚𝑦) +_s (𝑥𝑚𝑢)) -_s (𝑡𝑚𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ ( R ‘𝑥)∃𝑤 ∈ ( L ‘𝑦)𝑑 = (((𝑣𝑚𝑦) +_s (𝑥𝑚𝑤)) -_s (𝑣𝑚𝑤))}))))
cdivs 28183	class /_su
df-divs 28184	⊢ /_su = (𝑥 ∈ No , 𝑦 ∈ ( No ∖ { 0_s }) ↦ (℩𝑧 ∈ No (𝑦 ·_s 𝑧) = 𝑥))
cabss 28233	class abs_s
df-abss 28234	⊢ abs_s = (𝑥 ∈ No ↦ if( 0_s ≤s 𝑥, 𝑥, ( -u_s ‘𝑥)))
cons 28247	class On_s
df-ons 28248	⊢ On_s = {𝑥 ∈ No ∣ ( R ‘𝑥) = ∅}
cseqs 28279	class seq_s𝑀( + , 𝐹)
df-seqs 28280	⊢ seq_s𝑀( + , 𝐹) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +_s 1_s ), (𝑦 + (𝐹‘(𝑥 +_s 1_s )))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) “ ω)
cn0s 28308	class ℕ_0s
cnns 28309	class ℕ_s
df-n0s 28310	⊢ ℕ_0s = (rec((𝑥 ∈ V ↦ (𝑥 +_s 1_s )), 0_s ) “ ω)
df-nns 28311	⊢ ℕ_s = (ℕ_0s ∖ { 0_s })
czs 28374	class ℤ_s
df-zs 28375	⊢ ℤ_s = ( -_s “ (ℕ_s × ℕ_s))
c2s 28406	class 2_s
df-2s 28407	⊢ 2_s = ({ 1_s } \|s ∅)
cexps 28408	class ↑_s
df-exps 28409	⊢ ↑_s = (𝑥 ∈ No , 𝑦 ∈ ℤ_s ↦ if(𝑦 = 0_s , 1_s , if( 0_s <s 𝑦, (seq_s 1_s ( ·_s , (ℕ_s × {𝑥}))‘𝑦), ( 1_s /_su (seq_s 1_s ( ·_s , (ℕ_s × {𝑥}))‘( -u_s ‘𝑦))))))
cz12s 28410	class ℤ_s[1/2]
df-z12s 28411	⊢ ℤ_s[1/2] = {𝑥 ∣ ∃𝑦 ∈ ℤ_s ∃𝑧 ∈ ℕ_0s 𝑥 = (𝑦 /_su (2_s↑_s𝑧))}
creno 28485	class ℝ_s
df-reno 28486	⊢ ℝ_s = {𝑥 ∈ No ∣ (∃𝑛 ∈ ℕ_s (( -u_s ‘𝑛) <s 𝑥 ∧ 𝑥 <s 𝑛) ∧ 𝑥 = ({𝑦 ∣ ∃𝑛 ∈ ℕ_s 𝑦 = (𝑥 -_s ( 1_s /_su 𝑛))} \|s {𝑦 ∣ ∃𝑛 ∈ ℕ_s 𝑦 = (𝑥 +_s ( 1_s /_su 𝑛))}))}
cstrkg 28499	class TarskiG
cstrkgc 28500	class TarskiG_C
cstrkgb 28501	class TarskiG_B
cstrkgcb 28502	class TarskiG_CB
cstrkgld 28503	class Dim_TarskiG≥
cstrkge 28504	class TarskiG_E
citv 28505	class Itv
clng 28506	class LineG
df-itv 28507	⊢ Itv = Slot ;16
df-lng 28508	⊢ LineG = Slot ;17
df-trkgc 28520	⊢ TarskiG_C = {𝑓 ∣ [(Base‘𝑓) / 𝑝][(dist‘𝑓) / 𝑑](∀𝑥 ∈ 𝑝 ∀𝑦 ∈ 𝑝 (𝑥𝑑𝑦) = (𝑦𝑑𝑥) ∧ ∀𝑥 ∈ 𝑝 ∀𝑦 ∈ 𝑝 ∀𝑧 ∈ 𝑝 ((𝑥𝑑𝑦) = (𝑧𝑑𝑧) → 𝑥 = 𝑦))}
df-trkgb 28521	⊢ TarskiG_B = {𝑓 ∣ [(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](∀𝑥 ∈ 𝑝 ∀𝑦 ∈ 𝑝 (𝑦 ∈ (𝑥𝑖𝑥) → 𝑥 = 𝑦) ∧ ∀𝑥 ∈ 𝑝 ∀𝑦 ∈ 𝑝 ∀𝑧 ∈ 𝑝 ∀𝑢 ∈ 𝑝 ∀𝑣 ∈ 𝑝 ((𝑢 ∈ (𝑥𝑖𝑧) ∧ 𝑣 ∈ (𝑦𝑖𝑧)) → ∃𝑎 ∈ 𝑝 (𝑎 ∈ (𝑢𝑖𝑦) ∧ 𝑎 ∈ (𝑣𝑖𝑥))) ∧ ∀𝑠 ∈ 𝒫 𝑝∀𝑡 ∈ 𝒫 𝑝(∃𝑎 ∈ 𝑝 ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑥 ∈ (𝑎𝑖𝑦) → ∃𝑏 ∈ 𝑝 ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑏 ∈ (𝑥𝑖𝑦)))}
df-trkgcb 28522	⊢ TarskiG_CB = {𝑓 ∣ [(Base‘𝑓) / 𝑝][(dist‘𝑓) / 𝑑][(Itv‘𝑓) / 𝑖](∀𝑥 ∈ 𝑝 ∀𝑦 ∈ 𝑝 ∀𝑧 ∈ 𝑝 ∀𝑢 ∈ 𝑝 ∀𝑎 ∈ 𝑝 ∀𝑏 ∈ 𝑝 ∀𝑐 ∈ 𝑝 ∀𝑣 ∈ 𝑝 (((𝑥 ≠ 𝑦 ∧ 𝑦 ∈ (𝑥𝑖𝑧) ∧ 𝑏 ∈ (𝑎𝑖𝑐)) ∧ (((𝑥𝑑𝑦) = (𝑎𝑑𝑏) ∧ (𝑦𝑑𝑧) = (𝑏𝑑𝑐)) ∧ ((𝑥𝑑𝑢) = (𝑎𝑑𝑣) ∧ (𝑦𝑑𝑢) = (𝑏𝑑𝑣)))) → (𝑧𝑑𝑢) = (𝑐𝑑𝑣)) ∧ ∀𝑥 ∈ 𝑝 ∀𝑦 ∈ 𝑝 ∀𝑎 ∈ 𝑝 ∀𝑏 ∈ 𝑝 ∃𝑧 ∈ 𝑝 (𝑦 ∈ (𝑥𝑖𝑧) ∧ (𝑦𝑑𝑧) = (𝑎𝑑𝑏)))}
df-trkge 28523	⊢ TarskiG_E = {𝑓 ∣ [(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖]∀𝑥 ∈ 𝑝 ∀𝑦 ∈ 𝑝 ∀𝑧 ∈ 𝑝 ∀𝑢 ∈ 𝑝 ∀𝑣 ∈ 𝑝 ((𝑢 ∈ (𝑥𝑖𝑣) ∧ 𝑢 ∈ (𝑦𝑖𝑧) ∧ 𝑥 ≠ 𝑢) → ∃𝑎 ∈ 𝑝 ∃𝑏 ∈ 𝑝 (𝑦 ∈ (𝑥𝑖𝑎) ∧ 𝑧 ∈ (𝑥𝑖𝑏) ∧ 𝑣 ∈ (𝑎𝑖𝑏)))}
df-trkgld 28524	⊢ Dim_TarskiG≥ = {⟨𝑔, 𝑛⟩ ∣ [(Base‘𝑔) / 𝑝][(dist‘𝑔) / 𝑑][(Itv‘𝑔) / 𝑖]∃𝑓(𝑓:(1..^𝑛)–1-1→𝑝 ∧ ∃𝑥 ∈ 𝑝 ∃𝑦 ∈ 𝑝 ∃𝑧 ∈ 𝑝 (∀𝑗 ∈ (2..^𝑛)(((𝑓‘1)𝑑𝑥) = ((𝑓‘𝑗)𝑑𝑥) ∧ ((𝑓‘1)𝑑𝑦) = ((𝑓‘𝑗)𝑑𝑦) ∧ ((𝑓‘1)𝑑𝑧) = ((𝑓‘𝑗)𝑑𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))))}
df-trkg 28525	⊢ TarskiG = ((TarskiG_C ∩ TarskiG_B) ∩ (TarskiG_CB ∩ {𝑓 ∣ [(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥 ∈ 𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})}))
ccgrg 28582	class cgrG
df-cgrg 28583	⊢ cgrG = (𝑔 ∈ V ↦ {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ ((Base‘𝑔) ↑_pm ℝ) ∧ 𝑏 ∈ ((Base‘𝑔) ↑_pm ℝ)) ∧ (dom 𝑎 = dom 𝑏 ∧ ∀𝑖 ∈ dom 𝑎∀𝑗 ∈ dom 𝑎((𝑎‘𝑖)(dist‘𝑔)(𝑎‘𝑗)) = ((𝑏‘𝑖)(dist‘𝑔)(𝑏‘𝑗))))})
cismt 28604	class Ismt
df-ismt 28605	⊢ Ismt = (𝑔 ∈ V, ℎ ∈ V ↦ {𝑓 ∣ (𝑓:(Base‘𝑔)–1-1-onto→(Base‘ℎ) ∧ ∀𝑎 ∈ (Base‘𝑔)∀𝑏 ∈ (Base‘𝑔)((𝑓‘𝑎)(dist‘ℎ)(𝑓‘𝑏)) = (𝑎(dist‘𝑔)𝑏))})
cleg 28654	class ≤G
df-leg 28655	⊢ ≤G = (𝑔 ∈ V ↦ {⟨𝑒, 𝑓⟩ ∣ [(Base‘𝑔) / 𝑝][(dist‘𝑔) / 𝑑][(Itv‘𝑔) / 𝑖]∃𝑥 ∈ 𝑝 ∃𝑦 ∈ 𝑝 (𝑓 = (𝑥𝑑𝑦) ∧ ∃𝑧 ∈ 𝑝 (𝑧 ∈ (𝑥𝑖𝑦) ∧ 𝑒 = (𝑥𝑑𝑧)))})
chlg 28672	class hlG
df-hlg 28673	⊢ hlG = (𝑔 ∈ V ↦ (𝑐 ∈ (Base‘𝑔) ↦ {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (Base‘𝑔) ∧ 𝑏 ∈ (Base‘𝑔)) ∧ (𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐 ∧ (𝑎 ∈ (𝑐(Itv‘𝑔)𝑏) ∨ 𝑏 ∈ (𝑐(Itv‘𝑔)𝑎))))}))
cmir 28724	class pInvG
df-mir 28725	⊢ pInvG = (𝑔 ∈ V ↦ (𝑚 ∈ (Base‘𝑔) ↦ (𝑎 ∈ (Base‘𝑔) ↦ (℩𝑏 ∈ (Base‘𝑔)((𝑚(dist‘𝑔)𝑏) = (𝑚(dist‘𝑔)𝑎) ∧ 𝑚 ∈ (𝑏(Itv‘𝑔)𝑎))))))
crag 28765	class ∟G
df-rag 28766	⊢ ∟G = (𝑔 ∈ V ↦ {𝑤 ∈ Word (Base‘𝑔) ∣ ((♯‘𝑤) = 3 ∧ ((𝑤‘0)(dist‘𝑔)(𝑤‘2)) = ((𝑤‘0)(dist‘𝑔)(((pInvG‘𝑔)‘(𝑤‘1))‘(𝑤‘2))))})
cperpg 28767	class ⟂G
df-perpg 28768	⊢ ⟂G = (𝑔 ∈ V ↦ {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ ran (LineG‘𝑔) ∧ 𝑏 ∈ ran (LineG‘𝑔)) ∧ ∃𝑥 ∈ (𝑎 ∩ 𝑏)∀𝑢 ∈ 𝑎 ∀𝑣 ∈ 𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝑔))})
chpg 28829	class hpG
df-hpg 28830	⊢ hpG = (𝑔 ∈ V ↦ (𝑑 ∈ ran (LineG‘𝑔) ↦ {⟨𝑎, 𝑏⟩ ∣ [(Base‘𝑔) / 𝑝][(Itv‘𝑔) / 𝑖]∃𝑐 ∈ 𝑝 (((𝑎 ∈ (𝑝 ∖ 𝑑) ∧ 𝑐 ∈ (𝑝 ∖ 𝑑)) ∧ ∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑎𝑖𝑐)) ∧ ((𝑏 ∈ (𝑝 ∖ 𝑑) ∧ 𝑐 ∈ (𝑝 ∖ 𝑑)) ∧ ∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑏𝑖𝑐)))}))
cmid 28844	class midG
clmi 28845	class lInvG
df-mid 28846	⊢ midG = (𝑔 ∈ V ↦ (𝑎 ∈ (Base‘𝑔), 𝑏 ∈ (Base‘𝑔) ↦ (℩𝑚 ∈ (Base‘𝑔)𝑏 = (((pInvG‘𝑔)‘𝑚)‘𝑎))))
df-lmi 28847	⊢ lInvG = (𝑔 ∈ V ↦ (𝑚 ∈ ran (LineG‘𝑔) ↦ (𝑎 ∈ (Base‘𝑔) ↦ (℩𝑏 ∈ (Base‘𝑔)((𝑎(midG‘𝑔)𝑏) ∈ 𝑚 ∧ (𝑚(⟂G‘𝑔)(𝑎(LineG‘𝑔)𝑏) ∨ 𝑎 = 𝑏))))))
ccgra 28879	class cgrA
df-cgra 28880	⊢ cgrA = (𝑔 ∈ V ↦ {⟨𝑎, 𝑏⟩ ∣ [(Base‘𝑔) / 𝑝][(hlG‘𝑔) / 𝑘]((𝑎 ∈ (𝑝 ↑_m (0..^3)) ∧ 𝑏 ∈ (𝑝 ↑_m (0..^3))) ∧ ∃𝑥 ∈ 𝑝 ∃𝑦 ∈ 𝑝 (𝑎(cgrG‘𝑔)⟨“𝑥(𝑏‘1)𝑦”⟩ ∧ 𝑥(𝑘‘(𝑏‘1))(𝑏‘0) ∧ 𝑦(𝑘‘(𝑏‘1))(𝑏‘2)))})
cinag 28907	class inA
cleag 28908	class ≤_∠
df-inag 28909	⊢ inA = (𝑔 ∈ V ↦ {⟨𝑝, 𝑡⟩ ∣ ((𝑝 ∈ (Base‘𝑔) ∧ 𝑡 ∈ ((Base‘𝑔) ↑_m (0..^3))) ∧ (((𝑡‘0) ≠ (𝑡‘1) ∧ (𝑡‘2) ≠ (𝑡‘1) ∧ 𝑝 ≠ (𝑡‘1)) ∧ ∃𝑥 ∈ (Base‘𝑔)(𝑥 ∈ ((𝑡‘0)(Itv‘𝑔)(𝑡‘2)) ∧ (𝑥 = (𝑡‘1) ∨ 𝑥((hlG‘𝑔)‘(𝑡‘1))𝑝))))})
df-leag 28918	⊢ ≤_∠ = (𝑔 ∈ V ↦ {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ ((Base‘𝑔) ↑_m (0..^3)) ∧ 𝑏 ∈ ((Base‘𝑔) ↑_m (0..^3))) ∧ ∃𝑥 ∈ (Base‘𝑔)(𝑥(inA‘𝑔)⟨“(𝑏‘0)(𝑏‘1)(𝑏‘2)”⟩ ∧ ⟨“(𝑎‘0)(𝑎‘1)(𝑎‘2)”⟩(cgrA‘𝑔)⟨“(𝑏‘0)(𝑏‘1)𝑥”⟩))})
ceqlg 28937	class eqltrG
df-eqlg 28938	⊢ eqltrG = (𝑔 ∈ V ↦ {𝑥 ∈ ((Base‘𝑔) ↑_m (0..^3)) ∣ 𝑥(cgrG‘𝑔)⟨“(𝑥‘1)(𝑥‘2)(𝑥‘0)”⟩})
cttg 28945	class toTG
df-ttg 28946	⊢ toTG = (𝑤 ∈ V ↦ ⦋(𝑥 ∈ (Base‘𝑤), 𝑦 ∈ (Base‘𝑤) ↦ {𝑧 ∈ (Base‘𝑤) ∣ ∃𝑘 ∈ (0[,]1)(𝑧(-_g‘𝑤)𝑥) = (𝑘( ·_𝑠 ‘𝑤)(𝑦(-_g‘𝑤)𝑥))}) / 𝑖⦌((𝑤 sSet ⟨(Itv‘ndx), 𝑖⟩) sSet ⟨(LineG‘ndx), (𝑥 ∈ (Base‘𝑤), 𝑦 ∈ (Base‘𝑤) ↦ {𝑧 ∈ (Base‘𝑤) ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})⟩))
cee 28960	class 𝔼
cbtwn 28961	class Btwn
ccgr 28962	class Cgr
df-ee 28963	⊢ 𝔼 = (𝑛 ∈ ℕ ↦ (ℝ ↑_m (1...𝑛)))
df-btwn 28964	⊢ Btwn = ^◡{⟨⟨𝑥, 𝑧⟩, 𝑦⟩ ∣ ∃𝑛 ∈ ℕ ((𝑥 ∈ (𝔼‘𝑛) ∧ 𝑧 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛)) ∧ ∃𝑡 ∈ (0[,]1)∀𝑖 ∈ (1...𝑛)(𝑦‘𝑖) = (((1 − 𝑡) · (𝑥‘𝑖)) + (𝑡 · (𝑧‘𝑖))))}
df-cgr 28965	⊢ Cgr = {⟨𝑥, 𝑦⟩ ∣ ∃𝑛 ∈ ℕ ((𝑥 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑦 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛))) ∧ Σ𝑖 ∈ (1...𝑛)((((1^st ‘𝑥)‘𝑖) − ((2^nd ‘𝑥)‘𝑖))↑2) = Σ𝑖 ∈ (1...𝑛)((((1^st ‘𝑦)‘𝑖) − ((2^nd ‘𝑦)‘𝑖))↑2))}
ceeng 29050	class EEG
df-eeng 29051	⊢ EEG = (𝑛 ∈ ℕ ↦ ({⟨(Base‘ndx), (𝔼‘𝑛)⟩, ⟨(dist‘ndx), (𝑥 ∈ (𝔼‘𝑛), 𝑦 ∈ (𝔼‘𝑛) ↦ Σ𝑖 ∈ (1...𝑛)(((𝑥‘𝑖) − (𝑦‘𝑖))↑2))⟩} ∪ {⟨(Itv‘ndx), (𝑥 ∈ (𝔼‘𝑛), 𝑦 ∈ (𝔼‘𝑛) ↦ {𝑧 ∈ (𝔼‘𝑛) ∣ 𝑧 Btwn ⟨𝑥, 𝑦⟩})⟩, ⟨(LineG‘ndx), (𝑥 ∈ (𝔼‘𝑛), 𝑦 ∈ ((𝔼‘𝑛) ∖ {𝑥}) ↦ {𝑧 ∈ (𝔼‘𝑛) ∣ (𝑧 Btwn ⟨𝑥, 𝑦⟩ ∨ 𝑥 Btwn ⟨𝑧, 𝑦⟩ ∨ 𝑦 Btwn ⟨𝑥, 𝑧⟩)})⟩}))
cedgf 29061	class .ef
df-edgf 29062	⊢ .ef = Slot ;18
cvtx 29069	class Vtx
ciedg 29070	class iEdg
df-vtx 29071	⊢ Vtx = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1^st ‘𝑔), (Base‘𝑔)))
df-iedg 29072	⊢ iEdg = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (2^nd ‘𝑔), (.ef‘𝑔)))
cedg 29120	class Edg
df-edg 29121	⊢ Edg = (𝑔 ∈ V ↦ ran (iEdg‘𝑔))
cuhgr 29129	class UHGraph
cushgr 29130	class USHGraph
df-uhgr 29131	⊢ UHGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶(𝒫 𝑣 ∖ {∅})}
df-ushgr 29132	⊢ USHGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→(𝒫 𝑣 ∖ {∅})}
cupgr 29153	class UPGraph
cumgr 29154	class UMGraph
df-upgr 29155	⊢ UPGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}}
df-umgr 29156	⊢ UMGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2}}
cuspgr 29221	class USPGraph
cusgr 29222	class USGraph
df-uspgr 29223	⊢ USPGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}}
df-usgr 29224	⊢ USGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2}}
csubgr 29340	class SubGraph
df-subgr 29341	⊢ SubGraph = {⟨𝑠, 𝑔⟩ ∣ ((Vtx‘𝑠) ⊆ (Vtx‘𝑔) ∧ (iEdg‘𝑠) = ((iEdg‘𝑔) ↾ dom (iEdg‘𝑠)) ∧ (Edg‘𝑠) ⊆ 𝒫 (Vtx‘𝑠))}
cfusgr 29389	class FinUSGraph
df-fusgr 29390	⊢ FinUSGraph = {𝑔 ∈ USGraph ∣ (Vtx‘𝑔) ∈ Fin}
cnbgr 29405	class NeighbVtx
df-nbgr 29406	⊢ NeighbVtx = (𝑔 ∈ V, 𝑣 ∈ (Vtx‘𝑔) ↦ {𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣}) ∣ ∃𝑒 ∈ (Edg‘𝑔){𝑣, 𝑛} ⊆ 𝑒})
cuvtx 29458	class UnivVtx
df-uvtx 29459	⊢ UnivVtx = (𝑔 ∈ V ↦ {𝑣 ∈ (Vtx‘𝑔) ∣ ∀𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣})𝑛 ∈ (𝑔 NeighbVtx 𝑣)})
ccplgr 29482	class ComplGraph
ccusgr 29483	class ComplUSGraph
df-cplgr 29484	⊢ ComplGraph = {𝑔 ∣ (UnivVtx‘𝑔) = (Vtx‘𝑔)}
df-cusgr 29485	⊢ ComplUSGraph = (USGraph ∩ ComplGraph)
cvtxdg 29539	class VtxDeg
df-vtxdg 29540	⊢ VtxDeg = (𝑔 ∈ V ↦ ⦋(Vtx‘𝑔) / 𝑣⦌⦋(iEdg‘𝑔) / 𝑒⦌(𝑢 ∈ 𝑣 ↦ ((♯‘{𝑥 ∈ dom 𝑒 ∣ 𝑢 ∈ (𝑒‘𝑥)}) +_𝑒 (♯‘{𝑥 ∈ dom 𝑒 ∣ (𝑒‘𝑥) = {𝑢}}))))
crgr 29629	class RegGraph
crusgr 29630	class RegUSGraph
df-rgr 29631	⊢ RegGraph = {⟨𝑔, 𝑘⟩ ∣ (𝑘 ∈ ℕ₀^* ∧ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 𝑘)}
df-rusgr 29632	⊢ RegUSGraph = {⟨𝑔, 𝑘⟩ ∣ (𝑔 ∈ USGraph ∧ 𝑔 RegGraph 𝑘)}
cewlks 29669	class EdgWalks
cwlks 29670	class Walks
cwlkson 29671	class WalksOn
df-ewlks 29672	⊢ EdgWalks = (𝑔 ∈ V, 𝑠 ∈ ℕ₀^* ↦ {𝑓 ∣ [(iEdg‘𝑔) / 𝑖](𝑓 ∈ Word dom 𝑖 ∧ ∀𝑘 ∈ (1..^(♯‘𝑓))𝑠 ≤ (♯‘((𝑖‘(𝑓‘(𝑘 − 1))) ∩ (𝑖‘(𝑓‘𝑘)))))})
df-wlks 29673	⊢ Walks = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom (iEdg‘𝑔) ∧ 𝑝:(0...(♯‘𝑓))⟶(Vtx‘𝑔) ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))if-((𝑝‘𝑘) = (𝑝‘(𝑘 + 1)), ((iEdg‘𝑔)‘(𝑓‘𝑘)) = {(𝑝‘𝑘)}, {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))} ⊆ ((iEdg‘𝑔)‘(𝑓‘𝑘))))})
df-wlkson 29674	⊢ WalksOn = (𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝑔)𝑝 ∧ (𝑝‘0) = 𝑎 ∧ (𝑝‘(♯‘𝑓)) = 𝑏)}))
ctrls 29762	class Trails
ctrlson 29763	class TrailsOn
df-trls 29764	⊢ Trails = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝑔)𝑝 ∧ Fun ^◡𝑓)})
df-trlson 29765	⊢ TrailsOn = (𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑎(WalksOn‘𝑔)𝑏)𝑝 ∧ 𝑓(Trails‘𝑔)𝑝)}))
cpths 29783	class Paths
cspths 29784	class SPaths
cpthson 29785	class PathsOn
cspthson 29786	class SPathsOn
df-pths 29787	⊢ Paths = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝑔)𝑝 ∧ Fun ^◡(𝑝 ↾ (1..^(♯‘𝑓))) ∧ ((𝑝 “ {0, (♯‘𝑓)}) ∩ (𝑝 “ (1..^(♯‘𝑓)))) = ∅)})
df-spths 29788	⊢ SPaths = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝑔)𝑝 ∧ Fun ^◡𝑝)})
df-pthson 29789	⊢ PathsOn = (𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑎(TrailsOn‘𝑔)𝑏)𝑝 ∧ 𝑓(Paths‘𝑔)𝑝)}))
df-spthson 29790	⊢ SPathsOn = (𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑎(TrailsOn‘𝑔)𝑏)𝑝 ∧ 𝑓(SPaths‘𝑔)𝑝)}))
cclwlks 29843	class ClWalks
df-clwlks 29844	⊢ ClWalks = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝑔)𝑝 ∧ (𝑝‘0) = (𝑝‘(♯‘𝑓)))})
ccrcts 29857	class Circuits
ccycls 29858	class Cycles
df-crcts 29859	⊢ Circuits = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝑔)𝑝 ∧ (𝑝‘0) = (𝑝‘(♯‘𝑓)))})
df-cycls 29860	⊢ Cycles = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Paths‘𝑔)𝑝 ∧ (𝑝‘0) = (𝑝‘(♯‘𝑓)))})
cwwlks 29898	class WWalks
cwwlksn 29899	class WWalksN
cwwlksnon 29900	class WWalksNOn
cwwspthsn 29901	class WSPathsN
cwwspthsnon 29902	class WSPathsNOn
df-wwlks 29903	⊢ WWalks = (𝑔 ∈ V ↦ {𝑤 ∈ Word (Vtx‘𝑔) ∣ (𝑤 ≠ ∅ ∧ ∀𝑖 ∈ (0..^((♯‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝑔))})
df-wwlksn 29904	⊢ WWalksN = (𝑛 ∈ ℕ₀, 𝑔 ∈ V ↦ {𝑤 ∈ (WWalks‘𝑔) ∣ (♯‘𝑤) = (𝑛 + 1)})
df-wwlksnon 29905	⊢ WWalksNOn = (𝑛 ∈ ℕ₀, 𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {𝑤 ∈ (𝑛 WWalksN 𝑔) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤‘𝑛) = 𝑏)}))
df-wspthsn 29906	⊢ WSPathsN = (𝑛 ∈ ℕ₀, 𝑔 ∈ V ↦ {𝑤 ∈ (𝑛 WWalksN 𝑔) ∣ ∃𝑓 𝑓(SPaths‘𝑔)𝑤})
df-wspthsnon 29907	⊢ WSPathsNOn = (𝑛 ∈ ℕ₀, 𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {𝑤 ∈ (𝑎(𝑛 WWalksNOn 𝑔)𝑏) ∣ ∃𝑓 𝑓(𝑎(SPathsOn‘𝑔)𝑏)𝑤}))
cclwwlk 30056	class ClWWalks
df-clwwlk 30057	⊢ ClWWalks = (𝑔 ∈ V ↦ {𝑤 ∈ Word (Vtx‘𝑔) ∣ (𝑤 ≠ ∅ ∧ ∀𝑖 ∈ (0..^((♯‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝑔) ∧ {(lastS‘𝑤), (𝑤‘0)} ∈ (Edg‘𝑔))})
cclwwlkn 30099	class ClWWalksN
df-clwwlkn 30100	⊢ ClWWalksN = (𝑛 ∈ ℕ₀, 𝑔 ∈ V ↦ {𝑤 ∈ (ClWWalks‘𝑔) ∣ (♯‘𝑤) = 𝑛})
cclwwlknon 30162	class ClWWalksNOn
df-clwwlknon 30163	⊢ ClWWalksNOn = (𝑔 ∈ V ↦ (𝑣 ∈ (Vtx‘𝑔), 𝑛 ∈ ℕ₀ ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝑔) ∣ (𝑤‘0) = 𝑣}))
cconngr 30261	class ConnGraph
df-conngr 30262	⊢ ConnGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣]∀𝑘 ∈ 𝑣 ∀𝑛 ∈ 𝑣 ∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝}
ceupth 30272	class EulerPaths
df-eupth 30273	⊢ EulerPaths = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝑔)𝑝 ∧ 𝑓:(0..^(♯‘𝑓))–onto→dom (iEdg‘𝑔))})
cfrgr 30333	class FriendGraph
df-frgr 30334	⊢ FriendGraph = {𝑔 ∈ USGraph ∣ [(Vtx‘𝑔) / 𝑣][(Edg‘𝑔) / 𝑒]∀𝑘 ∈ 𝑣 ∀𝑙 ∈ (𝑣 ∖ {𝑘})∃!𝑥 ∈ 𝑣 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝑒}
ax-flt 30547	⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ (𝑋 ∈ ℕ ∧ 𝑌 ∈ ℕ ∧ 𝑍 ∈ ℕ)) → ((𝑋↑𝑁) + (𝑌↑𝑁)) ≠ (𝑍↑𝑁))
cplig 30549	class Plig
df-plig 30550	⊢ Plig = {𝑥 ∣ (∀𝑎 ∈ ∪ 𝑥∀𝑏 ∈ ∪ 𝑥(𝑎 ≠ 𝑏 → ∃!𝑙 ∈ 𝑥 (𝑎 ∈ 𝑙 ∧ 𝑏 ∈ 𝑙)) ∧ ∀𝑙 ∈ 𝑥 ∃𝑎 ∈ ∪ 𝑥∃𝑏 ∈ ∪ 𝑥(𝑎 ≠ 𝑏 ∧ 𝑎 ∈ 𝑙 ∧ 𝑏 ∈ 𝑙) ∧ ∃𝑎 ∈ ∪ 𝑥∃𝑏 ∈ ∪ 𝑥∃𝑐 ∈ ∪ 𝑥∀𝑙 ∈ 𝑥 ¬ (𝑎 ∈ 𝑙 ∧ 𝑏 ∈ 𝑙 ∧ 𝑐 ∈ 𝑙))}
cgr 30564	class GrpOp
cgi 30565	class GId
cgn 30566	class inv
cgs 30567	class /_𝑔
df-grpo 30568	⊢ GrpOp = {𝑔 ∣ ∃𝑡(𝑔:(𝑡 × 𝑡)⟶𝑡 ∧ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 ∀𝑧 ∈ 𝑡 ((𝑥𝑔𝑦)𝑔𝑧) = (𝑥𝑔(𝑦𝑔𝑧)) ∧ ∃𝑢 ∈ 𝑡 ∀𝑥 ∈ 𝑡 ((𝑢𝑔𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑡 (𝑦𝑔𝑥) = 𝑢))}
df-gid 30569	⊢ GId = (𝑔 ∈ V ↦ (℩𝑢 ∈ ran 𝑔∀𝑥 ∈ ran 𝑔((𝑢𝑔𝑥) = 𝑥 ∧ (𝑥𝑔𝑢) = 𝑥)))
df-ginv 30570	⊢ inv = (𝑔 ∈ GrpOp ↦ (𝑥 ∈ ran 𝑔 ↦ (℩𝑧 ∈ ran 𝑔(𝑧𝑔𝑥) = (GId‘𝑔))))
df-gdiv 30571	⊢ /_𝑔 = (𝑔 ∈ GrpOp ↦ (𝑥 ∈ ran 𝑔, 𝑦 ∈ ran 𝑔 ↦ (𝑥𝑔((inv‘𝑔)‘𝑦))))
cablo 30619	class AbelOp
df-ablo 30620	⊢ AbelOp = {𝑔 ∈ GrpOp ∣ ∀𝑥 ∈ ran 𝑔∀𝑦 ∈ ran 𝑔(𝑥𝑔𝑦) = (𝑦𝑔𝑥)}
cvc 30633	class CVec_OLD
df-vc 30634	⊢ CVec_OLD = {⟨𝑔, 𝑠⟩ ∣ (𝑔 ∈ AbelOp ∧ 𝑠:(ℂ × ran 𝑔)⟶ran 𝑔 ∧ ∀𝑥 ∈ ran 𝑔((1𝑠𝑥) = 𝑥 ∧ ∀𝑦 ∈ ℂ (∀𝑧 ∈ ran 𝑔(𝑦𝑠(𝑥𝑔𝑧)) = ((𝑦𝑠𝑥)𝑔(𝑦𝑠𝑧)) ∧ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑠𝑥) = ((𝑦𝑠𝑥)𝑔(𝑧𝑠𝑥)) ∧ ((𝑦 · 𝑧)𝑠𝑥) = (𝑦𝑠(𝑧𝑠𝑥))))))}
cnv 30659	class NrmCVec
cpv 30660	class +_𝑣
cba 30661	class BaseSet
cns 30662	class ·_𝑠OLD
cn0v 30663	class 0_vec
cnsb 30664	class −_𝑣
cnmcv 30665	class norm_CV
cims 30666	class IndMet
df-nv 30667	⊢ NrmCVec = {⟨⟨𝑔, 𝑠⟩, 𝑛⟩ ∣ (⟨𝑔, 𝑠⟩ ∈ CVec_OLD ∧ 𝑛:ran 𝑔⟶ℝ ∧ ∀𝑥 ∈ ran 𝑔(((𝑛‘𝑥) = 0 → 𝑥 = (GId‘𝑔)) ∧ ∀𝑦 ∈ ℂ (𝑛‘(𝑦𝑠𝑥)) = ((abs‘𝑦) · (𝑛‘𝑥)) ∧ ∀𝑦 ∈ ran 𝑔(𝑛‘(𝑥𝑔𝑦)) ≤ ((𝑛‘𝑥) + (𝑛‘𝑦))))}
df-va 30670	⊢ +_𝑣 = (1^st ∘ 1^st )
df-ba 30671	⊢ BaseSet = (𝑥 ∈ V ↦ ran ( +_𝑣 ‘𝑥))
df-sm 30672	⊢ ·_𝑠OLD = (2^nd ∘ 1^st )
df-0v 30673	⊢ 0_vec = (GId ∘ +_𝑣 )
df-vs 30674	⊢ −_𝑣 = ( /_𝑔 ∘ +_𝑣 )
df-nmcv 30675	⊢ norm_CV = 2^nd
df-ims 30676	⊢ IndMet = (𝑢 ∈ NrmCVec ↦ ((norm_CV‘𝑢) ∘ ( −_𝑣 ‘𝑢)))
cdip 30775	class ·_𝑖OLD
df-dip 30776	⊢ ·_𝑖OLD = (𝑢 ∈ NrmCVec ↦ (𝑥 ∈ (BaseSet‘𝑢), 𝑦 ∈ (BaseSet‘𝑢) ↦ (Σ𝑘 ∈ (1...4)((i↑𝑘) · (((norm_CV‘𝑢)‘(𝑥( +_𝑣 ‘𝑢)((i↑𝑘)( ·_𝑠OLD ‘𝑢)𝑦)))↑2)) / 4)))
css 30796	class SubSp
df-ssp 30797	⊢ SubSp = (𝑢 ∈ NrmCVec ↦ {𝑤 ∈ NrmCVec ∣ (( +_𝑣 ‘𝑤) ⊆ ( +_𝑣 ‘𝑢) ∧ ( ·_𝑠OLD ‘𝑤) ⊆ ( ·_𝑠OLD ‘𝑢) ∧ (norm_CV‘𝑤) ⊆ (norm_CV‘𝑢))})
clno 30815	class LnOp
cnmoo 30816	class normOp_OLD
cblo 30817	class BLnOp
c0o 30818	class 0_op
df-lno 30819	⊢ LnOp = (𝑢 ∈ NrmCVec, 𝑤 ∈ NrmCVec ↦ {𝑡 ∈ ((BaseSet‘𝑤) ↑_m (BaseSet‘𝑢)) ∣ ∀𝑥 ∈ ℂ ∀𝑦 ∈ (BaseSet‘𝑢)∀𝑧 ∈ (BaseSet‘𝑢)(𝑡‘((𝑥( ·_𝑠OLD ‘𝑢)𝑦)( +_𝑣 ‘𝑢)𝑧)) = ((𝑥( ·_𝑠OLD ‘𝑤)(𝑡‘𝑦))( +_𝑣 ‘𝑤)(𝑡‘𝑧))})
df-nmoo 30820	⊢ normOp_OLD = (𝑢 ∈ NrmCVec, 𝑤 ∈ NrmCVec ↦ (𝑡 ∈ ((BaseSet‘𝑤) ↑_m (BaseSet‘𝑢)) ↦ sup({𝑥 ∣ ∃𝑧 ∈ (BaseSet‘𝑢)(((norm_CV‘𝑢)‘𝑧) ≤ 1 ∧ 𝑥 = ((norm_CV‘𝑤)‘(𝑡‘𝑧)))}, ℝ^*, < )))
df-blo 30821	⊢ BLnOp = (𝑢 ∈ NrmCVec, 𝑤 ∈ NrmCVec ↦ {𝑡 ∈ (𝑢 LnOp 𝑤) ∣ ((𝑢 normOp_OLD 𝑤)‘𝑡) < +∞})
df-0o 30822	⊢ 0_op = (𝑢 ∈ NrmCVec, 𝑤 ∈ NrmCVec ↦ ((BaseSet‘𝑢) × {(0_vec‘𝑤)}))
caj 30823	class adj
chmo 30824	class HmOp
df-aj 30825	⊢ adj = (𝑢 ∈ NrmCVec, 𝑤 ∈ NrmCVec ↦ {⟨𝑡, 𝑠⟩ ∣ (𝑡:(BaseSet‘𝑢)⟶(BaseSet‘𝑤) ∧ 𝑠:(BaseSet‘𝑤)⟶(BaseSet‘𝑢) ∧ ∀𝑥 ∈ (BaseSet‘𝑢)∀𝑦 ∈ (BaseSet‘𝑤)((𝑡‘𝑥)(·_𝑖OLD‘𝑤)𝑦) = (𝑥(·_𝑖OLD‘𝑢)(𝑠‘𝑦)))})
df-hmo 30826	⊢ HmOp = (𝑢 ∈ NrmCVec ↦ {𝑡 ∈ dom (𝑢adj𝑢) ∣ ((𝑢adj𝑢)‘𝑡) = 𝑡})
ccphlo 30887	class CPreHil_OLD
df-ph 30888	⊢ CPreHil_OLD = (NrmCVec ∩ {⟨⟨𝑔, 𝑠⟩, 𝑛⟩ ∣ ∀𝑥 ∈ ran 𝑔∀𝑦 ∈ ran 𝑔(((𝑛‘(𝑥𝑔𝑦))↑2) + ((𝑛‘(𝑥𝑔(-1𝑠𝑦)))↑2)) = (2 · (((𝑛‘𝑥)↑2) + ((𝑛‘𝑦)↑2)))})
ccbn 30937	class CBan
df-cbn 30938	⊢ CBan = {𝑢 ∈ NrmCVec ∣ (IndMet‘𝑢) ∈ (CMet‘(BaseSet‘𝑢))}
chlo 30960	class CHil_OLD
df-hlo 30961	⊢ CHil_OLD = (CBan ∩ CPreHil_OLD)
The list of syntax, axioms (ax-) and definitions (df-) for the Hilbert Space Explorer starts here
chba 30994	class ℋ
cva 30995	class +_ℎ
csm 30996	class ·_ℎ
csp 30997	class ·_ih
cno 30998	class norm_ℎ
c0v 30999	class 0_ℎ
cmv 31000	class −_ℎ
ccauold 31001	class Cauchy
chli 31002	class ⇝_𝑣
csh 31003	class S_ℋ
cch 31004	class C_ℋ
cort 31005	class ⊥
cph 31006	class +_ℋ
cspn 31007	class span
chj 31008	class ∨_ℋ
chsup 31009	class ∨_ℋ
c0h 31010	class 0_ℋ
ccm 31011	class 𝐶_ℋ
cpjh 31012	class proj_ℎ
chos 31013	class +_op
chot 31014	class ·_op
chod 31015	class −_op
chfs 31016	class +_fn
chft 31017	class ·_fn
ch0o 31018	class 0_hop
chio 31019	class I_op
cnop 31020	class norm_op
ccop 31021	class ContOp
clo 31022	class LinOp
cbo 31023	class BndLinOp
cuo 31024	class UniOp
cho 31025	class HrmOp
cnmf 31026	class norm_fn
cnl 31027	class null
ccnfn 31028	class ContFn
clf 31029	class LinFn
cado 31030	class adj_ℎ
cbr 31031	class bra
ck 31032	class ketbra
cleo 31033	class ≤_op
cei 31034	class eigvec
cel 31035	class eigval
cspc 31036	class Lambda
cst 31037	class States
chst 31038	class CHStates
ccv 31039	class ⋖_ℋ
cat 31040	class HAtoms
cmd 31041	class 𝑀_ℋ
cdmd 31042	class 𝑀_ℋ^*
df-hnorm 31043	⊢ norm_ℎ = (𝑥 ∈ dom dom ·_ih ↦ (√‘(𝑥 ·_ih 𝑥)))
df-hba 31044	⊢ ℋ = (BaseSet‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
df-h0v 31045	⊢ 0_ℎ = (0_vec‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
df-hvsub 31046	⊢ −_ℎ = (𝑥 ∈ ℋ, 𝑦 ∈ ℋ ↦ (𝑥 +_ℎ (-1 ·_ℎ 𝑦)))
df-hlim 31047	⊢ ⇝_𝑣 = {⟨𝑓, 𝑤⟩ ∣ ((𝑓:ℕ⟶ ℋ ∧ 𝑤 ∈ ℋ) ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ_≥‘𝑦)(norm_ℎ‘((𝑓‘𝑧) −_ℎ 𝑤)) < 𝑥)}
df-hcau 31048	⊢ Cauchy = {𝑓 ∈ ( ℋ ↑_m ℕ) ∣ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ_≥‘𝑦)(norm_ℎ‘((𝑓‘𝑦) −_ℎ (𝑓‘𝑧))) < 𝑥}
ax-hilex 31074	⊢ ℋ ∈ V
ax-hfvadd 31075	⊢ +_ℎ :( ℋ × ℋ)⟶ ℋ
ax-hvcom 31076	⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 +_ℎ 𝐵) = (𝐵 +_ℎ 𝐴))
ax-hvass 31077	⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 +_ℎ 𝐵) +_ℎ 𝐶) = (𝐴 +_ℎ (𝐵 +_ℎ 𝐶)))
ax-hv0cl 31078	⊢ 0_ℎ ∈ ℋ
ax-hvaddid 31079	⊢ (𝐴 ∈ ℋ → (𝐴 +_ℎ 0_ℎ) = 𝐴)
ax-hfvmul 31080	⊢ ·_ℎ :(ℂ × ℋ)⟶ ℋ
ax-hvmulid 31081	⊢ (𝐴 ∈ ℋ → (1 ·_ℎ 𝐴) = 𝐴)
ax-hvmulass 31082	⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝐴 · 𝐵) ·_ℎ 𝐶) = (𝐴 ·_ℎ (𝐵 ·_ℎ 𝐶)))
ax-hvdistr1 31083	⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 ·_ℎ (𝐵 +_ℎ 𝐶)) = ((𝐴 ·_ℎ 𝐵) +_ℎ (𝐴 ·_ℎ 𝐶)))
ax-hvdistr2 31084	⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝐴 + 𝐵) ·_ℎ 𝐶) = ((𝐴 ·_ℎ 𝐶) +_ℎ (𝐵 ·_ℎ 𝐶)))
ax-hvmul0 31085	⊢ (𝐴 ∈ ℋ → (0 ·_ℎ 𝐴) = 0_ℎ)
ax-hfi 31154	⊢ ·_ih :( ℋ × ℋ)⟶ℂ
ax-his1 31157	⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·_ih 𝐵) = (∗‘(𝐵 ·_ih 𝐴)))
ax-his2 31158	⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 +_ℎ 𝐵) ·_ih 𝐶) = ((𝐴 ·_ih 𝐶) + (𝐵 ·_ih 𝐶)))
ax-his3 31159	⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 ·_ℎ 𝐵) ·_ih 𝐶) = (𝐴 · (𝐵 ·_ih 𝐶)))
ax-his4 31160	⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0_ℎ) → 0 < (𝐴 ·_ih 𝐴))
ax-hcompl 31277	⊢ (𝐹 ∈ Cauchy → ∃𝑥 ∈ ℋ 𝐹 ⇝_𝑣 𝑥)
df-sh 31282	⊢ S_ℋ = {ℎ ∈ 𝒫 ℋ ∣ (0_ℎ ∈ ℎ ∧ ( +_ℎ “ (ℎ × ℎ)) ⊆ ℎ ∧ ( ·_ℎ “ (ℂ × ℎ)) ⊆ ℎ)}
df-ch 31296	⊢ C_ℋ = {ℎ ∈ S_ℋ ∣ ( ⇝_𝑣 “ (ℎ ↑_m ℕ)) ⊆ ℎ}
df-oc 31327	⊢ ⊥ = (𝑥 ∈ 𝒫 ℋ ↦ {𝑦 ∈ ℋ ∣ ∀𝑧 ∈ 𝑥 (𝑦 ·_ih 𝑧) = 0})
df-ch0 31328	⊢ 0_ℋ = {0_ℎ}
df-shs 31383	⊢ +_ℋ = (𝑥 ∈ S_ℋ , 𝑦 ∈ S_ℋ ↦ ( +_ℎ “ (𝑥 × 𝑦)))
df-span 31384	⊢ span = (𝑥 ∈ 𝒫 ℋ ↦ ∩ {𝑦 ∈ S_ℋ ∣ 𝑥 ⊆ 𝑦})
df-chj 31385	⊢ ∨_ℋ = (𝑥 ∈ 𝒫 ℋ, 𝑦 ∈ 𝒫 ℋ ↦ (⊥‘(⊥‘(𝑥 ∪ 𝑦))))
df-chsup 31386	⊢ ∨_ℋ = (𝑥 ∈ 𝒫 𝒫 ℋ ↦ (⊥‘(⊥‘∪ 𝑥)))
df-pjh 31470	⊢ proj_ℎ = (ℎ ∈ C_ℋ ↦ (𝑥 ∈ ℋ ↦ (℩𝑧 ∈ ℎ ∃𝑦 ∈ (⊥‘ℎ)𝑥 = (𝑧 +_ℎ 𝑦))))
df-cm 31658	⊢ 𝐶_ℋ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ C_ℋ ∧ 𝑦 ∈ C_ℋ ) ∧ 𝑥 = ((𝑥 ∩ 𝑦) ∨_ℋ (𝑥 ∩ (⊥‘𝑦))))}
df-hosum 31805	⊢ +_op = (𝑓 ∈ ( ℋ ↑_m ℋ), 𝑔 ∈ ( ℋ ↑_m ℋ) ↦ (𝑥 ∈ ℋ ↦ ((𝑓‘𝑥) +_ℎ (𝑔‘𝑥))))
df-homul 31806	⊢ ·_op = (𝑓 ∈ ℂ, 𝑔 ∈ ( ℋ ↑_m ℋ) ↦ (𝑥 ∈ ℋ ↦ (𝑓 ·_ℎ (𝑔‘𝑥))))
df-hodif 31807	⊢ −_op = (𝑓 ∈ ( ℋ ↑_m ℋ), 𝑔 ∈ ( ℋ ↑_m ℋ) ↦ (𝑥 ∈ ℋ ↦ ((𝑓‘𝑥) −_ℎ (𝑔‘𝑥))))
df-hfsum 31808	⊢ +_fn = (𝑓 ∈ (ℂ ↑_m ℋ), 𝑔 ∈ (ℂ ↑_m ℋ) ↦ (𝑥 ∈ ℋ ↦ ((𝑓‘𝑥) + (𝑔‘𝑥))))
df-hfmul 31809	⊢ ·_fn = (𝑓 ∈ ℂ, 𝑔 ∈ (ℂ ↑_m ℋ) ↦ (𝑥 ∈ ℋ ↦ (𝑓 · (𝑔‘𝑥))))
df-h0op 31823	⊢ 0_hop = (proj_ℎ‘0_ℋ)
df-iop 31824	⊢ I_op = (proj_ℎ‘ ℋ)
df-nmop 31914	⊢ norm_op = (𝑡 ∈ ( ℋ ↑_m ℋ) ↦ sup({𝑥 ∣ ∃𝑧 ∈ ℋ ((norm_ℎ‘𝑧) ≤ 1 ∧ 𝑥 = (norm_ℎ‘(𝑡‘𝑧)))}, ℝ^*, < ))
df-cnop 31915	⊢ ContOp = {𝑡 ∈ ( ℋ ↑_m ℋ) ∣ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℝ⁺ ∃𝑧 ∈ ℝ⁺ ∀𝑤 ∈ ℋ ((norm_ℎ‘(𝑤 −_ℎ 𝑥)) < 𝑧 → (norm_ℎ‘((𝑡‘𝑤) −_ℎ (𝑡‘𝑥))) < 𝑦)}
df-lnop 31916	⊢ LinOp = {𝑡 ∈ ( ℋ ↑_m ℋ) ∣ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ (𝑡‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) = ((𝑥 ·_ℎ (𝑡‘𝑦)) +_ℎ (𝑡‘𝑧))}
df-bdop 31917	⊢ BndLinOp = {𝑡 ∈ LinOp ∣ (norm_op‘𝑡) < +∞}
df-unop 31918	⊢ UniOp = {𝑡 ∣ (𝑡: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑡‘𝑥) ·_ih (𝑡‘𝑦)) = (𝑥 ·_ih 𝑦))}
df-hmop 31919	⊢ HrmOp = {𝑡 ∈ ( ℋ ↑_m ℋ) ∣ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑡‘𝑦)) = ((𝑡‘𝑥) ·_ih 𝑦)}
df-nmfn 31920	⊢ norm_fn = (𝑡 ∈ (ℂ ↑_m ℋ) ↦ sup({𝑥 ∣ ∃𝑧 ∈ ℋ ((norm_ℎ‘𝑧) ≤ 1 ∧ 𝑥 = (abs‘(𝑡‘𝑧)))}, ℝ^*, < ))
df-nlfn 31921	⊢ null = (𝑡 ∈ (ℂ ↑_m ℋ) ↦ (^◡𝑡 “ {0}))
df-cnfn 31922	⊢ ContFn = {𝑡 ∈ (ℂ ↑_m ℋ) ∣ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℝ⁺ ∃𝑧 ∈ ℝ⁺ ∀𝑤 ∈ ℋ ((norm_ℎ‘(𝑤 −_ℎ 𝑥)) < 𝑧 → (abs‘((𝑡‘𝑤) − (𝑡‘𝑥))) < 𝑦)}
df-lnfn 31923	⊢ LinFn = {𝑡 ∈ (ℂ ↑_m ℋ) ∣ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ (𝑡‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) = ((𝑥 · (𝑡‘𝑦)) + (𝑡‘𝑧))}
df-adjh 31924	⊢ adj_ℎ = {⟨𝑡, 𝑢⟩ ∣ (𝑡: ℋ⟶ ℋ ∧ 𝑢: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑡‘𝑥) ·_ih 𝑦) = (𝑥 ·_ih (𝑢‘𝑦)))}
df-bra 31925	⊢ bra = (𝑥 ∈ ℋ ↦ (𝑦 ∈ ℋ ↦ (𝑦 ·_ih 𝑥)))
df-kb 31926	⊢ ketbra = (𝑥 ∈ ℋ, 𝑦 ∈ ℋ ↦ (𝑧 ∈ ℋ ↦ ((𝑧 ·_ih 𝑦) ·_ℎ 𝑥)))
df-leop 31927	⊢ ≤_op = {⟨𝑡, 𝑢⟩ ∣ ((𝑢 −_op 𝑡) ∈ HrmOp ∧ ∀𝑥 ∈ ℋ 0 ≤ (((𝑢 −_op 𝑡)‘𝑥) ·_ih 𝑥))}
df-eigvec 31928	⊢ eigvec = (𝑡 ∈ ( ℋ ↑_m ℋ) ↦ {𝑥 ∈ ( ℋ ∖ 0_ℋ) ∣ ∃𝑧 ∈ ℂ (𝑡‘𝑥) = (𝑧 ·_ℎ 𝑥)})
df-eigval 31929	⊢ eigval = (𝑡 ∈ ( ℋ ↑_m ℋ) ↦ (𝑥 ∈ (eigvec‘𝑡) ↦ (((𝑡‘𝑥) ·_ih 𝑥) / ((norm_ℎ‘𝑥)↑2))))
df-spec 31930	⊢ Lambda = (𝑡 ∈ ( ℋ ↑_m ℋ) ↦ {𝑥 ∈ ℂ ∣ ¬ (𝑡 −_op (𝑥 ·_op ( I ↾ ℋ))): ℋ–1-1→ ℋ})
df-st 32286	⊢ States = {𝑓 ∈ ((0[,]1) ↑_m C_ℋ ) ∣ ((𝑓‘ ℋ) = 1 ∧ ∀𝑥 ∈ C_ℋ ∀𝑦 ∈ C_ℋ (𝑥 ⊆ (⊥‘𝑦) → (𝑓‘(𝑥 ∨_ℋ 𝑦)) = ((𝑓‘𝑥) + (𝑓‘𝑦))))}
df-hst 32287	⊢ CHStates = {𝑓 ∈ ( ℋ ↑_m C_ℋ ) ∣ ((norm_ℎ‘(𝑓‘ ℋ)) = 1 ∧ ∀𝑥 ∈ C_ℋ ∀𝑦 ∈ C_ℋ (𝑥 ⊆ (⊥‘𝑦) → (((𝑓‘𝑥) ·_ih (𝑓‘𝑦)) = 0 ∧ (𝑓‘(𝑥 ∨_ℋ 𝑦)) = ((𝑓‘𝑥) +_ℎ (𝑓‘𝑦)))))}
df-cv 32354	⊢ ⋖_ℋ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ C_ℋ ∧ 𝑦 ∈ C_ℋ ) ∧ (𝑥 ⊊ 𝑦 ∧ ¬ ∃𝑧 ∈ C_ℋ (𝑥 ⊊ 𝑧 ∧ 𝑧 ⊊ 𝑦)))}
df-md 32355	⊢ 𝑀_ℋ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ C_ℋ ∧ 𝑦 ∈ C_ℋ ) ∧ ∀𝑧 ∈ C_ℋ (𝑧 ⊆ 𝑦 → ((𝑧 ∨_ℋ 𝑥) ∩ 𝑦) = (𝑧 ∨_ℋ (𝑥 ∩ 𝑦))))}
df-dmd 32356	⊢ 𝑀_ℋ^* = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ C_ℋ ∧ 𝑦 ∈ C_ℋ ) ∧ ∀𝑧 ∈ C_ℋ (𝑦 ⊆ 𝑧 → ((𝑧 ∩ 𝑥) ∨_ℋ 𝑦) = (𝑧 ∩ (𝑥 ∨_ℋ 𝑦))))}
df-at 32413	⊢ HAtoms = {𝑥 ∈ C_ℋ ∣ 0_ℋ ⋖_ℋ 𝑥}
The list of syntax, axioms (ax-) and definitions (df-) for the User Mathboxes starts here
w2reu 32552	wff ∃!𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵𝜑
df-2reu 32553	⊢ (∃!𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵𝜑 ↔ (∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ ∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑))
cind 32929	class 𝟭
df-ind 32930	⊢ 𝟭 = (𝑜 ∈ V ↦ (𝑎 ∈ 𝒫 𝑜 ↦ (𝑥 ∈ 𝑜 ↦ if(𝑥 ∈ 𝑎, 1, 0))))
cdp2 32952	class _𝐴𝐵
df-dp2 32953	⊢ _𝐴𝐵 = (𝐴 + (𝐵 / ;10))
cdp 32969	class .
df-dp 32970	⊢ . = (𝑥 ∈ ℕ₀, 𝑦 ∈ ℝ ↦ _𝑥𝑦)
cxdiv 32998	class /_𝑒
df-xdiv 32999	⊢ /_𝑒 = (𝑥 ∈ ℝ^, 𝑦 ∈ (ℝ ∖ {0}) ↦ (℩𝑧 ∈ ℝ^ (𝑦 ·_e 𝑧) = 𝑥))
cmnt 33060	class Monot
cmgc 33061	class MGalConn
df-mnt 33062	⊢ Monot = (𝑣 ∈ V, 𝑤 ∈ V ↦ ⦋(Base‘𝑣) / 𝑎⦌{𝑓 ∈ ((Base‘𝑤) ↑_m 𝑎) ∣ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑎 (𝑥(le‘𝑣)𝑦 → (𝑓‘𝑥)(le‘𝑤)(𝑓‘𝑦))})
df-mgc 33063	⊢ MGalConn = (𝑣 ∈ V, 𝑤 ∈ V ↦ ⦋(Base‘𝑣) / 𝑎⦌⦋(Base‘𝑤) / 𝑏⦌{⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑏 ↑_m 𝑎) ∧ 𝑔 ∈ (𝑎 ↑_m 𝑏)) ∧ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 ((𝑓‘𝑥)(le‘𝑤)𝑦 ↔ 𝑥(le‘𝑣)(𝑔‘𝑦)))})
ax-xrssca 33086	⊢ ℝ_fld = (Scalar‘ℝ^*_𝑠)
ax-xrsvsca 33087	⊢ ·_e = ( ·_𝑠 ‘ℝ^*_𝑠)
ctocyc 33188	class toCyc
df-tocyc 33189	⊢ toCyc = (𝑑 ∈ V ↦ (𝑤 ∈ {𝑢 ∈ Word 𝑑 ∣ 𝑢:dom 𝑢–1-1→𝑑} ↦ (( I ↾ (𝑑 ∖ ran 𝑤)) ∪ ((𝑤 cyclShift 1) ∘ ^◡𝑤))))
csgns 33240	class sgn_s
df-sgns 33241	⊢ sgn_s = (𝑟 ∈ V ↦ (𝑥 ∈ (Base‘𝑟) ↦ if(𝑥 = (0_g‘𝑟), 0, if((0_g‘𝑟)(lt‘𝑟)𝑥, 1, -1))))
cfxp 33245	class FixPts
df-fxp 33246	⊢ FixPts = (𝑏 ∈ V, 𝑎 ∈ V ↦ {𝑥 ∈ 𝑏 ∣ ∀𝑝 ∈ dom dom 𝑎(𝑝𝑎𝑥) = 𝑥})
cinftm 33258	class ⋘
carchi 33259	class Archi
df-inftm 33260	⊢ ⋘ = (𝑤 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (Base‘𝑤) ∧ 𝑦 ∈ (Base‘𝑤)) ∧ ((0_g‘𝑤)(lt‘𝑤)𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛(._g‘𝑤)𝑥)(lt‘𝑤)𝑦))})
df-archi 33261	⊢ Archi = {𝑤 ∣ (⋘‘𝑤) = ∅}
cslmd 33282	class SLMod
df-slmd 33283	⊢ SLMod = {𝑔 ∈ CMnd ∣ [(Base‘𝑔) / 𝑣][(+_g‘𝑔) / 𝑎][( ·_𝑠 ‘𝑔) / 𝑠][(Scalar‘𝑔) / 𝑓][(Base‘𝑓) / 𝑘][(+_g‘𝑓) / 𝑝][(._r‘𝑓) / 𝑡](𝑓 ∈ SRing ∧ ∀𝑞 ∈ 𝑘 ∀𝑟 ∈ 𝑘 ∀𝑥 ∈ 𝑣 ∀𝑤 ∈ 𝑣 (((𝑟𝑠𝑤) ∈ 𝑣 ∧ (𝑟𝑠(𝑤𝑎𝑥)) = ((𝑟𝑠𝑤)𝑎(𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤)𝑎(𝑟𝑠𝑤))) ∧ (((𝑞𝑡𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ((1_r‘𝑓)𝑠𝑤) = 𝑤 ∧ ((0_g‘𝑓)𝑠𝑤) = (0_g‘𝑔))))}
cerl 33335	class ~_RL
crloc 33336	class RLocal
df-erl 33337	⊢ ~_RL = (𝑟 ∈ V, 𝑠 ∈ V ↦ ⦋(._r‘𝑟) / 𝑥⦌⦋((Base‘𝑟) × 𝑠) / 𝑤⦌{⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ 𝑤 ∧ 𝑏 ∈ 𝑤) ∧ ∃𝑡 ∈ 𝑠 (𝑡𝑥(((1^st ‘𝑎)𝑥(2^nd ‘𝑏))(-_g‘𝑟)((1^st ‘𝑏)𝑥(2^nd ‘𝑎)))) = (0_g‘𝑟))})
df-rloc 33338	⊢ RLocal = (𝑟 ∈ V, 𝑠 ∈ V ↦ ⦋(._r‘𝑟) / 𝑥⦌⦋((Base‘𝑟) × 𝑠) / 𝑤⦌((({⟨(Base‘ndx), 𝑤⟩, ⟨(+_g‘ndx), (𝑎 ∈ 𝑤, 𝑏 ∈ 𝑤 ↦ ⟨(((1^st ‘𝑎)𝑥(2^nd ‘𝑏))(+_g‘𝑟)((1^st ‘𝑏)𝑥(2^nd ‘𝑎))), ((2^nd ‘𝑎)𝑥(2^nd ‘𝑏))⟩)⟩, ⟨(._r‘ndx), (𝑎 ∈ 𝑤, 𝑏 ∈ 𝑤 ↦ ⟨((1^st ‘𝑎)𝑥(1^st ‘𝑏)), ((2^nd ‘𝑎)𝑥(2^nd ‘𝑏))⟩)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑟)⟩, ⟨( ·_𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑟)), 𝑎 ∈ 𝑤 ↦ ⟨(𝑘( ·_𝑠 ‘𝑟)(1^st ‘𝑎)), (2^nd ‘𝑎)⟩)⟩, ⟨(·_𝑖‘ndx), ∅⟩}) ∪ {⟨(TopSet‘ndx), ((TopSet‘𝑟) ×_t ((TopSet‘𝑟) ↾_t 𝑠))⟩, ⟨(le‘ndx), {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ 𝑤 ∧ 𝑏 ∈ 𝑤) ∧ ((1^st ‘𝑎)𝑥(2^nd ‘𝑏))(le‘𝑟)((1^st ‘𝑏)𝑥(2^nd ‘𝑎)))}⟩, ⟨(dist‘ndx), (𝑎 ∈ 𝑤, 𝑏 ∈ 𝑤 ↦ (((1^st ‘𝑎)𝑥(2^nd ‘𝑏))(dist‘𝑟)((1^st ‘𝑏)𝑥(2^nd ‘𝑎))))⟩}) /_s (𝑟 ~_RL 𝑠)))
ceuf 33370	class EuclF
df-euf 33371	⊢ EuclF = Slot ;21
cedom 33374	class EDomn
df-edom 33375	⊢ EDomn = {𝑑 ∈ IDomn ∣ [(EuclF‘𝑑) / 𝑒][(Base‘𝑑) / 𝑣](Fun 𝑒 ∧ (𝑒 “ (𝑣 ∖ {(0_g‘𝑑)})) ⊆ (0[,)+∞) ∧ ∀𝑎 ∈ 𝑣 ∀𝑏 ∈ (𝑣 ∖ {(0_g‘𝑑)})∃𝑞 ∈ 𝑣 ∃𝑟 ∈ 𝑣 (𝑎 = ((𝑏(._r‘𝑑)𝑞)(+_g‘𝑑)𝑟) ∧ (𝑟 = (0_g‘𝑑) ∨ (𝑒‘𝑟) < (𝑒‘𝑏))))}
cfrac 33384	class Frac
df-frac 33385	⊢ Frac = (𝑟 ∈ V ↦ (𝑟 RLocal (RLReg‘𝑟)))
cfldgen 33392	class fldGen
df-fldgen 33393	⊢ fldGen = (𝑓 ∈ V, 𝑠 ∈ V ↦ ∩ {𝑎 ∈ (SubDRing‘𝑓) ∣ 𝑠 ⊆ 𝑎})
cresv 33407	class ↾_v
df-resv 33408	⊢ ↾_v = (𝑤 ∈ V, 𝑥 ∈ V ↦ if((Base‘(Scalar‘𝑤)) ⊆ 𝑥, 𝑤, (𝑤 sSet ⟨(Scalar‘ndx), ((Scalar‘𝑤) ↾_s 𝑥)⟩)))
cprmidl 33516	class PrmIdeal
df-prmidl 33517	⊢ PrmIdeal = (𝑟 ∈ Ring ↦ {𝑖 ∈ (LIdeal‘𝑟) ∣ (𝑖 ≠ (Base‘𝑟) ∧ ∀𝑎 ∈ (LIdeal‘𝑟)∀𝑏 ∈ (LIdeal‘𝑟)(∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 (𝑥(._r‘𝑟)𝑦) ∈ 𝑖 → (𝑎 ⊆ 𝑖 ∨ 𝑏 ⊆ 𝑖)))})
cmxidl 33540	class MaxIdeal
df-mxidl 33541	⊢ MaxIdeal = (𝑟 ∈ Ring ↦ {𝑖 ∈ (LIdeal‘𝑟) ∣ (𝑖 ≠ (Base‘𝑟) ∧ ∀𝑗 ∈ (LIdeal‘𝑟)(𝑖 ⊆ 𝑗 → (𝑗 = 𝑖 ∨ 𝑗 = (Base‘𝑟))))})
cidlsrg 33581	class IDLsrg
df-idlsrg 33582	⊢ IDLsrg = (𝑟 ∈ V ↦ ⦋(LIdeal‘𝑟) / 𝑏⦌({⟨(Base‘ndx), 𝑏⟩, ⟨(+_g‘ndx), (LSSum‘𝑟)⟩, ⟨(._r‘ndx), (𝑖 ∈ 𝑏, 𝑗 ∈ 𝑏 ↦ ((RSpan‘𝑟)‘(𝑖(LSSum‘(mulGrp‘𝑟))𝑗)))⟩} ∪ {⟨(TopSet‘ndx), ran (𝑖 ∈ 𝑏 ↦ {𝑗 ∈ 𝑏 ∣ ¬ 𝑖 ⊆ 𝑗})⟩, ⟨(le‘ndx), {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ 𝑏 ∧ 𝑖 ⊆ 𝑗)}⟩}))
cufd 33619	class UFD
df-ufd 33620	⊢ UFD = {𝑟 ∈ IDomn ∣ ∀𝑖 ∈ ((PrmIdeal‘𝑟) ∖ {{(0_g‘𝑟)}})(𝑖 ∩ (RPrime‘𝑟)) ≠ ∅}
cextv 33694	class extendVars
df-extv 33695	⊢ extendVars = (𝑖 ∈ V, 𝑟 ∈ V ↦ (𝑎 ∈ 𝑖 ↦ (𝑓 ∈ (Base‘((𝑖 ∖ {𝑎}) mPoly 𝑟)) ↦ (𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝑖) ∣ ℎ finSupp 0} ↦ if((𝑥‘𝑎) = 0, (𝑓‘(𝑥 ↾ (𝑖 ∖ {𝑎}))), (0_g‘𝑟))))))
csply 33713	class SymPoly
cesply 33714	class eSymPoly
df-sply 33715	⊢ SymPoly = (𝑖 ∈ V, 𝑟 ∈ V ↦ ((Base‘(𝑖 mPoly 𝑟))FixPts(𝑑 ∈ (Base‘(SymGrp‘𝑖)), 𝑓 ∈ (Base‘(𝑖 mPoly 𝑟)) ↦ (𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝑖) ∣ ℎ finSupp 0} ↦ (𝑓‘(𝑥 ∘ 𝑑))))))
df-esply 33716	⊢ eSymPoly = (𝑖 ∈ V, 𝑟 ∈ V ↦ (𝑘 ∈ ℕ₀ ↦ ((ℤRHom‘𝑟) ∘ ((𝟭‘{ℎ ∈ (ℕ₀ ↑_m 𝑖) ∣ ℎ finSupp 0})‘((𝟭‘𝑖) “ {𝑐 ∈ 𝒫 𝑖 ∣ (♯‘𝑐) = 𝑘})))))
cldim 33755	class dim
df-dim 33756	⊢ dim = (𝑓 ∈ V ↦ ∪ (♯ “ (LBasis‘𝑓)))
cfldext 33795	class /_FldExt
cfinext 33796	class /_FinExt
cextdg 33797	class [:]
df-fldext 33798	⊢ /_FldExt = {⟨𝑒, 𝑓⟩ ∣ ((𝑒 ∈ Field ∧ 𝑓 ∈ Field) ∧ (𝑓 = (𝑒 ↾_s (Base‘𝑓)) ∧ (Base‘𝑓) ∈ (SubRing‘𝑒)))}
df-extdg 33799	⊢ [:] = (𝑒 ∈ V, 𝑓 ∈ (/_FldExt “ {𝑒}) ↦ (dim‘((subringAlg ‘𝑒)‘(Base‘𝑓))))
df-finext 33800	⊢ /_FinExt = {⟨𝑒, 𝑓⟩ ∣ (𝑒/_FldExt𝑓 ∧ (𝑒[:]𝑓) ∈ ℕ₀)}
cirng 33840	class IntgRing
df-irng 33841	⊢ IntgRing = (𝑟 ∈ V, 𝑠 ∈ V ↦ ∪ 𝑓 ∈ (Monic_1p‘(𝑟 ↾_s 𝑠))(^◡((𝑟 evalSub₁ 𝑠)‘𝑓) “ {(0_g‘𝑟)}))
calgext 33852	class /_AlgExt
df-algext 33853	⊢ /_AlgExt = {⟨𝑒, 𝑓⟩ ∣ (𝑒/_FldExt𝑓 ∧ (𝑒 IntgRing (Base‘𝑓)) = (Base‘𝑒))}
cminply 33856	class minPoly
df-minply 33857	⊢ minPoly = (𝑒 ∈ V, 𝑓 ∈ V ↦ (𝑥 ∈ (Base‘𝑒) ↦ ((idlGen_1p‘(𝑒 ↾_s 𝑓))‘{𝑝 ∈ dom (𝑒 evalSub₁ 𝑓) ∣ (((𝑒 evalSub₁ 𝑓)‘𝑝)‘𝑥) = (0_g‘𝑒)})))
cconstr 33886	class Constr
df-constr 33887	⊢ Constr = ∪ (rec((𝑠 ∈ V ↦ {𝑥 ∈ ℂ ∣ (∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑡 ∈ ℝ ∃𝑟 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ 𝑥 = (𝑐 + (𝑟 · (𝑑 − 𝑐))) ∧ (ℑ‘((∗‘(𝑏 − 𝑎)) · (𝑑 − 𝑐))) ≠ 0) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 ∃𝑡 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ (abs‘(𝑥 − 𝑐)) = (abs‘(𝑒 − 𝑓))) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 (𝑎 ≠ 𝑑 ∧ (abs‘(𝑥 − 𝑎)) = (abs‘(𝑏 − 𝑐)) ∧ (abs‘(𝑥 − 𝑑)) = (abs‘(𝑒 − 𝑓))))}), {0, 1}) “ ω)
csmat 33950	class subMat1
df-smat 33951	⊢ subMat1 = (𝑚 ∈ V ↦ (𝑘 ∈ ℕ, 𝑙 ∈ ℕ ↦ (𝑚 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝑘, 𝑖, (𝑖 + 1)), if(𝑗 < 𝑙, 𝑗, (𝑗 + 1))⟩))))
clmat 33968	class litMat
df-lmat 33969	⊢ litMat = (𝑚 ∈ V ↦ (𝑖 ∈ (1...(♯‘𝑚)), 𝑗 ∈ (1...(♯‘(𝑚‘0))) ↦ ((𝑚‘(𝑖 − 1))‘(𝑗 − 1))))
ccref 33999	class CovHasRef𝐴
df-cref 34000	⊢ CovHasRef𝐴 = {𝑗 ∈ Top ∣ ∀𝑦 ∈ 𝒫 𝑗(∪ 𝑗 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑗 ∩ 𝐴)𝑧Ref𝑦)}
cldlf 34009	class Ldlf
df-ldlf 34010	⊢ Ldlf = CovHasRef{𝑥 ∣ 𝑥 ≼ ω}
cpcmp 34012	class Paracomp
df-pcmp 34013	⊢ Paracomp = {𝑗 ∣ 𝑗 ∈ CovHasRef(LocFin‘𝑗)}
crspec 34019	class Spec
df-rspec 34020	⊢ Spec = (𝑟 ∈ Ring ↦ ((IDLsrg‘𝑟) ↾_s (PrmIdeal‘𝑟)))
cmetid 34043	class ~_Met
cpstm 34044	class pstoMet
df-metid 34045	⊢ ~_Met = (𝑑 ∈ ∪ ran PsMet ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ dom dom 𝑑 ∧ 𝑦 ∈ dom dom 𝑑) ∧ (𝑥𝑑𝑦) = 0)})
df-pstm 34046	⊢ pstoMet = (𝑑 ∈ ∪ ran PsMet ↦ (𝑎 ∈ (dom dom 𝑑 / (~_Met‘𝑑)), 𝑏 ∈ (dom dom 𝑑 / (~_Met‘𝑑)) ↦ ∪ {𝑧 ∣ ∃𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝑧 = (𝑥𝑑𝑦)}))
chcmp 34113	class HCmp
df-hcmp 34114	⊢ HCmp = {⟨𝑢, 𝑤⟩ ∣ ((𝑢 ∈ ∪ ran UnifOn ∧ 𝑤 ∈ CUnifSp) ∧ ((UnifSt‘𝑤) ↾_t dom ∪ 𝑢) = 𝑢 ∧ ((cls‘(TopOpen‘𝑤))‘dom ∪ 𝑢) = (Base‘𝑤))}
cqqh 34127	class ℚHom
df-qqh 34128	⊢ ℚHom = (𝑟 ∈ V ↦ ran (𝑥 ∈ ℤ, 𝑦 ∈ (^◡(ℤRHom‘𝑟) “ (Unit‘𝑟)) ↦ ⟨(𝑥 / 𝑦), (((ℤRHom‘𝑟)‘𝑥)(/_r‘𝑟)((ℤRHom‘𝑟)‘𝑦))⟩))
crrh 34150	class ℝHom
crrext 34151	class ℝExt
df-rrh 34152	⊢ ℝHom = (𝑟 ∈ V ↦ (((topGen‘ran (,))CnExt(TopOpen‘𝑟))‘(ℚHom‘𝑟)))
df-rrext 34156	⊢ ℝExt = {𝑟 ∈ (NrmRing ∩ DivRing) ∣ (((ℤMod‘𝑟) ∈ NrmMod ∧ (chr‘𝑟) = 0) ∧ (𝑟 ∈ CUnifSp ∧ (UnifSt‘𝑟) = (metUnif‘((dist‘𝑟) ↾ ((Base‘𝑟) × (Base‘𝑟))))))}
cxrh 34173	class ℝ^*Hom
df-xrh 34174	⊢ ℝ^Hom = (𝑟 ∈ V ↦ (𝑥 ∈ ℝ^ ↦ if(𝑥 ∈ ℝ, ((ℝHom‘𝑟)‘𝑥), if(𝑥 = +∞, ((lub‘𝑟)‘((ℝHom‘𝑟) “ ℝ)), ((glb‘𝑟)‘((ℝHom‘𝑟) “ ℝ))))))
cmntop 34179	class ManTop
df-mntop 34180	⊢ ManTop = {⟨𝑛, 𝑗⟩ ∣ (𝑛 ∈ ℕ₀ ∧ (𝑗 ∈ 2^ndω ∧ 𝑗 ∈ Haus ∧ 𝑗 ∈ Locally [(TopOpen‘(𝔼_hil‘𝑛))] ≃ ))}
cesum 34184	class Σ^*𝑘 ∈ 𝐴𝐵
df-esum 34185	⊢ Σ^𝑘 ∈ 𝐴𝐵 = ∪ ((ℝ^_𝑠 ↾_s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵))
cofc 34252	class ∘_f/c 𝑅
df-ofc 34253	⊢ ∘_f/c 𝑅 = (𝑓 ∈ V, 𝑐 ∈ V ↦ (𝑥 ∈ dom 𝑓 ↦ ((𝑓‘𝑥)𝑅𝑐)))
csiga 34265	class sigAlgebra
df-siga 34266	⊢ sigAlgebra = (𝑜 ∈ V ↦ {𝑠 ∣ (𝑠 ⊆ 𝒫 𝑜 ∧ (𝑜 ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑜 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑠)))})
csigagen 34295	class sigaGen
df-sigagen 34296	⊢ sigaGen = (𝑥 ∈ V ↦ ∩ {𝑠 ∈ (sigAlgebra‘∪ 𝑥) ∣ 𝑥 ⊆ 𝑠})
cbrsiga 34338	class 𝔅_ℝ
df-brsiga 34339	⊢ 𝔅_ℝ = (sigaGen‘(topGen‘ran (,)))
csx 34345	class ×_s
df-sx 34346	⊢ ×_s = (𝑠 ∈ V, 𝑡 ∈ V ↦ (sigaGen‘ran (𝑥 ∈ 𝑠, 𝑦 ∈ 𝑡 ↦ (𝑥 × 𝑦))))
cmeas 34352	class measures
df-meas 34353	⊢ measures = (𝑠 ∈ ∪ ran sigAlgebra ↦ {𝑚 ∣ (𝑚:𝑠⟶(0[,]+∞) ∧ (𝑚‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → (𝑚‘∪ 𝑥) = Σ^*𝑦 ∈ 𝑥(𝑚‘𝑦)))})
cdde 34389	class δ
df-dde 34390	⊢ δ = (𝑎 ∈ 𝒫 ℝ ↦ if(0 ∈ 𝑎, 1, 0))
cae 34394	class a.e.
cfae 34395	class ~ a.e.
df-ae 34396	⊢ a.e. = {⟨𝑎, 𝑚⟩ ∣ (𝑚‘(∪ dom 𝑚 ∖ 𝑎)) = 0}
df-fae 34402	⊢ ~ a.e. = (𝑟 ∈ V, 𝑚 ∈ ∪ ran measures ↦ {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (dom 𝑟 ↑_m ∪ dom 𝑚) ∧ 𝑔 ∈ (dom 𝑟 ↑_m ∪ dom 𝑚)) ∧ {𝑥 ∈ ∪ dom 𝑚 ∣ (𝑓‘𝑥)𝑟(𝑔‘𝑥)}a.e.𝑚)})
cmbfm 34406	class MblFnM
df-mbfm 34407	⊢ MblFnM = (𝑠 ∈ ∪ ran sigAlgebra, 𝑡 ∈ ∪ ran sigAlgebra ↦ {𝑓 ∈ (∪ 𝑡 ↑_m ∪ 𝑠) ∣ ∀𝑥 ∈ 𝑡 (^◡𝑓 “ 𝑥) ∈ 𝑠})
coms 34448	class toOMeas
df-oms 34449	⊢ toOMeas = (𝑟 ∈ V ↦ (𝑎 ∈ 𝒫 ∪ dom 𝑟 ↦ inf(ran (𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑟 ∣ (𝑎 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω)} ↦ Σ^*𝑦 ∈ 𝑥(𝑟‘𝑦)), (0[,]+∞), < )))
ccarsg 34458	class toCaraSiga
df-carsg 34459	⊢ toCaraSiga = (𝑚 ∈ V ↦ {𝑎 ∈ 𝒫 ∪ dom 𝑚 ∣ ∀𝑒 ∈ 𝒫 ∪ dom 𝑚((𝑚‘(𝑒 ∩ 𝑎)) +_𝑒 (𝑚‘(𝑒 ∖ 𝑎))) = (𝑚‘𝑒)})
citgm 34484	class itgm
csitm 34485	class sitm
csitg 34486	class sitg
df-sitg 34487	⊢ sitg = (𝑤 ∈ V, 𝑚 ∈ ∪ ran measures ↦ (𝑓 ∈ {𝑔 ∈ (dom 𝑚MblFnM(sigaGen‘(TopOpen‘𝑤))) ∣ (ran 𝑔 ∈ Fin ∧ ∀𝑥 ∈ (ran 𝑔 ∖ {(0_g‘𝑤)})(𝑚‘(^◡𝑔 “ {𝑥})) ∈ (0[,)+∞))} ↦ (𝑤 Σ_g (𝑥 ∈ (ran 𝑓 ∖ {(0_g‘𝑤)}) ↦ (((ℝHom‘(Scalar‘𝑤))‘(𝑚‘(^◡𝑓 “ {𝑥})))( ·_𝑠 ‘𝑤)𝑥)))))
df-sitm 34488	⊢ sitm = (𝑤 ∈ V, 𝑚 ∈ ∪ ran measures ↦ (𝑓 ∈ dom (𝑤sitg𝑚), 𝑔 ∈ dom (𝑤sitg𝑚) ↦ (((ℝ^*_𝑠 ↾_s (0[,]+∞))sitg𝑚)‘(𝑓 ∘_f (dist‘𝑤)𝑔))))
df-itgm 34510	⊢ itgm = (𝑤 ∈ V, 𝑚 ∈ ∪ ran measures ↦ (((metUnif‘(𝑤sitm𝑚))CnExt(UnifSt‘𝑤))‘(𝑤sitg𝑚)))
csseq 34540	class seq_str
df-sseq 34541	⊢ seq_str = (𝑚 ∈ V, 𝑓 ∈ V ↦ (𝑚 ∪ (lastS ∘ seq(♯‘𝑚)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝑓‘𝑥)”⟩)), (ℕ₀ × {(𝑚 ++ ⟨“(𝑓‘𝑚)”⟩)})))))
cfib 34553	class Fibci
df-fib 34554	⊢ Fibci = (⟨“01”⟩seq_str(𝑤 ∈ (Word ℕ₀ ∩ (^◡♯ “ (ℤ_≥‘2))) ↦ ((𝑤‘((♯‘𝑤) − 2)) + (𝑤‘((♯‘𝑤) − 1)))))
cprb 34564	class Prob
df-prob 34565	⊢ Prob = {𝑝 ∈ ∪ ran measures ∣ (𝑝‘∪ dom 𝑝) = 1}
ccprob 34588	class cprob
df-cndprob 34589	⊢ cprob = (𝑝 ∈ Prob ↦ (𝑎 ∈ dom 𝑝, 𝑏 ∈ dom 𝑝 ↦ ((𝑝‘(𝑎 ∩ 𝑏)) / (𝑝‘𝑏))))
crrv 34597	class rRndVar
df-rrv 34598	⊢ rRndVar = (𝑝 ∈ Prob ↦ (dom 𝑝MblFnM𝔅_ℝ))
corvc 34613	class ∘_RV/𝑐𝑅
df-orvc 34614	⊢ ∘_RV/𝑐𝑅 = (𝑥 ∈ {𝑥 ∣ Fun 𝑥}, 𝑎 ∈ V ↦ (^◡𝑥 “ {𝑦 ∣ 𝑦𝑅𝑎}))
crepr 34765	class repr
df-repr 34766	⊢ repr = (𝑠 ∈ ℕ₀ ↦ (𝑏 ∈ 𝒫 ℕ, 𝑚 ∈ ℤ ↦ {𝑐 ∈ (𝑏 ↑_m (0..^𝑠)) ∣ Σ𝑎 ∈ (0..^𝑠)(𝑐‘𝑎) = 𝑚}))
cvts 34792	class vts
df-vts 34793	⊢ vts = (𝑙 ∈ (ℂ ↑_m ℕ), 𝑛 ∈ ℕ₀ ↦ (𝑥 ∈ ℂ ↦ Σ𝑎 ∈ (1...𝑛)((𝑙‘𝑎) · (exp‘((i · (2 · π)) · (𝑎 · 𝑥))))))
ax-hgt749 34801	⊢ ∀𝑛 ∈ {𝑧 ∈ ℤ ∣ ¬ 2 ∥ 𝑧} ((;10↑;27) ≤ 𝑛 → ∃ℎ ∈ ((0[,)+∞) ↑_m ℕ)∃𝑘 ∈ ((0[,)+∞) ↑_m ℕ)(∀𝑚 ∈ ℕ (𝑘‘𝑚) ≤ (1._0_7_9_9_55) ∧ ∀𝑚 ∈ ℕ (ℎ‘𝑚) ≤ (1._4_14) ∧ ((0._0_0_0_4_2_2_48) · (𝑛↑2)) ≤ ∫(0(,)1)(((((Λ ∘_f · ℎ)vts𝑛)‘𝑥) · ((((Λ ∘_f · 𝑘)vts𝑛)‘𝑥)↑2)) · (exp‘((i · (2 · π)) · (-𝑛 · 𝑥)))) d𝑥))
ax-ros335 34802	⊢ ∀𝑥 ∈ ℝ⁺ (ψ‘𝑥) < ((1._0_3_8_83) · 𝑥)
ax-ros336 34803	⊢ ∀𝑥 ∈ ℝ⁺ ((ψ‘𝑥) − (θ‘𝑥)) < ((1._4_2_62) · (√‘𝑥))
cstrkg2d 34821	class TarskiG_2D
df-trkg2d 34822	⊢ TarskiG_2D = {𝑓 ∣ [(Base‘𝑓) / 𝑝][(dist‘𝑓) / 𝑑][(Itv‘𝑓) / 𝑖](∃𝑥 ∈ 𝑝 ∃𝑦 ∈ 𝑝 ∃𝑧 ∈ 𝑝 ¬ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧)) ∧ ∀𝑥 ∈ 𝑝 ∀𝑦 ∈ 𝑝 ∀𝑧 ∈ 𝑝 ∀𝑢 ∈ 𝑝 ∀𝑣 ∈ 𝑝 ((((𝑥𝑑𝑢) = (𝑥𝑑𝑣) ∧ (𝑦𝑑𝑢) = (𝑦𝑑𝑣) ∧ (𝑧𝑑𝑢) = (𝑧𝑑𝑣)) ∧ 𝑢 ≠ 𝑣) → (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))))}
cafs 34826	class AFS
df-afs 34827	⊢ AFS = (𝑔 ∈ TarskiG ↦ {⟨𝑒, 𝑓⟩ ∣ [(Base‘𝑔) / 𝑝][(dist‘𝑔) / ℎ][(Itv‘𝑔) / 𝑖]∃𝑎 ∈ 𝑝 ∃𝑏 ∈ 𝑝 ∃𝑐 ∈ 𝑝 ∃𝑑 ∈ 𝑝 ∃𝑥 ∈ 𝑝 ∃𝑦 ∈ 𝑝 ∃𝑧 ∈ 𝑝 ∃𝑤 ∈ 𝑝 (𝑒 = ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ 𝑓 = ⟨⟨𝑥, 𝑦⟩, ⟨𝑧, 𝑤⟩⟩ ∧ ((𝑏 ∈ (𝑎𝑖𝑐) ∧ 𝑦 ∈ (𝑥𝑖𝑧)) ∧ ((𝑎ℎ𝑏) = (𝑥ℎ𝑦) ∧ (𝑏ℎ𝑐) = (𝑦ℎ𝑧)) ∧ ((𝑎ℎ𝑑) = (𝑥ℎ𝑤) ∧ (𝑏ℎ𝑑) = (𝑦ℎ𝑤))))})
clpad 34831	class leftpad
df-lpad 34832	⊢ leftpad = (𝑐 ∈ V, 𝑤 ∈ V ↦ (𝑙 ∈ ℕ₀ ↦ (((0..^(𝑙 − (♯‘𝑤))) × {𝑐}) ++ 𝑤)))
w-bnj17 34842	wff (𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃)
df-bnj17 34843	⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ ((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃))
c-bnj14 34844	class pred(𝑋, 𝐴, 𝑅)
df-bnj14 34845	⊢ pred(𝑋, 𝐴, 𝑅) = {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋}
w-bnj13 34846	wff 𝑅 Se 𝐴
df-bnj13 34847	⊢ (𝑅 Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 pred(𝑥, 𝐴, 𝑅) ∈ V)
w-bnj15 34848	wff 𝑅 FrSe 𝐴
df-bnj15 34849	⊢ (𝑅 FrSe 𝐴 ↔ (𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴))
c-bnj18 34850	class trCl(𝑋, 𝐴, 𝑅)
df-bnj18 34851	⊢ trCl(𝑋, 𝐴, 𝑅) = ∪ 𝑓 ∈ {𝑓 ∣ ∃𝑛 ∈ (ω ∖ {∅})(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))}∪ 𝑖 ∈ dom 𝑓(𝑓‘𝑖)
w-bnj19 34852	wff TrFo(𝐵, 𝐴, 𝑅)
df-bnj19 34853	⊢ ( TrFo(𝐵, 𝐴, 𝑅) ↔ ∀𝑥 ∈ 𝐵 pred(𝑥, 𝐴, 𝑅) ⊆ 𝐵)
ax-regs 35282	⊢ (∃𝑥𝜑 → ∃𝑦(∀𝑥(𝑥 = 𝑦 → 𝜑) ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ ∀𝑥(𝑥 = 𝑧 → 𝜑))))
cacycgr 35336	class AcyclicGraph
df-acycgr 35337	⊢ AcyclicGraph = {𝑔 ∣ ¬ ∃𝑓∃𝑝(𝑓(Cycles‘𝑔)𝑝 ∧ 𝑓 ≠ ∅)}
ax-7d 35353	⊢ (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑)
ax-8d 35354	⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 → 𝑦 = 𝑧))
ax-9d1 35355	⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑥
ax-9d2 35356	⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦
ax-10d 35357	⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
ax-11d 35358	⊢ (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
cretr 35411	class Retr
df-retr 35412	⊢ Retr = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ {𝑟 ∈ (𝑗 Cn 𝑘) ∣ ∃𝑠 ∈ (𝑘 Cn 𝑗)((𝑟 ∘ 𝑠)(𝑗 Htpy 𝑗)( I ↾ ∪ 𝑗)) ≠ ∅})
cpconn 35413	class PConn
csconn 35414	class SConn
df-pconn 35415	⊢ PConn = {𝑗 ∈ Top ∣ ∀𝑥 ∈ ∪ 𝑗∀𝑦 ∈ ∪ 𝑗∃𝑓 ∈ (II Cn 𝑗)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)}
df-sconn 35416	⊢ SConn = {𝑗 ∈ PConn ∣ ∀𝑓 ∈ (II Cn 𝑗)((𝑓‘0) = (𝑓‘1) → 𝑓( ≃_ph‘𝑗)((0[,]1) × {(𝑓‘0)}))}
ccvm 35449	class CovMap
df-cvm 35450	⊢ CovMap = (𝑐 ∈ Top, 𝑗 ∈ Top ↦ {𝑓 ∈ (𝑐 Cn 𝑗) ∣ ∀𝑥 ∈ ∪ 𝑗∃𝑘 ∈ 𝑗 (𝑥 ∈ 𝑘 ∧ ∃𝑠 ∈ (𝒫 𝑐 ∖ {∅})(∪ 𝑠 = (^◡𝑓 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝑓 ↾ 𝑢) ∈ ((𝑐 ↾_t 𝑢)Homeo(𝑗 ↾_t 𝑘)))))})
cgoe 35527	class ∈_𝑔
cgna 35528	class ⊼_𝑔
cgol 35529	class ∀_𝑔𝑁𝑈
csat 35530	class Sat
cfmla 35531	class Fmla
csate 35532	class Sat_∈
cprv 35533	class ⊧
df-goel 35534	⊢ ∈_𝑔 = (𝑥 ∈ (ω × ω) ↦ ⟨∅, 𝑥⟩)
df-gona 35535	⊢ ⊼_𝑔 = (𝑥 ∈ (V × V) ↦ ⟨1_o, 𝑥⟩)
df-goal 35536	⊢ ∀_𝑔𝑁𝑈 = ⟨2_o, ⟨𝑁, 𝑈⟩⟩
df-sat 35537	⊢ Sat = (𝑚 ∈ V, 𝑒 ∈ V ↦ (rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 (𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑦 = ((𝑚 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑚 ↑_m ω) ∣ ∀𝑧 ∈ 𝑚 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)}))})), {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈_𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑚 ↑_m ω) ∣ (𝑎‘𝑖)𝑒(𝑎‘𝑗)})}) ↾ suc ω))
df-sate 35538	⊢ Sat_∈ = (𝑚 ∈ V, 𝑢 ∈ V ↦ (((𝑚 Sat ( E ∩ (𝑚 × 𝑚)))‘ω)‘𝑢))
df-fmla 35539	⊢ Fmla = (𝑛 ∈ suc ω ↦ dom ((∅ Sat ∅)‘𝑛))
df-prv 35540	⊢ ⊧ = {⟨𝑚, 𝑢⟩ ∣ (𝑚 Sat_∈ 𝑢) = (𝑚 ↑_m ω)}
cgon 35626	class ¬_𝑔𝑈
cgoa 35627	class ∧_𝑔
cgoi 35628	class →_𝑔
cgoo 35629	class ∨_𝑔
cgob 35630	class ↔_𝑔
cgoq 35631	class =_𝑔
cgox 35632	class ∃_𝑔𝑁𝑈
df-gonot 35633	⊢ ¬_𝑔𝑈 = (𝑈⊼_𝑔𝑈)
df-goan 35634	⊢ ∧_𝑔 = (𝑢 ∈ V, 𝑣 ∈ V ↦ ¬_𝑔(𝑢⊼_𝑔𝑣))
df-goim 35635	⊢ →_𝑔 = (𝑢 ∈ V, 𝑣 ∈ V ↦ (𝑢⊼_𝑔¬_𝑔𝑣))
df-goor 35636	⊢ ∨_𝑔 = (𝑢 ∈ V, 𝑣 ∈ V ↦ (¬_𝑔𝑢 →_𝑔 𝑣))
df-gobi 35637	⊢ ↔_𝑔 = (𝑢 ∈ V, 𝑣 ∈ V ↦ ((𝑢 →_𝑔 𝑣)∧_𝑔(𝑣 →_𝑔 𝑢)))
df-goeq 35638	⊢ =_𝑔 = (𝑢 ∈ ω, 𝑣 ∈ ω ↦ ⦋suc (𝑢 ∪ 𝑣) / 𝑤⦌∀_𝑔𝑤((𝑤∈_𝑔𝑢) ↔_𝑔 (𝑤∈_𝑔𝑣)))
df-goex 35639	⊢ ∃_𝑔𝑁𝑈 = ¬_𝑔∀_𝑔𝑁¬_𝑔𝑈
cgze 35640	class AxExt
cgzr 35641	class AxRep
cgzp 35642	class AxPow
cgzu 35643	class AxUn
cgzg 35644	class AxReg
cgzi 35645	class AxInf
cgzf 35646	class ZF
df-gzext 35647	⊢ AxExt = (∀_𝑔2_o((2_o∈_𝑔∅) ↔_𝑔 (2_o∈_𝑔1_o)) →_𝑔 (∅=_𝑔1_o))
df-gzrep 35648	⊢ AxRep = (𝑢 ∈ (Fmla‘ω) ↦ (∀_𝑔3_o∃_𝑔1_o∀_𝑔2_o(∀_𝑔1_o𝑢 →_𝑔 (2_o=_𝑔1_o)) →_𝑔 ∀_𝑔1_o∀_𝑔2_o((2_o∈_𝑔1_o) ↔_𝑔 ∃_𝑔3_o((3_o∈_𝑔∅)∧_𝑔∀_𝑔1_o𝑢))))
df-gzpow 35649	⊢ AxPow = ∃_𝑔1_o∀_𝑔2_o(∀_𝑔1_o((1_o∈_𝑔2_o) ↔_𝑔 (1_o∈_𝑔∅)) →_𝑔 (2_o∈_𝑔1_o))
df-gzun 35650	⊢ AxUn = ∃_𝑔1_o∀_𝑔2_o(∃_𝑔1_o((2_o∈_𝑔1_o)∧_𝑔(1_o∈_𝑔∅)) →_𝑔 (2_o∈_𝑔1_o))
df-gzreg 35651	⊢ AxReg = (∃_𝑔1_o(1_o∈_𝑔∅) →_𝑔 ∃_𝑔1_o((1_o∈_𝑔∅)∧_𝑔∀_𝑔2_o((2_o∈_𝑔1_o) →_𝑔 ¬_𝑔(2_o∈_𝑔∅))))
df-gzinf 35652	⊢ AxInf = ∃_𝑔1_o((∅∈_𝑔1_o)∧_𝑔∀_𝑔2_o((2_o∈_𝑔1_o) →_𝑔 ∃_𝑔∅((2_o∈_𝑔∅)∧_𝑔(∅∈_𝑔1_o))))
df-gzf 35653	⊢ ZF = {𝑚 ∣ ((Tr 𝑚 ∧ 𝑚⊧AxExt ∧ 𝑚⊧AxPow) ∧ (𝑚⊧AxUn ∧ 𝑚⊧AxReg ∧ 𝑚⊧AxInf) ∧ ∀𝑢 ∈ (Fmla‘ω)𝑚⊧(AxRep‘𝑢))}
cmcn 35654	class mCN
cmvar 35655	class mVR
cmty 35656	class mType
cmvt 35657	class mVT
cmtc 35658	class mTC
cmax 35659	class mAx
cmrex 35660	class mREx
cmex 35661	class mEx
cmdv 35662	class mDV
cmvrs 35663	class mVars
cmrsub 35664	class mRSubst
cmsub 35665	class mSubst
cmvh 35666	class mVH
cmpst 35667	class mPreSt
cmsr 35668	class mStRed
cmsta 35669	class mStat
cmfs 35670	class mFS
cmcls 35671	class mCls
cmpps 35672	class mPPSt
cmthm 35673	class mThm
df-mcn 35674	⊢ mCN = Slot 1
df-mvar 35675	⊢ mVR = Slot 2
df-mty 35676	⊢ mType = Slot 3
df-mtc 35677	⊢ mTC = Slot 4
df-mmax 35678	⊢ mAx = Slot 5
df-mvt 35679	⊢ mVT = (𝑡 ∈ V ↦ ran (mType‘𝑡))
df-mrex 35680	⊢ mREx = (𝑡 ∈ V ↦ Word ((mCN‘𝑡) ∪ (mVR‘𝑡)))
df-mex 35681	⊢ mEx = (𝑡 ∈ V ↦ ((mTC‘𝑡) × (mREx‘𝑡)))
df-mdv 35682	⊢ mDV = (𝑡 ∈ V ↦ (((mVR‘𝑡) × (mVR‘𝑡)) ∖ I ))
df-mvrs 35683	⊢ mVars = (𝑡 ∈ V ↦ (𝑒 ∈ (mEx‘𝑡) ↦ (ran (2^nd ‘𝑒) ∩ (mVR‘𝑡))))
df-mrsub 35684	⊢ mRSubst = (𝑡 ∈ V ↦ (𝑓 ∈ ((mREx‘𝑡) ↑_pm (mVR‘𝑡)) ↦ (𝑒 ∈ (mREx‘𝑡) ↦ ((freeMnd‘((mCN‘𝑡) ∪ (mVR‘𝑡))) Σ_g ((𝑣 ∈ ((mCN‘𝑡) ∪ (mVR‘𝑡)) ↦ if(𝑣 ∈ dom 𝑓, (𝑓‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑒)))))
df-msub 35685	⊢ mSubst = (𝑡 ∈ V ↦ (𝑓 ∈ ((mREx‘𝑡) ↑_pm (mVR‘𝑡)) ↦ (𝑒 ∈ (mEx‘𝑡) ↦ ⟨(1^st ‘𝑒), (((mRSubst‘𝑡)‘𝑓)‘(2^nd ‘𝑒))⟩)))
df-mvh 35686	⊢ mVH = (𝑡 ∈ V ↦ (𝑣 ∈ (mVR‘𝑡) ↦ ⟨((mType‘𝑡)‘𝑣), ⟨“𝑣”⟩⟩))
df-mpst 35687	⊢ mPreSt = (𝑡 ∈ V ↦ (({𝑑 ∈ 𝒫 (mDV‘𝑡) ∣ ^◡𝑑 = 𝑑} × (𝒫 (mEx‘𝑡) ∩ Fin)) × (mEx‘𝑡)))
df-msr 35688	⊢ mStRed = (𝑡 ∈ V ↦ (𝑠 ∈ (mPreSt‘𝑡) ↦ ⦋(2^nd ‘(1^st ‘𝑠)) / ℎ⦌⦋(2^nd ‘𝑠) / 𝑎⦌⟨((1^st ‘(1^st ‘𝑠)) ∩ ⦋∪ ((mVars‘𝑡) “ (ℎ ∪ {𝑎})) / 𝑧⦌(𝑧 × 𝑧)), ℎ, 𝑎⟩))
df-msta 35689	⊢ mStat = (𝑡 ∈ V ↦ ran (mStRed‘𝑡))
df-mfs 35690	⊢ mFS = {𝑡 ∣ ((((mCN‘𝑡) ∩ (mVR‘𝑡)) = ∅ ∧ (mType‘𝑡):(mVR‘𝑡)⟶(mTC‘𝑡)) ∧ ((mAx‘𝑡) ⊆ (mStat‘𝑡) ∧ ∀𝑣 ∈ (mVT‘𝑡) ¬ (^◡(mType‘𝑡) “ {𝑣}) ∈ Fin))}
df-mcls 35691	⊢ mCls = (𝑡 ∈ V ↦ (𝑑 ∈ 𝒫 (mDV‘𝑡), ℎ ∈ 𝒫 (mEx‘𝑡) ↦ ∩ {𝑐 ∣ ((ℎ ∪ ran (mVH‘𝑡)) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑡) → ∀𝑠 ∈ ran (mSubst‘𝑡)(((𝑠 “ (𝑜 ∪ ran (mVH‘𝑡))) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → (((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑥))) × ((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑦)))) ⊆ 𝑑)) → (𝑠‘𝑝) ∈ 𝑐)))}))
df-mpps 35692	⊢ mPPSt = (𝑡 ∈ V ↦ {⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∣ (⟨𝑑, ℎ, 𝑎⟩ ∈ (mPreSt‘𝑡) ∧ 𝑎 ∈ (𝑑(mCls‘𝑡)ℎ))})
df-mthm 35693	⊢ mThm = (𝑡 ∈ V ↦ (^◡(mStRed‘𝑡) “ ((mStRed‘𝑡) “ (mPPSt‘𝑡))))
cm0s 35779	class m0St
cmsa 35780	class mSA
cmwgfs 35781	class mWGFS
cmsy 35782	class mSyn
cmesy 35783	class mESyn
cmgfs 35784	class mGFS
cmtree 35785	class mTree
cmst 35786	class mST
cmsax 35787	class mSAX
cmufs 35788	class mUFS
df-m0s 35789	⊢ m0St = (𝑎 ∈ V ↦ ⟨∅, ∅, 𝑎⟩)
df-msa 35790	⊢ mSA = (𝑡 ∈ V ↦ {𝑎 ∈ (mEx‘𝑡) ∣ ((m0St‘𝑎) ∈ (mAx‘𝑡) ∧ (1^st ‘𝑎) ∈ (mVT‘𝑡) ∧ Fun (^◡(2^nd ‘𝑎) ↾ (mVR‘𝑡)))})
df-mwgfs 35791	⊢ mWGFS = {𝑡 ∈ mFS ∣ ∀𝑑∀ℎ∀𝑎((⟨𝑑, ℎ, 𝑎⟩ ∈ (mAx‘𝑡) ∧ (1^st ‘𝑎) ∈ (mVT‘𝑡)) → ∃𝑠 ∈ ran (mSubst‘𝑡)𝑎 ∈ (𝑠 “ (mSA‘𝑡)))}
df-msyn 35792	⊢ mSyn = Slot 6
df-mesyn 35793	⊢ mESyn = (𝑡 ∈ V ↦ (𝑐 ∈ (mTC‘𝑡), 𝑒 ∈ (mREx‘𝑡) ↦ (((mSyn‘𝑡)‘𝑐)m0St𝑒)))
df-mgfs 35794	⊢ mGFS = {𝑡 ∈ mWGFS ∣ ((mSyn‘𝑡):(mTC‘𝑡)⟶(mVT‘𝑡) ∧ ∀𝑐 ∈ (mVT‘𝑡)((mSyn‘𝑡)‘𝑐) = 𝑐 ∧ ∀𝑑∀ℎ∀𝑎(⟨𝑑, ℎ, 𝑎⟩ ∈ (mAx‘𝑡) → ∀𝑒 ∈ (ℎ ∪ {𝑎})((mESyn‘𝑡)‘𝑒) ∈ (mPPSt‘𝑡)))}
df-mtree 35795	⊢ mTree = (𝑡 ∈ V ↦ (𝑑 ∈ 𝒫 (mDV‘𝑡), ℎ ∈ 𝒫 (mEx‘𝑡) ↦ ∩ {𝑟 ∣ (∀𝑒 ∈ ran (mVH‘𝑡)𝑒𝑟⟨(m0St‘𝑒), ∅⟩ ∧ ∀𝑒 ∈ ℎ 𝑒𝑟⟨((mStRed‘𝑡)‘⟨𝑑, ℎ, 𝑒⟩), ∅⟩ ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑡) → ∀𝑠 ∈ ran (mSubst‘𝑡)(∀𝑥∀𝑦(𝑥𝑚𝑦 → (((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑥))) × ((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑦)))) ⊆ 𝑑) → ({(𝑠‘𝑝)} × X𝑒 ∈ (𝑜 ∪ ((mVH‘𝑡) “ ∪ ((mVars‘𝑡) “ (𝑜 ∪ {𝑝}))))(𝑟 “ {(𝑠‘𝑒)})) ⊆ 𝑟)))}))
df-mst 35796	⊢ mST = (𝑡 ∈ V ↦ ((∅(mTree‘𝑡)∅) ↾ ((mEx‘𝑡) ↾ (mVT‘𝑡))))
df-msax 35797	⊢ mSAX = (𝑡 ∈ V ↦ (𝑝 ∈ (mSA‘𝑡) ↦ ((mVH‘𝑡) “ ((mVars‘𝑡)‘𝑝))))
df-mufs 35798	⊢ mUFS = {𝑡 ∈ mGFS ∣ Fun (mST‘𝑡)}
cmuv 35799	class mUV
cmvl 35800	class mVL
cmvsb 35801	class mVSubst
cmfsh 35802	class mFresh
cmfr 35803	class mFRel
cmevl 35804	class mEval
cmdl 35805	class mMdl
cusyn 35806	class mUSyn
cgmdl 35807	class mGMdl
cmitp 35808	class mItp
cmfitp 35809	class mFromItp
df-muv 35810	⊢ mUV = Slot 7
df-mfsh 35811	⊢ mFresh = Slot ;19
df-mevl 35812	⊢ mEval = Slot ;20
df-mvl 35813	⊢ mVL = (𝑡 ∈ V ↦ X𝑣 ∈ (mVR‘𝑡)((mUV‘𝑡) “ {((mType‘𝑡)‘𝑣)}))
df-mvsb 35814	⊢ mVSubst = (𝑡 ∈ V ↦ {⟨⟨𝑠, 𝑚⟩, 𝑥⟩ ∣ ((𝑠 ∈ ran (mSubst‘𝑡) ∧ 𝑚 ∈ (mVL‘𝑡)) ∧ ∀𝑣 ∈ (mVR‘𝑡)𝑚dom (mEval‘𝑡)(𝑠‘((mVH‘𝑡)‘𝑣)) ∧ 𝑥 = (𝑣 ∈ (mVR‘𝑡) ↦ (𝑚(mEval‘𝑡)(𝑠‘((mVH‘𝑡)‘𝑣)))))})
df-mfrel 35815	⊢ mFRel = (𝑡 ∈ V ↦ {𝑟 ∈ 𝒫 ((mUV‘𝑡) × (mUV‘𝑡)) ∣ (^◡𝑟 = 𝑟 ∧ ∀𝑐 ∈ (mVT‘𝑡)∀𝑤 ∈ (𝒫 (mUV‘𝑡) ∩ Fin)∃𝑣 ∈ ((mUV‘𝑡) “ {𝑐})𝑤 ⊆ (𝑟 “ {𝑣}))})
df-mdl 35816	⊢ mMdl = {𝑡 ∈ mFS ∣ [(mUV‘𝑡) / 𝑢][(mEx‘𝑡) / 𝑥][(mVL‘𝑡) / 𝑣][(mEval‘𝑡) / 𝑛][(mFresh‘𝑡) / 𝑓]((𝑢 ⊆ ((mTC‘𝑡) × V) ∧ 𝑓 ∈ (mFRel‘𝑡) ∧ 𝑛 ∈ (𝑢 ↑_pm (𝑣 × (mEx‘𝑡)))) ∧ ∀𝑚 ∈ 𝑣 ((∀𝑒 ∈ 𝑥 (𝑛 “ {⟨𝑚, 𝑒⟩}) ⊆ (𝑢 “ {(1^st ‘𝑒)}) ∧ ∀𝑦 ∈ (mVR‘𝑡)⟨𝑚, ((mVH‘𝑡)‘𝑦)⟩𝑛(𝑚‘𝑦) ∧ ∀𝑑∀ℎ∀𝑎(⟨𝑑, ℎ, 𝑎⟩ ∈ (mAx‘𝑡) → ((∀𝑦∀𝑧(𝑦𝑑𝑧 → (𝑚‘𝑦)𝑓(𝑚‘𝑧)) ∧ ℎ ⊆ (dom 𝑛 “ {𝑚})) → 𝑚dom 𝑛 𝑎))) ∧ (∀𝑠 ∈ ran (mSubst‘𝑡)∀𝑒 ∈ (mEx‘𝑡)∀𝑦(⟨𝑠, 𝑚⟩(mVSubst‘𝑡)𝑦 → (𝑛 “ {⟨𝑚, (𝑠‘𝑒)⟩}) = (𝑛 “ {⟨𝑦, 𝑒⟩})) ∧ ∀𝑝 ∈ 𝑣 ∀𝑒 ∈ 𝑥 ((𝑚 ↾ ((mVars‘𝑡)‘𝑒)) = (𝑝 ↾ ((mVars‘𝑡)‘𝑒)) → (𝑛 “ {⟨𝑚, 𝑒⟩}) = (𝑛 “ {⟨𝑝, 𝑒⟩})) ∧ ∀𝑦 ∈ 𝑢 ∀𝑒 ∈ 𝑥 ((𝑚 “ ((mVars‘𝑡)‘𝑒)) ⊆ (𝑓 “ {𝑦}) → (𝑛 “ {⟨𝑚, 𝑒⟩}) ⊆ (𝑓 “ {𝑦})))))}
df-musyn 35817	⊢ mUSyn = (𝑡 ∈ V ↦ (𝑣 ∈ (mUV‘𝑡) ↦ ⟨((mSyn‘𝑡)‘(1^st ‘𝑣)), (2^nd ‘𝑣)⟩))
df-gmdl 35818	⊢ mGMdl = {𝑡 ∈ (mGFS ∩ mMdl) ∣ (∀𝑐 ∈ (mTC‘𝑡)((mUV‘𝑡) “ {𝑐}) ⊆ ((mUV‘𝑡) “ {((mSyn‘𝑡)‘𝑐)}) ∧ ∀𝑣 ∈ (mUV‘𝑐)∀𝑤 ∈ (mUV‘𝑐)(𝑣(mFresh‘𝑡)𝑤 ↔ 𝑣(mFresh‘𝑡)((mUSyn‘𝑡)‘𝑤)) ∧ ∀𝑚 ∈ (mVL‘𝑡)∀𝑒 ∈ (mEx‘𝑡)((mEval‘𝑡) “ {⟨𝑚, 𝑒⟩}) = (((mEval‘𝑡) “ {⟨𝑚, ((mESyn‘𝑡)‘𝑒)⟩}) ∩ ((mUV‘𝑡) “ {(1^st ‘𝑒)})))}
df-mitp 35819	⊢ mItp = (𝑡 ∈ V ↦ (𝑎 ∈ (mSA‘𝑡) ↦ (𝑔 ∈ X𝑖 ∈ ((mVars‘𝑡)‘𝑎)((mUV‘𝑡) “ {((mType‘𝑡)‘𝑖)}) ↦ (℩𝑥∃𝑚 ∈ (mVL‘𝑡)(𝑔 = (𝑚 ↾ ((mVars‘𝑡)‘𝑎)) ∧ 𝑥 = (𝑚(mEval‘𝑡)𝑎))))))
df-mfitp 35820	⊢ mFromItp = (𝑡 ∈ V ↦ (𝑓 ∈ X𝑎 ∈ (mSA‘𝑡)(((mUV‘𝑡) “ {((1^st ‘𝑡)‘𝑎)}) ↑_m X𝑖 ∈ ((mVars‘𝑡)‘𝑎)((mUV‘𝑡) “ {((mType‘𝑡)‘𝑖)})) ↦ (℩𝑛 ∈ ((mUV‘𝑡) ↑_pm ((mVL‘𝑡) × (mEx‘𝑡)))∀𝑚 ∈ (mVL‘𝑡)(∀𝑣 ∈ (mVR‘𝑡)⟨𝑚, ((mVH‘𝑡)‘𝑣)⟩𝑛(𝑚‘𝑣) ∧ ∀𝑒∀𝑎∀𝑔(𝑒(mST‘𝑡)⟨𝑎, 𝑔⟩ → ⟨𝑚, 𝑒⟩𝑛(𝑓‘(𝑖 ∈ ((mVars‘𝑡)‘𝑎) ↦ (𝑚𝑛(𝑔‘((mVH‘𝑡)‘𝑖)))))) ∧ ∀𝑒 ∈ (mEx‘𝑡)(𝑛 “ {⟨𝑚, 𝑒⟩}) = ((𝑛 “ {⟨𝑚, ((mESyn‘𝑡)‘𝑒)⟩}) ∩ ((mUV‘𝑡) “ {(1^st ‘𝑒)}))))))
ccpms 35821	class cplMetSp
chlb 35822	class HomLimB
chlim 35823	class HomLim
cpfl 35824	class polyFld
csf1 35825	class splitFld₁
csf 35826	class splitFld
cpsl 35827	class polySplitLim
df-cplmet 35828	⊢ cplMetSp = (𝑤 ∈ V ↦ ⦋((𝑤 ↑_s ℕ) ↾_s (Cau‘(dist‘𝑤))) / 𝑟⦌⦋(Base‘𝑟) / 𝑣⦌⦋{⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝑣 ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ ℤ (𝑓 ↾ (ℤ_≥‘𝑗)):(ℤ_≥‘𝑗)⟶((𝑔‘𝑗)(ball‘(dist‘𝑤))𝑥))} / 𝑒⦌((𝑟 /_s 𝑒) sSet {⟨(dist‘ndx), {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ∃𝑝 ∈ 𝑣 ∃𝑞 ∈ 𝑣 ((𝑥 = [𝑝]𝑒 ∧ 𝑦 = [𝑞]𝑒) ∧ (𝑝 ∘_f (dist‘𝑟)𝑞) ⇝ 𝑧)}⟩}))
df-homlimb 35829	⊢ HomLimB = (𝑓 ∈ V ↦ ⦋∪ 𝑛 ∈ ℕ ({𝑛} × dom (𝑓‘𝑛)) / 𝑣⦌⦋∩ {𝑠 ∣ (𝑠 Er 𝑣 ∧ (𝑥 ∈ 𝑣 ↦ ⟨((1^st ‘𝑥) + 1), ((𝑓‘(1^st ‘𝑥))‘(2^nd ‘𝑥))⟩) ⊆ 𝑠)} / 𝑒⦌⟨(𝑣 / 𝑒), (𝑛 ∈ ℕ ↦ (𝑥 ∈ dom (𝑓‘𝑛) ↦ [⟨𝑛, 𝑥⟩]𝑒))⟩)
df-homlim 35830	⊢ HomLim = (𝑟 ∈ V, 𝑓 ∈ V ↦ ⦋( HomLimB ‘𝑓) / 𝑒⦌⦋(1^st ‘𝑒) / 𝑣⦌⦋(2^nd ‘𝑒) / 𝑔⦌({⟨(Base‘ndx), 𝑣⟩, ⟨(+_g‘ndx), ∪ 𝑛 ∈ ℕ ran (𝑥 ∈ dom (𝑔‘𝑛), 𝑦 ∈ dom (𝑔‘𝑛) ↦ ⟨⟨((𝑔‘𝑛)‘𝑥), ((𝑔‘𝑛)‘𝑦)⟩, ((𝑔‘𝑛)‘(𝑥(+_g‘(𝑟‘𝑛))𝑦))⟩)⟩, ⟨(._r‘ndx), ∪ 𝑛 ∈ ℕ ran (𝑥 ∈ dom (𝑔‘𝑛), 𝑦 ∈ dom (𝑔‘𝑛) ↦ ⟨⟨((𝑔‘𝑛)‘𝑥), ((𝑔‘𝑛)‘𝑦)⟩, ((𝑔‘𝑛)‘(𝑥(._r‘(𝑟‘𝑛))𝑦))⟩)⟩} ∪ {⟨(TopOpen‘ndx), {𝑠 ∈ 𝒫 𝑣 ∣ ∀𝑛 ∈ ℕ (^◡(𝑔‘𝑛) “ 𝑠) ∈ (TopOpen‘(𝑟‘𝑛))}⟩, ⟨(dist‘ndx), ∪ 𝑛 ∈ ℕ ran (𝑥 ∈ dom ((𝑔‘𝑛)‘𝑛), 𝑦 ∈ dom ((𝑔‘𝑛)‘𝑛) ↦ ⟨⟨((𝑔‘𝑛)‘𝑥), ((𝑔‘𝑛)‘𝑦)⟩, (𝑥(dist‘(𝑟‘𝑛))𝑦)⟩)⟩, ⟨(le‘ndx), ∪ 𝑛 ∈ ℕ (^◡(𝑔‘𝑛) ∘ ((le‘(𝑟‘𝑛)) ∘ (𝑔‘𝑛)))⟩}))
df-plfl 35831	⊢ polyFld = (𝑟 ∈ V, 𝑝 ∈ V ↦ ⦋(Poly₁‘𝑟) / 𝑠⦌⦋((RSpan‘𝑠)‘{𝑝}) / 𝑖⦌⦋(𝑐 ∈ (Base‘𝑟) ↦ [(𝑐( ·_𝑠 ‘𝑠)(1_r‘𝑠))](𝑠 ~_QG 𝑖)) / 𝑓⦌⟨⦋(𝑠 /_s (𝑠 ~_QG 𝑖)) / 𝑡⦌((𝑡 toNrmGrp (℩𝑛 ∈ (AbsVal‘𝑡)(𝑛 ∘ 𝑓) = (norm‘𝑟))) sSet ⟨(le‘ndx), ⦋(𝑧 ∈ (Base‘𝑡) ↦ (℩𝑞 ∈ 𝑧 (𝑞(rem_1p‘𝑟)𝑝) = 𝑞)) / 𝑔⦌(^◡𝑔 ∘ ((le‘𝑠) ∘ 𝑔))⟩), 𝑓⟩)
df-sfl1 35838	⊢ splitFld₁ = (𝑟 ∈ V, 𝑗 ∈ V ↦ (𝑝 ∈ (Poly₁‘𝑟) ↦ (rec((𝑠 ∈ V, 𝑓 ∈ V ↦ ⦋(Poly₁‘𝑠) / 𝑚⦌⦋{𝑔 ∈ ((Monic_1p‘𝑠) ∩ (Irred‘𝑚)) ∣ (𝑔(∥_r‘𝑚)(𝑝 ∘ 𝑓) ∧ 1 < (𝑠deg₁𝑔))} / 𝑏⦌if(((𝑝 ∘ 𝑓) = (0_g‘𝑚) ∨ 𝑏 = ∅), ⟨𝑠, 𝑓⟩, ⦋(glb‘𝑏) / ℎ⦌⦋(𝑠 polyFld ℎ) / 𝑡⦌⟨(1^st ‘𝑡), (𝑓 ∘ (2^nd ‘𝑡))⟩)), 𝑗)‘(card‘(1...(𝑟deg₁𝑝))))))
df-sfl 35839	⊢ splitFld = (𝑟 ∈ V, 𝑝 ∈ V ↦ (℩𝑥∃𝑓(𝑓 Isom < , (lt‘𝑟)((1...(♯‘𝑝)), 𝑝) ∧ 𝑥 = (seq0((𝑒 ∈ V, 𝑔 ∈ V ↦ ((𝑟 splitFld₁ 𝑒)‘𝑔)), (𝑓 ∪ {⟨0, ⟨𝑟, ( I ↾ (Base‘𝑟))⟩⟩}))‘(♯‘𝑝)))))
df-psl 35840	⊢ polySplitLim = (𝑟 ∈ V, 𝑝 ∈ ((𝒫 (Base‘𝑟) ∩ Fin) ↑_m ℕ) ↦ ⦋(1^st ∘ seq0((𝑔 ∈ V, 𝑞 ∈ V ↦ ⦋(1^st ‘𝑔) / 𝑒⦌⦋(1^st ‘𝑒) / 𝑠⦌⦋(𝑠 splitFld ran (𝑥 ∈ 𝑞 ↦ (𝑥 ∘ (2^nd ‘𝑔)))) / 𝑓⦌⟨𝑓, ((2^nd ‘𝑔) ∘ (2^nd ‘𝑓))⟩), (𝑝 ∪ {⟨0, ⟨⟨𝑟, ∅⟩, ( I ↾ (Base‘𝑟))⟩⟩}))) / 𝑓⦌((1^st ∘ (𝑓 shift 1)) HomLim (2^nd ∘ 𝑓)))
czr 35841	class ZRing
cgf 35842	class GF
cgfo 35843	class GF_∞
ceqp 35844	class ~Qp
crqp 35845	class /Qp
cqp 35846	class Qp
czp 35847	class Zp
cqpa 35848	class _Qp
ccp 35849	class Cp
df-zrng 35850	⊢ ZRing = (𝑟 ∈ V ↦ (𝑟 IntgRing ran (ℤRHom‘𝑟)))
df-gf 35851	⊢ GF = (𝑝 ∈ ℙ, 𝑛 ∈ ℕ ↦ ⦋(ℤ/nℤ‘𝑝) / 𝑟⦌(1^st ‘(𝑟 splitFld {⦋(Poly₁‘𝑟) / 𝑠⦌⦋(var₁‘𝑟) / 𝑥⦌(((𝑝↑𝑛)(._g‘(mulGrp‘𝑠))𝑥)(-_g‘𝑠)𝑥)})))
df-gfoo 35852	⊢ GF_∞ = (𝑝 ∈ ℙ ↦ ⦋(ℤ/nℤ‘𝑝) / 𝑟⦌(𝑟 polySplitLim (𝑛 ∈ ℕ ↦ {⦋(Poly₁‘𝑟) / 𝑠⦌⦋(var₁‘𝑟) / 𝑥⦌(((𝑝↑𝑛)(._g‘(mulGrp‘𝑠))𝑥)(-_g‘𝑠)𝑥)})))
df-eqp 35853	⊢ ~Qp = (𝑝 ∈ ℙ ↦ {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ (ℤ ↑_m ℤ) ∧ ∀𝑛 ∈ ℤ Σ𝑘 ∈ (ℤ_≥‘-𝑛)(((𝑓‘-𝑘) − (𝑔‘-𝑘)) / (𝑝↑(𝑘 + (𝑛 + 1)))) ∈ ℤ)})
df-rqp 35854	⊢ /Qp = (𝑝 ∈ ℙ ↦ (~Qp ∩ ⦋{𝑓 ∈ (ℤ ↑_m ℤ) ∣ ∃𝑥 ∈ ran ℤ_≥(^◡𝑓 “ (ℤ ∖ {0})) ⊆ 𝑥} / 𝑦⦌(𝑦 × (𝑦 ∩ (ℤ ↑_m (0...(𝑝 − 1)))))))
df-qp 35855	⊢ Qp = (𝑝 ∈ ℙ ↦ ⦋{ℎ ∈ (ℤ ↑_m (0...(𝑝 − 1))) ∣ ∃𝑥 ∈ ran ℤ_≥(^◡ℎ “ (ℤ ∖ {0})) ⊆ 𝑥} / 𝑏⦌(({⟨(Base‘ndx), 𝑏⟩, ⟨(+_g‘ndx), (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ ((/Qp‘𝑝)‘(𝑓 ∘_f + 𝑔)))⟩, ⟨(._r‘ndx), (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ ((/Qp‘𝑝)‘(𝑛 ∈ ℤ ↦ Σ𝑘 ∈ ℤ ((𝑓‘𝑘) · (𝑔‘(𝑛 − 𝑘))))))⟩} ∪ {⟨(le‘ndx), {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝑏 ∧ Σ𝑘 ∈ ℤ ((𝑓‘-𝑘) · ((𝑝 + 1)↑-𝑘)) < Σ𝑘 ∈ ℤ ((𝑔‘-𝑘) · ((𝑝 + 1)↑-𝑘)))}⟩}) toNrmGrp (𝑓 ∈ 𝑏 ↦ if(𝑓 = (ℤ × {0}), 0, (𝑝↑-inf((^◡𝑓 “ (ℤ ∖ {0})), ℝ, < ))))))
df-zp 35856	⊢ Zp = (ZRing ∘ Qp)
df-qpa 35857	⊢ _Qp = (𝑝 ∈ ℙ ↦ ⦋(Qp‘𝑝) / 𝑟⦌(𝑟 polySplitLim (𝑛 ∈ ℕ ↦ {𝑓 ∈ (Poly₁‘𝑟) ∣ ((𝑟deg₁𝑓) ≤ 𝑛 ∧ ∀𝑑 ∈ ran (coe₁‘𝑓)(^◡𝑑 “ (ℤ ∖ {0})) ⊆ (0...𝑛))})))
df-cp 35858	⊢ Cp = ( cplMetSp ∘ _Qp)
ccloneop 35889	class CloneOp
df-cloneop 35890	⊢ CloneOp = (𝑎 ∈ V ↦ {𝑥 ∣ ∃𝑛 ∈ (ω ∖ 1_o)𝑥 ∈ (𝑎 ↑_m (𝑎 ↑_m 𝑛))})
cprj 35891	class prj
df-prj 35892	⊢ prj = (𝑎 ∈ V ↦ (𝑛 ∈ (ω ∖ 1_o), 𝑖 ∈ 𝑛 ↦ (𝑥 ∈ (𝑎 ↑_m 𝑛) ↦ (𝑥‘𝑖))))
csuppos 35893	class suppos
df-suppos 35894	⊢ suppos = (𝑎 ∈ V ↦ (𝑛 ∈ (ω ∖ 1_o), 𝑚 ∈ (ω ∖ 1_o) ↦ (𝑓 ∈ (𝑎 ↑_m (𝑎 ↑_m 𝑛)), 𝑔 ∈ ((𝑎 ↑_m (𝑎 ↑_m 𝑚)) ↑_m 𝑛) ↦ (𝑥 ∈ (𝑎 ↑_m 𝑚) ↦ (𝑓‘(𝑖 ∈ 𝑛 ↦ ((𝑔‘𝑖)‘𝑥)))))))
cwsuc 36002	class wsuc(𝑅, 𝐴, 𝑋)
cwlim 36003	class WLim(𝑅, 𝐴)
df-wsuc 36004	⊢ wsuc(𝑅, 𝐴, 𝑋) = inf(Pred(^◡𝑅, 𝐴, 𝑋), 𝐴, 𝑅)
df-wlim 36005	⊢ WLim(𝑅, 𝐴) = {𝑥 ∈ 𝐴 ∣ (𝑥 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑥 = sup(Pred(𝑅, 𝐴, 𝑥), 𝐴, 𝑅))}
ctxp 36022	class (𝐴 ⊗ 𝐵)
cpprod 36023	class pprod(𝑅, 𝑆)
csset 36024	class SSet
ctrans 36025	class Trans
cbigcup 36026	class Bigcup
cfix 36027	class Fix 𝐴
climits 36028	class Limits
cfuns 36029	class Funs
csingle 36030	class Singleton
csingles 36031	class Singletons
cimage 36032	class Image𝐴
ccart 36033	class Cart
cimg 36034	class Img
cdomain 36035	class Domain
crange 36036	class Range
capply 36037	class Apply
ccup 36038	class Cup
ccap 36039	class Cap
csuccf 36040	class Succ
cfunpart 36041	class Funpart𝐹
cfullfn 36042	class FullFun𝐹
crestrict 36043	class Restrict
cub 36044	class UB𝑅
clb 36045	class LB𝑅
df-txp 36046	⊢ (𝐴 ⊗ 𝐵) = ((^◡(1^st ↾ (V × V)) ∘ 𝐴) ∩ (^◡(2^nd ↾ (V × V)) ∘ 𝐵))
df-pprod 36047	⊢ pprod(𝐴, 𝐵) = ((𝐴 ∘ (1^st ↾ (V × V))) ⊗ (𝐵 ∘ (2^nd ↾ (V × V))))
df-sset 36048	⊢ SSet = ((V × V) ∖ ran ( E ⊗ (V ∖ E )))
df-trans 36049	⊢ Trans = (V ∖ ran (( E ∘ E ) ∖ E ))
df-bigcup 36050	⊢ Bigcup = ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ E ) ⊗ V)))
df-fix 36051	⊢ Fix 𝐴 = dom (𝐴 ∩ I )
df-limits 36052	⊢ Limits = ((On ∩ Fix Bigcup ) ∖ {∅})
df-funs 36053	⊢ Funs = (𝒫 (V × V) ∖ Fix ( E ∘ ((1^st ⊗ ((V ∖ I ) ∘ 2^nd )) ∘ ^◡ E )))
df-singleton 36054	⊢ Singleton = ((V × V) ∖ ran ((V ⊗ E ) △ ( I ⊗ V)))
df-singles 36055	⊢ Singletons = ran Singleton
df-image 36056	⊢ Image𝐴 = ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ ^◡𝐴) ⊗ V)))
df-cart 36057	⊢ Cart = (((V × V) × V) ∖ ran ((V ⊗ E ) △ (pprod( E , E ) ⊗ V)))
df-img 36058	⊢ Img = (Image((2^nd ∘ 1^st ) ↾ (1^st ↾ (V × V))) ∘ Cart)
df-domain 36059	⊢ Domain = Image(1^st ↾ (V × V))
df-range 36060	⊢ Range = Image(2^nd ↾ (V × V))
df-cup 36061	⊢ Cup = (((V × V) × V) ∖ ran ((V ⊗ E ) △ (((^◡1^st ∘ E ) ∪ (^◡2^nd ∘ E )) ⊗ V)))
df-cap 36062	⊢ Cap = (((V × V) × V) ∖ ran ((V ⊗ E ) △ (((^◡1^st ∘ E ) ∩ (^◡2^nd ∘ E )) ⊗ V)))
df-restrict 36063	⊢ Restrict = (Cap ∘ (1^st ⊗ (Cart ∘ (2^nd ⊗ (Range ∘ 1^st )))))
df-succf 36064	⊢ Succ = (Cup ∘ ( I ⊗ Singleton))
df-apply 36065	⊢ Apply = (( Bigcup ∘ Bigcup ) ∘ (((V × V) ∖ ran ((V ⊗ E ) △ (( E ↾ Singletons ) ⊗ V))) ∘ ((Singleton ∘ Img) ∘ pprod( I , Singleton))))
df-funpart 36066	⊢ Funpart𝐹 = (𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))
df-fullfun 36067	⊢ FullFun𝐹 = (Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))
df-ub 36068	⊢ UB𝑅 = ((V × V) ∖ ((V ∖ 𝑅) ∘ ^◡ E ))
df-lb 36069	⊢ LB𝑅 = UB^◡𝑅
caltop 36150	class ⟪𝐴, 𝐵⟫
caltxp 36151	class (𝐴 ×× 𝐵)
df-altop 36152	⊢ ⟪𝐴, 𝐵⟫ = {{𝐴}, {𝐴, {𝐵}}}
df-altxp 36153	⊢ (𝐴 ×× 𝐵) = {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = ⟪𝑥, 𝑦⟫}
cofs 36176	class OuterFiveSeg
df-ofs 36177	⊢ OuterFiveSeg = {⟨𝑝, 𝑞⟩ ∣ ∃𝑛 ∈ ℕ ∃𝑎 ∈ (𝔼‘𝑛)∃𝑏 ∈ (𝔼‘𝑛)∃𝑐 ∈ (𝔼‘𝑛)∃𝑑 ∈ (𝔼‘𝑛)∃𝑥 ∈ (𝔼‘𝑛)∃𝑦 ∈ (𝔼‘𝑛)∃𝑧 ∈ (𝔼‘𝑛)∃𝑤 ∈ (𝔼‘𝑛)(𝑝 = ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ 𝑞 = ⟨⟨𝑥, 𝑦⟩, ⟨𝑧, 𝑤⟩⟩ ∧ ((𝑏 Btwn ⟨𝑎, 𝑐⟩ ∧ 𝑦 Btwn ⟨𝑥, 𝑧⟩) ∧ (⟨𝑎, 𝑏⟩Cgr⟨𝑥, 𝑦⟩ ∧ ⟨𝑏, 𝑐⟩Cgr⟨𝑦, 𝑧⟩) ∧ (⟨𝑎, 𝑑⟩Cgr⟨𝑥, 𝑤⟩ ∧ ⟨𝑏, 𝑑⟩Cgr⟨𝑦, 𝑤⟩)))}
ctransport 36223	class TransportTo
df-transport 36224	⊢ TransportTo = {⟨⟨𝑝, 𝑞⟩, 𝑥⟩ ∣ ∃𝑛 ∈ ℕ ((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1^st ‘𝑞) ≠ (2^nd ‘𝑞)) ∧ 𝑥 = (℩𝑟 ∈ (𝔼‘𝑛)((2^nd ‘𝑞) Btwn ⟨(1^st ‘𝑞), 𝑟⟩ ∧ ⟨(2^nd ‘𝑞), 𝑟⟩Cgr𝑝)))}
cifs 36229	class InnerFiveSeg
ccgr3 36230	class Cgr3
ccolin 36231	class Colinear
cfs 36232	class FiveSeg
df-colinear 36233	⊢ Colinear = ^◡{⟨⟨𝑏, 𝑐⟩, 𝑎⟩ ∣ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑐 ∈ (𝔼‘𝑛)) ∧ (𝑎 Btwn ⟨𝑏, 𝑐⟩ ∨ 𝑏 Btwn ⟨𝑐, 𝑎⟩ ∨ 𝑐 Btwn ⟨𝑎, 𝑏⟩))}
df-ifs 36234	⊢ InnerFiveSeg = {⟨𝑝, 𝑞⟩ ∣ ∃𝑛 ∈ ℕ ∃𝑎 ∈ (𝔼‘𝑛)∃𝑏 ∈ (𝔼‘𝑛)∃𝑐 ∈ (𝔼‘𝑛)∃𝑑 ∈ (𝔼‘𝑛)∃𝑥 ∈ (𝔼‘𝑛)∃𝑦 ∈ (𝔼‘𝑛)∃𝑧 ∈ (𝔼‘𝑛)∃𝑤 ∈ (𝔼‘𝑛)(𝑝 = ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ 𝑞 = ⟨⟨𝑥, 𝑦⟩, ⟨𝑧, 𝑤⟩⟩ ∧ ((𝑏 Btwn ⟨𝑎, 𝑐⟩ ∧ 𝑦 Btwn ⟨𝑥, 𝑧⟩) ∧ (⟨𝑎, 𝑐⟩Cgr⟨𝑥, 𝑧⟩ ∧ ⟨𝑏, 𝑐⟩Cgr⟨𝑦, 𝑧⟩) ∧ (⟨𝑎, 𝑑⟩Cgr⟨𝑥, 𝑤⟩ ∧ ⟨𝑐, 𝑑⟩Cgr⟨𝑧, 𝑤⟩)))}
df-cgr3 36235	⊢ Cgr3 = {⟨𝑝, 𝑞⟩ ∣ ∃𝑛 ∈ ℕ ∃𝑎 ∈ (𝔼‘𝑛)∃𝑏 ∈ (𝔼‘𝑛)∃𝑐 ∈ (𝔼‘𝑛)∃𝑑 ∈ (𝔼‘𝑛)∃𝑒 ∈ (𝔼‘𝑛)∃𝑓 ∈ (𝔼‘𝑛)(𝑝 = ⟨𝑎, ⟨𝑏, 𝑐⟩⟩ ∧ 𝑞 = ⟨𝑑, ⟨𝑒, 𝑓⟩⟩ ∧ (⟨𝑎, 𝑏⟩Cgr⟨𝑑, 𝑒⟩ ∧ ⟨𝑎, 𝑐⟩Cgr⟨𝑑, 𝑓⟩ ∧ ⟨𝑏, 𝑐⟩Cgr⟨𝑒, 𝑓⟩))}
df-fs 36236	⊢ FiveSeg = {⟨𝑝, 𝑞⟩ ∣ ∃𝑛 ∈ ℕ ∃𝑎 ∈ (𝔼‘𝑛)∃𝑏 ∈ (𝔼‘𝑛)∃𝑐 ∈ (𝔼‘𝑛)∃𝑑 ∈ (𝔼‘𝑛)∃𝑥 ∈ (𝔼‘𝑛)∃𝑦 ∈ (𝔼‘𝑛)∃𝑧 ∈ (𝔼‘𝑛)∃𝑤 ∈ (𝔼‘𝑛)(𝑝 = ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ 𝑞 = ⟨⟨𝑥, 𝑦⟩, ⟨𝑧, 𝑤⟩⟩ ∧ (𝑎 Colinear ⟨𝑏, 𝑐⟩ ∧ ⟨𝑎, ⟨𝑏, 𝑐⟩⟩Cgr3⟨𝑥, ⟨𝑦, 𝑧⟩⟩ ∧ (⟨𝑎, 𝑑⟩Cgr⟨𝑥, 𝑤⟩ ∧ ⟨𝑏, 𝑑⟩Cgr⟨𝑦, 𝑤⟩)))}
csegle 36300	class Seg_≤
df-segle 36301	⊢ Seg_≤ = {⟨𝑝, 𝑞⟩ ∣ ∃𝑛 ∈ ℕ ∃𝑎 ∈ (𝔼‘𝑛)∃𝑏 ∈ (𝔼‘𝑛)∃𝑐 ∈ (𝔼‘𝑛)∃𝑑 ∈ (𝔼‘𝑛)(𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝑞 = ⟨𝑐, 𝑑⟩ ∧ ∃𝑦 ∈ (𝔼‘𝑛)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩))}
coutsideof 36313	class OutsideOf
df-outsideof 36314	⊢ OutsideOf = ( Colinear ∖ Btwn )
cline2 36328	class Line
cray 36329	class Ray
clines2 36330	class LinesEE
df-line2 36331	⊢ Line = {⟨⟨𝑎, 𝑏⟩, 𝑙⟩ ∣ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎 ≠ 𝑏) ∧ 𝑙 = [⟨𝑎, 𝑏⟩]^◡ Colinear )}
df-ray 36332	⊢ Ray = {⟨⟨𝑝, 𝑎⟩, 𝑟⟩ ∣ ∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑎) ∧ 𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩})}
df-lines2 36333	⊢ LinesEE = ran Line
cfwddif 36352	class △
df-fwddif 36353	⊢ △ = (𝑓 ∈ (ℂ ↑_pm ℂ) ↦ (𝑥 ∈ {𝑦 ∈ dom 𝑓 ∣ (𝑦 + 1) ∈ dom 𝑓} ↦ ((𝑓‘(𝑥 + 1)) − (𝑓‘𝑥))))
cfwddifn 36354	class △ⁿ
df-fwddifn 36355	⊢ △ⁿ = (𝑛 ∈ ℕ₀, 𝑓 ∈ (ℂ ↑_pm ℂ) ↦ (𝑥 ∈ {𝑦 ∈ ℂ ∣ ∀𝑘 ∈ (0...𝑛)(𝑦 + 𝑘) ∈ dom 𝑓} ↦ Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · ((-1↑(𝑛 − 𝑘)) · (𝑓‘(𝑥 + 𝑘))))))
chf 36366	class Hf
df-hf 36367	⊢ Hf = ∪ (𝑅₁ “ ω)
cfne 36530	class Fne
df-fne 36531	⊢ Fne = {⟨𝑥, 𝑦⟩ ∣ (∪ 𝑥 = ∪ 𝑦 ∧ ∀𝑧 ∈ 𝑥 𝑧 ⊆ ∪ (𝑦 ∩ 𝒫 𝑧))}
w3nand 36591	wff (𝜑 ⊼ 𝜓 ⊼ 𝜒)
df-3nand 36592	⊢ ((𝜑 ⊼ 𝜓 ⊼ 𝜒) ↔ (𝜑 → (𝜓 → ¬ 𝜒)))
cgcdOLD 36653	class gcd_OLD (𝐴, 𝐵)
df-gcdOLD 36654	⊢ gcd_OLD (𝐴, 𝐵) = sup({𝑥 ∈ ℕ ∣ ((𝐴 / 𝑥) ∈ ℕ ∧ (𝐵 / 𝑥) ∈ ℕ)}, ℕ, < )
cprvb 36797	wff Prv 𝜑
ax-prv1 36798	⊢ 𝜑 ⇒ ⊢ Prv 𝜑
ax-prv2 36799	⊢ (Prv (𝜑 → 𝜓) → (Prv 𝜑 → Prv 𝜓))
ax-prv3 36800	⊢ (Prv 𝜑 → Prv Prv 𝜑)
wmoo 36851	wff ∃**𝑥𝜑
df-bj-mo 36852	⊢ (∃**𝑥𝜑 ↔ ∀𝑧∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))
wnnf 36924	wff Ⅎ'𝑥𝜑
df-bj-nnf 36925	⊢ (Ⅎ'𝑥𝜑 ↔ ((∃𝑥𝜑 → 𝜑) ∧ (𝜑 → ∀𝑥𝜑)))
bj-cgab 37134	class {𝐴 ∣ 𝑥 ∣ 𝜑}
df-bj-gab 37135	⊢ {𝐴 ∣ 𝑥 ∣ 𝜑} = {𝑦 ∣ ∃𝑥(𝐴 = 𝑦 ∧ 𝜑)}
wrnf 37142	wff Ⅎ𝑥 ∈ 𝐴𝜑
df-bj-rnf 37143	⊢ (Ⅎ𝑥 ∈ 𝐴𝜑 ↔ (∃𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 𝜑))
bj-csngl 37166	class sngl 𝐴
df-bj-sngl 37167	⊢ sngl 𝐴 = {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = {𝑦}}
bj-ctag 37175	class tag 𝐴
df-bj-tag 37176	⊢ tag 𝐴 = (sngl 𝐴 ∪ {∅})
bj-cproj 37191	class (𝐴 Proj 𝐵)
df-bj-proj 37192	⊢ (𝐴 Proj 𝐵) = {𝑥 ∣ {𝑥} ∈ (𝐵 “ {𝐴})}
bj-c1upl 37198	class ⦅𝐴⦆
df-bj-1upl 37199	⊢ ⦅𝐴⦆ = ({∅} × tag 𝐴)
bj-cpr1 37201	class pr₁ 𝐴
df-bj-pr1 37202	⊢ pr₁ 𝐴 = (∅ Proj 𝐴)
bj-c2uple 37211	class ⦅𝐴, 𝐵⦆
df-bj-2upl 37212	⊢ ⦅𝐴, 𝐵⦆ = (⦅𝐴⦆ ∪ ({1_o} × tag 𝐵))
bj-cpr2 37215	class pr₂ 𝐴
df-bj-pr2 37216	⊢ pr₂ 𝐴 = (1_o Proj 𝐴)
ax-bj-sn 37234	⊢ ∀𝑥∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ 𝑧 = 𝑥)
ax-bj-bun 37238	⊢ ∀𝑥∀𝑦∃𝑧∀𝑡(𝑡 ∈ 𝑧 ↔ (𝑡 ∈ 𝑥 ∨ 𝑡 ∈ 𝑦))
ax-bj-adj 37243	⊢ ∀𝑥∀𝑦∃𝑧∀𝑡(𝑡 ∈ 𝑧 ↔ (𝑡 ∈ 𝑥 ∨ 𝑡 = 𝑦))
celwise 37280	class elwise
df-elwise 37281	⊢ elwise = (𝑜 ∈ V ↦ (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑧 ∣ ∃𝑢 ∈ 𝑥 ∃𝑣 ∈ 𝑦 𝑧 = (𝑢𝑜𝑣)}))
cmoore 37304	class Moore
df-bj-moore 37305	⊢ Moore = {𝑥 ∣ ∀𝑦 ∈ 𝒫 𝑥(∪ 𝑥 ∩ ∩ 𝑦) ∈ 𝑥}
cmpt3 37321	class (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵, 𝑧 ∈ 𝐶 ↦ 𝐷)
df-bj-mpt3 37322	⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵, 𝑧 ∈ 𝐶 ↦ 𝐷) = {⟨𝑠, 𝑡⟩ ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 (𝑠 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝑡 = 𝐷)}
csethom 37323	class Set⟶
df-bj-sethom 37324	⊢ Set⟶ = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓 ∣ 𝑓:𝑥⟶𝑦})
ctophom 37325	class Top⟶
df-bj-tophom 37326	⊢ Top⟶ = (𝑥 ∈ TopSp, 𝑦 ∈ TopSp ↦ {𝑓 ∈ ((Base‘𝑥) Set⟶ (Base‘𝑦)) ∣ ∀𝑢 ∈ (TopOpen‘𝑦)(^◡𝑓 “ 𝑢) ∈ (TopOpen‘𝑥)})
cmgmhom 37327	class Mgm⟶
df-bj-mgmhom 37328	⊢ Mgm⟶ = (𝑥 ∈ Mgm, 𝑦 ∈ Mgm ↦ {𝑓 ∈ ((Base‘𝑥) Set⟶ (Base‘𝑦)) ∣ ∀𝑢 ∈ (Base‘𝑥)∀𝑣 ∈ (Base‘𝑥)(𝑓‘(𝑢(+_g‘𝑥)𝑣)) = ((𝑓‘𝑢)(+_g‘𝑦)(𝑓‘𝑣))})
ctopmgmhom 37329	class TopMgm⟶
df-bj-topmgmhom 37330	⊢ TopMgm⟶ = (𝑥 ∈ TopMnd, 𝑦 ∈ TopMnd ↦ ((𝑥 Top⟶ 𝑦) ∩ (𝑥 Mgm⟶ 𝑦)))
ccur- 37331	class curry_
df-bj-cur 37332	⊢ curry_ = (𝑥 ∈ V, 𝑦 ∈ V, 𝑧 ∈ V ↦ (𝑓 ∈ ((𝑥 × 𝑦) Set⟶ 𝑧) ↦ (𝑎 ∈ 𝑥 ↦ (𝑏 ∈ 𝑦 ↦ (𝑓‘⟨𝑎, 𝑏⟩)))))
cunc- 37333	class uncurry_
df-bj-unc 37334	⊢ uncurry_ = (𝑥 ∈ V, 𝑦 ∈ V, 𝑧 ∈ V ↦ (𝑓 ∈ (𝑥 Set⟶ (𝑦 Set⟶ 𝑧)) ↦ (𝑎 ∈ 𝑥, 𝑏 ∈ 𝑦 ↦ ((𝑓‘𝑎)‘𝑏))))
cstrset 37335	class [𝐵 / 𝐴]_struct𝑆
df-strset 37336	⊢ [𝐵 / 𝐴]_struct𝑆 = ((𝑆 ↾ (V ∖ {(𝐴‘ndx)})) ∪ {⟨(𝐴‘ndx), 𝐵⟩})
cdiag2 37373	class Id
df-bj-diag 37374	⊢ Id = (𝑥 ∈ V ↦ ( I ↾ 𝑥))
cimdir 37379	class 𝒫_*
df-imdir 37380	⊢ 𝒫_* = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑟 ∈ 𝒫 (𝑎 × 𝑏) ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝑎 ∧ 𝑦 ⊆ 𝑏) ∧ (𝑟 “ 𝑥) = 𝑦)}))
ciminv 37392	class 𝒫^*
df-iminv 37393	⊢ 𝒫^* = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑟 ∈ 𝒫 (𝑎 × 𝑏) ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝑎 ∧ 𝑦 ⊆ 𝑏) ∧ 𝑥 = (^◡𝑟 “ 𝑦))}))
cfractemp 37397	class {_R
df-bj-fractemp 37398	⊢ {_R = (𝑥 ∈ R ↦ (℩𝑦 ∈ R ((𝑦 = 0_R ∨ (0_R <_R 𝑦 ∧ 𝑦 <_R 1_R)) ∧ ∃𝑛 ∈ ω ([⟨{𝑧 ∈ Q ∣ 𝑧 <_Q ⟨suc 𝑛, 1_o⟩}, 1_P⟩] ~_R +_R 𝑦) = 𝑥)))
cinftyexpitau 37399	class +∞e^iτ
df-bj-inftyexpitau 37400	⊢ +∞e^iτ = (𝑥 ∈ ℝ ↦ ⟨({_R‘(1^st ‘𝑥)), {R}⟩)
cccinftyN 37401	class ℂ_∞N
df-bj-ccinftyN 37402	⊢ ℂ_∞N = ran +∞e^iτ
chalf 37404	class 1/2
df-bj-onehalf 37405	⊢ 1/2 = (℩𝑥 ∈ R (𝑥 +_R 𝑥) = 1_R)
cinftyexpi 37407	class +∞eⁱ
df-bj-inftyexpi 37408	⊢ +∞eⁱ = (𝑥 ∈ (-π(,]π) ↦ ⟨𝑥, ℂ⟩)
cccinfty 37412	class ℂ_∞
df-bj-ccinfty 37413	⊢ ℂ_∞ = ran +∞eⁱ
cccbar 37416	class ℂ̅
df-bj-ccbar 37417	⊢ ℂ̅ = (ℂ ∪ ℂ_∞)
cpinfty 37420	class +∞
df-bj-pinfty 37421	⊢ +∞ = (+∞eⁱ‘0)
cminfty 37424	class -∞
df-bj-minfty 37425	⊢ -∞ = (+∞eⁱ‘π)
crrbar 37429	class ℝ̅
df-bj-rrbar 37430	⊢ ℝ̅ = (ℝ ∪ {-∞, +∞})
cinfty 37431	class ∞
df-bj-infty 37432	⊢ ∞ = 𝒫 ∪ ℂ
ccchat 37433	class ℂ̂
df-bj-cchat 37434	⊢ ℂ̂ = (ℂ ∪ {∞})
crrhat 37435	class ℝ̂
df-bj-rrhat 37436	⊢ ℝ̂ = (ℝ ∪ {∞})
caddcc 37438	class +_ℂ̅
df-bj-addc 37439	⊢ +_ℂ̅ = (𝑥 ∈ (((ℂ × ℂ̅) ∪ (ℂ̅ × ℂ)) ∪ ((ℂ̂ × ℂ̂) ∪ ( I ↾ ℂ_∞))) ↦ if(((1^st ‘𝑥) = ∞ ∨ (2^nd ‘𝑥) = ∞), ∞, if((1^st ‘𝑥) ∈ ℂ, if((2^nd ‘𝑥) ∈ ℂ, ⟨((1^st ‘(1^st ‘𝑥)) +_R (1^st ‘(2^nd ‘𝑥))), ((2^nd ‘(1^st ‘𝑥)) +_R (2^nd ‘(2^nd ‘𝑥)))⟩, (2^nd ‘𝑥)), (1^st ‘𝑥))))
coppcc 37440	class -_ℂ̅
df-bj-oppc 37441	⊢ -_ℂ̅ = (𝑥 ∈ (ℂ̅ ∪ ℂ̂) ↦ if(𝑥 = ∞, ∞, if(𝑥 ∈ ℂ, (℩𝑦 ∈ ℂ (𝑥 +_ℂ̅ 𝑦) = 0), (+∞e^iτ‘(𝑥 +_ℂ̅ ⟨1/2, 0_R⟩)))))
cltxr 37442	class <_ℝ̅
df-bj-lt 37443	⊢ <_ℝ̅ = ({𝑥 ∈ (ℝ̅ × ℝ̅) ∣ ∃𝑦∃𝑧(((1^st ‘𝑥) = ⟨𝑦, 0_R⟩ ∧ (2^nd ‘𝑥) = ⟨𝑧, 0_R⟩) ∧ 𝑦 <_R 𝑧)} ∪ ((({-∞} × ℝ) ∪ (ℝ × {+∞})) ∪ ({-∞} × {+∞})))
carg 37444	class Arg
df-bj-arg 37445	⊢ Arg = (𝑥 ∈ (ℂ̅ ∖ {0}) ↦ if(𝑥 ∈ ℂ, (ℑ‘(log‘𝑥)), if(𝑥<_ℝ̅0, π, (((1^st ‘𝑥) / (2 · π)) − π))))
cmulc 37446	class ·_ℂ̅
df-bj-mulc 37447	⊢ ·_ℂ̅ = (𝑥 ∈ ((ℂ̅ × ℂ̅) ∪ (ℂ̂ × ℂ̂)) ↦ if(((1^st ‘𝑥) = 0 ∨ (2^nd ‘𝑥) = 0), 0, if(((1^st ‘𝑥) = ∞ ∨ (2^nd ‘𝑥) = ∞), ∞, if(𝑥 ∈ (ℂ × ℂ), ((1^st ‘𝑥) · (2^nd ‘𝑥)), (+∞e^iτ‘(((Arg‘(1^st ‘𝑥)) +_ℂ̅ (Arg‘(2^nd ‘𝑥))) / τ))))))
cinvc 37448	class ^-1_ℂ̅
df-bj-invc 37449	⊢ ^-1_ℂ̅ = (𝑥 ∈ (ℂ̅ ∪ ℂ̂) ↦ if(𝑥 = 0, ∞, if(𝑥 ∈ ℂ, (℩𝑦 ∈ ℂ (𝑥 ·_ℂ̅ 𝑦) = 1), 0)))
ciomnn 37450	class i_ω↪ℕ
df-bj-iomnn 37451	⊢ i_ω↪ℕ = ((𝑛 ∈ ω ↦ ⟨[⟨{𝑟 ∈ Q ∣ 𝑟 <_Q ⟨suc 𝑛, 1_o⟩}, 1_P⟩] ~_R , 0_R⟩) ∪ {⟨ω, +∞⟩})
cnnbar 37461	class ℕ̅
df-bj-nnbar 37462	⊢ ℕ̅ = (ℕ₀ ∪ {+∞})
czzbar 37463	class ℤ̅
df-bj-zzbar 37464	⊢ ℤ̅ = (ℤ ∪ {-∞, +∞})
czzhat 37465	class ℤ̂
df-bj-zzhat 37466	⊢ ℤ̂ = (ℤ ∪ {∞})
cdivc 37467	class ∥_ℂ
df-bj-divc 37468	⊢ ∥_ℂ = {⟨𝑥, 𝑦⟩ ∣ (⟨𝑥, 𝑦⟩ ∈ ((ℂ̅ × ℂ̅) ∪ (ℂ̂ × ℂ̂)) ∧ ∃𝑛 ∈ (ℤ̅ ∪ ℤ̂)(𝑛 ·_ℂ̅ 𝑥) = 𝑦)}
cfinsum 37484	class FinSum
df-bj-finsum 37485	⊢ FinSum = (𝑥 ∈ {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ CMnd ∧ ∃𝑡 ∈ Fin 𝑧:𝑡⟶(Base‘𝑦))} ↦ (℩𝑠∃𝑚 ∈ ℕ₀ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→dom (2^nd ‘𝑥) ∧ 𝑠 = (seq1((+_g‘(1^st ‘𝑥)), (𝑛 ∈ ℕ ↦ ((2^nd ‘𝑥)‘(𝑓‘𝑛))))‘𝑚))))
crrvec 37493	class ℝ-Vec
df-bj-rvec 37494	⊢ ℝ-Vec = (LMod ∩ (^◡Scalar “ {ℝ_fld}))
cend 37514	class End
df-bj-end 37515	⊢ End = (𝑐 ∈ Cat ↦ (𝑥 ∈ (Base‘𝑐) ↦ {⟨(Base‘ndx), (𝑥(Hom ‘𝑐)𝑥)⟩, ⟨(+_g‘ndx), (⟨𝑥, 𝑥⟩(comp‘𝑐)𝑥)⟩}))
cfinxp 37584	class (𝑈↑↑𝑁)
df-finxp 37585	⊢ (𝑈↑↑𝑁) = {𝑦 ∣ (𝑁 ∈ ω ∧ ∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1_o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1^st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨𝑁, 𝑦⟩)‘𝑁))}
ax-luk1 37620	⊢ ((𝜑 → 𝜓) → ((𝜓 → 𝜒) → (𝜑 → 𝜒)))
ax-luk2 37621	⊢ ((¬ 𝜑 → 𝜑) → 𝜑)
ax-luk3 37622	⊢ (𝜑 → (¬ 𝜑 → 𝜓))
ax-wl-13v 37694	⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
ax-wl-cleq 37703	⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
ax-wl-clel 37704	⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵))
ctotbnd 37963	class TotBnd
cbnd 37964	class Bnd
df-totbnd 37965	⊢ TotBnd = (𝑥 ∈ V ↦ {𝑚 ∈ (Met‘𝑥) ∣ ∀𝑑 ∈ ℝ⁺ ∃𝑣 ∈ Fin (∪ 𝑣 = 𝑥 ∧ ∀𝑏 ∈ 𝑣 ∃𝑦 ∈ 𝑥 𝑏 = (𝑦(ball‘𝑚)𝑑))})
df-bnd 37976	⊢ Bnd = (𝑥 ∈ V ↦ {𝑚 ∈ (Met‘𝑥) ∣ ∀𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ⁺ 𝑥 = (𝑦(ball‘𝑚)𝑟)})
cismty 37995	class Ismty
df-ismty 37996	⊢ Ismty = (𝑚 ∈ ∪ ran ∞Met, 𝑛 ∈ ∪ ran ∞Met ↦ {𝑓 ∣ (𝑓:dom dom 𝑚–1-1-onto→dom dom 𝑛 ∧ ∀𝑥 ∈ dom dom 𝑚∀𝑦 ∈ dom dom 𝑚(𝑥𝑚𝑦) = ((𝑓‘𝑥)𝑛(𝑓‘𝑦)))})
crrn 38022	class ℝⁿ
df-rrn 38023	⊢ ℝⁿ = (𝑖 ∈ Fin ↦ (𝑥 ∈ (ℝ ↑_m 𝑖), 𝑦 ∈ (ℝ ↑_m 𝑖) ↦ (√‘Σ𝑘 ∈ 𝑖 (((𝑥‘𝑘) − (𝑦‘𝑘))↑2))))
cass 38039	class Ass
df-ass 38040	⊢ Ass = {𝑔 ∣ ∀𝑥 ∈ dom dom 𝑔∀𝑦 ∈ dom dom 𝑔∀𝑧 ∈ dom dom 𝑔((𝑥𝑔𝑦)𝑔𝑧) = (𝑥𝑔(𝑦𝑔𝑧))}
cexid 38041	class ExId
df-exid 38042	⊢ ExId = {𝑔 ∣ ∃𝑥 ∈ dom dom 𝑔∀𝑦 ∈ dom dom 𝑔((𝑥𝑔𝑦) = 𝑦 ∧ (𝑦𝑔𝑥) = 𝑦)}
cmagm 38045	class Magma
df-mgmOLD 38046	⊢ Magma = {𝑔 ∣ ∃𝑡 𝑔:(𝑡 × 𝑡)⟶𝑡}
csem 38057	class SemiGrp
df-sgrOLD 38058	⊢ SemiGrp = (Magma ∩ Ass)
cmndo 38063	class MndOp
df-mndo 38064	⊢ MndOp = (SemiGrp ∩ ExId )
cghomOLD 38080	class GrpOpHom
df-ghomOLD 38081	⊢ GrpOpHom = (𝑔 ∈ GrpOp, ℎ ∈ GrpOp ↦ {𝑓 ∣ (𝑓:ran 𝑔⟶ran ℎ ∧ ∀𝑥 ∈ ran 𝑔∀𝑦 ∈ ran 𝑔((𝑓‘𝑥)ℎ(𝑓‘𝑦)) = (𝑓‘(𝑥𝑔𝑦)))})
crngo 38091	class RingOps
df-rngo 38092	⊢ RingOps = {⟨𝑔, ℎ⟩ ∣ ((𝑔 ∈ AbelOp ∧ ℎ:(ran 𝑔 × ran 𝑔)⟶ran 𝑔) ∧ (∀𝑥 ∈ ran 𝑔∀𝑦 ∈ ran 𝑔∀𝑧 ∈ ran 𝑔(((𝑥ℎ𝑦)ℎ𝑧) = (𝑥ℎ(𝑦ℎ𝑧)) ∧ (𝑥ℎ(𝑦𝑔𝑧)) = ((𝑥ℎ𝑦)𝑔(𝑥ℎ𝑧)) ∧ ((𝑥𝑔𝑦)ℎ𝑧) = ((𝑥ℎ𝑧)𝑔(𝑦ℎ𝑧))) ∧ ∃𝑥 ∈ ran 𝑔∀𝑦 ∈ ran 𝑔((𝑥ℎ𝑦) = 𝑦 ∧ (𝑦ℎ𝑥) = 𝑦)))}
cdrng 38145	class DivRingOps
df-drngo 38146	⊢ DivRingOps = {⟨𝑔, ℎ⟩ ∣ (⟨𝑔, ℎ⟩ ∈ RingOps ∧ (ℎ ↾ ((ran 𝑔 ∖ {(GId‘𝑔)}) × (ran 𝑔 ∖ {(GId‘𝑔)}))) ∈ GrpOp)}
crngohom 38157	class RingOpsHom
crngoiso 38158	class RingOpsIso
crisc 38159	class ≃_𝑟
df-rngohom 38160	⊢ RingOpsHom = (𝑟 ∈ RingOps, 𝑠 ∈ RingOps ↦ {𝑓 ∈ (ran (1^st ‘𝑠) ↑_m ran (1^st ‘𝑟)) ∣ ((𝑓‘(GId‘(2^nd ‘𝑟))) = (GId‘(2^nd ‘𝑠)) ∧ ∀𝑥 ∈ ran (1^st ‘𝑟)∀𝑦 ∈ ran (1^st ‘𝑟)((𝑓‘(𝑥(1^st ‘𝑟)𝑦)) = ((𝑓‘𝑥)(1^st ‘𝑠)(𝑓‘𝑦)) ∧ (𝑓‘(𝑥(2^nd ‘𝑟)𝑦)) = ((𝑓‘𝑥)(2^nd ‘𝑠)(𝑓‘𝑦))))})
df-rngoiso 38173	⊢ RingOpsIso = (𝑟 ∈ RingOps, 𝑠 ∈ RingOps ↦ {𝑓 ∈ (𝑟 RingOpsHom 𝑠) ∣ 𝑓:ran (1^st ‘𝑟)–1-1-onto→ran (1^st ‘𝑠)})
df-risc 38180	⊢ ≃_𝑟 = {⟨𝑟, 𝑠⟩ ∣ ((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑟 RingOpsIso 𝑠))}
ccm2 38186	class Com2
df-com2 38187	⊢ Com2 = {⟨𝑔, ℎ⟩ ∣ ∀𝑎 ∈ ran 𝑔∀𝑏 ∈ ran 𝑔(𝑎ℎ𝑏) = (𝑏ℎ𝑎)}
cfld 38188	class Fld
df-fld 38189	⊢ Fld = (DivRingOps ∩ Com2)
ccring 38190	class CRingOps
df-crngo 38191	⊢ CRingOps = (RingOps ∩ Com2)
cidl 38204	class Idl
cpridl 38205	class PrIdl
cmaxidl 38206	class MaxIdl
df-idl 38207	⊢ Idl = (𝑟 ∈ RingOps ↦ {𝑖 ∈ 𝒫 ran (1^st ‘𝑟) ∣ ((GId‘(1^st ‘𝑟)) ∈ 𝑖 ∧ ∀𝑥 ∈ 𝑖 (∀𝑦 ∈ 𝑖 (𝑥(1^st ‘𝑟)𝑦) ∈ 𝑖 ∧ ∀𝑧 ∈ ran (1^st ‘𝑟)((𝑧(2^nd ‘𝑟)𝑥) ∈ 𝑖 ∧ (𝑥(2^nd ‘𝑟)𝑧) ∈ 𝑖)))})
df-pridl 38208	⊢ PrIdl = (𝑟 ∈ RingOps ↦ {𝑖 ∈ (Idl‘𝑟) ∣ (𝑖 ≠ ran (1^st ‘𝑟) ∧ ∀𝑎 ∈ (Idl‘𝑟)∀𝑏 ∈ (Idl‘𝑟)(∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 (𝑥(2^nd ‘𝑟)𝑦) ∈ 𝑖 → (𝑎 ⊆ 𝑖 ∨ 𝑏 ⊆ 𝑖)))})
df-maxidl 38209	⊢ MaxIdl = (𝑟 ∈ RingOps ↦ {𝑖 ∈ (Idl‘𝑟) ∣ (𝑖 ≠ ran (1^st ‘𝑟) ∧ ∀𝑗 ∈ (Idl‘𝑟)(𝑖 ⊆ 𝑗 → (𝑗 = 𝑖 ∨ 𝑗 = ran (1^st ‘𝑟))))})
cprrng 38243	class PrRing
cdmn 38244	class Dmn
df-prrngo 38245	⊢ PrRing = {𝑟 ∈ RingOps ∣ {(GId‘(1^st ‘𝑟))} ∈ (PrIdl‘𝑟)}
df-dmn 38246	⊢ Dmn = (PrRing ∩ Com2)
cigen 38256	class IdlGen
df-igen 38257	⊢ IdlGen = (𝑟 ∈ RingOps, 𝑠 ∈ 𝒫 ran (1^st ‘𝑟) ↦ ∩ {𝑗 ∈ (Idl‘𝑟) ∣ 𝑠 ⊆ 𝑗})
cxrn 38371	class (𝐴 ⋉ 𝐵)
cadjliftmap 38372	class (𝑅 AdjLiftMap 𝐴)
cblockliftmap 38373	class (𝑅 BlockLiftMap 𝐴)
csucmap 38374	class SucMap
csuccl 38375	class Suc
cpre 38376	class pre 𝑁
cblockliftfix 38377	class BlockLiftFix
cshiftstable 38378	class (𝑆 ShiftStable 𝐹)
ccoss 38379	class ≀ 𝑅
ccoels 38380	class ∼ 𝐴
crels 38381	class Rels
cssr 38382	class S
crefs 38383	class Refs
crefrels 38384	class RefRels
wrefrel 38385	wff RefRel 𝑅
ccnvrefs 38386	class CnvRefs
ccnvrefrels 38387	class CnvRefRels
wcnvrefrel 38388	wff CnvRefRel 𝑅
csyms 38389	class Syms
csymrels 38390	class SymRels
wsymrel 38391	wff SymRel 𝑅
ctrs 38392	class Trs
ctrrels 38393	class TrRels
wtrrel 38394	wff TrRel 𝑅
ceqvrels 38395	class EqvRels
weqvrel 38396	wff EqvRel 𝑅
ccoeleqvrels 38397	class CoElEqvRels
wcoeleqvrel 38398	wff CoElEqvRel 𝐴
credunds 38399	class Redunds
wredund 38400	wff 𝐴 Redund ⟨𝐵, 𝐶⟩
wredundp 38401	wff redund (𝜑, 𝜓, 𝜒)
cdmqss 38402	class DomainQss
wdmqs 38403	wff 𝑅 DomainQs 𝐴
cers 38404	class Ers
werALTV 38405	wff 𝑅 ErALTV 𝐴
ccomembers 38406	class CoMembErs
wcomember 38407	wff CoMembEr 𝐴
cfunss 38408	class Funss
cfunsALTV 38409	class FunsALTV
wfunALTV 38410	wff FunALTV 𝐹
cdisjss 38411	class Disjss
cdisjs 38412	class Disjs
wdisjALTV 38413	wff Disj 𝑅
celdisjs 38414	class ElDisjs
weldisj 38415	wff ElDisj 𝐴
wantisymrel 38416	wff AntisymRel 𝑅
cparts 38417	class Parts
wpart 38418	wff 𝑅 Part 𝐴
cmembparts 38419	class MembParts
wmembpart 38420	wff MembPart 𝐴
df-xrn 38561	⊢ (𝐴 ⋉ 𝐵) = ((^◡(1^st ↾ (V × V)) ∘ 𝐴) ∩ (^◡(2^nd ↾ (V × V)) ∘ 𝐵))
df-rels 38621	⊢ Rels = 𝒫 (V × V)
df-adjliftmap 38627	⊢ (𝑅 AdjLiftMap 𝐴) = (𝑚 ∈ dom ((𝑅 ∪ ^◡ E ) ↾ 𝐴) ↦ [𝑚]((𝑅 ∪ ^◡ E ) ↾ 𝐴))
df-blockliftmap 38630	⊢ (𝑅 BlockLiftMap 𝐴) = (𝑚 ∈ dom (𝑅 ⋉ (^◡ E ↾ 𝐴)) ↦ [𝑚](𝑅 ⋉ (^◡ E ↾ 𝐴)))
df-sucmap 38632	⊢ SucMap = {⟨𝑚, 𝑛⟩ ∣ suc 𝑚 = 𝑛}
df-succl 38639	⊢ Suc = ran SucMap
df-pre 38645	⊢ pre 𝑁 = (℩𝑚𝑚 ∈ Pred( SucMap , dom SucMap , 𝑁))
df-blockliftfix 38651	⊢ BlockLiftFix = {⟨𝑟, 𝑎⟩ ∣ (𝑟 ∈ Rels ∧ (dom (𝑟 ⋉ (^◡ E ↾ 𝑎)) / (𝑟 ⋉ (^◡ E ↾ 𝑎))) = 𝑎)}
df-shiftstable 38652	⊢ (𝑆 ShiftStable 𝐹) = ((𝑆 ∘ 𝐹) ∩ 𝐹)
df-coss 38670	⊢ ≀ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)}
df-coels 38671	⊢ ∼ 𝐴 = ≀ (^◡ E ↾ 𝐴)
df-ssr 38747	⊢ S = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ⊆ 𝑦}
df-refs 38759	⊢ Refs = {𝑥 ∣ ( I ∩ (dom 𝑥 × ran 𝑥)) S (𝑥 ∩ (dom 𝑥 × ran 𝑥))}
df-refrels 38760	⊢ RefRels = ( Refs ∩ Rels )
df-refrel 38761	⊢ ( RefRel 𝑅 ↔ (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅))
df-cnvrefs 38774	⊢ CnvRefs = {𝑥 ∣ ( I ∩ (dom 𝑥 × ran 𝑥))^◡ S (𝑥 ∩ (dom 𝑥 × ran 𝑥))}
df-cnvrefrels 38775	⊢ CnvRefRels = ( CnvRefs ∩ Rels )
df-cnvrefrel 38776	⊢ ( CnvRefRel 𝑅 ↔ ((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ ( I ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅))
df-syms 38791	⊢ Syms = {𝑥 ∣ ^◡(𝑥 ∩ (dom 𝑥 × ran 𝑥)) S (𝑥 ∩ (dom 𝑥 × ran 𝑥))}
df-symrels 38792	⊢ SymRels = ( Syms ∩ Rels )
df-symrel 38793	⊢ ( SymRel 𝑅 ↔ (^◡(𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅))
df-trs 38825	⊢ Trs = {𝑥 ∣ ((𝑥 ∩ (dom 𝑥 × ran 𝑥)) ∘ (𝑥 ∩ (dom 𝑥 × ran 𝑥))) S (𝑥 ∩ (dom 𝑥 × ran 𝑥))}
df-trrels 38826	⊢ TrRels = ( Trs ∩ Rels )
df-trrel 38827	⊢ ( TrRel 𝑅 ↔ (((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∘ (𝑅 ∩ (dom 𝑅 × ran 𝑅))) ⊆ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅))
df-eqvrels 38837	⊢ EqvRels = (( RefRels ∩ SymRels ) ∩ TrRels )
df-eqvrel 38838	⊢ ( EqvRel 𝑅 ↔ ( RefRel 𝑅 ∧ SymRel 𝑅 ∧ TrRel 𝑅))
df-coeleqvrels 38839	⊢ CoElEqvRels = {𝑎 ∣ ≀ (^◡ E ↾ 𝑎) ∈ EqvRels }
df-coeleqvrel 38840	⊢ ( CoElEqvRel 𝐴 ↔ EqvRel ≀ (^◡ E ↾ 𝐴))
df-redunds 38876	⊢ Redunds = ^◡{⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ (𝑥 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑧) = (𝑦 ∩ 𝑧))}
df-redund 38877	⊢ (𝐴 Redund ⟨𝐵, 𝐶⟩ ↔ (𝐴 ⊆ 𝐵 ∧ (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐶)))
df-redundp 38878	⊢ ( redund (𝜑, 𝜓, 𝜒) ↔ ((𝜑 → 𝜓) ∧ ((𝜑 ∧ 𝜒) ↔ (𝜓 ∧ 𝜒))))
df-dmqss 38891	⊢ DomainQss = {⟨𝑥, 𝑦⟩ ∣ (dom 𝑥 / 𝑥) = 𝑦}
df-dmqs 38892	⊢ (𝑅 DomainQs 𝐴 ↔ (dom 𝑅 / 𝑅) = 𝐴)
df-ers 38918	⊢ Ers = ( DomainQss ↾ EqvRels )
df-erALTV 38919	⊢ (𝑅 ErALTV 𝐴 ↔ ( EqvRel 𝑅 ∧ 𝑅 DomainQs 𝐴))
df-comembers 38920	⊢ CoMembErs = {𝑎 ∣ ≀ (^◡ E ↾ 𝑎) Ers 𝑎}
df-comember 38921	⊢ ( CoMembEr 𝐴 ↔ ≀ (^◡ E ↾ 𝐴) ErALTV 𝐴)
df-funss 38935	⊢ Funss = {𝑥 ∣ ≀ 𝑥 ∈ CnvRefRels }
df-funsALTV 38936	⊢ FunsALTV = ( Funss ∩ Rels )
df-funALTV 38937	⊢ ( FunALTV 𝐹 ↔ ( CnvRefRel ≀ 𝐹 ∧ Rel 𝐹))
df-disjss 38958	⊢ Disjss = {𝑥 ∣ ≀ ^◡𝑥 ∈ CnvRefRels }
df-disjs 38959	⊢ Disjs = ( Disjss ∩ Rels )
df-disjALTV 38960	⊢ ( Disj 𝑅 ↔ ( CnvRefRel ≀ ^◡𝑅 ∧ Rel 𝑅))
df-eldisjs 38961	⊢ ElDisjs = {𝑎 ∣ (^◡ E ↾ 𝑎) ∈ Disjs }
df-eldisj 38962	⊢ ( ElDisj 𝐴 ↔ Disj (^◡ E ↾ 𝐴))
df-antisymrel 39015	⊢ ( AntisymRel 𝑅 ↔ ( CnvRefRel (𝑅 ∩ ^◡𝑅) ∧ Rel 𝑅))
df-parts 39020	⊢ Parts = ( DomainQss ↾ Disjs )
df-part 39021	⊢ (𝑅 Part 𝐴 ↔ ( Disj 𝑅 ∧ 𝑅 DomainQs 𝐴))
df-membparts 39022	⊢ MembParts = {𝑎 ∣ (^◡ E ↾ 𝑎) Parts 𝑎}
df-membpart 39023	⊢ ( MembPart 𝐴 ↔ (^◡ E ↾ 𝐴) Part 𝐴)
wprt 39127	wff Prt 𝐴
df-prt 39128	⊢ (Prt 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅))
ax-c5 39139	⊢ (∀𝑥𝜑 → 𝜑)
ax-c4 39140	⊢ (∀𝑥(∀𝑥𝜑 → 𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))
ax-c7 39141	⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜑 → 𝜑)
ax-c10 39142	⊢ (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → 𝜑)
ax-c11 39143	⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑))
ax-c11n 39144	⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
ax-c15 39145	⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))))
ax-c9 39146	⊢ (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))
ax-c14 39147	⊢ (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥 ∈ 𝑦 → ∀𝑧 𝑥 ∈ 𝑦)))
ax-c16 39148	⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))
ax-riotaBAD 39209	⊢ (℩𝑥 ∈ 𝐴 𝜑) = if(∃!𝑥 ∈ 𝐴 𝜑, (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)), (Undef‘{𝑥 ∣ 𝑥 ∈ 𝐴}))
clsa 39230	class LSAtoms
clsh 39231	class LSHyp
df-lsatoms 39232	⊢ LSAtoms = (𝑤 ∈ V ↦ ran (𝑣 ∈ ((Base‘𝑤) ∖ {(0_g‘𝑤)}) ↦ ((LSpan‘𝑤)‘{𝑣})))
df-lshyp 39233	⊢ LSHyp = (𝑤 ∈ V ↦ {𝑠 ∈ (LSubSp‘𝑤) ∣ (𝑠 ≠ (Base‘𝑤) ∧ ∃𝑣 ∈ (Base‘𝑤)((LSpan‘𝑤)‘(𝑠 ∪ {𝑣})) = (Base‘𝑤))})
clcv 39274	class ⋖_L
df-lcv 39275	⊢ ⋖_L = (𝑤 ∈ V ↦ {⟨𝑡, 𝑢⟩ ∣ ((𝑡 ∈ (LSubSp‘𝑤) ∧ 𝑢 ∈ (LSubSp‘𝑤)) ∧ (𝑡 ⊊ 𝑢 ∧ ¬ ∃𝑠 ∈ (LSubSp‘𝑤)(𝑡 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑢)))})
clfn 39313	class LFnl
df-lfl 39314	⊢ LFnl = (𝑤 ∈ V ↦ {𝑓 ∈ ((Base‘(Scalar‘𝑤)) ↑_m (Base‘𝑤)) ∣ ∀𝑟 ∈ (Base‘(Scalar‘𝑤))∀𝑥 ∈ (Base‘𝑤)∀𝑦 ∈ (Base‘𝑤)(𝑓‘((𝑟( ·_𝑠 ‘𝑤)𝑥)(+_g‘𝑤)𝑦)) = ((𝑟(._r‘(Scalar‘𝑤))(𝑓‘𝑥))(+_g‘(Scalar‘𝑤))(𝑓‘𝑦))})
clk 39341	class LKer
df-lkr 39342	⊢ LKer = (𝑤 ∈ V ↦ (𝑓 ∈ (LFnl‘𝑤) ↦ (^◡𝑓 “ {(0_g‘(Scalar‘𝑤))})))
cld 39379	class LDual
df-ldual 39380	⊢ LDual = (𝑣 ∈ V ↦ ({⟨(Base‘ndx), (LFnl‘𝑣)⟩, ⟨(+_g‘ndx), ( ∘_f (+_g‘(Scalar‘𝑣)) ↾ ((LFnl‘𝑣) × (LFnl‘𝑣)))⟩, ⟨(Scalar‘ndx), (opp_r‘(Scalar‘𝑣))⟩} ∪ {⟨( ·_𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑣)), 𝑓 ∈ (LFnl‘𝑣) ↦ (𝑓 ∘_f (._r‘(Scalar‘𝑣))((Base‘𝑣) × {𝑘})))⟩}))
cops 39428	class OP
ccmtN 39429	class cm
col 39430	class OL
coml 39431	class OML
df-oposet 39432	⊢ OP = {𝑝 ∈ Poset ∣ (((Base‘𝑝) ∈ dom (lub‘𝑝) ∧ (Base‘𝑝) ∈ dom (glb‘𝑝)) ∧ ∃𝑜(𝑜 = (oc‘𝑝) ∧ ∀𝑎 ∈ (Base‘𝑝)∀𝑏 ∈ (Base‘𝑝)(((𝑜‘𝑎) ∈ (Base‘𝑝) ∧ (𝑜‘(𝑜‘𝑎)) = 𝑎 ∧ (𝑎(le‘𝑝)𝑏 → (𝑜‘𝑏)(le‘𝑝)(𝑜‘𝑎))) ∧ (𝑎(join‘𝑝)(𝑜‘𝑎)) = (1.‘𝑝) ∧ (𝑎(meet‘𝑝)(𝑜‘𝑎)) = (0.‘𝑝))))}
df-cmtN 39433	⊢ cm = (𝑝 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑝) ∧ 𝑦 ∈ (Base‘𝑝) ∧ 𝑥 = ((𝑥(meet‘𝑝)𝑦)(join‘𝑝)(𝑥(meet‘𝑝)((oc‘𝑝)‘𝑦))))})
df-ol 39434	⊢ OL = (Lat ∩ OP)
df-oml 39435	⊢ OML = {𝑙 ∈ OL ∣ ∀𝑎 ∈ (Base‘𝑙)∀𝑏 ∈ (Base‘𝑙)(𝑎(le‘𝑙)𝑏 → 𝑏 = (𝑎(join‘𝑙)(𝑏(meet‘𝑙)((oc‘𝑙)‘𝑎))))}
ccvr 39518	class ⋖
catm 39519	class Atoms
cal 39520	class AtLat
clc 39521	class CvLat
df-covers 39522	⊢ ⋖ = (𝑝 ∈ V ↦ {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (Base‘𝑝) ∧ 𝑏 ∈ (Base‘𝑝)) ∧ 𝑎(lt‘𝑝)𝑏 ∧ ¬ ∃𝑧 ∈ (Base‘𝑝)(𝑎(lt‘𝑝)𝑧 ∧ 𝑧(lt‘𝑝)𝑏))})
df-ats 39523	⊢ Atoms = (𝑝 ∈ V ↦ {𝑎 ∈ (Base‘𝑝) ∣ (0.‘𝑝)( ⋖ ‘𝑝)𝑎})
df-atl 39554	⊢ AtLat = {𝑘 ∈ Lat ∣ ((Base‘𝑘) ∈ dom (glb‘𝑘) ∧ ∀𝑥 ∈ (Base‘𝑘)(𝑥 ≠ (0.‘𝑘) → ∃𝑝 ∈ (Atoms‘𝑘)𝑝(le‘𝑘)𝑥))}
df-cvlat 39578	⊢ CvLat = {𝑘 ∈ AtLat ∣ ∀𝑎 ∈ (Atoms‘𝑘)∀𝑏 ∈ (Atoms‘𝑘)∀𝑐 ∈ (Base‘𝑘)((¬ 𝑎(le‘𝑘)𝑐 ∧ 𝑎(le‘𝑘)(𝑐(join‘𝑘)𝑏)) → 𝑏(le‘𝑘)(𝑐(join‘𝑘)𝑎))}
chlt 39606	class HL
df-hlat 39607	⊢ HL = {𝑙 ∈ ((OML ∩ CLat) ∩ CvLat) ∣ (∀𝑎 ∈ (Atoms‘𝑙)∀𝑏 ∈ (Atoms‘𝑙)(𝑎 ≠ 𝑏 → ∃𝑐 ∈ (Atoms‘𝑙)(𝑐 ≠ 𝑎 ∧ 𝑐 ≠ 𝑏 ∧ 𝑐(le‘𝑙)(𝑎(join‘𝑙)𝑏))) ∧ ∃𝑎 ∈ (Base‘𝑙)∃𝑏 ∈ (Base‘𝑙)∃𝑐 ∈ (Base‘𝑙)(((0.‘𝑙)(lt‘𝑙)𝑎 ∧ 𝑎(lt‘𝑙)𝑏) ∧ (𝑏(lt‘𝑙)𝑐 ∧ 𝑐(lt‘𝑙)(1.‘𝑙))))}
clln 39747	class LLines
clpl 39748	class LPlanes
clvol 39749	class LVols
clines 39750	class Lines
cpointsN 39751	class Points
cpsubsp 39752	class PSubSp
cpmap 39753	class pmap
df-llines 39754	⊢ LLines = (𝑘 ∈ V ↦ {𝑥 ∈ (Base‘𝑘) ∣ ∃𝑝 ∈ (Atoms‘𝑘)𝑝( ⋖ ‘𝑘)𝑥})
df-lplanes 39755	⊢ LPlanes = (𝑘 ∈ V ↦ {𝑥 ∈ (Base‘𝑘) ∣ ∃𝑝 ∈ (LLines‘𝑘)𝑝( ⋖ ‘𝑘)𝑥})
df-lvols 39756	⊢ LVols = (𝑘 ∈ V ↦ {𝑥 ∈ (Base‘𝑘) ∣ ∃𝑝 ∈ (LPlanes‘𝑘)𝑝( ⋖ ‘𝑘)𝑥})
df-lines 39757	⊢ Lines = (𝑘 ∈ V ↦ {𝑠 ∣ ∃𝑞 ∈ (Atoms‘𝑘)∃𝑟 ∈ (Atoms‘𝑘)(𝑞 ≠ 𝑟 ∧ 𝑠 = {𝑝 ∈ (Atoms‘𝑘) ∣ 𝑝(le‘𝑘)(𝑞(join‘𝑘)𝑟)})})
df-pointsN 39758	⊢ Points = (𝑘 ∈ V ↦ {𝑞 ∣ ∃𝑝 ∈ (Atoms‘𝑘)𝑞 = {𝑝}})
df-psubsp 39759	⊢ PSubSp = (𝑘 ∈ V ↦ {𝑠 ∣ (𝑠 ⊆ (Atoms‘𝑘) ∧ ∀𝑝 ∈ 𝑠 ∀𝑞 ∈ 𝑠 ∀𝑟 ∈ (Atoms‘𝑘)(𝑟(le‘𝑘)(𝑝(join‘𝑘)𝑞) → 𝑟 ∈ 𝑠))})
df-pmap 39760	⊢ pmap = (𝑘 ∈ V ↦ (𝑎 ∈ (Base‘𝑘) ↦ {𝑝 ∈ (Atoms‘𝑘) ∣ 𝑝(le‘𝑘)𝑎}))
cpadd 40051	class +_𝑃
df-padd 40052	⊢ +_𝑃 = (𝑙 ∈ V ↦ (𝑚 ∈ 𝒫 (Atoms‘𝑙), 𝑛 ∈ 𝒫 (Atoms‘𝑙) ↦ ((𝑚 ∪ 𝑛) ∪ {𝑝 ∈ (Atoms‘𝑙) ∣ ∃𝑞 ∈ 𝑚 ∃𝑟 ∈ 𝑛 𝑝(le‘𝑙)(𝑞(join‘𝑙)𝑟)})))
cpclN 40143	class PCl
df-pclN 40144	⊢ PCl = (𝑘 ∈ V ↦ (𝑥 ∈ 𝒫 (Atoms‘𝑘) ↦ ∩ {𝑦 ∈ (PSubSp‘𝑘) ∣ 𝑥 ⊆ 𝑦}))
cpolN 40158	class ⊥_𝑃
df-polarityN 40159	⊢ ⊥_𝑃 = (𝑙 ∈ V ↦ (𝑚 ∈ 𝒫 (Atoms‘𝑙) ↦ ((Atoms‘𝑙) ∩ ∩ 𝑝 ∈ 𝑚 ((pmap‘𝑙)‘((oc‘𝑙)‘𝑝)))))
cpscN 40190	class PSubCl
df-psubclN 40191	⊢ PSubCl = (𝑘 ∈ V ↦ {𝑠 ∣ (𝑠 ⊆ (Atoms‘𝑘) ∧ ((⊥_𝑃‘𝑘)‘((⊥_𝑃‘𝑘)‘𝑠)) = 𝑠)})
clh 40240	class LHyp
claut 40241	class LAut
cwpointsN 40242	class WAtoms
cpautN 40243	class PAut
df-lhyp 40244	⊢ LHyp = (𝑘 ∈ V ↦ {𝑥 ∈ (Base‘𝑘) ∣ 𝑥( ⋖ ‘𝑘)(1.‘𝑘)})
df-laut 40245	⊢ LAut = (𝑘 ∈ V ↦ {𝑓 ∣ (𝑓:(Base‘𝑘)–1-1-onto→(Base‘𝑘) ∧ ∀𝑥 ∈ (Base‘𝑘)∀𝑦 ∈ (Base‘𝑘)(𝑥(le‘𝑘)𝑦 ↔ (𝑓‘𝑥)(le‘𝑘)(𝑓‘𝑦)))})
df-watsN 40246	⊢ WAtoms = (𝑘 ∈ V ↦ (𝑑 ∈ (Atoms‘𝑘) ↦ ((Atoms‘𝑘) ∖ ((⊥_𝑃‘𝑘)‘{𝑑}))))
df-pautN 40247	⊢ PAut = (𝑘 ∈ V ↦ {𝑓 ∣ (𝑓:(PSubSp‘𝑘)–1-1-onto→(PSubSp‘𝑘) ∧ ∀𝑥 ∈ (PSubSp‘𝑘)∀𝑦 ∈ (PSubSp‘𝑘)(𝑥 ⊆ 𝑦 ↔ (𝑓‘𝑥) ⊆ (𝑓‘𝑦)))})
cldil 40356	class LDil
cltrn 40357	class LTrn
cdilN 40358	class Dil
ctrnN 40359	class Trn
df-ldil 40360	⊢ LDil = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ {𝑓 ∈ (LAut‘𝑘) ∣ ∀𝑥 ∈ (Base‘𝑘)(𝑥(le‘𝑘)𝑤 → (𝑓‘𝑥) = 𝑥)}))
df-ltrn 40361	⊢ LTrn = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ {𝑓 ∈ ((LDil‘𝑘)‘𝑤) ∣ ∀𝑝 ∈ (Atoms‘𝑘)∀𝑞 ∈ (Atoms‘𝑘)((¬ 𝑝(le‘𝑘)𝑤 ∧ ¬ 𝑞(le‘𝑘)𝑤) → ((𝑝(join‘𝑘)(𝑓‘𝑝))(meet‘𝑘)𝑤) = ((𝑞(join‘𝑘)(𝑓‘𝑞))(meet‘𝑘)𝑤))}))
df-dilN 40362	⊢ Dil = (𝑘 ∈ V ↦ (𝑑 ∈ (Atoms‘𝑘) ↦ {𝑓 ∈ (PAut‘𝑘) ∣ ∀𝑥 ∈ (PSubSp‘𝑘)(𝑥 ⊆ ((WAtoms‘𝑘)‘𝑑) → (𝑓‘𝑥) = 𝑥)}))
df-trnN 40363	⊢ Trn = (𝑘 ∈ V ↦ (𝑑 ∈ (Atoms‘𝑘) ↦ {𝑓 ∈ ((Dil‘𝑘)‘𝑑) ∣ ∀𝑞 ∈ ((WAtoms‘𝑘)‘𝑑)∀𝑟 ∈ ((WAtoms‘𝑘)‘𝑑)((𝑞(+_𝑃‘𝑘)(𝑓‘𝑞)) ∩ ((⊥_𝑃‘𝑘)‘{𝑑})) = ((𝑟(+_𝑃‘𝑘)(𝑓‘𝑟)) ∩ ((⊥_𝑃‘𝑘)‘{𝑑}))}))
ctrl 40414	class trL
df-trl 40415	⊢ trL = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑓 ∈ ((LTrn‘𝑘)‘𝑤) ↦ (℩𝑥 ∈ (Base‘𝑘)∀𝑝 ∈ (Atoms‘𝑘)(¬ 𝑝(le‘𝑘)𝑤 → 𝑥 = ((𝑝(join‘𝑘)(𝑓‘𝑝))(meet‘𝑘)𝑤))))))
ctgrp 40998	class TGrp
df-tgrp 40999	⊢ TGrp = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ {⟨(Base‘ndx), ((LTrn‘𝑘)‘𝑤)⟩, ⟨(+_g‘ndx), (𝑓 ∈ ((LTrn‘𝑘)‘𝑤), 𝑔 ∈ ((LTrn‘𝑘)‘𝑤) ↦ (𝑓 ∘ 𝑔))⟩}))
ctendo 41008	class TEndo
cedring 41009	class EDRing
cedring-rN 41010	class EDRing_R
df-tendo 41011	⊢ TEndo = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ {𝑓 ∣ (𝑓:((LTrn‘𝑘)‘𝑤)⟶((LTrn‘𝑘)‘𝑤) ∧ ∀𝑥 ∈ ((LTrn‘𝑘)‘𝑤)∀𝑦 ∈ ((LTrn‘𝑘)‘𝑤)(𝑓‘(𝑥 ∘ 𝑦)) = ((𝑓‘𝑥) ∘ (𝑓‘𝑦)) ∧ ∀𝑥 ∈ ((LTrn‘𝑘)‘𝑤)(((trL‘𝑘)‘𝑤)‘(𝑓‘𝑥))(le‘𝑘)(((trL‘𝑘)‘𝑤)‘𝑥))}))
df-edring-rN 41012	⊢ EDRing_R = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ {⟨(Base‘ndx), ((TEndo‘𝑘)‘𝑤)⟩, ⟨(+_g‘ndx), (𝑠 ∈ ((TEndo‘𝑘)‘𝑤), 𝑡 ∈ ((TEndo‘𝑘)‘𝑤) ↦ (𝑓 ∈ ((LTrn‘𝑘)‘𝑤) ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓))))⟩, ⟨(._r‘ndx), (𝑠 ∈ ((TEndo‘𝑘)‘𝑤), 𝑡 ∈ ((TEndo‘𝑘)‘𝑤) ↦ (𝑡 ∘ 𝑠))⟩}))
df-edring 41013	⊢ EDRing = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ {⟨(Base‘ndx), ((TEndo‘𝑘)‘𝑤)⟩, ⟨(+_g‘ndx), (𝑠 ∈ ((TEndo‘𝑘)‘𝑤), 𝑡 ∈ ((TEndo‘𝑘)‘𝑤) ↦ (𝑓 ∈ ((LTrn‘𝑘)‘𝑤) ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓))))⟩, ⟨(._r‘ndx), (𝑠 ∈ ((TEndo‘𝑘)‘𝑤), 𝑡 ∈ ((TEndo‘𝑘)‘𝑤) ↦ (𝑠 ∘ 𝑡))⟩}))
cdveca 41258	class DVecA
df-dveca 41259	⊢ DVecA = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ ({⟨(Base‘ndx), ((LTrn‘𝑘)‘𝑤)⟩, ⟨(+_g‘ndx), (𝑓 ∈ ((LTrn‘𝑘)‘𝑤), 𝑔 ∈ ((LTrn‘𝑘)‘𝑤) ↦ (𝑓 ∘ 𝑔))⟩, ⟨(Scalar‘ndx), ((EDRing‘𝑘)‘𝑤)⟩} ∪ {⟨( ·_𝑠 ‘ndx), (𝑠 ∈ ((TEndo‘𝑘)‘𝑤), 𝑓 ∈ ((LTrn‘𝑘)‘𝑤) ↦ (𝑠‘𝑓))⟩})))
cdia 41284	class DIsoA
df-disoa 41285	⊢ DIsoA = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑥 ∈ {𝑦 ∈ (Base‘𝑘) ∣ 𝑦(le‘𝑘)𝑤} ↦ {𝑓 ∈ ((LTrn‘𝑘)‘𝑤) ∣ (((trL‘𝑘)‘𝑤)‘𝑓)(le‘𝑘)𝑥})))
cdvh 41334	class DVecH
df-dvech 41335	⊢ DVecH = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ ({⟨(Base‘ndx), (((LTrn‘𝑘)‘𝑤) × ((TEndo‘𝑘)‘𝑤))⟩, ⟨(+_g‘ndx), (𝑓 ∈ (((LTrn‘𝑘)‘𝑤) × ((TEndo‘𝑘)‘𝑤)), 𝑔 ∈ (((LTrn‘𝑘)‘𝑤) × ((TEndo‘𝑘)‘𝑤)) ↦ ⟨((1^st ‘𝑓) ∘ (1^st ‘𝑔)), (ℎ ∈ ((LTrn‘𝑘)‘𝑤) ↦ (((2^nd ‘𝑓)‘ℎ) ∘ ((2^nd ‘𝑔)‘ℎ)))⟩)⟩, ⟨(Scalar‘ndx), ((EDRing‘𝑘)‘𝑤)⟩} ∪ {⟨( ·_𝑠 ‘ndx), (𝑠 ∈ ((TEndo‘𝑘)‘𝑤), 𝑓 ∈ (((LTrn‘𝑘)‘𝑤) × ((TEndo‘𝑘)‘𝑤)) ↦ ⟨(𝑠‘(1^st ‘𝑓)), (𝑠 ∘ (2^nd ‘𝑓))⟩)⟩})))
cocaN 41375	class ocA
df-docaN 41376	⊢ ocA = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑥 ∈ 𝒫 ((LTrn‘𝑘)‘𝑤) ↦ (((DIsoA‘𝑘)‘𝑤)‘((((oc‘𝑘)‘(^◡((DIsoA‘𝑘)‘𝑤)‘∩ {𝑧 ∈ ran ((DIsoA‘𝑘)‘𝑤) ∣ 𝑥 ⊆ 𝑧}))(join‘𝑘)((oc‘𝑘)‘𝑤))(meet‘𝑘)𝑤)))))
cdjaN 41387	class vA
df-djaN 41388	⊢ vA = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑥 ∈ 𝒫 ((LTrn‘𝑘)‘𝑤), 𝑦 ∈ 𝒫 ((LTrn‘𝑘)‘𝑤) ↦ (((ocA‘𝑘)‘𝑤)‘((((ocA‘𝑘)‘𝑤)‘𝑥) ∩ (((ocA‘𝑘)‘𝑤)‘𝑦))))))
cdib 41394	class DIsoB
df-dib 41395	⊢ DIsoB = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑥 ∈ dom ((DIsoA‘𝑘)‘𝑤) ↦ ((((DIsoA‘𝑘)‘𝑤)‘𝑥) × {(𝑓 ∈ ((LTrn‘𝑘)‘𝑤) ↦ ( I ↾ (Base‘𝑘)))}))))
cdic 41428	class DIsoC
df-dic 41429	⊢ DIsoC = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑞 ∈ {𝑟 ∈ (Atoms‘𝑘) ∣ ¬ 𝑟(le‘𝑘)𝑤} ↦ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(℩𝑔 ∈ ((LTrn‘𝑘)‘𝑤)(𝑔‘((oc‘𝑘)‘𝑤)) = 𝑞)) ∧ 𝑠 ∈ ((TEndo‘𝑘)‘𝑤))})))
cdih 41484	class DIsoH
df-dih 41485	⊢ DIsoH = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑥 ∈ (Base‘𝑘) ↦ if(𝑥(le‘𝑘)𝑤, (((DIsoB‘𝑘)‘𝑤)‘𝑥), (℩𝑢 ∈ (LSubSp‘((DVecH‘𝑘)‘𝑤))∀𝑞 ∈ (Atoms‘𝑘)((¬ 𝑞(le‘𝑘)𝑤 ∧ (𝑞(join‘𝑘)(𝑥(meet‘𝑘)𝑤)) = 𝑥) → 𝑢 = ((((DIsoC‘𝑘)‘𝑤)‘𝑞)(LSSum‘((DVecH‘𝑘)‘𝑤))(((DIsoB‘𝑘)‘𝑤)‘(𝑥(meet‘𝑘)𝑤)))))))))
coch 41603	class ocH
df-doch 41604	⊢ ocH = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑥 ∈ 𝒫 (Base‘((DVecH‘𝑘)‘𝑤)) ↦ (((DIsoH‘𝑘)‘𝑤)‘((oc‘𝑘)‘((glb‘𝑘)‘{𝑦 ∈ (Base‘𝑘) ∣ 𝑥 ⊆ (((DIsoH‘𝑘)‘𝑤)‘𝑦)}))))))
cdjh 41650	class joinH
df-djh 41651	⊢ joinH = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑥 ∈ 𝒫 (Base‘((DVecH‘𝑘)‘𝑤)), 𝑦 ∈ 𝒫 (Base‘((DVecH‘𝑘)‘𝑤)) ↦ (((ocH‘𝑘)‘𝑤)‘((((ocH‘𝑘)‘𝑤)‘𝑥) ∩ (((ocH‘𝑘)‘𝑤)‘𝑦))))))
clpoN 41736	class LPol
df-lpolN 41737	⊢ LPol = (𝑤 ∈ V ↦ {𝑜 ∈ ((LSubSp‘𝑤) ↑_m 𝒫 (Base‘𝑤)) ∣ ((𝑜‘(Base‘𝑤)) = {(0_g‘𝑤)} ∧ ∀𝑥∀𝑦((𝑥 ⊆ (Base‘𝑤) ∧ 𝑦 ⊆ (Base‘𝑤) ∧ 𝑥 ⊆ 𝑦) → (𝑜‘𝑦) ⊆ (𝑜‘𝑥)) ∧ ∀𝑥 ∈ (LSAtoms‘𝑤)((𝑜‘𝑥) ∈ (LSHyp‘𝑤) ∧ (𝑜‘(𝑜‘𝑥)) = 𝑥))})
clcd 41842	class LCDual
df-lcdual 41843	⊢ LCDual = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ ((LDual‘((DVecH‘𝑘)‘𝑤)) ↾_s {𝑓 ∈ (LFnl‘((DVecH‘𝑘)‘𝑤)) ∣ (((ocH‘𝑘)‘𝑤)‘(((ocH‘𝑘)‘𝑤)‘((LKer‘((DVecH‘𝑘)‘𝑤))‘𝑓))) = ((LKer‘((DVecH‘𝑘)‘𝑤))‘𝑓)})))
cmpd 41880	class mapd
df-mapd 41881	⊢ mapd = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑠 ∈ (LSubSp‘((DVecH‘𝑘)‘𝑤)) ↦ {𝑓 ∈ (LFnl‘((DVecH‘𝑘)‘𝑤)) ∣ ((((ocH‘𝑘)‘𝑤)‘(((ocH‘𝑘)‘𝑤)‘((LKer‘((DVecH‘𝑘)‘𝑤))‘𝑓))) = ((LKer‘((DVecH‘𝑘)‘𝑤))‘𝑓) ∧ (((ocH‘𝑘)‘𝑤)‘((LKer‘((DVecH‘𝑘)‘𝑤))‘𝑓)) ⊆ 𝑠)})))
chvm 42012	class HVMap
df-hvmap 42013	⊢ HVMap = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑥 ∈ ((Base‘((DVecH‘𝑘)‘𝑤)) ∖ {(0_g‘((DVecH‘𝑘)‘𝑤))}) ↦ (𝑣 ∈ (Base‘((DVecH‘𝑘)‘𝑤)) ↦ (℩𝑗 ∈ (Base‘(Scalar‘((DVecH‘𝑘)‘𝑤)))∃𝑡 ∈ (((ocH‘𝑘)‘𝑤)‘{𝑥})𝑣 = (𝑡(+_g‘((DVecH‘𝑘)‘𝑤))(𝑗( ·_𝑠 ‘((DVecH‘𝑘)‘𝑤))𝑥)))))))
chdma1 42047	class HDMap1
chdma 42048	class HDMap
df-hdmap1 42049	⊢ HDMap1 = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ {𝑎 ∣ [((DVecH‘𝑘)‘𝑤) / 𝑢][(Base‘𝑢) / 𝑣][(LSpan‘𝑢) / 𝑛][((LCDual‘𝑘)‘𝑤) / 𝑐][(Base‘𝑐) / 𝑑][(LSpan‘𝑐) / 𝑗][((mapd‘𝑘)‘𝑤) / 𝑚]𝑎 ∈ (𝑥 ∈ ((𝑣 × 𝑑) × 𝑣) ↦ if((2^nd ‘𝑥) = (0_g‘𝑢), (0_g‘𝑐), (℩ℎ ∈ 𝑑 ((𝑚‘(𝑛‘{(2^nd ‘𝑥)})) = (𝑗‘{ℎ}) ∧ (𝑚‘(𝑛‘{((1^st ‘(1^st ‘𝑥))(-_g‘𝑢)(2^nd ‘𝑥))})) = (𝑗‘{((2^nd ‘(1^st ‘𝑥))(-_g‘𝑐)ℎ)})))))}))
df-hdmap 42050	⊢ HDMap = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ {𝑎 ∣ [⟨( I ↾ (Base‘𝑘)), ( I ↾ ((LTrn‘𝑘)‘𝑤))⟩ / 𝑒][((DVecH‘𝑘)‘𝑤) / 𝑢][(Base‘𝑢) / 𝑣][((HDMap1‘𝑘)‘𝑤) / 𝑖]𝑎 ∈ (𝑡 ∈ 𝑣 ↦ (℩𝑦 ∈ (Base‘((LCDual‘𝑘)‘𝑤))∀𝑧 ∈ 𝑣 (¬ 𝑧 ∈ (((LSpan‘𝑢)‘{𝑒}) ∪ ((LSpan‘𝑢)‘{𝑡})) → 𝑦 = (𝑖‘⟨𝑧, (𝑖‘⟨𝑒, (((HVMap‘𝑘)‘𝑤)‘𝑒), 𝑧⟩), 𝑡⟩))))}))
chg 42139	class HGMap
df-hgmap 42140	⊢ HGMap = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ {𝑎 ∣ [((DVecH‘𝑘)‘𝑤) / 𝑢][(Base‘(Scalar‘𝑢)) / 𝑏][((HDMap‘𝑘)‘𝑤) / 𝑚]𝑎 ∈ (𝑥 ∈ 𝑏 ↦ (℩𝑦 ∈ 𝑏 ∀𝑣 ∈ (Base‘𝑢)(𝑚‘(𝑥( ·_𝑠 ‘𝑢)𝑣)) = (𝑦( ·_𝑠 ‘((LCDual‘𝑘)‘𝑤))(𝑚‘𝑣))))}))
chlh 42188	class HLHil
df-hlhil 42189	⊢ HLHil = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ ⦋((DVecH‘𝑘)‘𝑤) / 𝑢⦌⦋(Base‘𝑢) / 𝑣⦌({⟨(Base‘ndx), 𝑣⟩, ⟨(+_g‘ndx), (+_g‘𝑢)⟩, ⟨(Scalar‘ndx), (((EDRing‘𝑘)‘𝑤) sSet ⟨(*_𝑟‘ndx), ((HGMap‘𝑘)‘𝑤)⟩)⟩} ∪ {⟨( ·_𝑠 ‘ndx), ( ·_𝑠 ‘𝑢)⟩, ⟨(·_𝑖‘ndx), (𝑥 ∈ 𝑣, 𝑦 ∈ 𝑣 ↦ ((((HDMap‘𝑘)‘𝑤)‘𝑦)‘𝑥))⟩})))
ccsrg 42218	class CSRing
df-csring 42219	⊢ CSRing = {𝑓 ∈ SRing ∣ (mulGrp‘𝑓) ∈ CMnd}
cprimroots 42341	class PrimRoots
df-primroots 42342	⊢ PrimRoots = (𝑟 ∈ CMnd, 𝑘 ∈ ℕ₀ ↦ ⦋(Base‘𝑟) / 𝑏⦌{𝑎 ∈ 𝑏 ∣ ((𝑘(._g‘𝑟)𝑎) = (0_g‘𝑟) ∧ ∀𝑙 ∈ ℕ₀ ((𝑙(._g‘𝑟)𝑎) = (0_g‘𝑟) → 𝑘 ∥ 𝑙))})
ax-exfinfld 42452	⊢ ∀𝑝 ∈ ℙ ∀𝑛 ∈ ℕ ∃𝑘 ∈ Field ((♯‘(Base‘𝑘)) = (𝑝↑𝑛) ∧ (chr‘𝑘) = 𝑝)
cresub 42616	class −_ℝ
df-resub 42617	⊢ −_ℝ = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (℩𝑧 ∈ ℝ (𝑦 + 𝑧) = 𝑥))
crediv 42691	class /_ℝ
df-rediv 42692	⊢ /_ℝ = (𝑥 ∈ ℝ, 𝑦 ∈ (ℝ ∖ {0}) ↦ (℩𝑧 ∈ ℝ (𝑦 · 𝑧) = 𝑥))
cprjsp 42840	class ℙ𝕣𝕠𝕛
df-prjsp 42841	⊢ ℙ𝕣𝕠𝕛 = (𝑣 ∈ LVec ↦ ⦋((Base‘𝑣) ∖ {(0_g‘𝑣)}) / 𝑏⦌(𝑏 / {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝑏 ∧ 𝑦 ∈ 𝑏) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑣))𝑥 = (𝑙( ·_𝑠 ‘𝑣)𝑦))}))
cprjspn 42853	class ℙ𝕣𝕠𝕛_n
df-prjspn 42854	⊢ ℙ𝕣𝕠𝕛_n = (𝑛 ∈ ℕ₀, 𝑘 ∈ DivRing ↦ (ℙ𝕣𝕠𝕛‘(𝑘 freeLMod (0...𝑛))))
cprjcrv 42868	class ℙ𝕣𝕠𝕛Crv
df-prjcrv 42869	⊢ ℙ𝕣𝕠𝕛Crv = (𝑛 ∈ ℕ₀, 𝑘 ∈ Field ↦ (𝑓 ∈ ∪ ran ((0...𝑛) mHomP 𝑘) ↦ {𝑝 ∈ (𝑛ℙ𝕣𝕠𝕛_n𝑘) ∣ ((((0...𝑛) eval 𝑘)‘𝑓) “ 𝑝) = {(0_g‘𝑘)}}))
cnacs 42940	class NoeACS
df-nacs 42941	⊢ NoeACS = (𝑥 ∈ V ↦ {𝑐 ∈ (ACS‘𝑥) ∣ ∀𝑠 ∈ 𝑐 ∃𝑔 ∈ (𝒫 𝑥 ∩ Fin)𝑠 = ((mrCls‘𝑐)‘𝑔)})
cmzpcl 42959	class mzPolyCld
cmzp 42960	class mzPoly
df-mzpcl 42961	⊢ mzPolyCld = (𝑣 ∈ V ↦ {𝑝 ∈ 𝒫 (ℤ ↑_m (ℤ ↑_m 𝑣)) ∣ ((∀𝑖 ∈ ℤ ((ℤ ↑_m 𝑣) × {𝑖}) ∈ 𝑝 ∧ ∀𝑗 ∈ 𝑣 (𝑥 ∈ (ℤ ↑_m 𝑣) ↦ (𝑥‘𝑗)) ∈ 𝑝) ∧ ∀𝑓 ∈ 𝑝 ∀𝑔 ∈ 𝑝 ((𝑓 ∘_f + 𝑔) ∈ 𝑝 ∧ (𝑓 ∘_f · 𝑔) ∈ 𝑝))})
df-mzp 42962	⊢ mzPoly = (𝑣 ∈ V ↦ ∩ (mzPolyCld‘𝑣))
cdioph 42993	class Dioph
df-dioph 42994	⊢ Dioph = (𝑛 ∈ ℕ₀ ↦ ran (𝑘 ∈ (ℤ_≥‘𝑛), 𝑝 ∈ (mzPoly‘(1...𝑘)) ↦ {𝑡 ∣ ∃𝑢 ∈ (ℕ₀ ↑_m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑛)) ∧ (𝑝‘𝑢) = 0)}))
csquarenn 43074	class ◻_NN
cpell1qr 43075	class Pell1QR
cpell1234qr 43076	class Pell1234QR
cpell14qr 43077	class Pell14QR
cpellfund 43078	class PellFund
df-squarenn 43079	⊢ ◻_NN = {𝑥 ∈ ℕ ∣ (√‘𝑥) ∈ ℚ}
df-pell1qr 43080	⊢ Pell1QR = (𝑥 ∈ (ℕ ∖ ◻_NN) ↦ {𝑦 ∈ ℝ ∣ ∃𝑧 ∈ ℕ₀ ∃𝑤 ∈ ℕ₀ (𝑦 = (𝑧 + ((√‘𝑥) · 𝑤)) ∧ ((𝑧↑2) − (𝑥 · (𝑤↑2))) = 1)})
df-pell14qr 43081	⊢ Pell14QR = (𝑥 ∈ (ℕ ∖ ◻_NN) ↦ {𝑦 ∈ ℝ ∣ ∃𝑧 ∈ ℕ₀ ∃𝑤 ∈ ℤ (𝑦 = (𝑧 + ((√‘𝑥) · 𝑤)) ∧ ((𝑧↑2) − (𝑥 · (𝑤↑2))) = 1)})
df-pell1234qr 43082	⊢ Pell1234QR = (𝑥 ∈ (ℕ ∖ ◻_NN) ↦ {𝑦 ∈ ℝ ∣ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ (𝑦 = (𝑧 + ((√‘𝑥) · 𝑤)) ∧ ((𝑧↑2) − (𝑥 · (𝑤↑2))) = 1)})
df-pellfund 43083	⊢ PellFund = (𝑥 ∈ (ℕ ∖ ◻_NN) ↦ inf({𝑧 ∈ (Pell14QR‘𝑥) ∣ 1 < 𝑧}, ℝ, < ))
crmx 43138	class X_rm
crmy 43139	class Y_rm
df-rmx 43140	⊢ X_rm = (𝑎 ∈ (ℤ_≥‘2), 𝑛 ∈ ℤ ↦ (1^st ‘(^◡(𝑏 ∈ (ℕ₀ × ℤ) ↦ ((1^st ‘𝑏) + ((√‘((𝑎↑2) − 1)) · (2^nd ‘𝑏))))‘((𝑎 + (√‘((𝑎↑2) − 1)))↑𝑛))))
df-rmy 43141	⊢ Y_rm = (𝑎 ∈ (ℤ_≥‘2), 𝑛 ∈ ℤ ↦ (2^nd ‘(^◡(𝑏 ∈ (ℕ₀ × ℤ) ↦ ((1^st ‘𝑏) + ((√‘((𝑎↑2) − 1)) · (2^nd ‘𝑏))))‘((𝑎 + (√‘((𝑎↑2) − 1)))↑𝑛))))
clfig 43305	class LFinGen
df-lfig 43306	⊢ LFinGen = {𝑤 ∈ LMod ∣ (Base‘𝑤) ∈ ((LSpan‘𝑤) “ (𝒫 (Base‘𝑤) ∩ Fin))}
clnm 43313	class LNoeM
df-lnm 43314	⊢ LNoeM = {𝑤 ∈ LMod ∣ ∀𝑖 ∈ (LSubSp‘𝑤)(𝑤 ↾_s 𝑖) ∈ LFinGen}
clnr 43347	class LNoeR
df-lnr 43348	⊢ LNoeR = {𝑎 ∈ Ring ∣ (ringLMod‘𝑎) ∈ LNoeM}
cldgis 43359	class ldgIdlSeq
df-ldgis 43360	⊢ ldgIdlSeq = (𝑟 ∈ V ↦ (𝑖 ∈ (LIdeal‘(Poly₁‘𝑟)) ↦ (𝑥 ∈ ℕ₀ ↦ {𝑗 ∣ ∃𝑘 ∈ 𝑖 (((deg₁‘𝑟)‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe₁‘𝑘)‘𝑥))})))
cmnc 43369	class Monic
cplylt 43370	class Poly_<
df-mnc 43371	⊢ Monic = (𝑠 ∈ 𝒫 ℂ ↦ {𝑝 ∈ (Poly‘𝑠) ∣ ((coeff‘𝑝)‘(deg‘𝑝)) = 1})
df-plylt 43372	⊢ Poly_< = (𝑠 ∈ 𝒫 ℂ, 𝑥 ∈ ℕ₀ ↦ {𝑝 ∈ (Poly‘𝑠) ∣ (𝑝 = 0_𝑝 ∨ (deg‘𝑝) < 𝑥)})
cdgraa 43378	class deg_AA
cmpaa 43379	class minPolyAA
df-dgraa 43380	⊢ deg_AA = (𝑥 ∈ 𝔸 ↦ inf({𝑑 ∈ ℕ ∣ ∃𝑝 ∈ ((Poly‘ℚ) ∖ {0_𝑝})((deg‘𝑝) = 𝑑 ∧ (𝑝‘𝑥) = 0)}, ℝ, < ))
df-mpaa 43381	⊢ minPolyAA = (𝑥 ∈ 𝔸 ↦ (℩𝑝 ∈ (Poly‘ℚ)((deg‘𝑝) = (deg_AA‘𝑥) ∧ (𝑝‘𝑥) = 0 ∧ ((coeff‘𝑝)‘(deg_AA‘𝑥)) = 1)))
citgo 43395	class IntgOver
cza 43396	class ℤ
df-itgo 43397	⊢ IntgOver = (𝑠 ∈ 𝒫 ℂ ↦ {𝑥 ∈ ℂ ∣ ∃𝑝 ∈ (Poly‘𝑠)((𝑝‘𝑥) = 0 ∧ ((coeff‘𝑝)‘(deg‘𝑝)) = 1)})
df-za 43398	⊢ ℤ = (IntgOver‘ℤ)
cmend 43409	class MEndo
df-mend 43410	⊢ MEndo = (𝑚 ∈ V ↦ ⦋(𝑚 LMHom 𝑚) / 𝑏⦌({⟨(Base‘ndx), 𝑏⟩, ⟨(+_g‘ndx), (𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ (𝑥 ∘_f (+_g‘𝑚)𝑦))⟩, ⟨(._r‘ndx), (𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ (𝑥 ∘ 𝑦))⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑚)⟩, ⟨( ·_𝑠 ‘ndx), (𝑥 ∈ (Base‘(Scalar‘𝑚)), 𝑦 ∈ 𝑏 ↦ (((Base‘𝑚) × {𝑥}) ∘_f ( ·_𝑠 ‘𝑚)𝑦))⟩}))
ccytp 43435	class CytP
df-cytp 43436	⊢ CytP = (𝑛 ∈ ℕ ↦ ((mulGrp‘(Poly₁‘ℂ_fld)) Σ_g (𝑟 ∈ (^◡(od‘((mulGrp‘ℂ_fld) ↾_s (ℂ ∖ {0}))) “ {𝑛}) ↦ ((var₁‘ℂ_fld)(-_g‘(Poly₁‘ℂ_fld))((algSc‘(Poly₁‘ℂ_fld))‘𝑟)))))
ctopsep 43444	class TopSep
ctoplnd 43445	class TopLnd
df-topsep 43446	⊢ TopSep = {𝑗 ∈ Top ∣ ∃𝑥 ∈ 𝒫 ∪ 𝑗(𝑥 ≼ ω ∧ ((cls‘𝑗)‘𝑥) = ∪ 𝑗)}
df-toplnd 43447	⊢ TopLnd = {𝑥 ∈ Top ∣ ∀𝑦 ∈ 𝒫 𝑥(∪ 𝑥 = ∪ 𝑦 → ∃𝑧 ∈ 𝒫 𝑥(𝑧 ≼ ω ∧ ∪ 𝑥 = ∪ 𝑧))}
crcl 43909	class r*
df-rcl 43910	⊢ r* = (𝑥 ∈ V ↦ ∩ {𝑧 ∣ (𝑥 ⊆ 𝑧 ∧ ( I ↾ (dom 𝑧 ∪ ran 𝑧)) ⊆ 𝑧)})
whe 44009	wff 𝑅 hereditary 𝐴
df-he 44010	⊢ (𝑅 hereditary 𝐴 ↔ (𝑅 “ 𝐴) ⊆ 𝐴)
ax-frege1 44027	⊢ (𝜑 → (𝜓 → 𝜑))
ax-frege2 44028	⊢ ((𝜑 → (𝜓 → 𝜒)) → ((𝜑 → 𝜓) → (𝜑 → 𝜒)))
ax-frege8 44046	⊢ ((𝜑 → (𝜓 → 𝜒)) → (𝜓 → (𝜑 → 𝜒)))
ax-frege28 44067	⊢ ((𝜑 → 𝜓) → (¬ 𝜓 → ¬ 𝜑))
ax-frege31 44071	⊢ (¬ ¬ 𝜑 → 𝜑)
ax-frege41 44082	⊢ (𝜑 → ¬ ¬ 𝜑)
ax-frege52a 44094	⊢ ((𝜑 ↔ 𝜓) → (if-(𝜑, 𝜃, 𝜒) → if-(𝜓, 𝜃, 𝜒)))
ax-frege54a 44099	⊢ (𝜑 ↔ 𝜑)
ax-frege58a 44112	⊢ ((𝜓 ∧ 𝜒) → if-(𝜑, 𝜓, 𝜒))
ax-frege52c 44125	⊢ (𝐴 = 𝐵 → ([𝐴 / 𝑥]𝜑 → [𝐵 / 𝑥]𝜑))
ax-frege54c 44129	⊢ 𝐴 = 𝐴
ax-frege58b 44138	⊢ (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑)
cmnring 44448	class MndRing
df-mnring 44449	⊢ MndRing = (𝑟 ∈ V, 𝑚 ∈ V ↦ ⦋(𝑟 freeLMod (Base‘𝑚)) / 𝑣⦌(𝑣 sSet ⟨(._r‘ndx), (𝑥 ∈ (Base‘𝑣), 𝑦 ∈ (Base‘𝑣) ↦ (𝑣 Σ_g (𝑎 ∈ (Base‘𝑚), 𝑏 ∈ (Base‘𝑚) ↦ (𝑖 ∈ (Base‘𝑚) ↦ if(𝑖 = (𝑎(+_g‘𝑚)𝑏), ((𝑥‘𝑎)(._r‘𝑟)(𝑦‘𝑏)), (0_g‘𝑟))))))⟩))
cscott 44472	class Scott 𝐴
df-scott 44473	⊢ Scott 𝐴 = {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)}
ccoll 44487	class (𝐹 Coll 𝐴)
df-coll 44488	⊢ (𝐹 Coll 𝐴) = ∪ 𝑥 ∈ 𝐴 Scott (𝐹 “ {𝑥})
cbcc 44573	class C_𝑐
df-bcc 44574	⊢ C_𝑐 = (𝑐 ∈ ℂ, 𝑘 ∈ ℕ₀ ↦ ((𝑐 FallFac 𝑘) / (!‘𝑘)))
cplusr 44693	class +_𝑟
cminusr 44694	class -_𝑟
ctimesr 44695	class ._𝑣
cptdfc 44696	class PtDf(𝐴, 𝐵)
crr3c 44697	class RR3
cline3 44698	class line3
df-addr 44699	⊢ +_𝑟 = (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑣 ∈ ℝ ↦ ((𝑥‘𝑣) + (𝑦‘𝑣))))
df-subr 44700	⊢ -_𝑟 = (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑣 ∈ ℝ ↦ ((𝑥‘𝑣) − (𝑦‘𝑣))))
df-mulv 44701	⊢ ._𝑣 = (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑣 ∈ ℝ ↦ (𝑥 · (𝑦‘𝑣))))
df-ptdf 44712	⊢ PtDf(𝐴, 𝐵) = (𝑥 ∈ ℝ ↦ (((𝑥._𝑣(𝐵-_𝑟𝐴)) +_𝑣 𝐴) “ {1, 2, 3}))
df-rr3 44713	⊢ RR3 = (ℝ ↑_m {1, 2, 3})
df-line3 44714	⊢ line3 = {𝑥 ∈ 𝒫 RR3 ∣ (2_o ≼ 𝑥 ∧ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑧 ≠ 𝑦 → ran PtDf(𝑦, 𝑧) = 𝑥))}
wvd1 44806	wff ( 𝜑 ▶ 𝜓 )
df-vd1 44807	⊢ (( 𝜑 ▶ 𝜓 ) ↔ (𝜑 → 𝜓))
wvd2 44814	wff ( 𝜑 , 𝜓 ▶ 𝜒 )
df-vd2 44815	⊢ (( 𝜑 , 𝜓 ▶ 𝜒 ) ↔ ((𝜑 ∧ 𝜓) → 𝜒))
wvhc2 44817	wff ( 𝜑 , 𝜓 )
df-vhc2 44818	⊢ (( 𝜑 , 𝜓 ) ↔ (𝜑 ∧ 𝜓))
wvd3 44824	wff ( 𝜑 , 𝜓 , 𝜒 ▶ 𝜃 )
wvhc3 44825	wff ( 𝜑 , 𝜓 , 𝜒 )
df-vhc3 44826	⊢ (( 𝜑 , 𝜓 , 𝜒 ) ↔ (𝜑 ∧ 𝜓 ∧ 𝜒))
df-vd3 44827	⊢ (( 𝜑 , 𝜓 , 𝜒 ▶ 𝜃 ) ↔ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃))
wrelp 45179	wff 𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵)
df-relp 45180	⊢ (𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵) ↔ (𝐻:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐻‘𝑥)𝑆(𝐻‘𝑦))))
clsi 45991	class lim inf
df-liminf 45992	⊢ lim inf = (𝑥 ∈ V ↦ sup(ran (𝑘 ∈ ℝ ↦ inf(((𝑥 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )), ℝ^*, < ))
clsxlim 46058	class ~~>*
df-xlim 46059	⊢ ~~>* = (⇝_𝑡‘(ordTop‘ ≤ ))
csalg 46548	class SAlg
df-salg 46549	⊢ SAlg = {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (∪ 𝑥 ∖ 𝑦) ∈ 𝑥 ∧ ∀𝑦 ∈ 𝒫 𝑥(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑥))}
csalon 46550	class SalOn
df-salon 46551	⊢ SalOn = (𝑥 ∈ V ↦ {𝑠 ∈ SAlg ∣ ∪ 𝑠 = 𝑥})
csalgen 46552	class SalGen
df-salgen 46553	⊢ SalGen = (𝑥 ∈ V ↦ ∩ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑥 ∧ 𝑥 ⊆ 𝑠)})
csumge0 46602	class Σ^{^}
df-sumge0 46603	⊢ Σ^{^} = (𝑥 ∈ V ↦ if(+∞ ∈ ran 𝑥, +∞, sup(ran (𝑦 ∈ (𝒫 dom 𝑥 ∩ Fin) ↦ Σ𝑤 ∈ 𝑦 (𝑥‘𝑤)), ℝ^*, < )))
cmea 46689	class Meas
df-mea 46690	⊢ Meas = {𝑥 ∣ (((𝑥:dom 𝑥⟶(0[,]+∞) ∧ dom 𝑥 ∈ SAlg) ∧ (𝑥‘∅) = 0) ∧ ∀𝑦 ∈ 𝒫 dom 𝑥((𝑦 ≼ ω ∧ Disj 𝑤 ∈ 𝑦 𝑤) → (𝑥‘∪ 𝑦) = (Σ^{^}‘(𝑥 ↾ 𝑦))))}
come 46729	class OutMeas
df-ome 46730	⊢ OutMeas = {𝑥 ∣ ((((𝑥:dom 𝑥⟶(0[,]+∞) ∧ dom 𝑥 = 𝒫 ∪ dom 𝑥) ∧ (𝑥‘∅) = 0) ∧ ∀𝑦 ∈ 𝒫 ∪ dom 𝑥∀𝑧 ∈ 𝒫 𝑦(𝑥‘𝑧) ≤ (𝑥‘𝑦)) ∧ ∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≼ ω → (𝑥‘∪ 𝑦) ≤ (Σ^{^}‘(𝑥 ↾ 𝑦))))}
ccaragen 46731	class CaraGen
df-caragen 46732	⊢ CaraGen = (𝑜 ∈ OutMeas ↦ {𝑒 ∈ 𝒫 ∪ dom 𝑜 ∣ ∀𝑎 ∈ 𝒫 ∪ dom 𝑜((𝑜‘(𝑎 ∩ 𝑒)) +_𝑒 (𝑜‘(𝑎 ∖ 𝑒))) = (𝑜‘𝑎)})
covoln 46776	class voln*
df-ovoln 46777	⊢ voln* = (𝑥 ∈ Fin ↦ (𝑦 ∈ 𝒫 (ℝ ↑_m 𝑥) ↦ if(𝑥 = ∅, 0, inf({𝑧 ∈ ℝ^* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑥) ↑_m ℕ)(𝑦 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑥 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑥 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))}, ℝ^*, < ))))
cvoln 46778	class voln
df-voln 46779	⊢ voln = (𝑥 ∈ Fin ↦ ((voln‘𝑥) ↾ (CaraGen‘(voln‘𝑥))))
csmblfn 46935	class SMblFn
df-smblfn 46936	⊢ SMblFn = (𝑠 ∈ SAlg ↦ {𝑓 ∈ (ℝ ↑_pm ∪ 𝑠) ∣ ∀𝑎 ∈ ℝ (^◡𝑓 “ (-∞(,)𝑎)) ∈ (𝑠 ↾_t dom 𝑓)})
caiota 47325	class (℩'𝑥𝜑)
df-aiota 47327	⊢ (℩'𝑥𝜑) = ∩ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}
wdfat 47358	wff 𝐹 defAt 𝐴
cafv 47359	class (𝐹'''𝐴)
caov 47360	class ((𝐴𝐹𝐵))
df-dfat 47361	⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
df-afv 47362	⊢ (𝐹'''𝐴) = (℩'𝑥𝐴𝐹𝑥)
df-aov 47363	⊢ ((𝐴𝐹𝐵)) = (𝐹'''⟨𝐴, 𝐵⟩)
cafv2 47450	class (𝐹''''𝐴)
df-afv2 47451	⊢ (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ∪ ran 𝐹)
cnelbr 47513	class _∉
df-nelbr 47514	⊢ _∉ = {⟨𝑥, 𝑦⟩ ∣ ¬ 𝑥 ∈ 𝑦}
ciccp 47655	class RePart
df-iccp 47656	⊢ RePart = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ^* ↑_m (0...𝑚)) ∣ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1))})
wich 47687	wff [𝑥⇄𝑦]𝜑
df-ich 47688	⊢ ([𝑥⇄𝑦]𝜑 ↔ ∀𝑥∀𝑦([𝑥 / 𝑎][𝑦 / 𝑥][𝑎 / 𝑦]𝜑 ↔ 𝜑))
cspr 47719	class Pairs
df-spr 47720	⊢ Pairs = (𝑣 ∈ V ↦ {𝑝 ∣ ∃𝑎 ∈ 𝑣 ∃𝑏 ∈ 𝑣 𝑝 = {𝑎, 𝑏}})
cprpr 47754	class Pairs_proper
df-prpr 47755	⊢ Pairs_proper = (𝑣 ∈ V ↦ {𝑝 ∣ ∃𝑎 ∈ 𝑣 ∃𝑏 ∈ 𝑣 (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})})
cfmtno 47769	class FermatNo
df-fmtno 47770	⊢ FermatNo = (𝑛 ∈ ℕ₀ ↦ ((2↑(2↑𝑛)) + 1))
ceven 47866	class Even
codd 47867	class Odd
df-even 47868	⊢ Even = {𝑧 ∈ ℤ ∣ (𝑧 / 2) ∈ ℤ}
df-odd 47869	⊢ Odd = {𝑧 ∈ ℤ ∣ ((𝑧 + 1) / 2) ∈ ℤ}
cfppr 47966	class FPPr
df-fppr 47967	⊢ FPPr = (𝑛 ∈ ℕ ↦ {𝑥 ∈ (ℤ_≥‘4) ∣ (𝑥 ∉ ℙ ∧ 𝑥 ∥ ((𝑛↑(𝑥 − 1)) − 1))})
cgbe 47987	class GoldbachEven
cgbow 47988	class GoldbachOddW
cgbo 47989	class GoldbachOdd
df-gbe 47990	⊢ GoldbachEven = {𝑧 ∈ Even ∣ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑧 = (𝑝 + 𝑞))}
df-gbow 47991	⊢ GoldbachOddW = {𝑧 ∈ Odd ∣ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑧 = ((𝑝 + 𝑞) + 𝑟)}
df-gbo 47992	⊢ GoldbachOdd = {𝑧 ∈ Odd ∣ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ ((𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑟 ∈ Odd ) ∧ 𝑧 = ((𝑝 + 𝑞) + 𝑟))}
ax-bgbltosilva 48052	⊢ ((𝑁 ∈ Even ∧ 4 < 𝑁 ∧ 𝑁 ≤ (4 · (;10↑;18))) → 𝑁 ∈ GoldbachEven )
ax-tgoldbachgt 48053	⊢ 𝑂 = {𝑧 ∈ ℤ ∣ ¬ 2 ∥ 𝑧} & ⊢ 𝐺 = {𝑧 ∈ 𝑂 ∣ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ ((𝑝 ∈ 𝑂 ∧ 𝑞 ∈ 𝑂 ∧ 𝑟 ∈ 𝑂) ∧ 𝑧 = ((𝑝 + 𝑞) + 𝑟))} ⇒ ⊢ ∃𝑚 ∈ ℕ (𝑚 ≤ (;10↑;27) ∧ ∀𝑛 ∈ 𝑂 (𝑚 < 𝑛 → 𝑛 ∈ 𝐺))
ax-hgprmladder 48056	⊢ ∃𝑑 ∈ (ℤ_≥‘3)∃𝑓 ∈ (RePart‘𝑑)(((𝑓‘0) = 7 ∧ (𝑓‘1) = ;13 ∧ (𝑓‘𝑑) = (;89 · (;10↑;29))) ∧ ∀𝑖 ∈ (0..^𝑑)((𝑓‘𝑖) ∈ (ℙ ∖ {2}) ∧ ((𝑓‘(𝑖 + 1)) − (𝑓‘𝑖)) < ((4 · (;10↑;18)) − 4) ∧ 4 < ((𝑓‘(𝑖 + 1)) − (𝑓‘𝑖))))
cclnbgr 48060	class ClNeighbVtx
df-clnbgr 48061	⊢ ClNeighbVtx = (𝑔 ∈ V, 𝑣 ∈ (Vtx‘𝑔) ↦ ({𝑣} ∪ {𝑛 ∈ (Vtx‘𝑔) ∣ ∃𝑒 ∈ (Edg‘𝑔){𝑣, 𝑛} ⊆ 𝑒}))
cisubgr 48102	class ISubGr
df-isubgr 48103	⊢ ISubGr = (𝑔 ∈ V, 𝑣 ∈ 𝒫 (Vtx‘𝑔) ↦ ⟨𝑣, ⦋(iEdg‘𝑔) / 𝑒⦌(𝑒 ↾ {𝑥 ∈ dom 𝑒 ∣ (𝑒‘𝑥) ⊆ 𝑣})⟩)
cgrisom 48116	class GraphIsom
cgrim 48117	class GraphIso
cgric 48118	class ≃_𝑔𝑟
df-grisom 48119	⊢ GraphIsom = (𝑥 ∈ V, 𝑦 ∈ V ↦ {⟨𝑓, 𝑔⟩ ∣ (𝑓:(Vtx‘𝑥)–1-1-onto→(Vtx‘𝑦) ∧ 𝑔:dom (iEdg‘𝑥)–1-1-onto→dom (iEdg‘𝑦) ∧ ∀𝑖 ∈ dom (iEdg‘𝑥)(𝑓 “ ((iEdg‘𝑥)‘𝑖)) = ((iEdg‘𝑦)‘(𝑔‘𝑖)))})
df-grim 48120	⊢ GraphIso = (𝑔 ∈ V, ℎ ∈ V ↦ {𝑓 ∣ (𝑓:(Vtx‘𝑔)–1-1-onto→(Vtx‘ℎ) ∧ ∃𝑗[(iEdg‘𝑔) / 𝑒][(iEdg‘ℎ) / 𝑑](𝑗:dom 𝑒–1-1-onto→dom 𝑑 ∧ ∀𝑖 ∈ dom 𝑒(𝑑‘(𝑗‘𝑖)) = (𝑓 “ (𝑒‘𝑖))))})
df-gric 48123	⊢ ≃_𝑔𝑟 = (^◡ GraphIso “ (V ∖ 1_o))
cgrtri 48179	class GrTriangles
df-grtri 48180	⊢ GrTriangles = (𝑔 ∈ V ↦ ⦋(Vtx‘𝑔) / 𝑣⦌⦋(Edg‘𝑔) / 𝑒⦌{𝑡 ∈ 𝒫 𝑣 ∣ ∃𝑓(𝑓:(0..^3)–1-1-onto→𝑡 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝑒 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝑒 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝑒))})
cstgr 48193	class StarGr
df-stgr 48194	⊢ StarGr = (𝑛 ∈ ℕ₀ ↦ {⟨(Base‘ndx), (0...𝑛)⟩, ⟨(.ef‘ndx), ( I ↾ {𝑒 ∈ 𝒫 (0...𝑛) ∣ ∃𝑥 ∈ (1...𝑛)𝑒 = {0, 𝑥}})⟩})
cgrlim 48218	class GraphLocIso
cgrlic 48219	class ≃_𝑙𝑔𝑟
df-grlim 48220	⊢ GraphLocIso = (𝑔 ∈ V, ℎ ∈ V ↦ {𝑓 ∣ (𝑓:(Vtx‘𝑔)–1-1-onto→(Vtx‘ℎ) ∧ ∀𝑣 ∈ (Vtx‘𝑔)(𝑔 ISubGr (𝑔 ClNeighbVtx 𝑣)) ≃_𝑔𝑟 (ℎ ISubGr (ℎ ClNeighbVtx (𝑓‘𝑣))))})
df-grlic 48223	⊢ ≃_𝑙𝑔𝑟 = (^◡ GraphLocIso “ (V ∖ 1_o))
cgpg 48282	class gPetersenGr
df-gpg 48283	⊢ gPetersenGr = (𝑛 ∈ ℕ, 𝑘 ∈ (1..^(⌈‘(𝑛 / 2))) ↦ {⟨(Base‘ndx), ({0, 1} × (0..^𝑛))⟩, ⟨(.ef‘ndx), ( I ↾ {𝑒 ∈ 𝒫 ({0, 1} × (0..^𝑛)) ∣ ∃𝑥 ∈ (0..^𝑛)(𝑒 = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑛)⟩} ∨ 𝑒 = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ 𝑒 = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 𝑘) mod 𝑛)⟩})})⟩})
cupwlks 48375	class UPWalks
df-upwlks 48376	⊢ UPWalks = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom (iEdg‘𝑔) ∧ 𝑝:(0...(♯‘𝑓))⟶(Vtx‘𝑔) ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))((iEdg‘𝑔)‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))})})
ccllaw 48425	class clLaw
casslaw 48426	class assLaw
ccomlaw 48427	class comLaw
df-cllaw 48428	⊢ clLaw = {⟨𝑜, 𝑚⟩ ∣ ∀𝑥 ∈ 𝑚 ∀𝑦 ∈ 𝑚 (𝑥𝑜𝑦) ∈ 𝑚}
df-comlaw 48429	⊢ comLaw = {⟨𝑜, 𝑚⟩ ∣ ∀𝑥 ∈ 𝑚 ∀𝑦 ∈ 𝑚 (𝑥𝑜𝑦) = (𝑦𝑜𝑥)}
df-asslaw 48430	⊢ assLaw = {⟨𝑜, 𝑚⟩ ∣ ∀𝑥 ∈ 𝑚 ∀𝑦 ∈ 𝑚 ∀𝑧 ∈ 𝑚 ((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧))}
cintop 48438	class intOp
cclintop 48439	class clIntOp
cassintop 48440	class assIntOp
df-intop 48441	⊢ intOp = (𝑚 ∈ V, 𝑛 ∈ V ↦ (𝑛 ↑_m (𝑚 × 𝑚)))
df-clintop 48442	⊢ clIntOp = (𝑚 ∈ V ↦ (𝑚 intOp 𝑚))
df-assintop 48443	⊢ assIntOp = (𝑚 ∈ V ↦ {𝑜 ∈ ( clIntOp ‘𝑚) ∣ 𝑜 assLaw 𝑚})
cmgm2 48457	class MgmALT
ccmgm2 48458	class CMgmALT
csgrp2 48459	class SGrpALT
ccsgrp2 48460	class CSGrpALT
df-mgm2 48461	⊢ MgmALT = {𝑚 ∣ (+_g‘𝑚) clLaw (Base‘𝑚)}
df-cmgm2 48462	⊢ CMgmALT = {𝑚 ∈ MgmALT ∣ (+_g‘𝑚) comLaw (Base‘𝑚)}
df-sgrp2 48463	⊢ SGrpALT = {𝑔 ∈ MgmALT ∣ (+_g‘𝑔) assLaw (Base‘𝑔)}
df-csgrp2 48464	⊢ CSGrpALT = {𝑔 ∈ SGrpALT ∣ (+_g‘𝑔) comLaw (Base‘𝑔)}
crngcALTV 48505	class RngCatALTV
df-rngcALTV 48506	⊢ RngCatALTV = (𝑢 ∈ V ↦ ⦋(𝑢 ∩ Rng) / 𝑏⦌{⟨(Base‘ndx), 𝑏⟩, ⟨(Hom ‘ndx), (𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ (𝑥 RngHom 𝑦))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝑏 × 𝑏), 𝑧 ∈ 𝑏 ↦ (𝑔 ∈ ((2^nd ‘𝑣) RngHom 𝑧), 𝑓 ∈ ((1^st ‘𝑣) RngHom (2^nd ‘𝑣)) ↦ (𝑔 ∘ 𝑓)))⟩})
cringcALTV 48529	class RingCatALTV
df-ringcALTV 48530	⊢ RingCatALTV = (𝑢 ∈ V ↦ ⦋(𝑢 ∩ Ring) / 𝑏⦌{⟨(Base‘ndx), 𝑏⟩, ⟨(Hom ‘ndx), (𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ (𝑥 RingHom 𝑦))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝑏 × 𝑏), 𝑧 ∈ 𝑏 ↦ (𝑔 ∈ ((2^nd ‘𝑣) RingHom 𝑧), 𝑓 ∈ ((1^st ‘𝑣) RingHom (2^nd ‘𝑣)) ↦ (𝑔 ∘ 𝑓)))⟩})
cdmatalt 48638	class DMatALT
cscmatalt 48639	class ScMatALT
df-dmatalt 48640	⊢ DMatALT = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ ⦋(𝑛 Mat 𝑟) / 𝑎⦌(𝑎 ↾_s {𝑚 ∈ (Base‘𝑎) ∣ ∀𝑖 ∈ 𝑛 ∀𝑗 ∈ 𝑛 (𝑖 ≠ 𝑗 → (𝑖𝑚𝑗) = (0_g‘𝑟))}))
df-scmatalt 48641	⊢ ScMatALT = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ ⦋(𝑛 Mat 𝑟) / 𝑎⦌(𝑎 ↾_s {𝑚 ∈ (Base‘𝑎) ∣ ∃𝑐 ∈ (Base‘𝑟)∀𝑖 ∈ 𝑛 ∀𝑗 ∈ 𝑛 (𝑖𝑚𝑗) = if(𝑖 = 𝑗, 𝑐, (0_g‘𝑟))}))
clinc 48646	class linC
clinco 48647	class LinCo
df-linc 48648	⊢ linC = (𝑚 ∈ V ↦ (𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑_m 𝑣), 𝑣 ∈ 𝒫 (Base‘𝑚) ↦ (𝑚 Σ_g (𝑥 ∈ 𝑣 ↦ ((𝑠‘𝑥)( ·_𝑠 ‘𝑚)𝑥)))))
df-lco 48649	⊢ LinCo = (𝑚 ∈ V, 𝑣 ∈ 𝒫 (Base‘𝑚) ↦ {𝑐 ∈ (Base‘𝑚) ∣ ∃𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑_m 𝑣)(𝑠 finSupp (0_g‘(Scalar‘𝑚)) ∧ 𝑐 = (𝑠( linC ‘𝑚)𝑣))})
clininds 48682	class linIndS
clindeps 48683	class linDepS
df-lininds 48684	⊢ linIndS = {⟨𝑠, 𝑚⟩ ∣ (𝑠 ∈ 𝒫 (Base‘𝑚) ∧ ∀𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑_m 𝑠)((𝑓 finSupp (0_g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)𝑠) = (0_g‘𝑚)) → ∀𝑥 ∈ 𝑠 (𝑓‘𝑥) = (0_g‘(Scalar‘𝑚))))}
df-lindeps 48686	⊢ linDepS = {⟨𝑠, 𝑚⟩ ∣ ¬ 𝑠 linIndS 𝑚}
cfdiv 48779	class /_f
df-fdiv 48780	⊢ /_f = (𝑓 ∈ V, 𝑔 ∈ V ↦ ((𝑓 ∘_f / 𝑔) ↾ (𝑔 supp 0)))
cbigo 48789	class Ο
df-bigo 48790	⊢ Ο = (𝑔 ∈ (ℝ ↑_pm ℝ) ↦ {𝑓 ∈ (ℝ ↑_pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(𝑓‘𝑦) ≤ (𝑚 · (𝑔‘𝑦))})
cblen 48811	class #_b
df-blen 48812	⊢ #_b = (𝑛 ∈ V ↦ if(𝑛 = 0, 1, ((⌊‘(2 log_b (abs‘𝑛))) + 1)))
cdig 48837	class digit
df-dig 48838	⊢ digit = (𝑏 ∈ ℕ ↦ (𝑘 ∈ ℤ, 𝑟 ∈ (0[,)+∞) ↦ ((⌊‘((𝑏↑-𝑘) · 𝑟)) mod 𝑏)))
cnaryf 48868	class -aryF
df-naryf 48869	⊢ -aryF = (𝑛 ∈ ℕ₀, 𝑥 ∈ V ↦ (𝑥 ↑_m (𝑥 ↑_m (0..^𝑛))))
citco 48899	class IterComp
cack 48900	class Ack
df-itco 48901	⊢ IterComp = (𝑓 ∈ V ↦ seq0((𝑔 ∈ V, 𝑗 ∈ V ↦ (𝑓 ∘ 𝑔)), (𝑖 ∈ ℕ₀ ↦ if(𝑖 = 0, ( I ↾ dom 𝑓), 𝑓))))
df-ack 48902	⊢ Ack = seq0((𝑓 ∈ V, 𝑗 ∈ V ↦ (𝑛 ∈ ℕ₀ ↦ (((IterComp‘𝑓)‘(𝑛 + 1))‘1))), (𝑖 ∈ ℕ₀ ↦ if(𝑖 = 0, (𝑛 ∈ ℕ₀ ↦ (𝑛 + 1)), 𝑖)))
cline 48969	class Line_M
csph 48970	class Sphere
df-line 48971	⊢ Line_M = (𝑤 ∈ V ↦ (𝑥 ∈ (Base‘𝑤), 𝑦 ∈ ((Base‘𝑤) ∖ {𝑥}) ↦ {𝑝 ∈ (Base‘𝑤) ∣ ∃𝑡 ∈ (Base‘(Scalar‘𝑤))𝑝 = ((((1_r‘(Scalar‘𝑤))(-_g‘(Scalar‘𝑤))𝑡)( ·_𝑠 ‘𝑤)𝑥)(+_g‘𝑤)(𝑡( ·_𝑠 ‘𝑤)𝑦))}))
df-sph 48972	⊢ Sphere = (𝑤 ∈ V ↦ (𝑥 ∈ (Base‘𝑤), 𝑟 ∈ (0[,]+∞) ↦ {𝑝 ∈ (Base‘𝑤) ∣ (𝑝(dist‘𝑤)𝑥) = 𝑟}))
coppf 49363	class oppFunc
df-oppf 49364	⊢ oppFunc = (𝑓 ∈ V, 𝑔 ∈ V ↦ if((Rel 𝑔 ∧ Rel dom 𝑔), ⟨𝑓, tpos 𝑔⟩, ∅))
cup 49414	class UP
df-up 49415	⊢ UP = (𝑑 ∈ V, 𝑒 ∈ V ↦ ⦋(Base‘𝑑) / 𝑏⦌⦋(Base‘𝑒) / 𝑐⦌⦋(Hom ‘𝑑) / ℎ⦌⦋(Hom ‘𝑒) / 𝑗⦌⦋(comp‘𝑒) / 𝑜⦌(𝑓 ∈ (𝑑 Func 𝑒), 𝑤 ∈ 𝑐 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝑏 ∧ 𝑚 ∈ (𝑤𝑗((1^st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝑏 ∀𝑔 ∈ (𝑤𝑗((1^st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥ℎ𝑦)𝑔 = (((𝑥(2^nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1^st ‘𝑓)‘𝑥)⟩𝑜((1^st ‘𝑓)‘𝑦))𝑚))}))
cswapf 49500	class swap_F
df-swapf 49501	⊢ swap_F = (𝑐 ∈ V, 𝑑 ∈ V ↦ ⦋(𝑐 ×_c 𝑑) / 𝑠⦌⦋(Base‘𝑠) / 𝑏⦌⦋(Hom ‘𝑠) / ℎ⦌⟨(𝑥 ∈ 𝑏 ↦ ∪ ^◡{𝑥}), (𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (𝑓 ∈ (𝑢ℎ𝑣) ↦ ∪ ^◡{𝑓}))⟩)
cfuco 49557	class ∘_F
df-fuco 49558	⊢ ∘_F = (𝑝 ∈ V, 𝑒 ∈ V ↦ ⦋(1^st ‘𝑝) / 𝑐⦌⦋(2^nd ‘𝑝) / 𝑑⦌⦋((𝑑 Func 𝑒) × (𝑐 Func 𝑑)) / 𝑤⦌⟨( ∘_func ↾ 𝑤), (𝑢 ∈ 𝑤, 𝑣 ∈ 𝑤 ↦ ⦋(1^st ‘(2^nd ‘𝑢)) / 𝑓⦌⦋(1^st ‘(1^st ‘𝑢)) / 𝑘⦌⦋(2^nd ‘(1^st ‘𝑢)) / 𝑙⦌⦋(1^st ‘(2^nd ‘𝑣)) / 𝑚⦌⦋(1^st ‘(1^st ‘𝑣)) / 𝑟⦌(𝑏 ∈ ((1^st ‘𝑢)(𝑑 Nat 𝑒)(1^st ‘𝑣)), 𝑎 ∈ ((2^nd ‘𝑢)(𝑐 Nat 𝑑)(2^nd ‘𝑣)) ↦ (𝑥 ∈ (Base‘𝑐) ↦ ((𝑏‘(𝑚‘𝑥))(⟨(𝑘‘(𝑓‘𝑥)), (𝑘‘(𝑚‘𝑥))⟩(comp‘𝑒)(𝑟‘(𝑚‘𝑥)))(((𝑓‘𝑥)𝑙(𝑚‘𝑥))‘(𝑎‘𝑥))))))⟩)
cprcof 49614	class −∘_F
df-prcof 49615	⊢ −∘_F = (𝑝 ∈ V, 𝑓 ∈ V ↦ ⦋(1^st ‘𝑝) / 𝑑⦌⦋(2^nd ‘𝑝) / 𝑒⦌⦋(𝑑 Func 𝑒) / 𝑏⦌⟨(𝑘 ∈ 𝑏 ↦ (𝑘 ∘_func 𝑓)), (𝑘 ∈ 𝑏, 𝑙 ∈ 𝑏 ↦ (𝑎 ∈ (𝑘(𝑑 Nat 𝑒)𝑙) ↦ (𝑎 ∘ (1^st ‘𝑓))))⟩)
cthinc 49658	class ThinCat
df-thinc 49659	⊢ ThinCat = {𝑐 ∈ Cat ∣ [(Base‘𝑐) / 𝑏][(Hom ‘𝑐) / ℎ]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∃*𝑓 𝑓 ∈ (𝑥ℎ𝑦)}
ctermc 49713	class TermCat
df-termc 49714	⊢ TermCat = {𝑐 ∈ ThinCat ∣ ∃𝑥(Base‘𝑐) = {𝑥}}
cprstc 49790	class ProsetToCat
df-prstc 49791	⊢ ProsetToCat = (𝑘 ∈ Proset ↦ ((𝑘 sSet ⟨(Hom ‘ndx), ((le‘𝑘) × {1_o})⟩) sSet ⟨(comp‘ndx), ∅⟩))
cmndtc 49818	class MndToCat
df-mndtc 49819	⊢ MndToCat = (𝑚 ∈ Mnd ↦ {⟨(Base‘ndx), {𝑚}⟩, ⟨(Hom ‘ndx), {⟨𝑚, 𝑚, (Base‘𝑚)⟩}⟩, ⟨(comp‘ndx), {⟨⟨𝑚, 𝑚, 𝑚⟩, (+_g‘𝑚)⟩}⟩})
clan 49846	class Lan
cran 49847	class Ran
df-lan 49848	⊢ Lan = (𝑝 ∈ (V × V), 𝑒 ∈ V ↦ ⦋(1^st ‘𝑝) / 𝑐⦌⦋(2^nd ‘𝑝) / 𝑑⦌(𝑓 ∈ (𝑐 Func 𝑑), 𝑥 ∈ (𝑐 Func 𝑒) ↦ ((⟨𝑑, 𝑒⟩ −∘_F 𝑓)((𝑑 FuncCat 𝑒) UP (𝑐 FuncCat 𝑒))𝑥)))
df-ran 49849	⊢ Ran = (𝑝 ∈ (V × V), 𝑒 ∈ V ↦ ⦋(1^st ‘𝑝) / 𝑐⦌⦋(2^nd ‘𝑝) / 𝑑⦌(𝑓 ∈ (𝑐 Func 𝑑), 𝑥 ∈ (𝑐 Func 𝑒) ↦ (( oppFunc ‘(⟨𝑑, 𝑒⟩ −∘_F 𝑓))((oppCat‘(𝑑 FuncCat 𝑒)) UP (oppCat‘(𝑐 FuncCat 𝑒)))𝑥)))
clmd 49884	class Limit
ccmd 49885	class Colimit
df-lmd 49886	⊢ Limit = (𝑐 ∈ V, 𝑑 ∈ V ↦ (𝑓 ∈ (𝑑 Func 𝑐) ↦ (( oppFunc ‘(𝑐Δ_func𝑑))((oppCat‘𝑐) UP (oppCat‘(𝑑 FuncCat 𝑐)))𝑓)))
df-cmd 49887	⊢ Colimit = (𝑐 ∈ V, 𝑑 ∈ V ↦ (𝑓 ∈ (𝑑 Func 𝑐) ↦ ((𝑐Δ_func𝑑)(𝑐 UP (𝑑 FuncCat 𝑐))𝑓)))
csetrecs 49924	class setrecs(𝐹)
df-setrecs 49925	⊢ setrecs(𝐹) = ∪ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)}
cpg 49950	class Pg
df-pg 49951	⊢ Pg = setrecs((𝑥 ∈ V ↦ (𝒫 𝑥 × 𝒫 𝑥)))
cge-real 49961	class ≥
cgt 49962	class >
df-gte 49963	⊢ ≥ = ^◡ ≤
df-gt 49964	⊢ > = ^◡ <
csinh 49971	class sinh
ccosh 49972	class cosh
ctanh 49973	class tanh
df-sinh 49974	⊢ sinh = (𝑥 ∈ ℂ ↦ ((sin‘(i · 𝑥)) / i))
df-cosh 49975	⊢ cosh = (𝑥 ∈ ℂ ↦ (cos‘(i · 𝑥)))
df-tanh 49976	⊢ tanh = (𝑥 ∈ (^◡cosh “ (ℂ ∖ {0})) ↦ ((tan‘(i · 𝑥)) / i))
csec 49982	class sec
ccsc 49983	class csc
ccot 49984	class cot
df-sec 49985	⊢ sec = (𝑥 ∈ {𝑦 ∈ ℂ ∣ (cos‘𝑦) ≠ 0} ↦ (1 / (cos‘𝑥)))
df-csc 49986	⊢ csc = (𝑥 ∈ {𝑦 ∈ ℂ ∣ (sin‘𝑦) ≠ 0} ↦ (1 / (sin‘𝑥)))
df-cot 49987	⊢ cot = (𝑥 ∈ {𝑦 ∈ ℂ ∣ (sin‘𝑦) ≠ 0} ↦ ((cos‘𝑥) / (sin‘𝑥)))
clog- 50006	class log_
df-logbALT 50007	⊢ log_ = (𝑏 ∈ (ℂ ∖ {0, 1}) ↦ (𝑥 ∈ (ℂ ∖ {0}) ↦ ((log‘𝑥) / (log‘𝑏))))
wreflexive 50008	wff 𝑅Reflexive𝐴
df-reflexive 50009	⊢ (𝑅Reflexive𝐴 ↔ (𝑅 ⊆ (𝐴 × 𝐴) ∧ ∀𝑥 ∈ 𝐴 𝑥𝑅𝑥))
wirreflexive 50010	wff 𝑅Irreflexive𝐴
df-irreflexive 50011	⊢ (𝑅Irreflexive𝐴 ↔ (𝑅 ⊆ (𝐴 × 𝐴) ∧ ∀𝑥 ∈ 𝐴 ¬ 𝑥𝑅𝑥))
walsi 50027	wff ∀!𝑥(𝜑 → 𝜓)
walsc 50028	wff ∀!𝑥 ∈ 𝐴𝜑
df-alsi 50029	⊢ (∀!𝑥(𝜑 → 𝜓) ↔ (∀𝑥(𝜑 → 𝜓) ∧ ∃𝑥𝜑))
df-alsc 50030	⊢ (∀!𝑥 ∈ 𝐴𝜑 ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑥 𝑥 ∈ 𝐴))

W3C validator