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	⊢ ([𝑡 / 𝑥]𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
wcel 2111	wff 𝐴 ∈ 𝐵
ax-8 2113	⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧))
ax-9 2121	⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦))
ax-10 2144	⊢ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
ax-11 2160	⊢ (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑)
ax-12 2180	⊢ (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
ax-13 2372	⊢ (¬ 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
wmo 2533	wff ∃*𝑥𝜑
df-mo 2535	⊢ (∃*𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))
weu 2563	wff ∃!𝑥𝜑
df-eu 2564	⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃*𝑥𝜑))
ax-ext 2703	⊢ (∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → 𝑥 = 𝑦)
cab 2709	class {𝑥 ∣ 𝜑}
df-clab 2710	⊢ (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ [𝑥 / 𝑦]𝜑)
df-cleq 2723	⊢ (𝑦 = 𝑧 ↔ ∀𝑢(𝑢 ∈ 𝑦 ↔ 𝑢 ∈ 𝑧)) & ⊢ (𝑡 = 𝑡 ↔ ∀𝑣(𝑣 ∈ 𝑡 ↔ 𝑣 ∈ 𝑡)) ⇒ ⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
df-clel 2806	⊢ (𝑦 ∈ 𝑧 ↔ ∃𝑢(𝑢 = 𝑦 ∧ 𝑢 ∈ 𝑧)) & ⊢ (𝑡 ∈ 𝑡 ↔ ∃𝑣(𝑣 = 𝑡 ∧ 𝑣 ∈ 𝑡)) ⇒ ⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵))
wnfc 2879	wff Ⅎ𝑥𝐴
df-nfc 2881	⊢ (Ⅎ𝑥𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴)
wne 2928	wff 𝐴 ≠ 𝐵
df-ne 2929	⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵)
wnel 3032	wff 𝐴 ∉ 𝐵
df-nel 3033	⊢ (𝐴 ∉ 𝐵 ↔ ¬ 𝐴 ∈ 𝐵)
wral 3047	wff ∀𝑥 ∈ 𝐴 𝜑
df-ral 3048	⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
wrex 3056	wff ∃𝑥 ∈ 𝐴 𝜑
df-rex 3057	⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
wreu 3344	wff ∃!𝑥 ∈ 𝐴 𝜑
wrmo 3345	wff ∃*𝑥 ∈ 𝐴 𝜑
df-rmo 3346	⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
df-reu 3347	⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
crab 3395	class {𝑥 ∈ 𝐴 ∣ 𝜑}
df-rab 3396	⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
cvv 3436	class V
df-v 3438	⊢ V = {𝑥 ∣ 𝑥 = 𝑥}
wcdeq 3722	wff CondEq(𝑥 = 𝑦 → 𝜑)
df-cdeq 3723	⊢ (CondEq(𝑥 = 𝑦 → 𝜑) ↔ (𝑥 = 𝑦 → 𝜑))
wsbc 3741	wff [𝐴 / 𝑥]𝜑
df-sbc 3742	⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑})
csb 3850	class ⦋𝐴 / 𝑥⦌𝐵
df-csb 3851	⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵}
cdif 3899	class (𝐴 ∖ 𝐵)
cun 3900	class (𝐴 ∪ 𝐵)
cin 3901	class (𝐴 ∩ 𝐵)
wss 3902	wff 𝐴 ⊆ 𝐵
wpss 3903	wff 𝐴 ⊊ 𝐵
df-dif 3905	⊢ (𝐴 ∖ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)}
df-un 3907	⊢ (𝐴 ∪ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)}
df-in 3909	⊢ (𝐴 ∩ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)}
df-ss 3919	⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
df-pss 3922	⊢ (𝐴 ⊊ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ 𝐵))
csymdif 4202	class (𝐴 △ 𝐵)
df-symdif 4203	⊢ (𝐴 △ 𝐵) = ((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐴))
c0 4283	class ∅
df-nul 4284	⊢ ∅ = (V ∖ V)
cif 4475	class if(𝜑, 𝐴, 𝐵)
df-if 4476	⊢ if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑))}
cpw 4550	class 𝒫 𝐴
df-pw 4552	⊢ 𝒫 𝐴 = {𝑥 ∣ 𝑥 ⊆ 𝐴}
csn 4576	class {𝐴}
df-sn 4577	⊢ {𝐴} = {𝑥 ∣ 𝑥 = 𝐴}
cpr 4578	class {𝐴, 𝐵}
df-pr 4579	⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
ctp 4580	class {𝐴, 𝐵, 𝐶}
df-tp 4581	⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
cop 4582	class ⟨𝐴, 𝐵⟩
df-op 4583	⊢ ⟨𝐴, 𝐵⟩ = {𝑥 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})}
cotp 4584	class ⟨𝐴, 𝐵, 𝐶⟩
df-ot 4585	⊢ ⟨𝐴, 𝐵, 𝐶⟩ = ⟨⟨𝐴, 𝐵⟩, 𝐶⟩
cuni 4859	class ∪ 𝐴
df-uni 4860	⊢ ∪ 𝐴 = {𝑥 ∣ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)}
cint 4897	class ∩ 𝐴
df-int 4898	⊢ ∩ 𝐴 = {𝑥 ∣ ∀𝑦(𝑦 ∈ 𝐴 → 𝑥 ∈ 𝑦)}
ciun 4941	class ∪ 𝑥 ∈ 𝐴 𝐵
ciin 4942	class ∩ 𝑥 ∈ 𝐴 𝐵
df-iun 4943	⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵}
df-iin 4944	⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵}
wdisj 5058	wff Disj 𝑥 ∈ 𝐴 𝐵
df-disj 5059	⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
wbr 5091	wff 𝐴𝑅𝐵
df-br 5092	⊢ (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
copab 5153	class {⟨𝑥, 𝑦⟩ ∣ 𝜑}
df-opab 5154	⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
cmpt 5172	class (𝑥 ∈ 𝐴 ↦ 𝐵)
df-mpt 5173	⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
wtr 5198	wff Tr 𝐴
df-tr 5199	⊢ (Tr 𝐴 ↔ ∪ 𝐴 ⊆ 𝐴)
ax-rep 5217	⊢ (∀𝑤∃𝑦∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦) → ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)))
ax-sep 5234	⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))
ax-nul 5244	⊢ ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥
ax-pow 5303	⊢ ∃𝑦∀𝑧(∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦)
ax-pr 5370	⊢ ∃𝑧∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧)
cid 5510	class I
df-id 5511	⊢ I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
cep 5515	class E
df-eprel 5516	⊢ E = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ 𝑦}
wpo 5522	wff 𝑅 Po 𝐴
wor 5523	wff 𝑅 Or 𝐴
df-po 5524	⊢ (𝑅 Po 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
df-so 5525	⊢ (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)))
wfr 5566	wff 𝑅 Fr 𝐴
wse 5567	wff 𝑅 Se 𝐴
wwe 5568	wff 𝑅 We 𝐴
df-fr 5569	⊢ (𝑅 Fr 𝐴 ↔ ∀𝑥((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦))
df-se 5570	⊢ (𝑅 Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V)
df-we 5571	⊢ (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴 ∧ 𝑅 Or 𝐴))
cxp 5614	class (𝐴 × 𝐵)
ccnv 5615	class ^◡𝐴
cdm 5616	class dom 𝐴
crn 5617	class ran 𝐴
cres 5618	class (𝐴 ↾ 𝐵)
cima 5619	class (𝐴 “ 𝐵)
ccom 5620	class (𝐴 ∘ 𝐵)
wrel 5621	wff Rel 𝐴
df-xp 5622	⊢ (𝐴 × 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)}
df-rel 5623	⊢ (Rel 𝐴 ↔ 𝐴 ⊆ (V × V))
df-cnv 5624	⊢ ^◡𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥}
df-co 5625	⊢ (𝐴 ∘ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)}
df-dm 5626	⊢ dom 𝐴 = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦}
df-rn 5627	⊢ ran 𝐴 = dom ^◡𝐴
df-res 5628	⊢ (𝐴 ↾ 𝐵) = (𝐴 ∩ (𝐵 × V))
df-ima 5629	⊢ (𝐴 “ 𝐵) = ran (𝐴 ↾ 𝐵)
cpred 6247	class Pred(𝑅, 𝐴, 𝑋)
df-pred 6248	⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (^◡𝑅 “ {𝑋}))
word 6305	wff Ord 𝐴
con0 6306	class On
wlim 6307	wff Lim 𝐴
csuc 6308	class suc 𝐴
df-ord 6309	⊢ (Ord 𝐴 ↔ (Tr 𝐴 ∧ E We 𝐴))
df-on 6310	⊢ On = {𝑥 ∣ Ord 𝑥}
df-lim 6311	⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴))
df-suc 6312	⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
cio 6435	class (℩𝑥𝜑)
df-iota 6437	⊢ (℩𝑥𝜑) = ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}
wfun 6475	wff Fun 𝐴
wfn 6476	wff 𝐴 Fn 𝐵
wf 6477	wff 𝐹:𝐴⟶𝐵
wf1 6478	wff 𝐹:𝐴–1-1→𝐵
wfo 6479	wff 𝐹:𝐴–onto→𝐵
wf1o 6480	wff 𝐹:𝐴–1-1-onto→𝐵
cfv 6481	class (𝐹‘𝐴)
wiso 6482	wff 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵)
df-fun 6483	⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ (𝐴 ∘ ^◡𝐴) ⊆ I ))
df-fn 6484	⊢ (𝐴 Fn 𝐵 ↔ (Fun 𝐴 ∧ dom 𝐴 = 𝐵))
df-f 6485	⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵))
df-f1 6486	⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ Fun ^◡𝐹))
df-fo 6487	⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵))
df-f1o 6488	⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹:𝐴–1-1→𝐵 ∧ 𝐹:𝐴–onto→𝐵))
df-fv 6489	⊢ (𝐹‘𝐴) = (℩𝑥𝐴𝐹𝑥)
df-isom 6490	⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐻:𝐴–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦))))
crio 7302	class (℩𝑥 ∈ 𝐴 𝜑)
df-riota 7303	⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
co 7346	class (𝐴𝐹𝐵)
coprab 7347	class {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
cmpo 7348	class (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
df-ov 7349	⊢ (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
df-oprab 7350	⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)}
df-mpo 7351	⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)}
cof 7608	class ∘_f 𝑅
cofr 7609	class ∘_r 𝑅
df-of 7610	⊢ ∘_f 𝑅 = (𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥))))
df-ofr 7611	⊢ ∘_r 𝑅 = {⟨𝑓, 𝑔⟩ ∣ ∀𝑥 ∈ (dom 𝑓 ∩ dom 𝑔)(𝑓‘𝑥)𝑅(𝑔‘𝑥)}
crpss 7655	class [⊊]
df-rpss 7656	⊢ [⊊] = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ⊊ 𝑦}
ax-un 7668	⊢ ∃𝑦∀𝑧(∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦)
com 7796	class ω
df-om 7797	⊢ ω = {𝑥 ∈ On ∣ ∀𝑦(Lim 𝑦 → 𝑥 ∈ 𝑦)}
c1st 7919	class 1^st
c2nd 7920	class 2^nd
df-1st 7921	⊢ 1^st = (𝑥 ∈ V ↦ ∪ dom {𝑥})
df-2nd 7922	⊢ 2^nd = (𝑥 ∈ V ↦ ∪ ran {𝑥})
csupp 8090	class supp
df-supp 8091	⊢ supp = (𝑥 ∈ V, 𝑧 ∈ V ↦ {𝑖 ∈ dom 𝑥 ∣ (𝑥 “ {𝑖}) ≠ {𝑧}})
ctpos 8155	class tpos 𝐹
df-tpos 8156	⊢ tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ (^◡dom 𝐹 ∪ {∅}) ↦ ∪ ^◡{𝑥}))
ccur 8195	class curry 𝐴
cunc 8196	class uncurry 𝐴
df-cur 8197	⊢ curry 𝐹 = (𝑥 ∈ dom dom 𝐹 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩𝐹𝑧})
df-unc 8198	⊢ uncurry 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(𝐹‘𝑥)𝑧}
cund 8202	class Undef
df-undef 8203	⊢ Undef = (𝑠 ∈ V ↦ 𝒫 ∪ 𝑠)
cfrecs 8210	class frecs(𝑅, 𝐴, 𝐹)
df-frecs 8211	⊢ 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 8241	class wrecs(𝑅, 𝐴, 𝐹)
df-wrecs 8242	⊢ wrecs(𝑅, 𝐴, 𝐹) = frecs(𝑅, 𝐴, (𝐹 ∘ 2^nd ))
wsmo 8265	wff Smo 𝐴
df-smo 8266	⊢ (Smo 𝐴 ↔ (𝐴:dom 𝐴⟶On ∧ Ord dom 𝐴 ∧ ∀𝑥 ∈ dom 𝐴∀𝑦 ∈ dom 𝐴(𝑥 ∈ 𝑦 → (𝐴‘𝑥) ∈ (𝐴‘𝑦))))
crecs 8290	class recs(𝐹)
df-recs 8291	⊢ recs(𝐹) = wrecs( E , On, 𝐹)
crdg 8328	class rec(𝐹, 𝐼)
df-rdg 8329	⊢ rec(𝐹, 𝐼) = recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔))))))
cseqom 8366	class seq_ω(𝐹, 𝐼)
df-seqom 8367	⊢ seq_ω(𝐹, 𝐼) = (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) “ ω)
c1o 8378	class 1_o
c2o 8379	class 2_o
c3o 8380	class 3_o
c4o 8381	class 4_o
coa 8382	class +_o
comu 8383	class ·_o
coe 8384	class ↑_o
df-1o 8385	⊢ 1_o = suc ∅
df-2o 8386	⊢ 2_o = suc 1_o
df-3o 8387	⊢ 3_o = suc 2_o
df-4o 8388	⊢ 4_o = suc 3_o
df-oadd 8389	⊢ +_o = (𝑥 ∈ On, 𝑦 ∈ On ↦ (rec((𝑧 ∈ V ↦ suc 𝑧), 𝑥)‘𝑦))
df-omul 8390	⊢ ·_o = (𝑥 ∈ On, 𝑦 ∈ On ↦ (rec((𝑧 ∈ V ↦ (𝑧 +_o 𝑥)), ∅)‘𝑦))
df-oexp 8391	⊢ ↑_o = (𝑥 ∈ On, 𝑦 ∈ On ↦ if(𝑥 = ∅, (1_o ∖ 𝑦), (rec((𝑧 ∈ V ↦ (𝑧 ·_o 𝑥)), 1_o)‘𝑦)))
cnadd 8580	class +no
df-nadd 8581	⊢ +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 8619	wff 𝑅 Er 𝐴
cec 8620	class [𝐴]𝑅
cqs 8621	class (𝐴 / 𝑅)
df-er 8622	⊢ (𝑅 Er 𝐴 ↔ (Rel 𝑅 ∧ dom 𝑅 = 𝐴 ∧ (^◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅))
df-ec 8624	⊢ [𝐴]𝑅 = (𝑅 “ {𝐴})
df-qs 8628	⊢ (𝐴 / 𝑅) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅}
cmap 8750	class ↑_m
cpm 8751	class ↑_pm
df-map 8752	⊢ ↑_m = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓 ∣ 𝑓:𝑦⟶𝑥})
df-pm 8753	⊢ ↑_pm = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓 ∈ 𝒫 (𝑦 × 𝑥) ∣ Fun 𝑓})
cixp 8821	class X𝑥 ∈ 𝐴 𝐵
df-ixp 8822	⊢ X𝑥 ∈ 𝐴 𝐵 = {𝑓 ∣ (𝑓 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵)}
cen 8866	class ≈
cdom 8867	class ≼
csdm 8868	class ≺
cfn 8869	class Fin
df-en 8870	⊢ ≈ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥–1-1-onto→𝑦}
df-dom 8871	⊢ ≼ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥–1-1→𝑦}
df-sdom 8872	⊢ ≺ = ( ≼ ∖ ≈ )
df-fin 8873	⊢ Fin = {𝑥 ∣ ∃𝑦 ∈ ω 𝑥 ≈ 𝑦}
cfsupp 9245	class finSupp
df-fsupp 9246	⊢ finSupp = {⟨𝑟, 𝑧⟩ ∣ (Fun 𝑟 ∧ (𝑟 supp 𝑧) ∈ Fin)}
cfi 9294	class fi
df-fi 9295	⊢ fi = (𝑥 ∈ V ↦ {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦})
csup 9324	class sup(𝐴, 𝐵, 𝑅)
cinf 9325	class inf(𝐴, 𝐵, 𝑅)
df-sup 9326	⊢ sup(𝐴, 𝐵, 𝑅) = ∪ {𝑥 ∈ 𝐵 ∣ (∀𝑦 ∈ 𝐴 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐵 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐴 𝑦𝑅𝑧))}
df-inf 9327	⊢ inf(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, ^◡𝑅)
coi 9395	class OrdIso(𝑅, 𝐴)
df-oi 9396	⊢ 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 9442	class har
df-har 9443	⊢ har = (𝑥 ∈ V ↦ {𝑦 ∈ On ∣ 𝑦 ≼ 𝑥})
cwdom 9450	class ≼^*
df-wdom 9451	⊢ ≼^* = {⟨𝑥, 𝑦⟩ ∣ (𝑥 = ∅ ∨ ∃𝑧 𝑧:𝑦–onto→𝑥)}
ax-reg 9478	⊢ (∃𝑦 𝑦 ∈ 𝑥 → ∃𝑦(𝑦 ∈ 𝑥 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑥)))
ax-inf 9528	⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦)))
ax-inf2 9531	⊢ ∃𝑥(∃𝑦(𝑦 ∈ 𝑥 ∧ ∀𝑧 ¬ 𝑧 ∈ 𝑦) ∧ ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑧 ∈ 𝑥 ∧ ∀𝑤(𝑤 ∈ 𝑧 ↔ (𝑤 ∈ 𝑦 ∨ 𝑤 = 𝑦)))))
ccnf 9551	class CNF
df-cnf 9552	⊢ 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 9597	class t++𝑅
df-ttrcl 9598	⊢ 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 9626	class TC
df-tc 9627	⊢ TC = (𝑥 ∈ V ↦ ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ Tr 𝑦)})
cr1 9652	class 𝑅₁
crnk 9653	class rank
df-r1 9654	⊢ 𝑅₁ = rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅)
df-rank 9655	⊢ rank = (𝑥 ∈ V ↦ ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅₁‘suc 𝑦)})
cdju 9788	class (𝐴 ⊔ 𝐵)
cinl 9789	class inl
cinr 9790	class inr
df-dju 9791	⊢ (𝐴 ⊔ 𝐵) = (({∅} × 𝐴) ∪ ({1_o} × 𝐵))
df-inl 9792	⊢ inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩)
df-inr 9793	⊢ inr = (𝑥 ∈ V ↦ ⟨1_o, 𝑥⟩)
ccrd 9825	class card
cale 9826	class ℵ
ccf 9827	class cf
wacn 9828	class AC 𝐴
df-card 9829	⊢ card = (𝑥 ∈ V ↦ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥})
df-aleph 9830	⊢ ℵ = rec(har, ω)
df-cf 9831	⊢ cf = (𝑥 ∈ On ↦ ∩ {𝑦 ∣ ∃𝑧(𝑦 = (card‘𝑧) ∧ (𝑧 ⊆ 𝑥 ∧ ∀𝑣 ∈ 𝑥 ∃𝑢 ∈ 𝑧 𝑣 ⊆ 𝑢))})
df-acn 9832	⊢ AC 𝐴 = {𝑥 ∣ (𝐴 ∈ V ∧ ∀𝑓 ∈ ((𝒫 𝑥 ∖ {∅}) ↑_m 𝐴)∃𝑔∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝑓‘𝑦))}
wac 10003	wff CHOICE
df-ac 10004	⊢ (CHOICE ↔ ∀𝑥∃𝑓(𝑓 ⊆ 𝑥 ∧ 𝑓 Fn dom 𝑥))
cfin1a 10166	class Fin^Ia
cfin2 10167	class Fin^II
cfin4 10168	class Fin^IV
cfin3 10169	class Fin^III
cfin5 10170	class Fin^V
cfin6 10171	class Fin^VI
cfin7 10172	class Fin^VII
df-fin1a 10173	⊢ Fin^Ia = {𝑥 ∣ ∀𝑦 ∈ 𝒫 𝑥(𝑦 ∈ Fin ∨ (𝑥 ∖ 𝑦) ∈ Fin)}
df-fin2 10174	⊢ Fin^II = {𝑥 ∣ ∀𝑦 ∈ 𝒫 𝒫 𝑥((𝑦 ≠ ∅ ∧ [⊊] Or 𝑦) → ∪ 𝑦 ∈ 𝑦)}
df-fin4 10175	⊢ Fin^IV = {𝑥 ∣ ¬ ∃𝑦(𝑦 ⊊ 𝑥 ∧ 𝑦 ≈ 𝑥)}
df-fin3 10176	⊢ Fin^III = {𝑥 ∣ 𝒫 𝑥 ∈ Fin^IV}
df-fin5 10177	⊢ Fin^V = {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 ≺ (𝑥 ⊔ 𝑥))}
df-fin6 10178	⊢ Fin^VI = {𝑥 ∣ (𝑥 ≺ 2_o ∨ 𝑥 ≺ (𝑥 × 𝑥))}
df-fin7 10179	⊢ Fin^VII = {𝑥 ∣ ¬ ∃𝑦 ∈ (On ∖ ω)𝑥 ≈ 𝑦}
ax-cc 10323	⊢ (𝑥 ≈ ω → ∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧))
ax-dc 10334	⊢ ((∃𝑦∃𝑧 𝑦𝑥𝑧 ∧ ran 𝑥 ⊆ dom 𝑥) → ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛))
ax-ac 10347	⊢ ∃𝑦∀𝑧∀𝑤((𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → ∃𝑣∀𝑢(∃𝑡((𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑡) ∧ (𝑢 ∈ 𝑡 ∧ 𝑡 ∈ 𝑦)) ↔ 𝑢 = 𝑣))
ax-ac2 10351	⊢ ∃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 10508	class GCH
df-gch 10509	⊢ GCH = (Fin ∪ {𝑥 ∣ ∀𝑦 ¬ (𝑥 ≺ 𝑦 ∧ 𝑦 ≺ 𝒫 𝑥)})
cwina 10570	class Inacc_w
cina 10571	class Inacc
df-wina 10572	⊢ Inacc_w = {𝑥 ∣ (𝑥 ≠ ∅ ∧ (cf‘𝑥) = 𝑥 ∧ ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝑥 𝑦 ≺ 𝑧)}
df-ina 10573	⊢ Inacc = {𝑥 ∣ (𝑥 ≠ ∅ ∧ (cf‘𝑥) = 𝑥 ∧ ∀𝑦 ∈ 𝑥 𝒫 𝑦 ≺ 𝑥)}
cwun 10588	class WUni
cwunm 10589	class wUniCl
df-wun 10590	⊢ WUni = {𝑢 ∣ (Tr 𝑢 ∧ 𝑢 ≠ ∅ ∧ ∀𝑥 ∈ 𝑢 (∪ 𝑥 ∈ 𝑢 ∧ 𝒫 𝑥 ∈ 𝑢 ∧ ∀𝑦 ∈ 𝑢 {𝑥, 𝑦} ∈ 𝑢))}
df-wunc 10591	⊢ wUniCl = (𝑥 ∈ V ↦ ∩ {𝑢 ∈ WUni ∣ 𝑥 ⊆ 𝑢})
ctsk 10636	class Tarski
df-tsk 10637	⊢ Tarski = {𝑦 ∣ (∀𝑧 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑦 ∧ ∃𝑤 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧 ≈ 𝑦 ∨ 𝑧 ∈ 𝑦))}
cgru 10678	class Univ
df-gru 10679	⊢ Univ = {𝑢 ∣ (Tr 𝑢 ∧ ∀𝑥 ∈ 𝑢 (𝒫 𝑥 ∈ 𝑢 ∧ ∀𝑦 ∈ 𝑢 {𝑥, 𝑦} ∈ 𝑢 ∧ ∀𝑦 ∈ (𝑢 ↑_m 𝑥)∪ ran 𝑦 ∈ 𝑢))}
ax-groth 10711	⊢ ∃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 10725	class tarskiMap
df-tskm 10726	⊢ tarskiMap = (𝑥 ∈ V ↦ ∩ {𝑦 ∈ Tarski ∣ 𝑥 ∈ 𝑦})
cnpi 10732	class N
cpli 10733	class +_N
cmi 10734	class ·_N
clti 10735	class <_N
cplpq 10736	class +_pQ
cmpq 10737	class ·_pQ
cltpq 10738	class <_pQ
ceq 10739	class ~_Q
cnq 10740	class Q
c1q 10741	class 1_Q
cerq 10742	class [Q]
cplq 10743	class +_Q
cmq 10744	class ·_Q
crq 10745	class *_Q
cltq 10746	class <_Q
cnp 10747	class P
c1p 10748	class 1_P
cpp 10749	class +_P
cmp 10750	class ·_P
cltp 10751	class <_P
cer 10752	class ~_R
cnr 10753	class R
c0r 10754	class 0_R
c1r 10755	class 1_R
cm1r 10756	class -1_R
cplr 10757	class +_R
cmr 10758	class ·_R
cltr 10759	class <_R
df-ni 10760	⊢ N = (ω ∖ {∅})
df-pli 10761	⊢ +_N = ( +_o ↾ (N × N))
df-mi 10762	⊢ ·_N = ( ·_o ↾ (N × N))
df-lti 10763	⊢ <_N = ( E ∩ (N × N))
df-plpq 10796	⊢ +_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 10797	⊢ ·_pQ = (𝑥 ∈ (N × N), 𝑦 ∈ (N × N) ↦ ⟨((1^st ‘𝑥) ·_N (1^st ‘𝑦)), ((2^nd ‘𝑥) ·_N (2^nd ‘𝑦))⟩)
df-ltpq 10798	⊢ <_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 10799	⊢ ~_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 10800	⊢ Q = {𝑥 ∈ (N × N) ∣ ∀𝑦 ∈ (N × N)(𝑥 ~_Q 𝑦 → ¬ (2^nd ‘𝑦) <_N (2^nd ‘𝑥))}
df-erq 10801	⊢ [Q] = ( ~_Q ∩ ((N × N) × Q))
df-plq 10802	⊢ +_Q = (([Q] ∘ +_pQ ) ↾ (Q × Q))
df-mq 10803	⊢ ·_Q = (([Q] ∘ ·_pQ ) ↾ (Q × Q))
df-1nq 10804	⊢ 1_Q = ⟨1_o, 1_o⟩
df-rq 10805	⊢ *_Q = (^◡ ·_Q “ {1_Q})
df-ltnq 10806	⊢ <_Q = ( <_pQ ∩ (Q × Q))
df-np 10869	⊢ P = {𝑥 ∣ ((∅ ⊊ 𝑥 ∧ 𝑥 ⊊ Q) ∧ ∀𝑦 ∈ 𝑥 (∀𝑧(𝑧 <_Q 𝑦 → 𝑧 ∈ 𝑥) ∧ ∃𝑧 ∈ 𝑥 𝑦 <_Q 𝑧))}
df-1p 10870	⊢ 1_P = {𝑥 ∣ 𝑥 <_Q 1_Q}
df-plp 10871	⊢ +_P = (𝑥 ∈ P, 𝑦 ∈ P ↦ {𝑤 ∣ ∃𝑣 ∈ 𝑥 ∃𝑢 ∈ 𝑦 𝑤 = (𝑣 +_Q 𝑢)})
df-mp 10872	⊢ ·_P = (𝑥 ∈ P, 𝑦 ∈ P ↦ {𝑤 ∣ ∃𝑣 ∈ 𝑥 ∃𝑢 ∈ 𝑦 𝑤 = (𝑣 ·_Q 𝑢)})
df-ltp 10873	⊢ <_P = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ 𝑥 ⊊ 𝑦)}
df-enr 10943	⊢ ~_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 10944	⊢ R = ((P × P) / ~_R )
df-plr 10945	⊢ +_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 10946	⊢ ·_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 10947	⊢ <_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 10948	⊢ 0_R = [⟨1_P, 1_P⟩] ~_R
df-1r 10949	⊢ 1_R = [⟨(1_P +_P 1_P), 1_P⟩] ~_R
df-m1r 10950	⊢ -1_R = [⟨1_P, (1_P +_P 1_P)⟩] ~_R
cc 11001	class ℂ
cr 11002	class ℝ
cc0 11003	class 0
c1 11004	class 1
ci 11005	class i
caddc 11006	class +
cltrr 11007	class <_ℝ
cmul 11008	class ·
df-c 11009	⊢ ℂ = (R × R)
df-0 11010	⊢ 0 = ⟨0_R, 0_R⟩
df-1 11011	⊢ 1 = ⟨1_R, 0_R⟩
df-i 11012	⊢ i = ⟨0_R, 1_R⟩
df-r 11013	⊢ ℝ = (R × {0_R})
df-add 11014	⊢ + = {⟨⟨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 11015	⊢ · = {⟨⟨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 11016	⊢ <_ℝ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ∃𝑧∃𝑤((𝑥 = ⟨𝑧, 0_R⟩ ∧ 𝑦 = ⟨𝑤, 0_R⟩) ∧ 𝑧 <_R 𝑤))}
ax-cnex 11059	⊢ ℂ ∈ V
ax-resscn 11060	⊢ ℝ ⊆ ℂ
ax-1cn 11061	⊢ 1 ∈ ℂ
ax-icn 11062	⊢ i ∈ ℂ
ax-addcl 11063	⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) ∈ ℂ)
ax-addrcl 11064	⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + 𝐵) ∈ ℝ)
ax-mulcl 11065	⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) ∈ ℂ)
ax-mulrcl 11066	⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 · 𝐵) ∈ ℝ)
ax-mulcom 11067	⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) = (𝐵 · 𝐴))
ax-addass 11068	⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶)))
ax-mulass 11069	⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)))
ax-distr 11070	⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶)))
ax-i2m1 11071	⊢ ((i · i) + 1) = 0
ax-1ne0 11072	⊢ 1 ≠ 0
ax-1rid 11073	⊢ (𝐴 ∈ ℝ → (𝐴 · 1) = 𝐴)
ax-rnegex 11074	⊢ (𝐴 ∈ ℝ → ∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0)
ax-rrecex 11075	⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → ∃𝑥 ∈ ℝ (𝐴 · 𝑥) = 1)
ax-cnre 11076	⊢ (𝐴 ∈ ℂ → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝐴 = (𝑥 + (i · 𝑦)))
ax-pre-lttri 11077	⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 <_ℝ 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 <_ℝ 𝐴)))
ax-pre-lttrn 11078	⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 <_ℝ 𝐵 ∧ 𝐵 <_ℝ 𝐶) → 𝐴 <_ℝ 𝐶))
ax-pre-ltadd 11079	⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 <_ℝ 𝐵 → (𝐶 + 𝐴) <_ℝ (𝐶 + 𝐵)))
ax-pre-mulgt0 11080	⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((0 <_ℝ 𝐴 ∧ 0 <_ℝ 𝐵) → 0 <_ℝ (𝐴 · 𝐵)))
ax-pre-sup 11081	⊢ ((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 11082	⊢ + :(ℂ × ℂ)⟶ℂ
ax-mulf 11083	⊢ · :(ℂ × ℂ)⟶ℂ
cpnf 11140	class +∞
cmnf 11141	class -∞
cxr 11142	class ℝ^*
clt 11143	class <
cle 11144	class ≤
df-pnf 11145	⊢ +∞ = 𝒫 ∪ ℂ
df-mnf 11146	⊢ -∞ = 𝒫 +∞
df-xr 11147	⊢ ℝ^* = (ℝ ∪ {+∞, -∞})
df-ltxr 11148	⊢ < = ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <_ℝ 𝑦)} ∪ (((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ)))
df-le 11149	⊢ ≤ = ((ℝ^* × ℝ^*) ∖ ^◡ < )
cmin 11341	class −
cneg 11342	class -𝐴
df-sub 11343	⊢ − = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥))
df-neg 11344	⊢ -𝐴 = (0 − 𝐴)
cdiv 11771	class /
df-div 11772	⊢ / = (𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥))
cn 12122	class ℕ
df-nn 12123	⊢ ℕ = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) “ ω)
c2 12177	class 2
c3 12178	class 3
c4 12179	class 4
c5 12180	class 5
c6 12181	class 6
c7 12182	class 7
c8 12183	class 8
c9 12184	class 9
df-2 12185	⊢ 2 = (1 + 1)
df-3 12186	⊢ 3 = (2 + 1)
df-4 12187	⊢ 4 = (3 + 1)
df-5 12188	⊢ 5 = (4 + 1)
df-6 12189	⊢ 6 = (5 + 1)
df-7 12190	⊢ 7 = (6 + 1)
df-8 12191	⊢ 8 = (7 + 1)
df-9 12192	⊢ 9 = (8 + 1)
cn0 12378	class ℕ₀
df-n0 12379	⊢ ℕ₀ = (ℕ ∪ {0})
cxnn0 12451	class ℕ₀^*
df-xnn0 12452	⊢ ℕ₀^* = (ℕ₀ ∪ {+∞})
cz 12465	class ℤ
df-z 12466	⊢ ℤ = {𝑛 ∈ ℝ ∣ (𝑛 = 0 ∨ 𝑛 ∈ ℕ ∨ -𝑛 ∈ ℕ)}
cdc 12585	class ;𝐴𝐵
df-dec 12586	⊢ ;𝐴𝐵 = (((9 + 1) · 𝐴) + 𝐵)
cuz 12729	class ℤ_≥
df-uz 12730	⊢ ℤ_≥ = (𝑗 ∈ ℤ ↦ {𝑘 ∈ ℤ ∣ 𝑗 ≤ 𝑘})
cq 12843	class ℚ
df-q 12844	⊢ ℚ = ( / “ (ℤ × ℕ))
crp 12887	class ℝ⁺
df-rp 12888	⊢ ℝ⁺ = {𝑥 ∈ ℝ ∣ 0 < 𝑥}
cxne 13005	class -_𝑒𝐴
cxad 13006	class +_𝑒
cxmu 13007	class ·_e
df-xneg 13008	⊢ -_𝑒𝐴 = if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴))
df-xadd 13009	⊢ +_𝑒 = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ if(𝑥 = +∞, if(𝑦 = -∞, 0, +∞), if(𝑥 = -∞, if(𝑦 = +∞, 0, -∞), if(𝑦 = +∞, +∞, if(𝑦 = -∞, -∞, (𝑥 + 𝑦))))))
df-xmul 13010	⊢ ·_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 13242	class (,)
cioc 13243	class (,]
cico 13244	class [,)
cicc 13245	class [,]
df-ioo 13246	⊢ (,) = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)})
df-ioc 13247	⊢ (,] = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 < 𝑧 ∧ 𝑧 ≤ 𝑦)})
df-ico 13248	⊢ [,) = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)})
df-icc 13249	⊢ [,] = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)})
cfz 13404	class ...
df-fz 13405	⊢ ... = (𝑚 ∈ ℤ, 𝑛 ∈ ℤ ↦ {𝑘 ∈ ℤ ∣ (𝑚 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛)})
cfzo 13551	class ..^
df-fzo 13552	⊢ ..^ = (𝑚 ∈ ℤ, 𝑛 ∈ ℤ ↦ (𝑚...(𝑛 − 1)))
cfl 13691	class ⌊
cceil 13692	class ⌈
df-fl 13693	⊢ ⌊ = (𝑥 ∈ ℝ ↦ (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))))
df-ceil 13694	⊢ ⌈ = (𝑥 ∈ ℝ ↦ -(⌊‘-𝑥))
cmo 13770	class mod
df-mod 13771	⊢ mod = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ⁺ ↦ (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦)))))
cseq 13905	class seq𝑀( + , 𝐹)
df-seq 13906	⊢ seq𝑀( + , 𝐹) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) “ ω)
cexp 13965	class ↑
df-exp 13966	⊢ ↑ = (𝑥 ∈ ℂ, 𝑦 ∈ ℤ ↦ if(𝑦 = 0, 1, if(0 < 𝑦, (seq1( · , (ℕ × {𝑥}))‘𝑦), (1 / (seq1( · , (ℕ × {𝑥}))‘-𝑦)))))
cfa 14177	class !
df-fac 14178	⊢ ! = ({⟨0, 1⟩} ∪ seq1( · , I ))
cbc 14206	class C
df-bc 14207	⊢ C = (𝑛 ∈ ℕ₀, 𝑘 ∈ ℤ ↦ if(𝑘 ∈ (0...𝑛), ((!‘𝑛) / ((!‘(𝑛 − 𝑘)) · (!‘𝑘))), 0))
chash 14234	class ♯
df-hash 14235	⊢ ♯ = (((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω) ∘ card) ∪ ((V ∖ Fin) × {+∞}))
cword 14417	class Word 𝑆
df-word 14418	⊢ Word 𝑆 = {𝑤 ∣ ∃𝑙 ∈ ℕ₀ 𝑤:(0..^𝑙)⟶𝑆}
clsw 14466	class lastS
df-lsw 14467	⊢ lastS = (𝑤 ∈ V ↦ (𝑤‘((♯‘𝑤) − 1)))
cconcat 14474	class ++
df-concat 14475	⊢ ++ = (𝑠 ∈ V, 𝑡 ∈ V ↦ (𝑥 ∈ (0..^((♯‘𝑠) + (♯‘𝑡))) ↦ if(𝑥 ∈ (0..^(♯‘𝑠)), (𝑠‘𝑥), (𝑡‘(𝑥 − (♯‘𝑠))))))
cs1 14500	class ⟨“𝐴”⟩
df-s1 14501	⊢ ⟨“𝐴”⟩ = {⟨0, ( I ‘𝐴)⟩}
csubstr 14545	class substr
df-substr 14546	⊢ 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 14575	class prefix
df-pfx 14576	⊢ prefix = (𝑠 ∈ V, 𝑙 ∈ ℕ₀ ↦ (𝑠 substr ⟨0, 𝑙⟩))
csplice 14653	class splice
df-splice 14654	⊢ splice = (𝑠 ∈ V, 𝑏 ∈ V ↦ (((𝑠 prefix (1^st ‘(1^st ‘𝑏))) ++ (2^nd ‘𝑏)) ++ (𝑠 substr ⟨(2^nd ‘(1^st ‘𝑏)), (♯‘𝑠)⟩)))
creverse 14662	class reverse
df-reverse 14663	⊢ reverse = (𝑠 ∈ V ↦ (𝑥 ∈ (0..^(♯‘𝑠)) ↦ (𝑠‘(((♯‘𝑠) − 1) − 𝑥))))
creps 14672	class repeatS
df-reps 14673	⊢ repeatS = (𝑠 ∈ V, 𝑛 ∈ ℕ₀ ↦ (𝑥 ∈ (0..^𝑛) ↦ 𝑠))
ccsh 14692	class cyclShift
df-csh 14693	⊢ 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 14745	class ⟨“𝐴𝐵”⟩
cs3 14746	class ⟨“𝐴𝐵𝐶”⟩
cs4 14747	class ⟨“𝐴𝐵𝐶𝐷”⟩
cs5 14748	class ⟨“𝐴𝐵𝐶𝐷𝐸”⟩
cs6 14749	class ⟨“𝐴𝐵𝐶𝐷𝐸𝐹”⟩
cs7 14750	class ⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”⟩
cs8 14751	class ⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺𝐻”⟩
df-s2 14752	⊢ ⟨“𝐴𝐵”⟩ = (⟨“𝐴”⟩ ++ ⟨“𝐵”⟩)
df-s3 14753	⊢ ⟨“𝐴𝐵𝐶”⟩ = (⟨“𝐴𝐵”⟩ ++ ⟨“𝐶”⟩)
df-s4 14754	⊢ ⟨“𝐴𝐵𝐶𝐷”⟩ = (⟨“𝐴𝐵𝐶”⟩ ++ ⟨“𝐷”⟩)
df-s5 14755	⊢ ⟨“𝐴𝐵𝐶𝐷𝐸”⟩ = (⟨“𝐴𝐵𝐶𝐷”⟩ ++ ⟨“𝐸”⟩)
df-s6 14756	⊢ ⟨“𝐴𝐵𝐶𝐷𝐸𝐹”⟩ = (⟨“𝐴𝐵𝐶𝐷𝐸”⟩ ++ ⟨“𝐹”⟩)
df-s7 14757	⊢ ⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”⟩ = (⟨“𝐴𝐵𝐶𝐷𝐸𝐹”⟩ ++ ⟨“𝐺”⟩)
df-s8 14758	⊢ ⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺𝐻”⟩ = (⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”⟩ ++ ⟨“𝐻”⟩)
ctcl 14889	class t+
crtcl 14890	class t*
df-trcl 14891	⊢ t+ = (𝑥 ∈ V ↦ ∩ {𝑧 ∣ (𝑥 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)})
df-rtrcl 14892	⊢ t* = (𝑥 ∈ V ↦ ∩ {𝑧 ∣ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧 ∧ 𝑥 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)})
crelexp 14923	class ↑_𝑟
df-relexp 14924	⊢ ↑_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 14959	class t*rec
df-rtrclrec 14960	⊢ t*rec = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ₀ (𝑟↑_𝑟𝑛))
cshi 14970	class shift
df-shft 14971	⊢ shift = (𝑓 ∈ V, 𝑥 ∈ ℂ ↦ {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ ℂ ∧ (𝑦 − 𝑥)𝑓𝑧)})
csgn 14990	class sgn
df-sgn 14991	⊢ sgn = (𝑥 ∈ ℝ^* ↦ if(𝑥 = 0, 0, if(𝑥 < 0, -1, 1)))
ccj 15000	class ∗
cre 15001	class ℜ
cim 15002	class ℑ
df-cj 15003	⊢ ∗ = (𝑥 ∈ ℂ ↦ (℩𝑦 ∈ ℂ ((𝑥 + 𝑦) ∈ ℝ ∧ (i · (𝑥 − 𝑦)) ∈ ℝ)))
df-re 15004	⊢ ℜ = (𝑥 ∈ ℂ ↦ ((𝑥 + (∗‘𝑥)) / 2))
df-im 15005	⊢ ℑ = (𝑥 ∈ ℂ ↦ (ℜ‘(𝑥 / i)))
csqrt 15137	class √
cabs 15138	class abs
df-sqrt 15139	⊢ √ = (𝑥 ∈ ℂ ↦ (℩𝑦 ∈ ℂ ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ⁺)))
df-abs 15140	⊢ abs = (𝑥 ∈ ℂ ↦ (√‘(𝑥 · (∗‘𝑥))))
clsp 15374	class lim sup
df-limsup 15375	⊢ lim sup = (𝑥 ∈ V ↦ inf(ran (𝑘 ∈ ℝ ↦ sup(((𝑥 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )), ℝ^*, < ))
cli 15388	class ⇝
crli 15389	class ⇝_𝑟
co1 15390	class 𝑂(1)
clo1 15391	class ≤𝑂(1)
df-clim 15392	⊢ ⇝ = {⟨𝑓, 𝑦⟩ ∣ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝑓‘𝑘) ∈ ℂ ∧ (abs‘((𝑓‘𝑘) − 𝑦)) < 𝑥))}
df-rlim 15393	⊢ ⇝_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 15394	⊢ 𝑂(1) = {𝑓 ∈ (ℂ ↑_pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(abs‘(𝑓‘𝑦)) ≤ 𝑚}
df-lo1 15395	⊢ ≤𝑂(1) = {𝑓 ∈ (ℝ ↑_pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(𝑓‘𝑦) ≤ 𝑚}
csu 15590	class Σ𝑘 ∈ 𝐴 𝐵
df-sum 15591	⊢ 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 15807	class ∏𝑘 ∈ 𝐴 𝐵
df-prod 15808	⊢ ∏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 15908	class FallFac
crisefac 15909	class RiseFac
df-risefac 15910	⊢ RiseFac = (𝑥 ∈ ℂ, 𝑛 ∈ ℕ₀ ↦ ∏𝑘 ∈ (0...(𝑛 − 1))(𝑥 + 𝑘))
df-fallfac 15911	⊢ FallFac = (𝑥 ∈ ℂ, 𝑛 ∈ ℕ₀ ↦ ∏𝑘 ∈ (0...(𝑛 − 1))(𝑥 − 𝑘))
cbp 15950	class BernPoly
df-bpoly 15951	⊢ 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 15965	class exp
ceu 15966	class e
csin 15967	class sin
ccos 15968	class cos
ctan 15969	class tan
cpi 15970	class π
df-ef 15971	⊢ exp = (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ ℕ₀ ((𝑥↑𝑘) / (!‘𝑘)))
df-e 15972	⊢ e = (exp‘1)
df-sin 15973	⊢ sin = (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) − (exp‘(-i · 𝑥))) / (2 · i)))
df-cos 15974	⊢ cos = (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥))) / 2))
df-tan 15975	⊢ tan = (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) ↦ ((sin‘𝑥) / (cos‘𝑥)))
df-pi 15976	⊢ π = inf((ℝ⁺ ∩ (^◡sin “ {0})), ℝ, < )
ctau 16108	class τ
df-tau 16109	⊢ τ = inf((ℝ⁺ ∩ (^◡cos “ {1})), ℝ, < )
cdvds 16160	class ∥
df-dvds 16161	⊢ ∥ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ ∃𝑛 ∈ ℤ (𝑛 · 𝑥) = 𝑦)}
cbits 16327	class bits
csad 16328	class sadd
csmu 16329	class smul
df-bits 16330	⊢ bits = (𝑛 ∈ ℤ ↦ {𝑚 ∈ ℕ₀ ∣ ¬ 2 ∥ (⌊‘(𝑛 / (2↑𝑚)))})
df-sad 16359	⊢ 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 16384	⊢ 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 16402	class gcd
df-gcd 16403	⊢ gcd = (𝑥 ∈ ℤ, 𝑦 ∈ ℤ ↦ if((𝑥 = 0 ∧ 𝑦 = 0), 0, sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑥 ∧ 𝑛 ∥ 𝑦)}, ℝ, < )))
clcm 16496	class lcm
clcmf 16497	class lcm
df-lcm 16498	⊢ lcm = (𝑥 ∈ ℤ, 𝑦 ∈ ℤ ↦ if((𝑥 = 0 ∨ 𝑦 = 0), 0, inf({𝑛 ∈ ℕ ∣ (𝑥 ∥ 𝑛 ∧ 𝑦 ∥ 𝑛)}, ℝ, < )))
df-lcmf 16499	⊢ lcm = (𝑧 ∈ 𝒫 ℤ ↦ if(0 ∈ 𝑧, 0, inf({𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑧 𝑚 ∥ 𝑛}, ℝ, < )))
cprime 16579	class ℙ
df-prm 16580	⊢ ℙ = {𝑝 ∈ ℕ ∣ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑝} ≈ 2_o}
cnumer 16641	class numer
cdenom 16642	class denom
df-numer 16643	⊢ 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 16644	⊢ denom = (𝑦 ∈ ℚ ↦ (2^nd ‘(℩𝑥 ∈ (ℤ × ℕ)(((1^st ‘𝑥) gcd (2^nd ‘𝑥)) = 1 ∧ 𝑦 = ((1^st ‘𝑥) / (2^nd ‘𝑥))))))
codz 16671	class od_ℤ
cphi 16672	class ϕ
df-odz 16673	⊢ od_ℤ = (𝑛 ∈ ℕ ↦ (𝑥 ∈ {𝑥 ∈ ℤ ∣ (𝑥 gcd 𝑛) = 1} ↦ inf({𝑚 ∈ ℕ ∣ 𝑛 ∥ ((𝑥↑𝑚) − 1)}, ℝ, < )))
df-phi 16674	⊢ ϕ = (𝑛 ∈ ℕ ↦ (♯‘{𝑥 ∈ (1...𝑛) ∣ (𝑥 gcd 𝑛) = 1}))
cpc 16745	class pCnt
df-pc 16746	⊢ 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 16838	class ℤ[i]
df-gz 16839	⊢ ℤ[i] = {𝑥 ∈ ℂ ∣ ((ℜ‘𝑥) ∈ ℤ ∧ (ℑ‘𝑥) ∈ ℤ)}
cvdwa 16874	class AP
cvdwm 16875	class MonoAP
cvdwp 16876	class PolyAP
df-vdwap 16877	⊢ AP = (𝑘 ∈ ℕ₀ ↦ (𝑎 ∈ ℕ, 𝑑 ∈ ℕ ↦ ran (𝑚 ∈ (0...(𝑘 − 1)) ↦ (𝑎 + (𝑚 · 𝑑)))))
df-vdwmc 16878	⊢ MonoAP = {⟨𝑘, 𝑓⟩ ∣ ∃𝑐(ran (AP‘𝑘) ∩ 𝒫 (^◡𝑓 “ {𝑐})) ≠ ∅}
df-vdwpc 16879	⊢ 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 16908	class Ramsey
df-ram 16910	⊢ 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 16940	class #_p
df-prmo 16941	⊢ #_p = (𝑛 ∈ ℕ₀ ↦ ∏𝑘 ∈ (1...𝑛)if(𝑘 ∈ ℙ, 𝑘, 1))
cstr 17054	class Struct
df-struct 17055	⊢ Struct = {⟨𝑓, 𝑥⟩ ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))}
csts 17071	class sSet
df-sets 17072	⊢ sSet = (𝑠 ∈ V, 𝑒 ∈ V ↦ ((𝑠 ↾ (V ∖ dom {𝑒})) ∪ {𝑒}))
cslot 17089	class Slot 𝐴
df-slot 17090	⊢ Slot 𝐴 = (𝑥 ∈ V ↦ (𝑥‘𝐴))
cnx 17101	class ndx
df-ndx 17102	⊢ ndx = ( I ↾ ℕ)
cbs 17117	class Base
df-base 17118	⊢ Base = Slot 1
cress 17138	class ↾_s
df-ress 17139	⊢ ↾_s = (𝑤 ∈ V, 𝑥 ∈ V ↦ if((Base‘𝑤) ⊆ 𝑥, 𝑤, (𝑤 sSet ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩)))
cplusg 17158	class +_g
cmulr 17159	class ._r
cstv 17160	class *_𝑟
csca 17161	class Scalar
cvsca 17162	class ·_𝑠
cip 17163	class ·_𝑖
cts 17164	class TopSet
cple 17165	class le
coc 17166	class oc
cds 17167	class dist
cunif 17168	class UnifSet
chom 17169	class Hom
cco 17170	class comp
df-plusg 17171	⊢ +_g = Slot 2
df-mulr 17172	⊢ ._r = Slot 3
df-starv 17173	⊢ *_𝑟 = Slot 4
df-sca 17174	⊢ Scalar = Slot 5
df-vsca 17175	⊢ ·_𝑠 = Slot 6
df-ip 17176	⊢ ·_𝑖 = Slot 8
df-tset 17177	⊢ TopSet = Slot 9
df-ple 17178	⊢ le = Slot ;10
df-ocomp 17179	⊢ oc = Slot ;11
df-ds 17180	⊢ dist = Slot ;12
df-unif 17181	⊢ UnifSet = Slot ;13
df-hom 17182	⊢ Hom = Slot ;14
df-cco 17183	⊢ comp = Slot ;15
crest 17321	class ↾_t
ctopn 17322	class TopOpen
df-rest 17323	⊢ ↾_t = (𝑗 ∈ V, 𝑥 ∈ V ↦ ran (𝑦 ∈ 𝑗 ↦ (𝑦 ∩ 𝑥)))
df-topn 17324	⊢ TopOpen = (𝑤 ∈ V ↦ ((TopSet‘𝑤) ↾_t (Base‘𝑤)))
ctg 17338	class topGen
cpt 17339	class ∏_t
c0g 17340	class 0_g
cgsu 17341	class Σ_g
df-0g 17342	⊢ 0_g = (𝑔 ∈ V ↦ (℩𝑒(𝑒 ∈ (Base‘𝑔) ∧ ∀𝑥 ∈ (Base‘𝑔)((𝑒(+_g‘𝑔)𝑥) = 𝑥 ∧ (𝑥(+_g‘𝑔)𝑒) = 𝑥))))
df-gsum 17343	⊢ 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 17344	⊢ topGen = (𝑥 ∈ V ↦ {𝑦 ∣ 𝑦 ⊆ ∪ (𝑥 ∩ 𝒫 𝑦)})
df-pt 17345	⊢ ∏_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 17346	class X_s
cpws 17347	class ↑_s
df-prds 17348	⊢ 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 17350	⊢ ↑_s = (𝑟 ∈ V, 𝑖 ∈ V ↦ ((Scalar‘𝑟)X_s(𝑖 × {𝑟})))
cordt 17400	class ordTop
cxrs 17401	class ℝ^*_𝑠
df-ordt 17402	⊢ ordTop = (𝑟 ∈ V ↦ (topGen‘(fi‘({dom 𝑟} ∪ ran ((𝑥 ∈ dom 𝑟 ↦ {𝑦 ∈ dom 𝑟 ∣ ¬ 𝑦𝑟𝑥}) ∪ (𝑥 ∈ dom 𝑟 ↦ {𝑦 ∈ dom 𝑟 ∣ ¬ 𝑥𝑟𝑦}))))))
df-xrs 17403	⊢ 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 17404	class qTop
cimas 17405	class “_s
cqus 17406	class /_s
cxps 17407	class ×_s
df-qtop 17408	⊢ qTop = (𝑗 ∈ V, 𝑓 ∈ V ↦ {𝑠 ∈ 𝒫 (𝑓 “ ∪ 𝑗) ∣ ((^◡𝑓 “ 𝑠) ∩ ∪ 𝑗) ∈ 𝑗})
df-imas 17409	⊢ “_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 17410	⊢ /_s = (𝑟 ∈ V, 𝑒 ∈ V ↦ ((𝑥 ∈ (Base‘𝑟) ↦ [𝑥]𝑒) “_s 𝑟))
df-xps 17411	⊢ ×_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 17481	class Moore
cmrc 17482	class mrCls
cmri 17483	class mrInd
cacs 17484	class ACS
df-mre 17485	⊢ Moore = (𝑥 ∈ V ↦ {𝑐 ∈ 𝒫 𝒫 𝑥 ∣ (𝑥 ∈ 𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝑐))})
df-mrc 17486	⊢ mrCls = (𝑐 ∈ ∪ ran Moore ↦ (𝑥 ∈ 𝒫 ∪ 𝑐 ↦ ∩ {𝑠 ∈ 𝑐 ∣ 𝑥 ⊆ 𝑠}))
df-mri 17487	⊢ mrInd = (𝑐 ∈ ∪ ran Moore ↦ {𝑠 ∈ 𝒫 ∪ 𝑐 ∣ ∀𝑥 ∈ 𝑠 ¬ 𝑥 ∈ ((mrCls‘𝑐)‘(𝑠 ∖ {𝑥}))})
df-acs 17488	⊢ ACS = (𝑥 ∈ V ↦ {𝑐 ∈ (Moore‘𝑥) ∣ ∃𝑓(𝑓:𝒫 𝑥⟶𝒫 𝑥 ∧ ∀𝑠 ∈ 𝒫 𝑥(𝑠 ∈ 𝑐 ↔ ∪ (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠))})
ccat 17567	class Cat
ccid 17568	class Id
chomf 17569	class Hom_f
ccomf 17570	class comp_f
df-cat 17571	⊢ 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 17572	⊢ 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 17573	⊢ Hom_f = (𝑐 ∈ V ↦ (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ (𝑥(Hom ‘𝑐)𝑦)))
df-comf 17574	⊢ 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 17614	class oppCat
df-oppc 17615	⊢ 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 17632	class Mono
cepi 17633	class Epi
df-mon 17634	⊢ 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 17635	⊢ Epi = (𝑐 ∈ Cat ↦ tpos (Mono‘(oppCat‘𝑐)))
csect 17648	class Sect
cinv 17649	class Inv
ciso 17650	class Iso
df-sect 17651	⊢ 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 17652	⊢ Inv = (𝑐 ∈ Cat ↦ (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ ((𝑥(Sect‘𝑐)𝑦) ∩ ^◡(𝑦(Sect‘𝑐)𝑥))))
df-iso 17653	⊢ Iso = (𝑐 ∈ Cat ↦ ((𝑥 ∈ V ↦ dom 𝑥) ∘ (Inv‘𝑐)))
ccic 17699	class ≃_𝑐
df-cic 17700	⊢ ≃_𝑐 = (𝑐 ∈ Cat ↦ ((Iso‘𝑐) supp ∅))
cssc 17711	class ⊆_cat
cresc 17712	class ↾_cat
csubc 17713	class Subcat
df-ssc 17714	⊢ ⊆_cat = {⟨ℎ, 𝑗⟩ ∣ ∃𝑡(𝑗 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝑗‘𝑥))}
df-resc 17715	⊢ ↾_cat = (𝑐 ∈ V, ℎ ∈ V ↦ ((𝑐 ↾_s dom dom ℎ) sSet ⟨(Hom ‘ndx), ℎ⟩))
df-subc 17716	⊢ 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 17758	class Func
cidfu 17759	class id_func
ccofu 17760	class ∘_func
cresf 17761	class ↾_f
df-func 17762	⊢ 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 17763	⊢ id_func = (𝑡 ∈ Cat ↦ ⦋(Base‘𝑡) / 𝑏⦌⟨( I ↾ 𝑏), (𝑧 ∈ (𝑏 × 𝑏) ↦ ( I ↾ ((Hom ‘𝑡)‘𝑧)))⟩)
df-cofu 17764	⊢ ∘_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 17765	⊢ ↾_f = (𝑓 ∈ V, ℎ ∈ V ↦ ⟨((1^st ‘𝑓) ↾ dom dom ℎ), (𝑥 ∈ dom ℎ ↦ (((2^nd ‘𝑓)‘𝑥) ↾ (ℎ‘𝑥)))⟩)
cful 17808	class Full
cfth 17809	class Faith
df-full 17810	⊢ 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 17811	⊢ Faith = (𝑐 ∈ Cat, 𝑑 ∈ Cat ↦ {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝑐 Func 𝑑)𝑔 ∧ ∀𝑥 ∈ (Base‘𝑐)∀𝑦 ∈ (Base‘𝑐)Fun ^◡(𝑥𝑔𝑦))})
cnat 17848	class Nat
cfuc 17849	class FuncCat
df-nat 17850	⊢ 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 17851	⊢ 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 17885	class InitO
ctermo 17886	class TermO
czeroo 17887	class ZeroO
df-inito 17888	⊢ InitO = (𝑐 ∈ Cat ↦ {𝑎 ∈ (Base‘𝑐) ∣ ∀𝑏 ∈ (Base‘𝑐)∃!ℎ ℎ ∈ (𝑎(Hom ‘𝑐)𝑏)})
df-termo 17889	⊢ TermO = (𝑐 ∈ Cat ↦ {𝑎 ∈ (Base‘𝑐) ∣ ∀𝑏 ∈ (Base‘𝑐)∃!ℎ ℎ ∈ (𝑏(Hom ‘𝑐)𝑎)})
df-zeroo 17890	⊢ ZeroO = (𝑐 ∈ Cat ↦ ((InitO‘𝑐) ∩ (TermO‘𝑐)))
cdoma 17924	class dom_a
ccoda 17925	class cod_a
carw 17926	class Arrow
choma 17927	class Hom_a
df-doma 17928	⊢ dom_a = (1^st ∘ 1^st )
df-coda 17929	⊢ cod_a = (2^nd ∘ 1^st )
df-homa 17930	⊢ Hom_a = (𝑐 ∈ Cat ↦ (𝑥 ∈ ((Base‘𝑐) × (Base‘𝑐)) ↦ ({𝑥} × ((Hom ‘𝑐)‘𝑥))))
df-arw 17931	⊢ Arrow = (𝑐 ∈ Cat ↦ ∪ ran (Hom_a‘𝑐))
cida 17957	class Id_a
ccoa 17958	class comp_a
df-ida 17959	⊢ Id_a = (𝑐 ∈ Cat ↦ (𝑥 ∈ (Base‘𝑐) ↦ ⟨𝑥, 𝑥, ((Id‘𝑐)‘𝑥)⟩))
df-coa 17960	⊢ 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 17979	class SetCat
df-setc 17980	⊢ 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 18002	class CatCat
df-catc 18003	⊢ 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 18025	class ExtStrCat
df-estrc 18026	⊢ 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 18071	class ×_c
c1stf 18072	class 1^st_F
c2ndf 18073	class 2^nd_F
cprf 18074	class ⟨,⟩_F
df-xpc 18075	⊢ ×_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 18076	⊢ 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 18077	⊢ 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 18078	⊢ ⟨,⟩_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 18112	class eval_F
ccurf 18113	class curry_F
cuncf 18114	class uncurry_F
cdiag 18115	class Δ_func
df-evlf 18116	⊢ 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 18117	⊢ 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 18118	⊢ 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 18119	⊢ Δ_func = (𝑐 ∈ Cat, 𝑑 ∈ Cat ↦ (⟨𝑐, 𝑑⟩ curry_F (𝑐 1^st_F 𝑑)))
chof 18151	class Hom_F
cyon 18152	class Yon
df-hof 18153	⊢ 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 18154	⊢ Yon = (𝑐 ∈ Cat ↦ (⟨𝑐, (oppCat‘𝑐)⟩ curry_F (Hom_F‘(oppCat‘𝑐))))
codu 18189	class ODual
df-odu 18190	⊢ ODual = (𝑤 ∈ V ↦ (𝑤 sSet ⟨(le‘ndx), ^◡(le‘𝑤)⟩))
cproset 18195	class Proset
cdrs 18196	class Dirset
df-proset 18197	⊢ Proset = {𝑓 ∣ [(Base‘𝑓) / 𝑏][(le‘𝑓) / 𝑟]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) → 𝑥𝑟𝑧))}
df-drs 18198	⊢ Dirset = {𝑓 ∈ Proset ∣ [(Base‘𝑓) / 𝑏][(le‘𝑓) / 𝑟](𝑏 ≠ ∅ ∧ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∃𝑧 ∈ 𝑏 (𝑥𝑟𝑧 ∧ 𝑦𝑟𝑧))}
cpo 18210	class Poset
cplt 18211	class lt
club 18212	class lub
cglb 18213	class glb
cjn 18214	class join
cmee 18215	class meet
df-poset 18216	⊢ 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 18231	⊢ lt = (𝑝 ∈ V ↦ ((le‘𝑝) ∖ I ))
df-lub 18247	⊢ 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 18248	⊢ 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 18249	⊢ join = (𝑝 ∈ V ↦ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦} (lub‘𝑝)𝑧})
df-meet 18250	⊢ meet = (𝑝 ∈ V ↦ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦} (glb‘𝑝)𝑧})
ctos 18317	class Toset
df-toset 18318	⊢ Toset = {𝑓 ∈ Poset ∣ [(Base‘𝑓) / 𝑏][(le‘𝑓) / 𝑟]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥𝑟𝑦 ∨ 𝑦𝑟𝑥)}
cp0 18324	class 0.
cp1 18325	class 1.
df-p0 18326	⊢ 0. = (𝑝 ∈ V ↦ ((glb‘𝑝)‘(Base‘𝑝)))
df-p1 18327	⊢ 1. = (𝑝 ∈ V ↦ ((lub‘𝑝)‘(Base‘𝑝)))
clat 18334	class Lat
df-lat 18335	⊢ Lat = {𝑝 ∈ Poset ∣ (dom (join‘𝑝) = ((Base‘𝑝) × (Base‘𝑝)) ∧ dom (meet‘𝑝) = ((Base‘𝑝) × (Base‘𝑝)))}
ccla 18401	class CLat
df-clat 18402	⊢ CLat = {𝑝 ∈ Poset ∣ (dom (lub‘𝑝) = 𝒫 (Base‘𝑝) ∧ dom (glb‘𝑝) = 𝒫 (Base‘𝑝))}
cdlat 18423	class DLat
df-dlat 18424	⊢ DLat = {𝑘 ∈ Lat ∣ [(Base‘𝑘) / 𝑏][(join‘𝑘) / 𝑗][(meet‘𝑘) / 𝑚]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 (𝑥𝑚(𝑦𝑗𝑧)) = ((𝑥𝑚𝑦)𝑗(𝑥𝑚𝑧))}
cipo 18430	class toInc
df-ipo 18431	⊢ 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 18467	class PosetRel
ctsr 18468	class TosetRel
df-ps 18469	⊢ PosetRel = {𝑟 ∣ (Rel 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟 ∧ (𝑟 ∩ ^◡𝑟) = ( I ↾ ∪ ∪ 𝑟))}
df-tsr 18470	⊢ TosetRel = {𝑟 ∈ PosetRel ∣ (dom 𝑟 × dom 𝑟) ⊆ (𝑟 ∪ ^◡𝑟)}
cdir 18497	class DirRel
ctail 18498	class tail
df-dir 18499	⊢ DirRel = {𝑟 ∣ ((Rel 𝑟 ∧ ( I ↾ ∪ ∪ 𝑟) ⊆ 𝑟) ∧ ((𝑟 ∘ 𝑟) ⊆ 𝑟 ∧ (∪ ∪ 𝑟 × ∪ ∪ 𝑟) ⊆ (^◡𝑟 ∘ 𝑟)))}
df-tail 18500	⊢ tail = (𝑟 ∈ DirRel ↦ (𝑥 ∈ ∪ ∪ 𝑟 ↦ (𝑟 “ {𝑥})))
cchn 18508	class ( < Chain 𝐴)
df-chn 18509	⊢ ( < Chain 𝐴) = {𝑐 ∈ Word 𝐴 ∣ ∀𝑛 ∈ (dom 𝑐 ∖ {0})(𝑐‘(𝑛 − 1)) < (𝑐‘𝑛)}
cplusf 18542	class +_𝑓
cmgm 18543	class Mgm
df-plusf 18544	⊢ +_𝑓 = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(+_g‘𝑔)𝑦)))
df-mgm 18545	⊢ Mgm = {𝑔 ∣ [(Base‘𝑔) / 𝑏][(+_g‘𝑔) / 𝑜]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥𝑜𝑦) ∈ 𝑏}
cmgmhm 18595	class MgmHom
csubmgm 18596	class SubMgm
df-mgmhm 18597	⊢ 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 18598	⊢ SubMgm = (𝑠 ∈ Mgm ↦ {𝑡 ∈ 𝒫 (Base‘𝑠) ∣ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (𝑥(+_g‘𝑠)𝑦) ∈ 𝑡})
csgrp 18623	class Smgrp
df-sgrp 18624	⊢ Smgrp = {𝑔 ∈ Mgm ∣ [(Base‘𝑔) / 𝑏][(+_g‘𝑔) / 𝑜]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧))}
cmnd 18639	class Mnd
df-mnd 18640	⊢ Mnd = {𝑔 ∈ Smgrp ∣ [(Base‘𝑔) / 𝑏][(+_g‘𝑔) / 𝑝]∃𝑒 ∈ 𝑏 ∀𝑥 ∈ 𝑏 ((𝑒𝑝𝑥) = 𝑥 ∧ (𝑥𝑝𝑒) = 𝑥)}
cmhm 18686	class MndHom
csubmnd 18687	class SubMnd
df-mhm 18688	⊢ 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 18689	⊢ SubMnd = (𝑠 ∈ Mnd ↦ {𝑡 ∈ 𝒫 (Base‘𝑠) ∣ ((0_g‘𝑠) ∈ 𝑡 ∧ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (𝑥(+_g‘𝑠)𝑦) ∈ 𝑡)})
cfrmd 18752	class freeMnd
cvrmd 18753	class var_FMnd
df-frmd 18754	⊢ freeMnd = (𝑖 ∈ V ↦ {⟨(Base‘ndx), Word 𝑖⟩, ⟨(+_g‘ndx), ( ++ ↾ (Word 𝑖 × Word 𝑖))⟩})
df-vrmd 18755	⊢ var_FMnd = (𝑖 ∈ V ↦ (𝑗 ∈ 𝑖 ↦ ⟨“𝑗”⟩))
cefmnd 18773	class EndoFMnd
df-efmnd 18774	⊢ 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 18843	class Grp
cminusg 18844	class inv_g
csg 18845	class -_g
df-grp 18846	⊢ Grp = {𝑔 ∈ Mnd ∣ ∀𝑎 ∈ (Base‘𝑔)∃𝑚 ∈ (Base‘𝑔)(𝑚(+_g‘𝑔)𝑎) = (0_g‘𝑔)}
df-minusg 18847	⊢ inv_g = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔) ↦ (℩𝑤 ∈ (Base‘𝑔)(𝑤(+_g‘𝑔)𝑥) = (0_g‘𝑔))))
df-sbg 18848	⊢ -_g = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(+_g‘𝑔)((inv_g‘𝑔)‘𝑦))))
cmg 18977	class ._g
df-mulg 18978	⊢ ._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 19030	class SubGrp
cnsg 19031	class NrmSGrp
cqg 19032	class ~_QG
df-subg 19033	⊢ SubGrp = (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾_s 𝑠) ∈ Grp})
df-nsg 19034	⊢ NrmSGrp = (𝑤 ∈ Grp ↦ {𝑠 ∈ (SubGrp‘𝑤) ∣ [(Base‘𝑤) / 𝑏][(+_g‘𝑤) / 𝑝]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ((𝑥𝑝𝑦) ∈ 𝑠 ↔ (𝑦𝑝𝑥) ∈ 𝑠)})
df-eqg 19035	⊢ ~_QG = (𝑟 ∈ V, 𝑖 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (Base‘𝑟) ∧ (((inv_g‘𝑟)‘𝑥)(+_g‘𝑟)𝑦) ∈ 𝑖)})
cghm 19122	class GrpHom
df-ghm 19123	⊢ 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 19167	class GrpIso
cgic 19168	class ≃_𝑔
df-gim 19169	⊢ GrpIso = (𝑠 ∈ Grp, 𝑡 ∈ Grp ↦ {𝑔 ∈ (𝑠 GrpHom 𝑡) ∣ 𝑔:(Base‘𝑠)–1-1-onto→(Base‘𝑡)})
df-gic 19170	⊢ ≃_𝑔 = (^◡ GrpIso “ (V ∖ 1_o))
cga 19199	class GrpAct
df-ga 19200	⊢ 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 19225	class Cntz
ccntr 19226	class Cntr
df-cntz 19227	⊢ Cntz = (𝑚 ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘𝑚) ↦ {𝑥 ∈ (Base‘𝑚) ∣ ∀𝑦 ∈ 𝑠 (𝑥(+_g‘𝑚)𝑦) = (𝑦(+_g‘𝑚)𝑥)}))
df-cntr 19228	⊢ Cntr = (𝑚 ∈ V ↦ ((Cntz‘𝑚)‘(Base‘𝑚)))
coppg 19255	class opp_g
df-oppg 19256	⊢ opp_g = (𝑤 ∈ V ↦ (𝑤 sSet ⟨(+_g‘ndx), tpos (+_g‘𝑤)⟩))
csymg 19279	class SymGrp
df-symg 19280	⊢ SymGrp = (𝑥 ∈ V ↦ ((EndoFMnd‘𝑥) ↾_s {ℎ ∣ ℎ:𝑥–1-1-onto→𝑥}))
cpmtr 19351	class pmTrsp
df-pmtr 19352	⊢ pmTrsp = (𝑑 ∈ V ↦ (𝑝 ∈ {𝑦 ∈ 𝒫 𝑑 ∣ 𝑦 ≈ 2_o} ↦ (𝑧 ∈ 𝑑 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))))
cpsgn 19399	class pmSgn
cevpm 19400	class pmEven
df-psgn 19401	⊢ 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 19402	⊢ pmEven = (𝑑 ∈ V ↦ (^◡(pmSgn‘𝑑) “ {1}))
cod 19434	class od
cgex 19435	class gEx
cpgp 19436	class pGrp
cslw 19437	class pSyl
df-od 19438	⊢ od = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔) ↦ ⦋{𝑛 ∈ ℕ ∣ (𝑛(._g‘𝑔)𝑥) = (0_g‘𝑔)} / 𝑖⦌if(𝑖 = ∅, 0, inf(𝑖, ℝ, < ))))
df-gex 19439	⊢ gEx = (𝑔 ∈ V ↦ ⦋{𝑛 ∈ ℕ ∣ ∀𝑥 ∈ (Base‘𝑔)(𝑛(._g‘𝑔)𝑥) = (0_g‘𝑔)} / 𝑖⦌if(𝑖 = ∅, 0, inf(𝑖, ℝ, < )))
df-pgp 19440	⊢ pGrp = {⟨𝑝, 𝑔⟩ ∣ ((𝑝 ∈ ℙ ∧ 𝑔 ∈ Grp) ∧ ∀𝑥 ∈ (Base‘𝑔)∃𝑛 ∈ ℕ₀ ((od‘𝑔)‘𝑥) = (𝑝↑𝑛))}
df-slw 19441	⊢ pSyl = (𝑝 ∈ ℙ, 𝑔 ∈ Grp ↦ {ℎ ∈ (SubGrp‘𝑔) ∣ ∀𝑘 ∈ (SubGrp‘𝑔)((ℎ ⊆ 𝑘 ∧ 𝑝 pGrp (𝑔 ↾_s 𝑘)) ↔ ℎ = 𝑘)})
clsm 19544	class LSSum
cpj1 19545	class proj₁
df-lsm 19546	⊢ LSSum = (𝑤 ∈ V ↦ (𝑡 ∈ 𝒫 (Base‘𝑤), 𝑢 ∈ 𝒫 (Base‘𝑤) ↦ ran (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥(+_g‘𝑤)𝑦))))
df-pj1 19547	⊢ 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 19616	class ~_FG
cfrgp 19617	class freeGrp
cvrgp 19618	class var_FGrp
df-efg 19619	⊢ ~_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 19620	⊢ freeGrp = (𝑖 ∈ V ↦ ((freeMnd‘(𝑖 × 2_o)) /_s ( ~_FG ‘𝑖)))
df-vrgp 19621	⊢ var_FGrp = (𝑖 ∈ V ↦ (𝑗 ∈ 𝑖 ↦ [⟨“⟨𝑗, ∅⟩”⟩]( ~_FG ‘𝑖)))
ccmn 19690	class CMnd
cabl 19691	class Abel
df-cmn 19692	⊢ CMnd = {𝑔 ∈ Mnd ∣ ∀𝑎 ∈ (Base‘𝑔)∀𝑏 ∈ (Base‘𝑔)(𝑎(+_g‘𝑔)𝑏) = (𝑏(+_g‘𝑔)𝑎)}
df-abl 19693	⊢ Abel = (Grp ∩ CMnd)
ccyg 19787	class CycGrp
df-cyg 19788	⊢ CycGrp = {𝑔 ∈ Grp ∣ ∃𝑥 ∈ (Base‘𝑔)ran (𝑛 ∈ ℤ ↦ (𝑛(._g‘𝑔)𝑥)) = (Base‘𝑔)}
cdprd 19905	class DProd
cdpj 19906	class dProj
df-dprd 19907	⊢ 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 19908	⊢ dProj = (𝑔 ∈ Grp, 𝑠 ∈ (dom DProd “ {𝑔}) ↦ (𝑖 ∈ dom 𝑠 ↦ ((𝑠‘𝑖)(proj₁‘𝑔)(𝑔 DProd (𝑠 ↾ (dom 𝑠 ∖ {𝑖}))))))
csimpg 20002	class SimpGrp
df-simpg 20003	⊢ SimpGrp = {𝑔 ∈ Grp ∣ (NrmSGrp‘𝑔) ≈ 2_o}
comnd 20029	class oMnd
cogrp 20030	class oGrp
df-omnd 20031	⊢ 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 20032	⊢ oGrp = (Grp ∩ oMnd)
cmgp 20056	class mulGrp
df-mgp 20057	⊢ mulGrp = (𝑤 ∈ V ↦ (𝑤 sSet ⟨(+_g‘ndx), (._r‘𝑤)⟩))
crng 20068	class Rng
df-rng 20069	⊢ 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 20097	class 1_r
df-ur 20098	⊢ 1_r = (0_g ∘ mulGrp)
csrg 20102	class SRing
df-srg 20103	⊢ 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 20149	class Ring
ccrg 20150	class CRing
df-ring 20151	⊢ 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 20152	⊢ CRing = {𝑓 ∈ Ring ∣ (mulGrp‘𝑓) ∈ CMnd}
coppr 20252	class opp_r
df-oppr 20253	⊢ opp_r = (𝑓 ∈ V ↦ (𝑓 sSet ⟨(._r‘ndx), tpos (._r‘𝑓)⟩))
cdsr 20270	class ∥_r
cui 20271	class Unit
cir 20272	class Irred
df-dvdsr 20273	⊢ ∥_r = (𝑤 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑤) ∧ ∃𝑧 ∈ (Base‘𝑤)(𝑧(._r‘𝑤)𝑥) = 𝑦)})
df-unit 20274	⊢ Unit = (𝑤 ∈ V ↦ (^◡((∥_r‘𝑤) ∩ (∥_r‘(opp_r‘𝑤))) “ {(1_r‘𝑤)}))
df-irred 20275	⊢ Irred = (𝑤 ∈ V ↦ ⦋((Base‘𝑤) ∖ (Unit‘𝑤)) / 𝑏⦌{𝑧 ∈ 𝑏 ∣ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥(._r‘𝑤)𝑦) ≠ 𝑧})
cinvr 20303	class inv_r
df-invr 20304	⊢ inv_r = (𝑟 ∈ V ↦ (inv_g‘((mulGrp‘𝑟) ↾_s (Unit‘𝑟))))
cdvr 20316	class /_r
df-dvr 20317	⊢ /_r = (𝑟 ∈ V ↦ (𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Unit‘𝑟) ↦ (𝑥(._r‘𝑟)((inv_r‘𝑟)‘𝑦))))
crpm 20348	class RPrime
df-rprm 20349	⊢ 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 20350	class RngHom
crngim 20351	class RngIso
df-rnghm 20352	⊢ 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 20353	⊢ RngIso = (𝑟 ∈ V, 𝑠 ∈ V ↦ {𝑓 ∈ (𝑟 RngHom 𝑠) ∣ ^◡𝑓 ∈ (𝑠 RngHom 𝑟)})
crh 20385	class RingHom
crs 20386	class RingIso
cric 20387	class ≃_𝑟
df-rhm 20388	⊢ 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 20389	⊢ RingIso = (𝑟 ∈ V, 𝑠 ∈ V ↦ {𝑓 ∈ (𝑟 RingHom 𝑠) ∣ ^◡𝑓 ∈ (𝑠 RingHom 𝑟)})
df-ric 20391	⊢ ≃_𝑟 = (^◡ RingIso “ (V ∖ 1_o))
cnzr 20425	class NzRing
df-nzr 20426	⊢ NzRing = {𝑟 ∈ Ring ∣ (1_r‘𝑟) ≠ (0_g‘𝑟)}
clring 20451	class LRing
df-lring 20452	⊢ LRing = {𝑟 ∈ NzRing ∣ ∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)((𝑥(+_g‘𝑟)𝑦) = (1_r‘𝑟) → (𝑥 ∈ (Unit‘𝑟) ∨ 𝑦 ∈ (Unit‘𝑟)))}
csubrng 20458	class SubRng
df-subrng 20459	⊢ SubRng = (𝑤 ∈ Rng ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾_s 𝑠) ∈ Rng})
csubrg 20482	class SubRing
df-subrg 20483	⊢ SubRing = (𝑤 ∈ Ring ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ ((𝑤 ↾_s 𝑠) ∈ Ring ∧ (1_r‘𝑤) ∈ 𝑠)})
crgspn 20523	class RingSpan
df-rgspn 20524	⊢ RingSpan = (𝑤 ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘𝑤) ↦ ∩ {𝑡 ∈ (SubRing‘𝑤) ∣ 𝑠 ⊆ 𝑡}))
crngc 20529	class RngCat
df-rngc 20530	⊢ RngCat = (𝑢 ∈ V ↦ ((ExtStrCat‘𝑢) ↾_cat ( RngHom ↾ ((𝑢 ∩ Rng) × (𝑢 ∩ Rng)))))
cringc 20558	class RingCat
df-ringc 20559	⊢ RingCat = (𝑢 ∈ V ↦ ((ExtStrCat‘𝑢) ↾_cat ( RingHom ↾ ((𝑢 ∩ Ring) × (𝑢 ∩ Ring)))))
crlreg 20604	class RLReg
cdomn 20605	class Domn
cidom 20606	class IDomn
df-rlreg 20607	⊢ RLReg = (𝑟 ∈ V ↦ {𝑥 ∈ (Base‘𝑟) ∣ ∀𝑦 ∈ (Base‘𝑟)((𝑥(._r‘𝑟)𝑦) = (0_g‘𝑟) → 𝑦 = (0_g‘𝑟))})
df-domn 20608	⊢ 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 20609	⊢ IDomn = (CRing ∩ Domn)
cdr 20642	class DivRing
cfield 20643	class Field
df-drng 20644	⊢ DivRing = {𝑟 ∈ Ring ∣ (Unit‘𝑟) = ((Base‘𝑟) ∖ {(0_g‘𝑟)})}
df-field 20645	⊢ Field = (DivRing ∩ CRing)
csdrg 20699	class SubDRing
df-sdrg 20700	⊢ SubDRing = (𝑤 ∈ DivRing ↦ {𝑠 ∈ (SubRing‘𝑤) ∣ (𝑤 ↾_s 𝑠) ∈ DivRing})
cabv 20721	class AbsVal
df-abv 20722	⊢ 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 20750	class *_rf
csr 20751	class *-Ring
df-staf 20752	⊢ _rf = (𝑓 ∈ V ↦ (𝑥 ∈ (Base‘𝑓) ↦ ((_𝑟‘𝑓)‘𝑥)))
df-srng 20753	⊢ -Ring = {𝑓 ∣ [(_rf‘𝑓) / 𝑖](𝑖 ∈ (𝑓 RingHom (opp_r‘𝑓)) ∧ 𝑖 = ^◡𝑖)}
corng 20770	class oRing
cofld 20771	class oField
df-orng 20772	⊢ 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 20773	⊢ oField = (Field ∩ oRing)
clmod 20791	class LMod
cscaf 20792	class ·_sf
df-lmod 20793	⊢ 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 20794	⊢ ·_sf = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘(Scalar‘𝑔)), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥( ·_𝑠 ‘𝑔)𝑦)))
clss 20862	class LSubSp
df-lss 20863	⊢ LSubSp = (𝑤 ∈ V ↦ {𝑠 ∈ (𝒫 (Base‘𝑤) ∖ {∅}) ∣ ∀𝑥 ∈ (Base‘(Scalar‘𝑤))∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑥( ·_𝑠 ‘𝑤)𝑎)(+_g‘𝑤)𝑏) ∈ 𝑠})
clspn 20902	class LSpan
df-lsp 20903	⊢ LSpan = (𝑤 ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘𝑤) ↦ ∩ {𝑡 ∈ (LSubSp‘𝑤) ∣ 𝑠 ⊆ 𝑡}))
clmhm 20951	class LMHom
clmim 20952	class LMIso
clmic 20953	class ≃_𝑚
df-lmhm 20954	⊢ 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 20955	⊢ LMIso = (𝑠 ∈ LMod, 𝑡 ∈ LMod ↦ {𝑔 ∈ (𝑠 LMHom 𝑡) ∣ 𝑔:(Base‘𝑠)–1-1-onto→(Base‘𝑡)})
df-lmic 20956	⊢ ≃_𝑚 = (^◡ LMIso “ (V ∖ 1_o))
clbs 21006	class LBasis
df-lbs 21007	⊢ 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 21034	class LVec
df-lvec 21035	⊢ LVec = {𝑓 ∈ LMod ∣ (Scalar‘𝑓) ∈ DivRing}
csra 21103	class subringAlg
crglmod 21104	class ringLMod
df-sra 21105	⊢ 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 21106	⊢ ringLMod = (𝑤 ∈ V ↦ ((subringAlg ‘𝑤)‘(Base‘𝑤)))
clidl 21141	class LIdeal
crsp 21142	class RSpan
df-lidl 21143	⊢ LIdeal = (LSubSp ∘ ringLMod)
df-rsp 21144	⊢ RSpan = (LSpan ∘ ringLMod)
c2idl 21184	class 2Ideal
df-2idl 21185	⊢ 2Ideal = (𝑟 ∈ V ↦ ((LIdeal‘𝑟) ∩ (LIdeal‘(opp_r‘𝑟))))
clpidl 21255	class LPIdeal
clpir 21256	class LPIR
df-lpidl 21257	⊢ LPIdeal = (𝑤 ∈ Ring ↦ ∪ 𝑔 ∈ (Base‘𝑤){((RSpan‘𝑤)‘{𝑔})})
df-lpir 21258	⊢ LPIR = {𝑤 ∈ Ring ∣ (LIdeal‘𝑤) = (LPIdeal‘𝑤)}
cpid 21271	class PID
df-pid 21272	⊢ PID = (IDomn ∩ LPIR)
cpsmet 21273	class PsMet
cxmet 21274	class ∞Met
cmet 21275	class Met
cbl 21276	class ball
cfbas 21277	class fBas
cfg 21278	class filGen
cmopn 21279	class MetOpen
cmetu 21280	class metUnif
df-psmet 21281	⊢ PsMet = (𝑥 ∈ V ↦ {𝑑 ∈ (ℝ^* ↑_m (𝑥 × 𝑥)) ∣ ∀𝑦 ∈ 𝑥 ((𝑦𝑑𝑦) = 0 ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) +_𝑒 (𝑤𝑑𝑧)))})
df-xmet 21282	⊢ ∞Met = (𝑥 ∈ V ↦ {𝑑 ∈ (ℝ^* ↑_m (𝑥 × 𝑥)) ∣ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (((𝑦𝑑𝑧) = 0 ↔ 𝑦 = 𝑧) ∧ ∀𝑤 ∈ 𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) +_𝑒 (𝑤𝑑𝑧)))})
df-met 21283	⊢ Met = (𝑥 ∈ V ↦ {𝑑 ∈ (ℝ ↑_m (𝑥 × 𝑥)) ∣ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (((𝑦𝑑𝑧) = 0 ↔ 𝑦 = 𝑧) ∧ ∀𝑤 ∈ 𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) + (𝑤𝑑𝑧)))})
df-bl 21284	⊢ ball = (𝑑 ∈ V ↦ (𝑥 ∈ dom dom 𝑑, 𝑧 ∈ ℝ^* ↦ {𝑦 ∈ dom dom 𝑑 ∣ (𝑥𝑑𝑦) < 𝑧}))
df-mopn 21285	⊢ MetOpen = (𝑑 ∈ ∪ ran ∞Met ↦ (topGen‘ran (ball‘𝑑)))
df-fbas 21286	⊢ fBas = (𝑤 ∈ V ↦ {𝑥 ∈ 𝒫 𝒫 𝑤 ∣ (𝑥 ≠ ∅ ∧ ∅ ∉ 𝑥 ∧ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑥 ∩ 𝒫 (𝑦 ∩ 𝑧)) ≠ ∅)})
df-fg 21287	⊢ filGen = (𝑤 ∈ V, 𝑥 ∈ (fBas‘𝑤) ↦ {𝑦 ∈ 𝒫 𝑤 ∣ (𝑥 ∩ 𝒫 𝑦) ≠ ∅})
df-metu 21288	⊢ metUnif = (𝑑 ∈ ∪ ran PsMet ↦ ((dom dom 𝑑 × dom dom 𝑑)filGenran (𝑎 ∈ ℝ⁺ ↦ (^◡𝑑 “ (0[,)𝑎)))))
ccnfld 21289	class ℂ_fld
df-cnfld 21290	⊢ 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 21381	class ℤ_ring
df-zring 21382	⊢ ℤ_ring = (ℂ_fld ↾_s ℤ)
czrh 21434	class ℤRHom
czlm 21435	class ℤMod
cchr 21436	class chr
czn 21437	class ℤ/nℤ
df-zrh 21438	⊢ ℤRHom = (𝑟 ∈ V ↦ ∪ (ℤ_ring RingHom 𝑟))
df-zlm 21439	⊢ ℤMod = (𝑔 ∈ V ↦ ((𝑔 sSet ⟨(Scalar‘ndx), ℤ_ring⟩) sSet ⟨( ·_𝑠 ‘ndx), (._g‘𝑔)⟩))
df-chr 21440	⊢ chr = (𝑔 ∈ V ↦ ((od‘𝑔)‘(1_r‘𝑔)))
df-zn 21441	⊢ 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 21539	class ℝ_fld
df-refld 21540	⊢ ℝ_fld = (ℂ_fld ↾_s ℝ)
cphl 21559	class PreHil
cipf 21560	class ·_if
df-phl 21561	⊢ 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 21562	⊢ ·_if = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(·_𝑖‘𝑔)𝑦)))
cocv 21595	class ocv
ccss 21596	class ClSubSp
cthl 21597	class toHL
df-ocv 21598	⊢ ocv = (ℎ ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘ℎ) ↦ {𝑥 ∈ (Base‘ℎ) ∣ ∀𝑦 ∈ 𝑠 (𝑥(·_𝑖‘ℎ)𝑦) = (0_g‘(Scalar‘ℎ))}))
df-css 21599	⊢ ClSubSp = (ℎ ∈ V ↦ {𝑠 ∣ 𝑠 = ((ocv‘ℎ)‘((ocv‘ℎ)‘𝑠))})
df-thl 21600	⊢ toHL = (ℎ ∈ V ↦ ((toInc‘(ClSubSp‘ℎ)) sSet ⟨(oc‘ndx), (ocv‘ℎ)⟩))
cpj 21635	class proj
chil 21636	class Hil
cobs 21637	class OBasis
df-pj 21638	⊢ proj = (ℎ ∈ V ↦ ((𝑥 ∈ (LSubSp‘ℎ) ↦ (𝑥(proj₁‘ℎ)((ocv‘ℎ)‘𝑥))) ∩ (V × ((Base‘ℎ) ↑_m (Base‘ℎ)))))
df-hil 21639	⊢ Hil = {ℎ ∈ PreHil ∣ dom (proj‘ℎ) = (ClSubSp‘ℎ)}
df-obs 21640	⊢ 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 21666	class ⊕_m
df-dsmm 21667	⊢ ⊕_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 21681	class freeLMod
df-frlm 21682	⊢ freeLMod = (𝑟 ∈ V, 𝑖 ∈ V ↦ (𝑟 ⊕_m (𝑖 × {(ringLMod‘𝑟)})))
cuvc 21717	class unitVec
df-uvc 21718	⊢ unitVec = (𝑟 ∈ V, 𝑖 ∈ V ↦ (𝑗 ∈ 𝑖 ↦ (𝑘 ∈ 𝑖 ↦ if(𝑘 = 𝑗, (1_r‘𝑟), (0_g‘𝑟)))))
clindf 21739	class LIndF
clinds 21740	class LIndS
df-lindf 21741	⊢ 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 21742	⊢ LIndS = (𝑤 ∈ V ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ ( I ↾ 𝑠) LIndF 𝑤})
casa 21785	class AssAlg
casp 21786	class AlgSpan
cascl 21787	class algSc
df-assa 21788	⊢ 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 21789	⊢ AlgSpan = (𝑤 ∈ AssAlg ↦ (𝑠 ∈ 𝒫 (Base‘𝑤) ↦ ∩ {𝑡 ∈ ((SubRing‘𝑤) ∩ (LSubSp‘𝑤)) ∣ 𝑠 ⊆ 𝑡}))
df-ascl 21790	⊢ algSc = (𝑤 ∈ V ↦ (𝑥 ∈ (Base‘(Scalar‘𝑤)) ↦ (𝑥( ·_𝑠 ‘𝑤)(1_r‘𝑤))))
cmps 21839	class mPwSer
cmvr 21840	class mVar
cmpl 21841	class mPoly
cltb 21842	class <_bag
copws 21843	class ordPwSer
df-psr 21844	⊢ 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 21845	⊢ 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 21846	⊢ mPoly = (𝑖 ∈ V, 𝑟 ∈ V ↦ ⦋(𝑖 mPwSer 𝑟) / 𝑤⦌(𝑤 ↾_s {𝑓 ∈ (Base‘𝑤) ∣ 𝑓 finSupp (0_g‘𝑟)}))
df-ltbag 21847	⊢ <_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 21848	⊢ 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 22005	class evalSub
cevl 22006	class eval
df-evls 22007	⊢ 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 22008	⊢ eval = (𝑖 ∈ V, 𝑟 ∈ V ↦ ((𝑖 evalSub 𝑟)‘(Base‘𝑟)))
cslv 22041	class selectVars
cmhp 22042	class mHomP
cpsd 22043	class mPSDer
cai 22044	class AlgInd
df-selv 22045	⊢ 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 22049	⊢ 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 22069	⊢ 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 22084	⊢ 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 22085	class PwSer₁
cv1 22086	class var₁
cpl1 22087	class Poly₁
cco1 22088	class coe₁
ctp1 22089	class toPoly₁
df-psr1 22090	⊢ PwSer₁ = (𝑟 ∈ V ↦ ((1_o ordPwSer 𝑟)‘∅))
df-vr1 22091	⊢ var₁ = (𝑟 ∈ V ↦ ((1_o mVar 𝑟)‘∅))
df-ply1 22092	⊢ Poly₁ = (𝑟 ∈ V ↦ ((PwSer₁‘𝑟) ↾_s (Base‘(1_o mPoly 𝑟))))
df-coe1 22093	⊢ coe₁ = (𝑓 ∈ V ↦ (𝑛 ∈ ℕ₀ ↦ (𝑓‘(1_o × {𝑛}))))
df-toply1 22094	⊢ toPoly₁ = (𝑓 ∈ V ↦ (𝑛 ∈ (ℕ₀ ↑_m 1_o) ↦ (𝑓‘(𝑛‘∅))))
ces1 22226	class evalSub₁
ce1 22227	class eval₁
df-evls1 22228	⊢ 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 22229	⊢ eval₁ = (𝑟 ∈ V ↦ ⦋(Base‘𝑟) / 𝑏⦌((𝑥 ∈ (𝑏 ↑_m (𝑏 ↑_m 1_o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝑏 ↦ (1_o × {𝑦})))) ∘ (1_o eval 𝑟)))
cmmul 22303	class maMul
df-mamu 22304	⊢ 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 22320	class Mat
df-mat 22321	⊢ Mat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ ((𝑟 freeLMod (𝑛 × 𝑛)) sSet ⟨(._r‘ndx), (𝑟 maMul ⟨𝑛, 𝑛, 𝑛⟩)⟩))
cdmat 22401	class DMat
cscmat 22402	class ScMat
df-dmat 22403	⊢ DMat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ {𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ∣ ∀𝑖 ∈ 𝑛 ∀𝑗 ∈ 𝑛 (𝑖 ≠ 𝑗 → (𝑖𝑚𝑗) = (0_g‘𝑟))})
df-scmat 22404	⊢ ScMat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ ⦋(𝑛 Mat 𝑟) / 𝑎⦌{𝑚 ∈ (Base‘𝑎) ∣ ∃𝑐 ∈ (Base‘𝑟)𝑚 = (𝑐( ·_𝑠 ‘𝑎)(1_r‘𝑎))})
cmvmul 22453	class maVecMul
df-mvmul 22454	⊢ 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 22469	class matRRep
cmatrepV 22470	class matRepV
df-marrep 22471	⊢ 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 22472	⊢ 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 22489	class subMat
df-subma 22490	⊢ subMat = (𝑛 ∈ V, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑘 ∈ 𝑛, 𝑙 ∈ 𝑛 ↦ (𝑖 ∈ (𝑛 ∖ {𝑘}), 𝑗 ∈ (𝑛 ∖ {𝑙}) ↦ (𝑖𝑚𝑗)))))
cmdat 22497	class maDet
df-mdet 22498	⊢ 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 22545	class maAdju
cminmar1 22546	class minMatR1
df-madu 22547	⊢ 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 22548	⊢ 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 22616	class ConstPolyMat
cmat2pmat 22617	class matToPolyMat
ccpmat2mat 22618	class cPolyMatToMat
df-cpmat 22619	⊢ 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 22620	⊢ matToPolyMat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑥 ∈ 𝑛, 𝑦 ∈ 𝑛 ↦ ((algSc‘(Poly₁‘𝑟))‘(𝑥𝑚𝑦)))))
df-cpmat2mat 22621	⊢ cPolyMatToMat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ (𝑚 ∈ (𝑛 ConstPolyMat 𝑟) ↦ (𝑥 ∈ 𝑛, 𝑦 ∈ 𝑛 ↦ ((coe₁‘(𝑥𝑚𝑦))‘0))))
cdecpmat 22675	class decompPMat
df-decpmat 22676	⊢ decompPMat = (𝑚 ∈ V, 𝑘 ∈ ℕ₀ ↦ (𝑖 ∈ dom dom 𝑚, 𝑗 ∈ dom dom 𝑚 ↦ ((coe₁‘(𝑖𝑚𝑗))‘𝑘)))
cpm2mp 22705	class pMatToMatPoly
df-pm2mp 22706	⊢ 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 22739	class CharPlyMat
df-chpmat 22740	⊢ 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 22806	class Top
df-top 22807	⊢ Top = {𝑥 ∣ (∀𝑦 ∈ 𝒫 𝑥∪ 𝑦 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑦 ∩ 𝑧) ∈ 𝑥)}
ctopon 22823	class TopOn
df-topon 22824	⊢ TopOn = (𝑏 ∈ V ↦ {𝑗 ∈ Top ∣ 𝑏 = ∪ 𝑗})
ctps 22845	class TopSp
df-topsp 22846	⊢ TopSp = {𝑓 ∣ (TopOpen‘𝑓) ∈ (TopOn‘(Base‘𝑓))}
ctb 22858	class TopBases
df-bases 22859	⊢ TopBases = {𝑥 ∣ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑦 ∩ 𝑧) ⊆ ∪ (𝑥 ∩ 𝒫 (𝑦 ∩ 𝑧))}
ccld 22929	class Clsd
cnt 22930	class int
ccl 22931	class cls
df-cld 22932	⊢ Clsd = (𝑗 ∈ Top ↦ {𝑥 ∈ 𝒫 ∪ 𝑗 ∣ (∪ 𝑗 ∖ 𝑥) ∈ 𝑗})
df-ntr 22933	⊢ int = (𝑗 ∈ Top ↦ (𝑥 ∈ 𝒫 ∪ 𝑗 ↦ ∪ (𝑗 ∩ 𝒫 𝑥)))
df-cls 22934	⊢ cls = (𝑗 ∈ Top ↦ (𝑥 ∈ 𝒫 ∪ 𝑗 ↦ ∩ {𝑦 ∈ (Clsd‘𝑗) ∣ 𝑥 ⊆ 𝑦}))
cnei 23010	class nei
df-nei 23011	⊢ nei = (𝑗 ∈ Top ↦ (𝑥 ∈ 𝒫 ∪ 𝑗 ↦ {𝑦 ∈ 𝒫 ∪ 𝑗 ∣ ∃𝑔 ∈ 𝑗 (𝑥 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑦)}))
clp 23047	class limPt
cperf 23048	class Perf
df-lp 23049	⊢ limPt = (𝑗 ∈ Top ↦ (𝑥 ∈ 𝒫 ∪ 𝑗 ↦ {𝑦 ∣ 𝑦 ∈ ((cls‘𝑗)‘(𝑥 ∖ {𝑦}))}))
df-perf 23050	⊢ Perf = {𝑗 ∈ Top ∣ ((limPt‘𝑗)‘∪ 𝑗) = ∪ 𝑗}
ccn 23137	class Cn
ccnp 23138	class CnP
clm 23139	class ⇝_𝑡
df-cn 23140	⊢ Cn = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ {𝑓 ∈ (∪ 𝑘 ↑_m ∪ 𝑗) ∣ ∀𝑦 ∈ 𝑘 (^◡𝑓 “ 𝑦) ∈ 𝑗})
df-cnp 23141	⊢ 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 23142	⊢ ⇝_𝑡 = (𝑗 ∈ Top ↦ {⟨𝑓, 𝑥⟩ ∣ (𝑓 ∈ (∪ 𝑗 ↑_pm ℂ) ∧ 𝑥 ∈ ∪ 𝑗 ∧ ∀𝑢 ∈ 𝑗 (𝑥 ∈ 𝑢 → ∃𝑦 ∈ ran ℤ_≥(𝑓 ↾ 𝑦):𝑦⟶𝑢))})
ct0 23219	class Kol2
ct1 23220	class Fre
cha 23221	class Haus
creg 23222	class Reg
cnrm 23223	class Nrm
ccnrm 23224	class CNrm
cpnrm 23225	class PNrm
df-t0 23226	⊢ Kol2 = {𝑗 ∈ Top ∣ ∀𝑥 ∈ ∪ 𝑗∀𝑦 ∈ ∪ 𝑗(∀𝑜 ∈ 𝑗 (𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜) → 𝑥 = 𝑦)}
df-t1 23227	⊢ Fre = {𝑥 ∈ Top ∣ ∀𝑎 ∈ ∪ 𝑥{𝑎} ∈ (Clsd‘𝑥)}
df-haus 23228	⊢ Haus = {𝑗 ∈ Top ∣ ∀𝑥 ∈ ∪ 𝑗∀𝑦 ∈ ∪ 𝑗(𝑥 ≠ 𝑦 → ∃𝑛 ∈ 𝑗 ∃𝑚 ∈ 𝑗 (𝑥 ∈ 𝑛 ∧ 𝑦 ∈ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅))}
df-reg 23229	⊢ Reg = {𝑗 ∈ Top ∣ ∀𝑥 ∈ 𝑗 ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝑗 (𝑦 ∈ 𝑧 ∧ ((cls‘𝑗)‘𝑧) ⊆ 𝑥)}
df-nrm 23230	⊢ Nrm = {𝑗 ∈ Top ∣ ∀𝑥 ∈ 𝑗 ∀𝑦 ∈ ((Clsd‘𝑗) ∩ 𝒫 𝑥)∃𝑧 ∈ 𝑗 (𝑦 ⊆ 𝑧 ∧ ((cls‘𝑗)‘𝑧) ⊆ 𝑥)}
df-cnrm 23231	⊢ CNrm = {𝑗 ∈ Top ∣ ∀𝑥 ∈ 𝒫 ∪ 𝑗(𝑗 ↾_t 𝑥) ∈ Nrm}
df-pnrm 23232	⊢ PNrm = {𝑗 ∈ Nrm ∣ (Clsd‘𝑗) ⊆ ran (𝑓 ∈ (𝑗 ↑_m ℕ) ↦ ∩ ran 𝑓)}
ccmp 23299	class Comp
df-cmp 23300	⊢ Comp = {𝑥 ∈ Top ∣ ∀𝑦 ∈ 𝒫 𝑥(∪ 𝑥 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝑥 = ∪ 𝑧)}
cconn 23324	class Conn
df-conn 23325	⊢ Conn = {𝑗 ∈ Top ∣ (𝑗 ∩ (Clsd‘𝑗)) = {∅, ∪ 𝑗}}
c1stc 23350	class 1^stω
c2ndc 23351	class 2^ndω
df-1stc 23352	⊢ 1^stω = {𝑗 ∈ Top ∣ ∀𝑥 ∈ ∪ 𝑗∃𝑦 ∈ 𝒫 𝑗(𝑦 ≼ ω ∧ ∀𝑧 ∈ 𝑗 (𝑥 ∈ 𝑧 → 𝑥 ∈ ∪ (𝑦 ∩ 𝒫 𝑧)))}
df-2ndc 23353	⊢ 2^ndω = {𝑗 ∣ ∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝑗)}
clly 23377	class Locally 𝐴
cnlly 23378	class 𝑛-Locally 𝐴
df-lly 23379	⊢ Locally 𝐴 = {𝑗 ∈ Top ∣ ∀𝑥 ∈ 𝑗 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝑗 ↾_t 𝑢) ∈ 𝐴)}
df-nlly 23380	⊢ 𝑛-Locally 𝐴 = {𝑗 ∈ Top ∣ ∀𝑥 ∈ 𝑗 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗 ↾_t 𝑢) ∈ 𝐴}
cref 23415	class Ref
cptfin 23416	class PtFin
clocfin 23417	class LocFin
df-ref 23418	⊢ Ref = {⟨𝑥, 𝑦⟩ ∣ (∪ 𝑦 = ∪ 𝑥 ∧ ∀𝑧 ∈ 𝑥 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)}
df-ptfin 23419	⊢ PtFin = {𝑥 ∣ ∀𝑦 ∈ ∪ 𝑥{𝑧 ∈ 𝑥 ∣ 𝑦 ∈ 𝑧} ∈ Fin}
df-locfin 23420	⊢ LocFin = (𝑥 ∈ Top ↦ {𝑦 ∣ (∪ 𝑥 = ∪ 𝑦 ∧ ∀𝑝 ∈ ∪ 𝑥∃𝑛 ∈ 𝑥 (𝑝 ∈ 𝑛 ∧ {𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))})
ckgen 23446	class 𝑘Gen
df-kgen 23447	⊢ 𝑘Gen = (𝑗 ∈ Top ↦ {𝑥 ∈ 𝒫 ∪ 𝑗 ∣ ∀𝑘 ∈ 𝒫 ∪ 𝑗((𝑗 ↾_t 𝑘) ∈ Comp → (𝑥 ∩ 𝑘) ∈ (𝑗 ↾_t 𝑘))})
ctx 23473	class ×_t
cxko 23474	class ↑_ko
df-tx 23475	⊢ ×_t = (𝑟 ∈ V, 𝑠 ∈ V ↦ (topGen‘ran (𝑥 ∈ 𝑟, 𝑦 ∈ 𝑠 ↦ (𝑥 × 𝑦))))
df-xko 23476	⊢ ↑_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 23606	class KQ
df-kq 23607	⊢ KQ = (𝑗 ∈ Top ↦ (𝑗 qTop (𝑥 ∈ ∪ 𝑗 ↦ {𝑦 ∈ 𝑗 ∣ 𝑥 ∈ 𝑦})))
chmeo 23666	class Homeo
chmph 23667	class ≃
df-hmeo 23668	⊢ Homeo = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ {𝑓 ∈ (𝑗 Cn 𝑘) ∣ ^◡𝑓 ∈ (𝑘 Cn 𝑗)})
df-hmph 23669	⊢ ≃ = (^◡Homeo “ (V ∖ 1_o))
cfil 23758	class Fil
df-fil 23759	⊢ Fil = (𝑧 ∈ V ↦ {𝑓 ∈ (fBas‘𝑧) ∣ ∀𝑥 ∈ 𝒫 𝑧((𝑓 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥 ∈ 𝑓)})
cufil 23812	class UFil
cufl 23813	class UFL
df-ufil 23814	⊢ UFil = (𝑔 ∈ V ↦ {𝑓 ∈ (Fil‘𝑔) ∣ ∀𝑥 ∈ 𝒫 𝑔(𝑥 ∈ 𝑓 ∨ (𝑔 ∖ 𝑥) ∈ 𝑓)})
df-ufl 23815	⊢ UFL = {𝑥 ∣ ∀𝑓 ∈ (Fil‘𝑥)∃𝑔 ∈ (UFil‘𝑥)𝑓 ⊆ 𝑔}
cfm 23846	class FilMap
cflim 23847	class fLim
cflf 23848	class fLimf
cfcls 23849	class fClus
cfcf 23850	class fClusf
df-fm 23851	⊢ FilMap = (𝑥 ∈ V, 𝑓 ∈ V ↦ (𝑦 ∈ (fBas‘dom 𝑓) ↦ (𝑥filGenran (𝑡 ∈ 𝑦 ↦ (𝑓 “ 𝑡)))))
df-flim 23852	⊢ fLim = (𝑗 ∈ Top, 𝑓 ∈ ∪ ran Fil ↦ {𝑥 ∈ ∪ 𝑗 ∣ (((nei‘𝑗)‘{𝑥}) ⊆ 𝑓 ∧ 𝑓 ⊆ 𝒫 ∪ 𝑗)})
df-flf 23853	⊢ fLimf = (𝑥 ∈ Top, 𝑦 ∈ ∪ ran Fil ↦ (𝑓 ∈ (∪ 𝑥 ↑_m ∪ 𝑦) ↦ (𝑥 fLim ((∪ 𝑥 FilMap 𝑓)‘𝑦))))
df-fcls 23854	⊢ fClus = (𝑗 ∈ Top, 𝑓 ∈ ∪ ran Fil ↦ if(∪ 𝑗 = ∪ 𝑓, ∩ 𝑥 ∈ 𝑓 ((cls‘𝑗)‘𝑥), ∅))
df-fcf 23855	⊢ fClusf = (𝑗 ∈ Top, 𝑓 ∈ ∪ ran Fil ↦ (𝑔 ∈ (∪ 𝑗 ↑_m ∪ 𝑓) ↦ (𝑗 fClus ((∪ 𝑗 FilMap 𝑔)‘𝑓))))
ccnext 23972	class CnExt
df-cnext 23973	⊢ 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 23983	class TopMnd
ctgp 23984	class TopGrp
df-tmd 23985	⊢ TopMnd = {𝑓 ∈ (Mnd ∩ TopSp) ∣ [(TopOpen‘𝑓) / 𝑗](+_𝑓‘𝑓) ∈ ((𝑗 ×_t 𝑗) Cn 𝑗)}
df-tgp 23986	⊢ TopGrp = {𝑓 ∈ (Grp ∩ TopMnd) ∣ [(TopOpen‘𝑓) / 𝑗](inv_g‘𝑓) ∈ (𝑗 Cn 𝑗)}
ctsu 24039	class tsums
df-tsms 24040	⊢ 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 24069	class TopRing
ctdrg 24070	class TopDRing
ctlm 24071	class TopMod
ctvc 24072	class TopVec
df-trg 24073	⊢ TopRing = {𝑟 ∈ (TopGrp ∩ Ring) ∣ (mulGrp‘𝑟) ∈ TopMnd}
df-tdrg 24074	⊢ TopDRing = {𝑟 ∈ (TopRing ∩ DivRing) ∣ ((mulGrp‘𝑟) ↾_s (Unit‘𝑟)) ∈ TopGrp}
df-tlm 24075	⊢ TopMod = {𝑤 ∈ (TopMnd ∩ LMod) ∣ ((Scalar‘𝑤) ∈ TopRing ∧ ( ·_sf ‘𝑤) ∈ (((TopOpen‘(Scalar‘𝑤)) ×_t (TopOpen‘𝑤)) Cn (TopOpen‘𝑤)))}
df-tvc 24076	⊢ TopVec = {𝑤 ∈ TopMod ∣ (Scalar‘𝑤) ∈ TopDRing}
cust 24113	class UnifOn
df-ust 24114	⊢ 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 24143	class unifTop
df-utop 24144	⊢ unifTop = (𝑢 ∈ ∪ ran UnifOn ↦ {𝑎 ∈ 𝒫 dom ∪ 𝑢 ∣ ∀𝑥 ∈ 𝑎 ∃𝑣 ∈ 𝑢 (𝑣 “ {𝑥}) ⊆ 𝑎})
cuss 24166	class UnifSt
cusp 24167	class UnifSp
ctus 24168	class toUnifSp
df-uss 24169	⊢ UnifSt = (𝑓 ∈ V ↦ ((UnifSet‘𝑓) ↾_t ((Base‘𝑓) × (Base‘𝑓))))
df-usp 24170	⊢ UnifSp = {𝑓 ∣ ((UnifSt‘𝑓) ∈ (UnifOn‘(Base‘𝑓)) ∧ (TopOpen‘𝑓) = (unifTop‘(UnifSt‘𝑓)))}
df-tus 24171	⊢ toUnifSp = (𝑢 ∈ ∪ ran UnifOn ↦ ({⟨(Base‘ndx), dom ∪ 𝑢⟩, ⟨(UnifSet‘ndx), 𝑢⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑢)⟩))
cucn 24187	class Cn_u
df-ucn 24188	⊢ 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 24198	class CauFil_u
df-cfilu 24199	⊢ CauFil_u = (𝑢 ∈ ∪ ran UnifOn ↦ {𝑓 ∈ (fBas‘dom ∪ 𝑢) ∣ ∀𝑣 ∈ 𝑢 ∃𝑎 ∈ 𝑓 (𝑎 × 𝑎) ⊆ 𝑣})
ccusp 24209	class CUnifSp
df-cusp 24210	⊢ CUnifSp = {𝑤 ∈ UnifSp ∣ ∀𝑐 ∈ (Fil‘(Base‘𝑤))(𝑐 ∈ (CauFil_u‘(UnifSt‘𝑤)) → ((TopOpen‘𝑤) fLim 𝑐) ≠ ∅)}
cxms 24230	class ∞MetSp
cms 24231	class MetSp
ctms 24232	class toMetSp
df-xms 24233	⊢ ∞MetSp = {𝑓 ∈ TopSp ∣ (TopOpen‘𝑓) = (MetOpen‘((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))))}
df-ms 24234	⊢ MetSp = {𝑓 ∈ ∞MetSp ∣ ((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))) ∈ (Met‘(Base‘𝑓))}
df-tms 24235	⊢ toMetSp = (𝑑 ∈ ∪ ran ∞Met ↦ ({⟨(Base‘ndx), dom dom 𝑑⟩, ⟨(dist‘ndx), 𝑑⟩} sSet ⟨(TopSet‘ndx), (MetOpen‘𝑑)⟩))
cnm 24489	class norm
cngp 24490	class NrmGrp
ctng 24491	class toNrmGrp
cnrg 24492	class NrmRing
cnlm 24493	class NrmMod
cnvc 24494	class NrmVec
df-nm 24495	⊢ norm = (𝑤 ∈ V ↦ (𝑥 ∈ (Base‘𝑤) ↦ (𝑥(dist‘𝑤)(0_g‘𝑤))))
df-ngp 24496	⊢ NrmGrp = {𝑔 ∈ (Grp ∩ MetSp) ∣ ((norm‘𝑔) ∘ (-_g‘𝑔)) ⊆ (dist‘𝑔)}
df-tng 24497	⊢ toNrmGrp = (𝑔 ∈ V, 𝑓 ∈ V ↦ ((𝑔 sSet ⟨(dist‘ndx), (𝑓 ∘ (-_g‘𝑔))⟩) sSet ⟨(TopSet‘ndx), (MetOpen‘(𝑓 ∘ (-_g‘𝑔)))⟩))
df-nrg 24498	⊢ NrmRing = {𝑤 ∈ NrmGrp ∣ (norm‘𝑤) ∈ (AbsVal‘𝑤)}
df-nlm 24499	⊢ 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 24500	⊢ NrmVec = (NrmMod ∩ LVec)
cnmo 24618	class normOp
cnghm 24619	class NGHom
cnmhm 24620	class NMHom
df-nmo 24621	⊢ 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 24622	⊢ NGHom = (𝑠 ∈ NrmGrp, 𝑡 ∈ NrmGrp ↦ (^◡(𝑠 normOp 𝑡) “ ℝ))
df-nmhm 24623	⊢ NMHom = (𝑠 ∈ NrmMod, 𝑡 ∈ NrmMod ↦ ((𝑠 LMHom 𝑡) ∩ (𝑠 NGHom 𝑡)))
cii 24793	class II
ccncf 24794	class –cn→
df-ii 24795	⊢ II = (MetOpen‘((abs ∘ − ) ↾ ((0[,]1) × (0[,]1))))
df-cncf 24796	⊢ –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 24891	class Htpy
cphtpy 24892	class PHtpy
cphtpc 24893	class ≃_ph
df-htpy 24894	⊢ 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 24895	⊢ 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 24916	⊢ ≃_ph = (𝑥 ∈ Top ↦ {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ (II Cn 𝑥) ∧ (𝑓(PHtpy‘𝑥)𝑔) ≠ ∅)})
cpco 24925	class *_𝑝
comi 24926	class Ω₁
comn 24927	class Ω_𝑛
cpi1 24928	class π₁
cpin 24929	class π_n
df-pco 24930	⊢ *_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 24931	⊢ 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 24932	⊢ 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 24933	⊢ π₁ = (𝑗 ∈ Top, 𝑦 ∈ ∪ 𝑗 ↦ ((𝑗 Ω₁ 𝑦) /_s ( ≃_ph‘𝑗)))
df-pin 24934	⊢ 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 24987	class ℂMod
df-clm 24988	⊢ ℂMod = {𝑤 ∈ LMod ∣ [(Scalar‘𝑤) / 𝑓][(Base‘𝑓) / 𝑘](𝑓 = (ℂ_fld ↾_s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂ_fld))}
ccvs 25048	class ℂVec
df-cvs 25049	⊢ ℂVec = (ℂMod ∩ LVec)
ccph 25091	class ℂPreHil
ctcph 25092	class toℂPreHil
df-cph 25093	⊢ 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 25094	⊢ toℂPreHil = (𝑤 ∈ V ↦ (𝑤 toNrmGrp (𝑥 ∈ (Base‘𝑤) ↦ (√‘(𝑥(·_𝑖‘𝑤)𝑥)))))
ccfil 25177	class CauFil
ccau 25178	class Cau
ccmet 25179	class CMet
df-cfil 25180	⊢ CauFil = (𝑑 ∈ ∪ ran ∞Met ↦ {𝑓 ∈ (Fil‘dom dom 𝑑) ∣ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ 𝑓 (𝑑 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)})
df-cau 25181	⊢ 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 25182	⊢ CMet = (𝑥 ∈ V ↦ {𝑑 ∈ (Met‘𝑥) ∣ ∀𝑓 ∈ (CauFil‘𝑑)((MetOpen‘𝑑) fLim 𝑓) ≠ ∅})
ccms 25257	class CMetSp
cbn 25258	class Ban
chl 25259	class ℂHil
df-cms 25260	⊢ CMetSp = {𝑤 ∈ MetSp ∣ [(Base‘𝑤) / 𝑏]((dist‘𝑤) ↾ (𝑏 × 𝑏)) ∈ (CMet‘𝑏)}
df-bn 25261	⊢ Ban = {𝑤 ∈ (NrmVec ∩ CMetSp) ∣ (Scalar‘𝑤) ∈ CMetSp}
df-hl 25262	⊢ ℂHil = (Ban ∩ ℂPreHil)
crrx 25308	class ℝ^
cehl 25309	class 𝔼_hil
df-rrx 25310	⊢ ℝ^ = (𝑖 ∈ V ↦ (toℂPreHil‘(ℝ_fld freeLMod 𝑖)))
df-ehl 25311	⊢ 𝔼_hil = (𝑛 ∈ ℕ₀ ↦ (ℝ^‘(1...𝑛)))
covol 25388	class vol*
cvol 25389	class vol
df-ovol 25390	⊢ 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 25391	⊢ vol = (vol* ↾ {𝑥 ∣ ∀𝑦 ∈ (^◡vol* “ ℝ)(vol‘𝑦) = ((vol‘(𝑦 ∩ 𝑥)) + (vol*‘(𝑦 ∖ 𝑥)))})
cmbf 25540	class MblFn
citg1 25541	class ∫₁
citg2 25542	class ∫₂
cibl 25543	class 𝐿¹
citg 25544	class ∫𝐴𝐵 d𝑥
df-mbf 25545	⊢ MblFn = {𝑓 ∈ (ℂ ↑_pm ℝ) ∣ ∀𝑥 ∈ ran (,)((^◡(ℜ ∘ 𝑓) “ 𝑥) ∈ dom vol ∧ (^◡(ℑ ∘ 𝑓) “ 𝑥) ∈ dom vol)}
df-itg1 25546	⊢ ∫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 25547	⊢ ∫₂ = (𝑓 ∈ ((0[,]+∞) ↑_m ℝ) ↦ sup({𝑥 ∣ ∃𝑔 ∈ dom ∫₁(𝑔 ∘_r ≤ 𝑓 ∧ 𝑥 = (∫₁‘𝑔))}, ℝ^*, < ))
df-ibl 25548	⊢ 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 25549	⊢ ∫𝐴𝐵 d𝑥 = Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫₂‘(𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐵 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0))))
c0p 25595	class 0_𝑝
df-0p 25596	⊢ 0_𝑝 = (ℂ × {0})
cdit 25772	class ⨜[𝐴 → 𝐵]𝐶 d𝑥
df-ditg 25773	⊢ ⨜[𝐴 → 𝐵]𝐶 d𝑥 = if(𝐴 ≤ 𝐵, ∫(𝐴(,)𝐵)𝐶 d𝑥, -∫(𝐵(,)𝐴)𝐶 d𝑥)
climc 25788	class lim_ℂ
cdv 25789	class D
cdvn 25790	class D^𝑛
ccpn 25791	class 𝓑C^𝑛
df-limc 25792	⊢ 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 25793	⊢ 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 25794	⊢ D^𝑛 = (𝑠 ∈ 𝒫 ℂ, 𝑓 ∈ (ℂ ↑_pm 𝑠) ↦ seq0(((𝑥 ∈ V ↦ (𝑠 D 𝑥)) ∘ 1^st ), (ℕ₀ × {𝑓})))
df-cpn 25795	⊢ 𝓑C^𝑛 = (𝑠 ∈ 𝒫 ℂ ↦ (𝑥 ∈ ℕ₀ ↦ {𝑓 ∈ (ℂ ↑_pm 𝑠) ∣ ((𝑠 D^𝑛 𝑓)‘𝑥) ∈ (dom 𝑓–cn→ℂ)}))
cmdg 25983	class mDeg
cdg1 25984	class deg₁
df-mdeg 25985	⊢ 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 25986	⊢ deg₁ = (𝑟 ∈ V ↦ (1_o mDeg 𝑟))
cmn1 26056	class Monic_1p
cuc1p 26057	class Unic_1p
cq1p 26058	class quot_1p
cr1p 26059	class rem_1p
cig1p 26060	class idlGen_1p
df-mon1 26061	⊢ Monic_1p = (𝑟 ∈ V ↦ {𝑓 ∈ (Base‘(Poly₁‘𝑟)) ∣ (𝑓 ≠ (0_g‘(Poly₁‘𝑟)) ∧ ((coe₁‘𝑓)‘((deg₁‘𝑟)‘𝑓)) = (1_r‘𝑟))})
df-uc1p 26062	⊢ Unic_1p = (𝑟 ∈ V ↦ {𝑓 ∈ (Base‘(Poly₁‘𝑟)) ∣ (𝑓 ≠ (0_g‘(Poly₁‘𝑟)) ∧ ((coe₁‘𝑓)‘((deg₁‘𝑟)‘𝑓)) ∈ (Unit‘𝑟))})
df-q1p 26063	⊢ 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 26064	⊢ 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 26065	⊢ 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 26114	class Poly
cidp 26115	class X_p
ccoe 26116	class coeff
cdgr 26117	class deg
df-ply 26118	⊢ Poly = (𝑥 ∈ 𝒫 ℂ ↦ {𝑓 ∣ ∃𝑛 ∈ ℕ₀ ∃𝑎 ∈ ((𝑥 ∪ {0}) ↑_m ℕ₀)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))})
df-idp 26119	⊢ X_p = ( I ↾ ℂ)
df-coe 26120	⊢ 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 26121	⊢ deg = (𝑓 ∈ (Poly‘ℂ) ↦ sup((^◡(coeff‘𝑓) “ (ℂ ∖ {0})), ℕ₀, < ))
cquot 26223	class quot
df-quot 26224	⊢ 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 26247	class 𝔸
df-aa 26248	⊢ 𝔸 = ∪ 𝑓 ∈ ((Poly‘ℤ) ∖ {0_𝑝})(^◡𝑓 “ {0})
ctayl 26285	class Tayl
cana 26286	class Ana
df-tayl 26287	⊢ 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 26288	⊢ 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 26310	class ⇝_𝑢
df-ulm 26311	⊢ ⇝_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 26488	class log
ccxp 26489	class ↑_𝑐
df-log 26490	⊢ log = ^◡(exp ↾ (^◡ℑ “ (-π(,]π)))
df-cxp 26491	⊢ ↑_𝑐 = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ if(𝑥 = 0, if(𝑦 = 0, 1, 0), (exp‘(𝑦 · (log‘𝑥)))))
clogb 26699	class log_b
df-logb 26700	⊢ log_b = (𝑥 ∈ (ℂ ∖ {0, 1}), 𝑦 ∈ (ℂ ∖ {0}) ↦ ((log‘𝑦) / (log‘𝑥)))
casin 26797	class arcsin
cacos 26798	class arccos
catan 26799	class arctan
df-asin 26800	⊢ arcsin = (𝑥 ∈ ℂ ↦ (-i · (log‘((i · 𝑥) + (√‘(1 − (𝑥↑2)))))))
df-acos 26801	⊢ arccos = (𝑥 ∈ ℂ ↦ ((π / 2) − (arcsin‘𝑥)))
df-atan 26802	⊢ arctan = (𝑥 ∈ (ℂ ∖ {-i, i}) ↦ ((i / 2) · ((log‘(1 − (i · 𝑥))) − (log‘(1 + (i · 𝑥))))))
carea 26890	class area
df-area 26891	⊢ area = (𝑠 ∈ {𝑡 ∈ 𝒫 (ℝ × ℝ) ∣ (∀𝑥 ∈ ℝ (𝑡 “ {𝑥}) ∈ (^◡vol “ ℝ) ∧ (𝑥 ∈ ℝ ↦ (vol‘(𝑡 “ {𝑥}))) ∈ 𝐿¹)} ↦ ∫ℝ(vol‘(𝑠 “ {𝑥})) d𝑥)
cem 26927	class γ
df-em 26928	⊢ γ = Σ𝑘 ∈ ℕ ((1 / 𝑘) − (log‘(1 + (1 / 𝑘))))
czeta 26948	class ζ
df-zeta 26949	⊢ 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 26951	class log Γ
cgam 26952	class Γ
cigam 26953	class 1/Γ
df-lgam 26954	⊢ log Γ = (𝑧 ∈ (ℂ ∖ (ℤ ∖ ℕ)) ↦ (Σ𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))) − (log‘𝑧)))
df-gam 26955	⊢ Γ = (exp ∘ log Γ)
df-igam 26956	⊢ 1/Γ = (𝑥 ∈ ℂ ↦ if(𝑥 ∈ (ℤ ∖ ℕ), 0, (1 / (Γ‘𝑥))))
ccht 27026	class θ
cvma 27027	class Λ
cchp 27028	class ψ
cppi 27029	class π
cmu 27030	class μ
csgm 27031	class σ
df-cht 27032	⊢ θ = (𝑥 ∈ ℝ ↦ Σ𝑝 ∈ ((0[,]𝑥) ∩ ℙ)(log‘𝑝))
df-vma 27033	⊢ Λ = (𝑥 ∈ ℕ ↦ ⦋{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥} / 𝑠⦌if((♯‘𝑠) = 1, (log‘∪ 𝑠), 0))
df-chp 27034	⊢ ψ = (𝑥 ∈ ℝ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))(Λ‘𝑛))
df-ppi 27035	⊢ π = (𝑥 ∈ ℝ ↦ (♯‘((0[,]𝑥) ∩ ℙ)))
df-mu 27036	⊢ μ = (𝑥 ∈ ℕ ↦ if(∃𝑝 ∈ ℙ (𝑝↑2) ∥ 𝑥, 0, (-1↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥}))))
df-sgm 27037	⊢ σ = (𝑥 ∈ ℂ, 𝑛 ∈ ℕ ↦ Σ𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝑛} (𝑘↑_𝑐𝑥))
cdchr 27168	class DChr
df-dchr 27169	⊢ 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 27230	class /_L
df-lgs 27231	⊢ /_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 27576	class No
cslt 27577	class <s
cbday 27578	class bday
df-no 27579	⊢ No = {𝑓 ∣ ∃𝑎 ∈ On 𝑓:𝑎⟶{1_o, 2_o}}
df-slt 27580	⊢ <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 27581	⊢ bday = (𝑥 ∈ No ↦ dom 𝑥)
csle 27681	class ≤s
df-sle 27682	⊢ ≤s = (( No × No ) ∖ ^◡ <s )
csslt 27718	class <<s
df-sslt 27719	⊢ <<s = {⟨𝑎, 𝑏⟩ ∣ (𝑎 ⊆ No ∧ 𝑏 ⊆ No ∧ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 𝑥 <s 𝑦)}
cscut 27720	class \|s
df-scut 27721	⊢ \|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 27764	class 0_s
c1s 27765	class 1_s
df-0s 27766	⊢ 0_s = (∅ \|s ∅)
df-1s 27767	⊢ 1_s = ({ 0_s } \|s ∅)
cmade 27781	class M
cold 27782	class O
cnew 27783	class N
cleft 27784	class L
cright 27785	class R
df-made 27786	⊢ M = recs((𝑓 ∈ V ↦ ( \|s “ (𝒫 ∪ ran 𝑓 × 𝒫 ∪ ran 𝑓))))
df-old 27787	⊢ O = (𝑥 ∈ On ↦ ∪ ( M “ 𝑥))
df-new 27788	⊢ N = (𝑥 ∈ On ↦ (( M ‘𝑥) ∖ ( O ‘𝑥)))
df-left 27789	⊢ L = (𝑥 ∈ No ↦ {𝑦 ∈ ( O ‘( bday ‘𝑥)) ∣ 𝑦 <s 𝑥})
df-right 27790	⊢ R = (𝑥 ∈ No ↦ {𝑦 ∈ ( O ‘( bday ‘𝑥)) ∣ 𝑥 <s 𝑦})
cnorec 27878	class norec (𝐹)
df-norec 27879	⊢ norec (𝐹) = frecs({⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}, No , 𝐹)
cnorec2 27889	class norec2 (𝐹)
df-norec2 27890	⊢ 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 27900	class +_s
df-adds 27901	⊢ +_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 27959	class -u_s
csubs 27960	class -_s
df-negs 27961	⊢ -u_s = norec ((𝑥 ∈ V, 𝑛 ∈ V ↦ ((𝑛 “ ( R ‘𝑥)) \|s (𝑛 “ ( L ‘𝑥)))))
df-subs 27962	⊢ -_s = (𝑥 ∈ No , 𝑦 ∈ No ↦ (𝑥 +_s ( -u_s ‘𝑦)))
cmuls 28043	class ·_s
df-muls 28044	⊢ ·_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 28124	class /_su
df-divs 28125	⊢ /_su = (𝑥 ∈ No , 𝑦 ∈ ( No ∖ { 0_s }) ↦ (℩𝑧 ∈ No (𝑦 ·_s 𝑧) = 𝑥))
cabss 28173	class abs_s
df-abss 28174	⊢ abs_s = (𝑥 ∈ No ↦ if( 0_s ≤s 𝑥, 𝑥, ( -u_s ‘𝑥)))
cons 28186	class On_s
df-ons 28187	⊢ On_s = {𝑥 ∈ No ∣ ( R ‘𝑥) = ∅}
cseqs 28211	class seq_s𝑀( + , 𝐹)
df-seqs 28212	⊢ seq_s𝑀( + , 𝐹) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +_s 1_s ), (𝑦 + (𝐹‘(𝑥 +_s 1_s )))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) “ ω)
cnn0s 28240	class ℕ_0s
cnns 28241	class ℕ_s
df-n0s 28242	⊢ ℕ_0s = (rec((𝑥 ∈ V ↦ (𝑥 +_s 1_s )), 0_s ) “ ω)
df-nns 28243	⊢ ℕ_s = (ℕ_0s ∖ { 0_s })
czs 28300	class ℤ_s
df-zs 28301	⊢ ℤ_s = ( -_s “ (ℕ_s × ℕ_s))
c2s 28331	class 2_s
df-2s 28332	⊢ 2_s = ({ 1_s } \|s ∅)
cexps 28333	class ↑_s
df-exps 28334	⊢ ↑_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))))))
czs12 28335	class ℤ_s[1/2]
df-zs12 28336	⊢ ℤ_s[1/2] = {𝑥 ∣ ∃𝑦 ∈ ℤ_s ∃𝑧 ∈ ℕ_0s 𝑥 = (𝑦 /_su (2_s↑_s𝑧))}
creno 28393	class ℝ_s
df-reno 28394	⊢ 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 28403	class TarskiG
cstrkgc 28404	class TarskiG_C
cstrkgb 28405	class TarskiG_B
cstrkgcb 28406	class TarskiG_CB
cstrkgld 28407	class Dim_TarskiG≥
cstrkge 28408	class TarskiG_E
citv 28409	class Itv
clng 28410	class LineG
df-itv 28411	⊢ Itv = Slot ;16
df-lng 28412	⊢ LineG = Slot ;17
df-trkgc 28424	⊢ 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 28425	⊢ 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 28426	⊢ 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 28427	⊢ 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 28428	⊢ 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 28429	⊢ 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 28486	class cgrG
df-cgrg 28487	⊢ 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 28508	class Ismt
df-ismt 28509	⊢ 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 28558	class ≤G
df-leg 28559	⊢ ≤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 28576	class hlG
df-hlg 28577	⊢ 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 28628	class pInvG
df-mir 28629	⊢ 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 28669	class ∟G
df-rag 28670	⊢ ∟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 28671	class ⟂G
df-perpg 28672	⊢ ⟂G = (𝑔 ∈ V ↦ {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ ran (LineG‘𝑔) ∧ 𝑏 ∈ ran (LineG‘𝑔)) ∧ ∃𝑥 ∈ (𝑎 ∩ 𝑏)∀𝑢 ∈ 𝑎 ∀𝑣 ∈ 𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝑔))})
chpg 28733	class hpG
df-hpg 28734	⊢ 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 28748	class midG
clmi 28749	class lInvG
df-mid 28750	⊢ midG = (𝑔 ∈ V ↦ (𝑎 ∈ (Base‘𝑔), 𝑏 ∈ (Base‘𝑔) ↦ (℩𝑚 ∈ (Base‘𝑔)𝑏 = (((pInvG‘𝑔)‘𝑚)‘𝑎))))
df-lmi 28751	⊢ 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 28783	class cgrA
df-cgra 28784	⊢ 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 28811	class inA
cleag 28812	class ≤_∠
df-inag 28813	⊢ 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 28822	⊢ ≤_∠ = (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 28841	class eqltrG
df-eqlg 28842	⊢ eqltrG = (𝑔 ∈ V ↦ {𝑥 ∈ ((Base‘𝑔) ↑_m (0..^3)) ∣ 𝑥(cgrG‘𝑔)⟨“(𝑥‘1)(𝑥‘2)(𝑥‘0)”⟩})
cttg 28849	class toTG
df-ttg 28850	⊢ 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 28864	class 𝔼
cbtwn 28865	class Btwn
ccgr 28866	class Cgr
df-ee 28867	⊢ 𝔼 = (𝑛 ∈ ℕ ↦ (ℝ ↑_m (1...𝑛)))
df-btwn 28868	⊢ 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 28869	⊢ 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 28953	class EEG
df-eeng 28954	⊢ 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 28964	class .ef
df-edgf 28965	⊢ .ef = Slot ;18
cvtx 28972	class Vtx
ciedg 28973	class iEdg
df-vtx 28974	⊢ Vtx = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1^st ‘𝑔), (Base‘𝑔)))
df-iedg 28975	⊢ iEdg = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (2^nd ‘𝑔), (.ef‘𝑔)))
cedg 29023	class Edg
df-edg 29024	⊢ Edg = (𝑔 ∈ V ↦ ran (iEdg‘𝑔))
cuhgr 29032	class UHGraph
cushgr 29033	class USHGraph
df-uhgr 29034	⊢ UHGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶(𝒫 𝑣 ∖ {∅})}
df-ushgr 29035	⊢ USHGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→(𝒫 𝑣 ∖ {∅})}
cupgr 29056	class UPGraph
cumgr 29057	class UMGraph
df-upgr 29058	⊢ UPGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}}
df-umgr 29059	⊢ UMGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2}}
cuspgr 29124	class USPGraph
cusgr 29125	class USGraph
df-uspgr 29126	⊢ USPGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}}
df-usgr 29127	⊢ USGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2}}
csubgr 29243	class SubGraph
df-subgr 29244	⊢ SubGraph = {⟨𝑠, 𝑔⟩ ∣ ((Vtx‘𝑠) ⊆ (Vtx‘𝑔) ∧ (iEdg‘𝑠) = ((iEdg‘𝑔) ↾ dom (iEdg‘𝑠)) ∧ (Edg‘𝑠) ⊆ 𝒫 (Vtx‘𝑠))}
cfusgr 29292	class FinUSGraph
df-fusgr 29293	⊢ FinUSGraph = {𝑔 ∈ USGraph ∣ (Vtx‘𝑔) ∈ Fin}
cnbgr 29308	class NeighbVtx
df-nbgr 29309	⊢ NeighbVtx = (𝑔 ∈ V, 𝑣 ∈ (Vtx‘𝑔) ↦ {𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣}) ∣ ∃𝑒 ∈ (Edg‘𝑔){𝑣, 𝑛} ⊆ 𝑒})
cuvtx 29361	class UnivVtx
df-uvtx 29362	⊢ UnivVtx = (𝑔 ∈ V ↦ {𝑣 ∈ (Vtx‘𝑔) ∣ ∀𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣})𝑛 ∈ (𝑔 NeighbVtx 𝑣)})
ccplgr 29385	class ComplGraph
ccusgr 29386	class ComplUSGraph
df-cplgr 29387	⊢ ComplGraph = {𝑔 ∣ (UnivVtx‘𝑔) = (Vtx‘𝑔)}
df-cusgr 29388	⊢ ComplUSGraph = (USGraph ∩ ComplGraph)
cvtxdg 29442	class VtxDeg
df-vtxdg 29443	⊢ 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 29532	class RegGraph
crusgr 29533	class RegUSGraph
df-rgr 29534	⊢ RegGraph = {⟨𝑔, 𝑘⟩ ∣ (𝑘 ∈ ℕ₀^* ∧ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 𝑘)}
df-rusgr 29535	⊢ RegUSGraph = {⟨𝑔, 𝑘⟩ ∣ (𝑔 ∈ USGraph ∧ 𝑔 RegGraph 𝑘)}
cewlks 29572	class EdgWalks
cwlks 29573	class Walks
cwlkson 29574	class WalksOn
df-ewlks 29575	⊢ 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 29576	⊢ 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 29577	⊢ WalksOn = (𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝑔)𝑝 ∧ (𝑝‘0) = 𝑎 ∧ (𝑝‘(♯‘𝑓)) = 𝑏)}))
ctrls 29665	class Trails
ctrlson 29666	class TrailsOn
df-trls 29667	⊢ Trails = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝑔)𝑝 ∧ Fun ^◡𝑓)})
df-trlson 29668	⊢ TrailsOn = (𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑎(WalksOn‘𝑔)𝑏)𝑝 ∧ 𝑓(Trails‘𝑔)𝑝)}))
cpths 29686	class Paths
cspths 29687	class SPaths
cpthson 29688	class PathsOn
cspthson 29689	class SPathsOn
df-pths 29690	⊢ Paths = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝑔)𝑝 ∧ Fun ^◡(𝑝 ↾ (1..^(♯‘𝑓))) ∧ ((𝑝 “ {0, (♯‘𝑓)}) ∩ (𝑝 “ (1..^(♯‘𝑓)))) = ∅)})
df-spths 29691	⊢ SPaths = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝑔)𝑝 ∧ Fun ^◡𝑝)})
df-pthson 29692	⊢ PathsOn = (𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑎(TrailsOn‘𝑔)𝑏)𝑝 ∧ 𝑓(Paths‘𝑔)𝑝)}))
df-spthson 29693	⊢ SPathsOn = (𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑎(TrailsOn‘𝑔)𝑏)𝑝 ∧ 𝑓(SPaths‘𝑔)𝑝)}))
cclwlks 29746	class ClWalks
df-clwlks 29747	⊢ ClWalks = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝑔)𝑝 ∧ (𝑝‘0) = (𝑝‘(♯‘𝑓)))})
ccrcts 29760	class Circuits
ccycls 29761	class Cycles
df-crcts 29762	⊢ Circuits = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝑔)𝑝 ∧ (𝑝‘0) = (𝑝‘(♯‘𝑓)))})
df-cycls 29763	⊢ Cycles = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Paths‘𝑔)𝑝 ∧ (𝑝‘0) = (𝑝‘(♯‘𝑓)))})
cwwlks 29801	class WWalks
cwwlksn 29802	class WWalksN
cwwlksnon 29803	class WWalksNOn
cwwspthsn 29804	class WSPathsN
cwwspthsnon 29805	class WSPathsNOn
df-wwlks 29806	⊢ WWalks = (𝑔 ∈ V ↦ {𝑤 ∈ Word (Vtx‘𝑔) ∣ (𝑤 ≠ ∅ ∧ ∀𝑖 ∈ (0..^((♯‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝑔))})
df-wwlksn 29807	⊢ WWalksN = (𝑛 ∈ ℕ₀, 𝑔 ∈ V ↦ {𝑤 ∈ (WWalks‘𝑔) ∣ (♯‘𝑤) = (𝑛 + 1)})
df-wwlksnon 29808	⊢ WWalksNOn = (𝑛 ∈ ℕ₀, 𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {𝑤 ∈ (𝑛 WWalksN 𝑔) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤‘𝑛) = 𝑏)}))
df-wspthsn 29809	⊢ WSPathsN = (𝑛 ∈ ℕ₀, 𝑔 ∈ V ↦ {𝑤 ∈ (𝑛 WWalksN 𝑔) ∣ ∃𝑓 𝑓(SPaths‘𝑔)𝑤})
df-wspthsnon 29810	⊢ 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 29956	class ClWWalks
df-clwwlk 29957	⊢ 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 29999	class ClWWalksN
df-clwwlkn 30000	⊢ ClWWalksN = (𝑛 ∈ ℕ₀, 𝑔 ∈ V ↦ {𝑤 ∈ (ClWWalks‘𝑔) ∣ (♯‘𝑤) = 𝑛})
cclwwlknon 30062	class ClWWalksNOn
df-clwwlknon 30063	⊢ ClWWalksNOn = (𝑔 ∈ V ↦ (𝑣 ∈ (Vtx‘𝑔), 𝑛 ∈ ℕ₀ ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝑔) ∣ (𝑤‘0) = 𝑣}))
cconngr 30161	class ConnGraph
df-conngr 30162	⊢ ConnGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣]∀𝑘 ∈ 𝑣 ∀𝑛 ∈ 𝑣 ∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝}
ceupth 30172	class EulerPaths
df-eupth 30173	⊢ EulerPaths = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝑔)𝑝 ∧ 𝑓:(0..^(♯‘𝑓))–onto→dom (iEdg‘𝑔))})
cfrgr 30233	class FriendGraph
df-frgr 30234	⊢ FriendGraph = {𝑔 ∈ USGraph ∣ [(Vtx‘𝑔) / 𝑣][(Edg‘𝑔) / 𝑒]∀𝑘 ∈ 𝑣 ∀𝑙 ∈ (𝑣 ∖ {𝑘})∃!𝑥 ∈ 𝑣 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝑒}
ax-flt 30447	⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ (𝑋 ∈ ℕ ∧ 𝑌 ∈ ℕ ∧ 𝑍 ∈ ℕ)) → ((𝑋↑𝑁) + (𝑌↑𝑁)) ≠ (𝑍↑𝑁))
cplig 30449	class Plig
df-plig 30450	⊢ 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 30464	class GrpOp
cgi 30465	class GId
cgn 30466	class inv
cgs 30467	class /_𝑔
df-grpo 30468	⊢ 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 30469	⊢ GId = (𝑔 ∈ V ↦ (℩𝑢 ∈ ran 𝑔∀𝑥 ∈ ran 𝑔((𝑢𝑔𝑥) = 𝑥 ∧ (𝑥𝑔𝑢) = 𝑥)))
df-ginv 30470	⊢ inv = (𝑔 ∈ GrpOp ↦ (𝑥 ∈ ran 𝑔 ↦ (℩𝑧 ∈ ran 𝑔(𝑧𝑔𝑥) = (GId‘𝑔))))
df-gdiv 30471	⊢ /_𝑔 = (𝑔 ∈ GrpOp ↦ (𝑥 ∈ ran 𝑔, 𝑦 ∈ ran 𝑔 ↦ (𝑥𝑔((inv‘𝑔)‘𝑦))))
cablo 30519	class AbelOp
df-ablo 30520	⊢ AbelOp = {𝑔 ∈ GrpOp ∣ ∀𝑥 ∈ ran 𝑔∀𝑦 ∈ ran 𝑔(𝑥𝑔𝑦) = (𝑦𝑔𝑥)}
cvc 30533	class CVec_OLD
df-vc 30534	⊢ 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 30559	class NrmCVec
cpv 30560	class +_𝑣
cba 30561	class BaseSet
cns 30562	class ·_𝑠OLD
cn0v 30563	class 0_vec
cnsb 30564	class −_𝑣
cnmcv 30565	class norm_CV
cims 30566	class IndMet
df-nv 30567	⊢ 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 30570	⊢ +_𝑣 = (1^st ∘ 1^st )
df-ba 30571	⊢ BaseSet = (𝑥 ∈ V ↦ ran ( +_𝑣 ‘𝑥))
df-sm 30572	⊢ ·_𝑠OLD = (2^nd ∘ 1^st )
df-0v 30573	⊢ 0_vec = (GId ∘ +_𝑣 )
df-vs 30574	⊢ −_𝑣 = ( /_𝑔 ∘ +_𝑣 )
df-nmcv 30575	⊢ norm_CV = 2^nd
df-ims 30576	⊢ IndMet = (𝑢 ∈ NrmCVec ↦ ((norm_CV‘𝑢) ∘ ( −_𝑣 ‘𝑢)))
cdip 30675	class ·_𝑖OLD
df-dip 30676	⊢ ·_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 30696	class SubSp
df-ssp 30697	⊢ SubSp = (𝑢 ∈ NrmCVec ↦ {𝑤 ∈ NrmCVec ∣ (( +_𝑣 ‘𝑤) ⊆ ( +_𝑣 ‘𝑢) ∧ ( ·_𝑠OLD ‘𝑤) ⊆ ( ·_𝑠OLD ‘𝑢) ∧ (norm_CV‘𝑤) ⊆ (norm_CV‘𝑢))})
clno 30715	class LnOp
cnmoo 30716	class normOp_OLD
cblo 30717	class BLnOp
c0o 30718	class 0_op
df-lno 30719	⊢ 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 30720	⊢ 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 30721	⊢ BLnOp = (𝑢 ∈ NrmCVec, 𝑤 ∈ NrmCVec ↦ {𝑡 ∈ (𝑢 LnOp 𝑤) ∣ ((𝑢 normOp_OLD 𝑤)‘𝑡) < +∞})
df-0o 30722	⊢ 0_op = (𝑢 ∈ NrmCVec, 𝑤 ∈ NrmCVec ↦ ((BaseSet‘𝑢) × {(0_vec‘𝑤)}))
caj 30723	class adj
chmo 30724	class HmOp
df-aj 30725	⊢ 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 30726	⊢ HmOp = (𝑢 ∈ NrmCVec ↦ {𝑡 ∈ dom (𝑢adj𝑢) ∣ ((𝑢adj𝑢)‘𝑡) = 𝑡})
ccphlo 30787	class CPreHil_OLD
df-ph 30788	⊢ 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 30837	class CBan
df-cbn 30838	⊢ CBan = {𝑢 ∈ NrmCVec ∣ (IndMet‘𝑢) ∈ (CMet‘(BaseSet‘𝑢))}
chlo 30860	class CHil_OLD
df-hlo 30861	⊢ CHil_OLD = (CBan ∩ CPreHil_OLD)
The list of syntax, axioms (ax-) and definitions (df-) for the Hilbert Space Explorer starts here
chba 30894	class ℋ
cva 30895	class +_ℎ
csm 30896	class ·_ℎ
csp 30897	class ·_ih
cno 30898	class norm_ℎ
c0v 30899	class 0_ℎ
cmv 30900	class −_ℎ
ccauold 30901	class Cauchy
chli 30902	class ⇝_𝑣
csh 30903	class S_ℋ
cch 30904	class C_ℋ
cort 30905	class ⊥
cph 30906	class +_ℋ
cspn 30907	class span
chj 30908	class ∨_ℋ
chsup 30909	class ∨_ℋ
c0h 30910	class 0_ℋ
ccm 30911	class 𝐶_ℋ
cpjh 30912	class proj_ℎ
chos 30913	class +_op
chot 30914	class ·_op
chod 30915	class −_op
chfs 30916	class +_fn
chft 30917	class ·_fn
ch0o 30918	class 0_hop
chio 30919	class I_op
cnop 30920	class norm_op
ccop 30921	class ContOp
clo 30922	class LinOp
cbo 30923	class BndLinOp
cuo 30924	class UniOp
cho 30925	class HrmOp
cnmf 30926	class norm_fn
cnl 30927	class null
ccnfn 30928	class ContFn
clf 30929	class LinFn
cado 30930	class adj_ℎ
cbr 30931	class bra
ck 30932	class ketbra
cleo 30933	class ≤_op
cei 30934	class eigvec
cel 30935	class eigval
cspc 30936	class Lambda
cst 30937	class States
chst 30938	class CHStates
ccv 30939	class ⋖_ℋ
cat 30940	class HAtoms
cmd 30941	class 𝑀_ℋ
cdmd 30942	class 𝑀_ℋ^*
df-hnorm 30943	⊢ norm_ℎ = (𝑥 ∈ dom dom ·_ih ↦ (√‘(𝑥 ·_ih 𝑥)))
df-hba 30944	⊢ ℋ = (BaseSet‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
df-h0v 30945	⊢ 0_ℎ = (0_vec‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
df-hvsub 30946	⊢ −_ℎ = (𝑥 ∈ ℋ, 𝑦 ∈ ℋ ↦ (𝑥 +_ℎ (-1 ·_ℎ 𝑦)))
df-hlim 30947	⊢ ⇝_𝑣 = {⟨𝑓, 𝑤⟩ ∣ ((𝑓:ℕ⟶ ℋ ∧ 𝑤 ∈ ℋ) ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ_≥‘𝑦)(norm_ℎ‘((𝑓‘𝑧) −_ℎ 𝑤)) < 𝑥)}
df-hcau 30948	⊢ Cauchy = {𝑓 ∈ ( ℋ ↑_m ℕ) ∣ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ_≥‘𝑦)(norm_ℎ‘((𝑓‘𝑦) −_ℎ (𝑓‘𝑧))) < 𝑥}
ax-hilex 30974	⊢ ℋ ∈ V
ax-hfvadd 30975	⊢ +_ℎ :( ℋ × ℋ)⟶ ℋ
ax-hvcom 30976	⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 +_ℎ 𝐵) = (𝐵 +_ℎ 𝐴))
ax-hvass 30977	⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 +_ℎ 𝐵) +_ℎ 𝐶) = (𝐴 +_ℎ (𝐵 +_ℎ 𝐶)))
ax-hv0cl 30978	⊢ 0_ℎ ∈ ℋ
ax-hvaddid 30979	⊢ (𝐴 ∈ ℋ → (𝐴 +_ℎ 0_ℎ) = 𝐴)
ax-hfvmul 30980	⊢ ·_ℎ :(ℂ × ℋ)⟶ ℋ
ax-hvmulid 30981	⊢ (𝐴 ∈ ℋ → (1 ·_ℎ 𝐴) = 𝐴)
ax-hvmulass 30982	⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝐴 · 𝐵) ·_ℎ 𝐶) = (𝐴 ·_ℎ (𝐵 ·_ℎ 𝐶)))
ax-hvdistr1 30983	⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 ·_ℎ (𝐵 +_ℎ 𝐶)) = ((𝐴 ·_ℎ 𝐵) +_ℎ (𝐴 ·_ℎ 𝐶)))
ax-hvdistr2 30984	⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝐴 + 𝐵) ·_ℎ 𝐶) = ((𝐴 ·_ℎ 𝐶) +_ℎ (𝐵 ·_ℎ 𝐶)))
ax-hvmul0 30985	⊢ (𝐴 ∈ ℋ → (0 ·_ℎ 𝐴) = 0_ℎ)
ax-hfi 31054	⊢ ·_ih :( ℋ × ℋ)⟶ℂ
ax-his1 31057	⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·_ih 𝐵) = (∗‘(𝐵 ·_ih 𝐴)))
ax-his2 31058	⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 +_ℎ 𝐵) ·_ih 𝐶) = ((𝐴 ·_ih 𝐶) + (𝐵 ·_ih 𝐶)))
ax-his3 31059	⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 ·_ℎ 𝐵) ·_ih 𝐶) = (𝐴 · (𝐵 ·_ih 𝐶)))
ax-his4 31060	⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0_ℎ) → 0 < (𝐴 ·_ih 𝐴))
ax-hcompl 31177	⊢ (𝐹 ∈ Cauchy → ∃𝑥 ∈ ℋ 𝐹 ⇝_𝑣 𝑥)
df-sh 31182	⊢ S_ℋ = {ℎ ∈ 𝒫 ℋ ∣ (0_ℎ ∈ ℎ ∧ ( +_ℎ “ (ℎ × ℎ)) ⊆ ℎ ∧ ( ·_ℎ “ (ℂ × ℎ)) ⊆ ℎ)}
df-ch 31196	⊢ C_ℋ = {ℎ ∈ S_ℋ ∣ ( ⇝_𝑣 “ (ℎ ↑_m ℕ)) ⊆ ℎ}
df-oc 31227	⊢ ⊥ = (𝑥 ∈ 𝒫 ℋ ↦ {𝑦 ∈ ℋ ∣ ∀𝑧 ∈ 𝑥 (𝑦 ·_ih 𝑧) = 0})
df-ch0 31228	⊢ 0_ℋ = {0_ℎ}
df-shs 31283	⊢ +_ℋ = (𝑥 ∈ S_ℋ , 𝑦 ∈ S_ℋ ↦ ( +_ℎ “ (𝑥 × 𝑦)))
df-span 31284	⊢ span = (𝑥 ∈ 𝒫 ℋ ↦ ∩ {𝑦 ∈ S_ℋ ∣ 𝑥 ⊆ 𝑦})
df-chj 31285	⊢ ∨_ℋ = (𝑥 ∈ 𝒫 ℋ, 𝑦 ∈ 𝒫 ℋ ↦ (⊥‘(⊥‘(𝑥 ∪ 𝑦))))
df-chsup 31286	⊢ ∨_ℋ = (𝑥 ∈ 𝒫 𝒫 ℋ ↦ (⊥‘(⊥‘∪ 𝑥)))
df-pjh 31370	⊢ proj_ℎ = (ℎ ∈ C_ℋ ↦ (𝑥 ∈ ℋ ↦ (℩𝑧 ∈ ℎ ∃𝑦 ∈ (⊥‘ℎ)𝑥 = (𝑧 +_ℎ 𝑦))))
df-cm 31558	⊢ 𝐶_ℋ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ C_ℋ ∧ 𝑦 ∈ C_ℋ ) ∧ 𝑥 = ((𝑥 ∩ 𝑦) ∨_ℋ (𝑥 ∩ (⊥‘𝑦))))}
df-hosum 31705	⊢ +_op = (𝑓 ∈ ( ℋ ↑_m ℋ), 𝑔 ∈ ( ℋ ↑_m ℋ) ↦ (𝑥 ∈ ℋ ↦ ((𝑓‘𝑥) +_ℎ (𝑔‘𝑥))))
df-homul 31706	⊢ ·_op = (𝑓 ∈ ℂ, 𝑔 ∈ ( ℋ ↑_m ℋ) ↦ (𝑥 ∈ ℋ ↦ (𝑓 ·_ℎ (𝑔‘𝑥))))
df-hodif 31707	⊢ −_op = (𝑓 ∈ ( ℋ ↑_m ℋ), 𝑔 ∈ ( ℋ ↑_m ℋ) ↦ (𝑥 ∈ ℋ ↦ ((𝑓‘𝑥) −_ℎ (𝑔‘𝑥))))
df-hfsum 31708	⊢ +_fn = (𝑓 ∈ (ℂ ↑_m ℋ), 𝑔 ∈ (ℂ ↑_m ℋ) ↦ (𝑥 ∈ ℋ ↦ ((𝑓‘𝑥) + (𝑔‘𝑥))))
df-hfmul 31709	⊢ ·_fn = (𝑓 ∈ ℂ, 𝑔 ∈ (ℂ ↑_m ℋ) ↦ (𝑥 ∈ ℋ ↦ (𝑓 · (𝑔‘𝑥))))
df-h0op 31723	⊢ 0_hop = (proj_ℎ‘0_ℋ)
df-iop 31724	⊢ I_op = (proj_ℎ‘ ℋ)
df-nmop 31814	⊢ norm_op = (𝑡 ∈ ( ℋ ↑_m ℋ) ↦ sup({𝑥 ∣ ∃𝑧 ∈ ℋ ((norm_ℎ‘𝑧) ≤ 1 ∧ 𝑥 = (norm_ℎ‘(𝑡‘𝑧)))}, ℝ^*, < ))
df-cnop 31815	⊢ 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 31816	⊢ 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 31817	⊢ BndLinOp = {𝑡 ∈ LinOp ∣ (norm_op‘𝑡) < +∞}
df-unop 31818	⊢ UniOp = {𝑡 ∣ (𝑡: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑡‘𝑥) ·_ih (𝑡‘𝑦)) = (𝑥 ·_ih 𝑦))}
df-hmop 31819	⊢ HrmOp = {𝑡 ∈ ( ℋ ↑_m ℋ) ∣ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑡‘𝑦)) = ((𝑡‘𝑥) ·_ih 𝑦)}
df-nmfn 31820	⊢ norm_fn = (𝑡 ∈ (ℂ ↑_m ℋ) ↦ sup({𝑥 ∣ ∃𝑧 ∈ ℋ ((norm_ℎ‘𝑧) ≤ 1 ∧ 𝑥 = (abs‘(𝑡‘𝑧)))}, ℝ^*, < ))
df-nlfn 31821	⊢ null = (𝑡 ∈ (ℂ ↑_m ℋ) ↦ (^◡𝑡 “ {0}))
df-cnfn 31822	⊢ 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 31823	⊢ LinFn = {𝑡 ∈ (ℂ ↑_m ℋ) ∣ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ (𝑡‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) = ((𝑥 · (𝑡‘𝑦)) + (𝑡‘𝑧))}
df-adjh 31824	⊢ adj_ℎ = {⟨𝑡, 𝑢⟩ ∣ (𝑡: ℋ⟶ ℋ ∧ 𝑢: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑡‘𝑥) ·_ih 𝑦) = (𝑥 ·_ih (𝑢‘𝑦)))}
df-bra 31825	⊢ bra = (𝑥 ∈ ℋ ↦ (𝑦 ∈ ℋ ↦ (𝑦 ·_ih 𝑥)))
df-kb 31826	⊢ ketbra = (𝑥 ∈ ℋ, 𝑦 ∈ ℋ ↦ (𝑧 ∈ ℋ ↦ ((𝑧 ·_ih 𝑦) ·_ℎ 𝑥)))
df-leop 31827	⊢ ≤_op = {⟨𝑡, 𝑢⟩ ∣ ((𝑢 −_op 𝑡) ∈ HrmOp ∧ ∀𝑥 ∈ ℋ 0 ≤ (((𝑢 −_op 𝑡)‘𝑥) ·_ih 𝑥))}
df-eigvec 31828	⊢ eigvec = (𝑡 ∈ ( ℋ ↑_m ℋ) ↦ {𝑥 ∈ ( ℋ ∖ 0_ℋ) ∣ ∃𝑧 ∈ ℂ (𝑡‘𝑥) = (𝑧 ·_ℎ 𝑥)})
df-eigval 31829	⊢ eigval = (𝑡 ∈ ( ℋ ↑_m ℋ) ↦ (𝑥 ∈ (eigvec‘𝑡) ↦ (((𝑡‘𝑥) ·_ih 𝑥) / ((norm_ℎ‘𝑥)↑2))))
df-spec 31830	⊢ Lambda = (𝑡 ∈ ( ℋ ↑_m ℋ) ↦ {𝑥 ∈ ℂ ∣ ¬ (𝑡 −_op (𝑥 ·_op ( I ↾ ℋ))): ℋ–1-1→ ℋ})
df-st 32186	⊢ 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 32187	⊢ 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 32254	⊢ ⋖_ℋ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ C_ℋ ∧ 𝑦 ∈ C_ℋ ) ∧ (𝑥 ⊊ 𝑦 ∧ ¬ ∃𝑧 ∈ C_ℋ (𝑥 ⊊ 𝑧 ∧ 𝑧 ⊊ 𝑦)))}
df-md 32255	⊢ 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 32256	⊢ 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 32313	⊢ HAtoms = {𝑥 ∈ C_ℋ ∣ 0_ℋ ⋖_ℋ 𝑥}
The list of syntax, axioms (ax-) and definitions (df-) for the User Mathboxes starts here
w2reu 32452	wff ∃!𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵𝜑
df-2reu 32453	⊢ (∃!𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵𝜑 ↔ (∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ ∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑))
cind 32826	class 𝟭
df-ind 32827	⊢ 𝟭 = (𝑜 ∈ V ↦ (𝑎 ∈ 𝒫 𝑜 ↦ (𝑥 ∈ 𝑜 ↦ if(𝑥 ∈ 𝑎, 1, 0))))
cdp2 32846	class _𝐴𝐵
df-dp2 32847	⊢ _𝐴𝐵 = (𝐴 + (𝐵 / ;10))
cdp 32863	class .
df-dp 32864	⊢ . = (𝑥 ∈ ℕ₀, 𝑦 ∈ ℝ ↦ _𝑥𝑦)
cxdiv 32892	class /_𝑒
df-xdiv 32893	⊢ /_𝑒 = (𝑥 ∈ ℝ^, 𝑦 ∈ (ℝ ∖ {0}) ↦ (℩𝑧 ∈ ℝ^ (𝑦 ·_e 𝑧) = 𝑥))
cmnt 32954	class Monot
cmgc 32955	class MGalConn
df-mnt 32956	⊢ 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 32957	⊢ 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 32980	⊢ ℝ_fld = (Scalar‘ℝ^*_𝑠)
ax-xrsvsca 32981	⊢ ·_e = ( ·_𝑠 ‘ℝ^*_𝑠)
ctocyc 33070	class toCyc
df-tocyc 33071	⊢ toCyc = (𝑑 ∈ V ↦ (𝑤 ∈ {𝑢 ∈ Word 𝑑 ∣ 𝑢:dom 𝑢–1-1→𝑑} ↦ (( I ↾ (𝑑 ∖ ran 𝑤)) ∪ ((𝑤 cyclShift 1) ∘ ^◡𝑤))))
csgns 33122	class sgn_s
df-sgns 33123	⊢ sgn_s = (𝑟 ∈ V ↦ (𝑥 ∈ (Base‘𝑟) ↦ if(𝑥 = (0_g‘𝑟), 0, if((0_g‘𝑟)(lt‘𝑟)𝑥, 1, -1))))
cfxp 33127	class FixPts
df-fxp 33128	⊢ FixPts = (𝑏 ∈ V, 𝑎 ∈ V ↦ {𝑥 ∈ 𝑏 ∣ ∀𝑝 ∈ dom dom 𝑎(𝑝𝑎𝑥) = 𝑥})
cinftm 33140	class ⋘
carchi 33141	class Archi
df-inftm 33142	⊢ ⋘ = (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 33143	⊢ Archi = {𝑤 ∣ (⋘‘𝑤) = ∅}
cslmd 33164	class SLMod
df-slmd 33165	⊢ 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 33215	class ~_RL
crloc 33216	class RLocal
df-erl 33217	⊢ ~_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 33218	⊢ 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 33249	class EuclF
df-euf 33250	⊢ EuclF = Slot ;21
cedom 33253	class EDomn
df-edom 33254	⊢ 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 33263	class Frac
df-frac 33264	⊢ Frac = (𝑟 ∈ V ↦ (𝑟 RLocal (RLReg‘𝑟)))
cfldgen 33271	class fldGen
df-fldgen 33272	⊢ fldGen = (𝑓 ∈ V, 𝑠 ∈ V ↦ ∩ {𝑎 ∈ (SubDRing‘𝑓) ∣ 𝑠 ⊆ 𝑎})
cresv 33286	class ↾_v
df-resv 33287	⊢ ↾_v = (𝑤 ∈ V, 𝑥 ∈ V ↦ if((Base‘(Scalar‘𝑤)) ⊆ 𝑥, 𝑤, (𝑤 sSet ⟨(Scalar‘ndx), ((Scalar‘𝑤) ↾_s 𝑥)⟩)))
cprmidl 33395	class PrmIdeal
df-prmidl 33396	⊢ 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 33419	class MaxIdeal
df-mxidl 33420	⊢ MaxIdeal = (𝑟 ∈ Ring ↦ {𝑖 ∈ (LIdeal‘𝑟) ∣ (𝑖 ≠ (Base‘𝑟) ∧ ∀𝑗 ∈ (LIdeal‘𝑟)(𝑖 ⊆ 𝑗 → (𝑗 = 𝑖 ∨ 𝑗 = (Base‘𝑟))))})
cidlsrg 33460	class IDLsrg
df-idlsrg 33461	⊢ 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 33498	class UFD
df-ufd 33499	⊢ UFD = {𝑟 ∈ IDomn ∣ ∀𝑖 ∈ ((PrmIdeal‘𝑟) ∖ {{(0_g‘𝑟)}})(𝑖 ∩ (RPrime‘𝑟)) ≠ ∅}
csply 33573	class SymPoly
cesply 33574	class eSymPoly
df-sply 33575	⊢ 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 33576	⊢ 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 33606	class dim
df-dim 33607	⊢ dim = (𝑓 ∈ V ↦ ∪ (♯ “ (LBasis‘𝑓)))
cfldext 33646	class /_FldExt
cfinext 33647	class /_FinExt
cextdg 33648	class [:]
df-fldext 33649	⊢ /_FldExt = {⟨𝑒, 𝑓⟩ ∣ ((𝑒 ∈ Field ∧ 𝑓 ∈ Field) ∧ (𝑓 = (𝑒 ↾_s (Base‘𝑓)) ∧ (Base‘𝑓) ∈ (SubRing‘𝑒)))}
df-extdg 33650	⊢ [:] = (𝑒 ∈ V, 𝑓 ∈ (/_FldExt “ {𝑒}) ↦ (dim‘((subringAlg ‘𝑒)‘(Base‘𝑓))))
df-finext 33651	⊢ /_FinExt = {⟨𝑒, 𝑓⟩ ∣ (𝑒/_FldExt𝑓 ∧ (𝑒[:]𝑓) ∈ ℕ₀)}
cirng 33691	class IntgRing
df-irng 33692	⊢ IntgRing = (𝑟 ∈ V, 𝑠 ∈ V ↦ ∪ 𝑓 ∈ (Monic_1p‘(𝑟 ↾_s 𝑠))(^◡((𝑟 evalSub₁ 𝑠)‘𝑓) “ {(0_g‘𝑟)}))
calgext 33703	class /_AlgExt
df-algext 33704	⊢ /_AlgExt = {⟨𝑒, 𝑓⟩ ∣ (𝑒/_FldExt𝑓 ∧ (𝑒 IntgRing (Base‘𝑓)) = (Base‘𝑒))}
cminply 33707	class minPoly
df-minply 33708	⊢ 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 33737	class Constr
df-constr 33738	⊢ 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 33801	class subMat1
df-smat 33802	⊢ 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 33819	class litMat
df-lmat 33820	⊢ litMat = (𝑚 ∈ V ↦ (𝑖 ∈ (1...(♯‘𝑚)), 𝑗 ∈ (1...(♯‘(𝑚‘0))) ↦ ((𝑚‘(𝑖 − 1))‘(𝑗 − 1))))
ccref 33850	class CovHasRef𝐴
df-cref 33851	⊢ CovHasRef𝐴 = {𝑗 ∈ Top ∣ ∀𝑦 ∈ 𝒫 𝑗(∪ 𝑗 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑗 ∩ 𝐴)𝑧Ref𝑦)}
cldlf 33860	class Ldlf
df-ldlf 33861	⊢ Ldlf = CovHasRef{𝑥 ∣ 𝑥 ≼ ω}
cpcmp 33863	class Paracomp
df-pcmp 33864	⊢ Paracomp = {𝑗 ∣ 𝑗 ∈ CovHasRef(LocFin‘𝑗)}
crspec 33870	class Spec
df-rspec 33871	⊢ Spec = (𝑟 ∈ Ring ↦ ((IDLsrg‘𝑟) ↾_s (PrmIdeal‘𝑟)))
cmetid 33894	class ~_Met
cpstm 33895	class pstoMet
df-metid 33896	⊢ ~_Met = (𝑑 ∈ ∪ ran PsMet ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ dom dom 𝑑 ∧ 𝑦 ∈ dom dom 𝑑) ∧ (𝑥𝑑𝑦) = 0)})
df-pstm 33897	⊢ pstoMet = (𝑑 ∈ ∪ ran PsMet ↦ (𝑎 ∈ (dom dom 𝑑 / (~_Met‘𝑑)), 𝑏 ∈ (dom dom 𝑑 / (~_Met‘𝑑)) ↦ ∪ {𝑧 ∣ ∃𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝑧 = (𝑥𝑑𝑦)}))
chcmp 33964	class HCmp
df-hcmp 33965	⊢ HCmp = {⟨𝑢, 𝑤⟩ ∣ ((𝑢 ∈ ∪ ran UnifOn ∧ 𝑤 ∈ CUnifSp) ∧ ((UnifSt‘𝑤) ↾_t dom ∪ 𝑢) = 𝑢 ∧ ((cls‘(TopOpen‘𝑤))‘dom ∪ 𝑢) = (Base‘𝑤))}
cqqh 33978	class ℚHom
df-qqh 33979	⊢ ℚHom = (𝑟 ∈ V ↦ ran (𝑥 ∈ ℤ, 𝑦 ∈ (^◡(ℤRHom‘𝑟) “ (Unit‘𝑟)) ↦ ⟨(𝑥 / 𝑦), (((ℤRHom‘𝑟)‘𝑥)(/_r‘𝑟)((ℤRHom‘𝑟)‘𝑦))⟩))
crrh 34001	class ℝHom
crrext 34002	class ℝExt
df-rrh 34003	⊢ ℝHom = (𝑟 ∈ V ↦ (((topGen‘ran (,))CnExt(TopOpen‘𝑟))‘(ℚHom‘𝑟)))
df-rrext 34007	⊢ ℝExt = {𝑟 ∈ (NrmRing ∩ DivRing) ∣ (((ℤMod‘𝑟) ∈ NrmMod ∧ (chr‘𝑟) = 0) ∧ (𝑟 ∈ CUnifSp ∧ (UnifSt‘𝑟) = (metUnif‘((dist‘𝑟) ↾ ((Base‘𝑟) × (Base‘𝑟))))))}
cxrh 34024	class ℝ^*Hom
df-xrh 34025	⊢ 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 34030	class ManTop
df-mntop 34031	⊢ ManTop = {⟨𝑛, 𝑗⟩ ∣ (𝑛 ∈ ℕ₀ ∧ (𝑗 ∈ 2^ndω ∧ 𝑗 ∈ Haus ∧ 𝑗 ∈ Locally [(TopOpen‘(𝔼_hil‘𝑛))] ≃ ))}
cesum 34035	class Σ^*𝑘 ∈ 𝐴𝐵
df-esum 34036	⊢ Σ^𝑘 ∈ 𝐴𝐵 = ∪ ((ℝ^_𝑠 ↾_s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵))
cofc 34103	class ∘_f/c 𝑅
df-ofc 34104	⊢ ∘_f/c 𝑅 = (𝑓 ∈ V, 𝑐 ∈ V ↦ (𝑥 ∈ dom 𝑓 ↦ ((𝑓‘𝑥)𝑅𝑐)))
csiga 34116	class sigAlgebra
df-siga 34117	⊢ sigAlgebra = (𝑜 ∈ V ↦ {𝑠 ∣ (𝑠 ⊆ 𝒫 𝑜 ∧ (𝑜 ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑜 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑠)))})
csigagen 34146	class sigaGen
df-sigagen 34147	⊢ sigaGen = (𝑥 ∈ V ↦ ∩ {𝑠 ∈ (sigAlgebra‘∪ 𝑥) ∣ 𝑥 ⊆ 𝑠})
cbrsiga 34189	class 𝔅_ℝ
df-brsiga 34190	⊢ 𝔅_ℝ = (sigaGen‘(topGen‘ran (,)))
csx 34196	class ×_s
df-sx 34197	⊢ ×_s = (𝑠 ∈ V, 𝑡 ∈ V ↦ (sigaGen‘ran (𝑥 ∈ 𝑠, 𝑦 ∈ 𝑡 ↦ (𝑥 × 𝑦))))
cmeas 34203	class measures
df-meas 34204	⊢ 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 34240	class δ
df-dde 34241	⊢ δ = (𝑎 ∈ 𝒫 ℝ ↦ if(0 ∈ 𝑎, 1, 0))
cae 34245	class a.e.
cfae 34246	class ~ a.e.
df-ae 34247	⊢ a.e. = {⟨𝑎, 𝑚⟩ ∣ (𝑚‘(∪ dom 𝑚 ∖ 𝑎)) = 0}
df-fae 34253	⊢ ~ 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 34257	class MblFnM
df-mbfm 34258	⊢ MblFnM = (𝑠 ∈ ∪ ran sigAlgebra, 𝑡 ∈ ∪ ran sigAlgebra ↦ {𝑓 ∈ (∪ 𝑡 ↑_m ∪ 𝑠) ∣ ∀𝑥 ∈ 𝑡 (^◡𝑓 “ 𝑥) ∈ 𝑠})
coms 34299	class toOMeas
df-oms 34300	⊢ 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 34309	class toCaraSiga
df-carsg 34310	⊢ toCaraSiga = (𝑚 ∈ V ↦ {𝑎 ∈ 𝒫 ∪ dom 𝑚 ∣ ∀𝑒 ∈ 𝒫 ∪ dom 𝑚((𝑚‘(𝑒 ∩ 𝑎)) +_𝑒 (𝑚‘(𝑒 ∖ 𝑎))) = (𝑚‘𝑒)})
citgm 34335	class itgm
csitm 34336	class sitm
csitg 34337	class sitg
df-sitg 34338	⊢ 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 34339	⊢ sitm = (𝑤 ∈ V, 𝑚 ∈ ∪ ran measures ↦ (𝑓 ∈ dom (𝑤sitg𝑚), 𝑔 ∈ dom (𝑤sitg𝑚) ↦ (((ℝ^*_𝑠 ↾_s (0[,]+∞))sitg𝑚)‘(𝑓 ∘_f (dist‘𝑤)𝑔))))
df-itgm 34361	⊢ itgm = (𝑤 ∈ V, 𝑚 ∈ ∪ ran measures ↦ (((metUnif‘(𝑤sitm𝑚))CnExt(UnifSt‘𝑤))‘(𝑤sitg𝑚)))
csseq 34391	class seq_str
df-sseq 34392	⊢ seq_str = (𝑚 ∈ V, 𝑓 ∈ V ↦ (𝑚 ∪ (lastS ∘ seq(♯‘𝑚)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝑓‘𝑥)”⟩)), (ℕ₀ × {(𝑚 ++ ⟨“(𝑓‘𝑚)”⟩)})))))
cfib 34404	class Fibci
df-fib 34405	⊢ Fibci = (⟨“01”⟩seq_str(𝑤 ∈ (Word ℕ₀ ∩ (^◡♯ “ (ℤ_≥‘2))) ↦ ((𝑤‘((♯‘𝑤) − 2)) + (𝑤‘((♯‘𝑤) − 1)))))
cprb 34415	class Prob
df-prob 34416	⊢ Prob = {𝑝 ∈ ∪ ran measures ∣ (𝑝‘∪ dom 𝑝) = 1}
ccprob 34439	class cprob
df-cndprob 34440	⊢ cprob = (𝑝 ∈ Prob ↦ (𝑎 ∈ dom 𝑝, 𝑏 ∈ dom 𝑝 ↦ ((𝑝‘(𝑎 ∩ 𝑏)) / (𝑝‘𝑏))))
crrv 34448	class rRndVar
df-rrv 34449	⊢ rRndVar = (𝑝 ∈ Prob ↦ (dom 𝑝MblFnM𝔅_ℝ))
corvc 34464	class ∘_RV/𝑐𝑅
df-orvc 34465	⊢ ∘_RV/𝑐𝑅 = (𝑥 ∈ {𝑥 ∣ Fun 𝑥}, 𝑎 ∈ V ↦ (^◡𝑥 “ {𝑦 ∣ 𝑦𝑅𝑎}))
crepr 34616	class repr
df-repr 34617	⊢ repr = (𝑠 ∈ ℕ₀ ↦ (𝑏 ∈ 𝒫 ℕ, 𝑚 ∈ ℤ ↦ {𝑐 ∈ (𝑏 ↑_m (0..^𝑠)) ∣ Σ𝑎 ∈ (0..^𝑠)(𝑐‘𝑎) = 𝑚}))
cvts 34643	class vts
df-vts 34644	⊢ vts = (𝑙 ∈ (ℂ ↑_m ℕ), 𝑛 ∈ ℕ₀ ↦ (𝑥 ∈ ℂ ↦ Σ𝑎 ∈ (1...𝑛)((𝑙‘𝑎) · (exp‘((i · (2 · π)) · (𝑎 · 𝑥))))))
ax-hgt749 34652	⊢ ∀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 34653	⊢ ∀𝑥 ∈ ℝ⁺ (ψ‘𝑥) < ((1._0_3_8_83) · 𝑥)
ax-ros336 34654	⊢ ∀𝑥 ∈ ℝ⁺ ((ψ‘𝑥) − (θ‘𝑥)) < ((1._4_2_62) · (√‘𝑥))
cstrkg2d 34672	class TarskiG_2D
df-trkg2d 34673	⊢ 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 34677	class AFS
df-afs 34678	⊢ 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 34682	class leftpad
df-lpad 34683	⊢ leftpad = (𝑐 ∈ V, 𝑤 ∈ V ↦ (𝑙 ∈ ℕ₀ ↦ (((0..^(𝑙 − (♯‘𝑤))) × {𝑐}) ++ 𝑤)))
w-bnj17 34693	wff (𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃)
df-bnj17 34694	⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ ((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃))
c-bnj14 34695	class pred(𝑋, 𝐴, 𝑅)
df-bnj14 34696	⊢ pred(𝑋, 𝐴, 𝑅) = {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋}
w-bnj13 34697	wff 𝑅 Se 𝐴
df-bnj13 34698	⊢ (𝑅 Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 pred(𝑥, 𝐴, 𝑅) ∈ V)
w-bnj15 34699	wff 𝑅 FrSe 𝐴
df-bnj15 34700	⊢ (𝑅 FrSe 𝐴 ↔ (𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴))
c-bnj18 34701	class trCl(𝑋, 𝐴, 𝑅)
df-bnj18 34702	⊢ 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 34703	wff TrFo(𝐵, 𝐴, 𝑅)
df-bnj19 34704	⊢ ( TrFo(𝐵, 𝐴, 𝑅) ↔ ∀𝑥 ∈ 𝐵 pred(𝑥, 𝐴, 𝑅) ⊆ 𝐵)
ax-regs 35112	⊢ (∃𝑥𝜑 → ∃𝑦(∀𝑥(𝑥 = 𝑦 → 𝜑) ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ ∀𝑥(𝑥 = 𝑧 → 𝜑))))
cacycgr 35174	class AcyclicGraph
df-acycgr 35175	⊢ AcyclicGraph = {𝑔 ∣ ¬ ∃𝑓∃𝑝(𝑓(Cycles‘𝑔)𝑝 ∧ 𝑓 ≠ ∅)}
ax-7d 35191	⊢ (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑)
ax-8d 35192	⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 → 𝑦 = 𝑧))
ax-9d1 35193	⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑥
ax-9d2 35194	⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦
ax-10d 35195	⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
ax-11d 35196	⊢ (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
cretr 35249	class Retr
df-retr 35250	⊢ Retr = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ {𝑟 ∈ (𝑗 Cn 𝑘) ∣ ∃𝑠 ∈ (𝑘 Cn 𝑗)((𝑟 ∘ 𝑠)(𝑗 Htpy 𝑗)( I ↾ ∪ 𝑗)) ≠ ∅})
cpconn 35251	class PConn
csconn 35252	class SConn
df-pconn 35253	⊢ PConn = {𝑗 ∈ Top ∣ ∀𝑥 ∈ ∪ 𝑗∀𝑦 ∈ ∪ 𝑗∃𝑓 ∈ (II Cn 𝑗)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)}
df-sconn 35254	⊢ SConn = {𝑗 ∈ PConn ∣ ∀𝑓 ∈ (II Cn 𝑗)((𝑓‘0) = (𝑓‘1) → 𝑓( ≃_ph‘𝑗)((0[,]1) × {(𝑓‘0)}))}
ccvm 35287	class CovMap
df-cvm 35288	⊢ 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 35365	class ∈_𝑔
cgna 35366	class ⊼_𝑔
cgol 35367	class ∀_𝑔𝑁𝑈
csat 35368	class Sat
cfmla 35369	class Fmla
csate 35370	class Sat_∈
cprv 35371	class ⊧
df-goel 35372	⊢ ∈_𝑔 = (𝑥 ∈ (ω × ω) ↦ ⟨∅, 𝑥⟩)
df-gona 35373	⊢ ⊼_𝑔 = (𝑥 ∈ (V × V) ↦ ⟨1_o, 𝑥⟩)
df-goal 35374	⊢ ∀_𝑔𝑁𝑈 = ⟨2_o, ⟨𝑁, 𝑈⟩⟩
df-sat 35375	⊢ 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 35376	⊢ Sat_∈ = (𝑚 ∈ V, 𝑢 ∈ V ↦ (((𝑚 Sat ( E ∩ (𝑚 × 𝑚)))‘ω)‘𝑢))
df-fmla 35377	⊢ Fmla = (𝑛 ∈ suc ω ↦ dom ((∅ Sat ∅)‘𝑛))
df-prv 35378	⊢ ⊧ = {⟨𝑚, 𝑢⟩ ∣ (𝑚 Sat_∈ 𝑢) = (𝑚 ↑_m ω)}
cgon 35464	class ¬_𝑔𝑈
cgoa 35465	class ∧_𝑔
cgoi 35466	class →_𝑔
cgoo 35467	class ∨_𝑔
cgob 35468	class ↔_𝑔
cgoq 35469	class =_𝑔
cgox 35470	class ∃_𝑔𝑁𝑈
df-gonot 35471	⊢ ¬_𝑔𝑈 = (𝑈⊼_𝑔𝑈)
df-goan 35472	⊢ ∧_𝑔 = (𝑢 ∈ V, 𝑣 ∈ V ↦ ¬_𝑔(𝑢⊼_𝑔𝑣))
df-goim 35473	⊢ →_𝑔 = (𝑢 ∈ V, 𝑣 ∈ V ↦ (𝑢⊼_𝑔¬_𝑔𝑣))
df-goor 35474	⊢ ∨_𝑔 = (𝑢 ∈ V, 𝑣 ∈ V ↦ (¬_𝑔𝑢 →_𝑔 𝑣))
df-gobi 35475	⊢ ↔_𝑔 = (𝑢 ∈ V, 𝑣 ∈ V ↦ ((𝑢 →_𝑔 𝑣)∧_𝑔(𝑣 →_𝑔 𝑢)))
df-goeq 35476	⊢ =_𝑔 = (𝑢 ∈ ω, 𝑣 ∈ ω ↦ ⦋suc (𝑢 ∪ 𝑣) / 𝑤⦌∀_𝑔𝑤((𝑤∈_𝑔𝑢) ↔_𝑔 (𝑤∈_𝑔𝑣)))
df-goex 35477	⊢ ∃_𝑔𝑁𝑈 = ¬_𝑔∀_𝑔𝑁¬_𝑔𝑈
cgze 35478	class AxExt
cgzr 35479	class AxRep
cgzp 35480	class AxPow
cgzu 35481	class AxUn
cgzg 35482	class AxReg
cgzi 35483	class AxInf
cgzf 35484	class ZF
df-gzext 35485	⊢ AxExt = (∀_𝑔2_o((2_o∈_𝑔∅) ↔_𝑔 (2_o∈_𝑔1_o)) →_𝑔 (∅=_𝑔1_o))
df-gzrep 35486	⊢ 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 35487	⊢ AxPow = ∃_𝑔1_o∀_𝑔2_o(∀_𝑔1_o((1_o∈_𝑔2_o) ↔_𝑔 (1_o∈_𝑔∅)) →_𝑔 (2_o∈_𝑔1_o))
df-gzun 35488	⊢ AxUn = ∃_𝑔1_o∀_𝑔2_o(∃_𝑔1_o((2_o∈_𝑔1_o)∧_𝑔(1_o∈_𝑔∅)) →_𝑔 (2_o∈_𝑔1_o))
df-gzreg 35489	⊢ 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 35490	⊢ AxInf = ∃_𝑔1_o((∅∈_𝑔1_o)∧_𝑔∀_𝑔2_o((2_o∈_𝑔1_o) →_𝑔 ∃_𝑔∅((2_o∈_𝑔∅)∧_𝑔(∅∈_𝑔1_o))))
df-gzf 35491	⊢ ZF = {𝑚 ∣ ((Tr 𝑚 ∧ 𝑚⊧AxExt ∧ 𝑚⊧AxPow) ∧ (𝑚⊧AxUn ∧ 𝑚⊧AxReg ∧ 𝑚⊧AxInf) ∧ ∀𝑢 ∈ (Fmla‘ω)𝑚⊧(AxRep‘𝑢))}
cmcn 35492	class mCN
cmvar 35493	class mVR
cmty 35494	class mType
cmvt 35495	class mVT
cmtc 35496	class mTC
cmax 35497	class mAx
cmrex 35498	class mREx
cmex 35499	class mEx
cmdv 35500	class mDV
cmvrs 35501	class mVars
cmrsub 35502	class mRSubst
cmsub 35503	class mSubst
cmvh 35504	class mVH
cmpst 35505	class mPreSt
cmsr 35506	class mStRed
cmsta 35507	class mStat
cmfs 35508	class mFS
cmcls 35509	class mCls
cmpps 35510	class mPPSt
cmthm 35511	class mThm
df-mcn 35512	⊢ mCN = Slot 1
df-mvar 35513	⊢ mVR = Slot 2
df-mty 35514	⊢ mType = Slot 3
df-mtc 35515	⊢ mTC = Slot 4
df-mmax 35516	⊢ mAx = Slot 5
df-mvt 35517	⊢ mVT = (𝑡 ∈ V ↦ ran (mType‘𝑡))
df-mrex 35518	⊢ mREx = (𝑡 ∈ V ↦ Word ((mCN‘𝑡) ∪ (mVR‘𝑡)))
df-mex 35519	⊢ mEx = (𝑡 ∈ V ↦ ((mTC‘𝑡) × (mREx‘𝑡)))
df-mdv 35520	⊢ mDV = (𝑡 ∈ V ↦ (((mVR‘𝑡) × (mVR‘𝑡)) ∖ I ))
df-mvrs 35521	⊢ mVars = (𝑡 ∈ V ↦ (𝑒 ∈ (mEx‘𝑡) ↦ (ran (2^nd ‘𝑒) ∩ (mVR‘𝑡))))
df-mrsub 35522	⊢ 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 35523	⊢ mSubst = (𝑡 ∈ V ↦ (𝑓 ∈ ((mREx‘𝑡) ↑_pm (mVR‘𝑡)) ↦ (𝑒 ∈ (mEx‘𝑡) ↦ ⟨(1^st ‘𝑒), (((mRSubst‘𝑡)‘𝑓)‘(2^nd ‘𝑒))⟩)))
df-mvh 35524	⊢ mVH = (𝑡 ∈ V ↦ (𝑣 ∈ (mVR‘𝑡) ↦ ⟨((mType‘𝑡)‘𝑣), ⟨“𝑣”⟩⟩))
df-mpst 35525	⊢ mPreSt = (𝑡 ∈ V ↦ (({𝑑 ∈ 𝒫 (mDV‘𝑡) ∣ ^◡𝑑 = 𝑑} × (𝒫 (mEx‘𝑡) ∩ Fin)) × (mEx‘𝑡)))
df-msr 35526	⊢ 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 35527	⊢ mStat = (𝑡 ∈ V ↦ ran (mStRed‘𝑡))
df-mfs 35528	⊢ 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 35529	⊢ 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 35530	⊢ mPPSt = (𝑡 ∈ V ↦ {⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∣ (⟨𝑑, ℎ, 𝑎⟩ ∈ (mPreSt‘𝑡) ∧ 𝑎 ∈ (𝑑(mCls‘𝑡)ℎ))})
df-mthm 35531	⊢ mThm = (𝑡 ∈ V ↦ (^◡(mStRed‘𝑡) “ ((mStRed‘𝑡) “ (mPPSt‘𝑡))))
cm0s 35617	class m0St
cmsa 35618	class mSA
cmwgfs 35619	class mWGFS
cmsy 35620	class mSyn
cmesy 35621	class mESyn
cmgfs 35622	class mGFS
cmtree 35623	class mTree
cmst 35624	class mST
cmsax 35625	class mSAX
cmufs 35626	class mUFS
df-m0s 35627	⊢ m0St = (𝑎 ∈ V ↦ ⟨∅, ∅, 𝑎⟩)
df-msa 35628	⊢ mSA = (𝑡 ∈ V ↦ {𝑎 ∈ (mEx‘𝑡) ∣ ((m0St‘𝑎) ∈ (mAx‘𝑡) ∧ (1^st ‘𝑎) ∈ (mVT‘𝑡) ∧ Fun (^◡(2^nd ‘𝑎) ↾ (mVR‘𝑡)))})
df-mwgfs 35629	⊢ 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 35630	⊢ mSyn = Slot 6
df-mesyn 35631	⊢ mESyn = (𝑡 ∈ V ↦ (𝑐 ∈ (mTC‘𝑡), 𝑒 ∈ (mREx‘𝑡) ↦ (((mSyn‘𝑡)‘𝑐)m0St𝑒)))
df-mgfs 35632	⊢ 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 35633	⊢ 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 35634	⊢ mST = (𝑡 ∈ V ↦ ((∅(mTree‘𝑡)∅) ↾ ((mEx‘𝑡) ↾ (mVT‘𝑡))))
df-msax 35635	⊢ mSAX = (𝑡 ∈ V ↦ (𝑝 ∈ (mSA‘𝑡) ↦ ((mVH‘𝑡) “ ((mVars‘𝑡)‘𝑝))))
df-mufs 35636	⊢ mUFS = {𝑡 ∈ mGFS ∣ Fun (mST‘𝑡)}
cmuv 35637	class mUV
cmvl 35638	class mVL
cmvsb 35639	class mVSubst
cmfsh 35640	class mFresh
cmfr 35641	class mFRel
cmevl 35642	class mEval
cmdl 35643	class mMdl
cusyn 35644	class mUSyn
cgmdl 35645	class mGMdl
cmitp 35646	class mItp
cmfitp 35647	class mFromItp
df-muv 35648	⊢ mUV = Slot 7
df-mfsh 35649	⊢ mFresh = Slot ;19
df-mevl 35650	⊢ mEval = Slot ;20
df-mvl 35651	⊢ mVL = (𝑡 ∈ V ↦ X𝑣 ∈ (mVR‘𝑡)((mUV‘𝑡) “ {((mType‘𝑡)‘𝑣)}))
df-mvsb 35652	⊢ 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 35653	⊢ 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 35654	⊢ 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 35655	⊢ mUSyn = (𝑡 ∈ V ↦ (𝑣 ∈ (mUV‘𝑡) ↦ ⟨((mSyn‘𝑡)‘(1^st ‘𝑣)), (2^nd ‘𝑣)⟩))
df-gmdl 35656	⊢ 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 35657	⊢ 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 35658	⊢ 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 35659	class cplMetSp
chlb 35660	class HomLimB
chlim 35661	class HomLim
cpfl 35662	class polyFld
csf1 35663	class splitFld₁
csf 35664	class splitFld
cpsl 35665	class polySplitLim
df-cplmet 35666	⊢ 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 35667	⊢ 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 35668	⊢ 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 35669	⊢ 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 35676	⊢ 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 35677	⊢ 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 35678	⊢ 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 35679	class ZRing
cgf 35680	class GF
cgfo 35681	class GF_∞
ceqp 35682	class ~Qp
crqp 35683	class /Qp
cqp 35684	class Qp
czp 35685	class Zp
cqpa 35686	class _Qp
ccp 35687	class Cp
df-zrng 35688	⊢ ZRing = (𝑟 ∈ V ↦ (𝑟 IntgRing ran (ℤRHom‘𝑟)))
df-gf 35689	⊢ 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 35690	⊢ 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 35691	⊢ ~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 35692	⊢ /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 35693	⊢ 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 35694	⊢ Zp = (ZRing ∘ Qp)
df-qpa 35695	⊢ _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 35696	⊢ Cp = ( cplMetSp ∘ _Qp)
ccloneop 35727	class CloneOp
df-cloneop 35728	⊢ CloneOp = (𝑎 ∈ V ↦ {𝑥 ∣ ∃𝑛 ∈ (ω ∖ 1_o)𝑥 ∈ (𝑎 ↑_m (𝑎 ↑_m 𝑛))})
cprj 35729	class prj
df-prj 35730	⊢ prj = (𝑎 ∈ V ↦ (𝑛 ∈ (ω ∖ 1_o), 𝑖 ∈ 𝑛 ↦ (𝑥 ∈ (𝑎 ↑_m 𝑛) ↦ (𝑥‘𝑖))))
csuppos 35731	class suppos
df-suppos 35732	⊢ 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 35843	class wsuc(𝑅, 𝐴, 𝑋)
cwlim 35844	class WLim(𝑅, 𝐴)
df-wsuc 35845	⊢ wsuc(𝑅, 𝐴, 𝑋) = inf(Pred(^◡𝑅, 𝐴, 𝑋), 𝐴, 𝑅)
df-wlim 35846	⊢ WLim(𝑅, 𝐴) = {𝑥 ∈ 𝐴 ∣ (𝑥 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑥 = sup(Pred(𝑅, 𝐴, 𝑥), 𝐴, 𝑅))}
ctxp 35863	class (𝐴 ⊗ 𝐵)
cpprod 35864	class pprod(𝑅, 𝑆)
csset 35865	class SSet
ctrans 35866	class Trans
cbigcup 35867	class Bigcup
cfix 35868	class Fix 𝐴
climits 35869	class Limits
cfuns 35870	class Funs
csingle 35871	class Singleton
csingles 35872	class Singletons
cimage 35873	class Image𝐴
ccart 35874	class Cart
cimg 35875	class Img
cdomain 35876	class Domain
crange 35877	class Range
capply 35878	class Apply
ccup 35879	class Cup
ccap 35880	class Cap
csuccf 35881	class Succ
cfunpart 35882	class Funpart𝐹
cfullfn 35883	class FullFun𝐹
crestrict 35884	class Restrict
cub 35885	class UB𝑅
clb 35886	class LB𝑅
df-txp 35887	⊢ (𝐴 ⊗ 𝐵) = ((^◡(1^st ↾ (V × V)) ∘ 𝐴) ∩ (^◡(2^nd ↾ (V × V)) ∘ 𝐵))
df-pprod 35888	⊢ pprod(𝐴, 𝐵) = ((𝐴 ∘ (1^st ↾ (V × V))) ⊗ (𝐵 ∘ (2^nd ↾ (V × V))))
df-sset 35889	⊢ SSet = ((V × V) ∖ ran ( E ⊗ (V ∖ E )))
df-trans 35890	⊢ Trans = (V ∖ ran (( E ∘ E ) ∖ E ))
df-bigcup 35891	⊢ Bigcup = ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ E ) ⊗ V)))
df-fix 35892	⊢ Fix 𝐴 = dom (𝐴 ∩ I )
df-limits 35893	⊢ Limits = ((On ∩ Fix Bigcup ) ∖ {∅})
df-funs 35894	⊢ Funs = (𝒫 (V × V) ∖ Fix ( E ∘ ((1^st ⊗ ((V ∖ I ) ∘ 2^nd )) ∘ ^◡ E )))
df-singleton 35895	⊢ Singleton = ((V × V) ∖ ran ((V ⊗ E ) △ ( I ⊗ V)))
df-singles 35896	⊢ Singletons = ran Singleton
df-image 35897	⊢ Image𝐴 = ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ ^◡𝐴) ⊗ V)))
df-cart 35898	⊢ Cart = (((V × V) × V) ∖ ran ((V ⊗ E ) △ (pprod( E , E ) ⊗ V)))
df-img 35899	⊢ Img = (Image((2^nd ∘ 1^st ) ↾ (1^st ↾ (V × V))) ∘ Cart)
df-domain 35900	⊢ Domain = Image(1^st ↾ (V × V))
df-range 35901	⊢ Range = Image(2^nd ↾ (V × V))
df-cup 35902	⊢ Cup = (((V × V) × V) ∖ ran ((V ⊗ E ) △ (((^◡1^st ∘ E ) ∪ (^◡2^nd ∘ E )) ⊗ V)))
df-cap 35903	⊢ Cap = (((V × V) × V) ∖ ran ((V ⊗ E ) △ (((^◡1^st ∘ E ) ∩ (^◡2^nd ∘ E )) ⊗ V)))
df-restrict 35904	⊢ Restrict = (Cap ∘ (1^st ⊗ (Cart ∘ (2^nd ⊗ (Range ∘ 1^st )))))
df-succf 35905	⊢ Succ = (Cup ∘ ( I ⊗ Singleton))
df-apply 35906	⊢ Apply = (( Bigcup ∘ Bigcup ) ∘ (((V × V) ∖ ran ((V ⊗ E ) △ (( E ↾ Singletons ) ⊗ V))) ∘ ((Singleton ∘ Img) ∘ pprod( I , Singleton))))
df-funpart 35907	⊢ Funpart𝐹 = (𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))
df-fullfun 35908	⊢ FullFun𝐹 = (Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))
df-ub 35909	⊢ UB𝑅 = ((V × V) ∖ ((V ∖ 𝑅) ∘ ^◡ E ))
df-lb 35910	⊢ LB𝑅 = UB^◡𝑅
caltop 35989	class ⟪𝐴, 𝐵⟫
caltxp 35990	class (𝐴 ×× 𝐵)
df-altop 35991	⊢ ⟪𝐴, 𝐵⟫ = {{𝐴}, {𝐴, {𝐵}}}
df-altxp 35992	⊢ (𝐴 ×× 𝐵) = {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = ⟪𝑥, 𝑦⟫}
cofs 36015	class OuterFiveSeg
df-ofs 36016	⊢ 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 36062	class TransportTo
df-transport 36063	⊢ 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 36068	class InnerFiveSeg
ccgr3 36069	class Cgr3
ccolin 36070	class Colinear
cfs 36071	class FiveSeg
df-colinear 36072	⊢ 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 36073	⊢ 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 36074	⊢ 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 36075	⊢ 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 36139	class Seg_≤
df-segle 36140	⊢ 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 36152	class OutsideOf
df-outsideof 36153	⊢ OutsideOf = ( Colinear ∖ Btwn )
cline2 36167	class Line
cray 36168	class Ray
clines2 36169	class LinesEE
df-line2 36170	⊢ Line = {⟨⟨𝑎, 𝑏⟩, 𝑙⟩ ∣ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎 ≠ 𝑏) ∧ 𝑙 = [⟨𝑎, 𝑏⟩]^◡ Colinear )}
df-ray 36171	⊢ Ray = {⟨⟨𝑝, 𝑎⟩, 𝑟⟩ ∣ ∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑎) ∧ 𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩})}
df-lines2 36172	⊢ LinesEE = ran Line
cfwddif 36191	class △
df-fwddif 36192	⊢ △ = (𝑓 ∈ (ℂ ↑_pm ℂ) ↦ (𝑥 ∈ {𝑦 ∈ dom 𝑓 ∣ (𝑦 + 1) ∈ dom 𝑓} ↦ ((𝑓‘(𝑥 + 1)) − (𝑓‘𝑥))))
cfwddifn 36193	class △ⁿ
df-fwddifn 36194	⊢ △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 36205	class Hf
df-hf 36206	⊢ Hf = ∪ (𝑅₁ “ ω)
cfne 36369	class Fne
df-fne 36370	⊢ Fne = {⟨𝑥, 𝑦⟩ ∣ (∪ 𝑥 = ∪ 𝑦 ∧ ∀𝑧 ∈ 𝑥 𝑧 ⊆ ∪ (𝑦 ∩ 𝒫 𝑧))}
w3nand 36430	wff (𝜑 ⊼ 𝜓 ⊼ 𝜒)
df-3nand 36431	⊢ ((𝜑 ⊼ 𝜓 ⊼ 𝜒) ↔ (𝜑 → (𝜓 → ¬ 𝜒)))
cgcdOLD 36492	class gcd_OLD (𝐴, 𝐵)
df-gcdOLD 36493	⊢ gcd_OLD (𝐴, 𝐵) = sup({𝑥 ∈ ℕ ∣ ((𝐴 / 𝑥) ∈ ℕ ∧ (𝐵 / 𝑥) ∈ ℕ)}, ℕ, < )
cprvb 36630	wff Prv 𝜑
ax-prv1 36631	⊢ 𝜑 ⇒ ⊢ Prv 𝜑
ax-prv2 36632	⊢ (Prv (𝜑 → 𝜓) → (Prv 𝜑 → Prv 𝜓))
ax-prv3 36633	⊢ (Prv 𝜑 → Prv Prv 𝜑)
wmoo 36684	wff ∃**𝑥𝜑
df-bj-mo 36685	⊢ (∃**𝑥𝜑 ↔ ∀𝑧∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))
wnnf 36756	wff Ⅎ'𝑥𝜑
df-bj-nnf 36757	⊢ (Ⅎ'𝑥𝜑 ↔ ((∃𝑥𝜑 → 𝜑) ∧ (𝜑 → ∀𝑥𝜑)))
bj-cgab 36966	class {𝐴 ∣ 𝑥 ∣ 𝜑}
df-bj-gab 36967	⊢ {𝐴 ∣ 𝑥 ∣ 𝜑} = {𝑦 ∣ ∃𝑥(𝐴 = 𝑦 ∧ 𝜑)}
wrnf 36974	wff Ⅎ𝑥 ∈ 𝐴𝜑
df-bj-rnf 36975	⊢ (Ⅎ𝑥 ∈ 𝐴𝜑 ↔ (∃𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 𝜑))
bj-csngl 36998	class sngl 𝐴
df-bj-sngl 36999	⊢ sngl 𝐴 = {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = {𝑦}}
bj-ctag 37007	class tag 𝐴
df-bj-tag 37008	⊢ tag 𝐴 = (sngl 𝐴 ∪ {∅})
bj-cproj 37023	class (𝐴 Proj 𝐵)
df-bj-proj 37024	⊢ (𝐴 Proj 𝐵) = {𝑥 ∣ {𝑥} ∈ (𝐵 “ {𝐴})}
bj-c1upl 37030	class ⦅𝐴⦆
df-bj-1upl 37031	⊢ ⦅𝐴⦆ = ({∅} × tag 𝐴)
bj-cpr1 37033	class pr₁ 𝐴
df-bj-pr1 37034	⊢ pr₁ 𝐴 = (∅ Proj 𝐴)
bj-c2uple 37043	class ⦅𝐴, 𝐵⦆
df-bj-2upl 37044	⊢ ⦅𝐴, 𝐵⦆ = (⦅𝐴⦆ ∪ ({1_o} × tag 𝐵))
bj-cpr2 37047	class pr₂ 𝐴
df-bj-pr2 37048	⊢ pr₂ 𝐴 = (1_o Proj 𝐴)
ax-bj-sn 37066	⊢ ∀𝑥∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ 𝑧 = 𝑥)
ax-bj-bun 37070	⊢ ∀𝑥∀𝑦∃𝑧∀𝑡(𝑡 ∈ 𝑧 ↔ (𝑡 ∈ 𝑥 ∨ 𝑡 ∈ 𝑦))
ax-bj-adj 37075	⊢ ∀𝑥∀𝑦∃𝑧∀𝑡(𝑡 ∈ 𝑧 ↔ (𝑡 ∈ 𝑥 ∨ 𝑡 = 𝑦))
celwise 37112	class elwise
df-elwise 37113	⊢ elwise = (𝑜 ∈ V ↦ (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑧 ∣ ∃𝑢 ∈ 𝑥 ∃𝑣 ∈ 𝑦 𝑧 = (𝑢𝑜𝑣)}))
cmoore 37136	class Moore
df-bj-moore 37137	⊢ Moore = {𝑥 ∣ ∀𝑦 ∈ 𝒫 𝑥(∪ 𝑥 ∩ ∩ 𝑦) ∈ 𝑥}
cmpt3 37153	class (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵, 𝑧 ∈ 𝐶 ↦ 𝐷)
df-bj-mpt3 37154	⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵, 𝑧 ∈ 𝐶 ↦ 𝐷) = {⟨𝑠, 𝑡⟩ ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 (𝑠 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝑡 = 𝐷)}
csethom 37155	class Set⟶
df-bj-sethom 37156	⊢ Set⟶ = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓 ∣ 𝑓:𝑥⟶𝑦})
ctophom 37157	class Top⟶
df-bj-tophom 37158	⊢ Top⟶ = (𝑥 ∈ TopSp, 𝑦 ∈ TopSp ↦ {𝑓 ∈ ((Base‘𝑥) Set⟶ (Base‘𝑦)) ∣ ∀𝑢 ∈ (TopOpen‘𝑦)(^◡𝑓 “ 𝑢) ∈ (TopOpen‘𝑥)})
cmgmhom 37159	class Mgm⟶
df-bj-mgmhom 37160	⊢ 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 37161	class TopMgm⟶
df-bj-topmgmhom 37162	⊢ TopMgm⟶ = (𝑥 ∈ TopMnd, 𝑦 ∈ TopMnd ↦ ((𝑥 Top⟶ 𝑦) ∩ (𝑥 Mgm⟶ 𝑦)))
ccur- 37163	class curry_
df-bj-cur 37164	⊢ curry_ = (𝑥 ∈ V, 𝑦 ∈ V, 𝑧 ∈ V ↦ (𝑓 ∈ ((𝑥 × 𝑦) Set⟶ 𝑧) ↦ (𝑎 ∈ 𝑥 ↦ (𝑏 ∈ 𝑦 ↦ (𝑓‘⟨𝑎, 𝑏⟩)))))
cunc- 37165	class uncurry_
df-bj-unc 37166	⊢ uncurry_ = (𝑥 ∈ V, 𝑦 ∈ V, 𝑧 ∈ V ↦ (𝑓 ∈ (𝑥 Set⟶ (𝑦 Set⟶ 𝑧)) ↦ (𝑎 ∈ 𝑥, 𝑏 ∈ 𝑦 ↦ ((𝑓‘𝑎)‘𝑏))))
cstrset 37167	class [𝐵 / 𝐴]_struct𝑆
df-strset 37168	⊢ [𝐵 / 𝐴]_struct𝑆 = ((𝑆 ↾ (V ∖ {(𝐴‘ndx)})) ∪ {⟨(𝐴‘ndx), 𝐵⟩})
cdiag2 37205	class Id
df-bj-diag 37206	⊢ Id = (𝑥 ∈ V ↦ ( I ↾ 𝑥))
cimdir 37211	class 𝒫_*
df-imdir 37212	⊢ 𝒫_* = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑟 ∈ 𝒫 (𝑎 × 𝑏) ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝑎 ∧ 𝑦 ⊆ 𝑏) ∧ (𝑟 “ 𝑥) = 𝑦)}))
ciminv 37224	class 𝒫^*
df-iminv 37225	⊢ 𝒫^* = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑟 ∈ 𝒫 (𝑎 × 𝑏) ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝑎 ∧ 𝑦 ⊆ 𝑏) ∧ 𝑥 = (^◡𝑟 “ 𝑦))}))
cfractemp 37229	class {_R
df-bj-fractemp 37230	⊢ {_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 37231	class +∞e^iτ
df-bj-inftyexpitau 37232	⊢ +∞e^iτ = (𝑥 ∈ ℝ ↦ ⟨({_R‘(1^st ‘𝑥)), {R}⟩)
cccinftyN 37233	class ℂ_∞N
df-bj-ccinftyN 37234	⊢ ℂ_∞N = ran +∞e^iτ
chalf 37236	class 1/2
df-bj-onehalf 37237	⊢ 1/2 = (℩𝑥 ∈ R (𝑥 +_R 𝑥) = 1_R)
cinftyexpi 37239	class +∞eⁱ
df-bj-inftyexpi 37240	⊢ +∞eⁱ = (𝑥 ∈ (-π(,]π) ↦ ⟨𝑥, ℂ⟩)
cccinfty 37244	class ℂ_∞
df-bj-ccinfty 37245	⊢ ℂ_∞ = ran +∞eⁱ
cccbar 37248	class ℂ̅
df-bj-ccbar 37249	⊢ ℂ̅ = (ℂ ∪ ℂ_∞)
cpinfty 37252	class +∞
df-bj-pinfty 37253	⊢ +∞ = (+∞eⁱ‘0)
cminfty 37256	class -∞
df-bj-minfty 37257	⊢ -∞ = (+∞eⁱ‘π)
crrbar 37261	class ℝ̅
df-bj-rrbar 37262	⊢ ℝ̅ = (ℝ ∪ {-∞, +∞})
cinfty 37263	class ∞
df-bj-infty 37264	⊢ ∞ = 𝒫 ∪ ℂ
ccchat 37265	class ℂ̂
df-bj-cchat 37266	⊢ ℂ̂ = (ℂ ∪ {∞})
crrhat 37267	class ℝ̂
df-bj-rrhat 37268	⊢ ℝ̂ = (ℝ ∪ {∞})
caddcc 37270	class +_ℂ̅
df-bj-addc 37271	⊢ +_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 37272	class -_ℂ̅
df-bj-oppc 37273	⊢ -_ℂ̅ = (𝑥 ∈ (ℂ̅ ∪ ℂ̂) ↦ if(𝑥 = ∞, ∞, if(𝑥 ∈ ℂ, (℩𝑦 ∈ ℂ (𝑥 +_ℂ̅ 𝑦) = 0), (+∞e^iτ‘(𝑥 +_ℂ̅ ⟨1/2, 0_R⟩)))))
cltxr 37274	class <_ℝ̅
df-bj-lt 37275	⊢ <_ℝ̅ = ({𝑥 ∈ (ℝ̅ × ℝ̅) ∣ ∃𝑦∃𝑧(((1^st ‘𝑥) = ⟨𝑦, 0_R⟩ ∧ (2^nd ‘𝑥) = ⟨𝑧, 0_R⟩) ∧ 𝑦 <_R 𝑧)} ∪ ((({-∞} × ℝ) ∪ (ℝ × {+∞})) ∪ ({-∞} × {+∞})))
carg 37276	class Arg
df-bj-arg 37277	⊢ Arg = (𝑥 ∈ (ℂ̅ ∖ {0}) ↦ if(𝑥 ∈ ℂ, (ℑ‘(log‘𝑥)), if(𝑥<_ℝ̅0, π, (((1^st ‘𝑥) / (2 · π)) − π))))
cmulc 37278	class ·_ℂ̅
df-bj-mulc 37279	⊢ ·_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 37280	class ^-1_ℂ̅
df-bj-invc 37281	⊢ ^-1_ℂ̅ = (𝑥 ∈ (ℂ̅ ∪ ℂ̂) ↦ if(𝑥 = 0, ∞, if(𝑥 ∈ ℂ, (℩𝑦 ∈ ℂ (𝑥 ·_ℂ̅ 𝑦) = 1), 0)))
ciomnn 37282	class i_ω↪ℕ
df-bj-iomnn 37283	⊢ i_ω↪ℕ = ((𝑛 ∈ ω ↦ ⟨[⟨{𝑟 ∈ Q ∣ 𝑟 <_Q ⟨suc 𝑛, 1_o⟩}, 1_P⟩] ~_R , 0_R⟩) ∪ {⟨ω, +∞⟩})
cnnbar 37293	class ℕ̅
df-bj-nnbar 37294	⊢ ℕ̅ = (ℕ₀ ∪ {+∞})
czzbar 37295	class ℤ̅
df-bj-zzbar 37296	⊢ ℤ̅ = (ℤ ∪ {-∞, +∞})
czzhat 37297	class ℤ̂
df-bj-zzhat 37298	⊢ ℤ̂ = (ℤ ∪ {∞})
cdivc 37299	class ∥_ℂ
df-bj-divc 37300	⊢ ∥_ℂ = {⟨𝑥, 𝑦⟩ ∣ (⟨𝑥, 𝑦⟩ ∈ ((ℂ̅ × ℂ̅) ∪ (ℂ̂ × ℂ̂)) ∧ ∃𝑛 ∈ (ℤ̅ ∪ ℤ̂)(𝑛 ·_ℂ̅ 𝑥) = 𝑦)}
cfinsum 37316	class FinSum
df-bj-finsum 37317	⊢ 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 37325	class ℝ-Vec
df-bj-rvec 37326	⊢ ℝ-Vec = (LMod ∩ (^◡Scalar “ {ℝ_fld}))
cend 37346	class End
df-bj-end 37347	⊢ End = (𝑐 ∈ Cat ↦ (𝑥 ∈ (Base‘𝑐) ↦ {⟨(Base‘ndx), (𝑥(Hom ‘𝑐)𝑥)⟩, ⟨(+_g‘ndx), (⟨𝑥, 𝑥⟩(comp‘𝑐)𝑥)⟩}))
cfinxp 37416	class (𝑈↑↑𝑁)
df-finxp 37417	⊢ (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 37452	⊢ ((𝜑 → 𝜓) → ((𝜓 → 𝜒) → (𝜑 → 𝜒)))
ax-luk2 37453	⊢ ((¬ 𝜑 → 𝜑) → 𝜑)
ax-luk3 37454	⊢ (𝜑 → (¬ 𝜑 → 𝜓))
ax-wl-13v 37526	⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
ax-wl-cleq 37535	⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
ax-wl-clel 37536	⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵))
ax-wl-11v 37617	⊢ (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑)
ctotbnd 37805	class TotBnd
cbnd 37806	class Bnd
df-totbnd 37807	⊢ TotBnd = (𝑥 ∈ V ↦ {𝑚 ∈ (Met‘𝑥) ∣ ∀𝑑 ∈ ℝ⁺ ∃𝑣 ∈ Fin (∪ 𝑣 = 𝑥 ∧ ∀𝑏 ∈ 𝑣 ∃𝑦 ∈ 𝑥 𝑏 = (𝑦(ball‘𝑚)𝑑))})
df-bnd 37818	⊢ Bnd = (𝑥 ∈ V ↦ {𝑚 ∈ (Met‘𝑥) ∣ ∀𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ⁺ 𝑥 = (𝑦(ball‘𝑚)𝑟)})
cismty 37837	class Ismty
df-ismty 37838	⊢ 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 37864	class ℝⁿ
df-rrn 37865	⊢ ℝⁿ = (𝑖 ∈ Fin ↦ (𝑥 ∈ (ℝ ↑_m 𝑖), 𝑦 ∈ (ℝ ↑_m 𝑖) ↦ (√‘Σ𝑘 ∈ 𝑖 (((𝑥‘𝑘) − (𝑦‘𝑘))↑2))))
cass 37881	class Ass
df-ass 37882	⊢ Ass = {𝑔 ∣ ∀𝑥 ∈ dom dom 𝑔∀𝑦 ∈ dom dom 𝑔∀𝑧 ∈ dom dom 𝑔((𝑥𝑔𝑦)𝑔𝑧) = (𝑥𝑔(𝑦𝑔𝑧))}
cexid 37883	class ExId
df-exid 37884	⊢ ExId = {𝑔 ∣ ∃𝑥 ∈ dom dom 𝑔∀𝑦 ∈ dom dom 𝑔((𝑥𝑔𝑦) = 𝑦 ∧ (𝑦𝑔𝑥) = 𝑦)}
cmagm 37887	class Magma
df-mgmOLD 37888	⊢ Magma = {𝑔 ∣ ∃𝑡 𝑔:(𝑡 × 𝑡)⟶𝑡}
csem 37899	class SemiGrp
df-sgrOLD 37900	⊢ SemiGrp = (Magma ∩ Ass)
cmndo 37905	class MndOp
df-mndo 37906	⊢ MndOp = (SemiGrp ∩ ExId )
cghomOLD 37922	class GrpOpHom
df-ghomOLD 37923	⊢ 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 37933	class RingOps
df-rngo 37934	⊢ 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 37987	class DivRingOps
df-drngo 37988	⊢ DivRingOps = {⟨𝑔, ℎ⟩ ∣ (⟨𝑔, ℎ⟩ ∈ RingOps ∧ (ℎ ↾ ((ran 𝑔 ∖ {(GId‘𝑔)}) × (ran 𝑔 ∖ {(GId‘𝑔)}))) ∈ GrpOp)}
crngohom 37999	class RingOpsHom
crngoiso 38000	class RingOpsIso
crisc 38001	class ≃_𝑟
df-rngohom 38002	⊢ 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 38015	⊢ RingOpsIso = (𝑟 ∈ RingOps, 𝑠 ∈ RingOps ↦ {𝑓 ∈ (𝑟 RingOpsHom 𝑠) ∣ 𝑓:ran (1^st ‘𝑟)–1-1-onto→ran (1^st ‘𝑠)})
df-risc 38022	⊢ ≃_𝑟 = {⟨𝑟, 𝑠⟩ ∣ ((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑟 RingOpsIso 𝑠))}
ccm2 38028	class Com2
df-com2 38029	⊢ Com2 = {⟨𝑔, ℎ⟩ ∣ ∀𝑎 ∈ ran 𝑔∀𝑏 ∈ ran 𝑔(𝑎ℎ𝑏) = (𝑏ℎ𝑎)}
cfld 38030	class Fld
df-fld 38031	⊢ Fld = (DivRingOps ∩ Com2)
ccring 38032	class CRingOps
df-crngo 38033	⊢ CRingOps = (RingOps ∩ Com2)
cidl 38046	class Idl
cpridl 38047	class PrIdl
cmaxidl 38048	class MaxIdl
df-idl 38049	⊢ 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 38050	⊢ 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 38051	⊢ MaxIdl = (𝑟 ∈ RingOps ↦ {𝑖 ∈ (Idl‘𝑟) ∣ (𝑖 ≠ ran (1^st ‘𝑟) ∧ ∀𝑗 ∈ (Idl‘𝑟)(𝑖 ⊆ 𝑗 → (𝑗 = 𝑖 ∨ 𝑗 = ran (1^st ‘𝑟))))})
cprrng 38085	class PrRing
cdmn 38086	class Dmn
df-prrngo 38087	⊢ PrRing = {𝑟 ∈ RingOps ∣ {(GId‘(1^st ‘𝑟))} ∈ (PrIdl‘𝑟)}
df-dmn 38088	⊢ Dmn = (PrRing ∩ Com2)
cigen 38098	class IdlGen
df-igen 38099	⊢ IdlGen = (𝑟 ∈ RingOps, 𝑠 ∈ 𝒫 ran (1^st ‘𝑟) ↦ ∩ {𝑗 ∈ (Idl‘𝑟) ∣ 𝑠 ⊆ 𝑗})
cxrn 38213	class (𝐴 ⋉ 𝐵)
ccoss 38214	class ≀ 𝑅
ccoels 38215	class ∼ 𝐴
crels 38216	class Rels
cssr 38217	class S
crefs 38218	class Refs
crefrels 38219	class RefRels
wrefrel 38220	wff RefRel 𝑅
ccnvrefs 38221	class CnvRefs
ccnvrefrels 38222	class CnvRefRels
wcnvrefrel 38223	wff CnvRefRel 𝑅
csyms 38224	class Syms
csymrels 38225	class SymRels
wsymrel 38226	wff SymRel 𝑅
ctrs 38227	class Trs
ctrrels 38228	class TrRels
wtrrel 38229	wff TrRel 𝑅
ceqvrels 38230	class EqvRels
weqvrel 38231	wff EqvRel 𝑅
ccoeleqvrels 38232	class CoElEqvRels
wcoeleqvrel 38233	wff CoElEqvRel 𝐴
credunds 38234	class Redunds
wredund 38235	wff 𝐴 Redund ⟨𝐵, 𝐶⟩
wredundp 38236	wff redund (𝜑, 𝜓, 𝜒)
cdmqss 38237	class DomainQss
wdmqs 38238	wff 𝑅 DomainQs 𝐴
cers 38239	class Ers
werALTV 38240	wff 𝑅 ErALTV 𝐴
ccomembers 38241	class CoMembErs
wcomember 38242	wff CoMembEr 𝐴
cfunss 38243	class Funss
cfunsALTV 38244	class FunsALTV
wfunALTV 38245	wff FunALTV 𝐹
cdisjss 38246	class Disjss
cdisjs 38247	class Disjs
wdisjALTV 38248	wff Disj 𝑅
celdisjs 38249	class ElDisjs
weldisj 38250	wff ElDisj 𝐴
wantisymrel 38251	wff AntisymRel 𝑅
cparts 38252	class Parts
wpart 38253	wff 𝑅 Part 𝐴
cmembparts 38254	class MembParts
wmembpart 38255	wff MembPart 𝐴
df-xrn 38398	⊢ (𝐴 ⋉ 𝐵) = ((^◡(1^st ↾ (V × V)) ∘ 𝐴) ∩ (^◡(2^nd ↾ (V × V)) ∘ 𝐵))
df-coss 38447	⊢ ≀ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)}
df-coels 38448	⊢ ∼ 𝐴 = ≀ (^◡ E ↾ 𝐴)
df-rels 38521	⊢ Rels = 𝒫 (V × V)
df-ssr 38534	⊢ S = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ⊆ 𝑦}
df-refs 38546	⊢ Refs = {𝑥 ∣ ( I ∩ (dom 𝑥 × ran 𝑥)) S (𝑥 ∩ (dom 𝑥 × ran 𝑥))}
df-refrels 38547	⊢ RefRels = ( Refs ∩ Rels )
df-refrel 38548	⊢ ( RefRel 𝑅 ↔ (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅))
df-cnvrefs 38561	⊢ CnvRefs = {𝑥 ∣ ( I ∩ (dom 𝑥 × ran 𝑥))^◡ S (𝑥 ∩ (dom 𝑥 × ran 𝑥))}
df-cnvrefrels 38562	⊢ CnvRefRels = ( CnvRefs ∩ Rels )
df-cnvrefrel 38563	⊢ ( CnvRefRel 𝑅 ↔ ((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ ( I ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅))
df-syms 38578	⊢ Syms = {𝑥 ∣ ^◡(𝑥 ∩ (dom 𝑥 × ran 𝑥)) S (𝑥 ∩ (dom 𝑥 × ran 𝑥))}
df-symrels 38579	⊢ SymRels = ( Syms ∩ Rels )
df-symrel 38580	⊢ ( SymRel 𝑅 ↔ (^◡(𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅))
df-trs 38608	⊢ Trs = {𝑥 ∣ ((𝑥 ∩ (dom 𝑥 × ran 𝑥)) ∘ (𝑥 ∩ (dom 𝑥 × ran 𝑥))) S (𝑥 ∩ (dom 𝑥 × ran 𝑥))}
df-trrels 38609	⊢ TrRels = ( Trs ∩ Rels )
df-trrel 38610	⊢ ( TrRel 𝑅 ↔ (((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∘ (𝑅 ∩ (dom 𝑅 × ran 𝑅))) ⊆ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅))
df-eqvrels 38620	⊢ EqvRels = (( RefRels ∩ SymRels ) ∩ TrRels )
df-eqvrel 38621	⊢ ( EqvRel 𝑅 ↔ ( RefRel 𝑅 ∧ SymRel 𝑅 ∧ TrRel 𝑅))
df-coeleqvrels 38622	⊢ CoElEqvRels = {𝑎 ∣ ≀ (^◡ E ↾ 𝑎) ∈ EqvRels }
df-coeleqvrel 38623	⊢ ( CoElEqvRel 𝐴 ↔ EqvRel ≀ (^◡ E ↾ 𝐴))
df-redunds 38659	⊢ Redunds = ^◡{⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ (𝑥 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑧) = (𝑦 ∩ 𝑧))}
df-redund 38660	⊢ (𝐴 Redund ⟨𝐵, 𝐶⟩ ↔ (𝐴 ⊆ 𝐵 ∧ (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐶)))
df-redundp 38661	⊢ ( redund (𝜑, 𝜓, 𝜒) ↔ ((𝜑 → 𝜓) ∧ ((𝜑 ∧ 𝜒) ↔ (𝜓 ∧ 𝜒))))
df-dmqss 38674	⊢ DomainQss = {⟨𝑥, 𝑦⟩ ∣ (dom 𝑥 / 𝑥) = 𝑦}
df-dmqs 38675	⊢ (𝑅 DomainQs 𝐴 ↔ (dom 𝑅 / 𝑅) = 𝐴)
df-ers 38700	⊢ Ers = ( DomainQss ↾ EqvRels )
df-erALTV 38701	⊢ (𝑅 ErALTV 𝐴 ↔ ( EqvRel 𝑅 ∧ 𝑅 DomainQs 𝐴))
df-comembers 38702	⊢ CoMembErs = {𝑎 ∣ ≀ (^◡ E ↾ 𝑎) Ers 𝑎}
df-comember 38703	⊢ ( CoMembEr 𝐴 ↔ ≀ (^◡ E ↾ 𝐴) ErALTV 𝐴)
df-funss 38717	⊢ Funss = {𝑥 ∣ ≀ 𝑥 ∈ CnvRefRels }
df-funsALTV 38718	⊢ FunsALTV = ( Funss ∩ Rels )
df-funALTV 38719	⊢ ( FunALTV 𝐹 ↔ ( CnvRefRel ≀ 𝐹 ∧ Rel 𝐹))
df-disjss 38740	⊢ Disjss = {𝑥 ∣ ≀ ^◡𝑥 ∈ CnvRefRels }
df-disjs 38741	⊢ Disjs = ( Disjss ∩ Rels )
df-disjALTV 38742	⊢ ( Disj 𝑅 ↔ ( CnvRefRel ≀ ^◡𝑅 ∧ Rel 𝑅))
df-eldisjs 38743	⊢ ElDisjs = {𝑎 ∣ (^◡ E ↾ 𝑎) ∈ Disjs }
df-eldisj 38744	⊢ ( ElDisj 𝐴 ↔ Disj (^◡ E ↾ 𝐴))
df-antisymrel 38797	⊢ ( AntisymRel 𝑅 ↔ ( CnvRefRel (𝑅 ∩ ^◡𝑅) ∧ Rel 𝑅))
df-parts 38802	⊢ Parts = ( DomainQss ↾ Disjs )
df-part 38803	⊢ (𝑅 Part 𝐴 ↔ ( Disj 𝑅 ∧ 𝑅 DomainQs 𝐴))
df-membparts 38804	⊢ MembParts = {𝑎 ∣ (^◡ E ↾ 𝑎) Parts 𝑎}
df-membpart 38805	⊢ ( MembPart 𝐴 ↔ (^◡ E ↾ 𝐴) Part 𝐴)
wprt 38909	wff Prt 𝐴
df-prt 38910	⊢ (Prt 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅))
ax-c5 38921	⊢ (∀𝑥𝜑 → 𝜑)
ax-c4 38922	⊢ (∀𝑥(∀𝑥𝜑 → 𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))
ax-c7 38923	⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜑 → 𝜑)
ax-c10 38924	⊢ (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → 𝜑)
ax-c11 38925	⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑))
ax-c11n 38926	⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
ax-c15 38927	⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))))
ax-c9 38928	⊢ (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))
ax-c14 38929	⊢ (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥 ∈ 𝑦 → ∀𝑧 𝑥 ∈ 𝑦)))
ax-c16 38930	⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))
ax-riotaBAD 38991	⊢ (℩𝑥 ∈ 𝐴 𝜑) = if(∃!𝑥 ∈ 𝐴 𝜑, (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)), (Undef‘{𝑥 ∣ 𝑥 ∈ 𝐴}))
clsa 39012	class LSAtoms
clsh 39013	class LSHyp
df-lsatoms 39014	⊢ LSAtoms = (𝑤 ∈ V ↦ ran (𝑣 ∈ ((Base‘𝑤) ∖ {(0_g‘𝑤)}) ↦ ((LSpan‘𝑤)‘{𝑣})))
df-lshyp 39015	⊢ LSHyp = (𝑤 ∈ V ↦ {𝑠 ∈ (LSubSp‘𝑤) ∣ (𝑠 ≠ (Base‘𝑤) ∧ ∃𝑣 ∈ (Base‘𝑤)((LSpan‘𝑤)‘(𝑠 ∪ {𝑣})) = (Base‘𝑤))})
clcv 39056	class ⋖_L
df-lcv 39057	⊢ ⋖_L = (𝑤 ∈ V ↦ {⟨𝑡, 𝑢⟩ ∣ ((𝑡 ∈ (LSubSp‘𝑤) ∧ 𝑢 ∈ (LSubSp‘𝑤)) ∧ (𝑡 ⊊ 𝑢 ∧ ¬ ∃𝑠 ∈ (LSubSp‘𝑤)(𝑡 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑢)))})
clfn 39095	class LFnl
df-lfl 39096	⊢ 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 39123	class LKer
df-lkr 39124	⊢ LKer = (𝑤 ∈ V ↦ (𝑓 ∈ (LFnl‘𝑤) ↦ (^◡𝑓 “ {(0_g‘(Scalar‘𝑤))})))
cld 39161	class LDual
df-ldual 39162	⊢ 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 39210	class OP
ccmtN 39211	class cm
col 39212	class OL
coml 39213	class OML
df-oposet 39214	⊢ 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 39215	⊢ 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 39216	⊢ OL = (Lat ∩ OP)
df-oml 39217	⊢ OML = {𝑙 ∈ OL ∣ ∀𝑎 ∈ (Base‘𝑙)∀𝑏 ∈ (Base‘𝑙)(𝑎(le‘𝑙)𝑏 → 𝑏 = (𝑎(join‘𝑙)(𝑏(meet‘𝑙)((oc‘𝑙)‘𝑎))))}
ccvr 39300	class ⋖
catm 39301	class Atoms
cal 39302	class AtLat
clc 39303	class CvLat
df-covers 39304	⊢ ⋖ = (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 39305	⊢ Atoms = (𝑝 ∈ V ↦ {𝑎 ∈ (Base‘𝑝) ∣ (0.‘𝑝)( ⋖ ‘𝑝)𝑎})
df-atl 39336	⊢ AtLat = {𝑘 ∈ Lat ∣ ((Base‘𝑘) ∈ dom (glb‘𝑘) ∧ ∀𝑥 ∈ (Base‘𝑘)(𝑥 ≠ (0.‘𝑘) → ∃𝑝 ∈ (Atoms‘𝑘)𝑝(le‘𝑘)𝑥))}
df-cvlat 39360	⊢ 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 39388	class HL
df-hlat 39389	⊢ 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 39529	class LLines
clpl 39530	class LPlanes
clvol 39531	class LVols
clines 39532	class Lines
cpointsN 39533	class Points
cpsubsp 39534	class PSubSp
cpmap 39535	class pmap
df-llines 39536	⊢ LLines = (𝑘 ∈ V ↦ {𝑥 ∈ (Base‘𝑘) ∣ ∃𝑝 ∈ (Atoms‘𝑘)𝑝( ⋖ ‘𝑘)𝑥})
df-lplanes 39537	⊢ LPlanes = (𝑘 ∈ V ↦ {𝑥 ∈ (Base‘𝑘) ∣ ∃𝑝 ∈ (LLines‘𝑘)𝑝( ⋖ ‘𝑘)𝑥})
df-lvols 39538	⊢ LVols = (𝑘 ∈ V ↦ {𝑥 ∈ (Base‘𝑘) ∣ ∃𝑝 ∈ (LPlanes‘𝑘)𝑝( ⋖ ‘𝑘)𝑥})
df-lines 39539	⊢ 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 39540	⊢ Points = (𝑘 ∈ V ↦ {𝑞 ∣ ∃𝑝 ∈ (Atoms‘𝑘)𝑞 = {𝑝}})
df-psubsp 39541	⊢ 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 39542	⊢ pmap = (𝑘 ∈ V ↦ (𝑎 ∈ (Base‘𝑘) ↦ {𝑝 ∈ (Atoms‘𝑘) ∣ 𝑝(le‘𝑘)𝑎}))
cpadd 39833	class +_𝑃
df-padd 39834	⊢ +_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 39925	class PCl
df-pclN 39926	⊢ PCl = (𝑘 ∈ V ↦ (𝑥 ∈ 𝒫 (Atoms‘𝑘) ↦ ∩ {𝑦 ∈ (PSubSp‘𝑘) ∣ 𝑥 ⊆ 𝑦}))
cpolN 39940	class ⊥_𝑃
df-polarityN 39941	⊢ ⊥_𝑃 = (𝑙 ∈ V ↦ (𝑚 ∈ 𝒫 (Atoms‘𝑙) ↦ ((Atoms‘𝑙) ∩ ∩ 𝑝 ∈ 𝑚 ((pmap‘𝑙)‘((oc‘𝑙)‘𝑝)))))
cpscN 39972	class PSubCl
df-psubclN 39973	⊢ PSubCl = (𝑘 ∈ V ↦ {𝑠 ∣ (𝑠 ⊆ (Atoms‘𝑘) ∧ ((⊥_𝑃‘𝑘)‘((⊥_𝑃‘𝑘)‘𝑠)) = 𝑠)})
clh 40022	class LHyp
claut 40023	class LAut
cwpointsN 40024	class WAtoms
cpautN 40025	class PAut
df-lhyp 40026	⊢ LHyp = (𝑘 ∈ V ↦ {𝑥 ∈ (Base‘𝑘) ∣ 𝑥( ⋖ ‘𝑘)(1.‘𝑘)})
df-laut 40027	⊢ 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 40028	⊢ WAtoms = (𝑘 ∈ V ↦ (𝑑 ∈ (Atoms‘𝑘) ↦ ((Atoms‘𝑘) ∖ ((⊥_𝑃‘𝑘)‘{𝑑}))))
df-pautN 40029	⊢ 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 40138	class LDil
cltrn 40139	class LTrn
cdilN 40140	class Dil
ctrnN 40141	class Trn
df-ldil 40142	⊢ LDil = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ {𝑓 ∈ (LAut‘𝑘) ∣ ∀𝑥 ∈ (Base‘𝑘)(𝑥(le‘𝑘)𝑤 → (𝑓‘𝑥) = 𝑥)}))
df-ltrn 40143	⊢ 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 40144	⊢ Dil = (𝑘 ∈ V ↦ (𝑑 ∈ (Atoms‘𝑘) ↦ {𝑓 ∈ (PAut‘𝑘) ∣ ∀𝑥 ∈ (PSubSp‘𝑘)(𝑥 ⊆ ((WAtoms‘𝑘)‘𝑑) → (𝑓‘𝑥) = 𝑥)}))
df-trnN 40145	⊢ 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 40196	class trL
df-trl 40197	⊢ 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 40780	class TGrp
df-tgrp 40781	⊢ TGrp = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ {⟨(Base‘ndx), ((LTrn‘𝑘)‘𝑤)⟩, ⟨(+_g‘ndx), (𝑓 ∈ ((LTrn‘𝑘)‘𝑤), 𝑔 ∈ ((LTrn‘𝑘)‘𝑤) ↦ (𝑓 ∘ 𝑔))⟩}))
ctendo 40790	class TEndo
cedring 40791	class EDRing
cedring-rN 40792	class EDRing_R
df-tendo 40793	⊢ 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 40794	⊢ 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 40795	⊢ 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 41040	class DVecA
df-dveca 41041	⊢ 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 41066	class DIsoA
df-disoa 41067	⊢ 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 41116	class DVecH
df-dvech 41117	⊢ 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 41157	class ocA
df-docaN 41158	⊢ 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 41169	class vA
df-djaN 41170	⊢ 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 41176	class DIsoB
df-dib 41177	⊢ DIsoB = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑥 ∈ dom ((DIsoA‘𝑘)‘𝑤) ↦ ((((DIsoA‘𝑘)‘𝑤)‘𝑥) × {(𝑓 ∈ ((LTrn‘𝑘)‘𝑤) ↦ ( I ↾ (Base‘𝑘)))}))))
cdic 41210	class DIsoC
df-dic 41211	⊢ 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 41266	class DIsoH
df-dih 41267	⊢ 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 41385	class ocH
df-doch 41386	⊢ 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 41432	class joinH
df-djh 41433	⊢ 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 41518	class LPol
df-lpolN 41519	⊢ 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 41624	class LCDual
df-lcdual 41625	⊢ 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 41662	class mapd
df-mapd 41663	⊢ 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 41794	class HVMap
df-hvmap 41795	⊢ 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 41829	class HDMap1
chdma 41830	class HDMap
df-hdmap1 41831	⊢ 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 41832	⊢ 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 41921	class HGMap
df-hgmap 41922	⊢ 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 41970	class HLHil
df-hlhil 41971	⊢ 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 42000	class CSRing
df-csring 42001	⊢ CSRing = {𝑓 ∈ SRing ∣ (mulGrp‘𝑓) ∈ CMnd}
cprimroots 42123	class PrimRoots
df-primroots 42124	⊢ 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 42234	⊢ ∀𝑝 ∈ ℙ ∀𝑛 ∈ ℕ ∃𝑘 ∈ Field ((♯‘(Base‘𝑘)) = (𝑝↑𝑛) ∧ (chr‘𝑘) = 𝑝)
cresub 42397	class −_ℝ
df-resub 42398	⊢ −_ℝ = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (℩𝑧 ∈ ℝ (𝑦 + 𝑧) = 𝑥))
crediv 42472	class /_ℝ
df-rediv 42473	⊢ /_ℝ = (𝑥 ∈ ℝ, 𝑦 ∈ (ℝ ∖ {0}) ↦ (℩𝑧 ∈ ℝ (𝑦 · 𝑧) = 𝑥))
cprjsp 42633	class ℙ𝕣𝕠𝕛
df-prjsp 42634	⊢ 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 42646	class ℙ𝕣𝕠𝕛_n
df-prjspn 42647	⊢ ℙ𝕣𝕠𝕛_n = (𝑛 ∈ ℕ₀, 𝑘 ∈ DivRing ↦ (ℙ𝕣𝕠𝕛‘(𝑘 freeLMod (0...𝑛))))
cprjcrv 42661	class ℙ𝕣𝕠𝕛Crv
df-prjcrv 42662	⊢ ℙ𝕣𝕠𝕛Crv = (𝑛 ∈ ℕ₀, 𝑘 ∈ Field ↦ (𝑓 ∈ ∪ ran ((0...𝑛) mHomP 𝑘) ↦ {𝑝 ∈ (𝑛ℙ𝕣𝕠𝕛_n𝑘) ∣ ((((0...𝑛) eval 𝑘)‘𝑓) “ 𝑝) = {(0_g‘𝑘)}}))
cnacs 42734	class NoeACS
df-nacs 42735	⊢ NoeACS = (𝑥 ∈ V ↦ {𝑐 ∈ (ACS‘𝑥) ∣ ∀𝑠 ∈ 𝑐 ∃𝑔 ∈ (𝒫 𝑥 ∩ Fin)𝑠 = ((mrCls‘𝑐)‘𝑔)})
cmzpcl 42753	class mzPolyCld
cmzp 42754	class mzPoly
df-mzpcl 42755	⊢ 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 42756	⊢ mzPoly = (𝑣 ∈ V ↦ ∩ (mzPolyCld‘𝑣))
cdioph 42787	class Dioph
df-dioph 42788	⊢ 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 42868	class ◻_NN
cpell1qr 42869	class Pell1QR
cpell1234qr 42870	class Pell1234QR
cpell14qr 42871	class Pell14QR
cpellfund 42872	class PellFund
df-squarenn 42873	⊢ ◻_NN = {𝑥 ∈ ℕ ∣ (√‘𝑥) ∈ ℚ}
df-pell1qr 42874	⊢ Pell1QR = (𝑥 ∈ (ℕ ∖ ◻_NN) ↦ {𝑦 ∈ ℝ ∣ ∃𝑧 ∈ ℕ₀ ∃𝑤 ∈ ℕ₀ (𝑦 = (𝑧 + ((√‘𝑥) · 𝑤)) ∧ ((𝑧↑2) − (𝑥 · (𝑤↑2))) = 1)})
df-pell14qr 42875	⊢ Pell14QR = (𝑥 ∈ (ℕ ∖ ◻_NN) ↦ {𝑦 ∈ ℝ ∣ ∃𝑧 ∈ ℕ₀ ∃𝑤 ∈ ℤ (𝑦 = (𝑧 + ((√‘𝑥) · 𝑤)) ∧ ((𝑧↑2) − (𝑥 · (𝑤↑2))) = 1)})
df-pell1234qr 42876	⊢ Pell1234QR = (𝑥 ∈ (ℕ ∖ ◻_NN) ↦ {𝑦 ∈ ℝ ∣ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ (𝑦 = (𝑧 + ((√‘𝑥) · 𝑤)) ∧ ((𝑧↑2) − (𝑥 · (𝑤↑2))) = 1)})
df-pellfund 42877	⊢ PellFund = (𝑥 ∈ (ℕ ∖ ◻_NN) ↦ inf({𝑧 ∈ (Pell14QR‘𝑥) ∣ 1 < 𝑧}, ℝ, < ))
crmx 42932	class X_rm
crmy 42933	class Y_rm
df-rmx 42934	⊢ 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 42935	⊢ 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 43099	class LFinGen
df-lfig 43100	⊢ LFinGen = {𝑤 ∈ LMod ∣ (Base‘𝑤) ∈ ((LSpan‘𝑤) “ (𝒫 (Base‘𝑤) ∩ Fin))}
clnm 43107	class LNoeM
df-lnm 43108	⊢ LNoeM = {𝑤 ∈ LMod ∣ ∀𝑖 ∈ (LSubSp‘𝑤)(𝑤 ↾_s 𝑖) ∈ LFinGen}
clnr 43141	class LNoeR
df-lnr 43142	⊢ LNoeR = {𝑎 ∈ Ring ∣ (ringLMod‘𝑎) ∈ LNoeM}
cldgis 43153	class ldgIdlSeq
df-ldgis 43154	⊢ ldgIdlSeq = (𝑟 ∈ V ↦ (𝑖 ∈ (LIdeal‘(Poly₁‘𝑟)) ↦ (𝑥 ∈ ℕ₀ ↦ {𝑗 ∣ ∃𝑘 ∈ 𝑖 (((deg₁‘𝑟)‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe₁‘𝑘)‘𝑥))})))
cmnc 43163	class Monic
cplylt 43164	class Poly_<
df-mnc 43165	⊢ Monic = (𝑠 ∈ 𝒫 ℂ ↦ {𝑝 ∈ (Poly‘𝑠) ∣ ((coeff‘𝑝)‘(deg‘𝑝)) = 1})
df-plylt 43166	⊢ Poly_< = (𝑠 ∈ 𝒫 ℂ, 𝑥 ∈ ℕ₀ ↦ {𝑝 ∈ (Poly‘𝑠) ∣ (𝑝 = 0_𝑝 ∨ (deg‘𝑝) < 𝑥)})
cdgraa 43172	class deg_AA
cmpaa 43173	class minPolyAA
df-dgraa 43174	⊢ deg_AA = (𝑥 ∈ 𝔸 ↦ inf({𝑑 ∈ ℕ ∣ ∃𝑝 ∈ ((Poly‘ℚ) ∖ {0_𝑝})((deg‘𝑝) = 𝑑 ∧ (𝑝‘𝑥) = 0)}, ℝ, < ))
df-mpaa 43175	⊢ minPolyAA = (𝑥 ∈ 𝔸 ↦ (℩𝑝 ∈ (Poly‘ℚ)((deg‘𝑝) = (deg_AA‘𝑥) ∧ (𝑝‘𝑥) = 0 ∧ ((coeff‘𝑝)‘(deg_AA‘𝑥)) = 1)))
citgo 43189	class IntgOver
cza 43190	class ℤ
df-itgo 43191	⊢ IntgOver = (𝑠 ∈ 𝒫 ℂ ↦ {𝑥 ∈ ℂ ∣ ∃𝑝 ∈ (Poly‘𝑠)((𝑝‘𝑥) = 0 ∧ ((coeff‘𝑝)‘(deg‘𝑝)) = 1)})
df-za 43192	⊢ ℤ = (IntgOver‘ℤ)
cmend 43203	class MEndo
df-mend 43204	⊢ 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 43229	class CytP
df-cytp 43230	⊢ 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 43238	class TopSep
ctoplnd 43239	class TopLnd
df-topsep 43240	⊢ TopSep = {𝑗 ∈ Top ∣ ∃𝑥 ∈ 𝒫 ∪ 𝑗(𝑥 ≼ ω ∧ ((cls‘𝑗)‘𝑥) = ∪ 𝑗)}
df-toplnd 43241	⊢ TopLnd = {𝑥 ∈ Top ∣ ∀𝑦 ∈ 𝒫 𝑥(∪ 𝑥 = ∪ 𝑦 → ∃𝑧 ∈ 𝒫 𝑥(𝑧 ≼ ω ∧ ∪ 𝑥 = ∪ 𝑧))}
crcl 43704	class r*
df-rcl 43705	⊢ r* = (𝑥 ∈ V ↦ ∩ {𝑧 ∣ (𝑥 ⊆ 𝑧 ∧ ( I ↾ (dom 𝑧 ∪ ran 𝑧)) ⊆ 𝑧)})
whe 43804	wff 𝑅 hereditary 𝐴
df-he 43805	⊢ (𝑅 hereditary 𝐴 ↔ (𝑅 “ 𝐴) ⊆ 𝐴)
ax-frege1 43822	⊢ (𝜑 → (𝜓 → 𝜑))
ax-frege2 43823	⊢ ((𝜑 → (𝜓 → 𝜒)) → ((𝜑 → 𝜓) → (𝜑 → 𝜒)))
ax-frege8 43841	⊢ ((𝜑 → (𝜓 → 𝜒)) → (𝜓 → (𝜑 → 𝜒)))
ax-frege28 43862	⊢ ((𝜑 → 𝜓) → (¬ 𝜓 → ¬ 𝜑))
ax-frege31 43866	⊢ (¬ ¬ 𝜑 → 𝜑)
ax-frege41 43877	⊢ (𝜑 → ¬ ¬ 𝜑)
ax-frege52a 43889	⊢ ((𝜑 ↔ 𝜓) → (if-(𝜑, 𝜃, 𝜒) → if-(𝜓, 𝜃, 𝜒)))
ax-frege54a 43894	⊢ (𝜑 ↔ 𝜑)
ax-frege58a 43907	⊢ ((𝜓 ∧ 𝜒) → if-(𝜑, 𝜓, 𝜒))
ax-frege52c 43920	⊢ (𝐴 = 𝐵 → ([𝐴 / 𝑥]𝜑 → [𝐵 / 𝑥]𝜑))
ax-frege54c 43924	⊢ 𝐴 = 𝐴
ax-frege58b 43933	⊢ (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑)
cmnring 44243	class MndRing
df-mnring 44244	⊢ 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 44267	class Scott 𝐴
df-scott 44268	⊢ Scott 𝐴 = {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)}
ccoll 44282	class (𝐹 Coll 𝐴)
df-coll 44283	⊢ (𝐹 Coll 𝐴) = ∪ 𝑥 ∈ 𝐴 Scott (𝐹 “ {𝑥})
cbcc 44368	class C_𝑐
df-bcc 44369	⊢ C_𝑐 = (𝑐 ∈ ℂ, 𝑘 ∈ ℕ₀ ↦ ((𝑐 FallFac 𝑘) / (!‘𝑘)))
cplusr 44488	class +_𝑟
cminusr 44489	class -_𝑟
ctimesr 44490	class ._𝑣
cptdfc 44491	class PtDf(𝐴, 𝐵)
crr3c 44492	class RR3
cline3 44493	class line3
df-addr 44494	⊢ +_𝑟 = (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑣 ∈ ℝ ↦ ((𝑥‘𝑣) + (𝑦‘𝑣))))
df-subr 44495	⊢ -_𝑟 = (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑣 ∈ ℝ ↦ ((𝑥‘𝑣) − (𝑦‘𝑣))))
df-mulv 44496	⊢ ._𝑣 = (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑣 ∈ ℝ ↦ (𝑥 · (𝑦‘𝑣))))
df-ptdf 44507	⊢ PtDf(𝐴, 𝐵) = (𝑥 ∈ ℝ ↦ (((𝑥._𝑣(𝐵-_𝑟𝐴)) +_𝑣 𝐴) “ {1, 2, 3}))
df-rr3 44508	⊢ RR3 = (ℝ ↑_m {1, 2, 3})
df-line3 44509	⊢ line3 = {𝑥 ∈ 𝒫 RR3 ∣ (2_o ≼ 𝑥 ∧ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑧 ≠ 𝑦 → ran PtDf(𝑦, 𝑧) = 𝑥))}
wvd1 44601	wff ( 𝜑 ▶ 𝜓 )
df-vd1 44602	⊢ (( 𝜑 ▶ 𝜓 ) ↔ (𝜑 → 𝜓))
wvd2 44609	wff ( 𝜑 , 𝜓 ▶ 𝜒 )
df-vd2 44610	⊢ (( 𝜑 , 𝜓 ▶ 𝜒 ) ↔ ((𝜑 ∧ 𝜓) → 𝜒))
wvhc2 44612	wff ( 𝜑 , 𝜓 )
df-vhc2 44613	⊢ (( 𝜑 , 𝜓 ) ↔ (𝜑 ∧ 𝜓))
wvd3 44619	wff ( 𝜑 , 𝜓 , 𝜒 ▶ 𝜃 )
wvhc3 44620	wff ( 𝜑 , 𝜓 , 𝜒 )
df-vhc3 44621	⊢ (( 𝜑 , 𝜓 , 𝜒 ) ↔ (𝜑 ∧ 𝜓 ∧ 𝜒))
df-vd3 44622	⊢ (( 𝜑 , 𝜓 , 𝜒 ▶ 𝜃 ) ↔ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃))
wrelp 44974	wff 𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵)
df-relp 44975	⊢ (𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵) ↔ (𝐻:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐻‘𝑥)𝑆(𝐻‘𝑦))))
clsi 45788	class lim inf
df-liminf 45789	⊢ lim inf = (𝑥 ∈ V ↦ sup(ran (𝑘 ∈ ℝ ↦ inf(((𝑥 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )), ℝ^*, < ))
clsxlim 45855	class ~~>*
df-xlim 45856	⊢ ~~>* = (⇝_𝑡‘(ordTop‘ ≤ ))
csalg 46345	class SAlg
df-salg 46346	⊢ SAlg = {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (∪ 𝑥 ∖ 𝑦) ∈ 𝑥 ∧ ∀𝑦 ∈ 𝒫 𝑥(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑥))}
csalon 46347	class SalOn
df-salon 46348	⊢ SalOn = (𝑥 ∈ V ↦ {𝑠 ∈ SAlg ∣ ∪ 𝑠 = 𝑥})
csalgen 46349	class SalGen
df-salgen 46350	⊢ SalGen = (𝑥 ∈ V ↦ ∩ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑥 ∧ 𝑥 ⊆ 𝑠)})
csumge0 46399	class Σ^{^}
df-sumge0 46400	⊢ Σ^{^} = (𝑥 ∈ V ↦ if(+∞ ∈ ran 𝑥, +∞, sup(ran (𝑦 ∈ (𝒫 dom 𝑥 ∩ Fin) ↦ Σ𝑤 ∈ 𝑦 (𝑥‘𝑤)), ℝ^*, < )))
cmea 46486	class Meas
df-mea 46487	⊢ 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 46526	class OutMeas
df-ome 46527	⊢ 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 46528	class CaraGen
df-caragen 46529	⊢ CaraGen = (𝑜 ∈ OutMeas ↦ {𝑒 ∈ 𝒫 ∪ dom 𝑜 ∣ ∀𝑎 ∈ 𝒫 ∪ dom 𝑜((𝑜‘(𝑎 ∩ 𝑒)) +_𝑒 (𝑜‘(𝑎 ∖ 𝑒))) = (𝑜‘𝑎)})
covoln 46573	class voln*
df-ovoln 46574	⊢ 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 46575	class voln
df-voln 46576	⊢ voln = (𝑥 ∈ Fin ↦ ((voln‘𝑥) ↾ (CaraGen‘(voln‘𝑥))))
csmblfn 46732	class SMblFn
df-smblfn 46733	⊢ SMblFn = (𝑠 ∈ SAlg ↦ {𝑓 ∈ (ℝ ↑_pm ∪ 𝑠) ∣ ∀𝑎 ∈ ℝ (^◡𝑓 “ (-∞(,)𝑎)) ∈ (𝑠 ↾_t dom 𝑓)})
caiota 47113	class (℩'𝑥𝜑)
df-aiota 47115	⊢ (℩'𝑥𝜑) = ∩ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}
wdfat 47146	wff 𝐹 defAt 𝐴
cafv 47147	class (𝐹'''𝐴)
caov 47148	class ((𝐴𝐹𝐵))
df-dfat 47149	⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
df-afv 47150	⊢ (𝐹'''𝐴) = (℩'𝑥𝐴𝐹𝑥)
df-aov 47151	⊢ ((𝐴𝐹𝐵)) = (𝐹'''⟨𝐴, 𝐵⟩)
cafv2 47238	class (𝐹''''𝐴)
df-afv2 47239	⊢ (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ∪ ran 𝐹)
cnelbr 47301	class _∉
df-nelbr 47302	⊢ _∉ = {⟨𝑥, 𝑦⟩ ∣ ¬ 𝑥 ∈ 𝑦}
ciccp 47443	class RePart
df-iccp 47444	⊢ RePart = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ^* ↑_m (0...𝑚)) ∣ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1))})
wich 47475	wff [𝑥⇄𝑦]𝜑
df-ich 47476	⊢ ([𝑥⇄𝑦]𝜑 ↔ ∀𝑥∀𝑦([𝑥 / 𝑎][𝑦 / 𝑥][𝑎 / 𝑦]𝜑 ↔ 𝜑))
cspr 47507	class Pairs
df-spr 47508	⊢ Pairs = (𝑣 ∈ V ↦ {𝑝 ∣ ∃𝑎 ∈ 𝑣 ∃𝑏 ∈ 𝑣 𝑝 = {𝑎, 𝑏}})
cprpr 47542	class Pairs_proper
df-prpr 47543	⊢ Pairs_proper = (𝑣 ∈ V ↦ {𝑝 ∣ ∃𝑎 ∈ 𝑣 ∃𝑏 ∈ 𝑣 (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})})
cfmtno 47557	class FermatNo
df-fmtno 47558	⊢ FermatNo = (𝑛 ∈ ℕ₀ ↦ ((2↑(2↑𝑛)) + 1))
ceven 47654	class Even
codd 47655	class Odd
df-even 47656	⊢ Even = {𝑧 ∈ ℤ ∣ (𝑧 / 2) ∈ ℤ}
df-odd 47657	⊢ Odd = {𝑧 ∈ ℤ ∣ ((𝑧 + 1) / 2) ∈ ℤ}
cfppr 47754	class FPPr
df-fppr 47755	⊢ FPPr = (𝑛 ∈ ℕ ↦ {𝑥 ∈ (ℤ_≥‘4) ∣ (𝑥 ∉ ℙ ∧ 𝑥 ∥ ((𝑛↑(𝑥 − 1)) − 1))})
cgbe 47775	class GoldbachEven
cgbow 47776	class GoldbachOddW
cgbo 47777	class GoldbachOdd
df-gbe 47778	⊢ GoldbachEven = {𝑧 ∈ Even ∣ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑧 = (𝑝 + 𝑞))}
df-gbow 47779	⊢ GoldbachOddW = {𝑧 ∈ Odd ∣ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑧 = ((𝑝 + 𝑞) + 𝑟)}
df-gbo 47780	⊢ GoldbachOdd = {𝑧 ∈ Odd ∣ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ ((𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑟 ∈ Odd ) ∧ 𝑧 = ((𝑝 + 𝑞) + 𝑟))}
ax-bgbltosilva 47840	⊢ ((𝑁 ∈ Even ∧ 4 < 𝑁 ∧ 𝑁 ≤ (4 · (;10↑;18))) → 𝑁 ∈ GoldbachEven )
ax-tgoldbachgt 47841	⊢ 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 47844	⊢ ∃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 47848	class ClNeighbVtx
df-clnbgr 47849	⊢ ClNeighbVtx = (𝑔 ∈ V, 𝑣 ∈ (Vtx‘𝑔) ↦ ({𝑣} ∪ {𝑛 ∈ (Vtx‘𝑔) ∣ ∃𝑒 ∈ (Edg‘𝑔){𝑣, 𝑛} ⊆ 𝑒}))
cisubgr 47890	class ISubGr
df-isubgr 47891	⊢ ISubGr = (𝑔 ∈ V, 𝑣 ∈ 𝒫 (Vtx‘𝑔) ↦ ⟨𝑣, ⦋(iEdg‘𝑔) / 𝑒⦌(𝑒 ↾ {𝑥 ∈ dom 𝑒 ∣ (𝑒‘𝑥) ⊆ 𝑣})⟩)
cgrisom 47904	class GraphIsom
cgrim 47905	class GraphIso
cgric 47906	class ≃_𝑔𝑟
df-grisom 47907	⊢ 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 47908	⊢ 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 47911	⊢ ≃_𝑔𝑟 = (^◡ GraphIso “ (V ∖ 1_o))
cgrtri 47967	class GrTriangles
df-grtri 47968	⊢ 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 47981	class StarGr
df-stgr 47982	⊢ StarGr = (𝑛 ∈ ℕ₀ ↦ {⟨(Base‘ndx), (0...𝑛)⟩, ⟨(.ef‘ndx), ( I ↾ {𝑒 ∈ 𝒫 (0...𝑛) ∣ ∃𝑥 ∈ (1...𝑛)𝑒 = {0, 𝑥}})⟩})
cgrlim 48006	class GraphLocIso
cgrlic 48007	class ≃_𝑙𝑔𝑟
df-grlim 48008	⊢ 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 48011	⊢ ≃_𝑙𝑔𝑟 = (^◡ GraphLocIso “ (V ∖ 1_o))
cgpg 48070	class gPetersenGr
df-gpg 48071	⊢ 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 48163	class UPWalks
df-upwlks 48164	⊢ 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 48213	class clLaw
casslaw 48214	class assLaw
ccomlaw 48215	class comLaw
df-cllaw 48216	⊢ clLaw = {⟨𝑜, 𝑚⟩ ∣ ∀𝑥 ∈ 𝑚 ∀𝑦 ∈ 𝑚 (𝑥𝑜𝑦) ∈ 𝑚}
df-comlaw 48217	⊢ comLaw = {⟨𝑜, 𝑚⟩ ∣ ∀𝑥 ∈ 𝑚 ∀𝑦 ∈ 𝑚 (𝑥𝑜𝑦) = (𝑦𝑜𝑥)}
df-asslaw 48218	⊢ assLaw = {⟨𝑜, 𝑚⟩ ∣ ∀𝑥 ∈ 𝑚 ∀𝑦 ∈ 𝑚 ∀𝑧 ∈ 𝑚 ((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧))}
cintop 48226	class intOp
cclintop 48227	class clIntOp
cassintop 48228	class assIntOp
df-intop 48229	⊢ intOp = (𝑚 ∈ V, 𝑛 ∈ V ↦ (𝑛 ↑_m (𝑚 × 𝑚)))
df-clintop 48230	⊢ clIntOp = (𝑚 ∈ V ↦ (𝑚 intOp 𝑚))
df-assintop 48231	⊢ assIntOp = (𝑚 ∈ V ↦ {𝑜 ∈ ( clIntOp ‘𝑚) ∣ 𝑜 assLaw 𝑚})
cmgm2 48245	class MgmALT
ccmgm2 48246	class CMgmALT
csgrp2 48247	class SGrpALT
ccsgrp2 48248	class CSGrpALT
df-mgm2 48249	⊢ MgmALT = {𝑚 ∣ (+_g‘𝑚) clLaw (Base‘𝑚)}
df-cmgm2 48250	⊢ CMgmALT = {𝑚 ∈ MgmALT ∣ (+_g‘𝑚) comLaw (Base‘𝑚)}
df-sgrp2 48251	⊢ SGrpALT = {𝑔 ∈ MgmALT ∣ (+_g‘𝑔) assLaw (Base‘𝑔)}
df-csgrp2 48252	⊢ CSGrpALT = {𝑔 ∈ SGrpALT ∣ (+_g‘𝑔) comLaw (Base‘𝑔)}
crngcALTV 48293	class RngCatALTV
df-rngcALTV 48294	⊢ 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 48317	class RingCatALTV
df-ringcALTV 48318	⊢ 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 48427	class DMatALT
cscmatalt 48428	class ScMatALT
df-dmatalt 48429	⊢ 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 48430	⊢ 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 48435	class linC
clinco 48436	class LinCo
df-linc 48437	⊢ 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 48438	⊢ 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 48471	class linIndS
clindeps 48472	class linDepS
df-lininds 48473	⊢ 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 48475	⊢ linDepS = {⟨𝑠, 𝑚⟩ ∣ ¬ 𝑠 linIndS 𝑚}
cfdiv 48568	class /_f
df-fdiv 48569	⊢ /_f = (𝑓 ∈ V, 𝑔 ∈ V ↦ ((𝑓 ∘_f / 𝑔) ↾ (𝑔 supp 0)))
cbigo 48578	class Ο
df-bigo 48579	⊢ Ο = (𝑔 ∈ (ℝ ↑_pm ℝ) ↦ {𝑓 ∈ (ℝ ↑_pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(𝑓‘𝑦) ≤ (𝑚 · (𝑔‘𝑦))})
cblen 48600	class #_b
df-blen 48601	⊢ #_b = (𝑛 ∈ V ↦ if(𝑛 = 0, 1, ((⌊‘(2 log_b (abs‘𝑛))) + 1)))
cdig 48626	class digit
df-dig 48627	⊢ digit = (𝑏 ∈ ℕ ↦ (𝑘 ∈ ℤ, 𝑟 ∈ (0[,)+∞) ↦ ((⌊‘((𝑏↑-𝑘) · 𝑟)) mod 𝑏)))
cnaryf 48657	class -aryF
df-naryf 48658	⊢ -aryF = (𝑛 ∈ ℕ₀, 𝑥 ∈ V ↦ (𝑥 ↑_m (𝑥 ↑_m (0..^𝑛))))
citco 48688	class IterComp
cack 48689	class Ack
df-itco 48690	⊢ IterComp = (𝑓 ∈ V ↦ seq0((𝑔 ∈ V, 𝑗 ∈ V ↦ (𝑓 ∘ 𝑔)), (𝑖 ∈ ℕ₀ ↦ if(𝑖 = 0, ( I ↾ dom 𝑓), 𝑓))))
df-ack 48691	⊢ Ack = seq0((𝑓 ∈ V, 𝑗 ∈ V ↦ (𝑛 ∈ ℕ₀ ↦ (((IterComp‘𝑓)‘(𝑛 + 1))‘1))), (𝑖 ∈ ℕ₀ ↦ if(𝑖 = 0, (𝑛 ∈ ℕ₀ ↦ (𝑛 + 1)), 𝑖)))
cline 48758	class Line_M
csph 48759	class Sphere
df-line 48760	⊢ 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 48761	⊢ Sphere = (𝑤 ∈ V ↦ (𝑥 ∈ (Base‘𝑤), 𝑟 ∈ (0[,]+∞) ↦ {𝑝 ∈ (Base‘𝑤) ∣ (𝑝(dist‘𝑤)𝑥) = 𝑟}))
coppf 49153	class oppFunc
df-oppf 49154	⊢ oppFunc = (𝑓 ∈ V, 𝑔 ∈ V ↦ if((Rel 𝑔 ∧ Rel dom 𝑔), ⟨𝑓, tpos 𝑔⟩, ∅))
cup 49204	class UP
df-up 49205	⊢ 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 49290	class swap_F
df-swapf 49291	⊢ 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 49347	class ∘_F
df-fuco 49348	⊢ ∘_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 49404	class −∘_F
df-prcof 49405	⊢ −∘_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 49448	class ThinCat
df-thinc 49449	⊢ ThinCat = {𝑐 ∈ Cat ∣ [(Base‘𝑐) / 𝑏][(Hom ‘𝑐) / ℎ]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∃*𝑓 𝑓 ∈ (𝑥ℎ𝑦)}
ctermc 49503	class TermCat
df-termc 49504	⊢ TermCat = {𝑐 ∈ ThinCat ∣ ∃𝑥(Base‘𝑐) = {𝑥}}
cprstc 49580	class ProsetToCat
df-prstc 49581	⊢ ProsetToCat = (𝑘 ∈ Proset ↦ ((𝑘 sSet ⟨(Hom ‘ndx), ((le‘𝑘) × {1_o})⟩) sSet ⟨(comp‘ndx), ∅⟩))
cmndtc 49608	class MndToCat
df-mndtc 49609	⊢ MndToCat = (𝑚 ∈ Mnd ↦ {⟨(Base‘ndx), {𝑚}⟩, ⟨(Hom ‘ndx), {⟨𝑚, 𝑚, (Base‘𝑚)⟩}⟩, ⟨(comp‘ndx), {⟨⟨𝑚, 𝑚, 𝑚⟩, (+_g‘𝑚)⟩}⟩})
clan 49636	class Lan
cran 49637	class Ran
df-lan 49638	⊢ 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 49639	⊢ 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 49674	class Limit
ccmd 49675	class Colimit
df-lmd 49676	⊢ Limit = (𝑐 ∈ V, 𝑑 ∈ V ↦ (𝑓 ∈ (𝑑 Func 𝑐) ↦ (( oppFunc ‘(𝑐Δ_func𝑑))((oppCat‘𝑐) UP (oppCat‘(𝑑 FuncCat 𝑐)))𝑓)))
df-cmd 49677	⊢ Colimit = (𝑐 ∈ V, 𝑑 ∈ V ↦ (𝑓 ∈ (𝑑 Func 𝑐) ↦ ((𝑐Δ_func𝑑)(𝑐 UP (𝑑 FuncCat 𝑐))𝑓)))
csetrecs 49714	class setrecs(𝐹)
df-setrecs 49715	⊢ setrecs(𝐹) = ∪ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)}
cpg 49740	class Pg
df-pg 49741	⊢ Pg = setrecs((𝑥 ∈ V ↦ (𝒫 𝑥 × 𝒫 𝑥)))
cge-real 49751	class ≥
cgt 49752	class >
df-gte 49753	⊢ ≥ = ^◡ ≤
df-gt 49754	⊢ > = ^◡ <
csinh 49761	class sinh
ccosh 49762	class cosh
ctanh 49763	class tanh
df-sinh 49764	⊢ sinh = (𝑥 ∈ ℂ ↦ ((sin‘(i · 𝑥)) / i))
df-cosh 49765	⊢ cosh = (𝑥 ∈ ℂ ↦ (cos‘(i · 𝑥)))
df-tanh 49766	⊢ tanh = (𝑥 ∈ (^◡cosh “ (ℂ ∖ {0})) ↦ ((tan‘(i · 𝑥)) / i))
csec 49772	class sec
ccsc 49773	class csc
ccot 49774	class cot
df-sec 49775	⊢ sec = (𝑥 ∈ {𝑦 ∈ ℂ ∣ (cos‘𝑦) ≠ 0} ↦ (1 / (cos‘𝑥)))
df-csc 49776	⊢ csc = (𝑥 ∈ {𝑦 ∈ ℂ ∣ (sin‘𝑦) ≠ 0} ↦ (1 / (sin‘𝑥)))
df-cot 49777	⊢ cot = (𝑥 ∈ {𝑦 ∈ ℂ ∣ (sin‘𝑦) ≠ 0} ↦ ((cos‘𝑥) / (sin‘𝑥)))
clog- 49796	class log_
df-logbALT 49797	⊢ log_ = (𝑏 ∈ (ℂ ∖ {0, 1}) ↦ (𝑥 ∈ (ℂ ∖ {0}) ↦ ((log‘𝑥) / (log‘𝑏))))
wreflexive 49798	wff 𝑅Reflexive𝐴
df-reflexive 49799	⊢ (𝑅Reflexive𝐴 ↔ (𝑅 ⊆ (𝐴 × 𝐴) ∧ ∀𝑥 ∈ 𝐴 𝑥𝑅𝑥))
wirreflexive 49800	wff 𝑅Irreflexive𝐴
df-irreflexive 49801	⊢ (𝑅Irreflexive𝐴 ↔ (𝑅 ⊆ (𝐴 × 𝐴) ∧ ∀𝑥 ∈ 𝐴 ¬ 𝑥𝑅𝑥))
walsi 49817	wff ∀!𝑥(𝜑 → 𝜓)
walsc 49818	wff ∀!𝑥 ∈ 𝐴𝜑
df-alsi 49819	⊢ (∀!𝑥(𝜑 → 𝜓) ↔ (∀𝑥(𝜑 → 𝜓) ∧ ∃𝑥𝜑))
df-alsc 49820	⊢ (∀!𝑥 ∈ 𝐴𝜑 ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑥 𝑥 ∈ 𝐴))

W3C validator