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 1491	wff (𝜑 ⊼ 𝜓)
df-nan 1492	⊢ ((𝜑 ⊼ 𝜓) ↔ ¬ (𝜑 ∧ 𝜓))
wxo 1511	wff (𝜑 ⊻ 𝜓)
df-xor 1512	⊢ ((𝜑 ⊻ 𝜓) ↔ ¬ (𝜑 ↔ 𝜓))
wnor 1528	wff (𝜑 ⊽ 𝜓)
df-nor 1529	⊢ ((𝜑 ⊽ 𝜓) ↔ ¬ (𝜑 ∨ 𝜓))
wal 1538	wff ∀𝑥𝜑
cv 1539	class 𝑥
wceq 1540	wff 𝐴 = 𝐵
wtru 1541	wff ⊤
df-tru 1543	⊢ (⊤ ↔ (∀𝑥 𝑥 = 𝑥 → ∀𝑥 𝑥 = 𝑥))
wfal 1552	wff ⊥
df-fal 1553	⊢ (⊥ ↔ ¬ ⊤)
whad 1593	wff hadd(𝜑, 𝜓, 𝜒)
df-had 1594	⊢ (hadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ⊻ 𝜓) ⊻ 𝜒))
wcad 1606	wff cadd(𝜑, 𝜓, 𝜒)
df-cad 1607	⊢ (cadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ∧ 𝜓) ∨ (𝜒 ∧ (𝜑 ⊻ 𝜓))))
wex 1779	wff ∃𝑥𝜑
df-ex 1780	⊢ (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑)
wnf 1783	wff Ⅎ𝑥𝜑
df-nf 1784	⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
ax-gen 1795	⊢ 𝜑 ⇒ ⊢ ∀𝑥𝜑
ax-4 1809	⊢ (∀𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))
ax-5 1910	⊢ (𝜑 → ∀𝑥𝜑)
ax-6 1967	⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦
ax-7 2008	⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 → 𝑦 = 𝑧))
wsb 2065	wff [𝑦 / 𝑥]𝜑
df-sb 2066	⊢ ([𝑡 / 𝑥]𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
wcel 2109	wff 𝐴 ∈ 𝐵
ax-8 2111	⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧))
ax-9 2119	⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦))
ax-10 2142	⊢ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
ax-11 2158	⊢ (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑)
ax-12 2178	⊢ (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
ax-13 2370	⊢ (¬ 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
wmo 2531	wff ∃*𝑥𝜑
df-mo 2533	⊢ (∃*𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))
weu 2561	wff ∃!𝑥𝜑
df-eu 2562	⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃*𝑥𝜑))
ax-ext 2701	⊢ (∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → 𝑥 = 𝑦)
cab 2707	class {𝑥 ∣ 𝜑}
df-clab 2708	⊢ (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ [𝑥 / 𝑦]𝜑)
df-cleq 2721	⊢ (𝑦 = 𝑧 ↔ ∀𝑢(𝑢 ∈ 𝑦 ↔ 𝑢 ∈ 𝑧)) & ⊢ (𝑡 = 𝑡 ↔ ∀𝑣(𝑣 ∈ 𝑡 ↔ 𝑣 ∈ 𝑡)) ⇒ ⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
df-clel 2803	⊢ (𝑦 ∈ 𝑧 ↔ ∃𝑢(𝑢 = 𝑦 ∧ 𝑢 ∈ 𝑧)) & ⊢ (𝑡 ∈ 𝑡 ↔ ∃𝑣(𝑣 = 𝑡 ∧ 𝑣 ∈ 𝑡)) ⇒ ⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵))
wnfc 2876	wff Ⅎ𝑥𝐴
df-nfc 2878	⊢ (Ⅎ𝑥𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴)
wne 2925	wff 𝐴 ≠ 𝐵
df-ne 2926	⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵)
wnel 3029	wff 𝐴 ∉ 𝐵
df-nel 3030	⊢ (𝐴 ∉ 𝐵 ↔ ¬ 𝐴 ∈ 𝐵)
wral 3044	wff ∀𝑥 ∈ 𝐴 𝜑
df-ral 3045	⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
wrex 3053	wff ∃𝑥 ∈ 𝐴 𝜑
df-rex 3054	⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
wreu 3343	wff ∃!𝑥 ∈ 𝐴 𝜑
wrmo 3344	wff ∃*𝑥 ∈ 𝐴 𝜑
df-rmo 3345	⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
df-reu 3346	⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
crab 3396	class {𝑥 ∈ 𝐴 ∣ 𝜑}
df-rab 3397	⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
cvv 3438	class V
df-v 3440	⊢ V = {𝑥 ∣ 𝑥 = 𝑥}
wcdeq 3725	wff CondEq(𝑥 = 𝑦 → 𝜑)
df-cdeq 3726	⊢ (CondEq(𝑥 = 𝑦 → 𝜑) ↔ (𝑥 = 𝑦 → 𝜑))
wsbc 3744	wff [𝐴 / 𝑥]𝜑
df-sbc 3745	⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑})
csb 3853	class ⦋𝐴 / 𝑥⦌𝐵
df-csb 3854	⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵}
cdif 3902	class (𝐴 ∖ 𝐵)
cun 3903	class (𝐴 ∪ 𝐵)
cin 3904	class (𝐴 ∩ 𝐵)
wss 3905	wff 𝐴 ⊆ 𝐵
wpss 3906	wff 𝐴 ⊊ 𝐵
df-dif 3908	⊢ (𝐴 ∖ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)}
df-un 3910	⊢ (𝐴 ∪ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)}
df-in 3912	⊢ (𝐴 ∩ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)}
df-ss 3922	⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
df-pss 3925	⊢ (𝐴 ⊊ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ 𝐵))
csymdif 4205	class (𝐴 △ 𝐵)
df-symdif 4206	⊢ (𝐴 △ 𝐵) = ((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐴))
c0 4286	class ∅
df-nul 4287	⊢ ∅ = (V ∖ V)
cif 4478	class if(𝜑, 𝐴, 𝐵)
df-if 4479	⊢ if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑))}
cpw 4553	class 𝒫 𝐴
df-pw 4555	⊢ 𝒫 𝐴 = {𝑥 ∣ 𝑥 ⊆ 𝐴}
csn 4579	class {𝐴}
df-sn 4580	⊢ {𝐴} = {𝑥 ∣ 𝑥 = 𝐴}
cpr 4581	class {𝐴, 𝐵}
df-pr 4582	⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
ctp 4583	class {𝐴, 𝐵, 𝐶}
df-tp 4584	⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
cop 4585	class ⟨𝐴, 𝐵⟩
df-op 4586	⊢ ⟨𝐴, 𝐵⟩ = {𝑥 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})}
cotp 4587	class ⟨𝐴, 𝐵, 𝐶⟩
df-ot 4588	⊢ ⟨𝐴, 𝐵, 𝐶⟩ = ⟨⟨𝐴, 𝐵⟩, 𝐶⟩
cuni 4861	class ∪ 𝐴
df-uni 4862	⊢ ∪ 𝐴 = {𝑥 ∣ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)}
cint 4899	class ∩ 𝐴
df-int 4900	⊢ ∩ 𝐴 = {𝑥 ∣ ∀𝑦(𝑦 ∈ 𝐴 → 𝑥 ∈ 𝑦)}
ciun 4944	class ∪ 𝑥 ∈ 𝐴 𝐵
ciin 4945	class ∩ 𝑥 ∈ 𝐴 𝐵
df-iun 4946	⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵}
df-iin 4947	⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵}
wdisj 5062	wff Disj 𝑥 ∈ 𝐴 𝐵
df-disj 5063	⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
wbr 5095	wff 𝐴𝑅𝐵
df-br 5096	⊢ (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
copab 5157	class {⟨𝑥, 𝑦⟩ ∣ 𝜑}
df-opab 5158	⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
cmpt 5176	class (𝑥 ∈ 𝐴 ↦ 𝐵)
df-mpt 5177	⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
wtr 5202	wff Tr 𝐴
df-tr 5203	⊢ (Tr 𝐴 ↔ ∪ 𝐴 ⊆ 𝐴)
ax-rep 5221	⊢ (∀𝑤∃𝑦∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦) → ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)))
ax-sep 5238	⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))
ax-nul 5248	⊢ ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥
ax-pow 5307	⊢ ∃𝑦∀𝑧(∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦)
ax-pr 5374	⊢ ∃𝑧∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧)
cid 5517	class I
df-id 5518	⊢ I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
cep 5522	class E
df-eprel 5523	⊢ E = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ 𝑦}
wpo 5529	wff 𝑅 Po 𝐴
wor 5530	wff 𝑅 Or 𝐴
df-po 5531	⊢ (𝑅 Po 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
df-so 5532	⊢ (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)))
wfr 5573	wff 𝑅 Fr 𝐴
wse 5574	wff 𝑅 Se 𝐴
wwe 5575	wff 𝑅 We 𝐴
df-fr 5576	⊢ (𝑅 Fr 𝐴 ↔ ∀𝑥((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦))
df-se 5577	⊢ (𝑅 Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V)
df-we 5578	⊢ (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴 ∧ 𝑅 Or 𝐴))
cxp 5621	class (𝐴 × 𝐵)
ccnv 5622	class ^◡𝐴
cdm 5623	class dom 𝐴
crn 5624	class ran 𝐴
cres 5625	class (𝐴 ↾ 𝐵)
cima 5626	class (𝐴 “ 𝐵)
ccom 5627	class (𝐴 ∘ 𝐵)
wrel 5628	wff Rel 𝐴
df-xp 5629	⊢ (𝐴 × 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)}
df-rel 5630	⊢ (Rel 𝐴 ↔ 𝐴 ⊆ (V × V))
df-cnv 5631	⊢ ^◡𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥}
df-co 5632	⊢ (𝐴 ∘ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)}
df-dm 5633	⊢ dom 𝐴 = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦}
df-rn 5634	⊢ ran 𝐴 = dom ^◡𝐴
df-res 5635	⊢ (𝐴 ↾ 𝐵) = (𝐴 ∩ (𝐵 × V))
df-ima 5636	⊢ (𝐴 “ 𝐵) = ran (𝐴 ↾ 𝐵)
cpred 6252	class Pred(𝑅, 𝐴, 𝑋)
df-pred 6253	⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (^◡𝑅 “ {𝑋}))
word 6310	wff Ord 𝐴
con0 6311	class On
wlim 6312	wff Lim 𝐴
csuc 6313	class suc 𝐴
df-ord 6314	⊢ (Ord 𝐴 ↔ (Tr 𝐴 ∧ E We 𝐴))
df-on 6315	⊢ On = {𝑥 ∣ Ord 𝑥}
df-lim 6316	⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴))
df-suc 6317	⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
cio 6440	class (℩𝑥𝜑)
df-iota 6442	⊢ (℩𝑥𝜑) = ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}
wfun 6480	wff Fun 𝐴
wfn 6481	wff 𝐴 Fn 𝐵
wf 6482	wff 𝐹:𝐴⟶𝐵
wf1 6483	wff 𝐹:𝐴–1-1→𝐵
wfo 6484	wff 𝐹:𝐴–onto→𝐵
wf1o 6485	wff 𝐹:𝐴–1-1-onto→𝐵
cfv 6486	class (𝐹‘𝐴)
wiso 6487	wff 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵)
df-fun 6488	⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ (𝐴 ∘ ^◡𝐴) ⊆ I ))
df-fn 6489	⊢ (𝐴 Fn 𝐵 ↔ (Fun 𝐴 ∧ dom 𝐴 = 𝐵))
df-f 6490	⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵))
df-f1 6491	⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ Fun ^◡𝐹))
df-fo 6492	⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵))
df-f1o 6493	⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹:𝐴–1-1→𝐵 ∧ 𝐹:𝐴–onto→𝐵))
df-fv 6494	⊢ (𝐹‘𝐴) = (℩𝑥𝐴𝐹𝑥)
df-isom 6495	⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐻:𝐴–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦))))
crio 7309	class (℩𝑥 ∈ 𝐴 𝜑)
df-riota 7310	⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
co 7353	class (𝐴𝐹𝐵)
coprab 7354	class {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
cmpo 7355	class (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
df-ov 7356	⊢ (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
df-oprab 7357	⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)}
df-mpo 7358	⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)}
cof 7615	class ∘_f 𝑅
cofr 7616	class ∘_r 𝑅
df-of 7617	⊢ ∘_f 𝑅 = (𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥))))
df-ofr 7618	⊢ ∘_r 𝑅 = {⟨𝑓, 𝑔⟩ ∣ ∀𝑥 ∈ (dom 𝑓 ∩ dom 𝑔)(𝑓‘𝑥)𝑅(𝑔‘𝑥)}
crpss 7662	class [⊊]
df-rpss 7663	⊢ [⊊] = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ⊊ 𝑦}
ax-un 7675	⊢ ∃𝑦∀𝑧(∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦)
com 7806	class ω
df-om 7807	⊢ ω = {𝑥 ∈ On ∣ ∀𝑦(Lim 𝑦 → 𝑥 ∈ 𝑦)}
c1st 7929	class 1^st
c2nd 7930	class 2^nd
df-1st 7931	⊢ 1^st = (𝑥 ∈ V ↦ ∪ dom {𝑥})
df-2nd 7932	⊢ 2^nd = (𝑥 ∈ V ↦ ∪ ran {𝑥})
csupp 8100	class supp
df-supp 8101	⊢ supp = (𝑥 ∈ V, 𝑧 ∈ V ↦ {𝑖 ∈ dom 𝑥 ∣ (𝑥 “ {𝑖}) ≠ {𝑧}})
ctpos 8165	class tpos 𝐹
df-tpos 8166	⊢ tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ (^◡dom 𝐹 ∪ {∅}) ↦ ∪ ^◡{𝑥}))
ccur 8205	class curry 𝐴
cunc 8206	class uncurry 𝐴
df-cur 8207	⊢ curry 𝐹 = (𝑥 ∈ dom dom 𝐹 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩𝐹𝑧})
df-unc 8208	⊢ uncurry 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(𝐹‘𝑥)𝑧}
cund 8212	class Undef
df-undef 8213	⊢ Undef = (𝑠 ∈ V ↦ 𝒫 ∪ 𝑠)
cfrecs 8220	class frecs(𝑅, 𝐴, 𝐹)
df-frecs 8221	⊢ 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 8251	class wrecs(𝑅, 𝐴, 𝐹)
df-wrecs 8252	⊢ wrecs(𝑅, 𝐴, 𝐹) = frecs(𝑅, 𝐴, (𝐹 ∘ 2^nd ))
wsmo 8275	wff Smo 𝐴
df-smo 8276	⊢ (Smo 𝐴 ↔ (𝐴:dom 𝐴⟶On ∧ Ord dom 𝐴 ∧ ∀𝑥 ∈ dom 𝐴∀𝑦 ∈ dom 𝐴(𝑥 ∈ 𝑦 → (𝐴‘𝑥) ∈ (𝐴‘𝑦))))
crecs 8300	class recs(𝐹)
df-recs 8301	⊢ recs(𝐹) = wrecs( E , On, 𝐹)
crdg 8338	class rec(𝐹, 𝐼)
df-rdg 8339	⊢ rec(𝐹, 𝐼) = recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔))))))
cseqom 8376	class seq_ω(𝐹, 𝐼)
df-seqom 8377	⊢ seq_ω(𝐹, 𝐼) = (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) “ ω)
c1o 8388	class 1_o
c2o 8389	class 2_o
c3o 8390	class 3_o
c4o 8391	class 4_o
coa 8392	class +_o
comu 8393	class ·_o
coe 8394	class ↑_o
df-1o 8395	⊢ 1_o = suc ∅
df-2o 8396	⊢ 2_o = suc 1_o
df-3o 8397	⊢ 3_o = suc 2_o
df-4o 8398	⊢ 4_o = suc 3_o
df-oadd 8399	⊢ +_o = (𝑥 ∈ On, 𝑦 ∈ On ↦ (rec((𝑧 ∈ V ↦ suc 𝑧), 𝑥)‘𝑦))
df-omul 8400	⊢ ·_o = (𝑥 ∈ On, 𝑦 ∈ On ↦ (rec((𝑧 ∈ V ↦ (𝑧 +_o 𝑥)), ∅)‘𝑦))
df-oexp 8401	⊢ ↑_o = (𝑥 ∈ On, 𝑦 ∈ On ↦ if(𝑥 = ∅, (1_o ∖ 𝑦), (rec((𝑧 ∈ V ↦ (𝑧 ·_o 𝑥)), 1_o)‘𝑦)))
cnadd 8590	class +no
df-nadd 8591	⊢ +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 8629	wff 𝑅 Er 𝐴
cec 8630	class [𝐴]𝑅
cqs 8631	class (𝐴 / 𝑅)
df-er 8632	⊢ (𝑅 Er 𝐴 ↔ (Rel 𝑅 ∧ dom 𝑅 = 𝐴 ∧ (^◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅))
df-ec 8634	⊢ [𝐴]𝑅 = (𝑅 “ {𝐴})
df-qs 8638	⊢ (𝐴 / 𝑅) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅}
cmap 8760	class ↑_m
cpm 8761	class ↑_pm
df-map 8762	⊢ ↑_m = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓 ∣ 𝑓:𝑦⟶𝑥})
df-pm 8763	⊢ ↑_pm = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓 ∈ 𝒫 (𝑦 × 𝑥) ∣ Fun 𝑓})
cixp 8831	class X𝑥 ∈ 𝐴 𝐵
df-ixp 8832	⊢ X𝑥 ∈ 𝐴 𝐵 = {𝑓 ∣ (𝑓 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵)}
cen 8876	class ≈
cdom 8877	class ≼
csdm 8878	class ≺
cfn 8879	class Fin
df-en 8880	⊢ ≈ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥–1-1-onto→𝑦}
df-dom 8881	⊢ ≼ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥–1-1→𝑦}
df-sdom 8882	⊢ ≺ = ( ≼ ∖ ≈ )
df-fin 8883	⊢ Fin = {𝑥 ∣ ∃𝑦 ∈ ω 𝑥 ≈ 𝑦}
cfsupp 9270	class finSupp
df-fsupp 9271	⊢ finSupp = {⟨𝑟, 𝑧⟩ ∣ (Fun 𝑟 ∧ (𝑟 supp 𝑧) ∈ Fin)}
cfi 9319	class fi
df-fi 9320	⊢ fi = (𝑥 ∈ V ↦ {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦})
csup 9349	class sup(𝐴, 𝐵, 𝑅)
cinf 9350	class inf(𝐴, 𝐵, 𝑅)
df-sup 9351	⊢ sup(𝐴, 𝐵, 𝑅) = ∪ {𝑥 ∈ 𝐵 ∣ (∀𝑦 ∈ 𝐴 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐵 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐴 𝑦𝑅𝑧))}
df-inf 9352	⊢ inf(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, ^◡𝑅)
coi 9420	class OrdIso(𝑅, 𝐴)
df-oi 9421	⊢ 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 9467	class har
df-har 9468	⊢ har = (𝑥 ∈ V ↦ {𝑦 ∈ On ∣ 𝑦 ≼ 𝑥})
cwdom 9475	class ≼^*
df-wdom 9476	⊢ ≼^* = {⟨𝑥, 𝑦⟩ ∣ (𝑥 = ∅ ∨ ∃𝑧 𝑧:𝑦–onto→𝑥)}
ax-reg 9503	⊢ (∃𝑦 𝑦 ∈ 𝑥 → ∃𝑦(𝑦 ∈ 𝑥 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑥)))
ax-inf 9553	⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦)))
ax-inf2 9556	⊢ ∃𝑥(∃𝑦(𝑦 ∈ 𝑥 ∧ ∀𝑧 ¬ 𝑧 ∈ 𝑦) ∧ ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑧 ∈ 𝑥 ∧ ∀𝑤(𝑤 ∈ 𝑧 ↔ (𝑤 ∈ 𝑦 ∨ 𝑤 = 𝑦)))))
ccnf 9576	class CNF
df-cnf 9577	⊢ 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 9622	class t++𝑅
df-ttrcl 9623	⊢ 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 9651	class TC
df-tc 9652	⊢ TC = (𝑥 ∈ V ↦ ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ Tr 𝑦)})
cr1 9677	class 𝑅₁
crnk 9678	class rank
df-r1 9679	⊢ 𝑅₁ = rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅)
df-rank 9680	⊢ rank = (𝑥 ∈ V ↦ ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅₁‘suc 𝑦)})
cdju 9813	class (𝐴 ⊔ 𝐵)
cinl 9814	class inl
cinr 9815	class inr
df-dju 9816	⊢ (𝐴 ⊔ 𝐵) = (({∅} × 𝐴) ∪ ({1_o} × 𝐵))
df-inl 9817	⊢ inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩)
df-inr 9818	⊢ inr = (𝑥 ∈ V ↦ ⟨1_o, 𝑥⟩)
ccrd 9850	class card
cale 9851	class ℵ
ccf 9852	class cf
wacn 9853	class AC 𝐴
df-card 9854	⊢ card = (𝑥 ∈ V ↦ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥})
df-aleph 9855	⊢ ℵ = rec(har, ω)
df-cf 9856	⊢ cf = (𝑥 ∈ On ↦ ∩ {𝑦 ∣ ∃𝑧(𝑦 = (card‘𝑧) ∧ (𝑧 ⊆ 𝑥 ∧ ∀𝑣 ∈ 𝑥 ∃𝑢 ∈ 𝑧 𝑣 ⊆ 𝑢))})
df-acn 9857	⊢ AC 𝐴 = {𝑥 ∣ (𝐴 ∈ V ∧ ∀𝑓 ∈ ((𝒫 𝑥 ∖ {∅}) ↑_m 𝐴)∃𝑔∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝑓‘𝑦))}
wac 10028	wff CHOICE
df-ac 10029	⊢ (CHOICE ↔ ∀𝑥∃𝑓(𝑓 ⊆ 𝑥 ∧ 𝑓 Fn dom 𝑥))
cfin1a 10191	class Fin^Ia
cfin2 10192	class Fin^II
cfin4 10193	class Fin^IV
cfin3 10194	class Fin^III
cfin5 10195	class Fin^V
cfin6 10196	class Fin^VI
cfin7 10197	class Fin^VII
df-fin1a 10198	⊢ Fin^Ia = {𝑥 ∣ ∀𝑦 ∈ 𝒫 𝑥(𝑦 ∈ Fin ∨ (𝑥 ∖ 𝑦) ∈ Fin)}
df-fin2 10199	⊢ Fin^II = {𝑥 ∣ ∀𝑦 ∈ 𝒫 𝒫 𝑥((𝑦 ≠ ∅ ∧ [⊊] Or 𝑦) → ∪ 𝑦 ∈ 𝑦)}
df-fin4 10200	⊢ Fin^IV = {𝑥 ∣ ¬ ∃𝑦(𝑦 ⊊ 𝑥 ∧ 𝑦 ≈ 𝑥)}
df-fin3 10201	⊢ Fin^III = {𝑥 ∣ 𝒫 𝑥 ∈ Fin^IV}
df-fin5 10202	⊢ Fin^V = {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 ≺ (𝑥 ⊔ 𝑥))}
df-fin6 10203	⊢ Fin^VI = {𝑥 ∣ (𝑥 ≺ 2_o ∨ 𝑥 ≺ (𝑥 × 𝑥))}
df-fin7 10204	⊢ Fin^VII = {𝑥 ∣ ¬ ∃𝑦 ∈ (On ∖ ω)𝑥 ≈ 𝑦}
ax-cc 10348	⊢ (𝑥 ≈ ω → ∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧))
ax-dc 10359	⊢ ((∃𝑦∃𝑧 𝑦𝑥𝑧 ∧ ran 𝑥 ⊆ dom 𝑥) → ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛))
ax-ac 10372	⊢ ∃𝑦∀𝑧∀𝑤((𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → ∃𝑣∀𝑢(∃𝑡((𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑡) ∧ (𝑢 ∈ 𝑡 ∧ 𝑡 ∈ 𝑦)) ↔ 𝑢 = 𝑣))
ax-ac2 10376	⊢ ∃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 10533	class GCH
df-gch 10534	⊢ GCH = (Fin ∪ {𝑥 ∣ ∀𝑦 ¬ (𝑥 ≺ 𝑦 ∧ 𝑦 ≺ 𝒫 𝑥)})
cwina 10595	class Inacc_w
cina 10596	class Inacc
df-wina 10597	⊢ Inacc_w = {𝑥 ∣ (𝑥 ≠ ∅ ∧ (cf‘𝑥) = 𝑥 ∧ ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝑥 𝑦 ≺ 𝑧)}
df-ina 10598	⊢ Inacc = {𝑥 ∣ (𝑥 ≠ ∅ ∧ (cf‘𝑥) = 𝑥 ∧ ∀𝑦 ∈ 𝑥 𝒫 𝑦 ≺ 𝑥)}
cwun 10613	class WUni
cwunm 10614	class wUniCl
df-wun 10615	⊢ WUni = {𝑢 ∣ (Tr 𝑢 ∧ 𝑢 ≠ ∅ ∧ ∀𝑥 ∈ 𝑢 (∪ 𝑥 ∈ 𝑢 ∧ 𝒫 𝑥 ∈ 𝑢 ∧ ∀𝑦 ∈ 𝑢 {𝑥, 𝑦} ∈ 𝑢))}
df-wunc 10616	⊢ wUniCl = (𝑥 ∈ V ↦ ∩ {𝑢 ∈ WUni ∣ 𝑥 ⊆ 𝑢})
ctsk 10661	class Tarski
df-tsk 10662	⊢ Tarski = {𝑦 ∣ (∀𝑧 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑦 ∧ ∃𝑤 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧 ≈ 𝑦 ∨ 𝑧 ∈ 𝑦))}
cgru 10703	class Univ
df-gru 10704	⊢ Univ = {𝑢 ∣ (Tr 𝑢 ∧ ∀𝑥 ∈ 𝑢 (𝒫 𝑥 ∈ 𝑢 ∧ ∀𝑦 ∈ 𝑢 {𝑥, 𝑦} ∈ 𝑢 ∧ ∀𝑦 ∈ (𝑢 ↑_m 𝑥)∪ ran 𝑦 ∈ 𝑢))}
ax-groth 10736	⊢ ∃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 10750	class tarskiMap
df-tskm 10751	⊢ tarskiMap = (𝑥 ∈ V ↦ ∩ {𝑦 ∈ Tarski ∣ 𝑥 ∈ 𝑦})
cnpi 10757	class N
cpli 10758	class +_N
cmi 10759	class ·_N
clti 10760	class <_N
cplpq 10761	class +_pQ
cmpq 10762	class ·_pQ
cltpq 10763	class <_pQ
ceq 10764	class ~_Q
cnq 10765	class Q
c1q 10766	class 1_Q
cerq 10767	class [Q]
cplq 10768	class +_Q
cmq 10769	class ·_Q
crq 10770	class *_Q
cltq 10771	class <_Q
cnp 10772	class P
c1p 10773	class 1_P
cpp 10774	class +_P
cmp 10775	class ·_P
cltp 10776	class <_P
cer 10777	class ~_R
cnr 10778	class R
c0r 10779	class 0_R
c1r 10780	class 1_R
cm1r 10781	class -1_R
cplr 10782	class +_R
cmr 10783	class ·_R
cltr 10784	class <_R
df-ni 10785	⊢ N = (ω ∖ {∅})
df-pli 10786	⊢ +_N = ( +_o ↾ (N × N))
df-mi 10787	⊢ ·_N = ( ·_o ↾ (N × N))
df-lti 10788	⊢ <_N = ( E ∩ (N × N))
df-plpq 10821	⊢ +_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 10822	⊢ ·_pQ = (𝑥 ∈ (N × N), 𝑦 ∈ (N × N) ↦ ⟨((1^st ‘𝑥) ·_N (1^st ‘𝑦)), ((2^nd ‘𝑥) ·_N (2^nd ‘𝑦))⟩)
df-ltpq 10823	⊢ <_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 10824	⊢ ~_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 10825	⊢ Q = {𝑥 ∈ (N × N) ∣ ∀𝑦 ∈ (N × N)(𝑥 ~_Q 𝑦 → ¬ (2^nd ‘𝑦) <_N (2^nd ‘𝑥))}
df-erq 10826	⊢ [Q] = ( ~_Q ∩ ((N × N) × Q))
df-plq 10827	⊢ +_Q = (([Q] ∘ +_pQ ) ↾ (Q × Q))
df-mq 10828	⊢ ·_Q = (([Q] ∘ ·_pQ ) ↾ (Q × Q))
df-1nq 10829	⊢ 1_Q = ⟨1_o, 1_o⟩
df-rq 10830	⊢ *_Q = (^◡ ·_Q “ {1_Q})
df-ltnq 10831	⊢ <_Q = ( <_pQ ∩ (Q × Q))
df-np 10894	⊢ P = {𝑥 ∣ ((∅ ⊊ 𝑥 ∧ 𝑥 ⊊ Q) ∧ ∀𝑦 ∈ 𝑥 (∀𝑧(𝑧 <_Q 𝑦 → 𝑧 ∈ 𝑥) ∧ ∃𝑧 ∈ 𝑥 𝑦 <_Q 𝑧))}
df-1p 10895	⊢ 1_P = {𝑥 ∣ 𝑥 <_Q 1_Q}
df-plp 10896	⊢ +_P = (𝑥 ∈ P, 𝑦 ∈ P ↦ {𝑤 ∣ ∃𝑣 ∈ 𝑥 ∃𝑢 ∈ 𝑦 𝑤 = (𝑣 +_Q 𝑢)})
df-mp 10897	⊢ ·_P = (𝑥 ∈ P, 𝑦 ∈ P ↦ {𝑤 ∣ ∃𝑣 ∈ 𝑥 ∃𝑢 ∈ 𝑦 𝑤 = (𝑣 ·_Q 𝑢)})
df-ltp 10898	⊢ <_P = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ 𝑥 ⊊ 𝑦)}
df-enr 10968	⊢ ~_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 10969	⊢ R = ((P × P) / ~_R )
df-plr 10970	⊢ +_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 10971	⊢ ·_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 10972	⊢ <_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 10973	⊢ 0_R = [⟨1_P, 1_P⟩] ~_R
df-1r 10974	⊢ 1_R = [⟨(1_P +_P 1_P), 1_P⟩] ~_R
df-m1r 10975	⊢ -1_R = [⟨1_P, (1_P +_P 1_P)⟩] ~_R
cc 11026	class ℂ
cr 11027	class ℝ
cc0 11028	class 0
c1 11029	class 1
ci 11030	class i
caddc 11031	class +
cltrr 11032	class <_ℝ
cmul 11033	class ·
df-c 11034	⊢ ℂ = (R × R)
df-0 11035	⊢ 0 = ⟨0_R, 0_R⟩
df-1 11036	⊢ 1 = ⟨1_R, 0_R⟩
df-i 11037	⊢ i = ⟨0_R, 1_R⟩
df-r 11038	⊢ ℝ = (R × {0_R})
df-add 11039	⊢ + = {⟨⟨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 11040	⊢ · = {⟨⟨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 11041	⊢ <_ℝ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ∃𝑧∃𝑤((𝑥 = ⟨𝑧, 0_R⟩ ∧ 𝑦 = ⟨𝑤, 0_R⟩) ∧ 𝑧 <_R 𝑤))}
ax-cnex 11084	⊢ ℂ ∈ V
ax-resscn 11085	⊢ ℝ ⊆ ℂ
ax-1cn 11086	⊢ 1 ∈ ℂ
ax-icn 11087	⊢ i ∈ ℂ
ax-addcl 11088	⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) ∈ ℂ)
ax-addrcl 11089	⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + 𝐵) ∈ ℝ)
ax-mulcl 11090	⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) ∈ ℂ)
ax-mulrcl 11091	⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 · 𝐵) ∈ ℝ)
ax-mulcom 11092	⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) = (𝐵 · 𝐴))
ax-addass 11093	⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶)))
ax-mulass 11094	⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)))
ax-distr 11095	⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶)))
ax-i2m1 11096	⊢ ((i · i) + 1) = 0
ax-1ne0 11097	⊢ 1 ≠ 0
ax-1rid 11098	⊢ (𝐴 ∈ ℝ → (𝐴 · 1) = 𝐴)
ax-rnegex 11099	⊢ (𝐴 ∈ ℝ → ∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0)
ax-rrecex 11100	⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → ∃𝑥 ∈ ℝ (𝐴 · 𝑥) = 1)
ax-cnre 11101	⊢ (𝐴 ∈ ℂ → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝐴 = (𝑥 + (i · 𝑦)))
ax-pre-lttri 11102	⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 <_ℝ 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 <_ℝ 𝐴)))
ax-pre-lttrn 11103	⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 <_ℝ 𝐵 ∧ 𝐵 <_ℝ 𝐶) → 𝐴 <_ℝ 𝐶))
ax-pre-ltadd 11104	⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 <_ℝ 𝐵 → (𝐶 + 𝐴) <_ℝ (𝐶 + 𝐵)))
ax-pre-mulgt0 11105	⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((0 <_ℝ 𝐴 ∧ 0 <_ℝ 𝐵) → 0 <_ℝ (𝐴 · 𝐵)))
ax-pre-sup 11106	⊢ ((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 11107	⊢ + :(ℂ × ℂ)⟶ℂ
ax-mulf 11108	⊢ · :(ℂ × ℂ)⟶ℂ
cpnf 11165	class +∞
cmnf 11166	class -∞
cxr 11167	class ℝ^*
clt 11168	class <
cle 11169	class ≤
df-pnf 11170	⊢ +∞ = 𝒫 ∪ ℂ
df-mnf 11171	⊢ -∞ = 𝒫 +∞
df-xr 11172	⊢ ℝ^* = (ℝ ∪ {+∞, -∞})
df-ltxr 11173	⊢ < = ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <_ℝ 𝑦)} ∪ (((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ)))
df-le 11174	⊢ ≤ = ((ℝ^* × ℝ^*) ∖ ^◡ < )
cmin 11365	class −
cneg 11366	class -𝐴
df-sub 11367	⊢ − = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥))
df-neg 11368	⊢ -𝐴 = (0 − 𝐴)
cdiv 11795	class /
df-div 11796	⊢ / = (𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥))
cn 12146	class ℕ
df-nn 12147	⊢ ℕ = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) “ ω)
c2 12201	class 2
c3 12202	class 3
c4 12203	class 4
c5 12204	class 5
c6 12205	class 6
c7 12206	class 7
c8 12207	class 8
c9 12208	class 9
df-2 12209	⊢ 2 = (1 + 1)
df-3 12210	⊢ 3 = (2 + 1)
df-4 12211	⊢ 4 = (3 + 1)
df-5 12212	⊢ 5 = (4 + 1)
df-6 12213	⊢ 6 = (5 + 1)
df-7 12214	⊢ 7 = (6 + 1)
df-8 12215	⊢ 8 = (7 + 1)
df-9 12216	⊢ 9 = (8 + 1)
cn0 12402	class ℕ₀
df-n0 12403	⊢ ℕ₀ = (ℕ ∪ {0})
cxnn0 12475	class ℕ₀^*
df-xnn0 12476	⊢ ℕ₀^* = (ℕ₀ ∪ {+∞})
cz 12489	class ℤ
df-z 12490	⊢ ℤ = {𝑛 ∈ ℝ ∣ (𝑛 = 0 ∨ 𝑛 ∈ ℕ ∨ -𝑛 ∈ ℕ)}
cdc 12609	class ;𝐴𝐵
df-dec 12610	⊢ ;𝐴𝐵 = (((9 + 1) · 𝐴) + 𝐵)
cuz 12753	class ℤ_≥
df-uz 12754	⊢ ℤ_≥ = (𝑗 ∈ ℤ ↦ {𝑘 ∈ ℤ ∣ 𝑗 ≤ 𝑘})
cq 12867	class ℚ
df-q 12868	⊢ ℚ = ( / “ (ℤ × ℕ))
crp 12911	class ℝ⁺
df-rp 12912	⊢ ℝ⁺ = {𝑥 ∈ ℝ ∣ 0 < 𝑥}
cxne 13029	class -_𝑒𝐴
cxad 13030	class +_𝑒
cxmu 13031	class ·_e
df-xneg 13032	⊢ -_𝑒𝐴 = if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴))
df-xadd 13033	⊢ +_𝑒 = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ if(𝑥 = +∞, if(𝑦 = -∞, 0, +∞), if(𝑥 = -∞, if(𝑦 = +∞, 0, -∞), if(𝑦 = +∞, +∞, if(𝑦 = -∞, -∞, (𝑥 + 𝑦))))))
df-xmul 13034	⊢ ·_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 13266	class (,)
cioc 13267	class (,]
cico 13268	class [,)
cicc 13269	class [,]
df-ioo 13270	⊢ (,) = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)})
df-ioc 13271	⊢ (,] = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 < 𝑧 ∧ 𝑧 ≤ 𝑦)})
df-ico 13272	⊢ [,) = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)})
df-icc 13273	⊢ [,] = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)})
cfz 13428	class ...
df-fz 13429	⊢ ... = (𝑚 ∈ ℤ, 𝑛 ∈ ℤ ↦ {𝑘 ∈ ℤ ∣ (𝑚 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛)})
cfzo 13575	class ..^
df-fzo 13576	⊢ ..^ = (𝑚 ∈ ℤ, 𝑛 ∈ ℤ ↦ (𝑚...(𝑛 − 1)))
cfl 13712	class ⌊
cceil 13713	class ⌈
df-fl 13714	⊢ ⌊ = (𝑥 ∈ ℝ ↦ (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))))
df-ceil 13715	⊢ ⌈ = (𝑥 ∈ ℝ ↦ -(⌊‘-𝑥))
cmo 13791	class mod
df-mod 13792	⊢ mod = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ⁺ ↦ (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦)))))
cseq 13926	class seq𝑀( + , 𝐹)
df-seq 13927	⊢ seq𝑀( + , 𝐹) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) “ ω)
cexp 13986	class ↑
df-exp 13987	⊢ ↑ = (𝑥 ∈ ℂ, 𝑦 ∈ ℤ ↦ if(𝑦 = 0, 1, if(0 < 𝑦, (seq1( · , (ℕ × {𝑥}))‘𝑦), (1 / (seq1( · , (ℕ × {𝑥}))‘-𝑦)))))
cfa 14198	class !
df-fac 14199	⊢ ! = ({⟨0, 1⟩} ∪ seq1( · , I ))
cbc 14227	class C
df-bc 14228	⊢ C = (𝑛 ∈ ℕ₀, 𝑘 ∈ ℤ ↦ if(𝑘 ∈ (0...𝑛), ((!‘𝑛) / ((!‘(𝑛 − 𝑘)) · (!‘𝑘))), 0))
chash 14255	class ♯
df-hash 14256	⊢ ♯ = (((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω) ∘ card) ∪ ((V ∖ Fin) × {+∞}))
cword 14438	class Word 𝑆
df-word 14439	⊢ Word 𝑆 = {𝑤 ∣ ∃𝑙 ∈ ℕ₀ 𝑤:(0..^𝑙)⟶𝑆}
clsw 14487	class lastS
df-lsw 14488	⊢ lastS = (𝑤 ∈ V ↦ (𝑤‘((♯‘𝑤) − 1)))
cconcat 14495	class ++
df-concat 14496	⊢ ++ = (𝑠 ∈ V, 𝑡 ∈ V ↦ (𝑥 ∈ (0..^((♯‘𝑠) + (♯‘𝑡))) ↦ if(𝑥 ∈ (0..^(♯‘𝑠)), (𝑠‘𝑥), (𝑡‘(𝑥 − (♯‘𝑠))))))
cs1 14520	class ⟨“𝐴”⟩
df-s1 14521	⊢ ⟨“𝐴”⟩ = {⟨0, ( I ‘𝐴)⟩}
csubstr 14565	class substr
df-substr 14566	⊢ 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 14595	class prefix
df-pfx 14596	⊢ prefix = (𝑠 ∈ V, 𝑙 ∈ ℕ₀ ↦ (𝑠 substr ⟨0, 𝑙⟩))
csplice 14673	class splice
df-splice 14674	⊢ splice = (𝑠 ∈ V, 𝑏 ∈ V ↦ (((𝑠 prefix (1^st ‘(1^st ‘𝑏))) ++ (2^nd ‘𝑏)) ++ (𝑠 substr ⟨(2^nd ‘(1^st ‘𝑏)), (♯‘𝑠)⟩)))
creverse 14682	class reverse
df-reverse 14683	⊢ reverse = (𝑠 ∈ V ↦ (𝑥 ∈ (0..^(♯‘𝑠)) ↦ (𝑠‘(((♯‘𝑠) − 1) − 𝑥))))
creps 14692	class repeatS
df-reps 14693	⊢ repeatS = (𝑠 ∈ V, 𝑛 ∈ ℕ₀ ↦ (𝑥 ∈ (0..^𝑛) ↦ 𝑠))
ccsh 14712	class cyclShift
df-csh 14713	⊢ 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 14766	class ⟨“𝐴𝐵”⟩
cs3 14767	class ⟨“𝐴𝐵𝐶”⟩
cs4 14768	class ⟨“𝐴𝐵𝐶𝐷”⟩
cs5 14769	class ⟨“𝐴𝐵𝐶𝐷𝐸”⟩
cs6 14770	class ⟨“𝐴𝐵𝐶𝐷𝐸𝐹”⟩
cs7 14771	class ⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”⟩
cs8 14772	class ⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺𝐻”⟩
df-s2 14773	⊢ ⟨“𝐴𝐵”⟩ = (⟨“𝐴”⟩ ++ ⟨“𝐵”⟩)
df-s3 14774	⊢ ⟨“𝐴𝐵𝐶”⟩ = (⟨“𝐴𝐵”⟩ ++ ⟨“𝐶”⟩)
df-s4 14775	⊢ ⟨“𝐴𝐵𝐶𝐷”⟩ = (⟨“𝐴𝐵𝐶”⟩ ++ ⟨“𝐷”⟩)
df-s5 14776	⊢ ⟨“𝐴𝐵𝐶𝐷𝐸”⟩ = (⟨“𝐴𝐵𝐶𝐷”⟩ ++ ⟨“𝐸”⟩)
df-s6 14777	⊢ ⟨“𝐴𝐵𝐶𝐷𝐸𝐹”⟩ = (⟨“𝐴𝐵𝐶𝐷𝐸”⟩ ++ ⟨“𝐹”⟩)
df-s7 14778	⊢ ⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”⟩ = (⟨“𝐴𝐵𝐶𝐷𝐸𝐹”⟩ ++ ⟨“𝐺”⟩)
df-s8 14779	⊢ ⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺𝐻”⟩ = (⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”⟩ ++ ⟨“𝐻”⟩)
ctcl 14910	class t+
crtcl 14911	class t*
df-trcl 14912	⊢ t+ = (𝑥 ∈ V ↦ ∩ {𝑧 ∣ (𝑥 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)})
df-rtrcl 14913	⊢ t* = (𝑥 ∈ V ↦ ∩ {𝑧 ∣ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧 ∧ 𝑥 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)})
crelexp 14944	class ↑_𝑟
df-relexp 14945	⊢ ↑_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 14980	class t*rec
df-rtrclrec 14981	⊢ t*rec = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ₀ (𝑟↑_𝑟𝑛))
cshi 14991	class shift
df-shft 14992	⊢ shift = (𝑓 ∈ V, 𝑥 ∈ ℂ ↦ {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ ℂ ∧ (𝑦 − 𝑥)𝑓𝑧)})
csgn 15011	class sgn
df-sgn 15012	⊢ sgn = (𝑥 ∈ ℝ^* ↦ if(𝑥 = 0, 0, if(𝑥 < 0, -1, 1)))
ccj 15021	class ∗
cre 15022	class ℜ
cim 15023	class ℑ
df-cj 15024	⊢ ∗ = (𝑥 ∈ ℂ ↦ (℩𝑦 ∈ ℂ ((𝑥 + 𝑦) ∈ ℝ ∧ (i · (𝑥 − 𝑦)) ∈ ℝ)))
df-re 15025	⊢ ℜ = (𝑥 ∈ ℂ ↦ ((𝑥 + (∗‘𝑥)) / 2))
df-im 15026	⊢ ℑ = (𝑥 ∈ ℂ ↦ (ℜ‘(𝑥 / i)))
csqrt 15158	class √
cabs 15159	class abs
df-sqrt 15160	⊢ √ = (𝑥 ∈ ℂ ↦ (℩𝑦 ∈ ℂ ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ⁺)))
df-abs 15161	⊢ abs = (𝑥 ∈ ℂ ↦ (√‘(𝑥 · (∗‘𝑥))))
clsp 15395	class lim sup
df-limsup 15396	⊢ lim sup = (𝑥 ∈ V ↦ inf(ran (𝑘 ∈ ℝ ↦ sup(((𝑥 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )), ℝ^*, < ))
cli 15409	class ⇝
crli 15410	class ⇝_𝑟
co1 15411	class 𝑂(1)
clo1 15412	class ≤𝑂(1)
df-clim 15413	⊢ ⇝ = {⟨𝑓, 𝑦⟩ ∣ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝑓‘𝑘) ∈ ℂ ∧ (abs‘((𝑓‘𝑘) − 𝑦)) < 𝑥))}
df-rlim 15414	⊢ ⇝_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 15415	⊢ 𝑂(1) = {𝑓 ∈ (ℂ ↑_pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(abs‘(𝑓‘𝑦)) ≤ 𝑚}
df-lo1 15416	⊢ ≤𝑂(1) = {𝑓 ∈ (ℝ ↑_pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(𝑓‘𝑦) ≤ 𝑚}
csu 15611	class Σ𝑘 ∈ 𝐴 𝐵
df-sum 15612	⊢ 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 15828	class ∏𝑘 ∈ 𝐴 𝐵
df-prod 15829	⊢ ∏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 15929	class FallFac
crisefac 15930	class RiseFac
df-risefac 15931	⊢ RiseFac = (𝑥 ∈ ℂ, 𝑛 ∈ ℕ₀ ↦ ∏𝑘 ∈ (0...(𝑛 − 1))(𝑥 + 𝑘))
df-fallfac 15932	⊢ FallFac = (𝑥 ∈ ℂ, 𝑛 ∈ ℕ₀ ↦ ∏𝑘 ∈ (0...(𝑛 − 1))(𝑥 − 𝑘))
cbp 15971	class BernPoly
df-bpoly 15972	⊢ 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 15986	class exp
ceu 15987	class e
csin 15988	class sin
ccos 15989	class cos
ctan 15990	class tan
cpi 15991	class π
df-ef 15992	⊢ exp = (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ ℕ₀ ((𝑥↑𝑘) / (!‘𝑘)))
df-e 15993	⊢ e = (exp‘1)
df-sin 15994	⊢ sin = (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) − (exp‘(-i · 𝑥))) / (2 · i)))
df-cos 15995	⊢ cos = (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥))) / 2))
df-tan 15996	⊢ tan = (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) ↦ ((sin‘𝑥) / (cos‘𝑥)))
df-pi 15997	⊢ π = inf((ℝ⁺ ∩ (^◡sin “ {0})), ℝ, < )
ctau 16129	class τ
df-tau 16130	⊢ τ = inf((ℝ⁺ ∩ (^◡cos “ {1})), ℝ, < )
cdvds 16181	class ∥
df-dvds 16182	⊢ ∥ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ ∃𝑛 ∈ ℤ (𝑛 · 𝑥) = 𝑦)}
cbits 16348	class bits
csad 16349	class sadd
csmu 16350	class smul
df-bits 16351	⊢ bits = (𝑛 ∈ ℤ ↦ {𝑚 ∈ ℕ₀ ∣ ¬ 2 ∥ (⌊‘(𝑛 / (2↑𝑚)))})
df-sad 16380	⊢ 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 16405	⊢ 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 16423	class gcd
df-gcd 16424	⊢ gcd = (𝑥 ∈ ℤ, 𝑦 ∈ ℤ ↦ if((𝑥 = 0 ∧ 𝑦 = 0), 0, sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑥 ∧ 𝑛 ∥ 𝑦)}, ℝ, < )))
clcm 16517	class lcm
clcmf 16518	class lcm
df-lcm 16519	⊢ lcm = (𝑥 ∈ ℤ, 𝑦 ∈ ℤ ↦ if((𝑥 = 0 ∨ 𝑦 = 0), 0, inf({𝑛 ∈ ℕ ∣ (𝑥 ∥ 𝑛 ∧ 𝑦 ∥ 𝑛)}, ℝ, < )))
df-lcmf 16520	⊢ lcm = (𝑧 ∈ 𝒫 ℤ ↦ if(0 ∈ 𝑧, 0, inf({𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑧 𝑚 ∥ 𝑛}, ℝ, < )))
cprime 16600	class ℙ
df-prm 16601	⊢ ℙ = {𝑝 ∈ ℕ ∣ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑝} ≈ 2_o}
cnumer 16662	class numer
cdenom 16663	class denom
df-numer 16664	⊢ 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 16665	⊢ denom = (𝑦 ∈ ℚ ↦ (2^nd ‘(℩𝑥 ∈ (ℤ × ℕ)(((1^st ‘𝑥) gcd (2^nd ‘𝑥)) = 1 ∧ 𝑦 = ((1^st ‘𝑥) / (2^nd ‘𝑥))))))
codz 16692	class od_ℤ
cphi 16693	class ϕ
df-odz 16694	⊢ od_ℤ = (𝑛 ∈ ℕ ↦ (𝑥 ∈ {𝑥 ∈ ℤ ∣ (𝑥 gcd 𝑛) = 1} ↦ inf({𝑚 ∈ ℕ ∣ 𝑛 ∥ ((𝑥↑𝑚) − 1)}, ℝ, < )))
df-phi 16695	⊢ ϕ = (𝑛 ∈ ℕ ↦ (♯‘{𝑥 ∈ (1...𝑛) ∣ (𝑥 gcd 𝑛) = 1}))
cpc 16766	class pCnt
df-pc 16767	⊢ 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 16859	class ℤ[i]
df-gz 16860	⊢ ℤ[i] = {𝑥 ∈ ℂ ∣ ((ℜ‘𝑥) ∈ ℤ ∧ (ℑ‘𝑥) ∈ ℤ)}
cvdwa 16895	class AP
cvdwm 16896	class MonoAP
cvdwp 16897	class PolyAP
df-vdwap 16898	⊢ AP = (𝑘 ∈ ℕ₀ ↦ (𝑎 ∈ ℕ, 𝑑 ∈ ℕ ↦ ran (𝑚 ∈ (0...(𝑘 − 1)) ↦ (𝑎 + (𝑚 · 𝑑)))))
df-vdwmc 16899	⊢ MonoAP = {⟨𝑘, 𝑓⟩ ∣ ∃𝑐(ran (AP‘𝑘) ∩ 𝒫 (^◡𝑓 “ {𝑐})) ≠ ∅}
df-vdwpc 16900	⊢ 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 16929	class Ramsey
df-ram 16931	⊢ 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 16961	class #_p
df-prmo 16962	⊢ #_p = (𝑛 ∈ ℕ₀ ↦ ∏𝑘 ∈ (1...𝑛)if(𝑘 ∈ ℙ, 𝑘, 1))
cstr 17075	class Struct
df-struct 17076	⊢ Struct = {⟨𝑓, 𝑥⟩ ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))}
csts 17092	class sSet
df-sets 17093	⊢ sSet = (𝑠 ∈ V, 𝑒 ∈ V ↦ ((𝑠 ↾ (V ∖ dom {𝑒})) ∪ {𝑒}))
cslot 17110	class Slot 𝐴
df-slot 17111	⊢ Slot 𝐴 = (𝑥 ∈ V ↦ (𝑥‘𝐴))
cnx 17122	class ndx
df-ndx 17123	⊢ ndx = ( I ↾ ℕ)
cbs 17138	class Base
df-base 17139	⊢ Base = Slot 1
cress 17159	class ↾_s
df-ress 17160	⊢ ↾_s = (𝑤 ∈ V, 𝑥 ∈ V ↦ if((Base‘𝑤) ⊆ 𝑥, 𝑤, (𝑤 sSet ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩)))
cplusg 17179	class +_g
cmulr 17180	class ._r
cstv 17181	class *_𝑟
csca 17182	class Scalar
cvsca 17183	class ·_𝑠
cip 17184	class ·_𝑖
cts 17185	class TopSet
cple 17186	class le
coc 17187	class oc
cds 17188	class dist
cunif 17189	class UnifSet
chom 17190	class Hom
cco 17191	class comp
df-plusg 17192	⊢ +_g = Slot 2
df-mulr 17193	⊢ ._r = Slot 3
df-starv 17194	⊢ *_𝑟 = Slot 4
df-sca 17195	⊢ Scalar = Slot 5
df-vsca 17196	⊢ ·_𝑠 = Slot 6
df-ip 17197	⊢ ·_𝑖 = Slot 8
df-tset 17198	⊢ TopSet = Slot 9
df-ple 17199	⊢ le = Slot ;10
df-ocomp 17200	⊢ oc = Slot ;11
df-ds 17201	⊢ dist = Slot ;12
df-unif 17202	⊢ UnifSet = Slot ;13
df-hom 17203	⊢ Hom = Slot ;14
df-cco 17204	⊢ comp = Slot ;15
crest 17342	class ↾_t
ctopn 17343	class TopOpen
df-rest 17344	⊢ ↾_t = (𝑗 ∈ V, 𝑥 ∈ V ↦ ran (𝑦 ∈ 𝑗 ↦ (𝑦 ∩ 𝑥)))
df-topn 17345	⊢ TopOpen = (𝑤 ∈ V ↦ ((TopSet‘𝑤) ↾_t (Base‘𝑤)))
ctg 17359	class topGen
cpt 17360	class ∏_t
c0g 17361	class 0_g
cgsu 17362	class Σ_g
df-0g 17363	⊢ 0_g = (𝑔 ∈ V ↦ (℩𝑒(𝑒 ∈ (Base‘𝑔) ∧ ∀𝑥 ∈ (Base‘𝑔)((𝑒(+_g‘𝑔)𝑥) = 𝑥 ∧ (𝑥(+_g‘𝑔)𝑒) = 𝑥))))
df-gsum 17364	⊢ 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 17365	⊢ topGen = (𝑥 ∈ V ↦ {𝑦 ∣ 𝑦 ⊆ ∪ (𝑥 ∩ 𝒫 𝑦)})
df-pt 17366	⊢ ∏_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 17367	class X_s
cpws 17368	class ↑_s
df-prds 17369	⊢ 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 17371	⊢ ↑_s = (𝑟 ∈ V, 𝑖 ∈ V ↦ ((Scalar‘𝑟)X_s(𝑖 × {𝑟})))
cordt 17421	class ordTop
cxrs 17422	class ℝ^*_𝑠
df-ordt 17423	⊢ ordTop = (𝑟 ∈ V ↦ (topGen‘(fi‘({dom 𝑟} ∪ ran ((𝑥 ∈ dom 𝑟 ↦ {𝑦 ∈ dom 𝑟 ∣ ¬ 𝑦𝑟𝑥}) ∪ (𝑥 ∈ dom 𝑟 ↦ {𝑦 ∈ dom 𝑟 ∣ ¬ 𝑥𝑟𝑦}))))))
df-xrs 17424	⊢ 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 17425	class qTop
cimas 17426	class “_s
cqus 17427	class /_s
cxps 17428	class ×_s
df-qtop 17429	⊢ qTop = (𝑗 ∈ V, 𝑓 ∈ V ↦ {𝑠 ∈ 𝒫 (𝑓 “ ∪ 𝑗) ∣ ((^◡𝑓 “ 𝑠) ∩ ∪ 𝑗) ∈ 𝑗})
df-imas 17430	⊢ “_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 17431	⊢ /_s = (𝑟 ∈ V, 𝑒 ∈ V ↦ ((𝑥 ∈ (Base‘𝑟) ↦ [𝑥]𝑒) “_s 𝑟))
df-xps 17432	⊢ ×_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 17502	class Moore
cmrc 17503	class mrCls
cmri 17504	class mrInd
cacs 17505	class ACS
df-mre 17506	⊢ Moore = (𝑥 ∈ V ↦ {𝑐 ∈ 𝒫 𝒫 𝑥 ∣ (𝑥 ∈ 𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝑐))})
df-mrc 17507	⊢ mrCls = (𝑐 ∈ ∪ ran Moore ↦ (𝑥 ∈ 𝒫 ∪ 𝑐 ↦ ∩ {𝑠 ∈ 𝑐 ∣ 𝑥 ⊆ 𝑠}))
df-mri 17508	⊢ mrInd = (𝑐 ∈ ∪ ran Moore ↦ {𝑠 ∈ 𝒫 ∪ 𝑐 ∣ ∀𝑥 ∈ 𝑠 ¬ 𝑥 ∈ ((mrCls‘𝑐)‘(𝑠 ∖ {𝑥}))})
df-acs 17509	⊢ ACS = (𝑥 ∈ V ↦ {𝑐 ∈ (Moore‘𝑥) ∣ ∃𝑓(𝑓:𝒫 𝑥⟶𝒫 𝑥 ∧ ∀𝑠 ∈ 𝒫 𝑥(𝑠 ∈ 𝑐 ↔ ∪ (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠))})
ccat 17588	class Cat
ccid 17589	class Id
chomf 17590	class Hom_f
ccomf 17591	class comp_f
df-cat 17592	⊢ 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 17593	⊢ 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 17594	⊢ Hom_f = (𝑐 ∈ V ↦ (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ (𝑥(Hom ‘𝑐)𝑦)))
df-comf 17595	⊢ 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 17635	class oppCat
df-oppc 17636	⊢ 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 17653	class Mono
cepi 17654	class Epi
df-mon 17655	⊢ 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 17656	⊢ Epi = (𝑐 ∈ Cat ↦ tpos (Mono‘(oppCat‘𝑐)))
csect 17669	class Sect
cinv 17670	class Inv
ciso 17671	class Iso
df-sect 17672	⊢ 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 17673	⊢ Inv = (𝑐 ∈ Cat ↦ (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ ((𝑥(Sect‘𝑐)𝑦) ∩ ^◡(𝑦(Sect‘𝑐)𝑥))))
df-iso 17674	⊢ Iso = (𝑐 ∈ Cat ↦ ((𝑥 ∈ V ↦ dom 𝑥) ∘ (Inv‘𝑐)))
ccic 17720	class ≃_𝑐
df-cic 17721	⊢ ≃_𝑐 = (𝑐 ∈ Cat ↦ ((Iso‘𝑐) supp ∅))
cssc 17732	class ⊆_cat
cresc 17733	class ↾_cat
csubc 17734	class Subcat
df-ssc 17735	⊢ ⊆_cat = {⟨ℎ, 𝑗⟩ ∣ ∃𝑡(𝑗 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝑗‘𝑥))}
df-resc 17736	⊢ ↾_cat = (𝑐 ∈ V, ℎ ∈ V ↦ ((𝑐 ↾_s dom dom ℎ) sSet ⟨(Hom ‘ndx), ℎ⟩))
df-subc 17737	⊢ 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 17779	class Func
cidfu 17780	class id_func
ccofu 17781	class ∘_func
cresf 17782	class ↾_f
df-func 17783	⊢ 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 17784	⊢ id_func = (𝑡 ∈ Cat ↦ ⦋(Base‘𝑡) / 𝑏⦌⟨( I ↾ 𝑏), (𝑧 ∈ (𝑏 × 𝑏) ↦ ( I ↾ ((Hom ‘𝑡)‘𝑧)))⟩)
df-cofu 17785	⊢ ∘_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 17786	⊢ ↾_f = (𝑓 ∈ V, ℎ ∈ V ↦ ⟨((1^st ‘𝑓) ↾ dom dom ℎ), (𝑥 ∈ dom ℎ ↦ (((2^nd ‘𝑓)‘𝑥) ↾ (ℎ‘𝑥)))⟩)
cful 17829	class Full
cfth 17830	class Faith
df-full 17831	⊢ 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 17832	⊢ Faith = (𝑐 ∈ Cat, 𝑑 ∈ Cat ↦ {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝑐 Func 𝑑)𝑔 ∧ ∀𝑥 ∈ (Base‘𝑐)∀𝑦 ∈ (Base‘𝑐)Fun ^◡(𝑥𝑔𝑦))})
cnat 17869	class Nat
cfuc 17870	class FuncCat
df-nat 17871	⊢ 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 17872	⊢ 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 17906	class InitO
ctermo 17907	class TermO
czeroo 17908	class ZeroO
df-inito 17909	⊢ InitO = (𝑐 ∈ Cat ↦ {𝑎 ∈ (Base‘𝑐) ∣ ∀𝑏 ∈ (Base‘𝑐)∃!ℎ ℎ ∈ (𝑎(Hom ‘𝑐)𝑏)})
df-termo 17910	⊢ TermO = (𝑐 ∈ Cat ↦ {𝑎 ∈ (Base‘𝑐) ∣ ∀𝑏 ∈ (Base‘𝑐)∃!ℎ ℎ ∈ (𝑏(Hom ‘𝑐)𝑎)})
df-zeroo 17911	⊢ ZeroO = (𝑐 ∈ Cat ↦ ((InitO‘𝑐) ∩ (TermO‘𝑐)))
cdoma 17945	class dom_a
ccoda 17946	class cod_a
carw 17947	class Arrow
choma 17948	class Hom_a
df-doma 17949	⊢ dom_a = (1^st ∘ 1^st )
df-coda 17950	⊢ cod_a = (2^nd ∘ 1^st )
df-homa 17951	⊢ Hom_a = (𝑐 ∈ Cat ↦ (𝑥 ∈ ((Base‘𝑐) × (Base‘𝑐)) ↦ ({𝑥} × ((Hom ‘𝑐)‘𝑥))))
df-arw 17952	⊢ Arrow = (𝑐 ∈ Cat ↦ ∪ ran (Hom_a‘𝑐))
cida 17978	class Id_a
ccoa 17979	class comp_a
df-ida 17980	⊢ Id_a = (𝑐 ∈ Cat ↦ (𝑥 ∈ (Base‘𝑐) ↦ ⟨𝑥, 𝑥, ((Id‘𝑐)‘𝑥)⟩))
df-coa 17981	⊢ 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 18000	class SetCat
df-setc 18001	⊢ 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 18023	class CatCat
df-catc 18024	⊢ 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 18046	class ExtStrCat
df-estrc 18047	⊢ 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 18092	class ×_c
c1stf 18093	class 1^st_F
c2ndf 18094	class 2^nd_F
cprf 18095	class ⟨,⟩_F
df-xpc 18096	⊢ ×_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 18097	⊢ 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 18098	⊢ 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 18099	⊢ ⟨,⟩_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 18133	class eval_F
ccurf 18134	class curry_F
cuncf 18135	class uncurry_F
cdiag 18136	class Δ_func
df-evlf 18137	⊢ 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 18138	⊢ 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 18139	⊢ 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 18140	⊢ Δ_func = (𝑐 ∈ Cat, 𝑑 ∈ Cat ↦ (⟨𝑐, 𝑑⟩ curry_F (𝑐 1^st_F 𝑑)))
chof 18172	class Hom_F
cyon 18173	class Yon
df-hof 18174	⊢ 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 18175	⊢ Yon = (𝑐 ∈ Cat ↦ (⟨𝑐, (oppCat‘𝑐)⟩ curry_F (Hom_F‘(oppCat‘𝑐))))
codu 18210	class ODual
df-odu 18211	⊢ ODual = (𝑤 ∈ V ↦ (𝑤 sSet ⟨(le‘ndx), ^◡(le‘𝑤)⟩))
cproset 18216	class Proset
cdrs 18217	class Dirset
df-proset 18218	⊢ Proset = {𝑓 ∣ [(Base‘𝑓) / 𝑏][(le‘𝑓) / 𝑟]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) → 𝑥𝑟𝑧))}
df-drs 18219	⊢ Dirset = {𝑓 ∈ Proset ∣ [(Base‘𝑓) / 𝑏][(le‘𝑓) / 𝑟](𝑏 ≠ ∅ ∧ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∃𝑧 ∈ 𝑏 (𝑥𝑟𝑧 ∧ 𝑦𝑟𝑧))}
cpo 18231	class Poset
cplt 18232	class lt
club 18233	class lub
cglb 18234	class glb
cjn 18235	class join
cmee 18236	class meet
df-poset 18237	⊢ 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 18252	⊢ lt = (𝑝 ∈ V ↦ ((le‘𝑝) ∖ I ))
df-lub 18268	⊢ 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 18269	⊢ 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 18270	⊢ join = (𝑝 ∈ V ↦ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦} (lub‘𝑝)𝑧})
df-meet 18271	⊢ meet = (𝑝 ∈ V ↦ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦} (glb‘𝑝)𝑧})
ctos 18338	class Toset
df-toset 18339	⊢ Toset = {𝑓 ∈ Poset ∣ [(Base‘𝑓) / 𝑏][(le‘𝑓) / 𝑟]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥𝑟𝑦 ∨ 𝑦𝑟𝑥)}
cp0 18345	class 0.
cp1 18346	class 1.
df-p0 18347	⊢ 0. = (𝑝 ∈ V ↦ ((glb‘𝑝)‘(Base‘𝑝)))
df-p1 18348	⊢ 1. = (𝑝 ∈ V ↦ ((lub‘𝑝)‘(Base‘𝑝)))
clat 18355	class Lat
df-lat 18356	⊢ Lat = {𝑝 ∈ Poset ∣ (dom (join‘𝑝) = ((Base‘𝑝) × (Base‘𝑝)) ∧ dom (meet‘𝑝) = ((Base‘𝑝) × (Base‘𝑝)))}
ccla 18422	class CLat
df-clat 18423	⊢ CLat = {𝑝 ∈ Poset ∣ (dom (lub‘𝑝) = 𝒫 (Base‘𝑝) ∧ dom (glb‘𝑝) = 𝒫 (Base‘𝑝))}
cdlat 18444	class DLat
df-dlat 18445	⊢ DLat = {𝑘 ∈ Lat ∣ [(Base‘𝑘) / 𝑏][(join‘𝑘) / 𝑗][(meet‘𝑘) / 𝑚]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 (𝑥𝑚(𝑦𝑗𝑧)) = ((𝑥𝑚𝑦)𝑗(𝑥𝑚𝑧))}
cipo 18451	class toInc
df-ipo 18452	⊢ 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 18488	class PosetRel
ctsr 18489	class TosetRel
df-ps 18490	⊢ PosetRel = {𝑟 ∣ (Rel 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟 ∧ (𝑟 ∩ ^◡𝑟) = ( I ↾ ∪ ∪ 𝑟))}
df-tsr 18491	⊢ TosetRel = {𝑟 ∈ PosetRel ∣ (dom 𝑟 × dom 𝑟) ⊆ (𝑟 ∪ ^◡𝑟)}
cdir 18518	class DirRel
ctail 18519	class tail
df-dir 18520	⊢ DirRel = {𝑟 ∣ ((Rel 𝑟 ∧ ( I ↾ ∪ ∪ 𝑟) ⊆ 𝑟) ∧ ((𝑟 ∘ 𝑟) ⊆ 𝑟 ∧ (∪ ∪ 𝑟 × ∪ ∪ 𝑟) ⊆ (^◡𝑟 ∘ 𝑟)))}
df-tail 18521	⊢ tail = (𝑟 ∈ DirRel ↦ (𝑥 ∈ ∪ ∪ 𝑟 ↦ (𝑟 “ {𝑥})))
cplusf 18529	class +_𝑓
cmgm 18530	class Mgm
df-plusf 18531	⊢ +_𝑓 = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(+_g‘𝑔)𝑦)))
df-mgm 18532	⊢ Mgm = {𝑔 ∣ [(Base‘𝑔) / 𝑏][(+_g‘𝑔) / 𝑜]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥𝑜𝑦) ∈ 𝑏}
cmgmhm 18582	class MgmHom
csubmgm 18583	class SubMgm
df-mgmhm 18584	⊢ 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 18585	⊢ SubMgm = (𝑠 ∈ Mgm ↦ {𝑡 ∈ 𝒫 (Base‘𝑠) ∣ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (𝑥(+_g‘𝑠)𝑦) ∈ 𝑡})
csgrp 18610	class Smgrp
df-sgrp 18611	⊢ Smgrp = {𝑔 ∈ Mgm ∣ [(Base‘𝑔) / 𝑏][(+_g‘𝑔) / 𝑜]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧))}
cmnd 18626	class Mnd
df-mnd 18627	⊢ Mnd = {𝑔 ∈ Smgrp ∣ [(Base‘𝑔) / 𝑏][(+_g‘𝑔) / 𝑝]∃𝑒 ∈ 𝑏 ∀𝑥 ∈ 𝑏 ((𝑒𝑝𝑥) = 𝑥 ∧ (𝑥𝑝𝑒) = 𝑥)}
cmhm 18673	class MndHom
csubmnd 18674	class SubMnd
df-mhm 18675	⊢ 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 18676	⊢ SubMnd = (𝑠 ∈ Mnd ↦ {𝑡 ∈ 𝒫 (Base‘𝑠) ∣ ((0_g‘𝑠) ∈ 𝑡 ∧ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (𝑥(+_g‘𝑠)𝑦) ∈ 𝑡)})
cfrmd 18739	class freeMnd
cvrmd 18740	class var_FMnd
df-frmd 18741	⊢ freeMnd = (𝑖 ∈ V ↦ {⟨(Base‘ndx), Word 𝑖⟩, ⟨(+_g‘ndx), ( ++ ↾ (Word 𝑖 × Word 𝑖))⟩})
df-vrmd 18742	⊢ var_FMnd = (𝑖 ∈ V ↦ (𝑗 ∈ 𝑖 ↦ ⟨“𝑗”⟩))
cefmnd 18760	class EndoFMnd
df-efmnd 18761	⊢ 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 18830	class Grp
cminusg 18831	class inv_g
csg 18832	class -_g
df-grp 18833	⊢ Grp = {𝑔 ∈ Mnd ∣ ∀𝑎 ∈ (Base‘𝑔)∃𝑚 ∈ (Base‘𝑔)(𝑚(+_g‘𝑔)𝑎) = (0_g‘𝑔)}
df-minusg 18834	⊢ inv_g = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔) ↦ (℩𝑤 ∈ (Base‘𝑔)(𝑤(+_g‘𝑔)𝑥) = (0_g‘𝑔))))
df-sbg 18835	⊢ -_g = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(+_g‘𝑔)((inv_g‘𝑔)‘𝑦))))
cmg 18964	class ._g
df-mulg 18965	⊢ ._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 19017	class SubGrp
cnsg 19018	class NrmSGrp
cqg 19019	class ~_QG
df-subg 19020	⊢ SubGrp = (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾_s 𝑠) ∈ Grp})
df-nsg 19021	⊢ NrmSGrp = (𝑤 ∈ Grp ↦ {𝑠 ∈ (SubGrp‘𝑤) ∣ [(Base‘𝑤) / 𝑏][(+_g‘𝑤) / 𝑝]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ((𝑥𝑝𝑦) ∈ 𝑠 ↔ (𝑦𝑝𝑥) ∈ 𝑠)})
df-eqg 19022	⊢ ~_QG = (𝑟 ∈ V, 𝑖 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (Base‘𝑟) ∧ (((inv_g‘𝑟)‘𝑥)(+_g‘𝑟)𝑦) ∈ 𝑖)})
cghm 19109	class GrpHom
df-ghm 19110	⊢ 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 19154	class GrpIso
cgic 19155	class ≃_𝑔
df-gim 19156	⊢ GrpIso = (𝑠 ∈ Grp, 𝑡 ∈ Grp ↦ {𝑔 ∈ (𝑠 GrpHom 𝑡) ∣ 𝑔:(Base‘𝑠)–1-1-onto→(Base‘𝑡)})
df-gic 19157	⊢ ≃_𝑔 = (^◡ GrpIso “ (V ∖ 1_o))
cga 19186	class GrpAct
df-ga 19187	⊢ 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 19212	class Cntz
ccntr 19213	class Cntr
df-cntz 19214	⊢ Cntz = (𝑚 ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘𝑚) ↦ {𝑥 ∈ (Base‘𝑚) ∣ ∀𝑦 ∈ 𝑠 (𝑥(+_g‘𝑚)𝑦) = (𝑦(+_g‘𝑚)𝑥)}))
df-cntr 19215	⊢ Cntr = (𝑚 ∈ V ↦ ((Cntz‘𝑚)‘(Base‘𝑚)))
coppg 19242	class opp_g
df-oppg 19243	⊢ opp_g = (𝑤 ∈ V ↦ (𝑤 sSet ⟨(+_g‘ndx), tpos (+_g‘𝑤)⟩))
csymg 19266	class SymGrp
df-symg 19267	⊢ SymGrp = (𝑥 ∈ V ↦ ((EndoFMnd‘𝑥) ↾_s {ℎ ∣ ℎ:𝑥–1-1-onto→𝑥}))
cpmtr 19338	class pmTrsp
df-pmtr 19339	⊢ pmTrsp = (𝑑 ∈ V ↦ (𝑝 ∈ {𝑦 ∈ 𝒫 𝑑 ∣ 𝑦 ≈ 2_o} ↦ (𝑧 ∈ 𝑑 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))))
cpsgn 19386	class pmSgn
cevpm 19387	class pmEven
df-psgn 19388	⊢ 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 19389	⊢ pmEven = (𝑑 ∈ V ↦ (^◡(pmSgn‘𝑑) “ {1}))
cod 19421	class od
cgex 19422	class gEx
cpgp 19423	class pGrp
cslw 19424	class pSyl
df-od 19425	⊢ od = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔) ↦ ⦋{𝑛 ∈ ℕ ∣ (𝑛(._g‘𝑔)𝑥) = (0_g‘𝑔)} / 𝑖⦌if(𝑖 = ∅, 0, inf(𝑖, ℝ, < ))))
df-gex 19426	⊢ gEx = (𝑔 ∈ V ↦ ⦋{𝑛 ∈ ℕ ∣ ∀𝑥 ∈ (Base‘𝑔)(𝑛(._g‘𝑔)𝑥) = (0_g‘𝑔)} / 𝑖⦌if(𝑖 = ∅, 0, inf(𝑖, ℝ, < )))
df-pgp 19427	⊢ pGrp = {⟨𝑝, 𝑔⟩ ∣ ((𝑝 ∈ ℙ ∧ 𝑔 ∈ Grp) ∧ ∀𝑥 ∈ (Base‘𝑔)∃𝑛 ∈ ℕ₀ ((od‘𝑔)‘𝑥) = (𝑝↑𝑛))}
df-slw 19428	⊢ pSyl = (𝑝 ∈ ℙ, 𝑔 ∈ Grp ↦ {ℎ ∈ (SubGrp‘𝑔) ∣ ∀𝑘 ∈ (SubGrp‘𝑔)((ℎ ⊆ 𝑘 ∧ 𝑝 pGrp (𝑔 ↾_s 𝑘)) ↔ ℎ = 𝑘)})
clsm 19531	class LSSum
cpj1 19532	class proj₁
df-lsm 19533	⊢ LSSum = (𝑤 ∈ V ↦ (𝑡 ∈ 𝒫 (Base‘𝑤), 𝑢 ∈ 𝒫 (Base‘𝑤) ↦ ran (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥(+_g‘𝑤)𝑦))))
df-pj1 19534	⊢ 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 19603	class ~_FG
cfrgp 19604	class freeGrp
cvrgp 19605	class var_FGrp
df-efg 19606	⊢ ~_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 19607	⊢ freeGrp = (𝑖 ∈ V ↦ ((freeMnd‘(𝑖 × 2_o)) /_s ( ~_FG ‘𝑖)))
df-vrgp 19608	⊢ var_FGrp = (𝑖 ∈ V ↦ (𝑗 ∈ 𝑖 ↦ [⟨“⟨𝑗, ∅⟩”⟩]( ~_FG ‘𝑖)))
ccmn 19677	class CMnd
cabl 19678	class Abel
df-cmn 19679	⊢ CMnd = {𝑔 ∈ Mnd ∣ ∀𝑎 ∈ (Base‘𝑔)∀𝑏 ∈ (Base‘𝑔)(𝑎(+_g‘𝑔)𝑏) = (𝑏(+_g‘𝑔)𝑎)}
df-abl 19680	⊢ Abel = (Grp ∩ CMnd)
ccyg 19774	class CycGrp
df-cyg 19775	⊢ CycGrp = {𝑔 ∈ Grp ∣ ∃𝑥 ∈ (Base‘𝑔)ran (𝑛 ∈ ℤ ↦ (𝑛(._g‘𝑔)𝑥)) = (Base‘𝑔)}
cdprd 19892	class DProd
cdpj 19893	class dProj
df-dprd 19894	⊢ 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 19895	⊢ dProj = (𝑔 ∈ Grp, 𝑠 ∈ (dom DProd “ {𝑔}) ↦ (𝑖 ∈ dom 𝑠 ↦ ((𝑠‘𝑖)(proj₁‘𝑔)(𝑔 DProd (𝑠 ↾ (dom 𝑠 ∖ {𝑖}))))))
csimpg 19989	class SimpGrp
df-simpg 19990	⊢ SimpGrp = {𝑔 ∈ Grp ∣ (NrmSGrp‘𝑔) ≈ 2_o}
comnd 20016	class oMnd
cogrp 20017	class oGrp
df-omnd 20018	⊢ 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 20019	⊢ oGrp = (Grp ∩ oMnd)
cmgp 20043	class mulGrp
df-mgp 20044	⊢ mulGrp = (𝑤 ∈ V ↦ (𝑤 sSet ⟨(+_g‘ndx), (._r‘𝑤)⟩))
crng 20055	class Rng
df-rng 20056	⊢ 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 20084	class 1_r
df-ur 20085	⊢ 1_r = (0_g ∘ mulGrp)
csrg 20089	class SRing
df-srg 20090	⊢ 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 20136	class Ring
ccrg 20137	class CRing
df-ring 20138	⊢ 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 20139	⊢ CRing = {𝑓 ∈ Ring ∣ (mulGrp‘𝑓) ∈ CMnd}
coppr 20239	class opp_r
df-oppr 20240	⊢ opp_r = (𝑓 ∈ V ↦ (𝑓 sSet ⟨(._r‘ndx), tpos (._r‘𝑓)⟩))
cdsr 20257	class ∥_r
cui 20258	class Unit
cir 20259	class Irred
df-dvdsr 20260	⊢ ∥_r = (𝑤 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑤) ∧ ∃𝑧 ∈ (Base‘𝑤)(𝑧(._r‘𝑤)𝑥) = 𝑦)})
df-unit 20261	⊢ Unit = (𝑤 ∈ V ↦ (^◡((∥_r‘𝑤) ∩ (∥_r‘(opp_r‘𝑤))) “ {(1_r‘𝑤)}))
df-irred 20262	⊢ Irred = (𝑤 ∈ V ↦ ⦋((Base‘𝑤) ∖ (Unit‘𝑤)) / 𝑏⦌{𝑧 ∈ 𝑏 ∣ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥(._r‘𝑤)𝑦) ≠ 𝑧})
cinvr 20290	class inv_r
df-invr 20291	⊢ inv_r = (𝑟 ∈ V ↦ (inv_g‘((mulGrp‘𝑟) ↾_s (Unit‘𝑟))))
cdvr 20303	class /_r
df-dvr 20304	⊢ /_r = (𝑟 ∈ V ↦ (𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Unit‘𝑟) ↦ (𝑥(._r‘𝑟)((inv_r‘𝑟)‘𝑦))))
crpm 20335	class RPrime
df-rprm 20336	⊢ 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 20337	class RngHom
crngim 20338	class RngIso
df-rnghm 20339	⊢ 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 20340	⊢ RngIso = (𝑟 ∈ V, 𝑠 ∈ V ↦ {𝑓 ∈ (𝑟 RngHom 𝑠) ∣ ^◡𝑓 ∈ (𝑠 RngHom 𝑟)})
crh 20372	class RingHom
crs 20373	class RingIso
cric 20374	class ≃_𝑟
df-rhm 20375	⊢ 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 20376	⊢ RingIso = (𝑟 ∈ V, 𝑠 ∈ V ↦ {𝑓 ∈ (𝑟 RingHom 𝑠) ∣ ^◡𝑓 ∈ (𝑠 RingHom 𝑟)})
df-ric 20378	⊢ ≃_𝑟 = (^◡ RingIso “ (V ∖ 1_o))
cnzr 20415	class NzRing
df-nzr 20416	⊢ NzRing = {𝑟 ∈ Ring ∣ (1_r‘𝑟) ≠ (0_g‘𝑟)}
clring 20441	class LRing
df-lring 20442	⊢ LRing = {𝑟 ∈ NzRing ∣ ∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)((𝑥(+_g‘𝑟)𝑦) = (1_r‘𝑟) → (𝑥 ∈ (Unit‘𝑟) ∨ 𝑦 ∈ (Unit‘𝑟)))}
csubrng 20448	class SubRng
df-subrng 20449	⊢ SubRng = (𝑤 ∈ Rng ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾_s 𝑠) ∈ Rng})
csubrg 20472	class SubRing
df-subrg 20473	⊢ SubRing = (𝑤 ∈ Ring ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ ((𝑤 ↾_s 𝑠) ∈ Ring ∧ (1_r‘𝑤) ∈ 𝑠)})
crgspn 20513	class RingSpan
df-rgspn 20514	⊢ RingSpan = (𝑤 ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘𝑤) ↦ ∩ {𝑡 ∈ (SubRing‘𝑤) ∣ 𝑠 ⊆ 𝑡}))
crngc 20519	class RngCat
df-rngc 20520	⊢ RngCat = (𝑢 ∈ V ↦ ((ExtStrCat‘𝑢) ↾_cat ( RngHom ↾ ((𝑢 ∩ Rng) × (𝑢 ∩ Rng)))))
cringc 20548	class RingCat
df-ringc 20549	⊢ RingCat = (𝑢 ∈ V ↦ ((ExtStrCat‘𝑢) ↾_cat ( RingHom ↾ ((𝑢 ∩ Ring) × (𝑢 ∩ Ring)))))
crlreg 20594	class RLReg
cdomn 20595	class Domn
cidom 20596	class IDomn
df-rlreg 20597	⊢ RLReg = (𝑟 ∈ V ↦ {𝑥 ∈ (Base‘𝑟) ∣ ∀𝑦 ∈ (Base‘𝑟)((𝑥(._r‘𝑟)𝑦) = (0_g‘𝑟) → 𝑦 = (0_g‘𝑟))})
df-domn 20598	⊢ 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 20599	⊢ IDomn = (CRing ∩ Domn)
cdr 20632	class DivRing
cfield 20633	class Field
df-drng 20634	⊢ DivRing = {𝑟 ∈ Ring ∣ (Unit‘𝑟) = ((Base‘𝑟) ∖ {(0_g‘𝑟)})}
df-field 20635	⊢ Field = (DivRing ∩ CRing)
csdrg 20689	class SubDRing
df-sdrg 20690	⊢ SubDRing = (𝑤 ∈ DivRing ↦ {𝑠 ∈ (SubRing‘𝑤) ∣ (𝑤 ↾_s 𝑠) ∈ DivRing})
cabv 20711	class AbsVal
df-abv 20712	⊢ 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 20740	class *_rf
csr 20741	class *-Ring
df-staf 20742	⊢ _rf = (𝑓 ∈ V ↦ (𝑥 ∈ (Base‘𝑓) ↦ ((_𝑟‘𝑓)‘𝑥)))
df-srng 20743	⊢ -Ring = {𝑓 ∣ [(_rf‘𝑓) / 𝑖](𝑖 ∈ (𝑓 RingHom (opp_r‘𝑓)) ∧ 𝑖 = ^◡𝑖)}
corng 20760	class oRing
cofld 20761	class oField
df-orng 20762	⊢ 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 20763	⊢ oField = (Field ∩ oRing)
clmod 20781	class LMod
cscaf 20782	class ·_sf
df-lmod 20783	⊢ 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 20784	⊢ ·_sf = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘(Scalar‘𝑔)), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥( ·_𝑠 ‘𝑔)𝑦)))
clss 20852	class LSubSp
df-lss 20853	⊢ LSubSp = (𝑤 ∈ V ↦ {𝑠 ∈ (𝒫 (Base‘𝑤) ∖ {∅}) ∣ ∀𝑥 ∈ (Base‘(Scalar‘𝑤))∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑥( ·_𝑠 ‘𝑤)𝑎)(+_g‘𝑤)𝑏) ∈ 𝑠})
clspn 20892	class LSpan
df-lsp 20893	⊢ LSpan = (𝑤 ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘𝑤) ↦ ∩ {𝑡 ∈ (LSubSp‘𝑤) ∣ 𝑠 ⊆ 𝑡}))
clmhm 20941	class LMHom
clmim 20942	class LMIso
clmic 20943	class ≃_𝑚
df-lmhm 20944	⊢ 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 20945	⊢ LMIso = (𝑠 ∈ LMod, 𝑡 ∈ LMod ↦ {𝑔 ∈ (𝑠 LMHom 𝑡) ∣ 𝑔:(Base‘𝑠)–1-1-onto→(Base‘𝑡)})
df-lmic 20946	⊢ ≃_𝑚 = (^◡ LMIso “ (V ∖ 1_o))
clbs 20996	class LBasis
df-lbs 20997	⊢ 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 21024	class LVec
df-lvec 21025	⊢ LVec = {𝑓 ∈ LMod ∣ (Scalar‘𝑓) ∈ DivRing}
csra 21093	class subringAlg
crglmod 21094	class ringLMod
df-sra 21095	⊢ 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 21096	⊢ ringLMod = (𝑤 ∈ V ↦ ((subringAlg ‘𝑤)‘(Base‘𝑤)))
clidl 21131	class LIdeal
crsp 21132	class RSpan
df-lidl 21133	⊢ LIdeal = (LSubSp ∘ ringLMod)
df-rsp 21134	⊢ RSpan = (LSpan ∘ ringLMod)
c2idl 21174	class 2Ideal
df-2idl 21175	⊢ 2Ideal = (𝑟 ∈ V ↦ ((LIdeal‘𝑟) ∩ (LIdeal‘(opp_r‘𝑟))))
clpidl 21245	class LPIdeal
clpir 21246	class LPIR
df-lpidl 21247	⊢ LPIdeal = (𝑤 ∈ Ring ↦ ∪ 𝑔 ∈ (Base‘𝑤){((RSpan‘𝑤)‘{𝑔})})
df-lpir 21248	⊢ LPIR = {𝑤 ∈ Ring ∣ (LIdeal‘𝑤) = (LPIdeal‘𝑤)}
cpid 21261	class PID
df-pid 21262	⊢ PID = (IDomn ∩ LPIR)
cpsmet 21263	class PsMet
cxmet 21264	class ∞Met
cmet 21265	class Met
cbl 21266	class ball
cfbas 21267	class fBas
cfg 21268	class filGen
cmopn 21269	class MetOpen
cmetu 21270	class metUnif
df-psmet 21271	⊢ PsMet = (𝑥 ∈ V ↦ {𝑑 ∈ (ℝ^* ↑_m (𝑥 × 𝑥)) ∣ ∀𝑦 ∈ 𝑥 ((𝑦𝑑𝑦) = 0 ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) +_𝑒 (𝑤𝑑𝑧)))})
df-xmet 21272	⊢ ∞Met = (𝑥 ∈ V ↦ {𝑑 ∈ (ℝ^* ↑_m (𝑥 × 𝑥)) ∣ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (((𝑦𝑑𝑧) = 0 ↔ 𝑦 = 𝑧) ∧ ∀𝑤 ∈ 𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) +_𝑒 (𝑤𝑑𝑧)))})
df-met 21273	⊢ Met = (𝑥 ∈ V ↦ {𝑑 ∈ (ℝ ↑_m (𝑥 × 𝑥)) ∣ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (((𝑦𝑑𝑧) = 0 ↔ 𝑦 = 𝑧) ∧ ∀𝑤 ∈ 𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) + (𝑤𝑑𝑧)))})
df-bl 21274	⊢ ball = (𝑑 ∈ V ↦ (𝑥 ∈ dom dom 𝑑, 𝑧 ∈ ℝ^* ↦ {𝑦 ∈ dom dom 𝑑 ∣ (𝑥𝑑𝑦) < 𝑧}))
df-mopn 21275	⊢ MetOpen = (𝑑 ∈ ∪ ran ∞Met ↦ (topGen‘ran (ball‘𝑑)))
df-fbas 21276	⊢ fBas = (𝑤 ∈ V ↦ {𝑥 ∈ 𝒫 𝒫 𝑤 ∣ (𝑥 ≠ ∅ ∧ ∅ ∉ 𝑥 ∧ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑥 ∩ 𝒫 (𝑦 ∩ 𝑧)) ≠ ∅)})
df-fg 21277	⊢ filGen = (𝑤 ∈ V, 𝑥 ∈ (fBas‘𝑤) ↦ {𝑦 ∈ 𝒫 𝑤 ∣ (𝑥 ∩ 𝒫 𝑦) ≠ ∅})
df-metu 21278	⊢ metUnif = (𝑑 ∈ ∪ ran PsMet ↦ ((dom dom 𝑑 × dom dom 𝑑)filGenran (𝑎 ∈ ℝ⁺ ↦ (^◡𝑑 “ (0[,)𝑎)))))
ccnfld 21279	class ℂ_fld
df-cnfld 21280	⊢ 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 21371	class ℤ_ring
df-zring 21372	⊢ ℤ_ring = (ℂ_fld ↾_s ℤ)
czrh 21424	class ℤRHom
czlm 21425	class ℤMod
cchr 21426	class chr
czn 21427	class ℤ/nℤ
df-zrh 21428	⊢ ℤRHom = (𝑟 ∈ V ↦ ∪ (ℤ_ring RingHom 𝑟))
df-zlm 21429	⊢ ℤMod = (𝑔 ∈ V ↦ ((𝑔 sSet ⟨(Scalar‘ndx), ℤ_ring⟩) sSet ⟨( ·_𝑠 ‘ndx), (._g‘𝑔)⟩))
df-chr 21430	⊢ chr = (𝑔 ∈ V ↦ ((od‘𝑔)‘(1_r‘𝑔)))
df-zn 21431	⊢ 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 21529	class ℝ_fld
df-refld 21530	⊢ ℝ_fld = (ℂ_fld ↾_s ℝ)
cphl 21549	class PreHil
cipf 21550	class ·_if
df-phl 21551	⊢ 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 21552	⊢ ·_if = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(·_𝑖‘𝑔)𝑦)))
cocv 21585	class ocv
ccss 21586	class ClSubSp
cthl 21587	class toHL
df-ocv 21588	⊢ ocv = (ℎ ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘ℎ) ↦ {𝑥 ∈ (Base‘ℎ) ∣ ∀𝑦 ∈ 𝑠 (𝑥(·_𝑖‘ℎ)𝑦) = (0_g‘(Scalar‘ℎ))}))
df-css 21589	⊢ ClSubSp = (ℎ ∈ V ↦ {𝑠 ∣ 𝑠 = ((ocv‘ℎ)‘((ocv‘ℎ)‘𝑠))})
df-thl 21590	⊢ toHL = (ℎ ∈ V ↦ ((toInc‘(ClSubSp‘ℎ)) sSet ⟨(oc‘ndx), (ocv‘ℎ)⟩))
cpj 21625	class proj
chil 21626	class Hil
cobs 21627	class OBasis
df-pj 21628	⊢ proj = (ℎ ∈ V ↦ ((𝑥 ∈ (LSubSp‘ℎ) ↦ (𝑥(proj₁‘ℎ)((ocv‘ℎ)‘𝑥))) ∩ (V × ((Base‘ℎ) ↑_m (Base‘ℎ)))))
df-hil 21629	⊢ Hil = {ℎ ∈ PreHil ∣ dom (proj‘ℎ) = (ClSubSp‘ℎ)}
df-obs 21630	⊢ 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 21656	class ⊕_m
df-dsmm 21657	⊢ ⊕_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 21671	class freeLMod
df-frlm 21672	⊢ freeLMod = (𝑟 ∈ V, 𝑖 ∈ V ↦ (𝑟 ⊕_m (𝑖 × {(ringLMod‘𝑟)})))
cuvc 21707	class unitVec
df-uvc 21708	⊢ unitVec = (𝑟 ∈ V, 𝑖 ∈ V ↦ (𝑗 ∈ 𝑖 ↦ (𝑘 ∈ 𝑖 ↦ if(𝑘 = 𝑗, (1_r‘𝑟), (0_g‘𝑟)))))
clindf 21729	class LIndF
clinds 21730	class LIndS
df-lindf 21731	⊢ 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 21732	⊢ LIndS = (𝑤 ∈ V ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ ( I ↾ 𝑠) LIndF 𝑤})
casa 21775	class AssAlg
casp 21776	class AlgSpan
cascl 21777	class algSc
df-assa 21778	⊢ 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 21779	⊢ AlgSpan = (𝑤 ∈ AssAlg ↦ (𝑠 ∈ 𝒫 (Base‘𝑤) ↦ ∩ {𝑡 ∈ ((SubRing‘𝑤) ∩ (LSubSp‘𝑤)) ∣ 𝑠 ⊆ 𝑡}))
df-ascl 21780	⊢ algSc = (𝑤 ∈ V ↦ (𝑥 ∈ (Base‘(Scalar‘𝑤)) ↦ (𝑥( ·_𝑠 ‘𝑤)(1_r‘𝑤))))
cmps 21829	class mPwSer
cmvr 21830	class mVar
cmpl 21831	class mPoly
cltb 21832	class <_bag
copws 21833	class ordPwSer
df-psr 21834	⊢ 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 21835	⊢ 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 21836	⊢ mPoly = (𝑖 ∈ V, 𝑟 ∈ V ↦ ⦋(𝑖 mPwSer 𝑟) / 𝑤⦌(𝑤 ↾_s {𝑓 ∈ (Base‘𝑤) ∣ 𝑓 finSupp (0_g‘𝑟)}))
df-ltbag 21837	⊢ <_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 21838	⊢ 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 21995	class evalSub
cevl 21996	class eval
df-evls 21997	⊢ 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 21998	⊢ eval = (𝑖 ∈ V, 𝑟 ∈ V ↦ ((𝑖 evalSub 𝑟)‘(Base‘𝑟)))
cslv 22031	class selectVars
cmhp 22032	class mHomP
cpsd 22033	class mPSDer
cai 22034	class AlgInd
df-selv 22035	⊢ 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 22039	⊢ 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 22059	⊢ 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 22074	⊢ 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 22075	class PwSer₁
cv1 22076	class var₁
cpl1 22077	class Poly₁
cco1 22078	class coe₁
ctp1 22079	class toPoly₁
df-psr1 22080	⊢ PwSer₁ = (𝑟 ∈ V ↦ ((1_o ordPwSer 𝑟)‘∅))
df-vr1 22081	⊢ var₁ = (𝑟 ∈ V ↦ ((1_o mVar 𝑟)‘∅))
df-ply1 22082	⊢ Poly₁ = (𝑟 ∈ V ↦ ((PwSer₁‘𝑟) ↾_s (Base‘(1_o mPoly 𝑟))))
df-coe1 22083	⊢ coe₁ = (𝑓 ∈ V ↦ (𝑛 ∈ ℕ₀ ↦ (𝑓‘(1_o × {𝑛}))))
df-toply1 22084	⊢ toPoly₁ = (𝑓 ∈ V ↦ (𝑛 ∈ (ℕ₀ ↑_m 1_o) ↦ (𝑓‘(𝑛‘∅))))
ces1 22216	class evalSub₁
ce1 22217	class eval₁
df-evls1 22218	⊢ 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 22219	⊢ eval₁ = (𝑟 ∈ V ↦ ⦋(Base‘𝑟) / 𝑏⦌((𝑥 ∈ (𝑏 ↑_m (𝑏 ↑_m 1_o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝑏 ↦ (1_o × {𝑦})))) ∘ (1_o eval 𝑟)))
cmmul 22293	class maMul
df-mamu 22294	⊢ 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 22310	class Mat
df-mat 22311	⊢ Mat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ ((𝑟 freeLMod (𝑛 × 𝑛)) sSet ⟨(._r‘ndx), (𝑟 maMul ⟨𝑛, 𝑛, 𝑛⟩)⟩))
cdmat 22391	class DMat
cscmat 22392	class ScMat
df-dmat 22393	⊢ DMat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ {𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ∣ ∀𝑖 ∈ 𝑛 ∀𝑗 ∈ 𝑛 (𝑖 ≠ 𝑗 → (𝑖𝑚𝑗) = (0_g‘𝑟))})
df-scmat 22394	⊢ ScMat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ ⦋(𝑛 Mat 𝑟) / 𝑎⦌{𝑚 ∈ (Base‘𝑎) ∣ ∃𝑐 ∈ (Base‘𝑟)𝑚 = (𝑐( ·_𝑠 ‘𝑎)(1_r‘𝑎))})
cmvmul 22443	class maVecMul
df-mvmul 22444	⊢ 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 22459	class matRRep
cmatrepV 22460	class matRepV
df-marrep 22461	⊢ 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 22462	⊢ 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 22479	class subMat
df-subma 22480	⊢ subMat = (𝑛 ∈ V, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑘 ∈ 𝑛, 𝑙 ∈ 𝑛 ↦ (𝑖 ∈ (𝑛 ∖ {𝑘}), 𝑗 ∈ (𝑛 ∖ {𝑙}) ↦ (𝑖𝑚𝑗)))))
cmdat 22487	class maDet
df-mdet 22488	⊢ 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 22535	class maAdju
cminmar1 22536	class minMatR1
df-madu 22537	⊢ 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 22538	⊢ 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 22606	class ConstPolyMat
cmat2pmat 22607	class matToPolyMat
ccpmat2mat 22608	class cPolyMatToMat
df-cpmat 22609	⊢ 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 22610	⊢ matToPolyMat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑥 ∈ 𝑛, 𝑦 ∈ 𝑛 ↦ ((algSc‘(Poly₁‘𝑟))‘(𝑥𝑚𝑦)))))
df-cpmat2mat 22611	⊢ cPolyMatToMat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ (𝑚 ∈ (𝑛 ConstPolyMat 𝑟) ↦ (𝑥 ∈ 𝑛, 𝑦 ∈ 𝑛 ↦ ((coe₁‘(𝑥𝑚𝑦))‘0))))
cdecpmat 22665	class decompPMat
df-decpmat 22666	⊢ decompPMat = (𝑚 ∈ V, 𝑘 ∈ ℕ₀ ↦ (𝑖 ∈ dom dom 𝑚, 𝑗 ∈ dom dom 𝑚 ↦ ((coe₁‘(𝑖𝑚𝑗))‘𝑘)))
cpm2mp 22695	class pMatToMatPoly
df-pm2mp 22696	⊢ 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 22729	class CharPlyMat
df-chpmat 22730	⊢ 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 22796	class Top
df-top 22797	⊢ Top = {𝑥 ∣ (∀𝑦 ∈ 𝒫 𝑥∪ 𝑦 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑦 ∩ 𝑧) ∈ 𝑥)}
ctopon 22813	class TopOn
df-topon 22814	⊢ TopOn = (𝑏 ∈ V ↦ {𝑗 ∈ Top ∣ 𝑏 = ∪ 𝑗})
ctps 22835	class TopSp
df-topsp 22836	⊢ TopSp = {𝑓 ∣ (TopOpen‘𝑓) ∈ (TopOn‘(Base‘𝑓))}
ctb 22848	class TopBases
df-bases 22849	⊢ TopBases = {𝑥 ∣ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑦 ∩ 𝑧) ⊆ ∪ (𝑥 ∩ 𝒫 (𝑦 ∩ 𝑧))}
ccld 22919	class Clsd
cnt 22920	class int
ccl 22921	class cls
df-cld 22922	⊢ Clsd = (𝑗 ∈ Top ↦ {𝑥 ∈ 𝒫 ∪ 𝑗 ∣ (∪ 𝑗 ∖ 𝑥) ∈ 𝑗})
df-ntr 22923	⊢ int = (𝑗 ∈ Top ↦ (𝑥 ∈ 𝒫 ∪ 𝑗 ↦ ∪ (𝑗 ∩ 𝒫 𝑥)))
df-cls 22924	⊢ cls = (𝑗 ∈ Top ↦ (𝑥 ∈ 𝒫 ∪ 𝑗 ↦ ∩ {𝑦 ∈ (Clsd‘𝑗) ∣ 𝑥 ⊆ 𝑦}))
cnei 23000	class nei
df-nei 23001	⊢ nei = (𝑗 ∈ Top ↦ (𝑥 ∈ 𝒫 ∪ 𝑗 ↦ {𝑦 ∈ 𝒫 ∪ 𝑗 ∣ ∃𝑔 ∈ 𝑗 (𝑥 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑦)}))
clp 23037	class limPt
cperf 23038	class Perf
df-lp 23039	⊢ limPt = (𝑗 ∈ Top ↦ (𝑥 ∈ 𝒫 ∪ 𝑗 ↦ {𝑦 ∣ 𝑦 ∈ ((cls‘𝑗)‘(𝑥 ∖ {𝑦}))}))
df-perf 23040	⊢ Perf = {𝑗 ∈ Top ∣ ((limPt‘𝑗)‘∪ 𝑗) = ∪ 𝑗}
ccn 23127	class Cn
ccnp 23128	class CnP
clm 23129	class ⇝_𝑡
df-cn 23130	⊢ Cn = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ {𝑓 ∈ (∪ 𝑘 ↑_m ∪ 𝑗) ∣ ∀𝑦 ∈ 𝑘 (^◡𝑓 “ 𝑦) ∈ 𝑗})
df-cnp 23131	⊢ 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 23132	⊢ ⇝_𝑡 = (𝑗 ∈ Top ↦ {⟨𝑓, 𝑥⟩ ∣ (𝑓 ∈ (∪ 𝑗 ↑_pm ℂ) ∧ 𝑥 ∈ ∪ 𝑗 ∧ ∀𝑢 ∈ 𝑗 (𝑥 ∈ 𝑢 → ∃𝑦 ∈ ran ℤ_≥(𝑓 ↾ 𝑦):𝑦⟶𝑢))})
ct0 23209	class Kol2
ct1 23210	class Fre
cha 23211	class Haus
creg 23212	class Reg
cnrm 23213	class Nrm
ccnrm 23214	class CNrm
cpnrm 23215	class PNrm
df-t0 23216	⊢ Kol2 = {𝑗 ∈ Top ∣ ∀𝑥 ∈ ∪ 𝑗∀𝑦 ∈ ∪ 𝑗(∀𝑜 ∈ 𝑗 (𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜) → 𝑥 = 𝑦)}
df-t1 23217	⊢ Fre = {𝑥 ∈ Top ∣ ∀𝑎 ∈ ∪ 𝑥{𝑎} ∈ (Clsd‘𝑥)}
df-haus 23218	⊢ Haus = {𝑗 ∈ Top ∣ ∀𝑥 ∈ ∪ 𝑗∀𝑦 ∈ ∪ 𝑗(𝑥 ≠ 𝑦 → ∃𝑛 ∈ 𝑗 ∃𝑚 ∈ 𝑗 (𝑥 ∈ 𝑛 ∧ 𝑦 ∈ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅))}
df-reg 23219	⊢ Reg = {𝑗 ∈ Top ∣ ∀𝑥 ∈ 𝑗 ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝑗 (𝑦 ∈ 𝑧 ∧ ((cls‘𝑗)‘𝑧) ⊆ 𝑥)}
df-nrm 23220	⊢ Nrm = {𝑗 ∈ Top ∣ ∀𝑥 ∈ 𝑗 ∀𝑦 ∈ ((Clsd‘𝑗) ∩ 𝒫 𝑥)∃𝑧 ∈ 𝑗 (𝑦 ⊆ 𝑧 ∧ ((cls‘𝑗)‘𝑧) ⊆ 𝑥)}
df-cnrm 23221	⊢ CNrm = {𝑗 ∈ Top ∣ ∀𝑥 ∈ 𝒫 ∪ 𝑗(𝑗 ↾_t 𝑥) ∈ Nrm}
df-pnrm 23222	⊢ PNrm = {𝑗 ∈ Nrm ∣ (Clsd‘𝑗) ⊆ ran (𝑓 ∈ (𝑗 ↑_m ℕ) ↦ ∩ ran 𝑓)}
ccmp 23289	class Comp
df-cmp 23290	⊢ Comp = {𝑥 ∈ Top ∣ ∀𝑦 ∈ 𝒫 𝑥(∪ 𝑥 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝑥 = ∪ 𝑧)}
cconn 23314	class Conn
df-conn 23315	⊢ Conn = {𝑗 ∈ Top ∣ (𝑗 ∩ (Clsd‘𝑗)) = {∅, ∪ 𝑗}}
c1stc 23340	class 1^stω
c2ndc 23341	class 2^ndω
df-1stc 23342	⊢ 1^stω = {𝑗 ∈ Top ∣ ∀𝑥 ∈ ∪ 𝑗∃𝑦 ∈ 𝒫 𝑗(𝑦 ≼ ω ∧ ∀𝑧 ∈ 𝑗 (𝑥 ∈ 𝑧 → 𝑥 ∈ ∪ (𝑦 ∩ 𝒫 𝑧)))}
df-2ndc 23343	⊢ 2^ndω = {𝑗 ∣ ∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝑗)}
clly 23367	class Locally 𝐴
cnlly 23368	class 𝑛-Locally 𝐴
df-lly 23369	⊢ Locally 𝐴 = {𝑗 ∈ Top ∣ ∀𝑥 ∈ 𝑗 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝑗 ↾_t 𝑢) ∈ 𝐴)}
df-nlly 23370	⊢ 𝑛-Locally 𝐴 = {𝑗 ∈ Top ∣ ∀𝑥 ∈ 𝑗 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗 ↾_t 𝑢) ∈ 𝐴}
cref 23405	class Ref
cptfin 23406	class PtFin
clocfin 23407	class LocFin
df-ref 23408	⊢ Ref = {⟨𝑥, 𝑦⟩ ∣ (∪ 𝑦 = ∪ 𝑥 ∧ ∀𝑧 ∈ 𝑥 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)}
df-ptfin 23409	⊢ PtFin = {𝑥 ∣ ∀𝑦 ∈ ∪ 𝑥{𝑧 ∈ 𝑥 ∣ 𝑦 ∈ 𝑧} ∈ Fin}
df-locfin 23410	⊢ LocFin = (𝑥 ∈ Top ↦ {𝑦 ∣ (∪ 𝑥 = ∪ 𝑦 ∧ ∀𝑝 ∈ ∪ 𝑥∃𝑛 ∈ 𝑥 (𝑝 ∈ 𝑛 ∧ {𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))})
ckgen 23436	class 𝑘Gen
df-kgen 23437	⊢ 𝑘Gen = (𝑗 ∈ Top ↦ {𝑥 ∈ 𝒫 ∪ 𝑗 ∣ ∀𝑘 ∈ 𝒫 ∪ 𝑗((𝑗 ↾_t 𝑘) ∈ Comp → (𝑥 ∩ 𝑘) ∈ (𝑗 ↾_t 𝑘))})
ctx 23463	class ×_t
cxko 23464	class ↑_ko
df-tx 23465	⊢ ×_t = (𝑟 ∈ V, 𝑠 ∈ V ↦ (topGen‘ran (𝑥 ∈ 𝑟, 𝑦 ∈ 𝑠 ↦ (𝑥 × 𝑦))))
df-xko 23466	⊢ ↑_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 23596	class KQ
df-kq 23597	⊢ KQ = (𝑗 ∈ Top ↦ (𝑗 qTop (𝑥 ∈ ∪ 𝑗 ↦ {𝑦 ∈ 𝑗 ∣ 𝑥 ∈ 𝑦})))
chmeo 23656	class Homeo
chmph 23657	class ≃
df-hmeo 23658	⊢ Homeo = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ {𝑓 ∈ (𝑗 Cn 𝑘) ∣ ^◡𝑓 ∈ (𝑘 Cn 𝑗)})
df-hmph 23659	⊢ ≃ = (^◡Homeo “ (V ∖ 1_o))
cfil 23748	class Fil
df-fil 23749	⊢ Fil = (𝑧 ∈ V ↦ {𝑓 ∈ (fBas‘𝑧) ∣ ∀𝑥 ∈ 𝒫 𝑧((𝑓 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥 ∈ 𝑓)})
cufil 23802	class UFil
cufl 23803	class UFL
df-ufil 23804	⊢ UFil = (𝑔 ∈ V ↦ {𝑓 ∈ (Fil‘𝑔) ∣ ∀𝑥 ∈ 𝒫 𝑔(𝑥 ∈ 𝑓 ∨ (𝑔 ∖ 𝑥) ∈ 𝑓)})
df-ufl 23805	⊢ UFL = {𝑥 ∣ ∀𝑓 ∈ (Fil‘𝑥)∃𝑔 ∈ (UFil‘𝑥)𝑓 ⊆ 𝑔}
cfm 23836	class FilMap
cflim 23837	class fLim
cflf 23838	class fLimf
cfcls 23839	class fClus
cfcf 23840	class fClusf
df-fm 23841	⊢ FilMap = (𝑥 ∈ V, 𝑓 ∈ V ↦ (𝑦 ∈ (fBas‘dom 𝑓) ↦ (𝑥filGenran (𝑡 ∈ 𝑦 ↦ (𝑓 “ 𝑡)))))
df-flim 23842	⊢ fLim = (𝑗 ∈ Top, 𝑓 ∈ ∪ ran Fil ↦ {𝑥 ∈ ∪ 𝑗 ∣ (((nei‘𝑗)‘{𝑥}) ⊆ 𝑓 ∧ 𝑓 ⊆ 𝒫 ∪ 𝑗)})
df-flf 23843	⊢ fLimf = (𝑥 ∈ Top, 𝑦 ∈ ∪ ran Fil ↦ (𝑓 ∈ (∪ 𝑥 ↑_m ∪ 𝑦) ↦ (𝑥 fLim ((∪ 𝑥 FilMap 𝑓)‘𝑦))))
df-fcls 23844	⊢ fClus = (𝑗 ∈ Top, 𝑓 ∈ ∪ ran Fil ↦ if(∪ 𝑗 = ∪ 𝑓, ∩ 𝑥 ∈ 𝑓 ((cls‘𝑗)‘𝑥), ∅))
df-fcf 23845	⊢ fClusf = (𝑗 ∈ Top, 𝑓 ∈ ∪ ran Fil ↦ (𝑔 ∈ (∪ 𝑗 ↑_m ∪ 𝑓) ↦ (𝑗 fClus ((∪ 𝑗 FilMap 𝑔)‘𝑓))))
ccnext 23962	class CnExt
df-cnext 23963	⊢ 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 23973	class TopMnd
ctgp 23974	class TopGrp
df-tmd 23975	⊢ TopMnd = {𝑓 ∈ (Mnd ∩ TopSp) ∣ [(TopOpen‘𝑓) / 𝑗](+_𝑓‘𝑓) ∈ ((𝑗 ×_t 𝑗) Cn 𝑗)}
df-tgp 23976	⊢ TopGrp = {𝑓 ∈ (Grp ∩ TopMnd) ∣ [(TopOpen‘𝑓) / 𝑗](inv_g‘𝑓) ∈ (𝑗 Cn 𝑗)}
ctsu 24029	class tsums
df-tsms 24030	⊢ 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 24059	class TopRing
ctdrg 24060	class TopDRing
ctlm 24061	class TopMod
ctvc 24062	class TopVec
df-trg 24063	⊢ TopRing = {𝑟 ∈ (TopGrp ∩ Ring) ∣ (mulGrp‘𝑟) ∈ TopMnd}
df-tdrg 24064	⊢ TopDRing = {𝑟 ∈ (TopRing ∩ DivRing) ∣ ((mulGrp‘𝑟) ↾_s (Unit‘𝑟)) ∈ TopGrp}
df-tlm 24065	⊢ TopMod = {𝑤 ∈ (TopMnd ∩ LMod) ∣ ((Scalar‘𝑤) ∈ TopRing ∧ ( ·_sf ‘𝑤) ∈ (((TopOpen‘(Scalar‘𝑤)) ×_t (TopOpen‘𝑤)) Cn (TopOpen‘𝑤)))}
df-tvc 24066	⊢ TopVec = {𝑤 ∈ TopMod ∣ (Scalar‘𝑤) ∈ TopDRing}
cust 24103	class UnifOn
df-ust 24104	⊢ 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 24134	class unifTop
df-utop 24135	⊢ unifTop = (𝑢 ∈ ∪ ran UnifOn ↦ {𝑎 ∈ 𝒫 dom ∪ 𝑢 ∣ ∀𝑥 ∈ 𝑎 ∃𝑣 ∈ 𝑢 (𝑣 “ {𝑥}) ⊆ 𝑎})
cuss 24157	class UnifSt
cusp 24158	class UnifSp
ctus 24159	class toUnifSp
df-uss 24160	⊢ UnifSt = (𝑓 ∈ V ↦ ((UnifSet‘𝑓) ↾_t ((Base‘𝑓) × (Base‘𝑓))))
df-usp 24161	⊢ UnifSp = {𝑓 ∣ ((UnifSt‘𝑓) ∈ (UnifOn‘(Base‘𝑓)) ∧ (TopOpen‘𝑓) = (unifTop‘(UnifSt‘𝑓)))}
df-tus 24162	⊢ toUnifSp = (𝑢 ∈ ∪ ran UnifOn ↦ ({⟨(Base‘ndx), dom ∪ 𝑢⟩, ⟨(UnifSet‘ndx), 𝑢⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑢)⟩))
cucn 24178	class Cn_u
df-ucn 24179	⊢ 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 24189	class CauFil_u
df-cfilu 24190	⊢ CauFil_u = (𝑢 ∈ ∪ ran UnifOn ↦ {𝑓 ∈ (fBas‘dom ∪ 𝑢) ∣ ∀𝑣 ∈ 𝑢 ∃𝑎 ∈ 𝑓 (𝑎 × 𝑎) ⊆ 𝑣})
ccusp 24200	class CUnifSp
df-cusp 24201	⊢ CUnifSp = {𝑤 ∈ UnifSp ∣ ∀𝑐 ∈ (Fil‘(Base‘𝑤))(𝑐 ∈ (CauFil_u‘(UnifSt‘𝑤)) → ((TopOpen‘𝑤) fLim 𝑐) ≠ ∅)}
cxms 24221	class ∞MetSp
cms 24222	class MetSp
ctms 24223	class toMetSp
df-xms 24224	⊢ ∞MetSp = {𝑓 ∈ TopSp ∣ (TopOpen‘𝑓) = (MetOpen‘((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))))}
df-ms 24225	⊢ MetSp = {𝑓 ∈ ∞MetSp ∣ ((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))) ∈ (Met‘(Base‘𝑓))}
df-tms 24226	⊢ toMetSp = (𝑑 ∈ ∪ ran ∞Met ↦ ({⟨(Base‘ndx), dom dom 𝑑⟩, ⟨(dist‘ndx), 𝑑⟩} sSet ⟨(TopSet‘ndx), (MetOpen‘𝑑)⟩))
cnm 24480	class norm
cngp 24481	class NrmGrp
ctng 24482	class toNrmGrp
cnrg 24483	class NrmRing
cnlm 24484	class NrmMod
cnvc 24485	class NrmVec
df-nm 24486	⊢ norm = (𝑤 ∈ V ↦ (𝑥 ∈ (Base‘𝑤) ↦ (𝑥(dist‘𝑤)(0_g‘𝑤))))
df-ngp 24487	⊢ NrmGrp = {𝑔 ∈ (Grp ∩ MetSp) ∣ ((norm‘𝑔) ∘ (-_g‘𝑔)) ⊆ (dist‘𝑔)}
df-tng 24488	⊢ toNrmGrp = (𝑔 ∈ V, 𝑓 ∈ V ↦ ((𝑔 sSet ⟨(dist‘ndx), (𝑓 ∘ (-_g‘𝑔))⟩) sSet ⟨(TopSet‘ndx), (MetOpen‘(𝑓 ∘ (-_g‘𝑔)))⟩))
df-nrg 24489	⊢ NrmRing = {𝑤 ∈ NrmGrp ∣ (norm‘𝑤) ∈ (AbsVal‘𝑤)}
df-nlm 24490	⊢ 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 24491	⊢ NrmVec = (NrmMod ∩ LVec)
cnmo 24609	class normOp
cnghm 24610	class NGHom
cnmhm 24611	class NMHom
df-nmo 24612	⊢ 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 24613	⊢ NGHom = (𝑠 ∈ NrmGrp, 𝑡 ∈ NrmGrp ↦ (^◡(𝑠 normOp 𝑡) “ ℝ))
df-nmhm 24614	⊢ NMHom = (𝑠 ∈ NrmMod, 𝑡 ∈ NrmMod ↦ ((𝑠 LMHom 𝑡) ∩ (𝑠 NGHom 𝑡)))
cii 24784	class II
ccncf 24785	class –cn→
df-ii 24786	⊢ II = (MetOpen‘((abs ∘ − ) ↾ ((0[,]1) × (0[,]1))))
df-cncf 24787	⊢ –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 24882	class Htpy
cphtpy 24883	class PHtpy
cphtpc 24884	class ≃_ph
df-htpy 24885	⊢ 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 24886	⊢ 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 24907	⊢ ≃_ph = (𝑥 ∈ Top ↦ {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ (II Cn 𝑥) ∧ (𝑓(PHtpy‘𝑥)𝑔) ≠ ∅)})
cpco 24916	class *_𝑝
comi 24917	class Ω₁
comn 24918	class Ω_𝑛
cpi1 24919	class π₁
cpin 24920	class π_n
df-pco 24921	⊢ *_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 24922	⊢ 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 24923	⊢ 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 24924	⊢ π₁ = (𝑗 ∈ Top, 𝑦 ∈ ∪ 𝑗 ↦ ((𝑗 Ω₁ 𝑦) /_s ( ≃_ph‘𝑗)))
df-pin 24925	⊢ 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 24978	class ℂMod
df-clm 24979	⊢ ℂMod = {𝑤 ∈ LMod ∣ [(Scalar‘𝑤) / 𝑓][(Base‘𝑓) / 𝑘](𝑓 = (ℂ_fld ↾_s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂ_fld))}
ccvs 25039	class ℂVec
df-cvs 25040	⊢ ℂVec = (ℂMod ∩ LVec)
ccph 25082	class ℂPreHil
ctcph 25083	class toℂPreHil
df-cph 25084	⊢ 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 25085	⊢ toℂPreHil = (𝑤 ∈ V ↦ (𝑤 toNrmGrp (𝑥 ∈ (Base‘𝑤) ↦ (√‘(𝑥(·_𝑖‘𝑤)𝑥)))))
ccfil 25168	class CauFil
ccau 25169	class Cau
ccmet 25170	class CMet
df-cfil 25171	⊢ CauFil = (𝑑 ∈ ∪ ran ∞Met ↦ {𝑓 ∈ (Fil‘dom dom 𝑑) ∣ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ 𝑓 (𝑑 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)})
df-cau 25172	⊢ 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 25173	⊢ CMet = (𝑥 ∈ V ↦ {𝑑 ∈ (Met‘𝑥) ∣ ∀𝑓 ∈ (CauFil‘𝑑)((MetOpen‘𝑑) fLim 𝑓) ≠ ∅})
ccms 25248	class CMetSp
cbn 25249	class Ban
chl 25250	class ℂHil
df-cms 25251	⊢ CMetSp = {𝑤 ∈ MetSp ∣ [(Base‘𝑤) / 𝑏]((dist‘𝑤) ↾ (𝑏 × 𝑏)) ∈ (CMet‘𝑏)}
df-bn 25252	⊢ Ban = {𝑤 ∈ (NrmVec ∩ CMetSp) ∣ (Scalar‘𝑤) ∈ CMetSp}
df-hl 25253	⊢ ℂHil = (Ban ∩ ℂPreHil)
crrx 25299	class ℝ^
cehl 25300	class 𝔼_hil
df-rrx 25301	⊢ ℝ^ = (𝑖 ∈ V ↦ (toℂPreHil‘(ℝ_fld freeLMod 𝑖)))
df-ehl 25302	⊢ 𝔼_hil = (𝑛 ∈ ℕ₀ ↦ (ℝ^‘(1...𝑛)))
covol 25379	class vol*
cvol 25380	class vol
df-ovol 25381	⊢ 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 25382	⊢ vol = (vol* ↾ {𝑥 ∣ ∀𝑦 ∈ (^◡vol* “ ℝ)(vol‘𝑦) = ((vol‘(𝑦 ∩ 𝑥)) + (vol*‘(𝑦 ∖ 𝑥)))})
cmbf 25531	class MblFn
citg1 25532	class ∫₁
citg2 25533	class ∫₂
cibl 25534	class 𝐿¹
citg 25535	class ∫𝐴𝐵 d𝑥
df-mbf 25536	⊢ MblFn = {𝑓 ∈ (ℂ ↑_pm ℝ) ∣ ∀𝑥 ∈ ran (,)((^◡(ℜ ∘ 𝑓) “ 𝑥) ∈ dom vol ∧ (^◡(ℑ ∘ 𝑓) “ 𝑥) ∈ dom vol)}
df-itg1 25537	⊢ ∫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 25538	⊢ ∫₂ = (𝑓 ∈ ((0[,]+∞) ↑_m ℝ) ↦ sup({𝑥 ∣ ∃𝑔 ∈ dom ∫₁(𝑔 ∘_r ≤ 𝑓 ∧ 𝑥 = (∫₁‘𝑔))}, ℝ^*, < ))
df-ibl 25539	⊢ 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 25540	⊢ ∫𝐴𝐵 d𝑥 = Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫₂‘(𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐵 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0))))
c0p 25586	class 0_𝑝
df-0p 25587	⊢ 0_𝑝 = (ℂ × {0})
cdit 25763	class ⨜[𝐴 → 𝐵]𝐶 d𝑥
df-ditg 25764	⊢ ⨜[𝐴 → 𝐵]𝐶 d𝑥 = if(𝐴 ≤ 𝐵, ∫(𝐴(,)𝐵)𝐶 d𝑥, -∫(𝐵(,)𝐴)𝐶 d𝑥)
climc 25779	class lim_ℂ
cdv 25780	class D
cdvn 25781	class D^𝑛
ccpn 25782	class 𝓑C^𝑛
df-limc 25783	⊢ 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 25784	⊢ 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 25785	⊢ D^𝑛 = (𝑠 ∈ 𝒫 ℂ, 𝑓 ∈ (ℂ ↑_pm 𝑠) ↦ seq0(((𝑥 ∈ V ↦ (𝑠 D 𝑥)) ∘ 1^st ), (ℕ₀ × {𝑓})))
df-cpn 25786	⊢ 𝓑C^𝑛 = (𝑠 ∈ 𝒫 ℂ ↦ (𝑥 ∈ ℕ₀ ↦ {𝑓 ∈ (ℂ ↑_pm 𝑠) ∣ ((𝑠 D^𝑛 𝑓)‘𝑥) ∈ (dom 𝑓–cn→ℂ)}))
cmdg 25974	class mDeg
cdg1 25975	class deg₁
df-mdeg 25976	⊢ 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 25977	⊢ deg₁ = (𝑟 ∈ V ↦ (1_o mDeg 𝑟))
cmn1 26047	class Monic_1p
cuc1p 26048	class Unic_1p
cq1p 26049	class quot_1p
cr1p 26050	class rem_1p
cig1p 26051	class idlGen_1p
df-mon1 26052	⊢ Monic_1p = (𝑟 ∈ V ↦ {𝑓 ∈ (Base‘(Poly₁‘𝑟)) ∣ (𝑓 ≠ (0_g‘(Poly₁‘𝑟)) ∧ ((coe₁‘𝑓)‘((deg₁‘𝑟)‘𝑓)) = (1_r‘𝑟))})
df-uc1p 26053	⊢ Unic_1p = (𝑟 ∈ V ↦ {𝑓 ∈ (Base‘(Poly₁‘𝑟)) ∣ (𝑓 ≠ (0_g‘(Poly₁‘𝑟)) ∧ ((coe₁‘𝑓)‘((deg₁‘𝑟)‘𝑓)) ∈ (Unit‘𝑟))})
df-q1p 26054	⊢ 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 26055	⊢ 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 26056	⊢ 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 26105	class Poly
cidp 26106	class X_p
ccoe 26107	class coeff
cdgr 26108	class deg
df-ply 26109	⊢ Poly = (𝑥 ∈ 𝒫 ℂ ↦ {𝑓 ∣ ∃𝑛 ∈ ℕ₀ ∃𝑎 ∈ ((𝑥 ∪ {0}) ↑_m ℕ₀)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))})
df-idp 26110	⊢ X_p = ( I ↾ ℂ)
df-coe 26111	⊢ 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 26112	⊢ deg = (𝑓 ∈ (Poly‘ℂ) ↦ sup((^◡(coeff‘𝑓) “ (ℂ ∖ {0})), ℕ₀, < ))
cquot 26214	class quot
df-quot 26215	⊢ 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 26238	class 𝔸
df-aa 26239	⊢ 𝔸 = ∪ 𝑓 ∈ ((Poly‘ℤ) ∖ {0_𝑝})(^◡𝑓 “ {0})
ctayl 26276	class Tayl
cana 26277	class Ana
df-tayl 26278	⊢ 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 26279	⊢ 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 26301	class ⇝_𝑢
df-ulm 26302	⊢ ⇝_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 26479	class log
ccxp 26480	class ↑_𝑐
df-log 26481	⊢ log = ^◡(exp ↾ (^◡ℑ “ (-π(,]π)))
df-cxp 26482	⊢ ↑_𝑐 = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ if(𝑥 = 0, if(𝑦 = 0, 1, 0), (exp‘(𝑦 · (log‘𝑥)))))
clogb 26690	class log_b
df-logb 26691	⊢ log_b = (𝑥 ∈ (ℂ ∖ {0, 1}), 𝑦 ∈ (ℂ ∖ {0}) ↦ ((log‘𝑦) / (log‘𝑥)))
casin 26788	class arcsin
cacos 26789	class arccos
catan 26790	class arctan
df-asin 26791	⊢ arcsin = (𝑥 ∈ ℂ ↦ (-i · (log‘((i · 𝑥) + (√‘(1 − (𝑥↑2)))))))
df-acos 26792	⊢ arccos = (𝑥 ∈ ℂ ↦ ((π / 2) − (arcsin‘𝑥)))
df-atan 26793	⊢ arctan = (𝑥 ∈ (ℂ ∖ {-i, i}) ↦ ((i / 2) · ((log‘(1 − (i · 𝑥))) − (log‘(1 + (i · 𝑥))))))
carea 26881	class area
df-area 26882	⊢ area = (𝑠 ∈ {𝑡 ∈ 𝒫 (ℝ × ℝ) ∣ (∀𝑥 ∈ ℝ (𝑡 “ {𝑥}) ∈ (^◡vol “ ℝ) ∧ (𝑥 ∈ ℝ ↦ (vol‘(𝑡 “ {𝑥}))) ∈ 𝐿¹)} ↦ ∫ℝ(vol‘(𝑠 “ {𝑥})) d𝑥)
cem 26918	class γ
df-em 26919	⊢ γ = Σ𝑘 ∈ ℕ ((1 / 𝑘) − (log‘(1 + (1 / 𝑘))))
czeta 26939	class ζ
df-zeta 26940	⊢ 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 26942	class log Γ
cgam 26943	class Γ
cigam 26944	class 1/Γ
df-lgam 26945	⊢ log Γ = (𝑧 ∈ (ℂ ∖ (ℤ ∖ ℕ)) ↦ (Σ𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))) − (log‘𝑧)))
df-gam 26946	⊢ Γ = (exp ∘ log Γ)
df-igam 26947	⊢ 1/Γ = (𝑥 ∈ ℂ ↦ if(𝑥 ∈ (ℤ ∖ ℕ), 0, (1 / (Γ‘𝑥))))
ccht 27017	class θ
cvma 27018	class Λ
cchp 27019	class ψ
cppi 27020	class π
cmu 27021	class μ
csgm 27022	class σ
df-cht 27023	⊢ θ = (𝑥 ∈ ℝ ↦ Σ𝑝 ∈ ((0[,]𝑥) ∩ ℙ)(log‘𝑝))
df-vma 27024	⊢ Λ = (𝑥 ∈ ℕ ↦ ⦋{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥} / 𝑠⦌if((♯‘𝑠) = 1, (log‘∪ 𝑠), 0))
df-chp 27025	⊢ ψ = (𝑥 ∈ ℝ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))(Λ‘𝑛))
df-ppi 27026	⊢ π = (𝑥 ∈ ℝ ↦ (♯‘((0[,]𝑥) ∩ ℙ)))
df-mu 27027	⊢ μ = (𝑥 ∈ ℕ ↦ if(∃𝑝 ∈ ℙ (𝑝↑2) ∥ 𝑥, 0, (-1↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥}))))
df-sgm 27028	⊢ σ = (𝑥 ∈ ℂ, 𝑛 ∈ ℕ ↦ Σ𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝑛} (𝑘↑_𝑐𝑥))
cdchr 27159	class DChr
df-dchr 27160	⊢ 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 27221	class /_L
df-lgs 27222	⊢ /_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 27567	class No
cslt 27568	class <s
cbday 27569	class bday
df-no 27570	⊢ No = {𝑓 ∣ ∃𝑎 ∈ On 𝑓:𝑎⟶{1_o, 2_o}}
df-slt 27571	⊢ <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 27572	⊢ bday = (𝑥 ∈ No ↦ dom 𝑥)
csle 27672	class ≤s
df-sle 27673	⊢ ≤s = (( No × No ) ∖ ^◡ <s )
csslt 27709	class <<s
df-sslt 27710	⊢ <<s = {⟨𝑎, 𝑏⟩ ∣ (𝑎 ⊆ No ∧ 𝑏 ⊆ No ∧ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 𝑥 <s 𝑦)}
cscut 27711	class \|s
df-scut 27712	⊢ \|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 27754	class 0_s
c1s 27755	class 1_s
df-0s 27756	⊢ 0_s = (∅ \|s ∅)
df-1s 27757	⊢ 1_s = ({ 0_s } \|s ∅)
cmade 27770	class M
cold 27771	class O
cnew 27772	class N
cleft 27773	class L
cright 27774	class R
df-made 27775	⊢ M = recs((𝑓 ∈ V ↦ ( \|s “ (𝒫 ∪ ran 𝑓 × 𝒫 ∪ ran 𝑓))))
df-old 27776	⊢ O = (𝑥 ∈ On ↦ ∪ ( M “ 𝑥))
df-new 27777	⊢ N = (𝑥 ∈ On ↦ (( M ‘𝑥) ∖ ( O ‘𝑥)))
df-left 27778	⊢ L = (𝑥 ∈ No ↦ {𝑦 ∈ ( O ‘( bday ‘𝑥)) ∣ 𝑦 <s 𝑥})
df-right 27779	⊢ R = (𝑥 ∈ No ↦ {𝑦 ∈ ( O ‘( bday ‘𝑥)) ∣ 𝑥 <s 𝑦})
cnorec 27867	class norec (𝐹)
df-norec 27868	⊢ norec (𝐹) = frecs({⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}, No , 𝐹)
cnorec2 27878	class norec2 (𝐹)
df-norec2 27879	⊢ 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 27889	class +_s
df-adds 27890	⊢ +_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 27948	class -u_s
csubs 27949	class -_s
df-negs 27950	⊢ -u_s = norec ((𝑥 ∈ V, 𝑛 ∈ V ↦ ((𝑛 “ ( R ‘𝑥)) \|s (𝑛 “ ( L ‘𝑥)))))
df-subs 27951	⊢ -_s = (𝑥 ∈ No , 𝑦 ∈ No ↦ (𝑥 +_s ( -u_s ‘𝑦)))
cmuls 28032	class ·_s
df-muls 28033	⊢ ·_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 28113	class /_su
df-divs 28114	⊢ /_su = (𝑥 ∈ No , 𝑦 ∈ ( No ∖ { 0_s }) ↦ (℩𝑧 ∈ No (𝑦 ·_s 𝑧) = 𝑥))
cabss 28162	class abs_s
df-abss 28163	⊢ abs_s = (𝑥 ∈ No ↦ if( 0_s ≤s 𝑥, 𝑥, ( -u_s ‘𝑥)))
cons 28175	class On_s
df-ons 28176	⊢ On_s = {𝑥 ∈ No ∣ ( R ‘𝑥) = ∅}
cseqs 28200	class seq_s𝑀( + , 𝐹)
df-seqs 28201	⊢ seq_s𝑀( + , 𝐹) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +_s 1_s ), (𝑦 + (𝐹‘(𝑥 +_s 1_s )))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) “ ω)
cnn0s 28229	class ℕ_0s
cnns 28230	class ℕ_s
df-n0s 28231	⊢ ℕ_0s = (rec((𝑥 ∈ V ↦ (𝑥 +_s 1_s )), 0_s ) “ ω)
df-nns 28232	⊢ ℕ_s = (ℕ_0s ∖ { 0_s })
czs 28289	class ℤ_s
df-zs 28290	⊢ ℤ_s = ( -_s “ (ℕ_s × ℕ_s))
c2s 28320	class 2_s
df-2s 28321	⊢ 2_s = ({ 1_s } \|s ∅)
cexps 28322	class ↑_s
df-exps 28323	⊢ ↑_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 28324	class ℤ_s[1/2]
df-zs12 28325	⊢ ℤ_s[1/2] = {𝑥 ∣ ∃𝑦 ∈ ℤ_s ∃𝑧 ∈ ℕ_0s 𝑥 = (𝑦 /_su (2_s↑_s𝑧))}
creno 28380	class ℝ_s
df-reno 28381	⊢ 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 28390	class TarskiG
cstrkgc 28391	class TarskiG_C
cstrkgb 28392	class TarskiG_B
cstrkgcb 28393	class TarskiG_CB
cstrkgld 28394	class Dim_TarskiG≥
cstrkge 28395	class TarskiG_E
citv 28396	class Itv
clng 28397	class LineG
df-itv 28398	⊢ Itv = Slot ;16
df-lng 28399	⊢ LineG = Slot ;17
df-trkgc 28411	⊢ 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 28412	⊢ 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 28413	⊢ 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 28414	⊢ 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 28415	⊢ 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 28416	⊢ 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 28473	class cgrG
df-cgrg 28474	⊢ 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 28495	class Ismt
df-ismt 28496	⊢ 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 28545	class ≤G
df-leg 28546	⊢ ≤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 28563	class hlG
df-hlg 28564	⊢ 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 28615	class pInvG
df-mir 28616	⊢ 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 28656	class ∟G
df-rag 28657	⊢ ∟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 28658	class ⟂G
df-perpg 28659	⊢ ⟂G = (𝑔 ∈ V ↦ {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ ran (LineG‘𝑔) ∧ 𝑏 ∈ ran (LineG‘𝑔)) ∧ ∃𝑥 ∈ (𝑎 ∩ 𝑏)∀𝑢 ∈ 𝑎 ∀𝑣 ∈ 𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝑔))})
chpg 28720	class hpG
df-hpg 28721	⊢ 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 28735	class midG
clmi 28736	class lInvG
df-mid 28737	⊢ midG = (𝑔 ∈ V ↦ (𝑎 ∈ (Base‘𝑔), 𝑏 ∈ (Base‘𝑔) ↦ (℩𝑚 ∈ (Base‘𝑔)𝑏 = (((pInvG‘𝑔)‘𝑚)‘𝑎))))
df-lmi 28738	⊢ 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 28770	class cgrA
df-cgra 28771	⊢ 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 28798	class inA
cleag 28799	class ≤_∠
df-inag 28800	⊢ 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 28809	⊢ ≤_∠ = (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 28828	class eqltrG
df-eqlg 28829	⊢ eqltrG = (𝑔 ∈ V ↦ {𝑥 ∈ ((Base‘𝑔) ↑_m (0..^3)) ∣ 𝑥(cgrG‘𝑔)⟨“(𝑥‘1)(𝑥‘2)(𝑥‘0)”⟩})
cttg 28836	class toTG
df-ttg 28837	⊢ 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 28851	class 𝔼
cbtwn 28852	class Btwn
ccgr 28853	class Cgr
df-ee 28854	⊢ 𝔼 = (𝑛 ∈ ℕ ↦ (ℝ ↑_m (1...𝑛)))
df-btwn 28855	⊢ 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 28856	⊢ 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 28940	class EEG
df-eeng 28941	⊢ 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 28951	class .ef
df-edgf 28952	⊢ .ef = Slot ;18
cvtx 28959	class Vtx
ciedg 28960	class iEdg
df-vtx 28961	⊢ Vtx = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1^st ‘𝑔), (Base‘𝑔)))
df-iedg 28962	⊢ iEdg = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (2^nd ‘𝑔), (.ef‘𝑔)))
cedg 29010	class Edg
df-edg 29011	⊢ Edg = (𝑔 ∈ V ↦ ran (iEdg‘𝑔))
cuhgr 29019	class UHGraph
cushgr 29020	class USHGraph
df-uhgr 29021	⊢ UHGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶(𝒫 𝑣 ∖ {∅})}
df-ushgr 29022	⊢ USHGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→(𝒫 𝑣 ∖ {∅})}
cupgr 29043	class UPGraph
cumgr 29044	class UMGraph
df-upgr 29045	⊢ UPGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}}
df-umgr 29046	⊢ UMGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2}}
cuspgr 29111	class USPGraph
cusgr 29112	class USGraph
df-uspgr 29113	⊢ USPGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}}
df-usgr 29114	⊢ USGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2}}
csubgr 29230	class SubGraph
df-subgr 29231	⊢ SubGraph = {⟨𝑠, 𝑔⟩ ∣ ((Vtx‘𝑠) ⊆ (Vtx‘𝑔) ∧ (iEdg‘𝑠) = ((iEdg‘𝑔) ↾ dom (iEdg‘𝑠)) ∧ (Edg‘𝑠) ⊆ 𝒫 (Vtx‘𝑠))}
cfusgr 29279	class FinUSGraph
df-fusgr 29280	⊢ FinUSGraph = {𝑔 ∈ USGraph ∣ (Vtx‘𝑔) ∈ Fin}
cnbgr 29295	class NeighbVtx
df-nbgr 29296	⊢ NeighbVtx = (𝑔 ∈ V, 𝑣 ∈ (Vtx‘𝑔) ↦ {𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣}) ∣ ∃𝑒 ∈ (Edg‘𝑔){𝑣, 𝑛} ⊆ 𝑒})
cuvtx 29348	class UnivVtx
df-uvtx 29349	⊢ UnivVtx = (𝑔 ∈ V ↦ {𝑣 ∈ (Vtx‘𝑔) ∣ ∀𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣})𝑛 ∈ (𝑔 NeighbVtx 𝑣)})
ccplgr 29372	class ComplGraph
ccusgr 29373	class ComplUSGraph
df-cplgr 29374	⊢ ComplGraph = {𝑔 ∣ (UnivVtx‘𝑔) = (Vtx‘𝑔)}
df-cusgr 29375	⊢ ComplUSGraph = (USGraph ∩ ComplGraph)
cvtxdg 29429	class VtxDeg
df-vtxdg 29430	⊢ 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 29519	class RegGraph
crusgr 29520	class RegUSGraph
df-rgr 29521	⊢ RegGraph = {⟨𝑔, 𝑘⟩ ∣ (𝑘 ∈ ℕ₀^* ∧ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 𝑘)}
df-rusgr 29522	⊢ RegUSGraph = {⟨𝑔, 𝑘⟩ ∣ (𝑔 ∈ USGraph ∧ 𝑔 RegGraph 𝑘)}
cewlks 29559	class EdgWalks
cwlks 29560	class Walks
cwlkson 29561	class WalksOn
df-ewlks 29562	⊢ 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 29563	⊢ 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 29564	⊢ WalksOn = (𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝑔)𝑝 ∧ (𝑝‘0) = 𝑎 ∧ (𝑝‘(♯‘𝑓)) = 𝑏)}))
ctrls 29652	class Trails
ctrlson 29653	class TrailsOn
df-trls 29654	⊢ Trails = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝑔)𝑝 ∧ Fun ^◡𝑓)})
df-trlson 29655	⊢ TrailsOn = (𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑎(WalksOn‘𝑔)𝑏)𝑝 ∧ 𝑓(Trails‘𝑔)𝑝)}))
cpths 29673	class Paths
cspths 29674	class SPaths
cpthson 29675	class PathsOn
cspthson 29676	class SPathsOn
df-pths 29677	⊢ Paths = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝑔)𝑝 ∧ Fun ^◡(𝑝 ↾ (1..^(♯‘𝑓))) ∧ ((𝑝 “ {0, (♯‘𝑓)}) ∩ (𝑝 “ (1..^(♯‘𝑓)))) = ∅)})
df-spths 29678	⊢ SPaths = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝑔)𝑝 ∧ Fun ^◡𝑝)})
df-pthson 29679	⊢ PathsOn = (𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑎(TrailsOn‘𝑔)𝑏)𝑝 ∧ 𝑓(Paths‘𝑔)𝑝)}))
df-spthson 29680	⊢ SPathsOn = (𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑎(TrailsOn‘𝑔)𝑏)𝑝 ∧ 𝑓(SPaths‘𝑔)𝑝)}))
cclwlks 29733	class ClWalks
df-clwlks 29734	⊢ ClWalks = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝑔)𝑝 ∧ (𝑝‘0) = (𝑝‘(♯‘𝑓)))})
ccrcts 29747	class Circuits
ccycls 29748	class Cycles
df-crcts 29749	⊢ Circuits = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝑔)𝑝 ∧ (𝑝‘0) = (𝑝‘(♯‘𝑓)))})
df-cycls 29750	⊢ Cycles = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Paths‘𝑔)𝑝 ∧ (𝑝‘0) = (𝑝‘(♯‘𝑓)))})
cwwlks 29788	class WWalks
cwwlksn 29789	class WWalksN
cwwlksnon 29790	class WWalksNOn
cwwspthsn 29791	class WSPathsN
cwwspthsnon 29792	class WSPathsNOn
df-wwlks 29793	⊢ WWalks = (𝑔 ∈ V ↦ {𝑤 ∈ Word (Vtx‘𝑔) ∣ (𝑤 ≠ ∅ ∧ ∀𝑖 ∈ (0..^((♯‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝑔))})
df-wwlksn 29794	⊢ WWalksN = (𝑛 ∈ ℕ₀, 𝑔 ∈ V ↦ {𝑤 ∈ (WWalks‘𝑔) ∣ (♯‘𝑤) = (𝑛 + 1)})
df-wwlksnon 29795	⊢ WWalksNOn = (𝑛 ∈ ℕ₀, 𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {𝑤 ∈ (𝑛 WWalksN 𝑔) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤‘𝑛) = 𝑏)}))
df-wspthsn 29796	⊢ WSPathsN = (𝑛 ∈ ℕ₀, 𝑔 ∈ V ↦ {𝑤 ∈ (𝑛 WWalksN 𝑔) ∣ ∃𝑓 𝑓(SPaths‘𝑔)𝑤})
df-wspthsnon 29797	⊢ 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 29943	class ClWWalks
df-clwwlk 29944	⊢ 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 29986	class ClWWalksN
df-clwwlkn 29987	⊢ ClWWalksN = (𝑛 ∈ ℕ₀, 𝑔 ∈ V ↦ {𝑤 ∈ (ClWWalks‘𝑔) ∣ (♯‘𝑤) = 𝑛})
cclwwlknon 30049	class ClWWalksNOn
df-clwwlknon 30050	⊢ ClWWalksNOn = (𝑔 ∈ V ↦ (𝑣 ∈ (Vtx‘𝑔), 𝑛 ∈ ℕ₀ ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝑔) ∣ (𝑤‘0) = 𝑣}))
cconngr 30148	class ConnGraph
df-conngr 30149	⊢ ConnGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣]∀𝑘 ∈ 𝑣 ∀𝑛 ∈ 𝑣 ∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝}
ceupth 30159	class EulerPaths
df-eupth 30160	⊢ EulerPaths = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝑔)𝑝 ∧ 𝑓:(0..^(♯‘𝑓))–onto→dom (iEdg‘𝑔))})
cfrgr 30220	class FriendGraph
df-frgr 30221	⊢ FriendGraph = {𝑔 ∈ USGraph ∣ [(Vtx‘𝑔) / 𝑣][(Edg‘𝑔) / 𝑒]∀𝑘 ∈ 𝑣 ∀𝑙 ∈ (𝑣 ∖ {𝑘})∃!𝑥 ∈ 𝑣 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝑒}
ax-flt 30434	⊢ ((𝑁 ∈ (ℤ_≥‘3) ∧ (𝑋 ∈ ℕ ∧ 𝑌 ∈ ℕ ∧ 𝑍 ∈ ℕ)) → ((𝑋↑𝑁) + (𝑌↑𝑁)) ≠ (𝑍↑𝑁))
cplig 30436	class Plig
df-plig 30437	⊢ 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 30451	class GrpOp
cgi 30452	class GId
cgn 30453	class inv
cgs 30454	class /_𝑔
df-grpo 30455	⊢ 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 30456	⊢ GId = (𝑔 ∈ V ↦ (℩𝑢 ∈ ran 𝑔∀𝑥 ∈ ran 𝑔((𝑢𝑔𝑥) = 𝑥 ∧ (𝑥𝑔𝑢) = 𝑥)))
df-ginv 30457	⊢ inv = (𝑔 ∈ GrpOp ↦ (𝑥 ∈ ran 𝑔 ↦ (℩𝑧 ∈ ran 𝑔(𝑧𝑔𝑥) = (GId‘𝑔))))
df-gdiv 30458	⊢ /_𝑔 = (𝑔 ∈ GrpOp ↦ (𝑥 ∈ ran 𝑔, 𝑦 ∈ ran 𝑔 ↦ (𝑥𝑔((inv‘𝑔)‘𝑦))))
cablo 30506	class AbelOp
df-ablo 30507	⊢ AbelOp = {𝑔 ∈ GrpOp ∣ ∀𝑥 ∈ ran 𝑔∀𝑦 ∈ ran 𝑔(𝑥𝑔𝑦) = (𝑦𝑔𝑥)}
cvc 30520	class CVec_OLD
df-vc 30521	⊢ 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 30546	class NrmCVec
cpv 30547	class +_𝑣
cba 30548	class BaseSet
cns 30549	class ·_𝑠OLD
cn0v 30550	class 0_vec
cnsb 30551	class −_𝑣
cnmcv 30552	class norm_CV
cims 30553	class IndMet
df-nv 30554	⊢ 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 30557	⊢ +_𝑣 = (1^st ∘ 1^st )
df-ba 30558	⊢ BaseSet = (𝑥 ∈ V ↦ ran ( +_𝑣 ‘𝑥))
df-sm 30559	⊢ ·_𝑠OLD = (2^nd ∘ 1^st )
df-0v 30560	⊢ 0_vec = (GId ∘ +_𝑣 )
df-vs 30561	⊢ −_𝑣 = ( /_𝑔 ∘ +_𝑣 )
df-nmcv 30562	⊢ norm_CV = 2^nd
df-ims 30563	⊢ IndMet = (𝑢 ∈ NrmCVec ↦ ((norm_CV‘𝑢) ∘ ( −_𝑣 ‘𝑢)))
cdip 30662	class ·_𝑖OLD
df-dip 30663	⊢ ·_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 30683	class SubSp
df-ssp 30684	⊢ SubSp = (𝑢 ∈ NrmCVec ↦ {𝑤 ∈ NrmCVec ∣ (( +_𝑣 ‘𝑤) ⊆ ( +_𝑣 ‘𝑢) ∧ ( ·_𝑠OLD ‘𝑤) ⊆ ( ·_𝑠OLD ‘𝑢) ∧ (norm_CV‘𝑤) ⊆ (norm_CV‘𝑢))})
clno 30702	class LnOp
cnmoo 30703	class normOp_OLD
cblo 30704	class BLnOp
c0o 30705	class 0_op
df-lno 30706	⊢ 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 30707	⊢ 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 30708	⊢ BLnOp = (𝑢 ∈ NrmCVec, 𝑤 ∈ NrmCVec ↦ {𝑡 ∈ (𝑢 LnOp 𝑤) ∣ ((𝑢 normOp_OLD 𝑤)‘𝑡) < +∞})
df-0o 30709	⊢ 0_op = (𝑢 ∈ NrmCVec, 𝑤 ∈ NrmCVec ↦ ((BaseSet‘𝑢) × {(0_vec‘𝑤)}))
caj 30710	class adj
chmo 30711	class HmOp
df-aj 30712	⊢ 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 30713	⊢ HmOp = (𝑢 ∈ NrmCVec ↦ {𝑡 ∈ dom (𝑢adj𝑢) ∣ ((𝑢adj𝑢)‘𝑡) = 𝑡})
ccphlo 30774	class CPreHil_OLD
df-ph 30775	⊢ 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 30824	class CBan
df-cbn 30825	⊢ CBan = {𝑢 ∈ NrmCVec ∣ (IndMet‘𝑢) ∈ (CMet‘(BaseSet‘𝑢))}
chlo 30847	class CHil_OLD
df-hlo 30848	⊢ CHil_OLD = (CBan ∩ CPreHil_OLD)
The list of syntax, axioms (ax-) and definitions (df-) for the Hilbert Space Explorer starts here
chba 30881	class ℋ
cva 30882	class +_ℎ
csm 30883	class ·_ℎ
csp 30884	class ·_ih
cno 30885	class norm_ℎ
c0v 30886	class 0_ℎ
cmv 30887	class −_ℎ
ccauold 30888	class Cauchy
chli 30889	class ⇝_𝑣
csh 30890	class S_ℋ
cch 30891	class C_ℋ
cort 30892	class ⊥
cph 30893	class +_ℋ
cspn 30894	class span
chj 30895	class ∨_ℋ
chsup 30896	class ∨_ℋ
c0h 30897	class 0_ℋ
ccm 30898	class 𝐶_ℋ
cpjh 30899	class proj_ℎ
chos 30900	class +_op
chot 30901	class ·_op
chod 30902	class −_op
chfs 30903	class +_fn
chft 30904	class ·_fn
ch0o 30905	class 0_hop
chio 30906	class I_op
cnop 30907	class norm_op
ccop 30908	class ContOp
clo 30909	class LinOp
cbo 30910	class BndLinOp
cuo 30911	class UniOp
cho 30912	class HrmOp
cnmf 30913	class norm_fn
cnl 30914	class null
ccnfn 30915	class ContFn
clf 30916	class LinFn
cado 30917	class adj_ℎ
cbr 30918	class bra
ck 30919	class ketbra
cleo 30920	class ≤_op
cei 30921	class eigvec
cel 30922	class eigval
cspc 30923	class Lambda
cst 30924	class States
chst 30925	class CHStates
ccv 30926	class ⋖_ℋ
cat 30927	class HAtoms
cmd 30928	class 𝑀_ℋ
cdmd 30929	class 𝑀_ℋ^*
df-hnorm 30930	⊢ norm_ℎ = (𝑥 ∈ dom dom ·_ih ↦ (√‘(𝑥 ·_ih 𝑥)))
df-hba 30931	⊢ ℋ = (BaseSet‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
df-h0v 30932	⊢ 0_ℎ = (0_vec‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
df-hvsub 30933	⊢ −_ℎ = (𝑥 ∈ ℋ, 𝑦 ∈ ℋ ↦ (𝑥 +_ℎ (-1 ·_ℎ 𝑦)))
df-hlim 30934	⊢ ⇝_𝑣 = {⟨𝑓, 𝑤⟩ ∣ ((𝑓:ℕ⟶ ℋ ∧ 𝑤 ∈ ℋ) ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ_≥‘𝑦)(norm_ℎ‘((𝑓‘𝑧) −_ℎ 𝑤)) < 𝑥)}
df-hcau 30935	⊢ Cauchy = {𝑓 ∈ ( ℋ ↑_m ℕ) ∣ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ_≥‘𝑦)(norm_ℎ‘((𝑓‘𝑦) −_ℎ (𝑓‘𝑧))) < 𝑥}
ax-hilex 30961	⊢ ℋ ∈ V
ax-hfvadd 30962	⊢ +_ℎ :( ℋ × ℋ)⟶ ℋ
ax-hvcom 30963	⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 +_ℎ 𝐵) = (𝐵 +_ℎ 𝐴))
ax-hvass 30964	⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 +_ℎ 𝐵) +_ℎ 𝐶) = (𝐴 +_ℎ (𝐵 +_ℎ 𝐶)))
ax-hv0cl 30965	⊢ 0_ℎ ∈ ℋ
ax-hvaddid 30966	⊢ (𝐴 ∈ ℋ → (𝐴 +_ℎ 0_ℎ) = 𝐴)
ax-hfvmul 30967	⊢ ·_ℎ :(ℂ × ℋ)⟶ ℋ
ax-hvmulid 30968	⊢ (𝐴 ∈ ℋ → (1 ·_ℎ 𝐴) = 𝐴)
ax-hvmulass 30969	⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝐴 · 𝐵) ·_ℎ 𝐶) = (𝐴 ·_ℎ (𝐵 ·_ℎ 𝐶)))
ax-hvdistr1 30970	⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 ·_ℎ (𝐵 +_ℎ 𝐶)) = ((𝐴 ·_ℎ 𝐵) +_ℎ (𝐴 ·_ℎ 𝐶)))
ax-hvdistr2 30971	⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝐴 + 𝐵) ·_ℎ 𝐶) = ((𝐴 ·_ℎ 𝐶) +_ℎ (𝐵 ·_ℎ 𝐶)))
ax-hvmul0 30972	⊢ (𝐴 ∈ ℋ → (0 ·_ℎ 𝐴) = 0_ℎ)
ax-hfi 31041	⊢ ·_ih :( ℋ × ℋ)⟶ℂ
ax-his1 31044	⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·_ih 𝐵) = (∗‘(𝐵 ·_ih 𝐴)))
ax-his2 31045	⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 +_ℎ 𝐵) ·_ih 𝐶) = ((𝐴 ·_ih 𝐶) + (𝐵 ·_ih 𝐶)))
ax-his3 31046	⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 ·_ℎ 𝐵) ·_ih 𝐶) = (𝐴 · (𝐵 ·_ih 𝐶)))
ax-his4 31047	⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0_ℎ) → 0 < (𝐴 ·_ih 𝐴))
ax-hcompl 31164	⊢ (𝐹 ∈ Cauchy → ∃𝑥 ∈ ℋ 𝐹 ⇝_𝑣 𝑥)
df-sh 31169	⊢ S_ℋ = {ℎ ∈ 𝒫 ℋ ∣ (0_ℎ ∈ ℎ ∧ ( +_ℎ “ (ℎ × ℎ)) ⊆ ℎ ∧ ( ·_ℎ “ (ℂ × ℎ)) ⊆ ℎ)}
df-ch 31183	⊢ C_ℋ = {ℎ ∈ S_ℋ ∣ ( ⇝_𝑣 “ (ℎ ↑_m ℕ)) ⊆ ℎ}
df-oc 31214	⊢ ⊥ = (𝑥 ∈ 𝒫 ℋ ↦ {𝑦 ∈ ℋ ∣ ∀𝑧 ∈ 𝑥 (𝑦 ·_ih 𝑧) = 0})
df-ch0 31215	⊢ 0_ℋ = {0_ℎ}
df-shs 31270	⊢ +_ℋ = (𝑥 ∈ S_ℋ , 𝑦 ∈ S_ℋ ↦ ( +_ℎ “ (𝑥 × 𝑦)))
df-span 31271	⊢ span = (𝑥 ∈ 𝒫 ℋ ↦ ∩ {𝑦 ∈ S_ℋ ∣ 𝑥 ⊆ 𝑦})
df-chj 31272	⊢ ∨_ℋ = (𝑥 ∈ 𝒫 ℋ, 𝑦 ∈ 𝒫 ℋ ↦ (⊥‘(⊥‘(𝑥 ∪ 𝑦))))
df-chsup 31273	⊢ ∨_ℋ = (𝑥 ∈ 𝒫 𝒫 ℋ ↦ (⊥‘(⊥‘∪ 𝑥)))
df-pjh 31357	⊢ proj_ℎ = (ℎ ∈ C_ℋ ↦ (𝑥 ∈ ℋ ↦ (℩𝑧 ∈ ℎ ∃𝑦 ∈ (⊥‘ℎ)𝑥 = (𝑧 +_ℎ 𝑦))))
df-cm 31545	⊢ 𝐶_ℋ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ C_ℋ ∧ 𝑦 ∈ C_ℋ ) ∧ 𝑥 = ((𝑥 ∩ 𝑦) ∨_ℋ (𝑥 ∩ (⊥‘𝑦))))}
df-hosum 31692	⊢ +_op = (𝑓 ∈ ( ℋ ↑_m ℋ), 𝑔 ∈ ( ℋ ↑_m ℋ) ↦ (𝑥 ∈ ℋ ↦ ((𝑓‘𝑥) +_ℎ (𝑔‘𝑥))))
df-homul 31693	⊢ ·_op = (𝑓 ∈ ℂ, 𝑔 ∈ ( ℋ ↑_m ℋ) ↦ (𝑥 ∈ ℋ ↦ (𝑓 ·_ℎ (𝑔‘𝑥))))
df-hodif 31694	⊢ −_op = (𝑓 ∈ ( ℋ ↑_m ℋ), 𝑔 ∈ ( ℋ ↑_m ℋ) ↦ (𝑥 ∈ ℋ ↦ ((𝑓‘𝑥) −_ℎ (𝑔‘𝑥))))
df-hfsum 31695	⊢ +_fn = (𝑓 ∈ (ℂ ↑_m ℋ), 𝑔 ∈ (ℂ ↑_m ℋ) ↦ (𝑥 ∈ ℋ ↦ ((𝑓‘𝑥) + (𝑔‘𝑥))))
df-hfmul 31696	⊢ ·_fn = (𝑓 ∈ ℂ, 𝑔 ∈ (ℂ ↑_m ℋ) ↦ (𝑥 ∈ ℋ ↦ (𝑓 · (𝑔‘𝑥))))
df-h0op 31710	⊢ 0_hop = (proj_ℎ‘0_ℋ)
df-iop 31711	⊢ I_op = (proj_ℎ‘ ℋ)
df-nmop 31801	⊢ norm_op = (𝑡 ∈ ( ℋ ↑_m ℋ) ↦ sup({𝑥 ∣ ∃𝑧 ∈ ℋ ((norm_ℎ‘𝑧) ≤ 1 ∧ 𝑥 = (norm_ℎ‘(𝑡‘𝑧)))}, ℝ^*, < ))
df-cnop 31802	⊢ 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 31803	⊢ 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 31804	⊢ BndLinOp = {𝑡 ∈ LinOp ∣ (norm_op‘𝑡) < +∞}
df-unop 31805	⊢ UniOp = {𝑡 ∣ (𝑡: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑡‘𝑥) ·_ih (𝑡‘𝑦)) = (𝑥 ·_ih 𝑦))}
df-hmop 31806	⊢ HrmOp = {𝑡 ∈ ( ℋ ↑_m ℋ) ∣ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑡‘𝑦)) = ((𝑡‘𝑥) ·_ih 𝑦)}
df-nmfn 31807	⊢ norm_fn = (𝑡 ∈ (ℂ ↑_m ℋ) ↦ sup({𝑥 ∣ ∃𝑧 ∈ ℋ ((norm_ℎ‘𝑧) ≤ 1 ∧ 𝑥 = (abs‘(𝑡‘𝑧)))}, ℝ^*, < ))
df-nlfn 31808	⊢ null = (𝑡 ∈ (ℂ ↑_m ℋ) ↦ (^◡𝑡 “ {0}))
df-cnfn 31809	⊢ 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 31810	⊢ LinFn = {𝑡 ∈ (ℂ ↑_m ℋ) ∣ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ (𝑡‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) = ((𝑥 · (𝑡‘𝑦)) + (𝑡‘𝑧))}
df-adjh 31811	⊢ adj_ℎ = {⟨𝑡, 𝑢⟩ ∣ (𝑡: ℋ⟶ ℋ ∧ 𝑢: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑡‘𝑥) ·_ih 𝑦) = (𝑥 ·_ih (𝑢‘𝑦)))}
df-bra 31812	⊢ bra = (𝑥 ∈ ℋ ↦ (𝑦 ∈ ℋ ↦ (𝑦 ·_ih 𝑥)))
df-kb 31813	⊢ ketbra = (𝑥 ∈ ℋ, 𝑦 ∈ ℋ ↦ (𝑧 ∈ ℋ ↦ ((𝑧 ·_ih 𝑦) ·_ℎ 𝑥)))
df-leop 31814	⊢ ≤_op = {⟨𝑡, 𝑢⟩ ∣ ((𝑢 −_op 𝑡) ∈ HrmOp ∧ ∀𝑥 ∈ ℋ 0 ≤ (((𝑢 −_op 𝑡)‘𝑥) ·_ih 𝑥))}
df-eigvec 31815	⊢ eigvec = (𝑡 ∈ ( ℋ ↑_m ℋ) ↦ {𝑥 ∈ ( ℋ ∖ 0_ℋ) ∣ ∃𝑧 ∈ ℂ (𝑡‘𝑥) = (𝑧 ·_ℎ 𝑥)})
df-eigval 31816	⊢ eigval = (𝑡 ∈ ( ℋ ↑_m ℋ) ↦ (𝑥 ∈ (eigvec‘𝑡) ↦ (((𝑡‘𝑥) ·_ih 𝑥) / ((norm_ℎ‘𝑥)↑2))))
df-spec 31817	⊢ Lambda = (𝑡 ∈ ( ℋ ↑_m ℋ) ↦ {𝑥 ∈ ℂ ∣ ¬ (𝑡 −_op (𝑥 ·_op ( I ↾ ℋ))): ℋ–1-1→ ℋ})
df-st 32173	⊢ 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 32174	⊢ 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 32241	⊢ ⋖_ℋ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ C_ℋ ∧ 𝑦 ∈ C_ℋ ) ∧ (𝑥 ⊊ 𝑦 ∧ ¬ ∃𝑧 ∈ C_ℋ (𝑥 ⊊ 𝑧 ∧ 𝑧 ⊊ 𝑦)))}
df-md 32242	⊢ 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 32243	⊢ 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 32300	⊢ HAtoms = {𝑥 ∈ C_ℋ ∣ 0_ℋ ⋖_ℋ 𝑥}
The list of syntax, axioms (ax-) and definitions (df-) for the User Mathboxes starts here
w2reu 32440	wff ∃!𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵𝜑
df-2reu 32441	⊢ (∃!𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵𝜑 ↔ (∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ ∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑))
cind 32806	class 𝟭
df-ind 32807	⊢ 𝟭 = (𝑜 ∈ V ↦ (𝑎 ∈ 𝒫 𝑜 ↦ (𝑥 ∈ 𝑜 ↦ if(𝑥 ∈ 𝑎, 1, 0))))
cdp2 32824	class _𝐴𝐵
df-dp2 32825	⊢ _𝐴𝐵 = (𝐴 + (𝐵 / ;10))
cdp 32841	class .
df-dp 32842	⊢ . = (𝑥 ∈ ℕ₀, 𝑦 ∈ ℝ ↦ _𝑥𝑦)
cxdiv 32870	class /_𝑒
df-xdiv 32871	⊢ /_𝑒 = (𝑥 ∈ ℝ^, 𝑦 ∈ (ℝ ∖ {0}) ↦ (℩𝑧 ∈ ℝ^ (𝑦 ·_e 𝑧) = 𝑥))
cmnt 32933	class Monot
cmgc 32934	class MGalConn
df-mnt 32935	⊢ 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 32936	⊢ 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)))})
cchn 32959	class ( < Chain𝐴)
df-chn 32960	⊢ ( < Chain𝐴) = {𝑐 ∈ Word 𝐴 ∣ ∀𝑛 ∈ (dom 𝑐 ∖ {0})(𝑐‘(𝑛 − 1)) < (𝑐‘𝑛)}
ax-xrssca 32971	⊢ ℝ_fld = (Scalar‘ℝ^*_𝑠)
ax-xrsvsca 32972	⊢ ·_e = ( ·_𝑠 ‘ℝ^*_𝑠)
ctocyc 33061	class toCyc
df-tocyc 33062	⊢ toCyc = (𝑑 ∈ V ↦ (𝑤 ∈ {𝑢 ∈ Word 𝑑 ∣ 𝑢:dom 𝑢–1-1→𝑑} ↦ (( I ↾ (𝑑 ∖ ran 𝑤)) ∪ ((𝑤 cyclShift 1) ∘ ^◡𝑤))))
csgns 33113	class sgn_s
df-sgns 33114	⊢ sgn_s = (𝑟 ∈ V ↦ (𝑥 ∈ (Base‘𝑟) ↦ if(𝑥 = (0_g‘𝑟), 0, if((0_g‘𝑟)(lt‘𝑟)𝑥, 1, -1))))
cfxp 33118	class FixPts
df-fxp 33119	⊢ FixPts = (𝑏 ∈ V, 𝑎 ∈ V ↦ {𝑥 ∈ 𝑏 ∣ ∀𝑝 ∈ dom dom 𝑎(𝑝𝑎𝑥) = 𝑥})
cinftm 33128	class ⋘
carchi 33129	class Archi
df-inftm 33130	⊢ ⋘ = (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 33131	⊢ Archi = {𝑤 ∣ (⋘‘𝑤) = ∅}
cslmd 33152	class SLMod
df-slmd 33153	⊢ 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 33203	class ~_RL
crloc 33204	class RLocal
df-erl 33205	⊢ ~_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 33206	⊢ 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 33237	class EuclF
df-euf 33238	⊢ EuclF = Slot ;21
cedom 33241	class EDomn
df-edom 33242	⊢ 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 33251	class Frac
df-frac 33252	⊢ Frac = (𝑟 ∈ V ↦ (𝑟 RLocal (RLReg‘𝑟)))
cfldgen 33259	class fldGen
df-fldgen 33260	⊢ fldGen = (𝑓 ∈ V, 𝑠 ∈ V ↦ ∩ {𝑎 ∈ (SubDRing‘𝑓) ∣ 𝑠 ⊆ 𝑎})
cresv 33274	class ↾_v
df-resv 33275	⊢ ↾_v = (𝑤 ∈ V, 𝑥 ∈ V ↦ if((Base‘(Scalar‘𝑤)) ⊆ 𝑥, 𝑤, (𝑤 sSet ⟨(Scalar‘ndx), ((Scalar‘𝑤) ↾_s 𝑥)⟩)))
cprmidl 33382	class PrmIdeal
df-prmidl 33383	⊢ 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 33406	class MaxIdeal
df-mxidl 33407	⊢ MaxIdeal = (𝑟 ∈ Ring ↦ {𝑖 ∈ (LIdeal‘𝑟) ∣ (𝑖 ≠ (Base‘𝑟) ∧ ∀𝑗 ∈ (LIdeal‘𝑟)(𝑖 ⊆ 𝑗 → (𝑗 = 𝑖 ∨ 𝑗 = (Base‘𝑟))))})
cidlsrg 33447	class IDLsrg
df-idlsrg 33448	⊢ 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 33485	class UFD
df-ufd 33486	⊢ UFD = {𝑟 ∈ IDomn ∣ ∀𝑖 ∈ ((PrmIdeal‘𝑟) ∖ {{(0_g‘𝑟)}})(𝑖 ∩ (RPrime‘𝑟)) ≠ ∅}
cldim 33570	class dim
df-dim 33571	⊢ dim = (𝑓 ∈ V ↦ ∪ (♯ “ (LBasis‘𝑓)))
cfldext 33610	class /_FldExt
cfinext 33611	class /_FinExt
cextdg 33612	class [:]
df-fldext 33613	⊢ /_FldExt = {⟨𝑒, 𝑓⟩ ∣ ((𝑒 ∈ Field ∧ 𝑓 ∈ Field) ∧ (𝑓 = (𝑒 ↾_s (Base‘𝑓)) ∧ (Base‘𝑓) ∈ (SubRing‘𝑒)))}
df-extdg 33614	⊢ [:] = (𝑒 ∈ V, 𝑓 ∈ (/_FldExt “ {𝑒}) ↦ (dim‘((subringAlg ‘𝑒)‘(Base‘𝑓))))
df-finext 33615	⊢ /_FinExt = {⟨𝑒, 𝑓⟩ ∣ (𝑒/_FldExt𝑓 ∧ (𝑒[:]𝑓) ∈ ℕ₀)}
cirng 33654	class IntgRing
df-irng 33655	⊢ IntgRing = (𝑟 ∈ V, 𝑠 ∈ V ↦ ∪ 𝑓 ∈ (Monic_1p‘(𝑟 ↾_s 𝑠))(^◡((𝑟 evalSub₁ 𝑠)‘𝑓) “ {(0_g‘𝑟)}))
calgext 33663	class /_AlgExt
df-algext 33664	⊢ /_AlgExt = {⟨𝑒, 𝑓⟩ ∣ (𝑒/_FldExt𝑓 ∧ (𝑒 IntgRing 𝑓) = (Base‘𝑒))}
cminply 33665	class minPoly
df-minply 33666	⊢ 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 33695	class Constr
df-constr 33696	⊢ 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 33759	class subMat1
df-smat 33760	⊢ 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 33777	class litMat
df-lmat 33778	⊢ litMat = (𝑚 ∈ V ↦ (𝑖 ∈ (1...(♯‘𝑚)), 𝑗 ∈ (1...(♯‘(𝑚‘0))) ↦ ((𝑚‘(𝑖 − 1))‘(𝑗 − 1))))
ccref 33808	class CovHasRef𝐴
df-cref 33809	⊢ CovHasRef𝐴 = {𝑗 ∈ Top ∣ ∀𝑦 ∈ 𝒫 𝑗(∪ 𝑗 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑗 ∩ 𝐴)𝑧Ref𝑦)}
cldlf 33818	class Ldlf
df-ldlf 33819	⊢ Ldlf = CovHasRef{𝑥 ∣ 𝑥 ≼ ω}
cpcmp 33821	class Paracomp
df-pcmp 33822	⊢ Paracomp = {𝑗 ∣ 𝑗 ∈ CovHasRef(LocFin‘𝑗)}
crspec 33828	class Spec
df-rspec 33829	⊢ Spec = (𝑟 ∈ Ring ↦ ((IDLsrg‘𝑟) ↾_s (PrmIdeal‘𝑟)))
cmetid 33852	class ~_Met
cpstm 33853	class pstoMet
df-metid 33854	⊢ ~_Met = (𝑑 ∈ ∪ ran PsMet ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ dom dom 𝑑 ∧ 𝑦 ∈ dom dom 𝑑) ∧ (𝑥𝑑𝑦) = 0)})
df-pstm 33855	⊢ pstoMet = (𝑑 ∈ ∪ ran PsMet ↦ (𝑎 ∈ (dom dom 𝑑 / (~_Met‘𝑑)), 𝑏 ∈ (dom dom 𝑑 / (~_Met‘𝑑)) ↦ ∪ {𝑧 ∣ ∃𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝑧 = (𝑥𝑑𝑦)}))
chcmp 33922	class HCmp
df-hcmp 33923	⊢ HCmp = {⟨𝑢, 𝑤⟩ ∣ ((𝑢 ∈ ∪ ran UnifOn ∧ 𝑤 ∈ CUnifSp) ∧ ((UnifSt‘𝑤) ↾_t dom ∪ 𝑢) = 𝑢 ∧ ((cls‘(TopOpen‘𝑤))‘dom ∪ 𝑢) = (Base‘𝑤))}
cqqh 33936	class ℚHom
df-qqh 33937	⊢ ℚHom = (𝑟 ∈ V ↦ ran (𝑥 ∈ ℤ, 𝑦 ∈ (^◡(ℤRHom‘𝑟) “ (Unit‘𝑟)) ↦ ⟨(𝑥 / 𝑦), (((ℤRHom‘𝑟)‘𝑥)(/_r‘𝑟)((ℤRHom‘𝑟)‘𝑦))⟩))
crrh 33959	class ℝHom
crrext 33960	class ℝExt
df-rrh 33961	⊢ ℝHom = (𝑟 ∈ V ↦ (((topGen‘ran (,))CnExt(TopOpen‘𝑟))‘(ℚHom‘𝑟)))
df-rrext 33965	⊢ ℝExt = {𝑟 ∈ (NrmRing ∩ DivRing) ∣ (((ℤMod‘𝑟) ∈ NrmMod ∧ (chr‘𝑟) = 0) ∧ (𝑟 ∈ CUnifSp ∧ (UnifSt‘𝑟) = (metUnif‘((dist‘𝑟) ↾ ((Base‘𝑟) × (Base‘𝑟))))))}
cxrh 33982	class ℝ^*Hom
df-xrh 33983	⊢ 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 33988	class ManTop
df-mntop 33989	⊢ ManTop = {⟨𝑛, 𝑗⟩ ∣ (𝑛 ∈ ℕ₀ ∧ (𝑗 ∈ 2^ndω ∧ 𝑗 ∈ Haus ∧ 𝑗 ∈ Locally [(TopOpen‘(𝔼_hil‘𝑛))] ≃ ))}
cesum 33993	class Σ^*𝑘 ∈ 𝐴𝐵
df-esum 33994	⊢ Σ^𝑘 ∈ 𝐴𝐵 = ∪ ((ℝ^_𝑠 ↾_s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵))
cofc 34061	class ∘_f/c 𝑅
df-ofc 34062	⊢ ∘_f/c 𝑅 = (𝑓 ∈ V, 𝑐 ∈ V ↦ (𝑥 ∈ dom 𝑓 ↦ ((𝑓‘𝑥)𝑅𝑐)))
csiga 34074	class sigAlgebra
df-siga 34075	⊢ sigAlgebra = (𝑜 ∈ V ↦ {𝑠 ∣ (𝑠 ⊆ 𝒫 𝑜 ∧ (𝑜 ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑜 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑠)))})
csigagen 34104	class sigaGen
df-sigagen 34105	⊢ sigaGen = (𝑥 ∈ V ↦ ∩ {𝑠 ∈ (sigAlgebra‘∪ 𝑥) ∣ 𝑥 ⊆ 𝑠})
cbrsiga 34147	class 𝔅_ℝ
df-brsiga 34148	⊢ 𝔅_ℝ = (sigaGen‘(topGen‘ran (,)))
csx 34154	class ×_s
df-sx 34155	⊢ ×_s = (𝑠 ∈ V, 𝑡 ∈ V ↦ (sigaGen‘ran (𝑥 ∈ 𝑠, 𝑦 ∈ 𝑡 ↦ (𝑥 × 𝑦))))
cmeas 34161	class measures
df-meas 34162	⊢ 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 34198	class δ
df-dde 34199	⊢ δ = (𝑎 ∈ 𝒫 ℝ ↦ if(0 ∈ 𝑎, 1, 0))
cae 34203	class a.e.
cfae 34204	class ~ a.e.
df-ae 34205	⊢ a.e. = {⟨𝑎, 𝑚⟩ ∣ (𝑚‘(∪ dom 𝑚 ∖ 𝑎)) = 0}
df-fae 34211	⊢ ~ 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 34215	class MblFnM
df-mbfm 34216	⊢ MblFnM = (𝑠 ∈ ∪ ran sigAlgebra, 𝑡 ∈ ∪ ran sigAlgebra ↦ {𝑓 ∈ (∪ 𝑡 ↑_m ∪ 𝑠) ∣ ∀𝑥 ∈ 𝑡 (^◡𝑓 “ 𝑥) ∈ 𝑠})
coms 34258	class toOMeas
df-oms 34259	⊢ 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 34268	class toCaraSiga
df-carsg 34269	⊢ toCaraSiga = (𝑚 ∈ V ↦ {𝑎 ∈ 𝒫 ∪ dom 𝑚 ∣ ∀𝑒 ∈ 𝒫 ∪ dom 𝑚((𝑚‘(𝑒 ∩ 𝑎)) +_𝑒 (𝑚‘(𝑒 ∖ 𝑎))) = (𝑚‘𝑒)})
citgm 34294	class itgm
csitm 34295	class sitm
csitg 34296	class sitg
df-sitg 34297	⊢ 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 34298	⊢ sitm = (𝑤 ∈ V, 𝑚 ∈ ∪ ran measures ↦ (𝑓 ∈ dom (𝑤sitg𝑚), 𝑔 ∈ dom (𝑤sitg𝑚) ↦ (((ℝ^*_𝑠 ↾_s (0[,]+∞))sitg𝑚)‘(𝑓 ∘_f (dist‘𝑤)𝑔))))
df-itgm 34320	⊢ itgm = (𝑤 ∈ V, 𝑚 ∈ ∪ ran measures ↦ (((metUnif‘(𝑤sitm𝑚))CnExt(UnifSt‘𝑤))‘(𝑤sitg𝑚)))
csseq 34350	class seq_str
df-sseq 34351	⊢ seq_str = (𝑚 ∈ V, 𝑓 ∈ V ↦ (𝑚 ∪ (lastS ∘ seq(♯‘𝑚)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝑓‘𝑥)”⟩)), (ℕ₀ × {(𝑚 ++ ⟨“(𝑓‘𝑚)”⟩)})))))
cfib 34363	class Fibci
df-fib 34364	⊢ Fibci = (⟨“01”⟩seq_str(𝑤 ∈ (Word ℕ₀ ∩ (^◡♯ “ (ℤ_≥‘2))) ↦ ((𝑤‘((♯‘𝑤) − 2)) + (𝑤‘((♯‘𝑤) − 1)))))
cprb 34374	class Prob
df-prob 34375	⊢ Prob = {𝑝 ∈ ∪ ran measures ∣ (𝑝‘∪ dom 𝑝) = 1}
ccprob 34398	class cprob
df-cndprob 34399	⊢ cprob = (𝑝 ∈ Prob ↦ (𝑎 ∈ dom 𝑝, 𝑏 ∈ dom 𝑝 ↦ ((𝑝‘(𝑎 ∩ 𝑏)) / (𝑝‘𝑏))))
crrv 34407	class rRndVar
df-rrv 34408	⊢ rRndVar = (𝑝 ∈ Prob ↦ (dom 𝑝MblFnM𝔅_ℝ))
corvc 34423	class ∘_RV/𝑐𝑅
df-orvc 34424	⊢ ∘_RV/𝑐𝑅 = (𝑥 ∈ {𝑥 ∣ Fun 𝑥}, 𝑎 ∈ V ↦ (^◡𝑥 “ {𝑦 ∣ 𝑦𝑅𝑎}))
crepr 34575	class repr
df-repr 34576	⊢ repr = (𝑠 ∈ ℕ₀ ↦ (𝑏 ∈ 𝒫 ℕ, 𝑚 ∈ ℤ ↦ {𝑐 ∈ (𝑏 ↑_m (0..^𝑠)) ∣ Σ𝑎 ∈ (0..^𝑠)(𝑐‘𝑎) = 𝑚}))
cvts 34602	class vts
df-vts 34603	⊢ vts = (𝑙 ∈ (ℂ ↑_m ℕ), 𝑛 ∈ ℕ₀ ↦ (𝑥 ∈ ℂ ↦ Σ𝑎 ∈ (1...𝑛)((𝑙‘𝑎) · (exp‘((i · (2 · π)) · (𝑎 · 𝑥))))))
ax-hgt749 34611	⊢ ∀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 34612	⊢ ∀𝑥 ∈ ℝ⁺ (ψ‘𝑥) < ((1._0_3_8_83) · 𝑥)
ax-ros336 34613	⊢ ∀𝑥 ∈ ℝ⁺ ((ψ‘𝑥) − (θ‘𝑥)) < ((1._4_2_62) · (√‘𝑥))
cstrkg2d 34631	class TarskiG_2D
df-trkg2d 34632	⊢ 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 34636	class AFS
df-afs 34637	⊢ 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 34641	class leftpad
df-lpad 34642	⊢ leftpad = (𝑐 ∈ V, 𝑤 ∈ V ↦ (𝑙 ∈ ℕ₀ ↦ (((0..^(𝑙 − (♯‘𝑤))) × {𝑐}) ++ 𝑤)))
w-bnj17 34652	wff (𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃)
df-bnj17 34653	⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ ((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃))
c-bnj14 34654	class pred(𝑋, 𝐴, 𝑅)
df-bnj14 34655	⊢ pred(𝑋, 𝐴, 𝑅) = {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋}
w-bnj13 34656	wff 𝑅 Se 𝐴
df-bnj13 34657	⊢ (𝑅 Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 pred(𝑥, 𝐴, 𝑅) ∈ V)
w-bnj15 34658	wff 𝑅 FrSe 𝐴
df-bnj15 34659	⊢ (𝑅 FrSe 𝐴 ↔ (𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴))
c-bnj18 34660	class trCl(𝑋, 𝐴, 𝑅)
df-bnj18 34661	⊢ 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 34662	wff TrFo(𝐵, 𝐴, 𝑅)
df-bnj19 34663	⊢ ( TrFo(𝐵, 𝐴, 𝑅) ↔ ∀𝑥 ∈ 𝐵 pred(𝑥, 𝐴, 𝑅) ⊆ 𝐵)
ax-regs 35060	⊢ (∃𝑥𝜑 → ∃𝑦(∀𝑥(𝑥 = 𝑦 → 𝜑) ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ ∀𝑥(𝑥 = 𝑧 → 𝜑))))
cacycgr 35114	class AcyclicGraph
df-acycgr 35115	⊢ AcyclicGraph = {𝑔 ∣ ¬ ∃𝑓∃𝑝(𝑓(Cycles‘𝑔)𝑝 ∧ 𝑓 ≠ ∅)}
ax-7d 35131	⊢ (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑)
ax-8d 35132	⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 → 𝑦 = 𝑧))
ax-9d1 35133	⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑥
ax-9d2 35134	⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦
ax-10d 35135	⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
ax-11d 35136	⊢ (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
cretr 35189	class Retr
df-retr 35190	⊢ Retr = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ {𝑟 ∈ (𝑗 Cn 𝑘) ∣ ∃𝑠 ∈ (𝑘 Cn 𝑗)((𝑟 ∘ 𝑠)(𝑗 Htpy 𝑗)( I ↾ ∪ 𝑗)) ≠ ∅})
cpconn 35191	class PConn
csconn 35192	class SConn
df-pconn 35193	⊢ PConn = {𝑗 ∈ Top ∣ ∀𝑥 ∈ ∪ 𝑗∀𝑦 ∈ ∪ 𝑗∃𝑓 ∈ (II Cn 𝑗)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)}
df-sconn 35194	⊢ SConn = {𝑗 ∈ PConn ∣ ∀𝑓 ∈ (II Cn 𝑗)((𝑓‘0) = (𝑓‘1) → 𝑓( ≃_ph‘𝑗)((0[,]1) × {(𝑓‘0)}))}
ccvm 35227	class CovMap
df-cvm 35228	⊢ 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 35305	class ∈_𝑔
cgna 35306	class ⊼_𝑔
cgol 35307	class ∀_𝑔𝑁𝑈
csat 35308	class Sat
cfmla 35309	class Fmla
csate 35310	class Sat_∈
cprv 35311	class ⊧
df-goel 35312	⊢ ∈_𝑔 = (𝑥 ∈ (ω × ω) ↦ ⟨∅, 𝑥⟩)
df-gona 35313	⊢ ⊼_𝑔 = (𝑥 ∈ (V × V) ↦ ⟨1_o, 𝑥⟩)
df-goal 35314	⊢ ∀_𝑔𝑁𝑈 = ⟨2_o, ⟨𝑁, 𝑈⟩⟩
df-sat 35315	⊢ 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 35316	⊢ Sat_∈ = (𝑚 ∈ V, 𝑢 ∈ V ↦ (((𝑚 Sat ( E ∩ (𝑚 × 𝑚)))‘ω)‘𝑢))
df-fmla 35317	⊢ Fmla = (𝑛 ∈ suc ω ↦ dom ((∅ Sat ∅)‘𝑛))
df-prv 35318	⊢ ⊧ = {⟨𝑚, 𝑢⟩ ∣ (𝑚 Sat_∈ 𝑢) = (𝑚 ↑_m ω)}
cgon 35404	class ¬_𝑔𝑈
cgoa 35405	class ∧_𝑔
cgoi 35406	class →_𝑔
cgoo 35407	class ∨_𝑔
cgob 35408	class ↔_𝑔
cgoq 35409	class =_𝑔
cgox 35410	class ∃_𝑔𝑁𝑈
df-gonot 35411	⊢ ¬_𝑔𝑈 = (𝑈⊼_𝑔𝑈)
df-goan 35412	⊢ ∧_𝑔 = (𝑢 ∈ V, 𝑣 ∈ V ↦ ¬_𝑔(𝑢⊼_𝑔𝑣))
df-goim 35413	⊢ →_𝑔 = (𝑢 ∈ V, 𝑣 ∈ V ↦ (𝑢⊼_𝑔¬_𝑔𝑣))
df-goor 35414	⊢ ∨_𝑔 = (𝑢 ∈ V, 𝑣 ∈ V ↦ (¬_𝑔𝑢 →_𝑔 𝑣))
df-gobi 35415	⊢ ↔_𝑔 = (𝑢 ∈ V, 𝑣 ∈ V ↦ ((𝑢 →_𝑔 𝑣)∧_𝑔(𝑣 →_𝑔 𝑢)))
df-goeq 35416	⊢ =_𝑔 = (𝑢 ∈ ω, 𝑣 ∈ ω ↦ ⦋suc (𝑢 ∪ 𝑣) / 𝑤⦌∀_𝑔𝑤((𝑤∈_𝑔𝑢) ↔_𝑔 (𝑤∈_𝑔𝑣)))
df-goex 35417	⊢ ∃_𝑔𝑁𝑈 = ¬_𝑔∀_𝑔𝑁¬_𝑔𝑈
cgze 35418	class AxExt
cgzr 35419	class AxRep
cgzp 35420	class AxPow
cgzu 35421	class AxUn
cgzg 35422	class AxReg
cgzi 35423	class AxInf
cgzf 35424	class ZF
df-gzext 35425	⊢ AxExt = (∀_𝑔2_o((2_o∈_𝑔∅) ↔_𝑔 (2_o∈_𝑔1_o)) →_𝑔 (∅=_𝑔1_o))
df-gzrep 35426	⊢ 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 35427	⊢ AxPow = ∃_𝑔1_o∀_𝑔2_o(∀_𝑔1_o((1_o∈_𝑔2_o) ↔_𝑔 (1_o∈_𝑔∅)) →_𝑔 (2_o∈_𝑔1_o))
df-gzun 35428	⊢ AxUn = ∃_𝑔1_o∀_𝑔2_o(∃_𝑔1_o((2_o∈_𝑔1_o)∧_𝑔(1_o∈_𝑔∅)) →_𝑔 (2_o∈_𝑔1_o))
df-gzreg 35429	⊢ 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 35430	⊢ AxInf = ∃_𝑔1_o((∅∈_𝑔1_o)∧_𝑔∀_𝑔2_o((2_o∈_𝑔1_o) →_𝑔 ∃_𝑔∅((2_o∈_𝑔∅)∧_𝑔(∅∈_𝑔1_o))))
df-gzf 35431	⊢ ZF = {𝑚 ∣ ((Tr 𝑚 ∧ 𝑚⊧AxExt ∧ 𝑚⊧AxPow) ∧ (𝑚⊧AxUn ∧ 𝑚⊧AxReg ∧ 𝑚⊧AxInf) ∧ ∀𝑢 ∈ (Fmla‘ω)𝑚⊧(AxRep‘𝑢))}
cmcn 35432	class mCN
cmvar 35433	class mVR
cmty 35434	class mType
cmvt 35435	class mVT
cmtc 35436	class mTC
cmax 35437	class mAx
cmrex 35438	class mREx
cmex 35439	class mEx
cmdv 35440	class mDV
cmvrs 35441	class mVars
cmrsub 35442	class mRSubst
cmsub 35443	class mSubst
cmvh 35444	class mVH
cmpst 35445	class mPreSt
cmsr 35446	class mStRed
cmsta 35447	class mStat
cmfs 35448	class mFS
cmcls 35449	class mCls
cmpps 35450	class mPPSt
cmthm 35451	class mThm
df-mcn 35452	⊢ mCN = Slot 1
df-mvar 35453	⊢ mVR = Slot 2
df-mty 35454	⊢ mType = Slot 3
df-mtc 35455	⊢ mTC = Slot 4
df-mmax 35456	⊢ mAx = Slot 5
df-mvt 35457	⊢ mVT = (𝑡 ∈ V ↦ ran (mType‘𝑡))
df-mrex 35458	⊢ mREx = (𝑡 ∈ V ↦ Word ((mCN‘𝑡) ∪ (mVR‘𝑡)))
df-mex 35459	⊢ mEx = (𝑡 ∈ V ↦ ((mTC‘𝑡) × (mREx‘𝑡)))
df-mdv 35460	⊢ mDV = (𝑡 ∈ V ↦ (((mVR‘𝑡) × (mVR‘𝑡)) ∖ I ))
df-mvrs 35461	⊢ mVars = (𝑡 ∈ V ↦ (𝑒 ∈ (mEx‘𝑡) ↦ (ran (2^nd ‘𝑒) ∩ (mVR‘𝑡))))
df-mrsub 35462	⊢ 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 35463	⊢ mSubst = (𝑡 ∈ V ↦ (𝑓 ∈ ((mREx‘𝑡) ↑_pm (mVR‘𝑡)) ↦ (𝑒 ∈ (mEx‘𝑡) ↦ ⟨(1^st ‘𝑒), (((mRSubst‘𝑡)‘𝑓)‘(2^nd ‘𝑒))⟩)))
df-mvh 35464	⊢ mVH = (𝑡 ∈ V ↦ (𝑣 ∈ (mVR‘𝑡) ↦ ⟨((mType‘𝑡)‘𝑣), ⟨“𝑣”⟩⟩))
df-mpst 35465	⊢ mPreSt = (𝑡 ∈ V ↦ (({𝑑 ∈ 𝒫 (mDV‘𝑡) ∣ ^◡𝑑 = 𝑑} × (𝒫 (mEx‘𝑡) ∩ Fin)) × (mEx‘𝑡)))
df-msr 35466	⊢ 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 35467	⊢ mStat = (𝑡 ∈ V ↦ ran (mStRed‘𝑡))
df-mfs 35468	⊢ 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 35469	⊢ 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 35470	⊢ mPPSt = (𝑡 ∈ V ↦ {⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∣ (⟨𝑑, ℎ, 𝑎⟩ ∈ (mPreSt‘𝑡) ∧ 𝑎 ∈ (𝑑(mCls‘𝑡)ℎ))})
df-mthm 35471	⊢ mThm = (𝑡 ∈ V ↦ (^◡(mStRed‘𝑡) “ ((mStRed‘𝑡) “ (mPPSt‘𝑡))))
cm0s 35557	class m0St
cmsa 35558	class mSA
cmwgfs 35559	class mWGFS
cmsy 35560	class mSyn
cmesy 35561	class mESyn
cmgfs 35562	class mGFS
cmtree 35563	class mTree
cmst 35564	class mST
cmsax 35565	class mSAX
cmufs 35566	class mUFS
df-m0s 35567	⊢ m0St = (𝑎 ∈ V ↦ ⟨∅, ∅, 𝑎⟩)
df-msa 35568	⊢ mSA = (𝑡 ∈ V ↦ {𝑎 ∈ (mEx‘𝑡) ∣ ((m0St‘𝑎) ∈ (mAx‘𝑡) ∧ (1^st ‘𝑎) ∈ (mVT‘𝑡) ∧ Fun (^◡(2^nd ‘𝑎) ↾ (mVR‘𝑡)))})
df-mwgfs 35569	⊢ 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 35570	⊢ mSyn = Slot 6
df-mesyn 35571	⊢ mESyn = (𝑡 ∈ V ↦ (𝑐 ∈ (mTC‘𝑡), 𝑒 ∈ (mREx‘𝑡) ↦ (((mSyn‘𝑡)‘𝑐)m0St𝑒)))
df-mgfs 35572	⊢ 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 35573	⊢ 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 35574	⊢ mST = (𝑡 ∈ V ↦ ((∅(mTree‘𝑡)∅) ↾ ((mEx‘𝑡) ↾ (mVT‘𝑡))))
df-msax 35575	⊢ mSAX = (𝑡 ∈ V ↦ (𝑝 ∈ (mSA‘𝑡) ↦ ((mVH‘𝑡) “ ((mVars‘𝑡)‘𝑝))))
df-mufs 35576	⊢ mUFS = {𝑡 ∈ mGFS ∣ Fun (mST‘𝑡)}
cmuv 35577	class mUV
cmvl 35578	class mVL
cmvsb 35579	class mVSubst
cmfsh 35580	class mFresh
cmfr 35581	class mFRel
cmevl 35582	class mEval
cmdl 35583	class mMdl
cusyn 35584	class mUSyn
cgmdl 35585	class mGMdl
cmitp 35586	class mItp
cmfitp 35587	class mFromItp
df-muv 35588	⊢ mUV = Slot 7
df-mfsh 35589	⊢ mFresh = Slot ;19
df-mevl 35590	⊢ mEval = Slot ;20
df-mvl 35591	⊢ mVL = (𝑡 ∈ V ↦ X𝑣 ∈ (mVR‘𝑡)((mUV‘𝑡) “ {((mType‘𝑡)‘𝑣)}))
df-mvsb 35592	⊢ 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 35593	⊢ 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 35594	⊢ 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 35595	⊢ mUSyn = (𝑡 ∈ V ↦ (𝑣 ∈ (mUV‘𝑡) ↦ ⟨((mSyn‘𝑡)‘(1^st ‘𝑣)), (2^nd ‘𝑣)⟩))
df-gmdl 35596	⊢ 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 35597	⊢ 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 35598	⊢ 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 35599	class cplMetSp
chlb 35600	class HomLimB
chlim 35601	class HomLim
cpfl 35602	class polyFld
csf1 35603	class splitFld₁
csf 35604	class splitFld
cpsl 35605	class polySplitLim
df-cplmet 35606	⊢ 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 35607	⊢ 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 35608	⊢ 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 35609	⊢ 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 35616	⊢ 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 35617	⊢ 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 35618	⊢ 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 35619	class ZRing
cgf 35620	class GF
cgfo 35621	class GF_∞
ceqp 35622	class ~Qp
crqp 35623	class /Qp
cqp 35624	class Qp
czp 35625	class Zp
cqpa 35626	class _Qp
ccp 35627	class Cp
df-zrng 35628	⊢ ZRing = (𝑟 ∈ V ↦ (𝑟 IntgRing ran (ℤRHom‘𝑟)))
df-gf 35629	⊢ 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 35630	⊢ 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 35631	⊢ ~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 35632	⊢ /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 35633	⊢ 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 35634	⊢ Zp = (ZRing ∘ Qp)
df-qpa 35635	⊢ _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 35636	⊢ Cp = ( cplMetSp ∘ _Qp)
ccloneop 35667	class CloneOp
df-cloneop 35668	⊢ CloneOp = (𝑎 ∈ V ↦ {𝑥 ∣ ∃𝑛 ∈ (ω ∖ 1_o)𝑥 ∈ (𝑎 ↑_m (𝑎 ↑_m 𝑛))})
cprj 35669	class prj
df-prj 35670	⊢ prj = (𝑎 ∈ V ↦ (𝑛 ∈ (ω ∖ 1_o), 𝑖 ∈ 𝑛 ↦ (𝑥 ∈ (𝑎 ↑_m 𝑛) ↦ (𝑥‘𝑖))))
csuppos 35671	class suppos
df-suppos 35672	⊢ 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 35783	class wsuc(𝑅, 𝐴, 𝑋)
cwlim 35784	class WLim(𝑅, 𝐴)
df-wsuc 35785	⊢ wsuc(𝑅, 𝐴, 𝑋) = inf(Pred(^◡𝑅, 𝐴, 𝑋), 𝐴, 𝑅)
df-wlim 35786	⊢ WLim(𝑅, 𝐴) = {𝑥 ∈ 𝐴 ∣ (𝑥 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑥 = sup(Pred(𝑅, 𝐴, 𝑥), 𝐴, 𝑅))}
ctxp 35803	class (𝐴 ⊗ 𝐵)
cpprod 35804	class pprod(𝑅, 𝑆)
csset 35805	class SSet
ctrans 35806	class Trans
cbigcup 35807	class Bigcup
cfix 35808	class Fix 𝐴
climits 35809	class Limits
cfuns 35810	class Funs
csingle 35811	class Singleton
csingles 35812	class Singletons
cimage 35813	class Image𝐴
ccart 35814	class Cart
cimg 35815	class Img
cdomain 35816	class Domain
crange 35817	class Range
capply 35818	class Apply
ccup 35819	class Cup
ccap 35820	class Cap
csuccf 35821	class Succ
cfunpart 35822	class Funpart𝐹
cfullfn 35823	class FullFun𝐹
crestrict 35824	class Restrict
cub 35825	class UB𝑅
clb 35826	class LB𝑅
df-txp 35827	⊢ (𝐴 ⊗ 𝐵) = ((^◡(1^st ↾ (V × V)) ∘ 𝐴) ∩ (^◡(2^nd ↾ (V × V)) ∘ 𝐵))
df-pprod 35828	⊢ pprod(𝐴, 𝐵) = ((𝐴 ∘ (1^st ↾ (V × V))) ⊗ (𝐵 ∘ (2^nd ↾ (V × V))))
df-sset 35829	⊢ SSet = ((V × V) ∖ ran ( E ⊗ (V ∖ E )))
df-trans 35830	⊢ Trans = (V ∖ ran (( E ∘ E ) ∖ E ))
df-bigcup 35831	⊢ Bigcup = ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ E ) ⊗ V)))
df-fix 35832	⊢ Fix 𝐴 = dom (𝐴 ∩ I )
df-limits 35833	⊢ Limits = ((On ∩ Fix Bigcup ) ∖ {∅})
df-funs 35834	⊢ Funs = (𝒫 (V × V) ∖ Fix ( E ∘ ((1^st ⊗ ((V ∖ I ) ∘ 2^nd )) ∘ ^◡ E )))
df-singleton 35835	⊢ Singleton = ((V × V) ∖ ran ((V ⊗ E ) △ ( I ⊗ V)))
df-singles 35836	⊢ Singletons = ran Singleton
df-image 35837	⊢ Image𝐴 = ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ ^◡𝐴) ⊗ V)))
df-cart 35838	⊢ Cart = (((V × V) × V) ∖ ran ((V ⊗ E ) △ (pprod( E , E ) ⊗ V)))
df-img 35839	⊢ Img = (Image((2^nd ∘ 1^st ) ↾ (1^st ↾ (V × V))) ∘ Cart)
df-domain 35840	⊢ Domain = Image(1^st ↾ (V × V))
df-range 35841	⊢ Range = Image(2^nd ↾ (V × V))
df-cup 35842	⊢ Cup = (((V × V) × V) ∖ ran ((V ⊗ E ) △ (((^◡1^st ∘ E ) ∪ (^◡2^nd ∘ E )) ⊗ V)))
df-cap 35843	⊢ Cap = (((V × V) × V) ∖ ran ((V ⊗ E ) △ (((^◡1^st ∘ E ) ∩ (^◡2^nd ∘ E )) ⊗ V)))
df-restrict 35844	⊢ Restrict = (Cap ∘ (1^st ⊗ (Cart ∘ (2^nd ⊗ (Range ∘ 1^st )))))
df-succf 35845	⊢ Succ = (Cup ∘ ( I ⊗ Singleton))
df-apply 35846	⊢ Apply = (( Bigcup ∘ Bigcup ) ∘ (((V × V) ∖ ran ((V ⊗ E ) △ (( E ↾ Singletons ) ⊗ V))) ∘ ((Singleton ∘ Img) ∘ pprod( I , Singleton))))
df-funpart 35847	⊢ Funpart𝐹 = (𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))
df-fullfun 35848	⊢ FullFun𝐹 = (Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))
df-ub 35849	⊢ UB𝑅 = ((V × V) ∖ ((V ∖ 𝑅) ∘ ^◡ E ))
df-lb 35850	⊢ LB𝑅 = UB^◡𝑅
caltop 35929	class ⟪𝐴, 𝐵⟫
caltxp 35930	class (𝐴 ×× 𝐵)
df-altop 35931	⊢ ⟪𝐴, 𝐵⟫ = {{𝐴}, {𝐴, {𝐵}}}
df-altxp 35932	⊢ (𝐴 ×× 𝐵) = {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = ⟪𝑥, 𝑦⟫}
cofs 35955	class OuterFiveSeg
df-ofs 35956	⊢ 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 36002	class TransportTo
df-transport 36003	⊢ 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 36008	class InnerFiveSeg
ccgr3 36009	class Cgr3
ccolin 36010	class Colinear
cfs 36011	class FiveSeg
df-colinear 36012	⊢ 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 36013	⊢ 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 36014	⊢ 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 36015	⊢ 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 36079	class Seg_≤
df-segle 36080	⊢ 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 36092	class OutsideOf
df-outsideof 36093	⊢ OutsideOf = ( Colinear ∖ Btwn )
cline2 36107	class Line
cray 36108	class Ray
clines2 36109	class LinesEE
df-line2 36110	⊢ Line = {⟨⟨𝑎, 𝑏⟩, 𝑙⟩ ∣ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎 ≠ 𝑏) ∧ 𝑙 = [⟨𝑎, 𝑏⟩]^◡ Colinear )}
df-ray 36111	⊢ Ray = {⟨⟨𝑝, 𝑎⟩, 𝑟⟩ ∣ ∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑎) ∧ 𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩})}
df-lines2 36112	⊢ LinesEE = ran Line
cfwddif 36131	class △
df-fwddif 36132	⊢ △ = (𝑓 ∈ (ℂ ↑_pm ℂ) ↦ (𝑥 ∈ {𝑦 ∈ dom 𝑓 ∣ (𝑦 + 1) ∈ dom 𝑓} ↦ ((𝑓‘(𝑥 + 1)) − (𝑓‘𝑥))))
cfwddifn 36133	class △ⁿ
df-fwddifn 36134	⊢ △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 36145	class Hf
df-hf 36146	⊢ Hf = ∪ (𝑅₁ “ ω)
cfne 36309	class Fne
df-fne 36310	⊢ Fne = {⟨𝑥, 𝑦⟩ ∣ (∪ 𝑥 = ∪ 𝑦 ∧ ∀𝑧 ∈ 𝑥 𝑧 ⊆ ∪ (𝑦 ∩ 𝒫 𝑧))}
w3nand 36370	wff (𝜑 ⊼ 𝜓 ⊼ 𝜒)
df-3nand 36371	⊢ ((𝜑 ⊼ 𝜓 ⊼ 𝜒) ↔ (𝜑 → (𝜓 → ¬ 𝜒)))
cgcdOLD 36432	class gcd_OLD (𝐴, 𝐵)
df-gcdOLD 36433	⊢ gcd_OLD (𝐴, 𝐵) = sup({𝑥 ∈ ℕ ∣ ((𝐴 / 𝑥) ∈ ℕ ∧ (𝐵 / 𝑥) ∈ ℕ)}, ℕ, < )
cprvb 36570	wff Prv 𝜑
ax-prv1 36571	⊢ 𝜑 ⇒ ⊢ Prv 𝜑
ax-prv2 36572	⊢ (Prv (𝜑 → 𝜓) → (Prv 𝜑 → Prv 𝜓))
ax-prv3 36573	⊢ (Prv 𝜑 → Prv Prv 𝜑)
wmoo 36624	wff ∃**𝑥𝜑
df-bj-mo 36625	⊢ (∃**𝑥𝜑 ↔ ∀𝑧∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))
wnnf 36696	wff Ⅎ'𝑥𝜑
df-bj-nnf 36697	⊢ (Ⅎ'𝑥𝜑 ↔ ((∃𝑥𝜑 → 𝜑) ∧ (𝜑 → ∀𝑥𝜑)))
bj-cgab 36906	class {𝐴 ∣ 𝑥 ∣ 𝜑}
df-bj-gab 36907	⊢ {𝐴 ∣ 𝑥 ∣ 𝜑} = {𝑦 ∣ ∃𝑥(𝐴 = 𝑦 ∧ 𝜑)}
wrnf 36914	wff Ⅎ𝑥 ∈ 𝐴𝜑
df-bj-rnf 36915	⊢ (Ⅎ𝑥 ∈ 𝐴𝜑 ↔ (∃𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 𝜑))
bj-csngl 36938	class sngl 𝐴
df-bj-sngl 36939	⊢ sngl 𝐴 = {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = {𝑦}}
bj-ctag 36947	class tag 𝐴
df-bj-tag 36948	⊢ tag 𝐴 = (sngl 𝐴 ∪ {∅})
bj-cproj 36963	class (𝐴 Proj 𝐵)
df-bj-proj 36964	⊢ (𝐴 Proj 𝐵) = {𝑥 ∣ {𝑥} ∈ (𝐵 “ {𝐴})}
bj-c1upl 36970	class ⦅𝐴⦆
df-bj-1upl 36971	⊢ ⦅𝐴⦆ = ({∅} × tag 𝐴)
bj-cpr1 36973	class pr₁ 𝐴
df-bj-pr1 36974	⊢ pr₁ 𝐴 = (∅ Proj 𝐴)
bj-c2uple 36983	class ⦅𝐴, 𝐵⦆
df-bj-2upl 36984	⊢ ⦅𝐴, 𝐵⦆ = (⦅𝐴⦆ ∪ ({1_o} × tag 𝐵))
bj-cpr2 36987	class pr₂ 𝐴
df-bj-pr2 36988	⊢ pr₂ 𝐴 = (1_o Proj 𝐴)
ax-bj-sn 37006	⊢ ∀𝑥∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ 𝑧 = 𝑥)
ax-bj-bun 37010	⊢ ∀𝑥∀𝑦∃𝑧∀𝑡(𝑡 ∈ 𝑧 ↔ (𝑡 ∈ 𝑥 ∨ 𝑡 ∈ 𝑦))
ax-bj-adj 37015	⊢ ∀𝑥∀𝑦∃𝑧∀𝑡(𝑡 ∈ 𝑧 ↔ (𝑡 ∈ 𝑥 ∨ 𝑡 = 𝑦))
celwise 37052	class elwise
df-elwise 37053	⊢ elwise = (𝑜 ∈ V ↦ (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑧 ∣ ∃𝑢 ∈ 𝑥 ∃𝑣 ∈ 𝑦 𝑧 = (𝑢𝑜𝑣)}))
cmoore 37076	class Moore
df-bj-moore 37077	⊢ Moore = {𝑥 ∣ ∀𝑦 ∈ 𝒫 𝑥(∪ 𝑥 ∩ ∩ 𝑦) ∈ 𝑥}
cmpt3 37093	class (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵, 𝑧 ∈ 𝐶 ↦ 𝐷)
df-bj-mpt3 37094	⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵, 𝑧 ∈ 𝐶 ↦ 𝐷) = {⟨𝑠, 𝑡⟩ ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 (𝑠 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝑡 = 𝐷)}
csethom 37095	class Set⟶
df-bj-sethom 37096	⊢ Set⟶ = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓 ∣ 𝑓:𝑥⟶𝑦})
ctophom 37097	class Top⟶
df-bj-tophom 37098	⊢ Top⟶ = (𝑥 ∈ TopSp, 𝑦 ∈ TopSp ↦ {𝑓 ∈ ((Base‘𝑥) Set⟶ (Base‘𝑦)) ∣ ∀𝑢 ∈ (TopOpen‘𝑦)(^◡𝑓 “ 𝑢) ∈ (TopOpen‘𝑥)})
cmgmhom 37099	class Mgm⟶
df-bj-mgmhom 37100	⊢ 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 37101	class TopMgm⟶
df-bj-topmgmhom 37102	⊢ TopMgm⟶ = (𝑥 ∈ TopMnd, 𝑦 ∈ TopMnd ↦ ((𝑥 Top⟶ 𝑦) ∩ (𝑥 Mgm⟶ 𝑦)))
ccur- 37103	class curry_
df-bj-cur 37104	⊢ curry_ = (𝑥 ∈ V, 𝑦 ∈ V, 𝑧 ∈ V ↦ (𝑓 ∈ ((𝑥 × 𝑦) Set⟶ 𝑧) ↦ (𝑎 ∈ 𝑥 ↦ (𝑏 ∈ 𝑦 ↦ (𝑓‘⟨𝑎, 𝑏⟩)))))
cunc- 37105	class uncurry_
df-bj-unc 37106	⊢ uncurry_ = (𝑥 ∈ V, 𝑦 ∈ V, 𝑧 ∈ V ↦ (𝑓 ∈ (𝑥 Set⟶ (𝑦 Set⟶ 𝑧)) ↦ (𝑎 ∈ 𝑥, 𝑏 ∈ 𝑦 ↦ ((𝑓‘𝑎)‘𝑏))))
cstrset 37107	class [𝐵 / 𝐴]_struct𝑆
df-strset 37108	⊢ [𝐵 / 𝐴]_struct𝑆 = ((𝑆 ↾ (V ∖ {(𝐴‘ndx)})) ∪ {⟨(𝐴‘ndx), 𝐵⟩})
cdiag2 37145	class Id
df-bj-diag 37146	⊢ Id = (𝑥 ∈ V ↦ ( I ↾ 𝑥))
cimdir 37151	class 𝒫_*
df-imdir 37152	⊢ 𝒫_* = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑟 ∈ 𝒫 (𝑎 × 𝑏) ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝑎 ∧ 𝑦 ⊆ 𝑏) ∧ (𝑟 “ 𝑥) = 𝑦)}))
ciminv 37164	class 𝒫^*
df-iminv 37165	⊢ 𝒫^* = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑟 ∈ 𝒫 (𝑎 × 𝑏) ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝑎 ∧ 𝑦 ⊆ 𝑏) ∧ 𝑥 = (^◡𝑟 “ 𝑦))}))
cfractemp 37169	class {_R
df-bj-fractemp 37170	⊢ {_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 37171	class +∞e^iτ
df-bj-inftyexpitau 37172	⊢ +∞e^iτ = (𝑥 ∈ ℝ ↦ ⟨({_R‘(1^st ‘𝑥)), {R}⟩)
cccinftyN 37173	class ℂ_∞N
df-bj-ccinftyN 37174	⊢ ℂ_∞N = ran +∞e^iτ
chalf 37176	class 1/2
df-bj-onehalf 37177	⊢ 1/2 = (℩𝑥 ∈ R (𝑥 +_R 𝑥) = 1_R)
cinftyexpi 37179	class +∞eⁱ
df-bj-inftyexpi 37180	⊢ +∞eⁱ = (𝑥 ∈ (-π(,]π) ↦ ⟨𝑥, ℂ⟩)
cccinfty 37184	class ℂ_∞
df-bj-ccinfty 37185	⊢ ℂ_∞ = ran +∞eⁱ
cccbar 37188	class ℂ̅
df-bj-ccbar 37189	⊢ ℂ̅ = (ℂ ∪ ℂ_∞)
cpinfty 37192	class +∞
df-bj-pinfty 37193	⊢ +∞ = (+∞eⁱ‘0)
cminfty 37196	class -∞
df-bj-minfty 37197	⊢ -∞ = (+∞eⁱ‘π)
crrbar 37201	class ℝ̅
df-bj-rrbar 37202	⊢ ℝ̅ = (ℝ ∪ {-∞, +∞})
cinfty 37203	class ∞
df-bj-infty 37204	⊢ ∞ = 𝒫 ∪ ℂ
ccchat 37205	class ℂ̂
df-bj-cchat 37206	⊢ ℂ̂ = (ℂ ∪ {∞})
crrhat 37207	class ℝ̂
df-bj-rrhat 37208	⊢ ℝ̂ = (ℝ ∪ {∞})
caddcc 37210	class +_ℂ̅
df-bj-addc 37211	⊢ +_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 37212	class -_ℂ̅
df-bj-oppc 37213	⊢ -_ℂ̅ = (𝑥 ∈ (ℂ̅ ∪ ℂ̂) ↦ if(𝑥 = ∞, ∞, if(𝑥 ∈ ℂ, (℩𝑦 ∈ ℂ (𝑥 +_ℂ̅ 𝑦) = 0), (+∞e^iτ‘(𝑥 +_ℂ̅ ⟨1/2, 0_R⟩)))))
cltxr 37214	class <_ℝ̅
df-bj-lt 37215	⊢ <_ℝ̅ = ({𝑥 ∈ (ℝ̅ × ℝ̅) ∣ ∃𝑦∃𝑧(((1^st ‘𝑥) = ⟨𝑦, 0_R⟩ ∧ (2^nd ‘𝑥) = ⟨𝑧, 0_R⟩) ∧ 𝑦 <_R 𝑧)} ∪ ((({-∞} × ℝ) ∪ (ℝ × {+∞})) ∪ ({-∞} × {+∞})))
carg 37216	class Arg
df-bj-arg 37217	⊢ Arg = (𝑥 ∈ (ℂ̅ ∖ {0}) ↦ if(𝑥 ∈ ℂ, (ℑ‘(log‘𝑥)), if(𝑥<_ℝ̅0, π, (((1^st ‘𝑥) / (2 · π)) − π))))
cmulc 37218	class ·_ℂ̅
df-bj-mulc 37219	⊢ ·_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 37220	class ^-1_ℂ̅
df-bj-invc 37221	⊢ ^-1_ℂ̅ = (𝑥 ∈ (ℂ̅ ∪ ℂ̂) ↦ if(𝑥 = 0, ∞, if(𝑥 ∈ ℂ, (℩𝑦 ∈ ℂ (𝑥 ·_ℂ̅ 𝑦) = 1), 0)))
ciomnn 37222	class i_ω↪ℕ
df-bj-iomnn 37223	⊢ i_ω↪ℕ = ((𝑛 ∈ ω ↦ ⟨[⟨{𝑟 ∈ Q ∣ 𝑟 <_Q ⟨suc 𝑛, 1_o⟩}, 1_P⟩] ~_R , 0_R⟩) ∪ {⟨ω, +∞⟩})
cnnbar 37233	class ℕ̅
df-bj-nnbar 37234	⊢ ℕ̅ = (ℕ₀ ∪ {+∞})
czzbar 37235	class ℤ̅
df-bj-zzbar 37236	⊢ ℤ̅ = (ℤ ∪ {-∞, +∞})
czzhat 37237	class ℤ̂
df-bj-zzhat 37238	⊢ ℤ̂ = (ℤ ∪ {∞})
cdivc 37239	class ∥_ℂ
df-bj-divc 37240	⊢ ∥_ℂ = {⟨𝑥, 𝑦⟩ ∣ (⟨𝑥, 𝑦⟩ ∈ ((ℂ̅ × ℂ̅) ∪ (ℂ̂ × ℂ̂)) ∧ ∃𝑛 ∈ (ℤ̅ ∪ ℤ̂)(𝑛 ·_ℂ̅ 𝑥) = 𝑦)}
cfinsum 37256	class FinSum
df-bj-finsum 37257	⊢ 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 37265	class ℝ-Vec
df-bj-rvec 37266	⊢ ℝ-Vec = (LMod ∩ (^◡Scalar “ {ℝ_fld}))
cend 37286	class End
df-bj-end 37287	⊢ End = (𝑐 ∈ Cat ↦ (𝑥 ∈ (Base‘𝑐) ↦ {⟨(Base‘ndx), (𝑥(Hom ‘𝑐)𝑥)⟩, ⟨(+_g‘ndx), (⟨𝑥, 𝑥⟩(comp‘𝑐)𝑥)⟩}))
cfinxp 37356	class (𝑈↑↑𝑁)
df-finxp 37357	⊢ (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 37392	⊢ ((𝜑 → 𝜓) → ((𝜓 → 𝜒) → (𝜑 → 𝜒)))
ax-luk2 37393	⊢ ((¬ 𝜑 → 𝜑) → 𝜑)
ax-luk3 37394	⊢ (𝜑 → (¬ 𝜑 → 𝜓))
ax-wl-13v 37466	⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
ax-wl-cleq 37475	⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
ax-wl-clel 37476	⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵))
ax-wl-11v 37557	⊢ (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑)
ctotbnd 37745	class TotBnd
cbnd 37746	class Bnd
df-totbnd 37747	⊢ TotBnd = (𝑥 ∈ V ↦ {𝑚 ∈ (Met‘𝑥) ∣ ∀𝑑 ∈ ℝ⁺ ∃𝑣 ∈ Fin (∪ 𝑣 = 𝑥 ∧ ∀𝑏 ∈ 𝑣 ∃𝑦 ∈ 𝑥 𝑏 = (𝑦(ball‘𝑚)𝑑))})
df-bnd 37758	⊢ Bnd = (𝑥 ∈ V ↦ {𝑚 ∈ (Met‘𝑥) ∣ ∀𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ⁺ 𝑥 = (𝑦(ball‘𝑚)𝑟)})
cismty 37777	class Ismty
df-ismty 37778	⊢ 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 37804	class ℝⁿ
df-rrn 37805	⊢ ℝⁿ = (𝑖 ∈ Fin ↦ (𝑥 ∈ (ℝ ↑_m 𝑖), 𝑦 ∈ (ℝ ↑_m 𝑖) ↦ (√‘Σ𝑘 ∈ 𝑖 (((𝑥‘𝑘) − (𝑦‘𝑘))↑2))))
cass 37821	class Ass
df-ass 37822	⊢ Ass = {𝑔 ∣ ∀𝑥 ∈ dom dom 𝑔∀𝑦 ∈ dom dom 𝑔∀𝑧 ∈ dom dom 𝑔((𝑥𝑔𝑦)𝑔𝑧) = (𝑥𝑔(𝑦𝑔𝑧))}
cexid 37823	class ExId
df-exid 37824	⊢ ExId = {𝑔 ∣ ∃𝑥 ∈ dom dom 𝑔∀𝑦 ∈ dom dom 𝑔((𝑥𝑔𝑦) = 𝑦 ∧ (𝑦𝑔𝑥) = 𝑦)}
cmagm 37827	class Magma
df-mgmOLD 37828	⊢ Magma = {𝑔 ∣ ∃𝑡 𝑔:(𝑡 × 𝑡)⟶𝑡}
csem 37839	class SemiGrp
df-sgrOLD 37840	⊢ SemiGrp = (Magma ∩ Ass)
cmndo 37845	class MndOp
df-mndo 37846	⊢ MndOp = (SemiGrp ∩ ExId )
cghomOLD 37862	class GrpOpHom
df-ghomOLD 37863	⊢ 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 37873	class RingOps
df-rngo 37874	⊢ 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 37927	class DivRingOps
df-drngo 37928	⊢ DivRingOps = {⟨𝑔, ℎ⟩ ∣ (⟨𝑔, ℎ⟩ ∈ RingOps ∧ (ℎ ↾ ((ran 𝑔 ∖ {(GId‘𝑔)}) × (ran 𝑔 ∖ {(GId‘𝑔)}))) ∈ GrpOp)}
crngohom 37939	class RingOpsHom
crngoiso 37940	class RingOpsIso
crisc 37941	class ≃_𝑟
df-rngohom 37942	⊢ 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 37955	⊢ RingOpsIso = (𝑟 ∈ RingOps, 𝑠 ∈ RingOps ↦ {𝑓 ∈ (𝑟 RingOpsHom 𝑠) ∣ 𝑓:ran (1^st ‘𝑟)–1-1-onto→ran (1^st ‘𝑠)})
df-risc 37962	⊢ ≃_𝑟 = {⟨𝑟, 𝑠⟩ ∣ ((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑟 RingOpsIso 𝑠))}
ccm2 37968	class Com2
df-com2 37969	⊢ Com2 = {⟨𝑔, ℎ⟩ ∣ ∀𝑎 ∈ ran 𝑔∀𝑏 ∈ ran 𝑔(𝑎ℎ𝑏) = (𝑏ℎ𝑎)}
cfld 37970	class Fld
df-fld 37971	⊢ Fld = (DivRingOps ∩ Com2)
ccring 37972	class CRingOps
df-crngo 37973	⊢ CRingOps = (RingOps ∩ Com2)
cidl 37986	class Idl
cpridl 37987	class PrIdl
cmaxidl 37988	class MaxIdl
df-idl 37989	⊢ 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 37990	⊢ 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 37991	⊢ MaxIdl = (𝑟 ∈ RingOps ↦ {𝑖 ∈ (Idl‘𝑟) ∣ (𝑖 ≠ ran (1^st ‘𝑟) ∧ ∀𝑗 ∈ (Idl‘𝑟)(𝑖 ⊆ 𝑗 → (𝑗 = 𝑖 ∨ 𝑗 = ran (1^st ‘𝑟))))})
cprrng 38025	class PrRing
cdmn 38026	class Dmn
df-prrngo 38027	⊢ PrRing = {𝑟 ∈ RingOps ∣ {(GId‘(1^st ‘𝑟))} ∈ (PrIdl‘𝑟)}
df-dmn 38028	⊢ Dmn = (PrRing ∩ Com2)
cigen 38038	class IdlGen
df-igen 38039	⊢ IdlGen = (𝑟 ∈ RingOps, 𝑠 ∈ 𝒫 ran (1^st ‘𝑟) ↦ ∩ {𝑗 ∈ (Idl‘𝑟) ∣ 𝑠 ⊆ 𝑗})
cxrn 38153	class (𝐴 ⋉ 𝐵)
ccoss 38154	class ≀ 𝑅
ccoels 38155	class ∼ 𝐴
crels 38156	class Rels
cssr 38157	class S
crefs 38158	class Refs
crefrels 38159	class RefRels
wrefrel 38160	wff RefRel 𝑅
ccnvrefs 38161	class CnvRefs
ccnvrefrels 38162	class CnvRefRels
wcnvrefrel 38163	wff CnvRefRel 𝑅
csyms 38164	class Syms
csymrels 38165	class SymRels
wsymrel 38166	wff SymRel 𝑅
ctrs 38167	class Trs
ctrrels 38168	class TrRels
wtrrel 38169	wff TrRel 𝑅
ceqvrels 38170	class EqvRels
weqvrel 38171	wff EqvRel 𝑅
ccoeleqvrels 38172	class CoElEqvRels
wcoeleqvrel 38173	wff CoElEqvRel 𝐴
credunds 38174	class Redunds
wredund 38175	wff 𝐴 Redund ⟨𝐵, 𝐶⟩
wredundp 38176	wff redund (𝜑, 𝜓, 𝜒)
cdmqss 38177	class DomainQss
wdmqs 38178	wff 𝑅 DomainQs 𝐴
cers 38179	class Ers
werALTV 38180	wff 𝑅 ErALTV 𝐴
ccomembers 38181	class CoMembErs
wcomember 38182	wff CoMembEr 𝐴
cfunss 38183	class Funss
cfunsALTV 38184	class FunsALTV
wfunALTV 38185	wff FunALTV 𝐹
cdisjss 38186	class Disjss
cdisjs 38187	class Disjs
wdisjALTV 38188	wff Disj 𝑅
celdisjs 38189	class ElDisjs
weldisj 38190	wff ElDisj 𝐴
wantisymrel 38191	wff AntisymRel 𝑅
cparts 38192	class Parts
wpart 38193	wff 𝑅 Part 𝐴
cmembparts 38194	class MembParts
wmembpart 38195	wff MembPart 𝐴
df-xrn 38338	⊢ (𝐴 ⋉ 𝐵) = ((^◡(1^st ↾ (V × V)) ∘ 𝐴) ∩ (^◡(2^nd ↾ (V × V)) ∘ 𝐵))
df-coss 38387	⊢ ≀ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)}
df-coels 38388	⊢ ∼ 𝐴 = ≀ (^◡ E ↾ 𝐴)
df-rels 38461	⊢ Rels = 𝒫 (V × V)
df-ssr 38474	⊢ S = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ⊆ 𝑦}
df-refs 38486	⊢ Refs = {𝑥 ∣ ( I ∩ (dom 𝑥 × ran 𝑥)) S (𝑥 ∩ (dom 𝑥 × ran 𝑥))}
df-refrels 38487	⊢ RefRels = ( Refs ∩ Rels )
df-refrel 38488	⊢ ( RefRel 𝑅 ↔ (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅))
df-cnvrefs 38501	⊢ CnvRefs = {𝑥 ∣ ( I ∩ (dom 𝑥 × ran 𝑥))^◡ S (𝑥 ∩ (dom 𝑥 × ran 𝑥))}
df-cnvrefrels 38502	⊢ CnvRefRels = ( CnvRefs ∩ Rels )
df-cnvrefrel 38503	⊢ ( CnvRefRel 𝑅 ↔ ((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ ( I ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅))
df-syms 38518	⊢ Syms = {𝑥 ∣ ^◡(𝑥 ∩ (dom 𝑥 × ran 𝑥)) S (𝑥 ∩ (dom 𝑥 × ran 𝑥))}
df-symrels 38519	⊢ SymRels = ( Syms ∩ Rels )
df-symrel 38520	⊢ ( SymRel 𝑅 ↔ (^◡(𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅))
df-trs 38548	⊢ Trs = {𝑥 ∣ ((𝑥 ∩ (dom 𝑥 × ran 𝑥)) ∘ (𝑥 ∩ (dom 𝑥 × ran 𝑥))) S (𝑥 ∩ (dom 𝑥 × ran 𝑥))}
df-trrels 38549	⊢ TrRels = ( Trs ∩ Rels )
df-trrel 38550	⊢ ( TrRel 𝑅 ↔ (((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∘ (𝑅 ∩ (dom 𝑅 × ran 𝑅))) ⊆ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅))
df-eqvrels 38560	⊢ EqvRels = (( RefRels ∩ SymRels ) ∩ TrRels )
df-eqvrel 38561	⊢ ( EqvRel 𝑅 ↔ ( RefRel 𝑅 ∧ SymRel 𝑅 ∧ TrRel 𝑅))
df-coeleqvrels 38562	⊢ CoElEqvRels = {𝑎 ∣ ≀ (^◡ E ↾ 𝑎) ∈ EqvRels }
df-coeleqvrel 38563	⊢ ( CoElEqvRel 𝐴 ↔ EqvRel ≀ (^◡ E ↾ 𝐴))
df-redunds 38599	⊢ Redunds = ^◡{⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ (𝑥 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑧) = (𝑦 ∩ 𝑧))}
df-redund 38600	⊢ (𝐴 Redund ⟨𝐵, 𝐶⟩ ↔ (𝐴 ⊆ 𝐵 ∧ (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐶)))
df-redundp 38601	⊢ ( redund (𝜑, 𝜓, 𝜒) ↔ ((𝜑 → 𝜓) ∧ ((𝜑 ∧ 𝜒) ↔ (𝜓 ∧ 𝜒))))
df-dmqss 38614	⊢ DomainQss = {⟨𝑥, 𝑦⟩ ∣ (dom 𝑥 / 𝑥) = 𝑦}
df-dmqs 38615	⊢ (𝑅 DomainQs 𝐴 ↔ (dom 𝑅 / 𝑅) = 𝐴)
df-ers 38640	⊢ Ers = ( DomainQss ↾ EqvRels )
df-erALTV 38641	⊢ (𝑅 ErALTV 𝐴 ↔ ( EqvRel 𝑅 ∧ 𝑅 DomainQs 𝐴))
df-comembers 38642	⊢ CoMembErs = {𝑎 ∣ ≀ (^◡ E ↾ 𝑎) Ers 𝑎}
df-comember 38643	⊢ ( CoMembEr 𝐴 ↔ ≀ (^◡ E ↾ 𝐴) ErALTV 𝐴)
df-funss 38657	⊢ Funss = {𝑥 ∣ ≀ 𝑥 ∈ CnvRefRels }
df-funsALTV 38658	⊢ FunsALTV = ( Funss ∩ Rels )
df-funALTV 38659	⊢ ( FunALTV 𝐹 ↔ ( CnvRefRel ≀ 𝐹 ∧ Rel 𝐹))
df-disjss 38680	⊢ Disjss = {𝑥 ∣ ≀ ^◡𝑥 ∈ CnvRefRels }
df-disjs 38681	⊢ Disjs = ( Disjss ∩ Rels )
df-disjALTV 38682	⊢ ( Disj 𝑅 ↔ ( CnvRefRel ≀ ^◡𝑅 ∧ Rel 𝑅))
df-eldisjs 38683	⊢ ElDisjs = {𝑎 ∣ (^◡ E ↾ 𝑎) ∈ Disjs }
df-eldisj 38684	⊢ ( ElDisj 𝐴 ↔ Disj (^◡ E ↾ 𝐴))
df-antisymrel 38737	⊢ ( AntisymRel 𝑅 ↔ ( CnvRefRel (𝑅 ∩ ^◡𝑅) ∧ Rel 𝑅))
df-parts 38742	⊢ Parts = ( DomainQss ↾ Disjs )
df-part 38743	⊢ (𝑅 Part 𝐴 ↔ ( Disj 𝑅 ∧ 𝑅 DomainQs 𝐴))
df-membparts 38744	⊢ MembParts = {𝑎 ∣ (^◡ E ↾ 𝑎) Parts 𝑎}
df-membpart 38745	⊢ ( MembPart 𝐴 ↔ (^◡ E ↾ 𝐴) Part 𝐴)
wprt 38849	wff Prt 𝐴
df-prt 38850	⊢ (Prt 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅))
ax-c5 38861	⊢ (∀𝑥𝜑 → 𝜑)
ax-c4 38862	⊢ (∀𝑥(∀𝑥𝜑 → 𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))
ax-c7 38863	⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜑 → 𝜑)
ax-c10 38864	⊢ (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → 𝜑)
ax-c11 38865	⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑))
ax-c11n 38866	⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
ax-c15 38867	⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))))
ax-c9 38868	⊢ (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))
ax-c14 38869	⊢ (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥 ∈ 𝑦 → ∀𝑧 𝑥 ∈ 𝑦)))
ax-c16 38870	⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))
ax-riotaBAD 38931	⊢ (℩𝑥 ∈ 𝐴 𝜑) = if(∃!𝑥 ∈ 𝐴 𝜑, (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)), (Undef‘{𝑥 ∣ 𝑥 ∈ 𝐴}))
clsa 38952	class LSAtoms
clsh 38953	class LSHyp
df-lsatoms 38954	⊢ LSAtoms = (𝑤 ∈ V ↦ ran (𝑣 ∈ ((Base‘𝑤) ∖ {(0_g‘𝑤)}) ↦ ((LSpan‘𝑤)‘{𝑣})))
df-lshyp 38955	⊢ LSHyp = (𝑤 ∈ V ↦ {𝑠 ∈ (LSubSp‘𝑤) ∣ (𝑠 ≠ (Base‘𝑤) ∧ ∃𝑣 ∈ (Base‘𝑤)((LSpan‘𝑤)‘(𝑠 ∪ {𝑣})) = (Base‘𝑤))})
clcv 38996	class ⋖_L
df-lcv 38997	⊢ ⋖_L = (𝑤 ∈ V ↦ {⟨𝑡, 𝑢⟩ ∣ ((𝑡 ∈ (LSubSp‘𝑤) ∧ 𝑢 ∈ (LSubSp‘𝑤)) ∧ (𝑡 ⊊ 𝑢 ∧ ¬ ∃𝑠 ∈ (LSubSp‘𝑤)(𝑡 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑢)))})
clfn 39035	class LFnl
df-lfl 39036	⊢ 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 39063	class LKer
df-lkr 39064	⊢ LKer = (𝑤 ∈ V ↦ (𝑓 ∈ (LFnl‘𝑤) ↦ (^◡𝑓 “ {(0_g‘(Scalar‘𝑤))})))
cld 39101	class LDual
df-ldual 39102	⊢ 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 39150	class OP
ccmtN 39151	class cm
col 39152	class OL
coml 39153	class OML
df-oposet 39154	⊢ 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 39155	⊢ 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 39156	⊢ OL = (Lat ∩ OP)
df-oml 39157	⊢ OML = {𝑙 ∈ OL ∣ ∀𝑎 ∈ (Base‘𝑙)∀𝑏 ∈ (Base‘𝑙)(𝑎(le‘𝑙)𝑏 → 𝑏 = (𝑎(join‘𝑙)(𝑏(meet‘𝑙)((oc‘𝑙)‘𝑎))))}
ccvr 39240	class ⋖
catm 39241	class Atoms
cal 39242	class AtLat
clc 39243	class CvLat
df-covers 39244	⊢ ⋖ = (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 39245	⊢ Atoms = (𝑝 ∈ V ↦ {𝑎 ∈ (Base‘𝑝) ∣ (0.‘𝑝)( ⋖ ‘𝑝)𝑎})
df-atl 39276	⊢ AtLat = {𝑘 ∈ Lat ∣ ((Base‘𝑘) ∈ dom (glb‘𝑘) ∧ ∀𝑥 ∈ (Base‘𝑘)(𝑥 ≠ (0.‘𝑘) → ∃𝑝 ∈ (Atoms‘𝑘)𝑝(le‘𝑘)𝑥))}
df-cvlat 39300	⊢ 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 39328	class HL
df-hlat 39329	⊢ 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 39470	class LLines
clpl 39471	class LPlanes
clvol 39472	class LVols
clines 39473	class Lines
cpointsN 39474	class Points
cpsubsp 39475	class PSubSp
cpmap 39476	class pmap
df-llines 39477	⊢ LLines = (𝑘 ∈ V ↦ {𝑥 ∈ (Base‘𝑘) ∣ ∃𝑝 ∈ (Atoms‘𝑘)𝑝( ⋖ ‘𝑘)𝑥})
df-lplanes 39478	⊢ LPlanes = (𝑘 ∈ V ↦ {𝑥 ∈ (Base‘𝑘) ∣ ∃𝑝 ∈ (LLines‘𝑘)𝑝( ⋖ ‘𝑘)𝑥})
df-lvols 39479	⊢ LVols = (𝑘 ∈ V ↦ {𝑥 ∈ (Base‘𝑘) ∣ ∃𝑝 ∈ (LPlanes‘𝑘)𝑝( ⋖ ‘𝑘)𝑥})
df-lines 39480	⊢ 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 39481	⊢ Points = (𝑘 ∈ V ↦ {𝑞 ∣ ∃𝑝 ∈ (Atoms‘𝑘)𝑞 = {𝑝}})
df-psubsp 39482	⊢ 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 39483	⊢ pmap = (𝑘 ∈ V ↦ (𝑎 ∈ (Base‘𝑘) ↦ {𝑝 ∈ (Atoms‘𝑘) ∣ 𝑝(le‘𝑘)𝑎}))
cpadd 39774	class +_𝑃
df-padd 39775	⊢ +_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 39866	class PCl
df-pclN 39867	⊢ PCl = (𝑘 ∈ V ↦ (𝑥 ∈ 𝒫 (Atoms‘𝑘) ↦ ∩ {𝑦 ∈ (PSubSp‘𝑘) ∣ 𝑥 ⊆ 𝑦}))
cpolN 39881	class ⊥_𝑃
df-polarityN 39882	⊢ ⊥_𝑃 = (𝑙 ∈ V ↦ (𝑚 ∈ 𝒫 (Atoms‘𝑙) ↦ ((Atoms‘𝑙) ∩ ∩ 𝑝 ∈ 𝑚 ((pmap‘𝑙)‘((oc‘𝑙)‘𝑝)))))
cpscN 39913	class PSubCl
df-psubclN 39914	⊢ PSubCl = (𝑘 ∈ V ↦ {𝑠 ∣ (𝑠 ⊆ (Atoms‘𝑘) ∧ ((⊥_𝑃‘𝑘)‘((⊥_𝑃‘𝑘)‘𝑠)) = 𝑠)})
clh 39963	class LHyp
claut 39964	class LAut
cwpointsN 39965	class WAtoms
cpautN 39966	class PAut
df-lhyp 39967	⊢ LHyp = (𝑘 ∈ V ↦ {𝑥 ∈ (Base‘𝑘) ∣ 𝑥( ⋖ ‘𝑘)(1.‘𝑘)})
df-laut 39968	⊢ 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 39969	⊢ WAtoms = (𝑘 ∈ V ↦ (𝑑 ∈ (Atoms‘𝑘) ↦ ((Atoms‘𝑘) ∖ ((⊥_𝑃‘𝑘)‘{𝑑}))))
df-pautN 39970	⊢ 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 40079	class LDil
cltrn 40080	class LTrn
cdilN 40081	class Dil
ctrnN 40082	class Trn
df-ldil 40083	⊢ LDil = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ {𝑓 ∈ (LAut‘𝑘) ∣ ∀𝑥 ∈ (Base‘𝑘)(𝑥(le‘𝑘)𝑤 → (𝑓‘𝑥) = 𝑥)}))
df-ltrn 40084	⊢ 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 40085	⊢ Dil = (𝑘 ∈ V ↦ (𝑑 ∈ (Atoms‘𝑘) ↦ {𝑓 ∈ (PAut‘𝑘) ∣ ∀𝑥 ∈ (PSubSp‘𝑘)(𝑥 ⊆ ((WAtoms‘𝑘)‘𝑑) → (𝑓‘𝑥) = 𝑥)}))
df-trnN 40086	⊢ 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 40137	class trL
df-trl 40138	⊢ 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 40721	class TGrp
df-tgrp 40722	⊢ TGrp = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ {⟨(Base‘ndx), ((LTrn‘𝑘)‘𝑤)⟩, ⟨(+_g‘ndx), (𝑓 ∈ ((LTrn‘𝑘)‘𝑤), 𝑔 ∈ ((LTrn‘𝑘)‘𝑤) ↦ (𝑓 ∘ 𝑔))⟩}))
ctendo 40731	class TEndo
cedring 40732	class EDRing
cedring-rN 40733	class EDRing_R
df-tendo 40734	⊢ 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 40735	⊢ 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 40736	⊢ 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 40981	class DVecA
df-dveca 40982	⊢ 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 41007	class DIsoA
df-disoa 41008	⊢ 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 41057	class DVecH
df-dvech 41058	⊢ 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 41098	class ocA
df-docaN 41099	⊢ 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 41110	class vA
df-djaN 41111	⊢ 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 41117	class DIsoB
df-dib 41118	⊢ DIsoB = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑥 ∈ dom ((DIsoA‘𝑘)‘𝑤) ↦ ((((DIsoA‘𝑘)‘𝑤)‘𝑥) × {(𝑓 ∈ ((LTrn‘𝑘)‘𝑤) ↦ ( I ↾ (Base‘𝑘)))}))))
cdic 41151	class DIsoC
df-dic 41152	⊢ 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 41207	class DIsoH
df-dih 41208	⊢ 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 41326	class ocH
df-doch 41327	⊢ 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 41373	class joinH
df-djh 41374	⊢ 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 41459	class LPol
df-lpolN 41460	⊢ 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 41565	class LCDual
df-lcdual 41566	⊢ 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 41603	class mapd
df-mapd 41604	⊢ 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 41735	class HVMap
df-hvmap 41736	⊢ 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 41770	class HDMap1
chdma 41771	class HDMap
df-hdmap1 41772	⊢ 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 41773	⊢ 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 41862	class HGMap
df-hgmap 41863	⊢ 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 41911	class HLHil
df-hlhil 41912	⊢ 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 41941	class CSRing
df-csring 41942	⊢ CSRing = {𝑓 ∈ SRing ∣ (mulGrp‘𝑓) ∈ CMnd}
cprimroots 42064	class PrimRoots
df-primroots 42065	⊢ 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 42175	⊢ ∀𝑝 ∈ ℙ ∀𝑛 ∈ ℕ ∃𝑘 ∈ Field ((♯‘(Base‘𝑘)) = (𝑝↑𝑛) ∧ (chr‘𝑘) = 𝑝)
cresub 42338	class −_ℝ
df-resub 42339	⊢ −_ℝ = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (℩𝑧 ∈ ℝ (𝑦 + 𝑧) = 𝑥))
crediv 42413	class /_ℝ
df-rediv 42414	⊢ /_ℝ = (𝑥 ∈ ℝ, 𝑦 ∈ (ℝ ∖ {0}) ↦ (℩𝑧 ∈ ℝ (𝑦 · 𝑧) = 𝑥))
cprjsp 42574	class ℙ𝕣𝕠𝕛
df-prjsp 42575	⊢ 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 42587	class ℙ𝕣𝕠𝕛_n
df-prjspn 42588	⊢ ℙ𝕣𝕠𝕛_n = (𝑛 ∈ ℕ₀, 𝑘 ∈ DivRing ↦ (ℙ𝕣𝕠𝕛‘(𝑘 freeLMod (0...𝑛))))
cprjcrv 42602	class ℙ𝕣𝕠𝕛Crv
df-prjcrv 42603	⊢ ℙ𝕣𝕠𝕛Crv = (𝑛 ∈ ℕ₀, 𝑘 ∈ Field ↦ (𝑓 ∈ ∪ ran ((0...𝑛) mHomP 𝑘) ↦ {𝑝 ∈ (𝑛ℙ𝕣𝕠𝕛_n𝑘) ∣ ((((0...𝑛) eval 𝑘)‘𝑓) “ 𝑝) = {(0_g‘𝑘)}}))
cnacs 42675	class NoeACS
df-nacs 42676	⊢ NoeACS = (𝑥 ∈ V ↦ {𝑐 ∈ (ACS‘𝑥) ∣ ∀𝑠 ∈ 𝑐 ∃𝑔 ∈ (𝒫 𝑥 ∩ Fin)𝑠 = ((mrCls‘𝑐)‘𝑔)})
cmzpcl 42694	class mzPolyCld
cmzp 42695	class mzPoly
df-mzpcl 42696	⊢ 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 42697	⊢ mzPoly = (𝑣 ∈ V ↦ ∩ (mzPolyCld‘𝑣))
cdioph 42728	class Dioph
df-dioph 42729	⊢ 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 42809	class ◻_NN
cpell1qr 42810	class Pell1QR
cpell1234qr 42811	class Pell1234QR
cpell14qr 42812	class Pell14QR
cpellfund 42813	class PellFund
df-squarenn 42814	⊢ ◻_NN = {𝑥 ∈ ℕ ∣ (√‘𝑥) ∈ ℚ}
df-pell1qr 42815	⊢ Pell1QR = (𝑥 ∈ (ℕ ∖ ◻_NN) ↦ {𝑦 ∈ ℝ ∣ ∃𝑧 ∈ ℕ₀ ∃𝑤 ∈ ℕ₀ (𝑦 = (𝑧 + ((√‘𝑥) · 𝑤)) ∧ ((𝑧↑2) − (𝑥 · (𝑤↑2))) = 1)})
df-pell14qr 42816	⊢ Pell14QR = (𝑥 ∈ (ℕ ∖ ◻_NN) ↦ {𝑦 ∈ ℝ ∣ ∃𝑧 ∈ ℕ₀ ∃𝑤 ∈ ℤ (𝑦 = (𝑧 + ((√‘𝑥) · 𝑤)) ∧ ((𝑧↑2) − (𝑥 · (𝑤↑2))) = 1)})
df-pell1234qr 42817	⊢ Pell1234QR = (𝑥 ∈ (ℕ ∖ ◻_NN) ↦ {𝑦 ∈ ℝ ∣ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ (𝑦 = (𝑧 + ((√‘𝑥) · 𝑤)) ∧ ((𝑧↑2) − (𝑥 · (𝑤↑2))) = 1)})
df-pellfund 42818	⊢ PellFund = (𝑥 ∈ (ℕ ∖ ◻_NN) ↦ inf({𝑧 ∈ (Pell14QR‘𝑥) ∣ 1 < 𝑧}, ℝ, < ))
crmx 42873	class X_rm
crmy 42874	class Y_rm
df-rmx 42875	⊢ 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 42876	⊢ 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 43040	class LFinGen
df-lfig 43041	⊢ LFinGen = {𝑤 ∈ LMod ∣ (Base‘𝑤) ∈ ((LSpan‘𝑤) “ (𝒫 (Base‘𝑤) ∩ Fin))}
clnm 43048	class LNoeM
df-lnm 43049	⊢ LNoeM = {𝑤 ∈ LMod ∣ ∀𝑖 ∈ (LSubSp‘𝑤)(𝑤 ↾_s 𝑖) ∈ LFinGen}
clnr 43082	class LNoeR
df-lnr 43083	⊢ LNoeR = {𝑎 ∈ Ring ∣ (ringLMod‘𝑎) ∈ LNoeM}
cldgis 43094	class ldgIdlSeq
df-ldgis 43095	⊢ ldgIdlSeq = (𝑟 ∈ V ↦ (𝑖 ∈ (LIdeal‘(Poly₁‘𝑟)) ↦ (𝑥 ∈ ℕ₀ ↦ {𝑗 ∣ ∃𝑘 ∈ 𝑖 (((deg₁‘𝑟)‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe₁‘𝑘)‘𝑥))})))
cmnc 43104	class Monic
cplylt 43105	class Poly_<
df-mnc 43106	⊢ Monic = (𝑠 ∈ 𝒫 ℂ ↦ {𝑝 ∈ (Poly‘𝑠) ∣ ((coeff‘𝑝)‘(deg‘𝑝)) = 1})
df-plylt 43107	⊢ Poly_< = (𝑠 ∈ 𝒫 ℂ, 𝑥 ∈ ℕ₀ ↦ {𝑝 ∈ (Poly‘𝑠) ∣ (𝑝 = 0_𝑝 ∨ (deg‘𝑝) < 𝑥)})
cdgraa 43113	class deg_AA
cmpaa 43114	class minPolyAA
df-dgraa 43115	⊢ deg_AA = (𝑥 ∈ 𝔸 ↦ inf({𝑑 ∈ ℕ ∣ ∃𝑝 ∈ ((Poly‘ℚ) ∖ {0_𝑝})((deg‘𝑝) = 𝑑 ∧ (𝑝‘𝑥) = 0)}, ℝ, < ))
df-mpaa 43116	⊢ minPolyAA = (𝑥 ∈ 𝔸 ↦ (℩𝑝 ∈ (Poly‘ℚ)((deg‘𝑝) = (deg_AA‘𝑥) ∧ (𝑝‘𝑥) = 0 ∧ ((coeff‘𝑝)‘(deg_AA‘𝑥)) = 1)))
citgo 43130	class IntgOver
cza 43131	class ℤ
df-itgo 43132	⊢ IntgOver = (𝑠 ∈ 𝒫 ℂ ↦ {𝑥 ∈ ℂ ∣ ∃𝑝 ∈ (Poly‘𝑠)((𝑝‘𝑥) = 0 ∧ ((coeff‘𝑝)‘(deg‘𝑝)) = 1)})
df-za 43133	⊢ ℤ = (IntgOver‘ℤ)
cmend 43144	class MEndo
df-mend 43145	⊢ 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 43170	class CytP
df-cytp 43171	⊢ 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 43179	class TopSep
ctoplnd 43180	class TopLnd
df-topsep 43181	⊢ TopSep = {𝑗 ∈ Top ∣ ∃𝑥 ∈ 𝒫 ∪ 𝑗(𝑥 ≼ ω ∧ ((cls‘𝑗)‘𝑥) = ∪ 𝑗)}
df-toplnd 43182	⊢ TopLnd = {𝑥 ∈ Top ∣ ∀𝑦 ∈ 𝒫 𝑥(∪ 𝑥 = ∪ 𝑦 → ∃𝑧 ∈ 𝒫 𝑥(𝑧 ≼ ω ∧ ∪ 𝑥 = ∪ 𝑧))}
crcl 43645	class r*
df-rcl 43646	⊢ r* = (𝑥 ∈ V ↦ ∩ {𝑧 ∣ (𝑥 ⊆ 𝑧 ∧ ( I ↾ (dom 𝑧 ∪ ran 𝑧)) ⊆ 𝑧)})
whe 43745	wff 𝑅 hereditary 𝐴
df-he 43746	⊢ (𝑅 hereditary 𝐴 ↔ (𝑅 “ 𝐴) ⊆ 𝐴)
ax-frege1 43763	⊢ (𝜑 → (𝜓 → 𝜑))
ax-frege2 43764	⊢ ((𝜑 → (𝜓 → 𝜒)) → ((𝜑 → 𝜓) → (𝜑 → 𝜒)))
ax-frege8 43782	⊢ ((𝜑 → (𝜓 → 𝜒)) → (𝜓 → (𝜑 → 𝜒)))
ax-frege28 43803	⊢ ((𝜑 → 𝜓) → (¬ 𝜓 → ¬ 𝜑))
ax-frege31 43807	⊢ (¬ ¬ 𝜑 → 𝜑)
ax-frege41 43818	⊢ (𝜑 → ¬ ¬ 𝜑)
ax-frege52a 43830	⊢ ((𝜑 ↔ 𝜓) → (if-(𝜑, 𝜃, 𝜒) → if-(𝜓, 𝜃, 𝜒)))
ax-frege54a 43835	⊢ (𝜑 ↔ 𝜑)
ax-frege58a 43848	⊢ ((𝜓 ∧ 𝜒) → if-(𝜑, 𝜓, 𝜒))
ax-frege52c 43861	⊢ (𝐴 = 𝐵 → ([𝐴 / 𝑥]𝜑 → [𝐵 / 𝑥]𝜑))
ax-frege54c 43865	⊢ 𝐴 = 𝐴
ax-frege58b 43874	⊢ (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑)
cmnring 44184	class MndRing
df-mnring 44185	⊢ 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 44208	class Scott 𝐴
df-scott 44209	⊢ Scott 𝐴 = {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)}
ccoll 44223	class (𝐹 Coll 𝐴)
df-coll 44224	⊢ (𝐹 Coll 𝐴) = ∪ 𝑥 ∈ 𝐴 Scott (𝐹 “ {𝑥})
cbcc 44309	class C_𝑐
df-bcc 44310	⊢ C_𝑐 = (𝑐 ∈ ℂ, 𝑘 ∈ ℕ₀ ↦ ((𝑐 FallFac 𝑘) / (!‘𝑘)))
cplusr 44430	class +_𝑟
cminusr 44431	class -_𝑟
ctimesr 44432	class ._𝑣
cptdfc 44433	class PtDf(𝐴, 𝐵)
crr3c 44434	class RR3
cline3 44435	class line3
df-addr 44436	⊢ +_𝑟 = (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑣 ∈ ℝ ↦ ((𝑥‘𝑣) + (𝑦‘𝑣))))
df-subr 44437	⊢ -_𝑟 = (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑣 ∈ ℝ ↦ ((𝑥‘𝑣) − (𝑦‘𝑣))))
df-mulv 44438	⊢ ._𝑣 = (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑣 ∈ ℝ ↦ (𝑥 · (𝑦‘𝑣))))
df-ptdf 44449	⊢ PtDf(𝐴, 𝐵) = (𝑥 ∈ ℝ ↦ (((𝑥._𝑣(𝐵-_𝑟𝐴)) +_𝑣 𝐴) “ {1, 2, 3}))
df-rr3 44450	⊢ RR3 = (ℝ ↑_m {1, 2, 3})
df-line3 44451	⊢ line3 = {𝑥 ∈ 𝒫 RR3 ∣ (2_o ≼ 𝑥 ∧ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑧 ≠ 𝑦 → ran PtDf(𝑦, 𝑧) = 𝑥))}
wvd1 44543	wff ( 𝜑 ▶ 𝜓 )
df-vd1 44544	⊢ (( 𝜑 ▶ 𝜓 ) ↔ (𝜑 → 𝜓))
wvd2 44551	wff ( 𝜑 , 𝜓 ▶ 𝜒 )
df-vd2 44552	⊢ (( 𝜑 , 𝜓 ▶ 𝜒 ) ↔ ((𝜑 ∧ 𝜓) → 𝜒))
wvhc2 44554	wff ( 𝜑 , 𝜓 )
df-vhc2 44555	⊢ (( 𝜑 , 𝜓 ) ↔ (𝜑 ∧ 𝜓))
wvd3 44561	wff ( 𝜑 , 𝜓 , 𝜒 ▶ 𝜃 )
wvhc3 44562	wff ( 𝜑 , 𝜓 , 𝜒 )
df-vhc3 44563	⊢ (( 𝜑 , 𝜓 , 𝜒 ) ↔ (𝜑 ∧ 𝜓 ∧ 𝜒))
df-vd3 44564	⊢ (( 𝜑 , 𝜓 , 𝜒 ▶ 𝜃 ) ↔ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃))
wrelp 44916	wff 𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵)
df-relp 44917	⊢ (𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵) ↔ (𝐻:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐻‘𝑥)𝑆(𝐻‘𝑦))))
clsi 45733	class lim inf
df-liminf 45734	⊢ lim inf = (𝑥 ∈ V ↦ sup(ran (𝑘 ∈ ℝ ↦ inf(((𝑥 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )), ℝ^*, < ))
clsxlim 45800	class ~~>*
df-xlim 45801	⊢ ~~>* = (⇝_𝑡‘(ordTop‘ ≤ ))
csalg 46290	class SAlg
df-salg 46291	⊢ SAlg = {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (∪ 𝑥 ∖ 𝑦) ∈ 𝑥 ∧ ∀𝑦 ∈ 𝒫 𝑥(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑥))}
csalon 46292	class SalOn
df-salon 46293	⊢ SalOn = (𝑥 ∈ V ↦ {𝑠 ∈ SAlg ∣ ∪ 𝑠 = 𝑥})
csalgen 46294	class SalGen
df-salgen 46295	⊢ SalGen = (𝑥 ∈ V ↦ ∩ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑥 ∧ 𝑥 ⊆ 𝑠)})
csumge0 46344	class Σ^{^}
df-sumge0 46345	⊢ Σ^{^} = (𝑥 ∈ V ↦ if(+∞ ∈ ran 𝑥, +∞, sup(ran (𝑦 ∈ (𝒫 dom 𝑥 ∩ Fin) ↦ Σ𝑤 ∈ 𝑦 (𝑥‘𝑤)), ℝ^*, < )))
cmea 46431	class Meas
df-mea 46432	⊢ 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 46471	class OutMeas
df-ome 46472	⊢ 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 46473	class CaraGen
df-caragen 46474	⊢ CaraGen = (𝑜 ∈ OutMeas ↦ {𝑒 ∈ 𝒫 ∪ dom 𝑜 ∣ ∀𝑎 ∈ 𝒫 ∪ dom 𝑜((𝑜‘(𝑎 ∩ 𝑒)) +_𝑒 (𝑜‘(𝑎 ∖ 𝑒))) = (𝑜‘𝑎)})
covoln 46518	class voln*
df-ovoln 46519	⊢ 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 46520	class voln
df-voln 46521	⊢ voln = (𝑥 ∈ Fin ↦ ((voln‘𝑥) ↾ (CaraGen‘(voln‘𝑥))))
csmblfn 46677	class SMblFn
df-smblfn 46678	⊢ SMblFn = (𝑠 ∈ SAlg ↦ {𝑓 ∈ (ℝ ↑_pm ∪ 𝑠) ∣ ∀𝑎 ∈ ℝ (^◡𝑓 “ (-∞(,)𝑎)) ∈ (𝑠 ↾_t dom 𝑓)})
cupword 46860	class UpWord 𝑆
df-upword 46861	⊢ UpWord 𝑆 = {𝑤 ∣ (𝑤 ∈ Word 𝑆 ∧ ∀𝑘 ∈ (0..^((♯‘𝑤) − 1))(𝑤‘𝑘) < (𝑤‘(𝑘 + 1)))}
caiota 47068	class (℩'𝑥𝜑)
df-aiota 47070	⊢ (℩'𝑥𝜑) = ∩ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}
wdfat 47101	wff 𝐹 defAt 𝐴
cafv 47102	class (𝐹'''𝐴)
caov 47103	class ((𝐴𝐹𝐵))
df-dfat 47104	⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
df-afv 47105	⊢ (𝐹'''𝐴) = (℩'𝑥𝐴𝐹𝑥)
df-aov 47106	⊢ ((𝐴𝐹𝐵)) = (𝐹'''⟨𝐴, 𝐵⟩)
cafv2 47193	class (𝐹''''𝐴)
df-afv2 47194	⊢ (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ∪ ran 𝐹)
cnelbr 47256	class _∉
df-nelbr 47257	⊢ _∉ = {⟨𝑥, 𝑦⟩ ∣ ¬ 𝑥 ∈ 𝑦}
ciccp 47398	class RePart
df-iccp 47399	⊢ RePart = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ^* ↑_m (0...𝑚)) ∣ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1))})
wich 47430	wff [𝑥⇄𝑦]𝜑
df-ich 47431	⊢ ([𝑥⇄𝑦]𝜑 ↔ ∀𝑥∀𝑦([𝑥 / 𝑎][𝑦 / 𝑥][𝑎 / 𝑦]𝜑 ↔ 𝜑))
cspr 47462	class Pairs
df-spr 47463	⊢ Pairs = (𝑣 ∈ V ↦ {𝑝 ∣ ∃𝑎 ∈ 𝑣 ∃𝑏 ∈ 𝑣 𝑝 = {𝑎, 𝑏}})
cprpr 47497	class Pairs_proper
df-prpr 47498	⊢ Pairs_proper = (𝑣 ∈ V ↦ {𝑝 ∣ ∃𝑎 ∈ 𝑣 ∃𝑏 ∈ 𝑣 (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})})
cfmtno 47512	class FermatNo
df-fmtno 47513	⊢ FermatNo = (𝑛 ∈ ℕ₀ ↦ ((2↑(2↑𝑛)) + 1))
ceven 47609	class Even
codd 47610	class Odd
df-even 47611	⊢ Even = {𝑧 ∈ ℤ ∣ (𝑧 / 2) ∈ ℤ}
df-odd 47612	⊢ Odd = {𝑧 ∈ ℤ ∣ ((𝑧 + 1) / 2) ∈ ℤ}
cfppr 47709	class FPPr
df-fppr 47710	⊢ FPPr = (𝑛 ∈ ℕ ↦ {𝑥 ∈ (ℤ_≥‘4) ∣ (𝑥 ∉ ℙ ∧ 𝑥 ∥ ((𝑛↑(𝑥 − 1)) − 1))})
cgbe 47730	class GoldbachEven
cgbow 47731	class GoldbachOddW
cgbo 47732	class GoldbachOdd
df-gbe 47733	⊢ GoldbachEven = {𝑧 ∈ Even ∣ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑧 = (𝑝 + 𝑞))}
df-gbow 47734	⊢ GoldbachOddW = {𝑧 ∈ Odd ∣ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑧 = ((𝑝 + 𝑞) + 𝑟)}
df-gbo 47735	⊢ GoldbachOdd = {𝑧 ∈ Odd ∣ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ ((𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑟 ∈ Odd ) ∧ 𝑧 = ((𝑝 + 𝑞) + 𝑟))}
ax-bgbltosilva 47795	⊢ ((𝑁 ∈ Even ∧ 4 < 𝑁 ∧ 𝑁 ≤ (4 · (;10↑;18))) → 𝑁 ∈ GoldbachEven )
ax-tgoldbachgt 47796	⊢ 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 47799	⊢ ∃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 47803	class ClNeighbVtx
df-clnbgr 47804	⊢ ClNeighbVtx = (𝑔 ∈ V, 𝑣 ∈ (Vtx‘𝑔) ↦ ({𝑣} ∪ {𝑛 ∈ (Vtx‘𝑔) ∣ ∃𝑒 ∈ (Edg‘𝑔){𝑣, 𝑛} ⊆ 𝑒}))
cisubgr 47845	class ISubGr
df-isubgr 47846	⊢ ISubGr = (𝑔 ∈ V, 𝑣 ∈ 𝒫 (Vtx‘𝑔) ↦ ⟨𝑣, ⦋(iEdg‘𝑔) / 𝑒⦌(𝑒 ↾ {𝑥 ∈ dom 𝑒 ∣ (𝑒‘𝑥) ⊆ 𝑣})⟩)
cgrisom 47859	class GraphIsom
cgrim 47860	class GraphIso
cgric 47861	class ≃_𝑔𝑟
df-grisom 47862	⊢ 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 47863	⊢ 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 47866	⊢ ≃_𝑔𝑟 = (^◡ GraphIso “ (V ∖ 1_o))
cgrtri 47922	class GrTriangles
df-grtri 47923	⊢ 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 47936	class StarGr
df-stgr 47937	⊢ StarGr = (𝑛 ∈ ℕ₀ ↦ {⟨(Base‘ndx), (0...𝑛)⟩, ⟨(.ef‘ndx), ( I ↾ {𝑒 ∈ 𝒫 (0...𝑛) ∣ ∃𝑥 ∈ (1...𝑛)𝑒 = {0, 𝑥}})⟩})
cgrlim 47961	class GraphLocIso
cgrlic 47962	class ≃_𝑙𝑔𝑟
df-grlim 47963	⊢ 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 47966	⊢ ≃_𝑙𝑔𝑟 = (^◡ GraphLocIso “ (V ∖ 1_o))
cgpg 48025	class gPetersenGr
df-gpg 48026	⊢ 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 48118	class UPWalks
df-upwlks 48119	⊢ 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 48168	class clLaw
casslaw 48169	class assLaw
ccomlaw 48170	class comLaw
df-cllaw 48171	⊢ clLaw = {⟨𝑜, 𝑚⟩ ∣ ∀𝑥 ∈ 𝑚 ∀𝑦 ∈ 𝑚 (𝑥𝑜𝑦) ∈ 𝑚}
df-comlaw 48172	⊢ comLaw = {⟨𝑜, 𝑚⟩ ∣ ∀𝑥 ∈ 𝑚 ∀𝑦 ∈ 𝑚 (𝑥𝑜𝑦) = (𝑦𝑜𝑥)}
df-asslaw 48173	⊢ assLaw = {⟨𝑜, 𝑚⟩ ∣ ∀𝑥 ∈ 𝑚 ∀𝑦 ∈ 𝑚 ∀𝑧 ∈ 𝑚 ((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧))}
cintop 48181	class intOp
cclintop 48182	class clIntOp
cassintop 48183	class assIntOp
df-intop 48184	⊢ intOp = (𝑚 ∈ V, 𝑛 ∈ V ↦ (𝑛 ↑_m (𝑚 × 𝑚)))
df-clintop 48185	⊢ clIntOp = (𝑚 ∈ V ↦ (𝑚 intOp 𝑚))
df-assintop 48186	⊢ assIntOp = (𝑚 ∈ V ↦ {𝑜 ∈ ( clIntOp ‘𝑚) ∣ 𝑜 assLaw 𝑚})
cmgm2 48200	class MgmALT
ccmgm2 48201	class CMgmALT
csgrp2 48202	class SGrpALT
ccsgrp2 48203	class CSGrpALT
df-mgm2 48204	⊢ MgmALT = {𝑚 ∣ (+_g‘𝑚) clLaw (Base‘𝑚)}
df-cmgm2 48205	⊢ CMgmALT = {𝑚 ∈ MgmALT ∣ (+_g‘𝑚) comLaw (Base‘𝑚)}
df-sgrp2 48206	⊢ SGrpALT = {𝑔 ∈ MgmALT ∣ (+_g‘𝑔) assLaw (Base‘𝑔)}
df-csgrp2 48207	⊢ CSGrpALT = {𝑔 ∈ SGrpALT ∣ (+_g‘𝑔) comLaw (Base‘𝑔)}
crngcALTV 48248	class RngCatALTV
df-rngcALTV 48249	⊢ 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 48272	class RingCatALTV
df-ringcALTV 48273	⊢ 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 48382	class DMatALT
cscmatalt 48383	class ScMatALT
df-dmatalt 48384	⊢ 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 48385	⊢ 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 48390	class linC
clinco 48391	class LinCo
df-linc 48392	⊢ 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 48393	⊢ 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 48426	class linIndS
clindeps 48427	class linDepS
df-lininds 48428	⊢ 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 48430	⊢ linDepS = {⟨𝑠, 𝑚⟩ ∣ ¬ 𝑠 linIndS 𝑚}
cfdiv 48523	class /_f
df-fdiv 48524	⊢ /_f = (𝑓 ∈ V, 𝑔 ∈ V ↦ ((𝑓 ∘_f / 𝑔) ↾ (𝑔 supp 0)))
cbigo 48533	class Ο
df-bigo 48534	⊢ Ο = (𝑔 ∈ (ℝ ↑_pm ℝ) ↦ {𝑓 ∈ (ℝ ↑_pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(𝑓‘𝑦) ≤ (𝑚 · (𝑔‘𝑦))})
cblen 48555	class #_b
df-blen 48556	⊢ #_b = (𝑛 ∈ V ↦ if(𝑛 = 0, 1, ((⌊‘(2 log_b (abs‘𝑛))) + 1)))
cdig 48581	class digit
df-dig 48582	⊢ digit = (𝑏 ∈ ℕ ↦ (𝑘 ∈ ℤ, 𝑟 ∈ (0[,)+∞) ↦ ((⌊‘((𝑏↑-𝑘) · 𝑟)) mod 𝑏)))
cnaryf 48612	class -aryF
df-naryf 48613	⊢ -aryF = (𝑛 ∈ ℕ₀, 𝑥 ∈ V ↦ (𝑥 ↑_m (𝑥 ↑_m (0..^𝑛))))
citco 48643	class IterComp
cack 48644	class Ack
df-itco 48645	⊢ IterComp = (𝑓 ∈ V ↦ seq0((𝑔 ∈ V, 𝑗 ∈ V ↦ (𝑓 ∘ 𝑔)), (𝑖 ∈ ℕ₀ ↦ if(𝑖 = 0, ( I ↾ dom 𝑓), 𝑓))))
df-ack 48646	⊢ Ack = seq0((𝑓 ∈ V, 𝑗 ∈ V ↦ (𝑛 ∈ ℕ₀ ↦ (((IterComp‘𝑓)‘(𝑛 + 1))‘1))), (𝑖 ∈ ℕ₀ ↦ if(𝑖 = 0, (𝑛 ∈ ℕ₀ ↦ (𝑛 + 1)), 𝑖)))
cline 48713	class Line_M
csph 48714	class Sphere
df-line 48715	⊢ 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 48716	⊢ Sphere = (𝑤 ∈ V ↦ (𝑥 ∈ (Base‘𝑤), 𝑟 ∈ (0[,]+∞) ↦ {𝑝 ∈ (Base‘𝑤) ∣ (𝑝(dist‘𝑤)𝑥) = 𝑟}))
coppf 49108	class oppFunc
df-oppf 49109	⊢ oppFunc = (𝑓 ∈ V, 𝑔 ∈ V ↦ if((Rel 𝑔 ∧ Rel dom 𝑔), ⟨𝑓, tpos 𝑔⟩, ∅))
cup 49159	class UP
df-up 49160	⊢ 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 49245	class swap_F
df-swapf 49246	⊢ 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 49302	class ∘_F
df-fuco 49303	⊢ ∘_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 49359	class −∘_F
df-prcof 49360	⊢ −∘_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 49403	class ThinCat
df-thinc 49404	⊢ ThinCat = {𝑐 ∈ Cat ∣ [(Base‘𝑐) / 𝑏][(Hom ‘𝑐) / ℎ]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∃*𝑓 𝑓 ∈ (𝑥ℎ𝑦)}
ctermc 49458	class TermCat
df-termc 49459	⊢ TermCat = {𝑐 ∈ ThinCat ∣ ∃𝑥(Base‘𝑐) = {𝑥}}
cprstc 49535	class ProsetToCat
df-prstc 49536	⊢ ProsetToCat = (𝑘 ∈ Proset ↦ ((𝑘 sSet ⟨(Hom ‘ndx), ((le‘𝑘) × {1_o})⟩) sSet ⟨(comp‘ndx), ∅⟩))
cmndtc 49563	class MndToCat
df-mndtc 49564	⊢ MndToCat = (𝑚 ∈ Mnd ↦ {⟨(Base‘ndx), {𝑚}⟩, ⟨(Hom ‘ndx), {⟨𝑚, 𝑚, (Base‘𝑚)⟩}⟩, ⟨(comp‘ndx), {⟨⟨𝑚, 𝑚, 𝑚⟩, (+_g‘𝑚)⟩}⟩})
clan 49591	class Lan
cran 49592	class Ran
df-lan 49593	⊢ 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 49594	⊢ 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 49629	class Limit
ccmd 49630	class Colimit
df-lmd 49631	⊢ Limit = (𝑐 ∈ V, 𝑑 ∈ V ↦ (𝑓 ∈ (𝑑 Func 𝑐) ↦ (( oppFunc ‘(𝑐Δ_func𝑑))((oppCat‘𝑐) UP (oppCat‘(𝑑 FuncCat 𝑐)))𝑓)))
df-cmd 49632	⊢ Colimit = (𝑐 ∈ V, 𝑑 ∈ V ↦ (𝑓 ∈ (𝑑 Func 𝑐) ↦ ((𝑐Δ_func𝑑)(𝑐 UP (𝑑 FuncCat 𝑐))𝑓)))
csetrecs 49669	class setrecs(𝐹)
df-setrecs 49670	⊢ setrecs(𝐹) = ∪ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)}
cpg 49695	class Pg
df-pg 49696	⊢ Pg = setrecs((𝑥 ∈ V ↦ (𝒫 𝑥 × 𝒫 𝑥)))
cge-real 49706	class ≥
cgt 49707	class >
df-gte 49708	⊢ ≥ = ^◡ ≤
df-gt 49709	⊢ > = ^◡ <
csinh 49716	class sinh
ccosh 49717	class cosh
ctanh 49718	class tanh
df-sinh 49719	⊢ sinh = (𝑥 ∈ ℂ ↦ ((sin‘(i · 𝑥)) / i))
df-cosh 49720	⊢ cosh = (𝑥 ∈ ℂ ↦ (cos‘(i · 𝑥)))
df-tanh 49721	⊢ tanh = (𝑥 ∈ (^◡cosh “ (ℂ ∖ {0})) ↦ ((tan‘(i · 𝑥)) / i))
csec 49727	class sec
ccsc 49728	class csc
ccot 49729	class cot
df-sec 49730	⊢ sec = (𝑥 ∈ {𝑦 ∈ ℂ ∣ (cos‘𝑦) ≠ 0} ↦ (1 / (cos‘𝑥)))
df-csc 49731	⊢ csc = (𝑥 ∈ {𝑦 ∈ ℂ ∣ (sin‘𝑦) ≠ 0} ↦ (1 / (sin‘𝑥)))
df-cot 49732	⊢ cot = (𝑥 ∈ {𝑦 ∈ ℂ ∣ (sin‘𝑦) ≠ 0} ↦ ((cos‘𝑥) / (sin‘𝑥)))
clog- 49751	class log_
df-logbALT 49752	⊢ log_ = (𝑏 ∈ (ℂ ∖ {0, 1}) ↦ (𝑥 ∈ (ℂ ∖ {0}) ↦ ((log‘𝑥) / (log‘𝑏))))
wreflexive 49753	wff 𝑅Reflexive𝐴
df-reflexive 49754	⊢ (𝑅Reflexive𝐴 ↔ (𝑅 ⊆ (𝐴 × 𝐴) ∧ ∀𝑥 ∈ 𝐴 𝑥𝑅𝑥))
wirreflexive 49755	wff 𝑅Irreflexive𝐴
df-irreflexive 49756	⊢ (𝑅Irreflexive𝐴 ↔ (𝑅 ⊆ (𝐴 × 𝐴) ∧ ∀𝑥 ∈ 𝐴 ¬ 𝑥𝑅𝑥))
walsi 49772	wff ∀!𝑥(𝜑 → 𝜓)
walsc 49773	wff ∀!𝑥 ∈ 𝐴𝜑
df-alsi 49774	⊢ (∀!𝑥(𝜑 → 𝜓) ↔ (∀𝑥(𝜑 → 𝜓) ∧ ∃𝑥𝜑))
df-alsc 49775	⊢ (∀!𝑥 ∈ 𝐴𝜑 ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑥 𝑥 ∈ 𝐴))

W3C validator