mmdefinitions - Metamath Proof Explorer

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 848	wff (𝜑 ∨ 𝜓)
df-or 849	⊢ ((𝜑 ∨ 𝜓) ↔ (¬ 𝜑 → 𝜓))
wif 1063	wff if-(𝜑, 𝜓, 𝜒)
df-ifp 1064	⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒)))
w3o 1086	wff (𝜑 ∨ 𝜓 ∨ 𝜒)
w3a 1087	wff (𝜑 ∧ 𝜓 ∧ 𝜒)
df-3or 1088	⊢ ((𝜑 ∨ 𝜓 ∨ 𝜒) ↔ ((𝜑 ∨ 𝜓) ∨ 𝜒))
df-3an 1089	⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ((𝜑 ∧ 𝜓) ∧ 𝜒))
wnan 1493	wff (𝜑 ⊼ 𝜓)
df-nan 1494	⊢ ((𝜑 ⊼ 𝜓) ↔ ¬ (𝜑 ∧ 𝜓))
wxo 1513	wff (𝜑 ⊻ 𝜓)
df-xor 1514	⊢ ((𝜑 ⊻ 𝜓) ↔ ¬ (𝜑 ↔ 𝜓))
wnor 1530	wff (𝜑 ⊽ 𝜓)
df-nor 1531	⊢ ((𝜑 ⊽ 𝜓) ↔ ¬ (𝜑 ∨ 𝜓))
wal 1540	wff ∀𝑥𝜑
cv 1541	class 𝑥
wceq 1542	wff 𝐴 = 𝐵
wtru 1543	wff ⊤
df-tru 1545	⊢ (⊤ ↔ (∀𝑥 𝑥 = 𝑥 → ∀𝑥 𝑥 = 𝑥))
wfal 1554	wff ⊥
df-fal 1555	⊢ (⊥ ↔ ¬ ⊤)
whad 1595	wff hadd(𝜑, 𝜓, 𝜒)
df-had 1596	⊢ (hadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ⊻ 𝜓) ⊻ 𝜒))
wcad 1608	wff cadd(𝜑, 𝜓, 𝜒)
df-cad 1609	⊢ (cadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ∧ 𝜓) ∨ (𝜒 ∧ (𝜑 ⊻ 𝜓))))
wex 1781	wff ∃𝑥𝜑
df-ex 1782	⊢ (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑)
wnf 1785	wff Ⅎ𝑥𝜑
df-nf 1786	⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
ax-gen 1797	⊢ 𝜑    ⇒   ⊢ ∀𝑥𝜑
ax-4 1811	⊢ (∀𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))
ax-5 1912	⊢ (𝜑 → ∀𝑥𝜑)
ax-6 1969	⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦
ax-7 2010	⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 → 𝑦 = 𝑧))
wsb 2068	wff [𝑦 / 𝑥]𝜑
df-sb 2069	⊢ (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) ↔ ∀𝑧(𝑧 = 𝑡 → ∀𝑥(𝑥 = 𝑧 → 𝜑)))    ⇒   ⊢ ([𝑡 / 𝑥]𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
wcel 2114	wff 𝐴 ∈ 𝐵
ax-8 2116	⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧))
ax-9 2124	⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦))
ax-10 2147	⊢ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
ax-11 2163	⊢ (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑)
ax-12 2185	⊢ (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
ax-13 2377	⊢ (¬ 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
wmo 2538	wff ∃*𝑥𝜑
df-mo 2540	⊢ (∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦) ↔ ∃𝑧∀𝑥(𝜑 → 𝑥 = 𝑧))    ⇒   ⊢ (∃*𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))
weu 2569	wff ∃!𝑥𝜑
df-eu 2570	⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃*𝑥𝜑))
ax-ext 2709	⊢ (∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → 𝑥 = 𝑦)
cab 2715	class {𝑥 ∣ 𝜑}
df-clab 2716	⊢ (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ [𝑥 / 𝑦]𝜑)
df-cleq 2729	⊢ (𝑦 = 𝑧 ↔ ∀𝑢(𝑢 ∈ 𝑦 ↔ 𝑢 ∈ 𝑧))    &   ⊢ (𝑡 = 𝑡 ↔ ∀𝑣(𝑣 ∈ 𝑡 ↔ 𝑣 ∈ 𝑡))    ⇒   ⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
df-clel 2812	⊢ (𝑦 ∈ 𝑧 ↔ ∃𝑢(𝑢 = 𝑦 ∧ 𝑢 ∈ 𝑧))    &   ⊢ (𝑡 ∈ 𝑡 ↔ ∃𝑣(𝑣 = 𝑡 ∧ 𝑣 ∈ 𝑡))    ⇒   ⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵))
wnfc 2884	wff Ⅎ𝑥𝐴
df-nfc 2886	⊢ (Ⅎ𝑥𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴)
wne 2933	wff 𝐴 ≠ 𝐵
df-ne 2934	⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵)
wnel 3037	wff 𝐴 ∉ 𝐵
df-nel 3038	⊢ (𝐴 ∉ 𝐵 ↔ ¬ 𝐴 ∈ 𝐵)
wral 3052	wff ∀𝑥 ∈ 𝐴 𝜑
df-ral 3053	⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
wrex 3062	wff ∃𝑥 ∈ 𝐴 𝜑
df-rex 3063	⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
wreu 3350	wff ∃!𝑥 ∈ 𝐴 𝜑
wrmo 3351	wff ∃*𝑥 ∈ 𝐴 𝜑
df-rmo 3352	⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
df-reu 3353	⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
crab 3401	class {𝑥 ∈ 𝐴 ∣ 𝜑}
df-rab 3402	⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
cvv 3442	class V
df-v 3444	⊢ V = {𝑥 ∣ 𝑥 = 𝑥}
wcdeq 3723	wff CondEq(𝑥 = 𝑦 → 𝜑)
df-cdeq 3724	⊢ (CondEq(𝑥 = 𝑦 → 𝜑) ↔ (𝑥 = 𝑦 → 𝜑))
wsbc 3742	wff [𝐴 / 𝑥]𝜑
df-sbc 3743	⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑})
csb 3851	class ⦋𝐴 / 𝑥⦌𝐵
df-csb 3852	⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵}
cdif 3900	class (𝐴 ∖ 𝐵)
cun 3901	class (𝐴 ∪ 𝐵)
cin 3902	class (𝐴 ∩ 𝐵)
wss 3903	wff 𝐴 ⊆ 𝐵
wpss 3904	wff 𝐴 ⊊ 𝐵
df-dif 3906	⊢ (𝐴 ∖ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)}
df-un 3908	⊢ (𝐴 ∪ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)}
df-in 3910	⊢ (𝐴 ∩ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)}
df-ss 3920	⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
df-pss 3923	⊢ (𝐴 ⊊ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ 𝐵))
csymdif 4206	class (𝐴 △ 𝐵)
df-symdif 4207	⊢ (𝐴 △ 𝐵) = ((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐴))
c0 4287	class ∅
df-nul 4288	⊢ ∅ = (V ∖ V)
cif 4481	class if(𝜑, 𝐴, 𝐵)
df-if 4482	⊢ if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑))}
cpw 4556	class 𝒫 𝐴
df-pw 4558	⊢ 𝒫 𝐴 = {𝑥 ∣ 𝑥 ⊆ 𝐴}
csn 4582	class {𝐴}
df-sn 4583	⊢ {𝐴} = {𝑥 ∣ 𝑥 = 𝐴}
cpr 4584	class {𝐴, 𝐵}
df-pr 4585	⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
ctp 4586	class {𝐴, 𝐵, 𝐶}
df-tp 4587	⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
cop 4588	class ⟨𝐴, 𝐵⟩
df-op 4589	⊢ ⟨𝐴, 𝐵⟩ = {𝑥 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})}
cotp 4590	class ⟨𝐴, 𝐵, 𝐶⟩
df-ot 4591	⊢ ⟨𝐴, 𝐵, 𝐶⟩ = ⟨⟨𝐴, 𝐵⟩, 𝐶⟩
cuni 4865	class ∪ 𝐴
df-uni 4866	⊢ ∪ 𝐴 = {𝑥 ∣ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)}
cint 4904	class ∩ 𝐴
df-int 4905	⊢ ∩ 𝐴 = {𝑥 ∣ ∀𝑦(𝑦 ∈ 𝐴 → 𝑥 ∈ 𝑦)}
ciun 4948	class ∪ 𝑥 ∈ 𝐴 𝐵
ciin 4949	class ∩ 𝑥 ∈ 𝐴 𝐵
df-iun 4950	⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵}
df-iin 4951	⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵}
wdisj 5067	wff Disj 𝑥 ∈ 𝐴 𝐵
df-disj 5068	⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
wbr 5100	wff 𝐴𝑅𝐵
df-br 5101	⊢ (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
copab 5162	class {⟨𝑥, 𝑦⟩ ∣ 𝜑}
df-opab 5163	⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
cmpt 5181	class (𝑥 ∈ 𝐴 ↦ 𝐵)
df-mpt 5182	⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
wtr 5207	wff Tr 𝐴
df-tr 5208	⊢ (Tr 𝐴 ↔ ∪ 𝐴 ⊆ 𝐴)
ax-rep 5226	⊢ (∀𝑤∃𝑦∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦) → ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)))
ax-sep 5243	⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))
ax-nul 5253	⊢ ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥
ax-pow 5312	⊢ ∃𝑦∀𝑧(∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦)
ax-pr 5379	⊢ ∃𝑧∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧)
cid 5526	class I
df-id 5527	⊢ I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
cep 5531	class E
df-eprel 5532	⊢ E = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ 𝑦}
wpo 5538	wff 𝑅 Po 𝐴
wor 5539	wff 𝑅 Or 𝐴
df-po 5540	⊢ (𝑅 Po 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
df-so 5541	⊢ (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)))
wfr 5582	wff 𝑅 Fr 𝐴
wse 5583	wff 𝑅 Se 𝐴
wwe 5584	wff 𝑅 We 𝐴
df-fr 5585	⊢ (𝑅 Fr 𝐴 ↔ ∀𝑥((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦))
df-se 5586	⊢ (𝑅 Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V)
df-we 5587	⊢ (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴 ∧ 𝑅 Or 𝐴))
cxp 5630	class (𝐴 × 𝐵)
ccnv 5631	class ◡𝐴
cdm 5632	class dom 𝐴
crn 5633	class ran 𝐴
cres 5634	class (𝐴 ↾ 𝐵)
cima 5635	class (𝐴 “ 𝐵)
ccom 5636	class (𝐴 ∘ 𝐵)
wrel 5637	wff Rel 𝐴
df-xp 5638	⊢ (𝐴 × 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)}
df-rel 5639	⊢ (Rel 𝐴 ↔ 𝐴 ⊆ (V × V))
df-cnv 5640	⊢ ◡𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥}
df-co 5641	⊢ (𝐴 ∘ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)}
df-dm 5642	⊢ dom 𝐴 = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦}
df-rn 5643	⊢ ran 𝐴 = dom ◡𝐴
df-res 5644	⊢ (𝐴 ↾ 𝐵) = (𝐴 ∩ (𝐵 × V))
df-ima 5645	⊢ (𝐴 “ 𝐵) = ran (𝐴 ↾ 𝐵)
cpred 6266	class Pred(𝑅, 𝐴, 𝑋)
df-pred 6267	⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋}))
word 6324	wff Ord 𝐴
con0 6325	class On
wlim 6326	wff Lim 𝐴
csuc 6327	class suc 𝐴
df-ord 6328	⊢ (Ord 𝐴 ↔ (Tr 𝐴 ∧ E We 𝐴))
df-on 6329	⊢ On = {𝑥 ∣ Ord 𝑥}
df-lim 6330	⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴))
df-suc 6331	⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
cio 6454	class (℩𝑥𝜑)
df-iota 6456	⊢ (℩𝑥𝜑) = ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}
wfun 6494	wff Fun 𝐴
wfn 6495	wff 𝐴 Fn 𝐵
wf 6496	wff 𝐹:𝐴⟶𝐵
wf1 6497	wff 𝐹:𝐴–1-1→𝐵
wfo 6498	wff 𝐹:𝐴–onto→𝐵
wf1o 6499	wff 𝐹:𝐴–1-1-onto→𝐵
cfv 6500	class (𝐹‘𝐴)
wiso 6501	wff 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵)
df-fun 6502	⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ (𝐴 ∘ ◡𝐴) ⊆ I ))
df-fn 6503	⊢ (𝐴 Fn 𝐵 ↔ (Fun 𝐴 ∧ dom 𝐴 = 𝐵))
df-f 6504	⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵))
df-f1 6505	⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ Fun ◡𝐹))
df-fo 6506	⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵))
df-f1o 6507	⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹:𝐴–1-1→𝐵 ∧ 𝐹:𝐴–onto→𝐵))
df-fv 6508	⊢ (𝐹‘𝐴) = (℩𝑥𝐴𝐹𝑥)
df-isom 6509	⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐻:𝐴–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦))))
crio 7324	class (℩𝑥 ∈ 𝐴 𝜑)
df-riota 7325	⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
co 7368	class (𝐴𝐹𝐵)
coprab 7369	class {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
cmpo 7370	class (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
df-ov 7371	⊢ (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
df-oprab 7372	⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)}
df-mpo 7373	⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)}
cof 7630	class ∘f 𝑅
cofr 7631	class ∘r 𝑅
df-of 7632	⊢ ∘f 𝑅 = (𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥))))
df-ofr 7633	⊢ ∘r 𝑅 = {⟨𝑓, 𝑔⟩ ∣ ∀𝑥 ∈ (dom 𝑓 ∩ dom 𝑔)(𝑓‘𝑥)𝑅(𝑔‘𝑥)}
crpss 7677	class [⊊]
df-rpss 7678	⊢ [⊊] = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ⊊ 𝑦}
ax-un 7690	⊢ ∃𝑦∀𝑧(∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦)
com 7818	class ω
df-om 7819	⊢ ω = {𝑥 ∈ On ∣ ∀𝑦(Lim 𝑦 → 𝑥 ∈ 𝑦)}
c1st 7941	class 1st
c2nd 7942	class 2nd
df-1st 7943	⊢ 1st = (𝑥 ∈ V ↦ ∪ dom {𝑥})
df-2nd 7944	⊢ 2nd = (𝑥 ∈ V ↦ ∪ ran {𝑥})
csupp 8112	class supp
df-supp 8113	⊢ supp = (𝑥 ∈ V, 𝑧 ∈ V ↦ {𝑖 ∈ dom 𝑥 ∣ (𝑥 “ {𝑖}) ≠ {𝑧}})
ctpos 8177	class tpos 𝐹
df-tpos 8178	⊢ tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}))
ccur 8217	class curry 𝐴
cunc 8218	class uncurry 𝐴
df-cur 8219	⊢ curry 𝐹 = (𝑥 ∈ dom dom 𝐹 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩𝐹𝑧})
df-unc 8220	⊢ uncurry 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(𝐹‘𝑥)𝑧}
cund 8224	class Undef
df-undef 8225	⊢ Undef = (𝑠 ∈ V ↦ 𝒫 ∪ 𝑠)
cfrecs 8232	class frecs(𝑅, 𝐴, 𝐹)
df-frecs 8233	⊢ 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 8263	class wrecs(𝑅, 𝐴, 𝐹)
df-wrecs 8264	⊢ wrecs(𝑅, 𝐴, 𝐹) = frecs(𝑅, 𝐴, (𝐹 ∘ 2nd ))
wsmo 8287	wff Smo 𝐴
df-smo 8288	⊢ (Smo 𝐴 ↔ (𝐴:dom 𝐴⟶On ∧ Ord dom 𝐴 ∧ ∀𝑥 ∈ dom 𝐴∀𝑦 ∈ dom 𝐴(𝑥 ∈ 𝑦 → (𝐴‘𝑥) ∈ (𝐴‘𝑦))))
crecs 8312	class recs(𝐹)
df-recs 8313	⊢ recs(𝐹) = wrecs( E , On, 𝐹)
crdg 8350	class rec(𝐹, 𝐼)
df-rdg 8351	⊢ rec(𝐹, 𝐼) = recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔))))))
cseqom 8388	class seqω(𝐹, 𝐼)
df-seqom 8389	⊢ seqω(𝐹, 𝐼) = (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) “ ω)
c1o 8400	class 1o
c2o 8401	class 2o
c3o 8402	class 3o
c4o 8403	class 4o
coa 8404	class +o
comu 8405	class ·o
coe 8406	class ↑o
df-1o 8407	⊢ 1o = suc ∅
df-2o 8408	⊢ 2o = suc 1o
df-3o 8409	⊢ 3o = suc 2o
df-4o 8410	⊢ 4o = suc 3o
df-oadd 8411	⊢ +o = (𝑥 ∈ On, 𝑦 ∈ On ↦ (rec((𝑧 ∈ V ↦ suc 𝑧), 𝑥)‘𝑦))
df-omul 8412	⊢ ·o = (𝑥 ∈ On, 𝑦 ∈ On ↦ (rec((𝑧 ∈ V ↦ (𝑧 +o 𝑥)), ∅)‘𝑦))
df-oexp 8413	⊢ ↑o = (𝑥 ∈ On, 𝑦 ∈ On ↦ if(𝑥 = ∅, (1o ∖ 𝑦), (rec((𝑧 ∈ V ↦ (𝑧 ·o 𝑥)), 1o)‘𝑦)))
cnadd 8603	class +no
df-nadd 8604	⊢ +no = frecs({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On) ∧ (((1st ‘𝑥) E (1st ‘𝑦) ∨ (1st ‘𝑥) = (1st ‘𝑦)) ∧ ((2nd ‘𝑥) E (2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))}, (On × On), (𝑧 ∈ V, 𝑎 ∈ V ↦ ∩ {𝑤 ∈ On ∣ ((𝑎 “ ({(1st ‘𝑧)} × (2nd ‘𝑧))) ⊆ 𝑤 ∧ (𝑎 “ ((1st ‘𝑧) × {(2nd ‘𝑧)})) ⊆ 𝑤)}))
wer 8642	wff 𝑅 Er 𝐴
cec 8643	class [𝐴]𝑅
cqs 8644	class (𝐴 / 𝑅)
df-er 8645	⊢ (𝑅 Er 𝐴 ↔ (Rel 𝑅 ∧ dom 𝑅 = 𝐴 ∧ (◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅))
df-ec 8647	⊢ [𝐴]𝑅 = (𝑅 “ {𝐴})
df-qs 8651	⊢ (𝐴 / 𝑅) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅}
cmap 8775	class ↑m
cpm 8776	class ↑pm
df-map 8777	⊢ ↑m = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓 ∣ 𝑓:𝑦⟶𝑥})
df-pm 8778	⊢ ↑pm = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓 ∈ 𝒫 (𝑦 × 𝑥) ∣ Fun 𝑓})
cixp 8847	class X𝑥 ∈ 𝐴 𝐵
df-ixp 8848	⊢ X𝑥 ∈ 𝐴 𝐵 = {𝑓 ∣ (𝑓 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵)}
cen 8892	class ≈
cdom 8893	class ≼
csdm 8894	class ≺
cfn 8895	class Fin
df-en 8896	⊢ ≈ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥–1-1-onto→𝑦}
df-dom 8897	⊢ ≼ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥–1-1→𝑦}
df-sdom 8898	⊢ ≺ = ( ≼ ∖ ≈ )
df-fin 8899	⊢ Fin = {𝑥 ∣ ∃𝑦 ∈ ω 𝑥 ≈ 𝑦}
cfsupp 9276	class finSupp
df-fsupp 9277	⊢ finSupp = {⟨𝑟, 𝑧⟩ ∣ (Fun 𝑟 ∧ (𝑟 supp 𝑧) ∈ Fin)}
cfi 9325	class fi
df-fi 9326	⊢ fi = (𝑥 ∈ V ↦ {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦})
csup 9355	class sup(𝐴, 𝐵, 𝑅)
cinf 9356	class inf(𝐴, 𝐵, 𝑅)
df-sup 9357	⊢ sup(𝐴, 𝐵, 𝑅) = ∪ {𝑥 ∈ 𝐵 ∣ (∀𝑦 ∈ 𝐴 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐵 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐴 𝑦𝑅𝑧))}
df-inf 9358	⊢ inf(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, ◡𝑅)
coi 9426	class OrdIso(𝑅, 𝐴)
df-oi 9427	⊢ 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 9473	class har
df-har 9474	⊢ har = (𝑥 ∈ V ↦ {𝑦 ∈ On ∣ 𝑦 ≼ 𝑥})
cwdom 9481	class ≼*
df-wdom 9482	⊢ ≼* = {⟨𝑥, 𝑦⟩ ∣ (𝑥 = ∅ ∨ ∃𝑧 𝑧:𝑦–onto→𝑥)}
ax-reg 9509	⊢ (∃𝑦 𝑦 ∈ 𝑥 → ∃𝑦(𝑦 ∈ 𝑥 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑥)))
ax-inf 9559	⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦)))
ax-inf2 9562	⊢ ∃𝑥(∃𝑦(𝑦 ∈ 𝑥 ∧ ∀𝑧 ¬ 𝑧 ∈ 𝑦) ∧ ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑧 ∈ 𝑥 ∧ ∀𝑤(𝑤 ∈ 𝑧 ↔ (𝑤 ∈ 𝑦 ∨ 𝑤 = 𝑦)))))
ccnf 9582	class CNF
df-cnf 9583	⊢ CNF = (𝑥 ∈ On, 𝑦 ∈ On ↦ (𝑓 ∈ {𝑔 ∈ (𝑥 ↑m 𝑦) ∣ 𝑔 finSupp ∅} ↦ ⦋OrdIso( E , (𝑓 supp ∅)) / ℎ⦌(seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝑥 ↑o (ℎ‘𝑘)) ·o (𝑓‘(ℎ‘𝑘))) +o 𝑧)), ∅)‘dom ℎ)))
cttrcl 9628	class t++𝑅
df-ttrcl 9629	⊢ t++𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓‘𝑛) = 𝑦) ∧ ∀𝑚 ∈ 𝑛 (𝑓‘𝑚)𝑅(𝑓‘suc 𝑚))}
ctc 9655	class TC
df-tc 9656	⊢ TC = (𝑥 ∈ V ↦ ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ Tr 𝑦)})
cr1 9686	class 𝑅1
crnk 9687	class rank
df-r1 9688	⊢ 𝑅1 = rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅)
df-rank 9689	⊢ rank = (𝑥 ∈ V ↦ ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)})
cdju 9822	class (𝐴 ⊔ 𝐵)
cinl 9823	class inl
cinr 9824	class inr
df-dju 9825	⊢ (𝐴 ⊔ 𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵))
df-inl 9826	⊢ inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩)
df-inr 9827	⊢ inr = (𝑥 ∈ V ↦ ⟨1o, 𝑥⟩)
ccrd 9859	class card
cale 9860	class ℵ
ccf 9861	class cf
wacn 9862	class AC 𝐴
df-card 9863	⊢ card = (𝑥 ∈ V ↦ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥})
df-aleph 9864	⊢ ℵ = rec(har, ω)
df-cf 9865	⊢ cf = (𝑥 ∈ On ↦ ∩ {𝑦 ∣ ∃𝑧(𝑦 = (card‘𝑧) ∧ (𝑧 ⊆ 𝑥 ∧ ∀𝑣 ∈ 𝑥 ∃𝑢 ∈ 𝑧 𝑣 ⊆ 𝑢))})
df-acn 9866	⊢ AC 𝐴 = {𝑥 ∣ (𝐴 ∈ V ∧ ∀𝑓 ∈ ((𝒫 𝑥 ∖ {∅}) ↑m 𝐴)∃𝑔∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝑓‘𝑦))}
wac 10037	wff CHOICE
df-ac 10038	⊢ (CHOICE ↔ ∀𝑥∃𝑓(𝑓 ⊆ 𝑥 ∧ 𝑓 Fn dom 𝑥))
cfin1a 10200	class FinIa
cfin2 10201	class FinII
cfin4 10202	class FinIV
cfin3 10203	class FinIII
cfin5 10204	class FinV
cfin6 10205	class FinVI
cfin7 10206	class FinVII
df-fin1a 10207	⊢ FinIa = {𝑥 ∣ ∀𝑦 ∈ 𝒫 𝑥(𝑦 ∈ Fin ∨ (𝑥 ∖ 𝑦) ∈ Fin)}
df-fin2 10208	⊢ FinII = {𝑥 ∣ ∀𝑦 ∈ 𝒫 𝒫 𝑥((𝑦 ≠ ∅ ∧ [⊊] Or 𝑦) → ∪ 𝑦 ∈ 𝑦)}
df-fin4 10209	⊢ FinIV = {𝑥 ∣ ¬ ∃𝑦(𝑦 ⊊ 𝑥 ∧ 𝑦 ≈ 𝑥)}
df-fin3 10210	⊢ FinIII = {𝑥 ∣ 𝒫 𝑥 ∈ FinIV}
df-fin5 10211	⊢ FinV = {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 ≺ (𝑥 ⊔ 𝑥))}
df-fin6 10212	⊢ FinVI = {𝑥 ∣ (𝑥 ≺ 2o ∨ 𝑥 ≺ (𝑥 × 𝑥))}
df-fin7 10213	⊢ FinVII = {𝑥 ∣ ¬ ∃𝑦 ∈ (On ∖ ω)𝑥 ≈ 𝑦}
ax-cc 10357	⊢ (𝑥 ≈ ω → ∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧))
ax-dc 10368	⊢ ((∃𝑦∃𝑧 𝑦𝑥𝑧 ∧ ran 𝑥 ⊆ dom 𝑥) → ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛))
ax-ac 10381	⊢ ∃𝑦∀𝑧∀𝑤((𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → ∃𝑣∀𝑢(∃𝑡((𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑡) ∧ (𝑢 ∈ 𝑡 ∧ 𝑡 ∈ 𝑦)) ↔ 𝑢 = 𝑣))
ax-ac2 10385	⊢ ∃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 10543	class GCH
df-gch 10544	⊢ GCH = (Fin ∪ {𝑥 ∣ ∀𝑦 ¬ (𝑥 ≺ 𝑦 ∧ 𝑦 ≺ 𝒫 𝑥)})
cwina 10605	class Inaccw
cina 10606	class Inacc
df-wina 10607	⊢ Inaccw = {𝑥 ∣ (𝑥 ≠ ∅ ∧ (cf‘𝑥) = 𝑥 ∧ ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝑥 𝑦 ≺ 𝑧)}
df-ina 10608	⊢ Inacc = {𝑥 ∣ (𝑥 ≠ ∅ ∧ (cf‘𝑥) = 𝑥 ∧ ∀𝑦 ∈ 𝑥 𝒫 𝑦 ≺ 𝑥)}
cwun 10623	class WUni
cwunm 10624	class wUniCl
df-wun 10625	⊢ WUni = {𝑢 ∣ (Tr 𝑢 ∧ 𝑢 ≠ ∅ ∧ ∀𝑥 ∈ 𝑢 (∪ 𝑥 ∈ 𝑢 ∧ 𝒫 𝑥 ∈ 𝑢 ∧ ∀𝑦 ∈ 𝑢 {𝑥, 𝑦} ∈ 𝑢))}
df-wunc 10626	⊢ wUniCl = (𝑥 ∈ V ↦ ∩ {𝑢 ∈ WUni ∣ 𝑥 ⊆ 𝑢})
ctsk 10671	class Tarski
df-tsk 10672	⊢ Tarski = {𝑦 ∣ (∀𝑧 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑦 ∧ ∃𝑤 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧 ≈ 𝑦 ∨ 𝑧 ∈ 𝑦))}
cgru 10713	class Univ
df-gru 10714	⊢ Univ = {𝑢 ∣ (Tr 𝑢 ∧ ∀𝑥 ∈ 𝑢 (𝒫 𝑥 ∈ 𝑢 ∧ ∀𝑦 ∈ 𝑢 {𝑥, 𝑦} ∈ 𝑢 ∧ ∀𝑦 ∈ (𝑢 ↑m 𝑥)∪ ran 𝑦 ∈ 𝑢))}
ax-groth 10746	⊢ ∃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 10760	class tarskiMap
df-tskm 10761	⊢ tarskiMap = (𝑥 ∈ V ↦ ∩ {𝑦 ∈ Tarski ∣ 𝑥 ∈ 𝑦})
cnpi 10767	class N
cpli 10768	class +N
cmi 10769	class ·N
clti 10770	class <N
cplpq 10771	class +pQ
cmpq 10772	class ·pQ
cltpq 10773	class <pQ
ceq 10774	class ~Q
cnq 10775	class Q
c1q 10776	class 1Q
cerq 10777	class [Q]
cplq 10778	class +Q
cmq 10779	class ·Q
crq 10780	class *Q
cltq 10781	class <Q
cnp 10782	class P
c1p 10783	class 1P
cpp 10784	class +P
cmp 10785	class ·P
cltp 10786	class <P
cer 10787	class ~R
cnr 10788	class R
c0r 10789	class 0R
c1r 10790	class 1R
cm1r 10791	class -1R
cplr 10792	class +R
cmr 10793	class ·R
cltr 10794	class <R
df-ni 10795	⊢ N = (ω ∖ {∅})
df-pli 10796	⊢ +N = ( +o ↾ (N × N))
df-mi 10797	⊢ ·N = ( ·o ↾ (N × N))
df-lti 10798	⊢ <N = ( E ∩ (N × N))
df-plpq 10831	⊢ +pQ = (𝑥 ∈ (N × N), 𝑦 ∈ (N × N) ↦ ⟨(((1st ‘𝑥) ·N (2nd ‘𝑦)) +N ((1st ‘𝑦) ·N (2nd ‘𝑥))), ((2nd ‘𝑥) ·N (2nd ‘𝑦))⟩)
df-mpq 10832	⊢ ·pQ = (𝑥 ∈ (N × N), 𝑦 ∈ (N × N) ↦ ⟨((1st ‘𝑥) ·N (1st ‘𝑦)), ((2nd ‘𝑥) ·N (2nd ‘𝑦))⟩)
df-ltpq 10833	⊢ <pQ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ((1st ‘𝑥) ·N (2nd ‘𝑦)) <N ((1st ‘𝑦) ·N (2nd ‘𝑥)))}
df-enq 10834	⊢ ~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 10835	⊢ Q = {𝑥 ∈ (N × N) ∣ ∀𝑦 ∈ (N × N)(𝑥 ~Q 𝑦 → ¬ (2nd ‘𝑦) <N (2nd ‘𝑥))}
df-erq 10836	⊢ [Q] = ( ~Q ∩ ((N × N) × Q))
df-plq 10837	⊢ +Q = (([Q] ∘ +pQ ) ↾ (Q × Q))
df-mq 10838	⊢ ·Q = (([Q] ∘ ·pQ ) ↾ (Q × Q))
df-1nq 10839	⊢ 1Q = ⟨1o, 1o⟩
df-rq 10840	⊢ *Q = (◡ ·Q “ {1Q})
df-ltnq 10841	⊢ <Q = ( <pQ ∩ (Q × Q))
df-np 10904	⊢ P = {𝑥 ∣ ((∅ ⊊ 𝑥 ∧ 𝑥 ⊊ Q) ∧ ∀𝑦 ∈ 𝑥 (∀𝑧(𝑧 <Q 𝑦 → 𝑧 ∈ 𝑥) ∧ ∃𝑧 ∈ 𝑥 𝑦 <Q 𝑧))}
df-1p 10905	⊢ 1P = {𝑥 ∣ 𝑥 <Q 1Q}
df-plp 10906	⊢ +P = (𝑥 ∈ P, 𝑦 ∈ P ↦ {𝑤 ∣ ∃𝑣 ∈ 𝑥 ∃𝑢 ∈ 𝑦 𝑤 = (𝑣 +Q 𝑢)})
df-mp 10907	⊢ ·P = (𝑥 ∈ P, 𝑦 ∈ P ↦ {𝑤 ∣ ∃𝑣 ∈ 𝑥 ∃𝑢 ∈ 𝑦 𝑤 = (𝑣 ·Q 𝑢)})
df-ltp 10908	⊢ <P = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ 𝑥 ⊊ 𝑦)}
df-enr 10978	⊢ ~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 10979	⊢ R = ((P × P) / ~R )
df-plr 10980	⊢ +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 10981	⊢ ·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 10982	⊢ <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 10983	⊢ 0R = [⟨1P, 1P⟩] ~R
df-1r 10984	⊢ 1R = [⟨(1P +P 1P), 1P⟩] ~R
df-m1r 10985	⊢ -1R = [⟨1P, (1P +P 1P)⟩] ~R
cc 11036	class ℂ
cr 11037	class ℝ
cc0 11038	class 0
c1 11039	class 1
ci 11040	class i
caddc 11041	class +
cltrr 11042	class <ℝ
cmul 11043	class ·
df-c 11044	⊢ ℂ = (R × R)
df-0 11045	⊢ 0 = ⟨0R, 0R⟩
df-1 11046	⊢ 1 = ⟨1R, 0R⟩
df-i 11047	⊢ i = ⟨0R, 1R⟩
df-r 11048	⊢ ℝ = (R × {0R})
df-add 11049	⊢ + = {⟨⟨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 11050	⊢ · = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨((𝑤 ·R 𝑢) +R (-1R ·R (𝑣 ·R 𝑓))), ((𝑣 ·R 𝑢) +R (𝑤 ·R 𝑓))⟩))}
df-lt 11051	⊢ <ℝ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ∃𝑧∃𝑤((𝑥 = ⟨𝑧, 0R⟩ ∧ 𝑦 = ⟨𝑤, 0R⟩) ∧ 𝑧 <R 𝑤))}
ax-cnex 11094	⊢ ℂ ∈ V
ax-resscn 11095	⊢ ℝ ⊆ ℂ
ax-1cn 11096	⊢ 1 ∈ ℂ
ax-icn 11097	⊢ i ∈ ℂ
ax-addcl 11098	⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) ∈ ℂ)
ax-addrcl 11099	⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + 𝐵) ∈ ℝ)
ax-mulcl 11100	⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) ∈ ℂ)
ax-mulrcl 11101	⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 · 𝐵) ∈ ℝ)
ax-mulcom 11102	⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) = (𝐵 · 𝐴))
ax-addass 11103	⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶)))
ax-mulass 11104	⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)))
ax-distr 11105	⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶)))
ax-i2m1 11106	⊢ ((i · i) + 1) = 0
ax-1ne0 11107	⊢ 1 ≠ 0
ax-1rid 11108	⊢ (𝐴 ∈ ℝ → (𝐴 · 1) = 𝐴)
ax-rnegex 11109	⊢ (𝐴 ∈ ℝ → ∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0)
ax-rrecex 11110	⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → ∃𝑥 ∈ ℝ (𝐴 · 𝑥) = 1)
ax-cnre 11111	⊢ (𝐴 ∈ ℂ → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝐴 = (𝑥 + (i · 𝑦)))
ax-pre-lttri 11112	⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 <ℝ 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 <ℝ 𝐴)))
ax-pre-lttrn 11113	⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 <ℝ 𝐵 ∧ 𝐵 <ℝ 𝐶) → 𝐴 <ℝ 𝐶))
ax-pre-ltadd 11114	⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 <ℝ 𝐵 → (𝐶 + 𝐴) <ℝ (𝐶 + 𝐵)))
ax-pre-mulgt0 11115	⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((0 <ℝ 𝐴 ∧ 0 <ℝ 𝐵) → 0 <ℝ (𝐴 · 𝐵)))
ax-pre-sup 11116	⊢ ((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 11117	⊢ + :(ℂ × ℂ)⟶ℂ
ax-mulf 11118	⊢ · :(ℂ × ℂ)⟶ℂ
cpnf 11175	class +∞
cmnf 11176	class -∞
cxr 11177	class ℝ*
clt 11178	class <
cle 11179	class ≤
df-pnf 11180	⊢ +∞ = 𝒫 ∪ ℂ
df-mnf 11181	⊢ -∞ = 𝒫 +∞
df-xr 11182	⊢ ℝ* = (ℝ ∪ {+∞, -∞})
df-ltxr 11183	⊢ < = ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)} ∪ (((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ)))
df-le 11184	⊢ ≤ = ((ℝ* × ℝ*) ∖ ◡ < )
cmin 11376	class −
cneg 11377	class -𝐴
df-sub 11378	⊢ − = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥))
df-neg 11379	⊢ -𝐴 = (0 − 𝐴)
cdiv 11806	class /
df-div 11807	⊢ / = (𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥))
cn 12157	class ℕ
df-nn 12158	⊢ ℕ = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) “ ω)
c2 12212	class 2
c3 12213	class 3
c4 12214	class 4
c5 12215	class 5
c6 12216	class 6
c7 12217	class 7
c8 12218	class 8
c9 12219	class 9
df-2 12220	⊢ 2 = (1 + 1)
df-3 12221	⊢ 3 = (2 + 1)
df-4 12222	⊢ 4 = (3 + 1)
df-5 12223	⊢ 5 = (4 + 1)
df-6 12224	⊢ 6 = (5 + 1)
df-7 12225	⊢ 7 = (6 + 1)
df-8 12226	⊢ 8 = (7 + 1)
df-9 12227	⊢ 9 = (8 + 1)
cn0 12413	class ℕ0
df-n0 12414	⊢ ℕ0 = (ℕ ∪ {0})
cxnn0 12486	class ℕ0*
df-xnn0 12487	⊢ ℕ0* = (ℕ0 ∪ {+∞})
cz 12500	class ℤ
df-z 12501	⊢ ℤ = {𝑛 ∈ ℝ ∣ (𝑛 = 0 ∨ 𝑛 ∈ ℕ ∨ -𝑛 ∈ ℕ)}
cdc 12619	class ;𝐴𝐵
df-dec 12620	⊢ ;𝐴𝐵 = (((9 + 1) · 𝐴) + 𝐵)
cuz 12763	class ℤ≥
df-uz 12764	⊢ ℤ≥ = (𝑗 ∈ ℤ ↦ {𝑘 ∈ ℤ ∣ 𝑗 ≤ 𝑘})
cq 12873	class ℚ
df-q 12874	⊢ ℚ = ( / “ (ℤ × ℕ))
crp 12917	class ℝ+
df-rp 12918	⊢ ℝ+ = {𝑥 ∈ ℝ ∣ 0 < 𝑥}
cxne 13035	class -𝑒𝐴
cxad 13036	class +𝑒
cxmu 13037	class ·e
df-xneg 13038	⊢ -𝑒𝐴 = if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴))
df-xadd 13039	⊢ +𝑒 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ if(𝑥 = +∞, if(𝑦 = -∞, 0, +∞), if(𝑥 = -∞, if(𝑦 = +∞, 0, -∞), if(𝑦 = +∞, +∞, if(𝑦 = -∞, -∞, (𝑥 + 𝑦))))))
df-xmul 13040	⊢ ·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 13273	class (,)
cioc 13274	class (,]
cico 13275	class [,)
cicc 13276	class [,]
df-ioo 13277	⊢ (,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)})
df-ioc 13278	⊢ (,] = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 ≤ 𝑦)})
df-ico 13279	⊢ [,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)})
df-icc 13280	⊢ [,] = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)})
cfz 13435	class ...
df-fz 13436	⊢ ... = (𝑚 ∈ ℤ, 𝑛 ∈ ℤ ↦ {𝑘 ∈ ℤ ∣ (𝑚 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛)})
cfzo 13582	class ..^
df-fzo 13583	⊢ ..^ = (𝑚 ∈ ℤ, 𝑛 ∈ ℤ ↦ (𝑚...(𝑛 − 1)))
cfl 13722	class ⌊
cceil 13723	class ⌈
df-fl 13724	⊢ ⌊ = (𝑥 ∈ ℝ ↦ (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))))
df-ceil 13725	⊢ ⌈ = (𝑥 ∈ ℝ ↦ -(⌊‘-𝑥))
cmo 13801	class mod
df-mod 13802	⊢ mod = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ+ ↦ (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦)))))
cseq 13936	class seq𝑀( + , 𝐹)
df-seq 13937	⊢ seq𝑀( + , 𝐹) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) “ ω)
cexp 13996	class ↑
df-exp 13997	⊢ ↑ = (𝑥 ∈ ℂ, 𝑦 ∈ ℤ ↦ if(𝑦 = 0, 1, if(0 < 𝑦, (seq1( · , (ℕ × {𝑥}))‘𝑦), (1 / (seq1( · , (ℕ × {𝑥}))‘-𝑦)))))
cfa 14208	class !
df-fac 14209	⊢ ! = ({⟨0, 1⟩} ∪ seq1( · , I ))
cbc 14237	class C
df-bc 14238	⊢ C = (𝑛 ∈ ℕ0, 𝑘 ∈ ℤ ↦ if(𝑘 ∈ (0...𝑛), ((!‘𝑛) / ((!‘(𝑛 − 𝑘)) · (!‘𝑘))), 0))
chash 14265	class ♯
df-hash 14266	⊢ ♯ = (((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω) ∘ card) ∪ ((V ∖ Fin) × {+∞}))
cword 14448	class Word 𝑆
df-word 14449	⊢ Word 𝑆 = {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆}
clsw 14497	class lastS
df-lsw 14498	⊢ lastS = (𝑤 ∈ V ↦ (𝑤‘((♯‘𝑤) − 1)))
cconcat 14505	class ++
df-concat 14506	⊢ ++ = (𝑠 ∈ V, 𝑡 ∈ V ↦ (𝑥 ∈ (0..^((♯‘𝑠) + (♯‘𝑡))) ↦ if(𝑥 ∈ (0..^(♯‘𝑠)), (𝑠‘𝑥), (𝑡‘(𝑥 − (♯‘𝑠))))))
cs1 14531	class ⟨“𝐴”⟩
df-s1 14532	⊢ ⟨“𝐴”⟩ = {⟨0, ( I ‘𝐴)⟩}
csubstr 14576	class substr
df-substr 14577	⊢ substr = (𝑠 ∈ V, 𝑏 ∈ (ℤ × ℤ) ↦ if(((1st ‘𝑏)..^(2nd ‘𝑏)) ⊆ dom 𝑠, (𝑥 ∈ (0..^((2nd ‘𝑏) − (1st ‘𝑏))) ↦ (𝑠‘(𝑥 + (1st ‘𝑏)))), ∅))
cpfx 14606	class prefix
df-pfx 14607	⊢ prefix = (𝑠 ∈ V, 𝑙 ∈ ℕ0 ↦ (𝑠 substr ⟨0, 𝑙⟩))
csplice 14684	class splice
df-splice 14685	⊢ splice = (𝑠 ∈ V, 𝑏 ∈ V ↦ (((𝑠 prefix (1st ‘(1st ‘𝑏))) ++ (2nd ‘𝑏)) ++ (𝑠 substr ⟨(2nd ‘(1st ‘𝑏)), (♯‘𝑠)⟩)))
creverse 14693	class reverse
df-reverse 14694	⊢ reverse = (𝑠 ∈ V ↦ (𝑥 ∈ (0..^(♯‘𝑠)) ↦ (𝑠‘(((♯‘𝑠) − 1) − 𝑥))))
creps 14703	class repeatS
df-reps 14704	⊢ repeatS = (𝑠 ∈ V, 𝑛 ∈ ℕ0 ↦ (𝑥 ∈ (0..^𝑛) ↦ 𝑠))
ccsh 14723	class cyclShift
df-csh 14724	⊢ 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 14776	class ⟨“𝐴𝐵”⟩
cs3 14777	class ⟨“𝐴𝐵𝐶”⟩
cs4 14778	class ⟨“𝐴𝐵𝐶𝐷”⟩
cs5 14779	class ⟨“𝐴𝐵𝐶𝐷𝐸”⟩
cs6 14780	class ⟨“𝐴𝐵𝐶𝐷𝐸𝐹”⟩
cs7 14781	class ⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”⟩
cs8 14782	class ⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺𝐻”⟩
df-s2 14783	⊢ ⟨“𝐴𝐵”⟩ = (⟨“𝐴”⟩ ++ ⟨“𝐵”⟩)
df-s3 14784	⊢ ⟨“𝐴𝐵𝐶”⟩ = (⟨“𝐴𝐵”⟩ ++ ⟨“𝐶”⟩)
df-s4 14785	⊢ ⟨“𝐴𝐵𝐶𝐷”⟩ = (⟨“𝐴𝐵𝐶”⟩ ++ ⟨“𝐷”⟩)
df-s5 14786	⊢ ⟨“𝐴𝐵𝐶𝐷𝐸”⟩ = (⟨“𝐴𝐵𝐶𝐷”⟩ ++ ⟨“𝐸”⟩)
df-s6 14787	⊢ ⟨“𝐴𝐵𝐶𝐷𝐸𝐹”⟩ = (⟨“𝐴𝐵𝐶𝐷𝐸”⟩ ++ ⟨“𝐹”⟩)
df-s7 14788	⊢ ⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”⟩ = (⟨“𝐴𝐵𝐶𝐷𝐸𝐹”⟩ ++ ⟨“𝐺”⟩)
df-s8 14789	⊢ ⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺𝐻”⟩ = (⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”⟩ ++ ⟨“𝐻”⟩)
ctcl 14920	class t+
crtcl 14921	class t*
df-trcl 14922	⊢ t+ = (𝑥 ∈ V ↦ ∩ {𝑧 ∣ (𝑥 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)})
df-rtrcl 14923	⊢ t* = (𝑥 ∈ V ↦ ∩ {𝑧 ∣ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧 ∧ 𝑥 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)})
crelexp 14954	class ↑𝑟
df-relexp 14955	⊢ ↑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 14990	class t*rec
df-rtrclrec 14991	⊢ t*rec = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))
cshi 15001	class shift
df-shft 15002	⊢ shift = (𝑓 ∈ V, 𝑥 ∈ ℂ ↦ {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ ℂ ∧ (𝑦 − 𝑥)𝑓𝑧)})
csgn 15021	class sgn
df-sgn 15022	⊢ sgn = (𝑥 ∈ ℝ* ↦ if(𝑥 = 0, 0, if(𝑥 < 0, -1, 1)))
ccj 15031	class ∗
cre 15032	class ℜ
cim 15033	class ℑ
df-cj 15034	⊢ ∗ = (𝑥 ∈ ℂ ↦ (℩𝑦 ∈ ℂ ((𝑥 + 𝑦) ∈ ℝ ∧ (i · (𝑥 − 𝑦)) ∈ ℝ)))
df-re 15035	⊢ ℜ = (𝑥 ∈ ℂ ↦ ((𝑥 + (∗‘𝑥)) / 2))
df-im 15036	⊢ ℑ = (𝑥 ∈ ℂ ↦ (ℜ‘(𝑥 / i)))
csqrt 15168	class √
cabs 15169	class abs
df-sqrt 15170	⊢ √ = (𝑥 ∈ ℂ ↦ (℩𝑦 ∈ ℂ ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+)))
df-abs 15171	⊢ abs = (𝑥 ∈ ℂ ↦ (√‘(𝑥 · (∗‘𝑥))))
clsp 15405	class lim sup
df-limsup 15406	⊢ lim sup = (𝑥 ∈ V ↦ inf(ran (𝑘 ∈ ℝ ↦ sup(((𝑥 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )), ℝ*, < ))
cli 15419	class ⇝
crli 15420	class ⇝𝑟
co1 15421	class 𝑂(1)
clo1 15422	class ≤𝑂(1)
df-clim 15423	⊢ ⇝ = {⟨𝑓, 𝑦⟩ ∣ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝑓‘𝑘) ∈ ℂ ∧ (abs‘((𝑓‘𝑘) − 𝑦)) < 𝑥))}
df-rlim 15424	⊢ ⇝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 15425	⊢ 𝑂(1) = {𝑓 ∈ (ℂ ↑pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(abs‘(𝑓‘𝑦)) ≤ 𝑚}
df-lo1 15426	⊢ ≤𝑂(1) = {𝑓 ∈ (ℝ ↑pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(𝑓‘𝑦) ≤ 𝑚}
csu 15621	class Σ𝑘 ∈ 𝐴 𝐵
df-sum 15622	⊢ 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 15838	class ∏𝑘 ∈ 𝐴 𝐵
df-prod 15839	⊢ ∏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 15939	class FallFac
crisefac 15940	class RiseFac
df-risefac 15941	⊢ RiseFac = (𝑥 ∈ ℂ, 𝑛 ∈ ℕ0 ↦ ∏𝑘 ∈ (0...(𝑛 − 1))(𝑥 + 𝑘))
df-fallfac 15942	⊢ FallFac = (𝑥 ∈ ℂ, 𝑛 ∈ ℕ0 ↦ ∏𝑘 ∈ (0...(𝑛 − 1))(𝑥 − 𝑘))
cbp 15981	class BernPoly
df-bpoly 15982	⊢ 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 15996	class exp
ceu 15997	class e
csin 15998	class sin
ccos 15999	class cos
ctan 16000	class tan
cpi 16001	class π
df-ef 16002	⊢ exp = (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ ℕ0 ((𝑥↑𝑘) / (!‘𝑘)))
df-e 16003	⊢ e = (exp‘1)
df-sin 16004	⊢ sin = (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) − (exp‘(-i · 𝑥))) / (2 · i)))
df-cos 16005	⊢ cos = (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥))) / 2))
df-tan 16006	⊢ tan = (𝑥 ∈ (◡cos “ (ℂ ∖ {0})) ↦ ((sin‘𝑥) / (cos‘𝑥)))
df-pi 16007	⊢ π = inf((ℝ+ ∩ (◡sin “ {0})), ℝ, < )
ctau 16139	class τ
df-tau 16140	⊢ τ = inf((ℝ+ ∩ (◡cos “ {1})), ℝ, < )
cdvds 16191	class ∥
df-dvds 16192	⊢ ∥ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ ∃𝑛 ∈ ℤ (𝑛 · 𝑥) = 𝑦)}
cbits 16358	class bits
csad 16359	class sadd
csmu 16360	class smul
df-bits 16361	⊢ bits = (𝑛 ∈ ℤ ↦ {𝑚 ∈ ℕ0 ∣ ¬ 2 ∥ (⌊‘(𝑛 / (2↑𝑚)))})
df-sad 16390	⊢ sadd = (𝑥 ∈ 𝒫 ℕ0, 𝑦 ∈ 𝒫 ℕ0 ↦ {𝑘 ∈ ℕ0 ∣ hadd(𝑘 ∈ 𝑥, 𝑘 ∈ 𝑦, ∅ ∈ (seq0((𝑐 ∈ 2o, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚 ∈ 𝑥, 𝑚 ∈ 𝑦, ∅ ∈ 𝑐), 1o, ∅)), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑘))})
df-smu 16415	⊢ 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 16433	class gcd
df-gcd 16434	⊢ gcd = (𝑥 ∈ ℤ, 𝑦 ∈ ℤ ↦ if((𝑥 = 0 ∧ 𝑦 = 0), 0, sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑥 ∧ 𝑛 ∥ 𝑦)}, ℝ, < )))
clcm 16527	class lcm
clcmf 16528	class lcm
df-lcm 16529	⊢ lcm = (𝑥 ∈ ℤ, 𝑦 ∈ ℤ ↦ if((𝑥 = 0 ∨ 𝑦 = 0), 0, inf({𝑛 ∈ ℕ ∣ (𝑥 ∥ 𝑛 ∧ 𝑦 ∥ 𝑛)}, ℝ, < )))
df-lcmf 16530	⊢ lcm = (𝑧 ∈ 𝒫 ℤ ↦ if(0 ∈ 𝑧, 0, inf({𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑧 𝑚 ∥ 𝑛}, ℝ, < )))
cprime 16610	class ℙ
df-prm 16611	⊢ ℙ = {𝑝 ∈ ℕ ∣ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑝} ≈ 2o}
cnumer 16672	class numer
cdenom 16673	class denom
df-numer 16674	⊢ numer = (𝑦 ∈ ℚ ↦ (1st ‘(℩𝑥 ∈ (ℤ × ℕ)(((1st ‘𝑥) gcd (2nd ‘𝑥)) = 1 ∧ 𝑦 = ((1st ‘𝑥) / (2nd ‘𝑥))))))
df-denom 16675	⊢ denom = (𝑦 ∈ ℚ ↦ (2nd ‘(℩𝑥 ∈ (ℤ × ℕ)(((1st ‘𝑥) gcd (2nd ‘𝑥)) = 1 ∧ 𝑦 = ((1st ‘𝑥) / (2nd ‘𝑥))))))
codz 16702	class odℤ
cphi 16703	class ϕ
df-odz 16704	⊢ odℤ = (𝑛 ∈ ℕ ↦ (𝑥 ∈ {𝑥 ∈ ℤ ∣ (𝑥 gcd 𝑛) = 1} ↦ inf({𝑚 ∈ ℕ ∣ 𝑛 ∥ ((𝑥↑𝑚) − 1)}, ℝ, < )))
df-phi 16705	⊢ ϕ = (𝑛 ∈ ℕ ↦ (♯‘{𝑥 ∈ (1...𝑛) ∣ (𝑥 gcd 𝑛) = 1}))
cpc 16776	class pCnt
df-pc 16777	⊢ 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 16869	class ℤ[i]
df-gz 16870	⊢ ℤ[i] = {𝑥 ∈ ℂ ∣ ((ℜ‘𝑥) ∈ ℤ ∧ (ℑ‘𝑥) ∈ ℤ)}
cvdwa 16905	class AP
cvdwm 16906	class MonoAP
cvdwp 16907	class PolyAP
df-vdwap 16908	⊢ AP = (𝑘 ∈ ℕ0 ↦ (𝑎 ∈ ℕ, 𝑑 ∈ ℕ ↦ ran (𝑚 ∈ (0...(𝑘 − 1)) ↦ (𝑎 + (𝑚 · 𝑑)))))
df-vdwmc 16909	⊢ MonoAP = {⟨𝑘, 𝑓⟩ ∣ ∃𝑐(ran (AP‘𝑘) ∩ 𝒫 (◡𝑓 “ {𝑐})) ≠ ∅}
df-vdwpc 16910	⊢ 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 16939	class Ramsey
df-ram 16941	⊢ 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 16971	class #p
df-prmo 16972	⊢ #p = (𝑛 ∈ ℕ0 ↦ ∏𝑘 ∈ (1...𝑛)if(𝑘 ∈ ℙ, 𝑘, 1))
cstr 17085	class Struct
df-struct 17086	⊢ Struct = {⟨𝑓, 𝑥⟩ ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))}
csts 17102	class sSet
df-sets 17103	⊢ sSet = (𝑠 ∈ V, 𝑒 ∈ V ↦ ((𝑠 ↾ (V ∖ dom {𝑒})) ∪ {𝑒}))
cslot 17120	class Slot 𝐴
df-slot 17121	⊢ Slot 𝐴 = (𝑥 ∈ V ↦ (𝑥‘𝐴))
cnx 17132	class ndx
df-ndx 17133	⊢ ndx = ( I ↾ ℕ)
cbs 17148	class Base
df-base 17149	⊢ Base = Slot 1
cress 17169	class ↾s
df-ress 17170	⊢ ↾s = (𝑤 ∈ V, 𝑥 ∈ V ↦ if((Base‘𝑤) ⊆ 𝑥, 𝑤, (𝑤 sSet ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩)))
cplusg 17189	class +g
cmulr 17190	class .r
cstv 17191	class *𝑟
csca 17192	class Scalar
cvsca 17193	class ·𝑠
cip 17194	class ·𝑖
cts 17195	class TopSet
cple 17196	class le
coc 17197	class oc
cds 17198	class dist
cunif 17199	class UnifSet
chom 17200	class Hom
cco 17201	class comp
df-plusg 17202	⊢ +g = Slot 2
df-mulr 17203	⊢ .r = Slot 3
df-starv 17204	⊢ *𝑟 = Slot 4
df-sca 17205	⊢ Scalar = Slot 5
df-vsca 17206	⊢ ·𝑠 = Slot 6
df-ip 17207	⊢ ·𝑖 = Slot 8
df-tset 17208	⊢ TopSet = Slot 9
df-ple 17209	⊢ le = Slot ;10
df-ocomp 17210	⊢ oc = Slot ;11
df-ds 17211	⊢ dist = Slot ;12
df-unif 17212	⊢ UnifSet = Slot ;13
df-hom 17213	⊢ Hom = Slot ;14
df-cco 17214	⊢ comp = Slot ;15
crest 17352	class ↾t
ctopn 17353	class TopOpen
df-rest 17354	⊢ ↾t = (𝑗 ∈ V, 𝑥 ∈ V ↦ ran (𝑦 ∈ 𝑗 ↦ (𝑦 ∩ 𝑥)))
df-topn 17355	⊢ TopOpen = (𝑤 ∈ V ↦ ((TopSet‘𝑤) ↾t (Base‘𝑤)))
ctg 17369	class topGen
cpt 17370	class ∏t
c0g 17371	class 0g
cgsu 17372	class Σg
df-0g 17373	⊢ 0g = (𝑔 ∈ V ↦ (℩𝑒(𝑒 ∈ (Base‘𝑔) ∧ ∀𝑥 ∈ (Base‘𝑔)((𝑒(+g‘𝑔)𝑥) = 𝑥 ∧ (𝑥(+g‘𝑔)𝑒) = 𝑥))))
df-gsum 17374	⊢ Σg = (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, (0g‘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 17375	⊢ topGen = (𝑥 ∈ V ↦ {𝑦 ∣ 𝑦 ⊆ ∪ (𝑥 ∩ 𝒫 𝑦)})
df-pt 17376	⊢ ∏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 17377	class Xs
cpws 17378	class ↑s
df-prds 17379	⊢ Xs = (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 Σg (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 ∈ ((2nd ‘𝑎)ℎ𝑐), 𝑒 ∈ (ℎ‘𝑎) ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑑‘𝑥)(⟨((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)⟩(comp‘(𝑟‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥)))))⟩})))
df-pws 17381	⊢ ↑s = (𝑟 ∈ V, 𝑖 ∈ V ↦ ((Scalar‘𝑟)Xs(𝑖 × {𝑟})))
cordt 17432	class ordTop
cxrs 17433	class ℝ*𝑠
df-ordt 17434	⊢ ordTop = (𝑟 ∈ V ↦ (topGen‘(fi‘({dom 𝑟} ∪ ran ((𝑥 ∈ dom 𝑟 ↦ {𝑦 ∈ dom 𝑟 ∣ ¬ 𝑦𝑟𝑥}) ∪ (𝑥 ∈ dom 𝑟 ↦ {𝑦 ∈ dom 𝑟 ∣ ¬ 𝑥𝑟𝑦}))))))
df-xrs 17435	⊢ 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 17436	class qTop
cimas 17437	class “s
cqus 17438	class /s
cxps 17439	class ×s
df-qtop 17440	⊢ qTop = (𝑗 ∈ V, 𝑓 ∈ V ↦ {𝑠 ∈ 𝒫 (𝑓 “ ∪ 𝑗) ∣ ((◡𝑓 “ 𝑠) ∩ ∪ 𝑗) ∈ 𝑗})
df-imas 17441	⊢ “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‘(1st ‘(ℎ‘1))) = 𝑥 ∧ (𝑓‘(2nd ‘(ℎ‘𝑛))) = 𝑦 ∧ ∀𝑖 ∈ (1...(𝑛 − 1))(𝑓‘(2nd ‘(ℎ‘𝑖))) = (𝑓‘(1st ‘(ℎ‘(𝑖 + 1)))))} ↦ (ℝ*𝑠 Σg ((dist‘𝑟) ∘ 𝑔))), ℝ*, < ))⟩}))
df-qus 17442	⊢ /s = (𝑟 ∈ V, 𝑒 ∈ V ↦ ((𝑥 ∈ (Base‘𝑟) ↦ [𝑥]𝑒) “s 𝑟))
df-xps 17443	⊢ ×s = (𝑟 ∈ V, 𝑠 ∈ V ↦ (◡(𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Base‘𝑠) ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}) “s ((Scalar‘𝑟)Xs{⟨∅, 𝑟⟩, ⟨1o, 𝑠⟩})))
cmre 17513	class Moore
cmrc 17514	class mrCls
cmri 17515	class mrInd
cacs 17516	class ACS
df-mre 17517	⊢ Moore = (𝑥 ∈ V ↦ {𝑐 ∈ 𝒫 𝒫 𝑥 ∣ (𝑥 ∈ 𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝑐))})
df-mrc 17518	⊢ mrCls = (𝑐 ∈ ∪ ran Moore ↦ (𝑥 ∈ 𝒫 ∪ 𝑐 ↦ ∩ {𝑠 ∈ 𝑐 ∣ 𝑥 ⊆ 𝑠}))
df-mri 17519	⊢ mrInd = (𝑐 ∈ ∪ ran Moore ↦ {𝑠 ∈ 𝒫 ∪ 𝑐 ∣ ∀𝑥 ∈ 𝑠 ¬ 𝑥 ∈ ((mrCls‘𝑐)‘(𝑠 ∖ {𝑥}))})
df-acs 17520	⊢ ACS = (𝑥 ∈ V ↦ {𝑐 ∈ (Moore‘𝑥) ∣ ∃𝑓(𝑓:𝒫 𝑥⟶𝒫 𝑥 ∧ ∀𝑠 ∈ 𝒫 𝑥(𝑠 ∈ 𝑐 ↔ ∪ (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠))})
ccat 17599	class Cat
ccid 17600	class Id
chomf 17601	class Homf
ccomf 17602	class compf
df-cat 17603	⊢ 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 17604	⊢ 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 17605	⊢ Homf = (𝑐 ∈ V ↦ (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ (𝑥(Hom ‘𝑐)𝑦)))
df-comf 17606	⊢ compf = (𝑐 ∈ V ↦ (𝑥 ∈ ((Base‘𝑐) × (Base‘𝑐)), 𝑦 ∈ (Base‘𝑐) ↦ (𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝑐)𝑦), 𝑓 ∈ ((Hom ‘𝑐)‘𝑥) ↦ (𝑔(𝑥(comp‘𝑐)𝑦)𝑓))))
coppc 17646	class oppCat
df-oppc 17647	⊢ oppCat = (𝑓 ∈ V ↦ ((𝑓 sSet ⟨(Hom ‘ndx), tpos (Hom ‘𝑓)⟩) sSet ⟨(comp‘ndx), (𝑢 ∈ ((Base‘𝑓) × (Base‘𝑓)), 𝑧 ∈ (Base‘𝑓) ↦ tpos (⟨𝑧, (2nd ‘𝑢)⟩(comp‘𝑓)(1st ‘𝑢)))⟩))
cmon 17664	class Mono
cepi 17665	class Epi
df-mon 17666	⊢ 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 17667	⊢ Epi = (𝑐 ∈ Cat ↦ tpos (Mono‘(oppCat‘𝑐)))
csect 17680	class Sect
cinv 17681	class Inv
ciso 17682	class Iso
df-sect 17683	⊢ 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 17684	⊢ Inv = (𝑐 ∈ Cat ↦ (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ ((𝑥(Sect‘𝑐)𝑦) ∩ ◡(𝑦(Sect‘𝑐)𝑥))))
df-iso 17685	⊢ Iso = (𝑐 ∈ Cat ↦ ((𝑥 ∈ V ↦ dom 𝑥) ∘ (Inv‘𝑐)))
ccic 17731	class ≃𝑐
df-cic 17732	⊢ ≃𝑐 = (𝑐 ∈ Cat ↦ ((Iso‘𝑐) supp ∅))
cssc 17743	class ⊆cat
cresc 17744	class ↾cat
csubc 17745	class Subcat
df-ssc 17746	⊢ ⊆cat = {⟨ℎ, 𝑗⟩ ∣ ∃𝑡(𝑗 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝑗‘𝑥))}
df-resc 17747	⊢ ↾cat = (𝑐 ∈ V, ℎ ∈ V ↦ ((𝑐 ↾s dom dom ℎ) sSet ⟨(Hom ‘ndx), ℎ⟩))
df-subc 17748	⊢ Subcat = (𝑐 ∈ Cat ↦ {ℎ ∣ (ℎ ⊆cat (Homf ‘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 17790	class Func
cidfu 17791	class idfunc
ccofu 17792	class ∘func
cresf 17793	class ↾f
df-func 17794	⊢ Func = (𝑡 ∈ Cat, 𝑢 ∈ Cat ↦ {⟨𝑓, 𝑔⟩ ∣ [(Base‘𝑡) / 𝑏](𝑓:𝑏⟶(Base‘𝑢) ∧ 𝑔 ∈ X𝑧 ∈ (𝑏 × 𝑏)(((𝑓‘(1st ‘𝑧))(Hom ‘𝑢)(𝑓‘(2nd ‘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 17795	⊢ idfunc = (𝑡 ∈ Cat ↦ ⦋(Base‘𝑡) / 𝑏⦌⟨( I ↾ 𝑏), (𝑧 ∈ (𝑏 × 𝑏) ↦ ( I ↾ ((Hom ‘𝑡)‘𝑧)))⟩)
df-cofu 17796	⊢ ∘func = (𝑔 ∈ V, 𝑓 ∈ V ↦ ⟨((1st ‘𝑔) ∘ (1st ‘𝑓)), (𝑥 ∈ dom dom (2nd ‘𝑓), 𝑦 ∈ dom dom (2nd ‘𝑓) ↦ ((((1st ‘𝑓)‘𝑥)(2nd ‘𝑔)((1st ‘𝑓)‘𝑦)) ∘ (𝑥(2nd ‘𝑓)𝑦)))⟩)
df-resf 17797	⊢ ↾f = (𝑓 ∈ V, ℎ ∈ V ↦ ⟨((1st ‘𝑓) ↾ dom dom ℎ), (𝑥 ∈ dom ℎ ↦ (((2nd ‘𝑓)‘𝑥) ↾ (ℎ‘𝑥)))⟩)
cful 17840	class Full
cfth 17841	class Faith
df-full 17842	⊢ 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 17843	⊢ Faith = (𝑐 ∈ Cat, 𝑑 ∈ Cat ↦ {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝑐 Func 𝑑)𝑔 ∧ ∀𝑥 ∈ (Base‘𝑐)∀𝑦 ∈ (Base‘𝑐)Fun ◡(𝑥𝑔𝑦))})
cnat 17880	class Nat
cfuc 17881	class FuncCat
df-nat 17882	⊢ Nat = (𝑡 ∈ Cat, 𝑢 ∈ Cat ↦ (𝑓 ∈ (𝑡 Func 𝑢), 𝑔 ∈ (𝑡 Func 𝑢) ↦ ⦋(1st ‘𝑓) / 𝑟⦌⦋(1st ‘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(2nd ‘𝑓)𝑦)‘ℎ)) = (((𝑥(2nd ‘𝑔)𝑦)‘ℎ)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝑢)(𝑠‘𝑦))(𝑎‘𝑥))}))
df-fuc 17883	⊢ 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) ↦ ⦋(1st ‘𝑣) / 𝑓⦌⦋(2nd ‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔(𝑡 Nat 𝑢)ℎ), 𝑎 ∈ (𝑓(𝑡 Nat 𝑢)𝑔) ↦ (𝑥 ∈ (Base‘𝑡) ↦ ((𝑏‘𝑥)(⟨((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)⟩(comp‘𝑢)((1st ‘ℎ)‘𝑥))(𝑎‘𝑥)))))⟩})
cinito 17917	class InitO
ctermo 17918	class TermO
czeroo 17919	class ZeroO
df-inito 17920	⊢ InitO = (𝑐 ∈ Cat ↦ {𝑎 ∈ (Base‘𝑐) ∣ ∀𝑏 ∈ (Base‘𝑐)∃!ℎ ℎ ∈ (𝑎(Hom ‘𝑐)𝑏)})
df-termo 17921	⊢ TermO = (𝑐 ∈ Cat ↦ {𝑎 ∈ (Base‘𝑐) ∣ ∀𝑏 ∈ (Base‘𝑐)∃!ℎ ℎ ∈ (𝑏(Hom ‘𝑐)𝑎)})
df-zeroo 17922	⊢ ZeroO = (𝑐 ∈ Cat ↦ ((InitO‘𝑐) ∩ (TermO‘𝑐)))
cdoma 17956	class doma
ccoda 17957	class coda
carw 17958	class Arrow
choma 17959	class Homa
df-doma 17960	⊢ doma = (1st ∘ 1st )
df-coda 17961	⊢ coda = (2nd ∘ 1st )
df-homa 17962	⊢ Homa = (𝑐 ∈ Cat ↦ (𝑥 ∈ ((Base‘𝑐) × (Base‘𝑐)) ↦ ({𝑥} × ((Hom ‘𝑐)‘𝑥))))
df-arw 17963	⊢ Arrow = (𝑐 ∈ Cat ↦ ∪ ran (Homa‘𝑐))
cida 17989	class Ida
ccoa 17990	class compa
df-ida 17991	⊢ Ida = (𝑐 ∈ Cat ↦ (𝑥 ∈ (Base‘𝑐) ↦ ⟨𝑥, 𝑥, ((Id‘𝑐)‘𝑥)⟩))
df-coa 17992	⊢ compa = (𝑐 ∈ Cat ↦ (𝑔 ∈ (Arrow‘𝑐), 𝑓 ∈ {ℎ ∈ (Arrow‘𝑐) ∣ (coda‘ℎ) = (doma‘𝑔)} ↦ ⟨(doma‘𝑓), (coda‘𝑔), ((2nd ‘𝑔)(⟨(doma‘𝑓), (doma‘𝑔)⟩(comp‘𝑐)(coda‘𝑔))(2nd ‘𝑓))⟩))
csetc 18011	class SetCat
df-setc 18012	⊢ 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 (2nd ‘𝑣)), 𝑓 ∈ ((2nd ‘𝑣) ↑m (1st ‘𝑣)) ↦ (𝑔 ∘ 𝑓)))⟩})
ccatc 18034	class CatCat
df-catc 18035	⊢ 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 ∈ ((2nd ‘𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔 ∘func 𝑓)))⟩})
cestrc 18057	class ExtStrCat
df-estrc 18058	⊢ 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‘(2nd ‘𝑣))), 𝑓 ∈ ((Base‘(2nd ‘𝑣)) ↑m (Base‘(1st ‘𝑣))) ↦ (𝑔 ∘ 𝑓)))⟩})
cxpc 18103	class ×c
c1stf 18104	class 1stF
c2ndf 18105	class 2ndF
cprf 18106	class ⟨,⟩F
df-xpc 18107	⊢ ×c = (𝑟 ∈ V, 𝑠 ∈ V ↦ ⦋((Base‘𝑟) × (Base‘𝑠)) / 𝑏⦌⦋(𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (((1st ‘𝑢)(Hom ‘𝑟)(1st ‘𝑣)) × ((2nd ‘𝑢)(Hom ‘𝑠)(2nd ‘𝑣)))) / ℎ⦌{⟨(Base‘ndx), 𝑏⟩, ⟨(Hom ‘ndx), ℎ⟩, ⟨(comp‘ndx), (𝑥 ∈ (𝑏 × 𝑏), 𝑦 ∈ 𝑏 ↦ (𝑔 ∈ ((2nd ‘𝑥)ℎ𝑦), 𝑓 ∈ (ℎ‘𝑥) ↦ ⟨((1st ‘𝑔)(⟨(1st ‘(1st ‘𝑥)), (1st ‘(2nd ‘𝑥))⟩(comp‘𝑟)(1st ‘𝑦))(1st ‘𝑓)), ((2nd ‘𝑔)(⟨(2nd ‘(1st ‘𝑥)), (2nd ‘(2nd ‘𝑥))⟩(comp‘𝑠)(2nd ‘𝑦))(2nd ‘𝑓))⟩))⟩})
df-1stf 18108	⊢ 1stF = (𝑟 ∈ Cat, 𝑠 ∈ Cat ↦ ⦋((Base‘𝑟) × (Base‘𝑠)) / 𝑏⦌⟨(1st ↾ 𝑏), (𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ (1st ↾ (𝑥(Hom ‘(𝑟 ×c 𝑠))𝑦)))⟩)
df-2ndf 18109	⊢ 2ndF = (𝑟 ∈ Cat, 𝑠 ∈ Cat ↦ ⦋((Base‘𝑟) × (Base‘𝑠)) / 𝑏⦌⟨(2nd ↾ 𝑏), (𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ (2nd ↾ (𝑥(Hom ‘(𝑟 ×c 𝑠))𝑦)))⟩)
df-prf 18110	⊢ ⟨,⟩F = (𝑓 ∈ V, 𝑔 ∈ V ↦ ⦋dom (1st ‘𝑓) / 𝑏⦌⟨(𝑥 ∈ 𝑏 ↦ ⟨((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)⟩), (𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ (ℎ ∈ dom (𝑥(2nd ‘𝑓)𝑦) ↦ ⟨((𝑥(2nd ‘𝑓)𝑦)‘ℎ), ((𝑥(2nd ‘𝑔)𝑦)‘ℎ)⟩))⟩)
cevlf 18144	class evalF
ccurf 18145	class curryF
cuncf 18146	class uncurryF
cdiag 18147	class Δfunc
df-evlf 18148	⊢ evalF = (𝑐 ∈ Cat, 𝑑 ∈ Cat ↦ ⟨(𝑓 ∈ (𝑐 Func 𝑑), 𝑥 ∈ (Base‘𝑐) ↦ ((1st ‘𝑓)‘𝑥)), (𝑥 ∈ ((𝑐 Func 𝑑) × (Base‘𝑐)), 𝑦 ∈ ((𝑐 Func 𝑑) × (Base‘𝑐)) ↦ ⦋(1st ‘𝑥) / 𝑚⦌⦋(1st ‘𝑦) / 𝑛⦌(𝑎 ∈ (𝑚(𝑐 Nat 𝑑)𝑛), 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝑐)(2nd ‘𝑦)) ↦ ((𝑎‘(2nd ‘𝑦))(⟨((1st ‘𝑚)‘(2nd ‘𝑥)), ((1st ‘𝑚)‘(2nd ‘𝑦))⟩(comp‘𝑑)((1st ‘𝑛)‘(2nd ‘𝑦)))(((2nd ‘𝑥)(2nd ‘𝑚)(2nd ‘𝑦))‘𝑔))))⟩)
df-curf 18149	⊢ curryF = (𝑒 ∈ V, 𝑓 ∈ V ↦ ⦋(1st ‘𝑒) / 𝑐⦌⦋(2nd ‘𝑒) / 𝑑⦌⟨(𝑥 ∈ (Base‘𝑐) ↦ ⟨(𝑦 ∈ (Base‘𝑑) ↦ (𝑥(1st ‘𝑓)𝑦)), (𝑦 ∈ (Base‘𝑑), 𝑧 ∈ (Base‘𝑑) ↦ (𝑔 ∈ (𝑦(Hom ‘𝑑)𝑧) ↦ (((Id‘𝑐)‘𝑥)(⟨𝑥, 𝑦⟩(2nd ‘𝑓)⟨𝑥, 𝑧⟩)𝑔)))⟩), (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ (𝑔 ∈ (𝑥(Hom ‘𝑐)𝑦) ↦ (𝑧 ∈ (Base‘𝑑) ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd ‘𝑓)⟨𝑦, 𝑧⟩)((Id‘𝑑)‘𝑧)))))⟩)
df-uncf 18150	⊢ uncurryF = (𝑐 ∈ V, 𝑓 ∈ V ↦ (((𝑐‘1) evalF (𝑐‘2)) ∘func ((𝑓 ∘func ((𝑐‘0) 1stF (𝑐‘1))) ⟨,⟩F ((𝑐‘0) 2ndF (𝑐‘1)))))
df-diag 18151	⊢ Δfunc = (𝑐 ∈ Cat, 𝑑 ∈ Cat ↦ (⟨𝑐, 𝑑⟩ curryF (𝑐 1stF 𝑑)))
chof 18183	class HomF
cyon 18184	class Yon
df-hof 18185	⊢ HomF = (𝑐 ∈ Cat ↦ ⟨(Homf ‘𝑐), ⦋(Base‘𝑐) / 𝑏⦌(𝑥 ∈ (𝑏 × 𝑏), 𝑦 ∈ (𝑏 × 𝑏) ↦ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝑐)(1st ‘𝑥)), 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝑐)(2nd ‘𝑦)) ↦ (ℎ ∈ ((Hom ‘𝑐)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝑐)(2nd ‘𝑦))ℎ)(⟨(1st ‘𝑦), (1st ‘𝑥)⟩(comp‘𝑐)(2nd ‘𝑦))𝑓))))⟩)
df-yon 18186	⊢ Yon = (𝑐 ∈ Cat ↦ (⟨𝑐, (oppCat‘𝑐)⟩ curryF (HomF‘(oppCat‘𝑐))))
codu 18221	class ODual
df-odu 18222	⊢ ODual = (𝑤 ∈ V ↦ (𝑤 sSet ⟨(le‘ndx), ◡(le‘𝑤)⟩))
cproset 18227	class Proset
cdrs 18228	class Dirset
df-proset 18229	⊢ Proset = {𝑓 ∣ [(Base‘𝑓) / 𝑏][(le‘𝑓) / 𝑟]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) → 𝑥𝑟𝑧))}
df-drs 18230	⊢ Dirset = {𝑓 ∈ Proset ∣ [(Base‘𝑓) / 𝑏][(le‘𝑓) / 𝑟](𝑏 ≠ ∅ ∧ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∃𝑧 ∈ 𝑏 (𝑥𝑟𝑧 ∧ 𝑦𝑟𝑧))}
cpo 18242	class Poset
cplt 18243	class lt
club 18244	class lub
cglb 18245	class glb
cjn 18246	class join
cmee 18247	class meet
df-poset 18248	⊢ 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 18263	⊢ lt = (𝑝 ∈ V ↦ ((le‘𝑝) ∖ I ))
df-lub 18279	⊢ 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 18280	⊢ 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 18281	⊢ join = (𝑝 ∈ V ↦ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦} (lub‘𝑝)𝑧})
df-meet 18282	⊢ meet = (𝑝 ∈ V ↦ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦} (glb‘𝑝)𝑧})
ctos 18349	class Toset
df-toset 18350	⊢ Toset = {𝑓 ∈ Poset ∣ [(Base‘𝑓) / 𝑏][(le‘𝑓) / 𝑟]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥𝑟𝑦 ∨ 𝑦𝑟𝑥)}
cp0 18356	class 0.
cp1 18357	class 1.
df-p0 18358	⊢ 0. = (𝑝 ∈ V ↦ ((glb‘𝑝)‘(Base‘𝑝)))
df-p1 18359	⊢ 1. = (𝑝 ∈ V ↦ ((lub‘𝑝)‘(Base‘𝑝)))
clat 18366	class Lat
df-lat 18367	⊢ Lat = {𝑝 ∈ Poset ∣ (dom (join‘𝑝) = ((Base‘𝑝) × (Base‘𝑝)) ∧ dom (meet‘𝑝) = ((Base‘𝑝) × (Base‘𝑝)))}
ccla 18433	class CLat
df-clat 18434	⊢ CLat = {𝑝 ∈ Poset ∣ (dom (lub‘𝑝) = 𝒫 (Base‘𝑝) ∧ dom (glb‘𝑝) = 𝒫 (Base‘𝑝))}
cdlat 18455	class DLat
df-dlat 18456	⊢ DLat = {𝑘 ∈ Lat ∣ [(Base‘𝑘) / 𝑏][(join‘𝑘) / 𝑗][(meet‘𝑘) / 𝑚]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 (𝑥𝑚(𝑦𝑗𝑧)) = ((𝑥𝑚𝑦)𝑗(𝑥𝑚𝑧))}
cipo 18462	class toInc
df-ipo 18463	⊢ 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 18499	class PosetRel
ctsr 18500	class TosetRel
df-ps 18501	⊢ PosetRel = {𝑟 ∣ (Rel 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟 ∧ (𝑟 ∩ ◡𝑟) = ( I ↾ ∪ ∪ 𝑟))}
df-tsr 18502	⊢ TosetRel = {𝑟 ∈ PosetRel ∣ (dom 𝑟 × dom 𝑟) ⊆ (𝑟 ∪ ◡𝑟)}
cdir 18529	class DirRel
ctail 18530	class tail
df-dir 18531	⊢ DirRel = {𝑟 ∣ ((Rel 𝑟 ∧ ( I ↾ ∪ ∪ 𝑟) ⊆ 𝑟) ∧ ((𝑟 ∘ 𝑟) ⊆ 𝑟 ∧ (∪ ∪ 𝑟 × ∪ ∪ 𝑟) ⊆ (◡𝑟 ∘ 𝑟)))}
df-tail 18532	⊢ tail = (𝑟 ∈ DirRel ↦ (𝑥 ∈ ∪ ∪ 𝑟 ↦ (𝑟 “ {𝑥})))
cchn 18540	class ( < Chain 𝐴)
df-chn 18541	⊢ ( < Chain 𝐴) = {𝑐 ∈ Word 𝐴 ∣ ∀𝑛 ∈ (dom 𝑐 ∖ {0})(𝑐‘(𝑛 − 1)) < (𝑐‘𝑛)}
cplusf 18574	class +𝑓
cmgm 18575	class Mgm
df-plusf 18576	⊢ +𝑓 = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(+g‘𝑔)𝑦)))
df-mgm 18577	⊢ Mgm = {𝑔 ∣ [(Base‘𝑔) / 𝑏][(+g‘𝑔) / 𝑜]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥𝑜𝑦) ∈ 𝑏}
cmgmhm 18627	class MgmHom
csubmgm 18628	class SubMgm
df-mgmhm 18629	⊢ 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 18630	⊢ SubMgm = (𝑠 ∈ Mgm ↦ {𝑡 ∈ 𝒫 (Base‘𝑠) ∣ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (𝑥(+g‘𝑠)𝑦) ∈ 𝑡})
csgrp 18655	class Smgrp
df-sgrp 18656	⊢ Smgrp = {𝑔 ∈ Mgm ∣ [(Base‘𝑔) / 𝑏][(+g‘𝑔) / 𝑜]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧))}
cmnd 18671	class Mnd
df-mnd 18672	⊢ Mnd = {𝑔 ∈ Smgrp ∣ [(Base‘𝑔) / 𝑏][(+g‘𝑔) / 𝑝]∃𝑒 ∈ 𝑏 ∀𝑥 ∈ 𝑏 ((𝑒𝑝𝑥) = 𝑥 ∧ (𝑥𝑝𝑒) = 𝑥)}
cmhm 18718	class MndHom
csubmnd 18719	class SubMnd
df-mhm 18720	⊢ 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‘(0g‘𝑠)) = (0g‘𝑡))})
df-submnd 18721	⊢ SubMnd = (𝑠 ∈ Mnd ↦ {𝑡 ∈ 𝒫 (Base‘𝑠) ∣ ((0g‘𝑠) ∈ 𝑡 ∧ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (𝑥(+g‘𝑠)𝑦) ∈ 𝑡)})
cfrmd 18784	class freeMnd
cvrmd 18785	class varFMnd
df-frmd 18786	⊢ freeMnd = (𝑖 ∈ V ↦ {⟨(Base‘ndx), Word 𝑖⟩, ⟨(+g‘ndx), ( ++ ↾ (Word 𝑖 × Word 𝑖))⟩})
df-vrmd 18787	⊢ varFMnd = (𝑖 ∈ V ↦ (𝑗 ∈ 𝑖 ↦ ⟨“𝑗”⟩))
cefmnd 18805	class EndoFMnd
df-efmnd 18806	⊢ 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 18875	class Grp
cminusg 18876	class invg
csg 18877	class -g
df-grp 18878	⊢ Grp = {𝑔 ∈ Mnd ∣ ∀𝑎 ∈ (Base‘𝑔)∃𝑚 ∈ (Base‘𝑔)(𝑚(+g‘𝑔)𝑎) = (0g‘𝑔)}
df-minusg 18879	⊢ invg = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔) ↦ (℩𝑤 ∈ (Base‘𝑔)(𝑤(+g‘𝑔)𝑥) = (0g‘𝑔))))
df-sbg 18880	⊢ -g = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(+g‘𝑔)((invg‘𝑔)‘𝑦))))
cmg 19009	class .g
df-mulg 19010	⊢ .g = (𝑔 ∈ V ↦ (𝑛 ∈ ℤ, 𝑥 ∈ (Base‘𝑔) ↦ if(𝑛 = 0, (0g‘𝑔), ⦋seq1((+g‘𝑔), (ℕ × {𝑥})) / 𝑠⦌if(0 < 𝑛, (𝑠‘𝑛), ((invg‘𝑔)‘(𝑠‘-𝑛))))))
csubg 19062	class SubGrp
cnsg 19063	class NrmSGrp
cqg 19064	class ~QG
df-subg 19065	⊢ SubGrp = (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp})
df-nsg 19066	⊢ NrmSGrp = (𝑤 ∈ Grp ↦ {𝑠 ∈ (SubGrp‘𝑤) ∣ [(Base‘𝑤) / 𝑏][(+g‘𝑤) / 𝑝]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ((𝑥𝑝𝑦) ∈ 𝑠 ↔ (𝑦𝑝𝑥) ∈ 𝑠)})
df-eqg 19067	⊢ ~QG = (𝑟 ∈ V, 𝑖 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (Base‘𝑟) ∧ (((invg‘𝑟)‘𝑥)(+g‘𝑟)𝑦) ∈ 𝑖)})
cghm 19153	class GrpHom
df-ghm 19154	⊢ 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 19198	class GrpIso
cgic 19199	class ≃𝑔
df-gim 19200	⊢ GrpIso = (𝑠 ∈ Grp, 𝑡 ∈ Grp ↦ {𝑔 ∈ (𝑠 GrpHom 𝑡) ∣ 𝑔:(Base‘𝑠)–1-1-onto→(Base‘𝑡)})
df-gic 19201	⊢ ≃𝑔 = (◡ GrpIso “ (V ∖ 1o))
cga 19230	class GrpAct
df-ga 19231	⊢ GrpAct = (𝑔 ∈ Grp, 𝑠 ∈ V ↦ ⦋(Base‘𝑔) / 𝑏⦌{𝑚 ∈ (𝑠 ↑m (𝑏 × 𝑠)) ∣ ∀𝑥 ∈ 𝑠 (((0g‘𝑔)𝑚𝑥) = 𝑥 ∧ ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ((𝑦(+g‘𝑔)𝑧)𝑚𝑥) = (𝑦𝑚(𝑧𝑚𝑥)))})
ccntz 19256	class Cntz
ccntr 19257	class Cntr
df-cntz 19258	⊢ Cntz = (𝑚 ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘𝑚) ↦ {𝑥 ∈ (Base‘𝑚) ∣ ∀𝑦 ∈ 𝑠 (𝑥(+g‘𝑚)𝑦) = (𝑦(+g‘𝑚)𝑥)}))
df-cntr 19259	⊢ Cntr = (𝑚 ∈ V ↦ ((Cntz‘𝑚)‘(Base‘𝑚)))
coppg 19286	class oppg
df-oppg 19287	⊢ oppg = (𝑤 ∈ V ↦ (𝑤 sSet ⟨(+g‘ndx), tpos (+g‘𝑤)⟩))
csymg 19310	class SymGrp
df-symg 19311	⊢ SymGrp = (𝑥 ∈ V ↦ ((EndoFMnd‘𝑥) ↾s {ℎ ∣ ℎ:𝑥–1-1-onto→𝑥}))
cpmtr 19382	class pmTrsp
df-pmtr 19383	⊢ pmTrsp = (𝑑 ∈ V ↦ (𝑝 ∈ {𝑦 ∈ 𝒫 𝑑 ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ 𝑑 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))))
cpsgn 19430	class pmSgn
cevpm 19431	class pmEven
df-psgn 19432	⊢ pmSgn = (𝑑 ∈ V ↦ (𝑥 ∈ {𝑝 ∈ (Base‘(SymGrp‘𝑑)) ∣ dom (𝑝 ∖ I ) ∈ Fin} ↦ (℩𝑠∃𝑤 ∈ Word ran (pmTrsp‘𝑑)(𝑥 = ((SymGrp‘𝑑) Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤))))))
df-evpm 19433	⊢ pmEven = (𝑑 ∈ V ↦ (◡(pmSgn‘𝑑) “ {1}))
cod 19465	class od
cgex 19466	class gEx
cpgp 19467	class pGrp
cslw 19468	class pSyl
df-od 19469	⊢ od = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔) ↦ ⦋{𝑛 ∈ ℕ ∣ (𝑛(.g‘𝑔)𝑥) = (0g‘𝑔)} / 𝑖⦌if(𝑖 = ∅, 0, inf(𝑖, ℝ, < ))))
df-gex 19470	⊢ gEx = (𝑔 ∈ V ↦ ⦋{𝑛 ∈ ℕ ∣ ∀𝑥 ∈ (Base‘𝑔)(𝑛(.g‘𝑔)𝑥) = (0g‘𝑔)} / 𝑖⦌if(𝑖 = ∅, 0, inf(𝑖, ℝ, < )))
df-pgp 19471	⊢ pGrp = {⟨𝑝, 𝑔⟩ ∣ ((𝑝 ∈ ℙ ∧ 𝑔 ∈ Grp) ∧ ∀𝑥 ∈ (Base‘𝑔)∃𝑛 ∈ ℕ0 ((od‘𝑔)‘𝑥) = (𝑝↑𝑛))}
df-slw 19472	⊢ pSyl = (𝑝 ∈ ℙ, 𝑔 ∈ Grp ↦ {ℎ ∈ (SubGrp‘𝑔) ∣ ∀𝑘 ∈ (SubGrp‘𝑔)((ℎ ⊆ 𝑘 ∧ 𝑝 pGrp (𝑔 ↾s 𝑘)) ↔ ℎ = 𝑘)})
clsm 19575	class LSSum
cpj1 19576	class proj1
df-lsm 19577	⊢ LSSum = (𝑤 ∈ V ↦ (𝑡 ∈ 𝒫 (Base‘𝑤), 𝑢 ∈ 𝒫 (Base‘𝑤) ↦ ran (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥(+g‘𝑤)𝑦))))
df-pj1 19578	⊢ 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 19647	class ~FG
cfrgp 19648	class freeGrp
cvrgp 19649	class varFGrp
df-efg 19650	⊢ ~FG = (𝑖 ∈ V ↦ ∩ {𝑟 ∣ (𝑟 Er Word (𝑖 × 2o) ∧ ∀𝑥 ∈ Word (𝑖 × 2o)∀𝑛 ∈ (0...(♯‘𝑥))∀𝑦 ∈ 𝑖 ∀𝑧 ∈ 2o 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1o ∖ 𝑧)⟩”⟩⟩))})
df-frgp 19651	⊢ freeGrp = (𝑖 ∈ V ↦ ((freeMnd‘(𝑖 × 2o)) /s ( ~FG ‘𝑖)))
df-vrgp 19652	⊢ varFGrp = (𝑖 ∈ V ↦ (𝑗 ∈ 𝑖 ↦ [⟨“⟨𝑗, ∅⟩”⟩]( ~FG ‘𝑖)))
ccmn 19721	class CMnd
cabl 19722	class Abel
df-cmn 19723	⊢ CMnd = {𝑔 ∈ Mnd ∣ ∀𝑎 ∈ (Base‘𝑔)∀𝑏 ∈ (Base‘𝑔)(𝑎(+g‘𝑔)𝑏) = (𝑏(+g‘𝑔)𝑎)}
df-abl 19724	⊢ Abel = (Grp ∩ CMnd)
ccyg 19818	class CycGrp
df-cyg 19819	⊢ CycGrp = {𝑔 ∈ Grp ∣ ∃𝑥 ∈ (Base‘𝑔)ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝑔)𝑥)) = (Base‘𝑔)}
cdprd 19936	class DProd
cdpj 19937	class dProj
df-dprd 19938	⊢ 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})))) = {(0g‘𝑔)}))} ↦ ran (𝑓 ∈ {ℎ ∈ X𝑥 ∈ dom 𝑠(𝑠‘𝑥) ∣ ℎ finSupp (0g‘𝑔)} ↦ (𝑔 Σg 𝑓)))
df-dpj 19939	⊢ dProj = (𝑔 ∈ Grp, 𝑠 ∈ (dom DProd “ {𝑔}) ↦ (𝑖 ∈ dom 𝑠 ↦ ((𝑠‘𝑖)(proj1‘𝑔)(𝑔 DProd (𝑠 ↾ (dom 𝑠 ∖ {𝑖}))))))
csimpg 20033	class SimpGrp
df-simpg 20034	⊢ SimpGrp = {𝑔 ∈ Grp ∣ (NrmSGrp‘𝑔) ≈ 2o}
comnd 20060	class oMnd
cogrp 20061	class oGrp
df-omnd 20062	⊢ 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 20063	⊢ oGrp = (Grp ∩ oMnd)
cmgp 20087	class mulGrp
df-mgp 20088	⊢ mulGrp = (𝑤 ∈ V ↦ (𝑤 sSet ⟨(+g‘ndx), (.r‘𝑤)⟩))
crng 20099	class Rng
df-rng 20100	⊢ 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 20128	class 1r
df-ur 20129	⊢ 1r = (0g ∘ mulGrp)
csrg 20133	class SRing
df-srg 20134	⊢ SRing = {𝑓 ∈ CMnd ∣ ((mulGrp‘𝑓) ∈ Mnd ∧ [(Base‘𝑓) / 𝑟][(+g‘𝑓) / 𝑝][(.r‘𝑓) / 𝑡][(0g‘𝑓) / 𝑛]∀𝑥 ∈ 𝑟 (∀𝑦 ∈ 𝑟 ∀𝑧 ∈ 𝑟 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))) ∧ ((𝑛𝑡𝑥) = 𝑛 ∧ (𝑥𝑡𝑛) = 𝑛)))}
crg 20180	class Ring
ccrg 20181	class CRing
df-ring 20182	⊢ 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 20183	⊢ CRing = {𝑓 ∈ Ring ∣ (mulGrp‘𝑓) ∈ CMnd}
coppr 20284	class oppr
df-oppr 20285	⊢ oppr = (𝑓 ∈ V ↦ (𝑓 sSet ⟨(.r‘ndx), tpos (.r‘𝑓)⟩))
cdsr 20302	class ∥r
cui 20303	class Unit
cir 20304	class Irred
df-dvdsr 20305	⊢ ∥r = (𝑤 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑤) ∧ ∃𝑧 ∈ (Base‘𝑤)(𝑧(.r‘𝑤)𝑥) = 𝑦)})
df-unit 20306	⊢ Unit = (𝑤 ∈ V ↦ (◡((∥r‘𝑤) ∩ (∥r‘(oppr‘𝑤))) “ {(1r‘𝑤)}))
df-irred 20307	⊢ Irred = (𝑤 ∈ V ↦ ⦋((Base‘𝑤) ∖ (Unit‘𝑤)) / 𝑏⦌{𝑧 ∈ 𝑏 ∣ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥(.r‘𝑤)𝑦) ≠ 𝑧})
cinvr 20335	class invr
df-invr 20336	⊢ invr = (𝑟 ∈ V ↦ (invg‘((mulGrp‘𝑟) ↾s (Unit‘𝑟))))
cdvr 20348	class /r
df-dvr 20349	⊢ /r = (𝑟 ∈ V ↦ (𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Unit‘𝑟) ↦ (𝑥(.r‘𝑟)((invr‘𝑟)‘𝑦))))
crpm 20380	class RPrime
df-rprm 20381	⊢ RPrime = (𝑤 ∈ V ↦ ⦋(Base‘𝑤) / 𝑏⦌{𝑝 ∈ (𝑏 ∖ ((Unit‘𝑤) ∪ {(0g‘𝑤)})) ∣ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 [(∥r‘𝑤) / 𝑑](𝑝𝑑(𝑥(.r‘𝑤)𝑦) → (𝑝𝑑𝑥 ∨ 𝑝𝑑𝑦))})
crnghm 20382	class RngHom
crngim 20383	class RngIso
df-rnghm 20384	⊢ 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 20385	⊢ RngIso = (𝑟 ∈ V, 𝑠 ∈ V ↦ {𝑓 ∈ (𝑟 RngHom 𝑠) ∣ ◡𝑓 ∈ (𝑠 RngHom 𝑟)})
crh 20417	class RingHom
crs 20418	class RingIso
cric 20419	class ≃𝑟
df-rhm 20420	⊢ RingHom = (𝑟 ∈ Ring, 𝑠 ∈ Ring ↦ ⦋(Base‘𝑟) / 𝑣⦌⦋(Base‘𝑠) / 𝑤⦌{𝑓 ∈ (𝑤 ↑m 𝑣) ∣ ((𝑓‘(1r‘𝑟)) = (1r‘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 20421	⊢ RingIso = (𝑟 ∈ V, 𝑠 ∈ V ↦ {𝑓 ∈ (𝑟 RingHom 𝑠) ∣ ◡𝑓 ∈ (𝑠 RingHom 𝑟)})
df-ric 20423	⊢ ≃𝑟 = (◡ RingIso “ (V ∖ 1o))
cnzr 20457	class NzRing
df-nzr 20458	⊢ NzRing = {𝑟 ∈ Ring ∣ (1r‘𝑟) ≠ (0g‘𝑟)}
clring 20483	class LRing
df-lring 20484	⊢ LRing = {𝑟 ∈ NzRing ∣ ∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)((𝑥(+g‘𝑟)𝑦) = (1r‘𝑟) → (𝑥 ∈ (Unit‘𝑟) ∨ 𝑦 ∈ (Unit‘𝑟)))}
csubrng 20490	class SubRng
df-subrng 20491	⊢ SubRng = (𝑤 ∈ Rng ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Rng})
csubrg 20514	class SubRing
df-subrg 20515	⊢ SubRing = (𝑤 ∈ Ring ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ ((𝑤 ↾s 𝑠) ∈ Ring ∧ (1r‘𝑤) ∈ 𝑠)})
crgspn 20555	class RingSpan
df-rgspn 20556	⊢ RingSpan = (𝑤 ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘𝑤) ↦ ∩ {𝑡 ∈ (SubRing‘𝑤) ∣ 𝑠 ⊆ 𝑡}))
crngc 20561	class RngCat
df-rngc 20562	⊢ RngCat = (𝑢 ∈ V ↦ ((ExtStrCat‘𝑢) ↾cat ( RngHom ↾ ((𝑢 ∩ Rng) × (𝑢 ∩ Rng)))))
cringc 20590	class RingCat
df-ringc 20591	⊢ RingCat = (𝑢 ∈ V ↦ ((ExtStrCat‘𝑢) ↾cat ( RingHom ↾ ((𝑢 ∩ Ring) × (𝑢 ∩ Ring)))))
crlreg 20636	class RLReg
cdomn 20637	class Domn
cidom 20638	class IDomn
df-rlreg 20639	⊢ RLReg = (𝑟 ∈ V ↦ {𝑥 ∈ (Base‘𝑟) ∣ ∀𝑦 ∈ (Base‘𝑟)((𝑥(.r‘𝑟)𝑦) = (0g‘𝑟) → 𝑦 = (0g‘𝑟))})
df-domn 20640	⊢ Domn = {𝑟 ∈ NzRing ∣ [(Base‘𝑟) / 𝑏][(0g‘𝑟) / 𝑧]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ((𝑥(.r‘𝑟)𝑦) = 𝑧 → (𝑥 = 𝑧 ∨ 𝑦 = 𝑧))}
df-idom 20641	⊢ IDomn = (CRing ∩ Domn)
cdr 20674	class DivRing
cfield 20675	class Field
df-drng 20676	⊢ DivRing = {𝑟 ∈ Ring ∣ (Unit‘𝑟) = ((Base‘𝑟) ∖ {(0g‘𝑟)})}
df-field 20677	⊢ Field = (DivRing ∩ CRing)
csdrg 20731	class SubDRing
df-sdrg 20732	⊢ SubDRing = (𝑤 ∈ DivRing ↦ {𝑠 ∈ (SubRing‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ DivRing})
cabv 20753	class AbsVal
df-abv 20754	⊢ AbsVal = (𝑟 ∈ Ring ↦ {𝑓 ∈ ((0[,)+∞) ↑m (Base‘𝑟)) ∣ ∀𝑥 ∈ (Base‘𝑟)(((𝑓‘𝑥) = 0 ↔ 𝑥 = (0g‘𝑟)) ∧ ∀𝑦 ∈ (Base‘𝑟)((𝑓‘(𝑥(.r‘𝑟)𝑦)) = ((𝑓‘𝑥) · (𝑓‘𝑦)) ∧ (𝑓‘(𝑥(+g‘𝑟)𝑦)) ≤ ((𝑓‘𝑥) + (𝑓‘𝑦))))})
cstf 20782	class *rf
csr 20783	class *-Ring
df-staf 20784	⊢ *rf = (𝑓 ∈ V ↦ (𝑥 ∈ (Base‘𝑓) ↦ ((*𝑟‘𝑓)‘𝑥)))
df-srng 20785	⊢ *-Ring = {𝑓 ∣ [(*rf‘𝑓) / 𝑖](𝑖 ∈ (𝑓 RingHom (oppr‘𝑓)) ∧ 𝑖 = ◡𝑖)}
corng 20802	class oRing
cofld 20803	class oField
df-orng 20804	⊢ oRing = {𝑟 ∈ (Ring ∩ oGrp) ∣ [(Base‘𝑟) / 𝑣][(0g‘𝑟) / 𝑧][(.r‘𝑟) / 𝑡][(le‘𝑟) / 𝑙]∀𝑎 ∈ 𝑣 ∀𝑏 ∈ 𝑣 ((𝑧𝑙𝑎 ∧ 𝑧𝑙𝑏) → 𝑧𝑙(𝑎𝑡𝑏))}
df-ofld 20805	⊢ oField = (Field ∩ oRing)
clmod 20823	class LMod
cscaf 20824	class ·sf
df-lmod 20825	⊢ 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)) ∧ ((1r‘𝑓)𝑠𝑤) = 𝑤)))}
df-scaf 20826	⊢ ·sf = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘(Scalar‘𝑔)), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥( ·𝑠 ‘𝑔)𝑦)))
clss 20894	class LSubSp
df-lss 20895	⊢ LSubSp = (𝑤 ∈ V ↦ {𝑠 ∈ (𝒫 (Base‘𝑤) ∖ {∅}) ∣ ∀𝑥 ∈ (Base‘(Scalar‘𝑤))∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑥( ·𝑠 ‘𝑤)𝑎)(+g‘𝑤)𝑏) ∈ 𝑠})
clspn 20934	class LSpan
df-lsp 20935	⊢ LSpan = (𝑤 ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘𝑤) ↦ ∩ {𝑡 ∈ (LSubSp‘𝑤) ∣ 𝑠 ⊆ 𝑡}))
clmhm 20983	class LMHom
clmim 20984	class LMIso
clmic 20985	class ≃𝑚
df-lmhm 20986	⊢ 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 20987	⊢ LMIso = (𝑠 ∈ LMod, 𝑡 ∈ LMod ↦ {𝑔 ∈ (𝑠 LMHom 𝑡) ∣ 𝑔:(Base‘𝑠)–1-1-onto→(Base‘𝑡)})
df-lmic 20988	⊢ ≃𝑚 = (◡ LMIso “ (V ∖ 1o))
clbs 21038	class LBasis
df-lbs 21039	⊢ LBasis = (𝑤 ∈ V ↦ {𝑏 ∈ 𝒫 (Base‘𝑤) ∣ [(LSpan‘𝑤) / 𝑛][(Scalar‘𝑤) / 𝑠]((𝑛‘𝑏) = (Base‘𝑤) ∧ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ ((Base‘𝑠) ∖ {(0g‘𝑠)}) ¬ (𝑦( ·𝑠 ‘𝑤)𝑥) ∈ (𝑛‘(𝑏 ∖ {𝑥})))})
clvec 21066	class LVec
df-lvec 21067	⊢ LVec = {𝑓 ∈ LMod ∣ (Scalar‘𝑓) ∈ DivRing}
csra 21135	class subringAlg
crglmod 21136	class ringLMod
df-sra 21137	⊢ 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 21138	⊢ ringLMod = (𝑤 ∈ V ↦ ((subringAlg ‘𝑤)‘(Base‘𝑤)))
clidl 21173	class LIdeal
crsp 21174	class RSpan
df-lidl 21175	⊢ LIdeal = (LSubSp ∘ ringLMod)
df-rsp 21176	⊢ RSpan = (LSpan ∘ ringLMod)
c2idl 21216	class 2Ideal
df-2idl 21217	⊢ 2Ideal = (𝑟 ∈ V ↦ ((LIdeal‘𝑟) ∩ (LIdeal‘(oppr‘𝑟))))
clpidl 21287	class LPIdeal
clpir 21288	class LPIR
df-lpidl 21289	⊢ LPIdeal = (𝑤 ∈ Ring ↦ ∪ 𝑔 ∈ (Base‘𝑤){((RSpan‘𝑤)‘{𝑔})})
df-lpir 21290	⊢ LPIR = {𝑤 ∈ Ring ∣ (LIdeal‘𝑤) = (LPIdeal‘𝑤)}
cpid 21303	class PID
df-pid 21304	⊢ PID = (IDomn ∩ LPIR)
cpsmet 21305	class PsMet
cxmet 21306	class ∞Met
cmet 21307	class Met
cbl 21308	class ball
cfbas 21309	class fBas
cfg 21310	class filGen
cmopn 21311	class MetOpen
cmetu 21312	class metUnif
df-psmet 21313	⊢ PsMet = (𝑥 ∈ V ↦ {𝑑 ∈ (ℝ* ↑m (𝑥 × 𝑥)) ∣ ∀𝑦 ∈ 𝑥 ((𝑦𝑑𝑦) = 0 ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) +𝑒 (𝑤𝑑𝑧)))})
df-xmet 21314	⊢ ∞Met = (𝑥 ∈ V ↦ {𝑑 ∈ (ℝ* ↑m (𝑥 × 𝑥)) ∣ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (((𝑦𝑑𝑧) = 0 ↔ 𝑦 = 𝑧) ∧ ∀𝑤 ∈ 𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) +𝑒 (𝑤𝑑𝑧)))})
df-met 21315	⊢ Met = (𝑥 ∈ V ↦ {𝑑 ∈ (ℝ ↑m (𝑥 × 𝑥)) ∣ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (((𝑦𝑑𝑧) = 0 ↔ 𝑦 = 𝑧) ∧ ∀𝑤 ∈ 𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) + (𝑤𝑑𝑧)))})
df-bl 21316	⊢ ball = (𝑑 ∈ V ↦ (𝑥 ∈ dom dom 𝑑, 𝑧 ∈ ℝ* ↦ {𝑦 ∈ dom dom 𝑑 ∣ (𝑥𝑑𝑦) < 𝑧}))
df-mopn 21317	⊢ MetOpen = (𝑑 ∈ ∪ ran ∞Met ↦ (topGen‘ran (ball‘𝑑)))
df-fbas 21318	⊢ fBas = (𝑤 ∈ V ↦ {𝑥 ∈ 𝒫 𝒫 𝑤 ∣ (𝑥 ≠ ∅ ∧ ∅ ∉ 𝑥 ∧ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑥 ∩ 𝒫 (𝑦 ∩ 𝑧)) ≠ ∅)})
df-fg 21319	⊢ filGen = (𝑤 ∈ V, 𝑥 ∈ (fBas‘𝑤) ↦ {𝑦 ∈ 𝒫 𝑤 ∣ (𝑥 ∩ 𝒫 𝑦) ≠ ∅})
df-metu 21320	⊢ metUnif = (𝑑 ∈ ∪ ran PsMet ↦ ((dom dom 𝑑 × dom dom 𝑑)filGenran (𝑎 ∈ ℝ+ ↦ (◡𝑑 “ (0[,)𝑎)))))
ccnfld 21321	class ℂfld
df-cnfld 21322	⊢ ℂfld = (({⟨(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 21413	class ℤring
df-zring 21414	⊢ ℤring = (ℂfld ↾s ℤ)
czrh 21466	class ℤRHom
czlm 21467	class ℤMod
cchr 21468	class chr
czn 21469	class ℤ/nℤ
df-zrh 21470	⊢ ℤRHom = (𝑟 ∈ V ↦ ∪ (ℤring RingHom 𝑟))
df-zlm 21471	⊢ ℤMod = (𝑔 ∈ V ↦ ((𝑔 sSet ⟨(Scalar‘ndx), ℤring⟩) sSet ⟨( ·𝑠 ‘ndx), (.g‘𝑔)⟩))
df-chr 21472	⊢ chr = (𝑔 ∈ V ↦ ((od‘𝑔)‘(1r‘𝑔)))
df-zn 21473	⊢ ℤ/nℤ = (𝑛 ∈ ℕ0 ↦ ⦋ℤring / 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 21571	class ℝfld
df-refld 21572	⊢ ℝfld = (ℂfld ↾s ℝ)
cphl 21591	class PreHil
cipf 21592	class ·if
df-phl 21593	⊢ PreHil = {𝑔 ∈ LVec ∣ [(Base‘𝑔) / 𝑣][(·𝑖‘𝑔) / ℎ][(Scalar‘𝑔) / 𝑓](𝑓 ∈ *-Ring ∧ ∀𝑥 ∈ 𝑣 ((𝑦 ∈ 𝑣 ↦ (𝑦ℎ𝑥)) ∈ (𝑔 LMHom (ringLMod‘𝑓)) ∧ ((𝑥ℎ𝑥) = (0g‘𝑓) → 𝑥 = (0g‘𝑔)) ∧ ∀𝑦 ∈ 𝑣 ((*𝑟‘𝑓)‘(𝑥ℎ𝑦)) = (𝑦ℎ𝑥)))}
df-ipf 21594	⊢ ·if = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(·𝑖‘𝑔)𝑦)))
cocv 21627	class ocv
ccss 21628	class ClSubSp
cthl 21629	class toHL
df-ocv 21630	⊢ ocv = (ℎ ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘ℎ) ↦ {𝑥 ∈ (Base‘ℎ) ∣ ∀𝑦 ∈ 𝑠 (𝑥(·𝑖‘ℎ)𝑦) = (0g‘(Scalar‘ℎ))}))
df-css 21631	⊢ ClSubSp = (ℎ ∈ V ↦ {𝑠 ∣ 𝑠 = ((ocv‘ℎ)‘((ocv‘ℎ)‘𝑠))})
df-thl 21632	⊢ toHL = (ℎ ∈ V ↦ ((toInc‘(ClSubSp‘ℎ)) sSet ⟨(oc‘ndx), (ocv‘ℎ)⟩))
cpj 21667	class proj
chil 21668	class Hil
cobs 21669	class OBasis
df-pj 21670	⊢ proj = (ℎ ∈ V ↦ ((𝑥 ∈ (LSubSp‘ℎ) ↦ (𝑥(proj1‘ℎ)((ocv‘ℎ)‘𝑥))) ∩ (V × ((Base‘ℎ) ↑m (Base‘ℎ)))))
df-hil 21671	⊢ Hil = {ℎ ∈ PreHil ∣ dom (proj‘ℎ) = (ClSubSp‘ℎ)}
df-obs 21672	⊢ OBasis = (ℎ ∈ PreHil ↦ {𝑏 ∈ 𝒫 (Base‘ℎ) ∣ (∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥(·𝑖‘ℎ)𝑦) = if(𝑥 = 𝑦, (1r‘(Scalar‘ℎ)), (0g‘(Scalar‘ℎ))) ∧ ((ocv‘ℎ)‘𝑏) = {(0g‘ℎ)})})
cdsmm 21698	class ⊕m
df-dsmm 21699	⊢ ⊕m = (𝑠 ∈ V, 𝑟 ∈ V ↦ ((𝑠Xs𝑟) ↾s {𝑓 ∈ X𝑥 ∈ dom 𝑟(Base‘(𝑟‘𝑥)) ∣ {𝑥 ∈ dom 𝑟 ∣ (𝑓‘𝑥) ≠ (0g‘(𝑟‘𝑥))} ∈ Fin}))
cfrlm 21713	class freeLMod
df-frlm 21714	⊢ freeLMod = (𝑟 ∈ V, 𝑖 ∈ V ↦ (𝑟 ⊕m (𝑖 × {(ringLMod‘𝑟)})))
cuvc 21749	class unitVec
df-uvc 21750	⊢ unitVec = (𝑟 ∈ V, 𝑖 ∈ V ↦ (𝑗 ∈ 𝑖 ↦ (𝑘 ∈ 𝑖 ↦ if(𝑘 = 𝑗, (1r‘𝑟), (0g‘𝑟)))))
clindf 21771	class LIndF
clinds 21772	class LIndS
df-lindf 21773	⊢ LIndF = {⟨𝑓, 𝑤⟩ ∣ (𝑓:dom 𝑓⟶(Base‘𝑤) ∧ [(Scalar‘𝑤) / 𝑠]∀𝑥 ∈ dom 𝑓∀𝑘 ∈ ((Base‘𝑠) ∖ {(0g‘𝑠)}) ¬ (𝑘( ·𝑠 ‘𝑤)(𝑓‘𝑥)) ∈ ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥}))))}
df-linds 21774	⊢ LIndS = (𝑤 ∈ V ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ ( I ↾ 𝑠) LIndF 𝑤})
casa 21817	class AssAlg
casp 21818	class AlgSpan
cascl 21819	class algSc
df-assa 21820	⊢ 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 21821	⊢ AlgSpan = (𝑤 ∈ AssAlg ↦ (𝑠 ∈ 𝒫 (Base‘𝑤) ↦ ∩ {𝑡 ∈ ((SubRing‘𝑤) ∩ (LSubSp‘𝑤)) ∣ 𝑠 ⊆ 𝑡}))
df-ascl 21822	⊢ algSc = (𝑤 ∈ V ↦ (𝑥 ∈ (Base‘(Scalar‘𝑤)) ↦ (𝑥( ·𝑠 ‘𝑤)(1r‘𝑤))))
cmps 21872	class mPwSer
cmvr 21873	class mVar
cmpl 21874	class mPoly
cltb 21875	class <bag
copws 21876	class ordPwSer
df-psr 21877	⊢ 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 Σg (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 21878	⊢ mVar = (𝑖 ∈ V, 𝑟 ∈ V ↦ (𝑥 ∈ 𝑖 ↦ (𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ if(𝑓 = (𝑦 ∈ 𝑖 ↦ if(𝑦 = 𝑥, 1, 0)), (1r‘𝑟), (0g‘𝑟)))))
df-mpl 21879	⊢ mPoly = (𝑖 ∈ V, 𝑟 ∈ V ↦ ⦋(𝑖 mPwSer 𝑟) / 𝑤⦌(𝑤 ↾s {𝑓 ∈ (Base‘𝑤) ∣ 𝑓 finSupp (0g‘𝑟)}))
df-ltbag 21880	⊢ <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 21881	⊢ 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 22039	class evalSub
cevl 22040	class eval
df-evls 22041	⊢ 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 22042	⊢ eval = (𝑖 ∈ V, 𝑟 ∈ V ↦ ((𝑖 evalSub 𝑟)‘(Base‘𝑟)))
cslv 22083	class selectVars
cmhp 22084	class mHomP
cpsd 22085	class mPSDer
cai 22086	class AlgInd
df-selv 22087	⊢ 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 22091	⊢ mHomP = (𝑖 ∈ V, 𝑟 ∈ V ↦ (𝑛 ∈ ℕ0 ↦ {𝑓 ∈ (Base‘(𝑖 mPoly 𝑟)) ∣ (𝑓 supp (0g‘𝑟)) ⊆ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑛}}))
df-psd 22111	⊢ 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 22126	⊢ 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 22127	class PwSer1
cv1 22128	class var1
cpl1 22129	class Poly1
cco1 22130	class coe1
ctp1 22131	class toPoly1
df-psr1 22132	⊢ PwSer1 = (𝑟 ∈ V ↦ ((1o ordPwSer 𝑟)‘∅))
df-vr1 22133	⊢ var1 = (𝑟 ∈ V ↦ ((1o mVar 𝑟)‘∅))
df-ply1 22134	⊢ Poly1 = (𝑟 ∈ V ↦ ((PwSer1‘𝑟) ↾s (Base‘(1o mPoly 𝑟))))
df-coe1 22135	⊢ coe1 = (𝑓 ∈ V ↦ (𝑛 ∈ ℕ0 ↦ (𝑓‘(1o × {𝑛}))))
df-toply1 22136	⊢ toPoly1 = (𝑓 ∈ V ↦ (𝑛 ∈ (ℕ0 ↑m 1o) ↦ (𝑓‘(𝑛‘∅))))
ces1 22269	class evalSub1
ce1 22270	class eval1
df-evls1 22271	⊢ evalSub1 = (𝑠 ∈ V, 𝑟 ∈ 𝒫 (Base‘𝑠) ↦ ⦋(Base‘𝑠) / 𝑏⦌((𝑥 ∈ (𝑏 ↑m (𝑏 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝑏 ↦ (1o × {𝑦})))) ∘ ((1o evalSub 𝑠)‘𝑟)))
df-evl1 22272	⊢ eval1 = (𝑟 ∈ V ↦ ⦋(Base‘𝑟) / 𝑏⦌((𝑥 ∈ (𝑏 ↑m (𝑏 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝑏 ↦ (1o × {𝑦})))) ∘ (1o eval 𝑟)))
cmmul 22346	class maMul
df-mamu 22347	⊢ maMul = (𝑟 ∈ V, 𝑜 ∈ V ↦ ⦋(1st ‘(1st ‘𝑜)) / 𝑚⦌⦋(2nd ‘(1st ‘𝑜)) / 𝑛⦌⦋(2nd ‘𝑜) / 𝑝⦌(𝑥 ∈ ((Base‘𝑟) ↑m (𝑚 × 𝑛)), 𝑦 ∈ ((Base‘𝑟) ↑m (𝑛 × 𝑝)) ↦ (𝑖 ∈ 𝑚, 𝑘 ∈ 𝑝 ↦ (𝑟 Σg (𝑗 ∈ 𝑛 ↦ ((𝑖𝑥𝑗)(.r‘𝑟)(𝑗𝑦𝑘)))))))
cmat 22363	class Mat
df-mat 22364	⊢ Mat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ ((𝑟 freeLMod (𝑛 × 𝑛)) sSet ⟨(.r‘ndx), (𝑟 maMul ⟨𝑛, 𝑛, 𝑛⟩)⟩))
cdmat 22444	class DMat
cscmat 22445	class ScMat
df-dmat 22446	⊢ DMat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ {𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ∣ ∀𝑖 ∈ 𝑛 ∀𝑗 ∈ 𝑛 (𝑖 ≠ 𝑗 → (𝑖𝑚𝑗) = (0g‘𝑟))})
df-scmat 22447	⊢ ScMat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ ⦋(𝑛 Mat 𝑟) / 𝑎⦌{𝑚 ∈ (Base‘𝑎) ∣ ∃𝑐 ∈ (Base‘𝑟)𝑚 = (𝑐( ·𝑠 ‘𝑎)(1r‘𝑎))})
cmvmul 22496	class maVecMul
df-mvmul 22497	⊢ maVecMul = (𝑟 ∈ V, 𝑜 ∈ V ↦ ⦋(1st ‘𝑜) / 𝑚⦌⦋(2nd ‘𝑜) / 𝑛⦌(𝑥 ∈ ((Base‘𝑟) ↑m (𝑚 × 𝑛)), 𝑦 ∈ ((Base‘𝑟) ↑m 𝑛) ↦ (𝑖 ∈ 𝑚 ↦ (𝑟 Σg (𝑗 ∈ 𝑛 ↦ ((𝑖𝑥𝑗)(.r‘𝑟)(𝑦‘𝑗)))))))
cmarrep 22512	class matRRep
cmatrepV 22513	class matRepV
df-marrep 22514	⊢ 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, (0g‘𝑟)), (𝑖𝑚𝑗))))))
df-marepv 22515	⊢ 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 22532	class subMat
df-subma 22533	⊢ subMat = (𝑛 ∈ V, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑘 ∈ 𝑛, 𝑙 ∈ 𝑛 ↦ (𝑖 ∈ (𝑛 ∖ {𝑘}), 𝑗 ∈ (𝑛 ∖ {𝑙}) ↦ (𝑖𝑚𝑗)))))
cmdat 22540	class maDet
df-mdet 22541	⊢ maDet = (𝑛 ∈ V, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑟 Σg (𝑝 ∈ (Base‘(SymGrp‘𝑛)) ↦ ((((ℤRHom‘𝑟) ∘ (pmSgn‘𝑛))‘𝑝)(.r‘𝑟)((mulGrp‘𝑟) Σg (𝑥 ∈ 𝑛 ↦ ((𝑝‘𝑥)𝑚𝑥))))))))
cmadu 22588	class maAdju
cminmar1 22589	class minMatR1
df-madu 22590	⊢ 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, (1r‘𝑟), (0g‘𝑟)), (𝑘𝑚𝑙)))))))
df-minmar1 22591	⊢ minMatR1 = (𝑛 ∈ V, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑘 ∈ 𝑛, 𝑙 ∈ 𝑛 ↦ (𝑖 ∈ 𝑛, 𝑗 ∈ 𝑛 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, (1r‘𝑟), (0g‘𝑟)), (𝑖𝑚𝑗))))))
ccpmat 22659	class ConstPolyMat
cmat2pmat 22660	class matToPolyMat
ccpmat2mat 22661	class cPolyMatToMat
df-cpmat 22662	⊢ ConstPolyMat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ {𝑚 ∈ (Base‘(𝑛 Mat (Poly1‘𝑟))) ∣ ∀𝑖 ∈ 𝑛 ∀𝑗 ∈ 𝑛 ∀𝑘 ∈ ℕ ((coe1‘(𝑖𝑚𝑗))‘𝑘) = (0g‘𝑟)})
df-mat2pmat 22663	⊢ matToPolyMat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑥 ∈ 𝑛, 𝑦 ∈ 𝑛 ↦ ((algSc‘(Poly1‘𝑟))‘(𝑥𝑚𝑦)))))
df-cpmat2mat 22664	⊢ cPolyMatToMat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ (𝑚 ∈ (𝑛 ConstPolyMat 𝑟) ↦ (𝑥 ∈ 𝑛, 𝑦 ∈ 𝑛 ↦ ((coe1‘(𝑥𝑚𝑦))‘0))))
cdecpmat 22718	class decompPMat
df-decpmat 22719	⊢ decompPMat = (𝑚 ∈ V, 𝑘 ∈ ℕ0 ↦ (𝑖 ∈ dom dom 𝑚, 𝑗 ∈ dom dom 𝑚 ↦ ((coe1‘(𝑖𝑚𝑗))‘𝑘)))
cpm2mp 22748	class pMatToMatPoly
df-pm2mp 22749	⊢ pMatToMatPoly = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat (Poly1‘𝑟))) ↦ ⦋(𝑛 Mat 𝑟) / 𝑎⦌⦋(Poly1‘𝑎) / 𝑞⦌(𝑞 Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘)( ·𝑠 ‘𝑞)(𝑘(.g‘(mulGrp‘𝑞))(var1‘𝑎)))))))
cchpmat 22782	class CharPlyMat
df-chpmat 22783	⊢ CharPlyMat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ ((𝑛 maDet (Poly1‘𝑟))‘(((var1‘𝑟)( ·𝑠 ‘(𝑛 Mat (Poly1‘𝑟)))(1r‘(𝑛 Mat (Poly1‘𝑟))))(-g‘(𝑛 Mat (Poly1‘𝑟)))((𝑛 matToPolyMat 𝑟)‘𝑚)))))
ctop 22849	class Top
df-top 22850	⊢ Top = {𝑥 ∣ (∀𝑦 ∈ 𝒫 𝑥∪ 𝑦 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑦 ∩ 𝑧) ∈ 𝑥)}
ctopon 22866	class TopOn
df-topon 22867	⊢ TopOn = (𝑏 ∈ V ↦ {𝑗 ∈ Top ∣ 𝑏 = ∪ 𝑗})
ctps 22888	class TopSp
df-topsp 22889	⊢ TopSp = {𝑓 ∣ (TopOpen‘𝑓) ∈ (TopOn‘(Base‘𝑓))}
ctb 22901	class TopBases
df-bases 22902	⊢ TopBases = {𝑥 ∣ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑦 ∩ 𝑧) ⊆ ∪ (𝑥 ∩ 𝒫 (𝑦 ∩ 𝑧))}
ccld 22972	class Clsd
cnt 22973	class int
ccl 22974	class cls
df-cld 22975	⊢ Clsd = (𝑗 ∈ Top ↦ {𝑥 ∈ 𝒫 ∪ 𝑗 ∣ (∪ 𝑗 ∖ 𝑥) ∈ 𝑗})
df-ntr 22976	⊢ int = (𝑗 ∈ Top ↦ (𝑥 ∈ 𝒫 ∪ 𝑗 ↦ ∪ (𝑗 ∩ 𝒫 𝑥)))
df-cls 22977	⊢ cls = (𝑗 ∈ Top ↦ (𝑥 ∈ 𝒫 ∪ 𝑗 ↦ ∩ {𝑦 ∈ (Clsd‘𝑗) ∣ 𝑥 ⊆ 𝑦}))
cnei 23053	class nei
df-nei 23054	⊢ nei = (𝑗 ∈ Top ↦ (𝑥 ∈ 𝒫 ∪ 𝑗 ↦ {𝑦 ∈ 𝒫 ∪ 𝑗 ∣ ∃𝑔 ∈ 𝑗 (𝑥 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑦)}))
clp 23090	class limPt
cperf 23091	class Perf
df-lp 23092	⊢ limPt = (𝑗 ∈ Top ↦ (𝑥 ∈ 𝒫 ∪ 𝑗 ↦ {𝑦 ∣ 𝑦 ∈ ((cls‘𝑗)‘(𝑥 ∖ {𝑦}))}))
df-perf 23093	⊢ Perf = {𝑗 ∈ Top ∣ ((limPt‘𝑗)‘∪ 𝑗) = ∪ 𝑗}
ccn 23180	class Cn
ccnp 23181	class CnP
clm 23182	class ⇝𝑡
df-cn 23183	⊢ Cn = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ {𝑓 ∈ (∪ 𝑘 ↑m ∪ 𝑗) ∣ ∀𝑦 ∈ 𝑘 (◡𝑓 “ 𝑦) ∈ 𝑗})
df-cnp 23184	⊢ 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 23185	⊢ ⇝𝑡 = (𝑗 ∈ Top ↦ {⟨𝑓, 𝑥⟩ ∣ (𝑓 ∈ (∪ 𝑗 ↑pm ℂ) ∧ 𝑥 ∈ ∪ 𝑗 ∧ ∀𝑢 ∈ 𝑗 (𝑥 ∈ 𝑢 → ∃𝑦 ∈ ran ℤ≥(𝑓 ↾ 𝑦):𝑦⟶𝑢))})
ct0 23262	class Kol2
ct1 23263	class Fre
cha 23264	class Haus
creg 23265	class Reg
cnrm 23266	class Nrm
ccnrm 23267	class CNrm
cpnrm 23268	class PNrm
df-t0 23269	⊢ Kol2 = {𝑗 ∈ Top ∣ ∀𝑥 ∈ ∪ 𝑗∀𝑦 ∈ ∪ 𝑗(∀𝑜 ∈ 𝑗 (𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜) → 𝑥 = 𝑦)}
df-t1 23270	⊢ Fre = {𝑥 ∈ Top ∣ ∀𝑎 ∈ ∪ 𝑥{𝑎} ∈ (Clsd‘𝑥)}
df-haus 23271	⊢ Haus = {𝑗 ∈ Top ∣ ∀𝑥 ∈ ∪ 𝑗∀𝑦 ∈ ∪ 𝑗(𝑥 ≠ 𝑦 → ∃𝑛 ∈ 𝑗 ∃𝑚 ∈ 𝑗 (𝑥 ∈ 𝑛 ∧ 𝑦 ∈ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅))}
df-reg 23272	⊢ Reg = {𝑗 ∈ Top ∣ ∀𝑥 ∈ 𝑗 ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝑗 (𝑦 ∈ 𝑧 ∧ ((cls‘𝑗)‘𝑧) ⊆ 𝑥)}
df-nrm 23273	⊢ Nrm = {𝑗 ∈ Top ∣ ∀𝑥 ∈ 𝑗 ∀𝑦 ∈ ((Clsd‘𝑗) ∩ 𝒫 𝑥)∃𝑧 ∈ 𝑗 (𝑦 ⊆ 𝑧 ∧ ((cls‘𝑗)‘𝑧) ⊆ 𝑥)}
df-cnrm 23274	⊢ CNrm = {𝑗 ∈ Top ∣ ∀𝑥 ∈ 𝒫 ∪ 𝑗(𝑗 ↾t 𝑥) ∈ Nrm}
df-pnrm 23275	⊢ PNrm = {𝑗 ∈ Nrm ∣ (Clsd‘𝑗) ⊆ ran (𝑓 ∈ (𝑗 ↑m ℕ) ↦ ∩ ran 𝑓)}
ccmp 23342	class Comp
df-cmp 23343	⊢ Comp = {𝑥 ∈ Top ∣ ∀𝑦 ∈ 𝒫 𝑥(∪ 𝑥 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝑥 = ∪ 𝑧)}
cconn 23367	class Conn
df-conn 23368	⊢ Conn = {𝑗 ∈ Top ∣ (𝑗 ∩ (Clsd‘𝑗)) = {∅, ∪ 𝑗}}
c1stc 23393	class 1stω
c2ndc 23394	class 2ndω
df-1stc 23395	⊢ 1stω = {𝑗 ∈ Top ∣ ∀𝑥 ∈ ∪ 𝑗∃𝑦 ∈ 𝒫 𝑗(𝑦 ≼ ω ∧ ∀𝑧 ∈ 𝑗 (𝑥 ∈ 𝑧 → 𝑥 ∈ ∪ (𝑦 ∩ 𝒫 𝑧)))}
df-2ndc 23396	⊢ 2ndω = {𝑗 ∣ ∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝑗)}
clly 23420	class Locally 𝐴
cnlly 23421	class 𝑛-Locally 𝐴
df-lly 23422	⊢ Locally 𝐴 = {𝑗 ∈ Top ∣ ∀𝑥 ∈ 𝑗 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝐴)}
df-nlly 23423	⊢ 𝑛-Locally 𝐴 = {𝑗 ∈ Top ∣ ∀𝑥 ∈ 𝑗 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗 ↾t 𝑢) ∈ 𝐴}
cref 23458	class Ref
cptfin 23459	class PtFin
clocfin 23460	class LocFin
df-ref 23461	⊢ Ref = {⟨𝑥, 𝑦⟩ ∣ (∪ 𝑦 = ∪ 𝑥 ∧ ∀𝑧 ∈ 𝑥 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)}
df-ptfin 23462	⊢ PtFin = {𝑥 ∣ ∀𝑦 ∈ ∪ 𝑥{𝑧 ∈ 𝑥 ∣ 𝑦 ∈ 𝑧} ∈ Fin}
df-locfin 23463	⊢ LocFin = (𝑥 ∈ Top ↦ {𝑦 ∣ (∪ 𝑥 = ∪ 𝑦 ∧ ∀𝑝 ∈ ∪ 𝑥∃𝑛 ∈ 𝑥 (𝑝 ∈ 𝑛 ∧ {𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))})
ckgen 23489	class 𝑘Gen
df-kgen 23490	⊢ 𝑘Gen = (𝑗 ∈ Top ↦ {𝑥 ∈ 𝒫 ∪ 𝑗 ∣ ∀𝑘 ∈ 𝒫 ∪ 𝑗((𝑗 ↾t 𝑘) ∈ Comp → (𝑥 ∩ 𝑘) ∈ (𝑗 ↾t 𝑘))})
ctx 23516	class ×t
cxko 23517	class ↑ko
df-tx 23518	⊢ ×t = (𝑟 ∈ V, 𝑠 ∈ V ↦ (topGen‘ran (𝑥 ∈ 𝑟, 𝑦 ∈ 𝑠 ↦ (𝑥 × 𝑦))))
df-xko 23519	⊢ ↑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 23649	class KQ
df-kq 23650	⊢ KQ = (𝑗 ∈ Top ↦ (𝑗 qTop (𝑥 ∈ ∪ 𝑗 ↦ {𝑦 ∈ 𝑗 ∣ 𝑥 ∈ 𝑦})))
chmeo 23709	class Homeo
chmph 23710	class ≃
df-hmeo 23711	⊢ Homeo = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ {𝑓 ∈ (𝑗 Cn 𝑘) ∣ ◡𝑓 ∈ (𝑘 Cn 𝑗)})
df-hmph 23712	⊢ ≃ = (◡Homeo “ (V ∖ 1o))
cfil 23801	class Fil
df-fil 23802	⊢ Fil = (𝑧 ∈ V ↦ {𝑓 ∈ (fBas‘𝑧) ∣ ∀𝑥 ∈ 𝒫 𝑧((𝑓 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥 ∈ 𝑓)})
cufil 23855	class UFil
cufl 23856	class UFL
df-ufil 23857	⊢ UFil = (𝑔 ∈ V ↦ {𝑓 ∈ (Fil‘𝑔) ∣ ∀𝑥 ∈ 𝒫 𝑔(𝑥 ∈ 𝑓 ∨ (𝑔 ∖ 𝑥) ∈ 𝑓)})
df-ufl 23858	⊢ UFL = {𝑥 ∣ ∀𝑓 ∈ (Fil‘𝑥)∃𝑔 ∈ (UFil‘𝑥)𝑓 ⊆ 𝑔}
cfm 23889	class FilMap
cflim 23890	class fLim
cflf 23891	class fLimf
cfcls 23892	class fClus
cfcf 23893	class fClusf
df-fm 23894	⊢ FilMap = (𝑥 ∈ V, 𝑓 ∈ V ↦ (𝑦 ∈ (fBas‘dom 𝑓) ↦ (𝑥filGenran (𝑡 ∈ 𝑦 ↦ (𝑓 “ 𝑡)))))
df-flim 23895	⊢ fLim = (𝑗 ∈ Top, 𝑓 ∈ ∪ ran Fil ↦ {𝑥 ∈ ∪ 𝑗 ∣ (((nei‘𝑗)‘{𝑥}) ⊆ 𝑓 ∧ 𝑓 ⊆ 𝒫 ∪ 𝑗)})
df-flf 23896	⊢ fLimf = (𝑥 ∈ Top, 𝑦 ∈ ∪ ran Fil ↦ (𝑓 ∈ (∪ 𝑥 ↑m ∪ 𝑦) ↦ (𝑥 fLim ((∪ 𝑥 FilMap 𝑓)‘𝑦))))
df-fcls 23897	⊢ fClus = (𝑗 ∈ Top, 𝑓 ∈ ∪ ran Fil ↦ if(∪ 𝑗 = ∪ 𝑓, ∩ 𝑥 ∈ 𝑓 ((cls‘𝑗)‘𝑥), ∅))
df-fcf 23898	⊢ fClusf = (𝑗 ∈ Top, 𝑓 ∈ ∪ ran Fil ↦ (𝑔 ∈ (∪ 𝑗 ↑m ∪ 𝑓) ↦ (𝑗 fClus ((∪ 𝑗 FilMap 𝑔)‘𝑓))))
ccnext 24015	class CnExt
df-cnext 24016	⊢ 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 24026	class TopMnd
ctgp 24027	class TopGrp
df-tmd 24028	⊢ TopMnd = {𝑓 ∈ (Mnd ∩ TopSp) ∣ [(TopOpen‘𝑓) / 𝑗](+𝑓‘𝑓) ∈ ((𝑗 ×t 𝑗) Cn 𝑗)}
df-tgp 24029	⊢ TopGrp = {𝑓 ∈ (Grp ∩ TopMnd) ∣ [(TopOpen‘𝑓) / 𝑗](invg‘𝑓) ∈ (𝑗 Cn 𝑗)}
ctsu 24082	class tsums
df-tsms 24083	⊢ tsums = (𝑤 ∈ V, 𝑓 ∈ V ↦ ⦋(𝒫 dom 𝑓 ∩ Fin) / 𝑠⦌(((TopOpen‘𝑤) fLimf (𝑠filGenran (𝑧 ∈ 𝑠 ↦ {𝑦 ∈ 𝑠 ∣ 𝑧 ⊆ 𝑦})))‘(𝑦 ∈ 𝑠 ↦ (𝑤 Σg (𝑓 ↾ 𝑦)))))
ctrg 24112	class TopRing
ctdrg 24113	class TopDRing
ctlm 24114	class TopMod
ctvc 24115	class TopVec
df-trg 24116	⊢ TopRing = {𝑟 ∈ (TopGrp ∩ Ring) ∣ (mulGrp‘𝑟) ∈ TopMnd}
df-tdrg 24117	⊢ TopDRing = {𝑟 ∈ (TopRing ∩ DivRing) ∣ ((mulGrp‘𝑟) ↾s (Unit‘𝑟)) ∈ TopGrp}
df-tlm 24118	⊢ TopMod = {𝑤 ∈ (TopMnd ∩ LMod) ∣ ((Scalar‘𝑤) ∈ TopRing ∧ ( ·sf ‘𝑤) ∈ (((TopOpen‘(Scalar‘𝑤)) ×t (TopOpen‘𝑤)) Cn (TopOpen‘𝑤)))}
df-tvc 24119	⊢ TopVec = {𝑤 ∈ TopMod ∣ (Scalar‘𝑤) ∈ TopDRing}
cust 24156	class UnifOn
df-ust 24157	⊢ 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 24186	class unifTop
df-utop 24187	⊢ unifTop = (𝑢 ∈ ∪ ran UnifOn ↦ {𝑎 ∈ 𝒫 dom ∪ 𝑢 ∣ ∀𝑥 ∈ 𝑎 ∃𝑣 ∈ 𝑢 (𝑣 “ {𝑥}) ⊆ 𝑎})
cuss 24209	class UnifSt
cusp 24210	class UnifSp
ctus 24211	class toUnifSp
df-uss 24212	⊢ UnifSt = (𝑓 ∈ V ↦ ((UnifSet‘𝑓) ↾t ((Base‘𝑓) × (Base‘𝑓))))
df-usp 24213	⊢ UnifSp = {𝑓 ∣ ((UnifSt‘𝑓) ∈ (UnifOn‘(Base‘𝑓)) ∧ (TopOpen‘𝑓) = (unifTop‘(UnifSt‘𝑓)))}
df-tus 24214	⊢ toUnifSp = (𝑢 ∈ ∪ ran UnifOn ↦ ({⟨(Base‘ndx), dom ∪ 𝑢⟩, ⟨(UnifSet‘ndx), 𝑢⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑢)⟩))
cucn 24230	class Cnu
df-ucn 24231	⊢ Cnu = (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 24241	class CauFilu
df-cfilu 24242	⊢ CauFilu = (𝑢 ∈ ∪ ran UnifOn ↦ {𝑓 ∈ (fBas‘dom ∪ 𝑢) ∣ ∀𝑣 ∈ 𝑢 ∃𝑎 ∈ 𝑓 (𝑎 × 𝑎) ⊆ 𝑣})
ccusp 24252	class CUnifSp
df-cusp 24253	⊢ CUnifSp = {𝑤 ∈ UnifSp ∣ ∀𝑐 ∈ (Fil‘(Base‘𝑤))(𝑐 ∈ (CauFilu‘(UnifSt‘𝑤)) → ((TopOpen‘𝑤) fLim 𝑐) ≠ ∅)}
cxms 24273	class ∞MetSp
cms 24274	class MetSp
ctms 24275	class toMetSp
df-xms 24276	⊢ ∞MetSp = {𝑓 ∈ TopSp ∣ (TopOpen‘𝑓) = (MetOpen‘((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))))}
df-ms 24277	⊢ MetSp = {𝑓 ∈ ∞MetSp ∣ ((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))) ∈ (Met‘(Base‘𝑓))}
df-tms 24278	⊢ toMetSp = (𝑑 ∈ ∪ ran ∞Met ↦ ({⟨(Base‘ndx), dom dom 𝑑⟩, ⟨(dist‘ndx), 𝑑⟩} sSet ⟨(TopSet‘ndx), (MetOpen‘𝑑)⟩))
cnm 24532	class norm
cngp 24533	class NrmGrp
ctng 24534	class toNrmGrp
cnrg 24535	class NrmRing
cnlm 24536	class NrmMod
cnvc 24537	class NrmVec
df-nm 24538	⊢ norm = (𝑤 ∈ V ↦ (𝑥 ∈ (Base‘𝑤) ↦ (𝑥(dist‘𝑤)(0g‘𝑤))))
df-ngp 24539	⊢ NrmGrp = {𝑔 ∈ (Grp ∩ MetSp) ∣ ((norm‘𝑔) ∘ (-g‘𝑔)) ⊆ (dist‘𝑔)}
df-tng 24540	⊢ toNrmGrp = (𝑔 ∈ V, 𝑓 ∈ V ↦ ((𝑔 sSet ⟨(dist‘ndx), (𝑓 ∘ (-g‘𝑔))⟩) sSet ⟨(TopSet‘ndx), (MetOpen‘(𝑓 ∘ (-g‘𝑔)))⟩))
df-nrg 24541	⊢ NrmRing = {𝑤 ∈ NrmGrp ∣ (norm‘𝑤) ∈ (AbsVal‘𝑤)}
df-nlm 24542	⊢ 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 24543	⊢ NrmVec = (NrmMod ∩ LVec)
cnmo 24661	class normOp
cnghm 24662	class NGHom
cnmhm 24663	class NMHom
df-nmo 24664	⊢ 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 24665	⊢ NGHom = (𝑠 ∈ NrmGrp, 𝑡 ∈ NrmGrp ↦ (◡(𝑠 normOp 𝑡) “ ℝ))
df-nmhm 24666	⊢ NMHom = (𝑠 ∈ NrmMod, 𝑡 ∈ NrmMod ↦ ((𝑠 LMHom 𝑡) ∩ (𝑠 NGHom 𝑡)))
cii 24836	class II
ccncf 24837	class –cn→
df-ii 24838	⊢ II = (MetOpen‘((abs ∘ − ) ↾ ((0[,]1) × (0[,]1))))
df-cncf 24839	⊢ –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 24934	class Htpy
cphtpy 24935	class PHtpy
cphtpc 24936	class ≃ph
df-htpy 24937	⊢ 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 24938	⊢ 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 24959	⊢ ≃ph = (𝑥 ∈ Top ↦ {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ (II Cn 𝑥) ∧ (𝑓(PHtpy‘𝑥)𝑔) ≠ ∅)})
cpco 24968	class *𝑝
comi 24969	class Ω1
comn 24970	class Ω𝑛
cpi1 24971	class π1
cpin 24972	class πn
df-pco 24973	⊢ *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 24974	⊢ 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 24975	⊢ Ω𝑛 = (𝑗 ∈ Top, 𝑦 ∈ ∪ 𝑗 ↦ seq0(((𝑥 ∈ V, 𝑝 ∈ V ↦ ⟨((TopOpen‘(1st ‘𝑥)) Ω1 (2nd ‘𝑥)), ((0[,]1) × {(2nd ‘𝑥)})⟩) ∘ 1st ), ⟨{⟨(Base‘ndx), ∪ 𝑗⟩, ⟨(TopSet‘ndx), 𝑗⟩}, 𝑦⟩))
df-pi1 24976	⊢ π1 = (𝑗 ∈ Top, 𝑦 ∈ ∪ 𝑗 ↦ ((𝑗 Ω1 𝑦) /s ( ≃ph‘𝑗)))
df-pin 24977	⊢ πn = (𝑗 ∈ Top, 𝑝 ∈ ∪ 𝑗 ↦ (𝑛 ∈ ℕ0 ↦ ((1st ‘((𝑗 Ω𝑛 𝑝)‘𝑛)) /s if(𝑛 = 0, {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 ∈ (II Cn 𝑗)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)}, ( ≃ph‘(TopOpen‘(1st ‘((𝑗 Ω𝑛 𝑝)‘(𝑛 − 1)))))))))
cclm 25030	class ℂMod
df-clm 25031	⊢ ℂMod = {𝑤 ∈ LMod ∣ [(Scalar‘𝑤) / 𝑓][(Base‘𝑓) / 𝑘](𝑓 = (ℂfld ↾s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))}
ccvs 25091	class ℂVec
df-cvs 25092	⊢ ℂVec = (ℂMod ∩ LVec)
ccph 25134	class ℂPreHil
ctcph 25135	class toℂPreHil
df-cph 25136	⊢ ℂPreHil = {𝑤 ∈ (PreHil ∩ NrmMod) ∣ [(Scalar‘𝑤) / 𝑓][(Base‘𝑓) / 𝑘](𝑓 = (ℂfld ↾s 𝑘) ∧ (√ “ (𝑘 ∩ (0[,)+∞))) ⊆ 𝑘 ∧ (norm‘𝑤) = (𝑥 ∈ (Base‘𝑤) ↦ (√‘(𝑥(·𝑖‘𝑤)𝑥))))}
df-tcph 25137	⊢ toℂPreHil = (𝑤 ∈ V ↦ (𝑤 toNrmGrp (𝑥 ∈ (Base‘𝑤) ↦ (√‘(𝑥(·𝑖‘𝑤)𝑥)))))
ccfil 25220	class CauFil
ccau 25221	class Cau
ccmet 25222	class CMet
df-cfil 25223	⊢ CauFil = (𝑑 ∈ ∪ ran ∞Met ↦ {𝑓 ∈ (Fil‘dom dom 𝑑) ∣ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ 𝑓 (𝑑 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)})
df-cau 25224	⊢ 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 25225	⊢ CMet = (𝑥 ∈ V ↦ {𝑑 ∈ (Met‘𝑥) ∣ ∀𝑓 ∈ (CauFil‘𝑑)((MetOpen‘𝑑) fLim 𝑓) ≠ ∅})
ccms 25300	class CMetSp
cbn 25301	class Ban
chl 25302	class ℂHil
df-cms 25303	⊢ CMetSp = {𝑤 ∈ MetSp ∣ [(Base‘𝑤) / 𝑏]((dist‘𝑤) ↾ (𝑏 × 𝑏)) ∈ (CMet‘𝑏)}
df-bn 25304	⊢ Ban = {𝑤 ∈ (NrmVec ∩ CMetSp) ∣ (Scalar‘𝑤) ∈ CMetSp}
df-hl 25305	⊢ ℂHil = (Ban ∩ ℂPreHil)
crrx 25351	class ℝ^
cehl 25352	class 𝔼hil
df-rrx 25353	⊢ ℝ^ = (𝑖 ∈ V ↦ (toℂPreHil‘(ℝfld freeLMod 𝑖)))
df-ehl 25354	⊢ 𝔼hil = (𝑛 ∈ ℕ0 ↦ (ℝ^‘(1...𝑛)))
covol 25431	class vol*
cvol 25432	class vol
df-ovol 25433	⊢ 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 25434	⊢ vol = (vol* ↾ {𝑥 ∣ ∀𝑦 ∈ (◡vol* “ ℝ)(vol*‘𝑦) = ((vol*‘(𝑦 ∩ 𝑥)) + (vol*‘(𝑦 ∖ 𝑥)))})
cmbf 25583	class MblFn
citg1 25584	class ∫1
citg2 25585	class ∫2
cibl 25586	class 𝐿1
citg 25587	class ∫𝐴𝐵 d𝑥
df-mbf 25588	⊢ MblFn = {𝑓 ∈ (ℂ ↑pm ℝ) ∣ ∀𝑥 ∈ ran (,)((◡(ℜ ∘ 𝑓) “ 𝑥) ∈ dom vol ∧ (◡(ℑ ∘ 𝑓) “ 𝑥) ∈ dom vol)}
df-itg1 25589	⊢ ∫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 25590	⊢ ∫2 = (𝑓 ∈ ((0[,]+∞) ↑m ℝ) ↦ sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔 ∘r ≤ 𝑓 ∧ 𝑥 = (∫1‘𝑔))}, ℝ*, < ))
df-ibl 25591	⊢ 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 25592	⊢ ∫𝐴𝐵 d𝑥 = Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐵 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0))))
c0p 25638	class 0𝑝
df-0p 25639	⊢ 0𝑝 = (ℂ × {0})
cdit 25815	class ⨜[𝐴 → 𝐵]𝐶 d𝑥
df-ditg 25816	⊢ ⨜[𝐴 → 𝐵]𝐶 d𝑥 = if(𝐴 ≤ 𝐵, ∫(𝐴(,)𝐵)𝐶 d𝑥, -∫(𝐵(,)𝐴)𝐶 d𝑥)
climc 25831	class limℂ
cdv 25832	class D
cdvn 25833	class D𝑛
ccpn 25834	class 𝓑C𝑛
df-limc 25835	⊢ limℂ = (𝑓 ∈ (ℂ ↑pm ℂ), 𝑥 ∈ ℂ ↦ {𝑦 ∣ [(TopOpen‘ℂfld) / 𝑗](𝑧 ∈ (dom 𝑓 ∪ {𝑥}) ↦ if(𝑧 = 𝑥, 𝑦, (𝑓‘𝑧))) ∈ (((𝑗 ↾t (dom 𝑓 ∪ {𝑥})) CnP 𝑗)‘𝑥)})
df-dv 25836	⊢ D = (𝑠 ∈ 𝒫 ℂ, 𝑓 ∈ (ℂ ↑pm 𝑠) ↦ ∪ 𝑥 ∈ ((int‘((TopOpen‘ℂfld) ↾t 𝑠))‘dom 𝑓)({𝑥} × ((𝑧 ∈ (dom 𝑓 ∖ {𝑥}) ↦ (((𝑓‘𝑧) − (𝑓‘𝑥)) / (𝑧 − 𝑥))) limℂ 𝑥)))
df-dvn 25837	⊢ D𝑛 = (𝑠 ∈ 𝒫 ℂ, 𝑓 ∈ (ℂ ↑pm 𝑠) ↦ seq0(((𝑥 ∈ V ↦ (𝑠 D 𝑥)) ∘ 1st ), (ℕ0 × {𝑓})))
df-cpn 25838	⊢ 𝓑C𝑛 = (𝑠 ∈ 𝒫 ℂ ↦ (𝑥 ∈ ℕ0 ↦ {𝑓 ∈ (ℂ ↑pm 𝑠) ∣ ((𝑠 D𝑛 𝑓)‘𝑥) ∈ (dom 𝑓–cn→ℂ)}))
cmdg 26026	class mDeg
cdg1 26027	class deg1
df-mdeg 26028	⊢ mDeg = (𝑖 ∈ V, 𝑟 ∈ V ↦ (𝑓 ∈ (Base‘(𝑖 mPoly 𝑟)) ↦ sup(ran (ℎ ∈ (𝑓 supp (0g‘𝑟)) ↦ (ℂfld Σg ℎ)), ℝ*, < )))
df-deg1 26029	⊢ deg1 = (𝑟 ∈ V ↦ (1o mDeg 𝑟))
cmn1 26099	class Monic1p
cuc1p 26100	class Unic1p
cq1p 26101	class quot1p
cr1p 26102	class rem1p
cig1p 26103	class idlGen1p
df-mon1 26104	⊢ Monic1p = (𝑟 ∈ V ↦ {𝑓 ∈ (Base‘(Poly1‘𝑟)) ∣ (𝑓 ≠ (0g‘(Poly1‘𝑟)) ∧ ((coe1‘𝑓)‘((deg1‘𝑟)‘𝑓)) = (1r‘𝑟))})
df-uc1p 26105	⊢ Unic1p = (𝑟 ∈ V ↦ {𝑓 ∈ (Base‘(Poly1‘𝑟)) ∣ (𝑓 ≠ (0g‘(Poly1‘𝑟)) ∧ ((coe1‘𝑓)‘((deg1‘𝑟)‘𝑓)) ∈ (Unit‘𝑟))})
df-q1p 26106	⊢ quot1p = (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 26107	⊢ rem1p = (𝑟 ∈ V ↦ ⦋(Base‘(Poly1‘𝑟)) / 𝑏⦌(𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (𝑓(-g‘(Poly1‘𝑟))((𝑓(quot1p‘𝑟)𝑔)(.r‘(Poly1‘𝑟))𝑔))))
df-ig1p 26108	⊢ idlGen1p = (𝑟 ∈ V ↦ (𝑖 ∈ (LIdeal‘(Poly1‘𝑟)) ↦ if(𝑖 = {(0g‘(Poly1‘𝑟))}, (0g‘(Poly1‘𝑟)), (℩𝑔 ∈ (𝑖 ∩ (Monic1p‘𝑟))((deg1‘𝑟)‘𝑔) = inf(((deg1‘𝑟) “ (𝑖 ∖ {(0g‘(Poly1‘𝑟))})), ℝ, < )))))
cply 26157	class Poly
cidp 26158	class Xp
ccoe 26159	class coeff
cdgr 26160	class deg
df-ply 26161	⊢ Poly = (𝑥 ∈ 𝒫 ℂ ↦ {𝑓 ∣ ∃𝑛 ∈ ℕ0 ∃𝑎 ∈ ((𝑥 ∪ {0}) ↑m ℕ0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))})
df-idp 26162	⊢ Xp = ( I ↾ ℂ)
df-coe 26163	⊢ 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 26164	⊢ deg = (𝑓 ∈ (Poly‘ℂ) ↦ sup((◡(coeff‘𝑓) “ (ℂ ∖ {0})), ℕ0, < ))
cquot 26266	class quot
df-quot 26267	⊢ quot = (𝑓 ∈ (Poly‘ℂ), 𝑔 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ↦ (℩𝑞 ∈ (Poly‘ℂ)[(𝑓 ∘f − (𝑔 ∘f · 𝑞)) / 𝑟](𝑟 = 0𝑝 ∨ (deg‘𝑟) < (deg‘𝑔))))
caa 26290	class 𝔸
df-aa 26291	⊢ 𝔸 = ∪ 𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝})(◡𝑓 “ {0})
ctayl 26328	class Tayl
cana 26329	class Ana
df-tayl 26330	⊢ Tayl = (𝑠 ∈ {ℝ, ℂ}, 𝑓 ∈ (ℂ ↑pm 𝑠) ↦ (𝑛 ∈ (ℕ0 ∪ {+∞}), 𝑎 ∈ ∩ 𝑘 ∈ ((0[,]𝑛) ∩ ℤ)dom ((𝑠 D𝑛 𝑓)‘𝑘) ↦ ∪ 𝑥 ∈ ℂ ({𝑥} × (ℂfld tsums (𝑘 ∈ ((0[,]𝑛) ∩ ℤ) ↦ (((((𝑠 D𝑛 𝑓)‘𝑘)‘𝑎) / (!‘𝑘)) · ((𝑥 − 𝑎)↑𝑘)))))))
df-ana 26331	⊢ Ana = (𝑠 ∈ {ℝ, ℂ} ↦ {𝑓 ∈ (ℂ ↑pm 𝑠) ∣ ∀𝑥 ∈ dom 𝑓 𝑥 ∈ ((int‘((TopOpen‘ℂfld) ↾t 𝑠))‘dom (𝑓 ∩ (+∞(𝑠 Tayl 𝑓)𝑥)))})
culm 26353	class ⇝𝑢
df-ulm 26354	⊢ ⇝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 26531	class log
ccxp 26532	class ↑𝑐
df-log 26533	⊢ log = ◡(exp ↾ (◡ℑ “ (-π(,]π)))
df-cxp 26534	⊢ ↑𝑐 = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ if(𝑥 = 0, if(𝑦 = 0, 1, 0), (exp‘(𝑦 · (log‘𝑥)))))
clogb 26742	class logb
df-logb 26743	⊢ logb = (𝑥 ∈ (ℂ ∖ {0, 1}), 𝑦 ∈ (ℂ ∖ {0}) ↦ ((log‘𝑦) / (log‘𝑥)))
casin 26840	class arcsin
cacos 26841	class arccos
catan 26842	class arctan
df-asin 26843	⊢ arcsin = (𝑥 ∈ ℂ ↦ (-i · (log‘((i · 𝑥) + (√‘(1 − (𝑥↑2)))))))
df-acos 26844	⊢ arccos = (𝑥 ∈ ℂ ↦ ((π / 2) − (arcsin‘𝑥)))
df-atan 26845	⊢ arctan = (𝑥 ∈ (ℂ ∖ {-i, i}) ↦ ((i / 2) · ((log‘(1 − (i · 𝑥))) − (log‘(1 + (i · 𝑥))))))
carea 26933	class area
df-area 26934	⊢ area = (𝑠 ∈ {𝑡 ∈ 𝒫 (ℝ × ℝ) ∣ (∀𝑥 ∈ ℝ (𝑡 “ {𝑥}) ∈ (◡vol “ ℝ) ∧ (𝑥 ∈ ℝ ↦ (vol‘(𝑡 “ {𝑥}))) ∈ 𝐿1)} ↦ ∫ℝ(vol‘(𝑠 “ {𝑥})) d𝑥)
cem 26970	class γ
df-em 26971	⊢ γ = Σ𝑘 ∈ ℕ ((1 / 𝑘) − (log‘(1 + (1 / 𝑘))))
czeta 26991	class ζ
df-zeta 26992	⊢ 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 26994	class log Γ
cgam 26995	class Γ
cigam 26996	class 1/Γ
df-lgam 26997	⊢ log Γ = (𝑧 ∈ (ℂ ∖ (ℤ ∖ ℕ)) ↦ (Σ𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))) − (log‘𝑧)))
df-gam 26998	⊢ Γ = (exp ∘ log Γ)
df-igam 26999	⊢ 1/Γ = (𝑥 ∈ ℂ ↦ if(𝑥 ∈ (ℤ ∖ ℕ), 0, (1 / (Γ‘𝑥))))
ccht 27069	class θ
cvma 27070	class Λ
cchp 27071	class ψ
cppi 27072	class π
cmu 27073	class μ
csgm 27074	class σ
df-cht 27075	⊢ θ = (𝑥 ∈ ℝ ↦ Σ𝑝 ∈ ((0[,]𝑥) ∩ ℙ)(log‘𝑝))
df-vma 27076	⊢ Λ = (𝑥 ∈ ℕ ↦ ⦋{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥} / 𝑠⦌if((♯‘𝑠) = 1, (log‘∪ 𝑠), 0))
df-chp 27077	⊢ ψ = (𝑥 ∈ ℝ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))(Λ‘𝑛))
df-ppi 27078	⊢ π = (𝑥 ∈ ℝ ↦ (♯‘((0[,]𝑥) ∩ ℙ)))
df-mu 27079	⊢ μ = (𝑥 ∈ ℕ ↦ if(∃𝑝 ∈ ℙ (𝑝↑2) ∥ 𝑥, 0, (-1↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥}))))
df-sgm 27080	⊢ σ = (𝑥 ∈ ℂ, 𝑛 ∈ ℕ ↦ Σ𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝑛} (𝑘↑𝑐𝑥))
cdchr 27211	class DChr
df-dchr 27212	⊢ DChr = (𝑛 ∈ ℕ ↦ ⦋(ℤ/nℤ‘𝑛) / 𝑧⦌⦋{𝑥 ∈ ((mulGrp‘𝑧) MndHom (mulGrp‘ℂfld)) ∣ (((Base‘𝑧) ∖ (Unit‘𝑧)) × {0}) ⊆ 𝑥} / 𝑏⦌{⟨(Base‘ndx), 𝑏⟩, ⟨(+g‘ndx), ( ∘f · ↾ (𝑏 × 𝑏))⟩})
clgs 27273	class /L
df-lgs 27274	⊢ /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 27619	class No
clts 27620	class <s
cbday 27621	class bday
df-no 27622	⊢ No = {𝑓 ∣ ∃𝑎 ∈ On 𝑓:𝑎⟶{1o, 2o}}
df-lts 27623	⊢ <s = {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ No ∧ 𝑔 ∈ No ) ∧ ∃𝑥 ∈ On (∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑔‘𝑦) ∧ (𝑓‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝑔‘𝑥)))}
df-bday 27624	⊢ bday = (𝑥 ∈ No ↦ dom 𝑥)
cles 27724	class ≤s
df-les 27725	⊢ ≤s = (( No × No ) ∖ ◡ <s )
cslts 27765	class <<s
df-slts 27766	⊢ <<s = {⟨𝑎, 𝑏⟩ ∣ (𝑎 ⊆ No ∧ 𝑏 ⊆ No ∧ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 𝑥 <s 𝑦)}
ccuts 27767	class |s
df-cuts 27768	⊢ |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 27813	class 0s
c1s 27814	class 1s
df-0s 27815	⊢ 0s = (∅ |s ∅)
df-1s 27816	⊢ 1s = ({ 0s } |s ∅)
cmade 27830	class M
cold 27831	class O
cnew 27832	class N
cleft 27833	class L
cright 27834	class R
df-made 27835	⊢ M = recs((𝑓 ∈ V ↦ ( |s “ (𝒫 ∪ ran 𝑓 × 𝒫 ∪ ran 𝑓))))
df-old 27836	⊢ O = (𝑥 ∈ On ↦ ∪ ( M “ 𝑥))
df-new 27837	⊢ N = (𝑥 ∈ On ↦ (( M ‘𝑥) ∖ ( O ‘𝑥)))
df-left 27838	⊢ L = (𝑥 ∈ No ↦ {𝑦 ∈ ( O ‘( bday ‘𝑥)) ∣ 𝑦 <s 𝑥})
df-right 27839	⊢ R = (𝑥 ∈ No ↦ {𝑦 ∈ ( O ‘( bday ‘𝑥)) ∣ 𝑥 <s 𝑦})
cnorec 27945	class norec (𝐹)
df-norec 27946	⊢ norec (𝐹) = frecs({⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}, No , 𝐹)
cnorec2 27956	class norec2 (𝐹)
df-norec2 27957	⊢ norec2 (𝐹) = frecs({⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ ( No × No ) ∧ 𝑏 ∈ ( No × No ) ∧ (((1st ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (1st ‘𝑏) ∨ (1st ‘𝑎) = (1st ‘𝑏)) ∧ ((2nd ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (2nd ‘𝑏) ∨ (2nd ‘𝑎) = (2nd ‘𝑏)) ∧ 𝑎 ≠ 𝑏))}, ( No × No ), 𝐹)
cadds 27967	class +s
df-adds 27968	⊢ +s = norec2 ((𝑥 ∈ V, 𝑎 ∈ V ↦ (({𝑦 ∣ ∃𝑙 ∈ ( L ‘(1st ‘𝑥))𝑦 = (𝑙𝑎(2nd ‘𝑥))} ∪ {𝑧 ∣ ∃𝑙 ∈ ( L ‘(2nd ‘𝑥))𝑧 = ((1st ‘𝑥)𝑎𝑙)}) |s ({𝑦 ∣ ∃𝑟 ∈ ( R ‘(1st ‘𝑥))𝑦 = (𝑟𝑎(2nd ‘𝑥))} ∪ {𝑧 ∣ ∃𝑟 ∈ ( R ‘(2nd ‘𝑥))𝑧 = ((1st ‘𝑥)𝑎𝑟)}))))
cnegs 28027	class -us
csubs 28028	class -s
df-negs 28029	⊢ -us = norec ((𝑥 ∈ V, 𝑛 ∈ V ↦ ((𝑛 “ ( R ‘𝑥)) |s (𝑛 “ ( L ‘𝑥)))))
df-subs 28030	⊢ -s = (𝑥 ∈ No , 𝑦 ∈ No ↦ (𝑥 +s ( -us ‘𝑦)))
cmuls 28114	class ·s
df-muls 28115	⊢ ·s = norec2 ((𝑧 ∈ V, 𝑚 ∈ V ↦ ⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd ‘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 28195	class /su
df-divs 28196	⊢ /su = (𝑥 ∈ No , 𝑦 ∈ ( No ∖ { 0s }) ↦ (℩𝑧 ∈ No (𝑦 ·s 𝑧) = 𝑥))
cabss 28245	class abss
df-abss 28246	⊢ abss = (𝑥 ∈ No ↦ if( 0s ≤s 𝑥, 𝑥, ( -us ‘𝑥)))
cons 28259	class Ons
df-ons 28260	⊢ Ons = {𝑥 ∈ No ∣ ( R ‘𝑥) = ∅}
cseqs 28291	class seqs𝑀( + , 𝐹)
df-seqs 28292	⊢ seqs𝑀( + , 𝐹) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑦 + (𝐹‘(𝑥 +s 1s )))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) “ ω)
cn0s 28320	class ℕ0s
cnns 28321	class ℕs
df-n0s 28322	⊢ ℕ0s = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 0s ) “ ω)
df-nns 28323	⊢ ℕs = (ℕ0s ∖ { 0s })
czs 28386	class ℤs
df-zs 28387	⊢ ℤs = ( -s “ (ℕs × ℕs))
c2s 28418	class 2s
df-2s 28419	⊢ 2s = ({ 1s } |s ∅)
cexps 28420	class ↑s
df-exps 28421	⊢ ↑s = (𝑥 ∈ No , 𝑦 ∈ ℤs ↦ if(𝑦 = 0s , 1s , if( 0s <s 𝑦, (seqs 1s ( ·s , (ℕs × {𝑥}))‘𝑦), ( 1s /su (seqs 1s ( ·s , (ℕs × {𝑥}))‘( -us ‘𝑦))))))
cz12s 28422	class ℤs[1/2]
df-z12s 28423	⊢ ℤs[1/2] = {𝑥 ∣ ∃𝑦 ∈ ℤs ∃𝑧 ∈ ℕ0s 𝑥 = (𝑦 /su (2s↑s𝑧))}
creno 28497	class ℝs
df-reno 28498	⊢ ℝs = {𝑥 ∈ No ∣ (∃𝑛 ∈ ℕs (( -us ‘𝑛) <s 𝑥 ∧ 𝑥 <s 𝑛) ∧ 𝑥 = ({𝑦 ∣ ∃𝑛 ∈ ℕs 𝑦 = (𝑥 -s ( 1s /su 𝑛))} |s {𝑦 ∣ ∃𝑛 ∈ ℕs 𝑦 = (𝑥 +s ( 1s /su 𝑛))}))}
cstrkg 28511	class TarskiG
cstrkgc 28512	class TarskiGC
cstrkgb 28513	class TarskiGB
cstrkgcb 28514	class TarskiGCB
cstrkgld 28515	class DimTarskiG≥
cstrkge 28516	class TarskiGE
citv 28517	class Itv
clng 28518	class LineG
df-itv 28519	⊢ Itv = Slot ;16
df-lng 28520	⊢ LineG = Slot ;17
df-trkgc 28532	⊢ TarskiGC = {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 28533	⊢ TarskiGB = {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 28534	⊢ TarskiGCB = {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 28535	⊢ TarskiGE = {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 28536	⊢ DimTarskiG≥ = {⟨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 28537	⊢ TarskiG = ((TarskiGC ∩ TarskiGB) ∩ (TarskiGCB ∩ {𝑓 ∣ [(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥 ∈ 𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})}))
ccgrg 28594	class cgrG
df-cgrg 28595	⊢ 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 28616	class Ismt
df-ismt 28617	⊢ 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 28666	class ≤G
df-leg 28667	⊢ ≤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 28684	class hlG
df-hlg 28685	⊢ 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 28736	class pInvG
df-mir 28737	⊢ 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 28777	class ∟G
df-rag 28778	⊢ ∟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 28779	class ⟂G
df-perpg 28780	⊢ ⟂G = (𝑔 ∈ V ↦ {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ ran (LineG‘𝑔) ∧ 𝑏 ∈ ran (LineG‘𝑔)) ∧ ∃𝑥 ∈ (𝑎 ∩ 𝑏)∀𝑢 ∈ 𝑎 ∀𝑣 ∈ 𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝑔))})
chpg 28841	class hpG
df-hpg 28842	⊢ 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 28856	class midG
clmi 28857	class lInvG
df-mid 28858	⊢ midG = (𝑔 ∈ V ↦ (𝑎 ∈ (Base‘𝑔), 𝑏 ∈ (Base‘𝑔) ↦ (℩𝑚 ∈ (Base‘𝑔)𝑏 = (((pInvG‘𝑔)‘𝑚)‘𝑎))))
df-lmi 28859	⊢ 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 28891	class cgrA
df-cgra 28892	⊢ 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 28919	class inA
cleag 28920	class ≤∠
df-inag 28921	⊢ 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 28930	⊢ ≤∠ = (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 28949	class eqltrG
df-eqlg 28950	⊢ eqltrG = (𝑔 ∈ V ↦ {𝑥 ∈ ((Base‘𝑔) ↑m (0..^3)) ∣ 𝑥(cgrG‘𝑔)⟨“(𝑥‘1)(𝑥‘2)(𝑥‘0)”⟩})
cttg 28957	class toTG
df-ttg 28958	⊢ 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 28972	class 𝔼
cbtwn 28973	class Btwn
ccgr 28974	class Cgr
df-ee 28975	⊢ 𝔼 = (𝑛 ∈ ℕ ↦ (ℝ ↑m (1...𝑛)))
df-btwn 28976	⊢ 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 28977	⊢ Cgr = {⟨𝑥, 𝑦⟩ ∣ ∃𝑛 ∈ ℕ ((𝑥 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑦 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛))) ∧ Σ𝑖 ∈ (1...𝑛)((((1st ‘𝑥)‘𝑖) − ((2nd ‘𝑥)‘𝑖))↑2) = Σ𝑖 ∈ (1...𝑛)((((1st ‘𝑦)‘𝑖) − ((2nd ‘𝑦)‘𝑖))↑2))}
ceeng 29062	class EEG
df-eeng 29063	⊢ 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 29073	class .ef
df-edgf 29074	⊢ .ef = Slot ;18
cvtx 29081	class Vtx
ciedg 29082	class iEdg
df-vtx 29083	⊢ Vtx = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔)))
df-iedg 29084	⊢ iEdg = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (2nd ‘𝑔), (.ef‘𝑔)))
cedg 29132	class Edg
df-edg 29133	⊢ Edg = (𝑔 ∈ V ↦ ran (iEdg‘𝑔))
cuhgr 29141	class UHGraph
cushgr 29142	class USHGraph
df-uhgr 29143	⊢ UHGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶(𝒫 𝑣 ∖ {∅})}
df-ushgr 29144	⊢ USHGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→(𝒫 𝑣 ∖ {∅})}
cupgr 29165	class UPGraph
cumgr 29166	class UMGraph
df-upgr 29167	⊢ UPGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}}
df-umgr 29168	⊢ UMGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2}}
cuspgr 29233	class USPGraph
cusgr 29234	class USGraph
df-uspgr 29235	⊢ USPGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}}
df-usgr 29236	⊢ USGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2}}
csubgr 29352	class SubGraph
df-subgr 29353	⊢ SubGraph = {⟨𝑠, 𝑔⟩ ∣ ((Vtx‘𝑠) ⊆ (Vtx‘𝑔) ∧ (iEdg‘𝑠) = ((iEdg‘𝑔) ↾ dom (iEdg‘𝑠)) ∧ (Edg‘𝑠) ⊆ 𝒫 (Vtx‘𝑠))}
cfusgr 29401	class FinUSGraph
df-fusgr 29402	⊢ FinUSGraph = {𝑔 ∈ USGraph ∣ (Vtx‘𝑔) ∈ Fin}
cnbgr 29417	class NeighbVtx
df-nbgr 29418	⊢ NeighbVtx = (𝑔 ∈ V, 𝑣 ∈ (Vtx‘𝑔) ↦ {𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣}) ∣ ∃𝑒 ∈ (Edg‘𝑔){𝑣, 𝑛} ⊆ 𝑒})
cuvtx 29470	class UnivVtx
df-uvtx 29471	⊢ UnivVtx = (𝑔 ∈ V ↦ {𝑣 ∈ (Vtx‘𝑔) ∣ ∀𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣})𝑛 ∈ (𝑔 NeighbVtx 𝑣)})
ccplgr 29494	class ComplGraph
ccusgr 29495	class ComplUSGraph
df-cplgr 29496	⊢ ComplGraph = {𝑔 ∣ (UnivVtx‘𝑔) = (Vtx‘𝑔)}
df-cusgr 29497	⊢ ComplUSGraph = (USGraph ∩ ComplGraph)
cvtxdg 29551	class VtxDeg
df-vtxdg 29552	⊢ 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 29641	class RegGraph
crusgr 29642	class RegUSGraph
df-rgr 29643	⊢ RegGraph = {⟨𝑔, 𝑘⟩ ∣ (𝑘 ∈ ℕ0* ∧ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 𝑘)}
df-rusgr 29644	⊢ RegUSGraph = {⟨𝑔, 𝑘⟩ ∣ (𝑔 ∈ USGraph ∧ 𝑔 RegGraph 𝑘)}
cewlks 29681	class EdgWalks
cwlks 29682	class Walks
cwlkson 29683	class WalksOn
df-ewlks 29684	⊢ 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 29685	⊢ 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 29686	⊢ WalksOn = (𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝑔)𝑝 ∧ (𝑝‘0) = 𝑎 ∧ (𝑝‘(♯‘𝑓)) = 𝑏)}))
ctrls 29774	class Trails
ctrlson 29775	class TrailsOn
df-trls 29776	⊢ Trails = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝑔)𝑝 ∧ Fun ◡𝑓)})
df-trlson 29777	⊢ TrailsOn = (𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑎(WalksOn‘𝑔)𝑏)𝑝 ∧ 𝑓(Trails‘𝑔)𝑝)}))
cpths 29795	class Paths
cspths 29796	class SPaths
cpthson 29797	class PathsOn
cspthson 29798	class SPathsOn
df-pths 29799	⊢ Paths = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝑔)𝑝 ∧ Fun ◡(𝑝 ↾ (1..^(♯‘𝑓))) ∧ ((𝑝 “ {0, (♯‘𝑓)}) ∩ (𝑝 “ (1..^(♯‘𝑓)))) = ∅)})
df-spths 29800	⊢ SPaths = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝑔)𝑝 ∧ Fun ◡𝑝)})
df-pthson 29801	⊢ PathsOn = (𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑎(TrailsOn‘𝑔)𝑏)𝑝 ∧ 𝑓(Paths‘𝑔)𝑝)}))
df-spthson 29802	⊢ SPathsOn = (𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑎(TrailsOn‘𝑔)𝑏)𝑝 ∧ 𝑓(SPaths‘𝑔)𝑝)}))
cclwlks 29855	class ClWalks
df-clwlks 29856	⊢ ClWalks = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝑔)𝑝 ∧ (𝑝‘0) = (𝑝‘(♯‘𝑓)))})
ccrcts 29869	class Circuits
ccycls 29870	class Cycles
df-crcts 29871	⊢ Circuits = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝑔)𝑝 ∧ (𝑝‘0) = (𝑝‘(♯‘𝑓)))})
df-cycls 29872	⊢ Cycles = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Paths‘𝑔)𝑝 ∧ (𝑝‘0) = (𝑝‘(♯‘𝑓)))})
cwwlks 29910	class WWalks
cwwlksn 29911	class WWalksN
cwwlksnon 29912	class WWalksNOn
cwwspthsn 29913	class WSPathsN
cwwspthsnon 29914	class WSPathsNOn
df-wwlks 29915	⊢ WWalks = (𝑔 ∈ V ↦ {𝑤 ∈ Word (Vtx‘𝑔) ∣ (𝑤 ≠ ∅ ∧ ∀𝑖 ∈ (0..^((♯‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝑔))})
df-wwlksn 29916	⊢ WWalksN = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ {𝑤 ∈ (WWalks‘𝑔) ∣ (♯‘𝑤) = (𝑛 + 1)})
df-wwlksnon 29917	⊢ WWalksNOn = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {𝑤 ∈ (𝑛 WWalksN 𝑔) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤‘𝑛) = 𝑏)}))
df-wspthsn 29918	⊢ WSPathsN = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ {𝑤 ∈ (𝑛 WWalksN 𝑔) ∣ ∃𝑓 𝑓(SPaths‘𝑔)𝑤})
df-wspthsnon 29919	⊢ 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 30068	class ClWWalks
df-clwwlk 30069	⊢ 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 30111	class ClWWalksN
df-clwwlkn 30112	⊢ ClWWalksN = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ {𝑤 ∈ (ClWWalks‘𝑔) ∣ (♯‘𝑤) = 𝑛})
cclwwlknon 30174	class ClWWalksNOn
df-clwwlknon 30175	⊢ ClWWalksNOn = (𝑔 ∈ V ↦ (𝑣 ∈ (Vtx‘𝑔), 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝑔) ∣ (𝑤‘0) = 𝑣}))
cconngr 30273	class ConnGraph
df-conngr 30274	⊢ ConnGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣]∀𝑘 ∈ 𝑣 ∀𝑛 ∈ 𝑣 ∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝}
ceupth 30284	class EulerPaths
df-eupth 30285	⊢ EulerPaths = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝑔)𝑝 ∧ 𝑓:(0..^(♯‘𝑓))–onto→dom (iEdg‘𝑔))})
cfrgr 30345	class FriendGraph
df-frgr 30346	⊢ FriendGraph = {𝑔 ∈ USGraph ∣ [(Vtx‘𝑔) / 𝑣][(Edg‘𝑔) / 𝑒]∀𝑘 ∈ 𝑣 ∀𝑙 ∈ (𝑣 ∖ {𝑘})∃!𝑥 ∈ 𝑣 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝑒}
ax-flt 30559	⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ (𝑋 ∈ ℕ ∧ 𝑌 ∈ ℕ ∧ 𝑍 ∈ ℕ)) → ((𝑋↑𝑁) + (𝑌↑𝑁)) ≠ (𝑍↑𝑁))
cplig 30561	class Plig
df-plig 30562	⊢ 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 30576	class GrpOp
cgi 30577	class GId
cgn 30578	class inv
cgs 30579	class /𝑔
df-grpo 30580	⊢ 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 30581	⊢ GId = (𝑔 ∈ V ↦ (℩𝑢 ∈ ran 𝑔∀𝑥 ∈ ran 𝑔((𝑢𝑔𝑥) = 𝑥 ∧ (𝑥𝑔𝑢) = 𝑥)))
df-ginv 30582	⊢ inv = (𝑔 ∈ GrpOp ↦ (𝑥 ∈ ran 𝑔 ↦ (℩𝑧 ∈ ran 𝑔(𝑧𝑔𝑥) = (GId‘𝑔))))
df-gdiv 30583	⊢ /𝑔 = (𝑔 ∈ GrpOp ↦ (𝑥 ∈ ran 𝑔, 𝑦 ∈ ran 𝑔 ↦ (𝑥𝑔((inv‘𝑔)‘𝑦))))
cablo 30631	class AbelOp
df-ablo 30632	⊢ AbelOp = {𝑔 ∈ GrpOp ∣ ∀𝑥 ∈ ran 𝑔∀𝑦 ∈ ran 𝑔(𝑥𝑔𝑦) = (𝑦𝑔𝑥)}
cvc 30645	class CVecOLD
df-vc 30646	⊢ CVecOLD = {⟨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 30671	class NrmCVec
cpv 30672	class +𝑣
cba 30673	class BaseSet
cns 30674	class ·𝑠OLD
cn0v 30675	class 0vec
cnsb 30676	class −𝑣
cnmcv 30677	class normCV
cims 30678	class IndMet
df-nv 30679	⊢ NrmCVec = {⟨⟨𝑔, 𝑠⟩, 𝑛⟩ ∣ (⟨𝑔, 𝑠⟩ ∈ CVecOLD ∧ 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 30682	⊢ +𝑣 = (1st ∘ 1st )
df-ba 30683	⊢ BaseSet = (𝑥 ∈ V ↦ ran ( +𝑣 ‘𝑥))
df-sm 30684	⊢ ·𝑠OLD = (2nd ∘ 1st )
df-0v 30685	⊢ 0vec = (GId ∘ +𝑣 )
df-vs 30686	⊢ −𝑣 = ( /𝑔 ∘ +𝑣 )
df-nmcv 30687	⊢ normCV = 2nd
df-ims 30688	⊢ IndMet = (𝑢 ∈ NrmCVec ↦ ((normCV‘𝑢) ∘ ( −𝑣 ‘𝑢)))
cdip 30787	class ·𝑖OLD
df-dip 30788	⊢ ·𝑖OLD = (𝑢 ∈ NrmCVec ↦ (𝑥 ∈ (BaseSet‘𝑢), 𝑦 ∈ (BaseSet‘𝑢) ↦ (Σ𝑘 ∈ (1...4)((i↑𝑘) · (((normCV‘𝑢)‘(𝑥( +𝑣 ‘𝑢)((i↑𝑘)( ·𝑠OLD ‘𝑢)𝑦)))↑2)) / 4)))
css 30808	class SubSp
df-ssp 30809	⊢ SubSp = (𝑢 ∈ NrmCVec ↦ {𝑤 ∈ NrmCVec ∣ (( +𝑣 ‘𝑤) ⊆ ( +𝑣 ‘𝑢) ∧ ( ·𝑠OLD ‘𝑤) ⊆ ( ·𝑠OLD ‘𝑢) ∧ (normCV‘𝑤) ⊆ (normCV‘𝑢))})
clno 30827	class LnOp
cnmoo 30828	class normOpOLD
cblo 30829	class BLnOp
c0o 30830	class 0op
df-lno 30831	⊢ 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 30832	⊢ normOpOLD = (𝑢 ∈ NrmCVec, 𝑤 ∈ NrmCVec ↦ (𝑡 ∈ ((BaseSet‘𝑤) ↑m (BaseSet‘𝑢)) ↦ sup({𝑥 ∣ ∃𝑧 ∈ (BaseSet‘𝑢)(((normCV‘𝑢)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV‘𝑤)‘(𝑡‘𝑧)))}, ℝ*, < )))
df-blo 30833	⊢ BLnOp = (𝑢 ∈ NrmCVec, 𝑤 ∈ NrmCVec ↦ {𝑡 ∈ (𝑢 LnOp 𝑤) ∣ ((𝑢 normOpOLD 𝑤)‘𝑡) < +∞})
df-0o 30834	⊢ 0op = (𝑢 ∈ NrmCVec, 𝑤 ∈ NrmCVec ↦ ((BaseSet‘𝑢) × {(0vec‘𝑤)}))
caj 30835	class adj
chmo 30836	class HmOp
df-aj 30837	⊢ 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 30838	⊢ HmOp = (𝑢 ∈ NrmCVec ↦ {𝑡 ∈ dom (𝑢adj𝑢) ∣ ((𝑢adj𝑢)‘𝑡) = 𝑡})
ccphlo 30899	class CPreHilOLD
df-ph 30900	⊢ CPreHilOLD = (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 30949	class CBan
df-cbn 30950	⊢ CBan = {𝑢 ∈ NrmCVec ∣ (IndMet‘𝑢) ∈ (CMet‘(BaseSet‘𝑢))}
chlo 30972	class CHilOLD
df-hlo 30973	⊢ CHilOLD = (CBan ∩ CPreHilOLD)
The list of syntax, axioms (ax-) and definitions (df-) for the Hilbert Space Explorer starts here
chba 31006	class ℋ
cva 31007	class +ℎ
csm 31008	class ·ℎ
csp 31009	class ·ih
cno 31010	class normℎ
c0v 31011	class 0ℎ
cmv 31012	class −ℎ
ccauold 31013	class Cauchy
chli 31014	class ⇝𝑣
csh 31015	class Sℋ
cch 31016	class Cℋ
cort 31017	class ⊥
cph 31018	class +ℋ
cspn 31019	class span
chj 31020	class ∨ℋ
chsup 31021	class ∨ℋ
c0h 31022	class 0ℋ
ccm 31023	class 𝐶ℋ
cpjh 31024	class projℎ
chos 31025	class +op
chot 31026	class ·op
chod 31027	class −op
chfs 31028	class +fn
chft 31029	class ·fn
ch0o 31030	class 0hop
chio 31031	class Iop
cnop 31032	class normop
ccop 31033	class ContOp
clo 31034	class LinOp
cbo 31035	class BndLinOp
cuo 31036	class UniOp
cho 31037	class HrmOp
cnmf 31038	class normfn
cnl 31039	class null
ccnfn 31040	class ContFn
clf 31041	class LinFn
cado 31042	class adjℎ
cbr 31043	class bra
ck 31044	class ketbra
cleo 31045	class ≤op
cei 31046	class eigvec
cel 31047	class eigval
cspc 31048	class Lambda
cst 31049	class States
chst 31050	class CHStates
ccv 31051	class ⋖ℋ
cat 31052	class HAtoms
cmd 31053	class 𝑀ℋ
cdmd 31054	class 𝑀ℋ*
df-hnorm 31055	⊢ normℎ = (𝑥 ∈ dom dom ·ih ↦ (√‘(𝑥 ·ih 𝑥)))
df-hba 31056	⊢ ℋ = (BaseSet‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩)
df-h0v 31057	⊢ 0ℎ = (0vec‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩)
df-hvsub 31058	⊢ −ℎ = (𝑥 ∈ ℋ, 𝑦 ∈ ℋ ↦ (𝑥 +ℎ (-1 ·ℎ 𝑦)))
df-hlim 31059	⊢ ⇝𝑣 = {⟨𝑓, 𝑤⟩ ∣ ((𝑓:ℕ⟶ ℋ ∧ 𝑤 ∈ ℋ) ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ≥‘𝑦)(normℎ‘((𝑓‘𝑧) −ℎ 𝑤)) < 𝑥)}
df-hcau 31060	⊢ Cauchy = {𝑓 ∈ ( ℋ ↑m ℕ) ∣ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ≥‘𝑦)(normℎ‘((𝑓‘𝑦) −ℎ (𝑓‘𝑧))) < 𝑥}
ax-hilex 31086	⊢ ℋ ∈ V
ax-hfvadd 31087	⊢ +ℎ :( ℋ × ℋ)⟶ ℋ
ax-hvcom 31088	⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 +ℎ 𝐵) = (𝐵 +ℎ 𝐴))
ax-hvass 31089	⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 +ℎ 𝐵) +ℎ 𝐶) = (𝐴 +ℎ (𝐵 +ℎ 𝐶)))
ax-hv0cl 31090	⊢ 0ℎ ∈ ℋ
ax-hvaddid 31091	⊢ (𝐴 ∈ ℋ → (𝐴 +ℎ 0ℎ) = 𝐴)
ax-hfvmul 31092	⊢ ·ℎ :(ℂ × ℋ)⟶ ℋ
ax-hvmulid 31093	⊢ (𝐴 ∈ ℋ → (1 ·ℎ 𝐴) = 𝐴)
ax-hvmulass 31094	⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝐴 · 𝐵) ·ℎ 𝐶) = (𝐴 ·ℎ (𝐵 ·ℎ 𝐶)))
ax-hvdistr1 31095	⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 ·ℎ (𝐵 +ℎ 𝐶)) = ((𝐴 ·ℎ 𝐵) +ℎ (𝐴 ·ℎ 𝐶)))
ax-hvdistr2 31096	⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝐴 + 𝐵) ·ℎ 𝐶) = ((𝐴 ·ℎ 𝐶) +ℎ (𝐵 ·ℎ 𝐶)))
ax-hvmul0 31097	⊢ (𝐴 ∈ ℋ → (0 ·ℎ 𝐴) = 0ℎ)
ax-hfi 31166	⊢ ·ih :( ℋ × ℋ)⟶ℂ
ax-his1 31169	⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ih 𝐵) = (∗‘(𝐵 ·ih 𝐴)))
ax-his2 31170	⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 +ℎ 𝐵) ·ih 𝐶) = ((𝐴 ·ih 𝐶) + (𝐵 ·ih 𝐶)))
ax-his3 31171	⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 ·ℎ 𝐵) ·ih 𝐶) = (𝐴 · (𝐵 ·ih 𝐶)))
ax-his4 31172	⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → 0 < (𝐴 ·ih 𝐴))
ax-hcompl 31289	⊢ (𝐹 ∈ Cauchy → ∃𝑥 ∈ ℋ 𝐹 ⇝𝑣 𝑥)
df-sh 31294	⊢ Sℋ = {ℎ ∈ 𝒫 ℋ ∣ (0ℎ ∈ ℎ ∧ ( +ℎ “ (ℎ × ℎ)) ⊆ ℎ ∧ ( ·ℎ “ (ℂ × ℎ)) ⊆ ℎ)}
df-ch 31308	⊢ Cℋ = {ℎ ∈ Sℋ ∣ ( ⇝𝑣 “ (ℎ ↑m ℕ)) ⊆ ℎ}
df-oc 31339	⊢ ⊥ = (𝑥 ∈ 𝒫 ℋ ↦ {𝑦 ∈ ℋ ∣ ∀𝑧 ∈ 𝑥 (𝑦 ·ih 𝑧) = 0})
df-ch0 31340	⊢ 0ℋ = {0ℎ}
df-shs 31395	⊢ +ℋ = (𝑥 ∈ Sℋ , 𝑦 ∈ Sℋ ↦ ( +ℎ “ (𝑥 × 𝑦)))
df-span 31396	⊢ span = (𝑥 ∈ 𝒫 ℋ ↦ ∩ {𝑦 ∈ Sℋ ∣ 𝑥 ⊆ 𝑦})
df-chj 31397	⊢ ∨ℋ = (𝑥 ∈ 𝒫 ℋ, 𝑦 ∈ 𝒫 ℋ ↦ (⊥‘(⊥‘(𝑥 ∪ 𝑦))))
df-chsup 31398	⊢ ∨ℋ = (𝑥 ∈ 𝒫 𝒫 ℋ ↦ (⊥‘(⊥‘∪ 𝑥)))
df-pjh 31482	⊢ projℎ = (ℎ ∈ Cℋ ↦ (𝑥 ∈ ℋ ↦ (℩𝑧 ∈ ℎ ∃𝑦 ∈ (⊥‘ℎ)𝑥 = (𝑧 +ℎ 𝑦))))
df-cm 31670	⊢ 𝐶ℋ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ Cℋ ∧ 𝑦 ∈ Cℋ ) ∧ 𝑥 = ((𝑥 ∩ 𝑦) ∨ℋ (𝑥 ∩ (⊥‘𝑦))))}
df-hosum 31817	⊢ +op = (𝑓 ∈ ( ℋ ↑m ℋ), 𝑔 ∈ ( ℋ ↑m ℋ) ↦ (𝑥 ∈ ℋ ↦ ((𝑓‘𝑥) +ℎ (𝑔‘𝑥))))
df-homul 31818	⊢ ·op = (𝑓 ∈ ℂ, 𝑔 ∈ ( ℋ ↑m ℋ) ↦ (𝑥 ∈ ℋ ↦ (𝑓 ·ℎ (𝑔‘𝑥))))
df-hodif 31819	⊢ −op = (𝑓 ∈ ( ℋ ↑m ℋ), 𝑔 ∈ ( ℋ ↑m ℋ) ↦ (𝑥 ∈ ℋ ↦ ((𝑓‘𝑥) −ℎ (𝑔‘𝑥))))
df-hfsum 31820	⊢ +fn = (𝑓 ∈ (ℂ ↑m ℋ), 𝑔 ∈ (ℂ ↑m ℋ) ↦ (𝑥 ∈ ℋ ↦ ((𝑓‘𝑥) + (𝑔‘𝑥))))
df-hfmul 31821	⊢ ·fn = (𝑓 ∈ ℂ, 𝑔 ∈ (ℂ ↑m ℋ) ↦ (𝑥 ∈ ℋ ↦ (𝑓 · (𝑔‘𝑥))))
df-h0op 31835	⊢ 0hop = (projℎ‘0ℋ)
df-iop 31836	⊢ Iop = (projℎ‘ ℋ)
df-nmop 31926	⊢ normop = (𝑡 ∈ ( ℋ ↑m ℋ) ↦ sup({𝑥 ∣ ∃𝑧 ∈ ℋ ((normℎ‘𝑧) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑡‘𝑧)))}, ℝ*, < ))
df-cnop 31927	⊢ ContOp = {𝑡 ∈ ( ℋ ↑m ℋ) ∣ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ ℋ ((normℎ‘(𝑤 −ℎ 𝑥)) < 𝑧 → (normℎ‘((𝑡‘𝑤) −ℎ (𝑡‘𝑥))) < 𝑦)}
df-lnop 31928	⊢ 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 31929	⊢ BndLinOp = {𝑡 ∈ LinOp ∣ (normop‘𝑡) < +∞}
df-unop 31930	⊢ UniOp = {𝑡 ∣ (𝑡: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑡‘𝑥) ·ih (𝑡‘𝑦)) = (𝑥 ·ih 𝑦))}
df-hmop 31931	⊢ HrmOp = {𝑡 ∈ ( ℋ ↑m ℋ) ∣ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑡‘𝑦)) = ((𝑡‘𝑥) ·ih 𝑦)}
df-nmfn 31932	⊢ normfn = (𝑡 ∈ (ℂ ↑m ℋ) ↦ sup({𝑥 ∣ ∃𝑧 ∈ ℋ ((normℎ‘𝑧) ≤ 1 ∧ 𝑥 = (abs‘(𝑡‘𝑧)))}, ℝ*, < ))
df-nlfn 31933	⊢ null = (𝑡 ∈ (ℂ ↑m ℋ) ↦ (◡𝑡 “ {0}))
df-cnfn 31934	⊢ ContFn = {𝑡 ∈ (ℂ ↑m ℋ) ∣ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ ℋ ((normℎ‘(𝑤 −ℎ 𝑥)) < 𝑧 → (abs‘((𝑡‘𝑤) − (𝑡‘𝑥))) < 𝑦)}
df-lnfn 31935	⊢ LinFn = {𝑡 ∈ (ℂ ↑m ℋ) ∣ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ (𝑡‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑥 · (𝑡‘𝑦)) + (𝑡‘𝑧))}
df-adjh 31936	⊢ adjℎ = {⟨𝑡, 𝑢⟩ ∣ (𝑡: ℋ⟶ ℋ ∧ 𝑢: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑡‘𝑥) ·ih 𝑦) = (𝑥 ·ih (𝑢‘𝑦)))}
df-bra 31937	⊢ bra = (𝑥 ∈ ℋ ↦ (𝑦 ∈ ℋ ↦ (𝑦 ·ih 𝑥)))
df-kb 31938	⊢ ketbra = (𝑥 ∈ ℋ, 𝑦 ∈ ℋ ↦ (𝑧 ∈ ℋ ↦ ((𝑧 ·ih 𝑦) ·ℎ 𝑥)))
df-leop 31939	⊢ ≤op = {⟨𝑡, 𝑢⟩ ∣ ((𝑢 −op 𝑡) ∈ HrmOp ∧ ∀𝑥 ∈ ℋ 0 ≤ (((𝑢 −op 𝑡)‘𝑥) ·ih 𝑥))}
df-eigvec 31940	⊢ eigvec = (𝑡 ∈ ( ℋ ↑m ℋ) ↦ {𝑥 ∈ ( ℋ ∖ 0ℋ) ∣ ∃𝑧 ∈ ℂ (𝑡‘𝑥) = (𝑧 ·ℎ 𝑥)})
df-eigval 31941	⊢ eigval = (𝑡 ∈ ( ℋ ↑m ℋ) ↦ (𝑥 ∈ (eigvec‘𝑡) ↦ (((𝑡‘𝑥) ·ih 𝑥) / ((normℎ‘𝑥)↑2))))
df-spec 31942	⊢ Lambda = (𝑡 ∈ ( ℋ ↑m ℋ) ↦ {𝑥 ∈ ℂ ∣ ¬ (𝑡 −op (𝑥 ·op ( I ↾ ℋ))): ℋ–1-1→ ℋ})
df-st 32298	⊢ States = {𝑓 ∈ ((0[,]1) ↑m Cℋ ) ∣ ((𝑓‘ ℋ) = 1 ∧ ∀𝑥 ∈ Cℋ ∀𝑦 ∈ Cℋ (𝑥 ⊆ (⊥‘𝑦) → (𝑓‘(𝑥 ∨ℋ 𝑦)) = ((𝑓‘𝑥) + (𝑓‘𝑦))))}
df-hst 32299	⊢ CHStates = {𝑓 ∈ ( ℋ ↑m Cℋ ) ∣ ((normℎ‘(𝑓‘ ℋ)) = 1 ∧ ∀𝑥 ∈ Cℋ ∀𝑦 ∈ Cℋ (𝑥 ⊆ (⊥‘𝑦) → (((𝑓‘𝑥) ·ih (𝑓‘𝑦)) = 0 ∧ (𝑓‘(𝑥 ∨ℋ 𝑦)) = ((𝑓‘𝑥) +ℎ (𝑓‘𝑦)))))}
df-cv 32366	⊢ ⋖ℋ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ Cℋ ∧ 𝑦 ∈ Cℋ ) ∧ (𝑥 ⊊ 𝑦 ∧ ¬ ∃𝑧 ∈ Cℋ (𝑥 ⊊ 𝑧 ∧ 𝑧 ⊊ 𝑦)))}
df-md 32367	⊢ 𝑀ℋ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ Cℋ ∧ 𝑦 ∈ Cℋ ) ∧ ∀𝑧 ∈ Cℋ (𝑧 ⊆ 𝑦 → ((𝑧 ∨ℋ 𝑥) ∩ 𝑦) = (𝑧 ∨ℋ (𝑥 ∩ 𝑦))))}
df-dmd 32368	⊢ 𝑀ℋ* = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ Cℋ ∧ 𝑦 ∈ Cℋ ) ∧ ∀𝑧 ∈ Cℋ (𝑦 ⊆ 𝑧 → ((𝑧 ∩ 𝑥) ∨ℋ 𝑦) = (𝑧 ∩ (𝑥 ∨ℋ 𝑦))))}
df-at 32425	⊢ HAtoms = {𝑥 ∈ Cℋ ∣ 0ℋ ⋖ℋ 𝑥}
The list of syntax, axioms (ax-) and definitions (df-) for the User Mathboxes starts here
w2reu 32563	wff ∃!𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵𝜑
df-2reu 32564	⊢ (∃!𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵𝜑 ↔ (∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ ∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑))
cind 32939	class 𝟭
df-ind 32940	⊢ 𝟭 = (𝑜 ∈ V ↦ (𝑎 ∈ 𝒫 𝑜 ↦ (𝑥 ∈ 𝑜 ↦ if(𝑥 ∈ 𝑎, 1, 0))))
cdp2 32962	class _𝐴𝐵
df-dp2 32963	⊢ _𝐴𝐵 = (𝐴 + (𝐵 / ;10))
cdp 32979	class .
df-dp 32980	⊢ . = (𝑥 ∈ ℕ0, 𝑦 ∈ ℝ ↦ _𝑥𝑦)
cxdiv 33008	class /𝑒
df-xdiv 33009	⊢ /𝑒 = (𝑥 ∈ ℝ*, 𝑦 ∈ (ℝ ∖ {0}) ↦ (℩𝑧 ∈ ℝ* (𝑦 ·e 𝑧) = 𝑥))
cmnt 33070	class Monot
cmgc 33071	class MGalConn
df-mnt 33072	⊢ 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 33073	⊢ MGalConn = (𝑣 ∈ V, 𝑤 ∈ V ↦ ⦋(Base‘𝑣) / 𝑎⦌⦋(Base‘𝑤) / 𝑏⦌{⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑏 ↑m 𝑎) ∧ 𝑔 ∈ (𝑎 ↑m 𝑏)) ∧ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 ((𝑓‘𝑥)(le‘𝑤)𝑦 ↔ 𝑥(le‘𝑣)(𝑔‘𝑦)))})
ax-xrssca 33096	⊢ ℝfld = (Scalar‘ℝ*𝑠)
ax-xrsvsca 33097	⊢ ·e = ( ·𝑠 ‘ℝ*𝑠)
ctocyc 33199	class toCyc
df-tocyc 33200	⊢ toCyc = (𝑑 ∈ V ↦ (𝑤 ∈ {𝑢 ∈ Word 𝑑 ∣ 𝑢:dom 𝑢–1-1→𝑑} ↦ (( I ↾ (𝑑 ∖ ran 𝑤)) ∪ ((𝑤 cyclShift 1) ∘ ◡𝑤))))
csgns 33251	class sgns
df-sgns 33252	⊢ sgns = (𝑟 ∈ V ↦ (𝑥 ∈ (Base‘𝑟) ↦ if(𝑥 = (0g‘𝑟), 0, if((0g‘𝑟)(lt‘𝑟)𝑥, 1, -1))))
cfxp 33256	class FixPts
df-fxp 33257	⊢ FixPts = (𝑏 ∈ V, 𝑎 ∈ V ↦ {𝑥 ∈ 𝑏 ∣ ∀𝑝 ∈ dom dom 𝑎(𝑝𝑎𝑥) = 𝑥})
cinftm 33269	class ⋘
carchi 33270	class Archi
df-inftm 33271	⊢ ⋘ = (𝑤 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (Base‘𝑤) ∧ 𝑦 ∈ (Base‘𝑤)) ∧ ((0g‘𝑤)(lt‘𝑤)𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛(.g‘𝑤)𝑥)(lt‘𝑤)𝑦))})
df-archi 33272	⊢ Archi = {𝑤 ∣ (⋘‘𝑤) = ∅}
cslmd 33293	class SLMod
df-slmd 33294	⊢ 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)) ∧ ((1r‘𝑓)𝑠𝑤) = 𝑤 ∧ ((0g‘𝑓)𝑠𝑤) = (0g‘𝑔))))}
cerl 33346	class ~RL
crloc 33347	class RLocal
df-erl 33348	⊢ ~RL = (𝑟 ∈ V, 𝑠 ∈ V ↦ ⦋(.r‘𝑟) / 𝑥⦌⦋((Base‘𝑟) × 𝑠) / 𝑤⦌{⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ 𝑤 ∧ 𝑏 ∈ 𝑤) ∧ ∃𝑡 ∈ 𝑠 (𝑡𝑥(((1st ‘𝑎)𝑥(2nd ‘𝑏))(-g‘𝑟)((1st ‘𝑏)𝑥(2nd ‘𝑎)))) = (0g‘𝑟))})
df-rloc 33349	⊢ RLocal = (𝑟 ∈ V, 𝑠 ∈ V ↦ ⦋(.r‘𝑟) / 𝑥⦌⦋((Base‘𝑟) × 𝑠) / 𝑤⦌((({⟨(Base‘ndx), 𝑤⟩, ⟨(+g‘ndx), (𝑎 ∈ 𝑤, 𝑏 ∈ 𝑤 ↦ ⟨(((1st ‘𝑎)𝑥(2nd ‘𝑏))(+g‘𝑟)((1st ‘𝑏)𝑥(2nd ‘𝑎))), ((2nd ‘𝑎)𝑥(2nd ‘𝑏))⟩)⟩, ⟨(.r‘ndx), (𝑎 ∈ 𝑤, 𝑏 ∈ 𝑤 ↦ ⟨((1st ‘𝑎)𝑥(1st ‘𝑏)), ((2nd ‘𝑎)𝑥(2nd ‘𝑏))⟩)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑟)⟩, ⟨( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑟)), 𝑎 ∈ 𝑤 ↦ ⟨(𝑘( ·𝑠 ‘𝑟)(1st ‘𝑎)), (2nd ‘𝑎)⟩)⟩, ⟨(·𝑖‘ndx), ∅⟩}) ∪ {⟨(TopSet‘ndx), ((TopSet‘𝑟) ×t ((TopSet‘𝑟) ↾t 𝑠))⟩, ⟨(le‘ndx), {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ 𝑤 ∧ 𝑏 ∈ 𝑤) ∧ ((1st ‘𝑎)𝑥(2nd ‘𝑏))(le‘𝑟)((1st ‘𝑏)𝑥(2nd ‘𝑎)))}⟩, ⟨(dist‘ndx), (𝑎 ∈ 𝑤, 𝑏 ∈ 𝑤 ↦ (((1st ‘𝑎)𝑥(2nd ‘𝑏))(dist‘𝑟)((1st ‘𝑏)𝑥(2nd ‘𝑎))))⟩}) /s (𝑟 ~RL 𝑠)))
ceuf 33381	class EuclF
df-euf 33382	⊢ EuclF = Slot ;21
cedom 33385	class EDomn
df-edom 33386	⊢ EDomn = {𝑑 ∈ IDomn ∣ [(EuclF‘𝑑) / 𝑒][(Base‘𝑑) / 𝑣](Fun 𝑒 ∧ (𝑒 “ (𝑣 ∖ {(0g‘𝑑)})) ⊆ (0[,)+∞) ∧ ∀𝑎 ∈ 𝑣 ∀𝑏 ∈ (𝑣 ∖ {(0g‘𝑑)})∃𝑞 ∈ 𝑣 ∃𝑟 ∈ 𝑣 (𝑎 = ((𝑏(.r‘𝑑)𝑞)(+g‘𝑑)𝑟) ∧ (𝑟 = (0g‘𝑑) ∨ (𝑒‘𝑟) < (𝑒‘𝑏))))}
cfrac 33395	class Frac
df-frac 33396	⊢ Frac = (𝑟 ∈ V ↦ (𝑟 RLocal (RLReg‘𝑟)))
cfldgen 33403	class fldGen
df-fldgen 33404	⊢ fldGen = (𝑓 ∈ V, 𝑠 ∈ V ↦ ∩ {𝑎 ∈ (SubDRing‘𝑓) ∣ 𝑠 ⊆ 𝑎})
cresv 33418	class ↾v
df-resv 33419	⊢ ↾v = (𝑤 ∈ V, 𝑥 ∈ V ↦ if((Base‘(Scalar‘𝑤)) ⊆ 𝑥, 𝑤, (𝑤 sSet ⟨(Scalar‘ndx), ((Scalar‘𝑤) ↾s 𝑥)⟩)))
cprmidl 33527	class PrmIdeal
df-prmidl 33528	⊢ 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 33551	class MaxIdeal
df-mxidl 33552	⊢ MaxIdeal = (𝑟 ∈ Ring ↦ {𝑖 ∈ (LIdeal‘𝑟) ∣ (𝑖 ≠ (Base‘𝑟) ∧ ∀𝑗 ∈ (LIdeal‘𝑟)(𝑖 ⊆ 𝑗 → (𝑗 = 𝑖 ∨ 𝑗 = (Base‘𝑟))))})
cidlsrg 33592	class IDLsrg
df-idlsrg 33593	⊢ 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 33630	class UFD
df-ufd 33631	⊢ UFD = {𝑟 ∈ IDomn ∣ ∀𝑖 ∈ ((PrmIdeal‘𝑟) ∖ {{(0g‘𝑟)}})(𝑖 ∩ (RPrime‘𝑟)) ≠ ∅}
cextv 33705	class extendVars
df-extv 33706	⊢ extendVars = (𝑖 ∈ V, 𝑟 ∈ V ↦ (𝑎 ∈ 𝑖 ↦ (𝑓 ∈ (Base‘((𝑖 ∖ {𝑎}) mPoly 𝑟)) ↦ (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝑖) ∣ ℎ finSupp 0} ↦ if((𝑥‘𝑎) = 0, (𝑓‘(𝑥 ↾ (𝑖 ∖ {𝑎}))), (0g‘𝑟))))))
csply 33731	class SymPoly
cesply 33732	class eSymPoly
df-sply 33733	⊢ SymPoly = (𝑖 ∈ V, 𝑟 ∈ V ↦ ((Base‘(𝑖 mPoly 𝑟))FixPts(𝑑 ∈ (Base‘(SymGrp‘𝑖)), 𝑓 ∈ (Base‘(𝑖 mPoly 𝑟)) ↦ (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝑖) ∣ ℎ finSupp 0} ↦ (𝑓‘(𝑥 ∘ 𝑑))))))
df-esply 33734	⊢ eSymPoly = (𝑖 ∈ V, 𝑟 ∈ V ↦ (𝑘 ∈ ℕ0 ↦ ((ℤRHom‘𝑟) ∘ ((𝟭‘{ℎ ∈ (ℕ0 ↑m 𝑖) ∣ ℎ finSupp 0})‘((𝟭‘𝑖) “ {𝑐 ∈ 𝒫 𝑖 ∣ (♯‘𝑐) = 𝑘})))))
cldim 33775	class dim
df-dim 33776	⊢ dim = (𝑓 ∈ V ↦ ∪ (♯ “ (LBasis‘𝑓)))
cfldext 33815	class /FldExt
cfinext 33816	class /FinExt
cextdg 33817	class [:]
df-fldext 33818	⊢ /FldExt = {⟨𝑒, 𝑓⟩ ∣ ((𝑒 ∈ Field ∧ 𝑓 ∈ Field) ∧ (𝑓 = (𝑒 ↾s (Base‘𝑓)) ∧ (Base‘𝑓) ∈ (SubRing‘𝑒)))}
df-extdg 33819	⊢ [:] = (𝑒 ∈ V, 𝑓 ∈ (/FldExt “ {𝑒}) ↦ (dim‘((subringAlg ‘𝑒)‘(Base‘𝑓))))
df-finext 33820	⊢ /FinExt = {⟨𝑒, 𝑓⟩ ∣ (𝑒/FldExt𝑓 ∧ (𝑒[:]𝑓) ∈ ℕ0)}
cirng 33860	class IntgRing
df-irng 33861	⊢ IntgRing = (𝑟 ∈ V, 𝑠 ∈ V ↦ ∪ 𝑓 ∈ (Monic1p‘(𝑟 ↾s 𝑠))(◡((𝑟 evalSub1 𝑠)‘𝑓) “ {(0g‘𝑟)}))
calgext 33872	class /AlgExt
df-algext 33873	⊢ /AlgExt = {⟨𝑒, 𝑓⟩ ∣ (𝑒/FldExt𝑓 ∧ (𝑒 IntgRing (Base‘𝑓)) = (Base‘𝑒))}
cminply 33876	class minPoly
df-minply 33877	⊢ minPoly = (𝑒 ∈ V, 𝑓 ∈ V ↦ (𝑥 ∈ (Base‘𝑒) ↦ ((idlGen1p‘(𝑒 ↾s 𝑓))‘{𝑝 ∈ dom (𝑒 evalSub1 𝑓) ∣ (((𝑒 evalSub1 𝑓)‘𝑝)‘𝑥) = (0g‘𝑒)})))
cconstr 33906	class Constr
df-constr 33907	⊢ 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 33970	class subMat1
df-smat 33971	⊢ 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 33988	class litMat
df-lmat 33989	⊢ litMat = (𝑚 ∈ V ↦ (𝑖 ∈ (1...(♯‘𝑚)), 𝑗 ∈ (1...(♯‘(𝑚‘0))) ↦ ((𝑚‘(𝑖 − 1))‘(𝑗 − 1))))
ccref 34019	class CovHasRef𝐴
df-cref 34020	⊢ CovHasRef𝐴 = {𝑗 ∈ Top ∣ ∀𝑦 ∈ 𝒫 𝑗(∪ 𝑗 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑗 ∩ 𝐴)𝑧Ref𝑦)}
cldlf 34029	class Ldlf
df-ldlf 34030	⊢ Ldlf = CovHasRef{𝑥 ∣ 𝑥 ≼ ω}
cpcmp 34032	class Paracomp
df-pcmp 34033	⊢ Paracomp = {𝑗 ∣ 𝑗 ∈ CovHasRef(LocFin‘𝑗)}
crspec 34039	class Spec
df-rspec 34040	⊢ Spec = (𝑟 ∈ Ring ↦ ((IDLsrg‘𝑟) ↾s (PrmIdeal‘𝑟)))
cmetid 34063	class ~Met
cpstm 34064	class pstoMet
df-metid 34065	⊢ ~Met = (𝑑 ∈ ∪ ran PsMet ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ dom dom 𝑑 ∧ 𝑦 ∈ dom dom 𝑑) ∧ (𝑥𝑑𝑦) = 0)})
df-pstm 34066	⊢ pstoMet = (𝑑 ∈ ∪ ran PsMet ↦ (𝑎 ∈ (dom dom 𝑑 / (~Met‘𝑑)), 𝑏 ∈ (dom dom 𝑑 / (~Met‘𝑑)) ↦ ∪ {𝑧 ∣ ∃𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝑧 = (𝑥𝑑𝑦)}))
chcmp 34133	class HCmp
df-hcmp 34134	⊢ HCmp = {⟨𝑢, 𝑤⟩ ∣ ((𝑢 ∈ ∪ ran UnifOn ∧ 𝑤 ∈ CUnifSp) ∧ ((UnifSt‘𝑤) ↾t dom ∪ 𝑢) = 𝑢 ∧ ((cls‘(TopOpen‘𝑤))‘dom ∪ 𝑢) = (Base‘𝑤))}
cqqh 34147	class ℚHom
df-qqh 34148	⊢ ℚHom = (𝑟 ∈ V ↦ ran (𝑥 ∈ ℤ, 𝑦 ∈ (◡(ℤRHom‘𝑟) “ (Unit‘𝑟)) ↦ ⟨(𝑥 / 𝑦), (((ℤRHom‘𝑟)‘𝑥)(/r‘𝑟)((ℤRHom‘𝑟)‘𝑦))⟩))
crrh 34170	class ℝHom
crrext 34171	class ℝExt
df-rrh 34172	⊢ ℝHom = (𝑟 ∈ V ↦ (((topGen‘ran (,))CnExt(TopOpen‘𝑟))‘(ℚHom‘𝑟)))
df-rrext 34176	⊢ ℝExt = {𝑟 ∈ (NrmRing ∩ DivRing) ∣ (((ℤMod‘𝑟) ∈ NrmMod ∧ (chr‘𝑟) = 0) ∧ (𝑟 ∈ CUnifSp ∧ (UnifSt‘𝑟) = (metUnif‘((dist‘𝑟) ↾ ((Base‘𝑟) × (Base‘𝑟))))))}
cxrh 34193	class ℝ*Hom
df-xrh 34194	⊢ 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 34199	class ManTop
df-mntop 34200	⊢ ManTop = {⟨𝑛, 𝑗⟩ ∣ (𝑛 ∈ ℕ0 ∧ (𝑗 ∈ 2ndω ∧ 𝑗 ∈ Haus ∧ 𝑗 ∈ Locally [(TopOpen‘(𝔼hil‘𝑛))] ≃ ))}
cesum 34204	class Σ*𝑘 ∈ 𝐴𝐵
df-esum 34205	⊢ Σ*𝑘 ∈ 𝐴𝐵 = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵))
cofc 34272	class ∘f/c 𝑅
df-ofc 34273	⊢ ∘f/c 𝑅 = (𝑓 ∈ V, 𝑐 ∈ V ↦ (𝑥 ∈ dom 𝑓 ↦ ((𝑓‘𝑥)𝑅𝑐)))
csiga 34285	class sigAlgebra
df-siga 34286	⊢ sigAlgebra = (𝑜 ∈ V ↦ {𝑠 ∣ (𝑠 ⊆ 𝒫 𝑜 ∧ (𝑜 ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑜 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑠)))})
csigagen 34315	class sigaGen
df-sigagen 34316	⊢ sigaGen = (𝑥 ∈ V ↦ ∩ {𝑠 ∈ (sigAlgebra‘∪ 𝑥) ∣ 𝑥 ⊆ 𝑠})
cbrsiga 34358	class 𝔅ℝ
df-brsiga 34359	⊢ 𝔅ℝ = (sigaGen‘(topGen‘ran (,)))
csx 34365	class ×s
df-sx 34366	⊢ ×s = (𝑠 ∈ V, 𝑡 ∈ V ↦ (sigaGen‘ran (𝑥 ∈ 𝑠, 𝑦 ∈ 𝑡 ↦ (𝑥 × 𝑦))))
cmeas 34372	class measures
df-meas 34373	⊢ 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 34409	class δ
df-dde 34410	⊢ δ = (𝑎 ∈ 𝒫 ℝ ↦ if(0 ∈ 𝑎, 1, 0))
cae 34414	class a.e.
cfae 34415	class ~ a.e.
df-ae 34416	⊢ a.e. = {⟨𝑎, 𝑚⟩ ∣ (𝑚‘(∪ dom 𝑚 ∖ 𝑎)) = 0}
df-fae 34422	⊢ ~ 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 34426	class MblFnM
df-mbfm 34427	⊢ MblFnM = (𝑠 ∈ ∪ ran sigAlgebra, 𝑡 ∈ ∪ ran sigAlgebra ↦ {𝑓 ∈ (∪ 𝑡 ↑m ∪ 𝑠) ∣ ∀𝑥 ∈ 𝑡 (◡𝑓 “ 𝑥) ∈ 𝑠})
coms 34468	class toOMeas
df-oms 34469	⊢ 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 34478	class toCaraSiga
df-carsg 34479	⊢ toCaraSiga = (𝑚 ∈ V ↦ {𝑎 ∈ 𝒫 ∪ dom 𝑚 ∣ ∀𝑒 ∈ 𝒫 ∪ dom 𝑚((𝑚‘(𝑒 ∩ 𝑎)) +𝑒 (𝑚‘(𝑒 ∖ 𝑎))) = (𝑚‘𝑒)})
citgm 34504	class itgm
csitm 34505	class sitm
csitg 34506	class sitg
df-sitg 34507	⊢ sitg = (𝑤 ∈ V, 𝑚 ∈ ∪ ran measures ↦ (𝑓 ∈ {𝑔 ∈ (dom 𝑚MblFnM(sigaGen‘(TopOpen‘𝑤))) ∣ (ran 𝑔 ∈ Fin ∧ ∀𝑥 ∈ (ran 𝑔 ∖ {(0g‘𝑤)})(𝑚‘(◡𝑔 “ {𝑥})) ∈ (0[,)+∞))} ↦ (𝑤 Σg (𝑥 ∈ (ran 𝑓 ∖ {(0g‘𝑤)}) ↦ (((ℝHom‘(Scalar‘𝑤))‘(𝑚‘(◡𝑓 “ {𝑥})))( ·𝑠 ‘𝑤)𝑥)))))
df-sitm 34508	⊢ sitm = (𝑤 ∈ V, 𝑚 ∈ ∪ ran measures ↦ (𝑓 ∈ dom (𝑤sitg𝑚), 𝑔 ∈ dom (𝑤sitg𝑚) ↦ (((ℝ*𝑠 ↾s (0[,]+∞))sitg𝑚)‘(𝑓 ∘f (dist‘𝑤)𝑔))))
df-itgm 34530	⊢ itgm = (𝑤 ∈ V, 𝑚 ∈ ∪ ran measures ↦ (((metUnif‘(𝑤sitm𝑚))CnExt(UnifSt‘𝑤))‘(𝑤sitg𝑚)))
csseq 34560	class seqstr
df-sseq 34561	⊢ seqstr = (𝑚 ∈ V, 𝑓 ∈ V ↦ (𝑚 ∪ (lastS ∘ seq(♯‘𝑚)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝑓‘𝑥)”⟩)), (ℕ0 × {(𝑚 ++ ⟨“(𝑓‘𝑚)”⟩)})))))
cfib 34573	class Fibci
df-fib 34574	⊢ Fibci = (⟨“01”⟩seqstr(𝑤 ∈ (Word ℕ0 ∩ (◡♯ “ (ℤ≥‘2))) ↦ ((𝑤‘((♯‘𝑤) − 2)) + (𝑤‘((♯‘𝑤) − 1)))))
cprb 34584	class Prob
df-prob 34585	⊢ Prob = {𝑝 ∈ ∪ ran measures ∣ (𝑝‘∪ dom 𝑝) = 1}
ccprob 34608	class cprob
df-cndprob 34609	⊢ cprob = (𝑝 ∈ Prob ↦ (𝑎 ∈ dom 𝑝, 𝑏 ∈ dom 𝑝 ↦ ((𝑝‘(𝑎 ∩ 𝑏)) / (𝑝‘𝑏))))
crrv 34617	class rRndVar
df-rrv 34618	⊢ rRndVar = (𝑝 ∈ Prob ↦ (dom 𝑝MblFnM𝔅ℝ))
corvc 34633	class ∘RV/𝑐𝑅
df-orvc 34634	⊢ ∘RV/𝑐𝑅 = (𝑥 ∈ {𝑥 ∣ Fun 𝑥}, 𝑎 ∈ V ↦ (◡𝑥 “ {𝑦 ∣ 𝑦𝑅𝑎}))
crepr 34785	class repr
df-repr 34786	⊢ repr = (𝑠 ∈ ℕ0 ↦ (𝑏 ∈ 𝒫 ℕ, 𝑚 ∈ ℤ ↦ {𝑐 ∈ (𝑏 ↑m (0..^𝑠)) ∣ Σ𝑎 ∈ (0..^𝑠)(𝑐‘𝑎) = 𝑚}))
cvts 34812	class vts
df-vts 34813	⊢ vts = (𝑙 ∈ (ℂ ↑m ℕ), 𝑛 ∈ ℕ0 ↦ (𝑥 ∈ ℂ ↦ Σ𝑎 ∈ (1...𝑛)((𝑙‘𝑎) · (exp‘((i · (2 · π)) · (𝑎 · 𝑥))))))
ax-hgt749 34821	⊢ ∀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 34822	⊢ ∀𝑥 ∈ ℝ+ (ψ‘𝑥) < ((1._0_3_8_83) · 𝑥)
ax-ros336 34823	⊢ ∀𝑥 ∈ ℝ+ ((ψ‘𝑥) − (θ‘𝑥)) < ((1._4_2_62) · (√‘𝑥))
cstrkg2d 34841	class TarskiG2D
df-trkg2d 34842	⊢ TarskiG2D = {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 34846	class AFS
df-afs 34847	⊢ 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 34851	class leftpad
df-lpad 34852	⊢ leftpad = (𝑐 ∈ V, 𝑤 ∈ V ↦ (𝑙 ∈ ℕ0 ↦ (((0..^(𝑙 − (♯‘𝑤))) × {𝑐}) ++ 𝑤)))
w-bnj17 34862	wff (𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃)
df-bnj17 34863	⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ ((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃))
c-bnj14 34864	class pred(𝑋, 𝐴, 𝑅)
df-bnj14 34865	⊢ pred(𝑋, 𝐴, 𝑅) = {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋}
w-bnj13 34866	wff 𝑅 Se 𝐴
df-bnj13 34867	⊢ (𝑅 Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 pred(𝑥, 𝐴, 𝑅) ∈ V)
w-bnj15 34868	wff 𝑅 FrSe 𝐴
df-bnj15 34869	⊢ (𝑅 FrSe 𝐴 ↔ (𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴))
c-bnj18 34870	class trCl(𝑋, 𝐴, 𝑅)
df-bnj18 34871	⊢ 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 34872	wff TrFo(𝐵, 𝐴, 𝑅)
df-bnj19 34873	⊢ ( TrFo(𝐵, 𝐴, 𝑅) ↔ ∀𝑥 ∈ 𝐵 pred(𝑥, 𝐴, 𝑅) ⊆ 𝐵)
ax-regs 35301	⊢ (∃𝑥𝜑 → ∃𝑦(∀𝑥(𝑥 = 𝑦 → 𝜑) ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ ∀𝑥(𝑥 = 𝑧 → 𝜑))))
cacycgr 35355	class AcyclicGraph
df-acycgr 35356	⊢ AcyclicGraph = {𝑔 ∣ ¬ ∃𝑓∃𝑝(𝑓(Cycles‘𝑔)𝑝 ∧ 𝑓 ≠ ∅)}
ax-7d 35372	⊢ (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑)
ax-8d 35373	⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 → 𝑦 = 𝑧))
ax-9d1 35374	⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑥
ax-9d2 35375	⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦
ax-10d 35376	⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
ax-11d 35377	⊢ (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
cretr 35430	class Retr
df-retr 35431	⊢ Retr = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ {𝑟 ∈ (𝑗 Cn 𝑘) ∣ ∃𝑠 ∈ (𝑘 Cn 𝑗)((𝑟 ∘ 𝑠)(𝑗 Htpy 𝑗)( I ↾ ∪ 𝑗)) ≠ ∅})
cpconn 35432	class PConn
csconn 35433	class SConn
df-pconn 35434	⊢ PConn = {𝑗 ∈ Top ∣ ∀𝑥 ∈ ∪ 𝑗∀𝑦 ∈ ∪ 𝑗∃𝑓 ∈ (II Cn 𝑗)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)}
df-sconn 35435	⊢ SConn = {𝑗 ∈ PConn ∣ ∀𝑓 ∈ (II Cn 𝑗)((𝑓‘0) = (𝑓‘1) → 𝑓( ≃ph‘𝑗)((0[,]1) × {(𝑓‘0)}))}
ccvm 35468	class CovMap
df-cvm 35469	⊢ 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 35546	class ∈𝑔
cgna 35547	class ⊼𝑔
cgol 35548	class ∀𝑔𝑁𝑈
csat 35549	class Sat
cfmla 35550	class Fmla
csate 35551	class Sat∈
cprv 35552	class ⊧
df-goel 35553	⊢ ∈𝑔 = (𝑥 ∈ (ω × ω) ↦ ⟨∅, 𝑥⟩)
df-gona 35554	⊢ ⊼𝑔 = (𝑥 ∈ (V × V) ↦ ⟨1o, 𝑥⟩)
df-goal 35555	⊢ ∀𝑔𝑁𝑈 = ⟨2o, ⟨𝑁, 𝑈⟩⟩
df-sat 35556	⊢ Sat = (𝑚 ∈ V, 𝑒 ∈ V ↦ (rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 (𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∧ 𝑦 = ((𝑚 ↑m ω) ∖ ((2nd ‘𝑢) ∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑚 ↑m ω) ∣ ∀𝑧 ∈ 𝑚 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}))})), {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑚 ↑m ω) ∣ (𝑎‘𝑖)𝑒(𝑎‘𝑗)})}) ↾ suc ω))
df-sate 35557	⊢ Sat∈ = (𝑚 ∈ V, 𝑢 ∈ V ↦ (((𝑚 Sat ( E ∩ (𝑚 × 𝑚)))‘ω)‘𝑢))
df-fmla 35558	⊢ Fmla = (𝑛 ∈ suc ω ↦ dom ((∅ Sat ∅)‘𝑛))
df-prv 35559	⊢ ⊧ = {⟨𝑚, 𝑢⟩ ∣ (𝑚 Sat∈ 𝑢) = (𝑚 ↑m ω)}
cgon 35645	class ¬𝑔𝑈
cgoa 35646	class ∧𝑔
cgoi 35647	class →𝑔
cgoo 35648	class ∨𝑔
cgob 35649	class ↔𝑔
cgoq 35650	class =𝑔
cgox 35651	class ∃𝑔𝑁𝑈
df-gonot 35652	⊢ ¬𝑔𝑈 = (𝑈⊼𝑔𝑈)
df-goan 35653	⊢ ∧𝑔 = (𝑢 ∈ V, 𝑣 ∈ V ↦ ¬𝑔(𝑢⊼𝑔𝑣))
df-goim 35654	⊢ →𝑔 = (𝑢 ∈ V, 𝑣 ∈ V ↦ (𝑢⊼𝑔¬𝑔𝑣))
df-goor 35655	⊢ ∨𝑔 = (𝑢 ∈ V, 𝑣 ∈ V ↦ (¬𝑔𝑢 →𝑔 𝑣))
df-gobi 35656	⊢ ↔𝑔 = (𝑢 ∈ V, 𝑣 ∈ V ↦ ((𝑢 →𝑔 𝑣)∧𝑔(𝑣 →𝑔 𝑢)))
df-goeq 35657	⊢ =𝑔 = (𝑢 ∈ ω, 𝑣 ∈ ω ↦ ⦋suc (𝑢 ∪ 𝑣) / 𝑤⦌∀𝑔𝑤((𝑤∈𝑔𝑢) ↔𝑔 (𝑤∈𝑔𝑣)))
df-goex 35658	⊢ ∃𝑔𝑁𝑈 = ¬𝑔∀𝑔𝑁¬𝑔𝑈
cgze 35659	class AxExt
cgzr 35660	class AxRep
cgzp 35661	class AxPow
cgzu 35662	class AxUn
cgzg 35663	class AxReg
cgzi 35664	class AxInf
cgzf 35665	class ZF
df-gzext 35666	⊢ AxExt = (∀𝑔2o((2o∈𝑔∅) ↔𝑔 (2o∈𝑔1o)) →𝑔 (∅=𝑔1o))
df-gzrep 35667	⊢ AxRep = (𝑢 ∈ (Fmla‘ω) ↦ (∀𝑔3o∃𝑔1o∀𝑔2o(∀𝑔1o𝑢 →𝑔 (2o=𝑔1o)) →𝑔 ∀𝑔1o∀𝑔2o((2o∈𝑔1o) ↔𝑔 ∃𝑔3o((3o∈𝑔∅)∧𝑔∀𝑔1o𝑢))))
df-gzpow 35668	⊢ AxPow = ∃𝑔1o∀𝑔2o(∀𝑔1o((1o∈𝑔2o) ↔𝑔 (1o∈𝑔∅)) →𝑔 (2o∈𝑔1o))
df-gzun 35669	⊢ AxUn = ∃𝑔1o∀𝑔2o(∃𝑔1o((2o∈𝑔1o)∧𝑔(1o∈𝑔∅)) →𝑔 (2o∈𝑔1o))
df-gzreg 35670	⊢ AxReg = (∃𝑔1o(1o∈𝑔∅) →𝑔 ∃𝑔1o((1o∈𝑔∅)∧𝑔∀𝑔2o((2o∈𝑔1o) →𝑔 ¬𝑔(2o∈𝑔∅))))
df-gzinf 35671	⊢ AxInf = ∃𝑔1o((∅∈𝑔1o)∧𝑔∀𝑔2o((2o∈𝑔1o) →𝑔 ∃𝑔∅((2o∈𝑔∅)∧𝑔(∅∈𝑔1o))))
df-gzf 35672	⊢ ZF = {𝑚 ∣ ((Tr 𝑚 ∧ 𝑚⊧AxExt ∧ 𝑚⊧AxPow) ∧ (𝑚⊧AxUn ∧ 𝑚⊧AxReg ∧ 𝑚⊧AxInf) ∧ ∀𝑢 ∈ (Fmla‘ω)𝑚⊧(AxRep‘𝑢))}
cmcn 35673	class mCN
cmvar 35674	class mVR
cmty 35675	class mType
cmvt 35676	class mVT
cmtc 35677	class mTC
cmax 35678	class mAx
cmrex 35679	class mREx
cmex 35680	class mEx
cmdv 35681	class mDV
cmvrs 35682	class mVars
cmrsub 35683	class mRSubst
cmsub 35684	class mSubst
cmvh 35685	class mVH
cmpst 35686	class mPreSt
cmsr 35687	class mStRed
cmsta 35688	class mStat
cmfs 35689	class mFS
cmcls 35690	class mCls
cmpps 35691	class mPPSt
cmthm 35692	class mThm
df-mcn 35693	⊢ mCN = Slot 1
df-mvar 35694	⊢ mVR = Slot 2
df-mty 35695	⊢ mType = Slot 3
df-mtc 35696	⊢ mTC = Slot 4
df-mmax 35697	⊢ mAx = Slot 5
df-mvt 35698	⊢ mVT = (𝑡 ∈ V ↦ ran (mType‘𝑡))
df-mrex 35699	⊢ mREx = (𝑡 ∈ V ↦ Word ((mCN‘𝑡) ∪ (mVR‘𝑡)))
df-mex 35700	⊢ mEx = (𝑡 ∈ V ↦ ((mTC‘𝑡) × (mREx‘𝑡)))
df-mdv 35701	⊢ mDV = (𝑡 ∈ V ↦ (((mVR‘𝑡) × (mVR‘𝑡)) ∖ I ))
df-mvrs 35702	⊢ mVars = (𝑡 ∈ V ↦ (𝑒 ∈ (mEx‘𝑡) ↦ (ran (2nd ‘𝑒) ∩ (mVR‘𝑡))))
df-mrsub 35703	⊢ mRSubst = (𝑡 ∈ V ↦ (𝑓 ∈ ((mREx‘𝑡) ↑pm (mVR‘𝑡)) ↦ (𝑒 ∈ (mREx‘𝑡) ↦ ((freeMnd‘((mCN‘𝑡) ∪ (mVR‘𝑡))) Σg ((𝑣 ∈ ((mCN‘𝑡) ∪ (mVR‘𝑡)) ↦ if(𝑣 ∈ dom 𝑓, (𝑓‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑒)))))
df-msub 35704	⊢ mSubst = (𝑡 ∈ V ↦ (𝑓 ∈ ((mREx‘𝑡) ↑pm (mVR‘𝑡)) ↦ (𝑒 ∈ (mEx‘𝑡) ↦ ⟨(1st ‘𝑒), (((mRSubst‘𝑡)‘𝑓)‘(2nd ‘𝑒))⟩)))
df-mvh 35705	⊢ mVH = (𝑡 ∈ V ↦ (𝑣 ∈ (mVR‘𝑡) ↦ ⟨((mType‘𝑡)‘𝑣), ⟨“𝑣”⟩⟩))
df-mpst 35706	⊢ mPreSt = (𝑡 ∈ V ↦ (({𝑑 ∈ 𝒫 (mDV‘𝑡) ∣ ◡𝑑 = 𝑑} × (𝒫 (mEx‘𝑡) ∩ Fin)) × (mEx‘𝑡)))
df-msr 35707	⊢ mStRed = (𝑡 ∈ V ↦ (𝑠 ∈ (mPreSt‘𝑡) ↦ ⦋(2nd ‘(1st ‘𝑠)) / ℎ⦌⦋(2nd ‘𝑠) / 𝑎⦌⟨((1st ‘(1st ‘𝑠)) ∩ ⦋∪ ((mVars‘𝑡) “ (ℎ ∪ {𝑎})) / 𝑧⦌(𝑧 × 𝑧)), ℎ, 𝑎⟩))
df-msta 35708	⊢ mStat = (𝑡 ∈ V ↦ ran (mStRed‘𝑡))
df-mfs 35709	⊢ 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 35710	⊢ 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 35711	⊢ mPPSt = (𝑡 ∈ V ↦ {⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∣ (⟨𝑑, ℎ, 𝑎⟩ ∈ (mPreSt‘𝑡) ∧ 𝑎 ∈ (𝑑(mCls‘𝑡)ℎ))})
df-mthm 35712	⊢ mThm = (𝑡 ∈ V ↦ (◡(mStRed‘𝑡) “ ((mStRed‘𝑡) “ (mPPSt‘𝑡))))
cm0s 35798	class m0St
cmsa 35799	class mSA
cmwgfs 35800	class mWGFS
cmsy 35801	class mSyn
cmesy 35802	class mESyn
cmgfs 35803	class mGFS
cmtree 35804	class mTree
cmst 35805	class mST
cmsax 35806	class mSAX
cmufs 35807	class mUFS
df-m0s 35808	⊢ m0St = (𝑎 ∈ V ↦ ⟨∅, ∅, 𝑎⟩)
df-msa 35809	⊢ mSA = (𝑡 ∈ V ↦ {𝑎 ∈ (mEx‘𝑡) ∣ ((m0St‘𝑎) ∈ (mAx‘𝑡) ∧ (1st ‘𝑎) ∈ (mVT‘𝑡) ∧ Fun (◡(2nd ‘𝑎) ↾ (mVR‘𝑡)))})
df-mwgfs 35810	⊢ mWGFS = {𝑡 ∈ mFS ∣ ∀𝑑∀ℎ∀𝑎((⟨𝑑, ℎ, 𝑎⟩ ∈ (mAx‘𝑡) ∧ (1st ‘𝑎) ∈ (mVT‘𝑡)) → ∃𝑠 ∈ ran (mSubst‘𝑡)𝑎 ∈ (𝑠 “ (mSA‘𝑡)))}
df-msyn 35811	⊢ mSyn = Slot 6
df-mesyn 35812	⊢ mESyn = (𝑡 ∈ V ↦ (𝑐 ∈ (mTC‘𝑡), 𝑒 ∈ (mREx‘𝑡) ↦ (((mSyn‘𝑡)‘𝑐)m0St𝑒)))
df-mgfs 35813	⊢ 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 35814	⊢ 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 35815	⊢ mST = (𝑡 ∈ V ↦ ((∅(mTree‘𝑡)∅) ↾ ((mEx‘𝑡) ↾ (mVT‘𝑡))))
df-msax 35816	⊢ mSAX = (𝑡 ∈ V ↦ (𝑝 ∈ (mSA‘𝑡) ↦ ((mVH‘𝑡) “ ((mVars‘𝑡)‘𝑝))))
df-mufs 35817	⊢ mUFS = {𝑡 ∈ mGFS ∣ Fun (mST‘𝑡)}
cmuv 35818	class mUV
cmvl 35819	class mVL
cmvsb 35820	class mVSubst
cmfsh 35821	class mFresh
cmfr 35822	class mFRel
cmevl 35823	class mEval
cmdl 35824	class mMdl
cusyn 35825	class mUSyn
cgmdl 35826	class mGMdl
cmitp 35827	class mItp
cmfitp 35828	class mFromItp
df-muv 35829	⊢ mUV = Slot 7
df-mfsh 35830	⊢ mFresh = Slot ;19
df-mevl 35831	⊢ mEval = Slot ;20
df-mvl 35832	⊢ mVL = (𝑡 ∈ V ↦ X𝑣 ∈ (mVR‘𝑡)((mUV‘𝑡) “ {((mType‘𝑡)‘𝑣)}))
df-mvsb 35833	⊢ 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 35834	⊢ 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 35835	⊢ 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 “ {(1st ‘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 35836	⊢ mUSyn = (𝑡 ∈ V ↦ (𝑣 ∈ (mUV‘𝑡) ↦ ⟨((mSyn‘𝑡)‘(1st ‘𝑣)), (2nd ‘𝑣)⟩))
df-gmdl 35837	⊢ 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) “ {(1st ‘𝑒)})))}
df-mitp 35838	⊢ 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 35839	⊢ mFromItp = (𝑡 ∈ V ↦ (𝑓 ∈ X𝑎 ∈ (mSA‘𝑡)(((mUV‘𝑡) “ {((1st ‘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) “ {(1st ‘𝑒)}))))))
ccpms 35840	class cplMetSp
chlb 35841	class HomLimB
chlim 35842	class HomLim
cpfl 35843	class polyFld
csf1 35844	class splitFld1
csf 35845	class splitFld
cpsl 35846	class polySplitLim
df-cplmet 35847	⊢ 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 35848	⊢ HomLimB = (𝑓 ∈ V ↦ ⦋∪ 𝑛 ∈ ℕ ({𝑛} × dom (𝑓‘𝑛)) / 𝑣⦌⦋∩ {𝑠 ∣ (𝑠 Er 𝑣 ∧ (𝑥 ∈ 𝑣 ↦ ⟨((1st ‘𝑥) + 1), ((𝑓‘(1st ‘𝑥))‘(2nd ‘𝑥))⟩) ⊆ 𝑠)} / 𝑒⦌⟨(𝑣 / 𝑒), (𝑛 ∈ ℕ ↦ (𝑥 ∈ dom (𝑓‘𝑛) ↦ [⟨𝑛, 𝑥⟩]𝑒))⟩)
df-homlim 35849	⊢ HomLim = (𝑟 ∈ V, 𝑓 ∈ V ↦ ⦋( HomLimB ‘𝑓) / 𝑒⦌⦋(1st ‘𝑒) / 𝑣⦌⦋(2nd ‘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 35850	⊢ polyFld = (𝑟 ∈ V, 𝑝 ∈ V ↦ ⦋(Poly1‘𝑟) / 𝑠⦌⦋((RSpan‘𝑠)‘{𝑝}) / 𝑖⦌⦋(𝑐 ∈ (Base‘𝑟) ↦ [(𝑐( ·𝑠 ‘𝑠)(1r‘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(rem1p‘𝑟)𝑝) = 𝑞)) / 𝑔⦌(◡𝑔 ∘ ((le‘𝑠) ∘ 𝑔))⟩), 𝑓⟩)
df-sfl1 35857	⊢ splitFld1 = (𝑟 ∈ V, 𝑗 ∈ V ↦ (𝑝 ∈ (Poly1‘𝑟) ↦ (rec((𝑠 ∈ V, 𝑓 ∈ V ↦ ⦋(Poly1‘𝑠) / 𝑚⦌⦋{𝑔 ∈ ((Monic1p‘𝑠) ∩ (Irred‘𝑚)) ∣ (𝑔(∥r‘𝑚)(𝑝 ∘ 𝑓) ∧ 1 < (𝑠deg1𝑔))} / 𝑏⦌if(((𝑝 ∘ 𝑓) = (0g‘𝑚) ∨ 𝑏 = ∅), ⟨𝑠, 𝑓⟩, ⦋(glb‘𝑏) / ℎ⦌⦋(𝑠 polyFld ℎ) / 𝑡⦌⟨(1st ‘𝑡), (𝑓 ∘ (2nd ‘𝑡))⟩)), 𝑗)‘(card‘(1...(𝑟deg1𝑝))))))
df-sfl 35858	⊢ 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 35859	⊢ polySplitLim = (𝑟 ∈ V, 𝑝 ∈ ((𝒫 (Base‘𝑟) ∩ Fin) ↑m ℕ) ↦ ⦋(1st ∘ seq0((𝑔 ∈ V, 𝑞 ∈ V ↦ ⦋(1st ‘𝑔) / 𝑒⦌⦋(1st ‘𝑒) / 𝑠⦌⦋(𝑠 splitFld ran (𝑥 ∈ 𝑞 ↦ (𝑥 ∘ (2nd ‘𝑔)))) / 𝑓⦌⟨𝑓, ((2nd ‘𝑔) ∘ (2nd ‘𝑓))⟩), (𝑝 ∪ {⟨0, ⟨⟨𝑟, ∅⟩, ( I ↾ (Base‘𝑟))⟩⟩}))) / 𝑓⦌((1st ∘ (𝑓 shift 1)) HomLim (2nd ∘ 𝑓)))
czr 35860	class ZRing
cgf 35861	class GF
cgfo 35862	class GF∞
ceqp 35863	class ~Qp
crqp 35864	class /Qp
cqp 35865	class Qp
czp 35866	class Zp
cqpa 35867	class _Qp
ccp 35868	class Cp
df-zrng 35869	⊢ ZRing = (𝑟 ∈ V ↦ (𝑟 IntgRing ran (ℤRHom‘𝑟)))
df-gf 35870	⊢ GF = (𝑝 ∈ ℙ, 𝑛 ∈ ℕ ↦ ⦋(ℤ/nℤ‘𝑝) / 𝑟⦌(1st ‘(𝑟 splitFld {⦋(Poly1‘𝑟) / 𝑠⦌⦋(var1‘𝑟) / 𝑥⦌(((𝑝↑𝑛)(.g‘(mulGrp‘𝑠))𝑥)(-g‘𝑠)𝑥)})))
df-gfoo 35871	⊢ 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 35872	⊢ ~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 35873	⊢ /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 35874	⊢ 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 35875	⊢ Zp = (ZRing ∘ Qp)
df-qpa 35876	⊢ _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 35877	⊢ Cp = ( cplMetSp ∘ _Qp)
ccloneop 35908	class CloneOp
df-cloneop 35909	⊢ CloneOp = (𝑎 ∈ V ↦ {𝑥 ∣ ∃𝑛 ∈ (ω ∖ 1o)𝑥 ∈ (𝑎 ↑m (𝑎 ↑m 𝑛))})
cprj 35910	class prj
df-prj 35911	⊢ prj = (𝑎 ∈ V ↦ (𝑛 ∈ (ω ∖ 1o), 𝑖 ∈ 𝑛 ↦ (𝑥 ∈ (𝑎 ↑m 𝑛) ↦ (𝑥‘𝑖))))
csuppos 35912	class suppos
df-suppos 35913	⊢ suppos = (𝑎 ∈ V ↦ (𝑛 ∈ (ω ∖ 1o), 𝑚 ∈ (ω ∖ 1o) ↦ (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 36021	class wsuc(𝑅, 𝐴, 𝑋)
cwlim 36022	class WLim(𝑅, 𝐴)
df-wsuc 36023	⊢ wsuc(𝑅, 𝐴, 𝑋) = inf(Pred(◡𝑅, 𝐴, 𝑋), 𝐴, 𝑅)
df-wlim 36024	⊢ WLim(𝑅, 𝐴) = {𝑥 ∈ 𝐴 ∣ (𝑥 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑥 = sup(Pred(𝑅, 𝐴, 𝑥), 𝐴, 𝑅))}
ctxp 36041	class (𝐴 ⊗ 𝐵)
cpprod 36042	class pprod(𝑅, 𝑆)
csset 36043	class SSet
ctrans 36044	class Trans
cbigcup 36045	class Bigcup
cfix 36046	class Fix 𝐴
climits 36047	class Limits
cfuns 36048	class Funs
csingle 36049	class Singleton
csingles 36050	class Singletons
cimage 36051	class Image𝐴
ccart 36052	class Cart
cimg 36053	class Img
cdomain 36054	class Domain
crange 36055	class Range
capply 36056	class Apply
ccup 36057	class Cup
ccap 36058	class Cap
csuccf 36059	class Succ
cfunpart 36060	class Funpart𝐹
cfullfn 36061	class FullFun𝐹
crestrict 36062	class Restrict
cub 36063	class UB𝑅
clb 36064	class LB𝑅
df-txp 36065	⊢ (𝐴 ⊗ 𝐵) = ((◡(1st ↾ (V × V)) ∘ 𝐴) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐵))
df-pprod 36066	⊢ pprod(𝐴, 𝐵) = ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V))))
df-sset 36067	⊢ SSet = ((V × V) ∖ ran ( E ⊗ (V ∖ E )))
df-trans 36068	⊢ Trans = (V ∖ ran (( E ∘ E ) ∖ E ))
df-bigcup 36069	⊢ Bigcup = ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ E ) ⊗ V)))
df-fix 36070	⊢ Fix 𝐴 = dom (𝐴 ∩ I )
df-limits 36071	⊢ Limits = ((On ∩ Fix Bigcup ) ∖ {∅})
df-funs 36072	⊢ Funs = (𝒫 (V × V) ∖ Fix ( E ∘ ((1st ⊗ ((V ∖ I ) ∘ 2nd )) ∘ ◡ E )))
df-singleton 36073	⊢ Singleton = ((V × V) ∖ ran ((V ⊗ E ) △ ( I ⊗ V)))
df-singles 36074	⊢ Singletons = ran Singleton
df-image 36075	⊢ Image𝐴 = ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ ◡𝐴) ⊗ V)))
df-cart 36076	⊢ Cart = (((V × V) × V) ∖ ran ((V ⊗ E ) △ (pprod( E , E ) ⊗ V)))
df-img 36077	⊢ Img = (Image((2nd ∘ 1st ) ↾ (1st ↾ (V × V))) ∘ Cart)
df-domain 36078	⊢ Domain = Image(1st ↾ (V × V))
df-range 36079	⊢ Range = Image(2nd ↾ (V × V))
df-cup 36080	⊢ Cup = (((V × V) × V) ∖ ran ((V ⊗ E ) △ (((◡1st ∘ E ) ∪ (◡2nd ∘ E )) ⊗ V)))
df-cap 36081	⊢ Cap = (((V × V) × V) ∖ ran ((V ⊗ E ) △ (((◡1st ∘ E ) ∩ (◡2nd ∘ E )) ⊗ V)))
df-restrict 36082	⊢ Restrict = (Cap ∘ (1st ⊗ (Cart ∘ (2nd ⊗ (Range ∘ 1st )))))
df-succf 36083	⊢ Succ = (Cup ∘ ( I ⊗ Singleton))
df-apply 36084	⊢ Apply = (( Bigcup ∘ Bigcup ) ∘ (((V × V) ∖ ran ((V ⊗ E ) △ (( E ↾ Singletons ) ⊗ V))) ∘ ((Singleton ∘ Img) ∘ pprod( I , Singleton))))
df-funpart 36085	⊢ Funpart𝐹 = (𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))
df-fullfun 36086	⊢ FullFun𝐹 = (Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))
df-ub 36087	⊢ UB𝑅 = ((V × V) ∖ ((V ∖ 𝑅) ∘ ◡ E ))
df-lb 36088	⊢ LB𝑅 = UB◡𝑅
caltop 36169	class ⟪𝐴, 𝐵⟫
caltxp 36170	class (𝐴 ×× 𝐵)
df-altop 36171	⊢ ⟪𝐴, 𝐵⟫ = {{𝐴}, {𝐴, {𝐵}}}
df-altxp 36172	⊢ (𝐴 ×× 𝐵) = {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = ⟪𝑥, 𝑦⟫}
cofs 36195	class OuterFiveSeg
df-ofs 36196	⊢ 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 36242	class TransportTo
df-transport 36243	⊢ TransportTo = {⟨⟨𝑝, 𝑞⟩, 𝑥⟩ ∣ ∃𝑛 ∈ ℕ ((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st ‘𝑞) ≠ (2nd ‘𝑞)) ∧ 𝑥 = (℩𝑟 ∈ (𝔼‘𝑛)((2nd ‘𝑞) Btwn ⟨(1st ‘𝑞), 𝑟⟩ ∧ ⟨(2nd ‘𝑞), 𝑟⟩Cgr𝑝)))}
cifs 36248	class InnerFiveSeg
ccgr3 36249	class Cgr3
ccolin 36250	class Colinear
cfs 36251	class FiveSeg
df-colinear 36252	⊢ 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 36253	⊢ 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 36254	⊢ 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 36255	⊢ 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 36319	class Seg≤
df-segle 36320	⊢ 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 36332	class OutsideOf
df-outsideof 36333	⊢ OutsideOf = ( Colinear ∖ Btwn )
cline2 36347	class Line
cray 36348	class Ray
clines2 36349	class LinesEE
df-line2 36350	⊢ Line = {⟨⟨𝑎, 𝑏⟩, 𝑙⟩ ∣ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎 ≠ 𝑏) ∧ 𝑙 = [⟨𝑎, 𝑏⟩]◡ Colinear )}
df-ray 36351	⊢ Ray = {⟨⟨𝑝, 𝑎⟩, 𝑟⟩ ∣ ∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑎) ∧ 𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩})}
df-lines2 36352	⊢ LinesEE = ran Line
cfwddif 36371	class △
df-fwddif 36372	⊢ △ = (𝑓 ∈ (ℂ ↑pm ℂ) ↦ (𝑥 ∈ {𝑦 ∈ dom 𝑓 ∣ (𝑦 + 1) ∈ dom 𝑓} ↦ ((𝑓‘(𝑥 + 1)) − (𝑓‘𝑥))))
cfwddifn 36373	class △n
df-fwddifn 36374	⊢ △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 36385	class Hf
df-hf 36386	⊢ Hf = ∪ (𝑅1 “ ω)
cfne 36549	class Fne
df-fne 36550	⊢ Fne = {⟨𝑥, 𝑦⟩ ∣ (∪ 𝑥 = ∪ 𝑦 ∧ ∀𝑧 ∈ 𝑥 𝑧 ⊆ ∪ (𝑦 ∩ 𝒫 𝑧))}
w3nand 36610	wff (𝜑 ⊼ 𝜓 ⊼ 𝜒)
df-3nand 36611	⊢ ((𝜑 ⊼ 𝜓 ⊼ 𝜒) ↔ (𝜑 → (𝜓 → ¬ 𝜒)))
cgcdOLD 36672	class gcdOLD (𝐴, 𝐵)
df-gcdOLD 36673	⊢ gcdOLD (𝐴, 𝐵) = sup({𝑥 ∈ ℕ ∣ ((𝐴 / 𝑥) ∈ ℕ ∧ (𝐵 / 𝑥) ∈ ℕ)}, ℕ, < )
cprvb 36818	wff Prv 𝜑
ax-prv1 36819	⊢ 𝜑    ⇒   ⊢ Prv 𝜑
ax-prv2 36820	⊢ (Prv (𝜑 → 𝜓) → (Prv 𝜑 → Prv 𝜓))
ax-prv3 36821	⊢ (Prv 𝜑 → Prv Prv 𝜑)
wmoo 36889	wff ∃**𝑥𝜑
df-bj-mo 36890	⊢ (∃**𝑥𝜑 ↔ ∀𝑧∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))
wnnf 36963	wff Ⅎ'𝑥𝜑
df-bj-nnf 36964	⊢ (Ⅎ'𝑥𝜑 ↔ ((∃𝑥𝜑 → 𝜑) ∧ (𝜑 → ∀𝑥𝜑)))
bj-cgab 37175	class {𝐴 ∣ 𝑥 ∣ 𝜑}
df-bj-gab 37176	⊢ {𝐴 ∣ 𝑥 ∣ 𝜑} = {𝑦 ∣ ∃𝑥(𝐴 = 𝑦 ∧ 𝜑)}
wrnf 37183	wff Ⅎ𝑥 ∈ 𝐴𝜑
df-bj-rnf 37184	⊢ (Ⅎ𝑥 ∈ 𝐴𝜑 ↔ (∃𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 𝜑))
bj-csngl 37207	class sngl 𝐴
df-bj-sngl 37208	⊢ sngl 𝐴 = {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = {𝑦}}
bj-ctag 37216	class tag 𝐴
df-bj-tag 37217	⊢ tag 𝐴 = (sngl 𝐴 ∪ {∅})
bj-cproj 37232	class (𝐴 Proj 𝐵)
df-bj-proj 37233	⊢ (𝐴 Proj 𝐵) = {𝑥 ∣ {𝑥} ∈ (𝐵 “ {𝐴})}
bj-c1upl 37239	class ⦅𝐴⦆
df-bj-1upl 37240	⊢ ⦅𝐴⦆ = ({∅} × tag 𝐴)
bj-cpr1 37242	class pr1 𝐴
df-bj-pr1 37243	⊢ pr1 𝐴 = (∅ Proj 𝐴)
bj-c2uple 37252	class ⦅𝐴, 𝐵⦆
df-bj-2upl 37253	⊢ ⦅𝐴, 𝐵⦆ = (⦅𝐴⦆ ∪ ({1o} × tag 𝐵))
bj-cpr2 37256	class pr2 𝐴
df-bj-pr2 37257	⊢ pr2 𝐴 = (1o Proj 𝐴)
ax-bj-sn 37275	⊢ ∀𝑥∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ 𝑧 = 𝑥)
ax-bj-bun 37279	⊢ ∀𝑥∀𝑦∃𝑧∀𝑡(𝑡 ∈ 𝑧 ↔ (𝑡 ∈ 𝑥 ∨ 𝑡 ∈ 𝑦))
ax-bj-adj 37284	⊢ ∀𝑥∀𝑦∃𝑧∀𝑡(𝑡 ∈ 𝑧 ↔ (𝑡 ∈ 𝑥 ∨ 𝑡 = 𝑦))
celwise 37326	class elwise
df-elwise 37327	⊢ elwise = (𝑜 ∈ V ↦ (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑧 ∣ ∃𝑢 ∈ 𝑥 ∃𝑣 ∈ 𝑦 𝑧 = (𝑢𝑜𝑣)}))
cmoore 37350	class Moore
df-bj-moore 37351	⊢ Moore = {𝑥 ∣ ∀𝑦 ∈ 𝒫 𝑥(∪ 𝑥 ∩ ∩ 𝑦) ∈ 𝑥}
cmpt3 37367	class (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵, 𝑧 ∈ 𝐶 ↦ 𝐷)
df-bj-mpt3 37368	⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵, 𝑧 ∈ 𝐶 ↦ 𝐷) = {⟨𝑠, 𝑡⟩ ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 (𝑠 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝑡 = 𝐷)}
csethom 37369	class Set⟶
df-bj-sethom 37370	⊢ Set⟶ = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓 ∣ 𝑓:𝑥⟶𝑦})
ctophom 37371	class Top⟶
df-bj-tophom 37372	⊢ Top⟶ = (𝑥 ∈ TopSp, 𝑦 ∈ TopSp ↦ {𝑓 ∈ ((Base‘𝑥) Set⟶ (Base‘𝑦)) ∣ ∀𝑢 ∈ (TopOpen‘𝑦)(◡𝑓 “ 𝑢) ∈ (TopOpen‘𝑥)})
cmgmhom 37373	class Mgm⟶
df-bj-mgmhom 37374	⊢ 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 37375	class TopMgm⟶
df-bj-topmgmhom 37376	⊢ TopMgm⟶ = (𝑥 ∈ TopMnd, 𝑦 ∈ TopMnd ↦ ((𝑥 Top⟶ 𝑦) ∩ (𝑥 Mgm⟶ 𝑦)))
ccur- 37377	class curry_
df-bj-cur 37378	⊢ curry_ = (𝑥 ∈ V, 𝑦 ∈ V, 𝑧 ∈ V ↦ (𝑓 ∈ ((𝑥 × 𝑦) Set⟶ 𝑧) ↦ (𝑎 ∈ 𝑥 ↦ (𝑏 ∈ 𝑦 ↦ (𝑓‘⟨𝑎, 𝑏⟩)))))
cunc- 37379	class uncurry_
df-bj-unc 37380	⊢ uncurry_ = (𝑥 ∈ V, 𝑦 ∈ V, 𝑧 ∈ V ↦ (𝑓 ∈ (𝑥 Set⟶ (𝑦 Set⟶ 𝑧)) ↦ (𝑎 ∈ 𝑥, 𝑏 ∈ 𝑦 ↦ ((𝑓‘𝑎)‘𝑏))))
cstrset 37381	class [𝐵 / 𝐴]struct𝑆
df-strset 37382	⊢ [𝐵 / 𝐴]struct𝑆 = ((𝑆 ↾ (V ∖ {(𝐴‘ndx)})) ∪ {⟨(𝐴‘ndx), 𝐵⟩})
cdiag2 37421	class Id
df-bj-diag 37422	⊢ Id = (𝑥 ∈ V ↦ ( I ↾ 𝑥))
cimdir 37427	class 𝒫*
df-imdir 37428	⊢ 𝒫* = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑟 ∈ 𝒫 (𝑎 × 𝑏) ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝑎 ∧ 𝑦 ⊆ 𝑏) ∧ (𝑟 “ 𝑥) = 𝑦)}))
ciminv 37440	class 𝒫*
df-iminv 37441	⊢ 𝒫* = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑟 ∈ 𝒫 (𝑎 × 𝑏) ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝑎 ∧ 𝑦 ⊆ 𝑏) ∧ 𝑥 = (◡𝑟 “ 𝑦))}))
cfractemp 37445	class {R
df-bj-fractemp 37446	⊢ {R = (𝑥 ∈ R ↦ (℩𝑦 ∈ R ((𝑦 = 0R ∨ (0R <R 𝑦 ∧ 𝑦 <R 1R)) ∧ ∃𝑛 ∈ ω ([⟨{𝑧 ∈ Q ∣ 𝑧 <Q ⟨suc 𝑛, 1o⟩}, 1P⟩] ~R +R 𝑦) = 𝑥)))
cinftyexpitau 37447	class +∞eiτ
df-bj-inftyexpitau 37448	⊢ +∞eiτ = (𝑥 ∈ ℝ ↦ ⟨({R‘(1st ‘𝑥)), {R}⟩)
cccinftyN 37449	class ℂ∞N
df-bj-ccinftyN 37450	⊢ ℂ∞N = ran +∞eiτ
chalf 37452	class 1/2
df-bj-onehalf 37453	⊢ 1/2 = (℩𝑥 ∈ R (𝑥 +R 𝑥) = 1R)
cinftyexpi 37455	class +∞ei
df-bj-inftyexpi 37456	⊢ +∞ei = (𝑥 ∈ (-π(,]π) ↦ ⟨𝑥, ℂ⟩)
cccinfty 37460	class ℂ∞
df-bj-ccinfty 37461	⊢ ℂ∞ = ran +∞ei
cccbar 37464	class ℂ̅
df-bj-ccbar 37465	⊢ ℂ̅ = (ℂ ∪ ℂ∞)
cpinfty 37468	class +∞
df-bj-pinfty 37469	⊢ +∞ = (+∞ei‘0)
cminfty 37472	class -∞
df-bj-minfty 37473	⊢ -∞ = (+∞ei‘π)
crrbar 37477	class ℝ̅
df-bj-rrbar 37478	⊢ ℝ̅ = (ℝ ∪ {-∞, +∞})
cinfty 37479	class ∞
df-bj-infty 37480	⊢ ∞ = 𝒫 ∪ ℂ
ccchat 37481	class ℂ̂
df-bj-cchat 37482	⊢ ℂ̂ = (ℂ ∪ {∞})
crrhat 37483	class ℝ̂
df-bj-rrhat 37484	⊢ ℝ̂ = (ℝ ∪ {∞})
caddcc 37486	class +ℂ̅
df-bj-addc 37487	⊢ +ℂ̅ = (𝑥 ∈ (((ℂ × ℂ̅) ∪ (ℂ̅ × ℂ)) ∪ ((ℂ̂ × ℂ̂) ∪ ( I ↾ ℂ∞))) ↦ if(((1st ‘𝑥) = ∞ ∨ (2nd ‘𝑥) = ∞), ∞, if((1st ‘𝑥) ∈ ℂ, if((2nd ‘𝑥) ∈ ℂ, ⟨((1st ‘(1st ‘𝑥)) +R (1st ‘(2nd ‘𝑥))), ((2nd ‘(1st ‘𝑥)) +R (2nd ‘(2nd ‘𝑥)))⟩, (2nd ‘𝑥)), (1st ‘𝑥))))
coppcc 37488	class -ℂ̅
df-bj-oppc 37489	⊢ -ℂ̅ = (𝑥 ∈ (ℂ̅ ∪ ℂ̂) ↦ if(𝑥 = ∞, ∞, if(𝑥 ∈ ℂ, (℩𝑦 ∈ ℂ (𝑥 +ℂ̅ 𝑦) = 0), (+∞eiτ‘(𝑥 +ℂ̅ ⟨1/2, 0R⟩)))))
cltxr 37490	class <ℝ̅
df-bj-lt 37491	⊢ <ℝ̅ = ({𝑥 ∈ (ℝ̅ × ℝ̅) ∣ ∃𝑦∃𝑧(((1st ‘𝑥) = ⟨𝑦, 0R⟩ ∧ (2nd ‘𝑥) = ⟨𝑧, 0R⟩) ∧ 𝑦 <R 𝑧)} ∪ ((({-∞} × ℝ) ∪ (ℝ × {+∞})) ∪ ({-∞} × {+∞})))
carg 37492	class Arg
df-bj-arg 37493	⊢ Arg = (𝑥 ∈ (ℂ̅ ∖ {0}) ↦ if(𝑥 ∈ ℂ, (ℑ‘(log‘𝑥)), if(𝑥<ℝ̅0, π, (((1st ‘𝑥) / (2 · π)) − π))))
cmulc 37494	class ·ℂ̅
df-bj-mulc 37495	⊢ ·ℂ̅ = (𝑥 ∈ ((ℂ̅ × ℂ̅) ∪ (ℂ̂ × ℂ̂)) ↦ if(((1st ‘𝑥) = 0 ∨ (2nd ‘𝑥) = 0), 0, if(((1st ‘𝑥) = ∞ ∨ (2nd ‘𝑥) = ∞), ∞, if(𝑥 ∈ (ℂ × ℂ), ((1st ‘𝑥) · (2nd ‘𝑥)), (+∞eiτ‘(((Arg‘(1st ‘𝑥)) +ℂ̅ (Arg‘(2nd ‘𝑥))) / τ))))))
cinvc 37496	class -1ℂ̅
df-bj-invc 37497	⊢ -1ℂ̅ = (𝑥 ∈ (ℂ̅ ∪ ℂ̂) ↦ if(𝑥 = 0, ∞, if(𝑥 ∈ ℂ, (℩𝑦 ∈ ℂ (𝑥 ·ℂ̅ 𝑦) = 1), 0)))
ciomnn 37498	class iω↪ℕ
df-bj-iomnn 37499	⊢ iω↪ℕ = ((𝑛 ∈ ω ↦ ⟨[⟨{𝑟 ∈ Q ∣ 𝑟 <Q ⟨suc 𝑛, 1o⟩}, 1P⟩] ~R , 0R⟩) ∪ {⟨ω, +∞⟩})
cnnbar 37509	class ℕ̅
df-bj-nnbar 37510	⊢ ℕ̅ = (ℕ0 ∪ {+∞})
czzbar 37511	class ℤ̅
df-bj-zzbar 37512	⊢ ℤ̅ = (ℤ ∪ {-∞, +∞})
czzhat 37513	class ℤ̂
df-bj-zzhat 37514	⊢ ℤ̂ = (ℤ ∪ {∞})
cdivc 37515	class ∥ℂ
df-bj-divc 37516	⊢ ∥ℂ = {⟨𝑥, 𝑦⟩ ∣ (⟨𝑥, 𝑦⟩ ∈ ((ℂ̅ × ℂ̅) ∪ (ℂ̂ × ℂ̂)) ∧ ∃𝑛 ∈ (ℤ̅ ∪ ℤ̂)(𝑛 ·ℂ̅ 𝑥) = 𝑦)}
cfinsum 37532	class FinSum
df-bj-finsum 37533	⊢ FinSum = (𝑥 ∈ {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ CMnd ∧ ∃𝑡 ∈ Fin 𝑧:𝑡⟶(Base‘𝑦))} ↦ (℩𝑠∃𝑚 ∈ ℕ0 ∃𝑓(𝑓:(1...𝑚)–1-1-onto→dom (2nd ‘𝑥) ∧ 𝑠 = (seq1((+g‘(1st ‘𝑥)), (𝑛 ∈ ℕ ↦ ((2nd ‘𝑥)‘(𝑓‘𝑛))))‘𝑚))))
crrvec 37541	class ℝ-Vec
df-bj-rvec 37542	⊢ ℝ-Vec = (LMod ∩ (◡Scalar “ {ℝfld}))
cend 37562	class End
df-bj-end 37563	⊢ End = (𝑐 ∈ Cat ↦ (𝑥 ∈ (Base‘𝑐) ↦ {⟨(Base‘ndx), (𝑥(Hom ‘𝑐)𝑥)⟩, ⟨(+g‘ndx), (⟨𝑥, 𝑥⟩(comp‘𝑐)𝑥)⟩}))
cfinxp 37632	class (𝑈↑↑𝑁)
df-finxp 37633	⊢ (𝑈↑↑𝑁) = {𝑦 ∣ (𝑁 ∈ ω ∧ ∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨𝑁, 𝑦⟩)‘𝑁))}
ax-luk1 37668	⊢ ((𝜑 → 𝜓) → ((𝜓 → 𝜒) → (𝜑 → 𝜒)))
ax-luk2 37669	⊢ ((¬ 𝜑 → 𝜑) → 𝜑)
ax-luk3 37670	⊢ (𝜑 → (¬ 𝜑 → 𝜓))
ax-wl-13v 37742	⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
ax-wl-cleq 37751	⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
ax-wl-clel 37752	⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵))
ctotbnd 38011	class TotBnd
cbnd 38012	class Bnd
df-totbnd 38013	⊢ TotBnd = (𝑥 ∈ V ↦ {𝑚 ∈ (Met‘𝑥) ∣ ∀𝑑 ∈ ℝ+ ∃𝑣 ∈ Fin (∪ 𝑣 = 𝑥 ∧ ∀𝑏 ∈ 𝑣 ∃𝑦 ∈ 𝑥 𝑏 = (𝑦(ball‘𝑚)𝑑))})
df-bnd 38024	⊢ Bnd = (𝑥 ∈ V ↦ {𝑚 ∈ (Met‘𝑥) ∣ ∀𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ 𝑥 = (𝑦(ball‘𝑚)𝑟)})
cismty 38043	class Ismty
df-ismty 38044	⊢ 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 38070	class ℝn
df-rrn 38071	⊢ ℝn = (𝑖 ∈ Fin ↦ (𝑥 ∈ (ℝ ↑m 𝑖), 𝑦 ∈ (ℝ ↑m 𝑖) ↦ (√‘Σ𝑘 ∈ 𝑖 (((𝑥‘𝑘) − (𝑦‘𝑘))↑2))))
cass 38087	class Ass
df-ass 38088	⊢ Ass = {𝑔 ∣ ∀𝑥 ∈ dom dom 𝑔∀𝑦 ∈ dom dom 𝑔∀𝑧 ∈ dom dom 𝑔((𝑥𝑔𝑦)𝑔𝑧) = (𝑥𝑔(𝑦𝑔𝑧))}
cexid 38089	class ExId
df-exid 38090	⊢ ExId = {𝑔 ∣ ∃𝑥 ∈ dom dom 𝑔∀𝑦 ∈ dom dom 𝑔((𝑥𝑔𝑦) = 𝑦 ∧ (𝑦𝑔𝑥) = 𝑦)}
cmagm 38093	class Magma
df-mgmOLD 38094	⊢ Magma = {𝑔 ∣ ∃𝑡 𝑔:(𝑡 × 𝑡)⟶𝑡}
csem 38105	class SemiGrp
df-sgrOLD 38106	⊢ SemiGrp = (Magma ∩ Ass)
cmndo 38111	class MndOp
df-mndo 38112	⊢ MndOp = (SemiGrp ∩ ExId )
cghomOLD 38128	class GrpOpHom
df-ghomOLD 38129	⊢ 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 38139	class RingOps
df-rngo 38140	⊢ 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 38193	class DivRingOps
df-drngo 38194	⊢ DivRingOps = {⟨𝑔, ℎ⟩ ∣ (⟨𝑔, ℎ⟩ ∈ RingOps ∧ (ℎ ↾ ((ran 𝑔 ∖ {(GId‘𝑔)}) × (ran 𝑔 ∖ {(GId‘𝑔)}))) ∈ GrpOp)}
crngohom 38205	class RingOpsHom
crngoiso 38206	class RingOpsIso
crisc 38207	class ≃𝑟
df-rngohom 38208	⊢ RingOpsHom = (𝑟 ∈ RingOps, 𝑠 ∈ RingOps ↦ {𝑓 ∈ (ran (1st ‘𝑠) ↑m ran (1st ‘𝑟)) ∣ ((𝑓‘(GId‘(2nd ‘𝑟))) = (GId‘(2nd ‘𝑠)) ∧ ∀𝑥 ∈ ran (1st ‘𝑟)∀𝑦 ∈ ran (1st ‘𝑟)((𝑓‘(𝑥(1st ‘𝑟)𝑦)) = ((𝑓‘𝑥)(1st ‘𝑠)(𝑓‘𝑦)) ∧ (𝑓‘(𝑥(2nd ‘𝑟)𝑦)) = ((𝑓‘𝑥)(2nd ‘𝑠)(𝑓‘𝑦))))})
df-rngoiso 38221	⊢ RingOpsIso = (𝑟 ∈ RingOps, 𝑠 ∈ RingOps ↦ {𝑓 ∈ (𝑟 RingOpsHom 𝑠) ∣ 𝑓:ran (1st ‘𝑟)–1-1-onto→ran (1st ‘𝑠)})
df-risc 38228	⊢ ≃𝑟 = {⟨𝑟, 𝑠⟩ ∣ ((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑟 RingOpsIso 𝑠))}
ccm2 38234	class Com2
df-com2 38235	⊢ Com2 = {⟨𝑔, ℎ⟩ ∣ ∀𝑎 ∈ ran 𝑔∀𝑏 ∈ ran 𝑔(𝑎ℎ𝑏) = (𝑏ℎ𝑎)}
cfld 38236	class Fld
df-fld 38237	⊢ Fld = (DivRingOps ∩ Com2)
ccring 38238	class CRingOps
df-crngo 38239	⊢ CRingOps = (RingOps ∩ Com2)
cidl 38252	class Idl
cpridl 38253	class PrIdl
cmaxidl 38254	class MaxIdl
df-idl 38255	⊢ Idl = (𝑟 ∈ RingOps ↦ {𝑖 ∈ 𝒫 ran (1st ‘𝑟) ∣ ((GId‘(1st ‘𝑟)) ∈ 𝑖 ∧ ∀𝑥 ∈ 𝑖 (∀𝑦 ∈ 𝑖 (𝑥(1st ‘𝑟)𝑦) ∈ 𝑖 ∧ ∀𝑧 ∈ ran (1st ‘𝑟)((𝑧(2nd ‘𝑟)𝑥) ∈ 𝑖 ∧ (𝑥(2nd ‘𝑟)𝑧) ∈ 𝑖)))})
df-pridl 38256	⊢ PrIdl = (𝑟 ∈ RingOps ↦ {𝑖 ∈ (Idl‘𝑟) ∣ (𝑖 ≠ ran (1st ‘𝑟) ∧ ∀𝑎 ∈ (Idl‘𝑟)∀𝑏 ∈ (Idl‘𝑟)(∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 (𝑥(2nd ‘𝑟)𝑦) ∈ 𝑖 → (𝑎 ⊆ 𝑖 ∨ 𝑏 ⊆ 𝑖)))})
df-maxidl 38257	⊢ MaxIdl = (𝑟 ∈ RingOps ↦ {𝑖 ∈ (Idl‘𝑟) ∣ (𝑖 ≠ ran (1st ‘𝑟) ∧ ∀𝑗 ∈ (Idl‘𝑟)(𝑖 ⊆ 𝑗 → (𝑗 = 𝑖 ∨ 𝑗 = ran (1st ‘𝑟))))})
cprrng 38291	class PrRing
cdmn 38292	class Dmn
df-prrngo 38293	⊢ PrRing = {𝑟 ∈ RingOps ∣ {(GId‘(1st ‘𝑟))} ∈ (PrIdl‘𝑟)}
df-dmn 38294	⊢ Dmn = (PrRing ∩ Com2)
cigen 38304	class IdlGen
df-igen 38305	⊢ IdlGen = (𝑟 ∈ RingOps, 𝑠 ∈ 𝒫 ran (1st ‘𝑟) ↦ ∩ {𝑗 ∈ (Idl‘𝑟) ∣ 𝑠 ⊆ 𝑗})
cxrn 38419	class (𝐴 ⋉ 𝐵)
cqmap 38420	class QMap 𝑅
cadjliftmap 38421	class (𝑅 AdjLiftMap 𝐴)
cblockliftmap 38422	class (𝑅 BlockLiftMap 𝐴)
csucmap 38423	class SucMap
csuccl 38424	class Suc
cpre 38425	class pre 𝑁
cblockliftfix 38426	class BlockLiftFix
cshiftstable 38427	class (𝑆 ShiftStable 𝐹)
ccoss 38428	class ≀ 𝑅
ccoels 38429	class ∼ 𝐴
crels 38430	class Rels
cssr 38431	class S
crefs 38432	class Refs
crefrels 38433	class RefRels
wrefrel 38434	wff RefRel 𝑅
ccnvrefs 38435	class CnvRefs
ccnvrefrels 38436	class CnvRefRels
wcnvrefrel 38437	wff CnvRefRel 𝑅
csyms 38438	class Syms
csymrels 38439	class SymRels
wsymrel 38440	wff SymRel 𝑅
ctrs 38441	class Trs
ctrrels 38442	class TrRels
wtrrel 38443	wff TrRel 𝑅
ceqvrels 38444	class EqvRels
weqvrel 38445	wff EqvRel 𝑅
ccoeleqvrels 38446	class CoElEqvRels
wcoeleqvrel 38447	wff CoElEqvRel 𝐴
credunds 38448	class Redunds
wredund 38449	wff 𝐴 Redund ⟨𝐵, 𝐶⟩
wredundp 38450	wff redund (𝜑, 𝜓, 𝜒)
cdmqss 38451	class DomainQss
wdmqs 38452	wff 𝑅 DomainQs 𝐴
cers 38453	class Ers
werALTV 38454	wff 𝑅 ErALTV 𝐴
cpeters 38455	class PetErs
cpet2ers 38456	class Pet2Ers
ccomembers 38457	class CoMembErs
wcomember 38458	wff CoMembEr 𝐴
cfunss 38459	class Funss
cfunsALTV 38460	class FunsALTV
wfunALTV 38461	wff FunALTV 𝐹
cdisjss 38462	class Disjss
cdisjs 38463	class Disjs
wdisjALTV 38464	wff Disj 𝑅
celdisjs 38465	class ElDisjs
weldisj 38466	wff ElDisj 𝐴
wantisymrel 38467	wff AntisymRel 𝑅
cparts 38468	class Parts
wpart 38469	wff 𝑅 Part 𝐴
cmembparts 38470	class MembParts
wmembpart 38471	wff MembPart 𝐴
cpetparts 38472	class PetParts
cpet2parts 38473	class Pet2Parts
df-xrn 38625	⊢ (𝐴 ⋉ 𝐵) = ((◡(1st ↾ (V × V)) ∘ 𝐴) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐵))
df-rels 38685	⊢ Rels = 𝒫 (V × V)
df-qmap 38691	⊢ QMap 𝑅 = (𝑥 ∈ dom 𝑅 ↦ [𝑥]𝑅)
df-adjliftmap 38700	⊢ (𝑅 AdjLiftMap 𝐴) = QMap ((𝑅 ∪ ◡ E ) ↾ 𝐴)
df-blockliftmap 38704	⊢ (𝑅 BlockLiftMap 𝐴) = QMap (𝑅 ⋉ (◡ E ↾ 𝐴))
df-sucmap 38707	⊢ SucMap = {⟨𝑚, 𝑛⟩ ∣ suc 𝑚 = 𝑛}
df-succl 38714	⊢ Suc = ran SucMap
df-pre 38720	⊢ pre 𝑁 = (℩𝑚𝑚 ∈ Pred( SucMap , dom SucMap , 𝑁))
df-blockliftfix 38726	⊢ BlockLiftFix = {⟨𝑟, 𝑎⟩ ∣ (dom (𝑟 ⋉ (◡ E ↾ 𝑎)) / (𝑟 ⋉ (◡ E ↾ 𝑎))) = 𝑎}
df-shiftstable 38727	⊢ (𝑆 ShiftStable 𝐹) = ((𝑆 ∘ 𝐹) ∩ 𝐹)
df-coss 38746	⊢ ≀ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)}
df-coels 38747	⊢ ∼ 𝐴 = ≀ (◡ E ↾ 𝐴)
df-ssr 38823	⊢ S = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ⊆ 𝑦}
df-refs 38835	⊢ Refs = {𝑥 ∣ ( I ∩ (dom 𝑥 × ran 𝑥)) S (𝑥 ∩ (dom 𝑥 × ran 𝑥))}
df-refrels 38836	⊢ RefRels = ( Refs ∩ Rels )
df-refrel 38837	⊢ ( RefRel 𝑅 ↔ (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅))
df-cnvrefs 38850	⊢ CnvRefs = {𝑥 ∣ ( I ∩ (dom 𝑥 × ran 𝑥))◡ S (𝑥 ∩ (dom 𝑥 × ran 𝑥))}
df-cnvrefrels 38851	⊢ CnvRefRels = ( CnvRefs ∩ Rels )
df-cnvrefrel 38852	⊢ ( CnvRefRel 𝑅 ↔ ((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ ( I ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅))
df-syms 38867	⊢ Syms = {𝑥 ∣ ◡(𝑥 ∩ (dom 𝑥 × ran 𝑥)) S (𝑥 ∩ (dom 𝑥 × ran 𝑥))}
df-symrels 38868	⊢ SymRels = ( Syms ∩ Rels )
df-symrel 38869	⊢ ( SymRel 𝑅 ↔ (◡(𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅))
df-trs 38901	⊢ Trs = {𝑥 ∣ ((𝑥 ∩ (dom 𝑥 × ran 𝑥)) ∘ (𝑥 ∩ (dom 𝑥 × ran 𝑥))) S (𝑥 ∩ (dom 𝑥 × ran 𝑥))}
df-trrels 38902	⊢ TrRels = ( Trs ∩ Rels )
df-trrel 38903	⊢ ( TrRel 𝑅 ↔ (((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∘ (𝑅 ∩ (dom 𝑅 × ran 𝑅))) ⊆ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅))
df-eqvrels 38913	⊢ EqvRels = (( RefRels ∩ SymRels ) ∩ TrRels )
df-eqvrel 38914	⊢ ( EqvRel 𝑅 ↔ ( RefRel 𝑅 ∧ SymRel 𝑅 ∧ TrRel 𝑅))
df-coeleqvrels 38915	⊢ CoElEqvRels = {𝑎 ∣ ≀ (◡ E ↾ 𝑎) ∈ EqvRels }
df-coeleqvrel 38916	⊢ ( CoElEqvRel 𝐴 ↔ EqvRel ≀ (◡ E ↾ 𝐴))
df-redunds 38952	⊢ Redunds = ◡{⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ (𝑥 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑧) = (𝑦 ∩ 𝑧))}
df-redund 38953	⊢ (𝐴 Redund ⟨𝐵, 𝐶⟩ ↔ (𝐴 ⊆ 𝐵 ∧ (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐶)))
df-redundp 38954	⊢ ( redund (𝜑, 𝜓, 𝜒) ↔ ((𝜑 → 𝜓) ∧ ((𝜑 ∧ 𝜒) ↔ (𝜓 ∧ 𝜒))))
df-dmqss 38967	⊢ DomainQss = {⟨𝑥, 𝑦⟩ ∣ (dom 𝑥 / 𝑥) = 𝑦}
df-dmqs 38968	⊢ (𝑅 DomainQs 𝐴 ↔ (dom 𝑅 / 𝑅) = 𝐴)
df-ers 38993	⊢ Ers = ( DomainQss ↾ EqvRels )
df-erALTV 38994	⊢ (𝑅 ErALTV 𝐴 ↔ ( EqvRel 𝑅 ∧ 𝑅 DomainQs 𝐴))
df-comembers 38995	⊢ CoMembErs = {𝑎 ∣ ≀ (◡ E ↾ 𝑎) Ers 𝑎}
df-comember 38996	⊢ ( CoMembEr 𝐴 ↔ ≀ (◡ E ↾ 𝐴) ErALTV 𝐴)
df-funss 39010	⊢ Funss = {𝑥 ∣ ≀ 𝑥 ∈ CnvRefRels }
df-funsALTV 39011	⊢ FunsALTV = ( Funss ∩ Rels )
df-funALTV 39012	⊢ ( FunALTV 𝐹 ↔ ( CnvRefRel ≀ 𝐹 ∧ Rel 𝐹))
df-disjss 39033	⊢ Disjss = {𝑥 ∣ ≀ ◡𝑥 ∈ CnvRefRels }
df-disjs 39034	⊢ Disjs = ( Disjss ∩ Rels )
df-disjALTV 39035	⊢ ( Disj 𝑅 ↔ ( CnvRefRel ≀ ◡𝑅 ∧ Rel 𝑅))
df-eldisjs 39036	⊢ ElDisjs = {𝑎 ∣ (◡ E ↾ 𝑎) ∈ Disjs }
df-eldisj 39037	⊢ ( ElDisj 𝐴 ↔ Disj (◡ E ↾ 𝐴))
df-antisymrel 39108	⊢ ( AntisymRel 𝑅 ↔ ( CnvRefRel (𝑅 ∩ ◡𝑅) ∧ Rel 𝑅))
df-parts 39113	⊢ Parts = ( DomainQss ↾ Disjs )
df-part 39114	⊢ (𝑅 Part 𝐴 ↔ ( Disj 𝑅 ∧ 𝑅 DomainQs 𝐴))
df-membparts 39115	⊢ MembParts = {𝑎 ∣ (◡ E ↾ 𝑎) Parts 𝑎}
df-membpart 39116	⊢ ( MembPart 𝐴 ↔ (◡ E ↾ 𝐴) Part 𝐴)
df-petparts 39213	⊢ PetParts = {⟨𝑟, 𝑛⟩ ∣ ((𝑟 ∈ Rels ∧ 𝑛 ∈ MembParts ) ∧ (𝑟 ⋉ (◡ E ↾ 𝑛)) Parts 𝑛)}
df-peters 39214	⊢ PetErs = {⟨𝑟, 𝑛⟩ ∣ ((𝑟 ∈ Rels ∧ 𝑛 ∈ CoMembErs ) ∧ ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) Ers 𝑛)}
df-pet2parts 39215	⊢ Pet2Parts = ( SucMap ShiftStable PetParts )
df-pet2ers 39216	⊢ Pet2Ers = ( SucMap ShiftStable PetErs )
wprt 39241	wff Prt 𝐴
df-prt 39242	⊢ (Prt 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅))
ax-c5 39253	⊢ (∀𝑥𝜑 → 𝜑)
ax-c4 39254	⊢ (∀𝑥(∀𝑥𝜑 → 𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))
ax-c7 39255	⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜑 → 𝜑)
ax-c10 39256	⊢ (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → 𝜑)
ax-c11 39257	⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑))
ax-c11n 39258	⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
ax-c15 39259	⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))))
ax-c9 39260	⊢ (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))
ax-c14 39261	⊢ (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥 ∈ 𝑦 → ∀𝑧 𝑥 ∈ 𝑦)))
ax-c16 39262	⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))
ax-riotaBAD 39323	⊢ (℩𝑥 ∈ 𝐴 𝜑) = if(∃!𝑥 ∈ 𝐴 𝜑, (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)), (Undef‘{𝑥 ∣ 𝑥 ∈ 𝐴}))
clsa 39344	class LSAtoms
clsh 39345	class LSHyp
df-lsatoms 39346	⊢ LSAtoms = (𝑤 ∈ V ↦ ran (𝑣 ∈ ((Base‘𝑤) ∖ {(0g‘𝑤)}) ↦ ((LSpan‘𝑤)‘{𝑣})))
df-lshyp 39347	⊢ LSHyp = (𝑤 ∈ V ↦ {𝑠 ∈ (LSubSp‘𝑤) ∣ (𝑠 ≠ (Base‘𝑤) ∧ ∃𝑣 ∈ (Base‘𝑤)((LSpan‘𝑤)‘(𝑠 ∪ {𝑣})) = (Base‘𝑤))})
clcv 39388	class ⋖L
df-lcv 39389	⊢ ⋖L = (𝑤 ∈ V ↦ {⟨𝑡, 𝑢⟩ ∣ ((𝑡 ∈ (LSubSp‘𝑤) ∧ 𝑢 ∈ (LSubSp‘𝑤)) ∧ (𝑡 ⊊ 𝑢 ∧ ¬ ∃𝑠 ∈ (LSubSp‘𝑤)(𝑡 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑢)))})
clfn 39427	class LFnl
df-lfl 39428	⊢ 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 39455	class LKer
df-lkr 39456	⊢ LKer = (𝑤 ∈ V ↦ (𝑓 ∈ (LFnl‘𝑤) ↦ (◡𝑓 “ {(0g‘(Scalar‘𝑤))})))
cld 39493	class LDual
df-ldual 39494	⊢ LDual = (𝑣 ∈ V ↦ ({⟨(Base‘ndx), (LFnl‘𝑣)⟩, ⟨(+g‘ndx), ( ∘f (+g‘(Scalar‘𝑣)) ↾ ((LFnl‘𝑣) × (LFnl‘𝑣)))⟩, ⟨(Scalar‘ndx), (oppr‘(Scalar‘𝑣))⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑣)), 𝑓 ∈ (LFnl‘𝑣) ↦ (𝑓 ∘f (.r‘(Scalar‘𝑣))((Base‘𝑣) × {𝑘})))⟩}))
cops 39542	class OP
ccmtN 39543	class cm
col 39544	class OL
coml 39545	class OML
df-oposet 39546	⊢ 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 39547	⊢ 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 39548	⊢ OL = (Lat ∩ OP)
df-oml 39549	⊢ OML = {𝑙 ∈ OL ∣ ∀𝑎 ∈ (Base‘𝑙)∀𝑏 ∈ (Base‘𝑙)(𝑎(le‘𝑙)𝑏 → 𝑏 = (𝑎(join‘𝑙)(𝑏(meet‘𝑙)((oc‘𝑙)‘𝑎))))}
ccvr 39632	class ⋖
catm 39633	class Atoms
cal 39634	class AtLat
clc 39635	class CvLat
df-covers 39636	⊢ ⋖ = (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 39637	⊢ Atoms = (𝑝 ∈ V ↦ {𝑎 ∈ (Base‘𝑝) ∣ (0.‘𝑝)( ⋖ ‘𝑝)𝑎})
df-atl 39668	⊢ AtLat = {𝑘 ∈ Lat ∣ ((Base‘𝑘) ∈ dom (glb‘𝑘) ∧ ∀𝑥 ∈ (Base‘𝑘)(𝑥 ≠ (0.‘𝑘) → ∃𝑝 ∈ (Atoms‘𝑘)𝑝(le‘𝑘)𝑥))}
df-cvlat 39692	⊢ 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 39720	class HL
df-hlat 39721	⊢ 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 39861	class LLines
clpl 39862	class LPlanes
clvol 39863	class LVols
clines 39864	class Lines
cpointsN 39865	class Points
cpsubsp 39866	class PSubSp
cpmap 39867	class pmap
df-llines 39868	⊢ LLines = (𝑘 ∈ V ↦ {𝑥 ∈ (Base‘𝑘) ∣ ∃𝑝 ∈ (Atoms‘𝑘)𝑝( ⋖ ‘𝑘)𝑥})
df-lplanes 39869	⊢ LPlanes = (𝑘 ∈ V ↦ {𝑥 ∈ (Base‘𝑘) ∣ ∃𝑝 ∈ (LLines‘𝑘)𝑝( ⋖ ‘𝑘)𝑥})
df-lvols 39870	⊢ LVols = (𝑘 ∈ V ↦ {𝑥 ∈ (Base‘𝑘) ∣ ∃𝑝 ∈ (LPlanes‘𝑘)𝑝( ⋖ ‘𝑘)𝑥})
df-lines 39871	⊢ 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 39872	⊢ Points = (𝑘 ∈ V ↦ {𝑞 ∣ ∃𝑝 ∈ (Atoms‘𝑘)𝑞 = {𝑝}})
df-psubsp 39873	⊢ 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 39874	⊢ pmap = (𝑘 ∈ V ↦ (𝑎 ∈ (Base‘𝑘) ↦ {𝑝 ∈ (Atoms‘𝑘) ∣ 𝑝(le‘𝑘)𝑎}))
cpadd 40165	class +𝑃
df-padd 40166	⊢ +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 40257	class PCl
df-pclN 40258	⊢ PCl = (𝑘 ∈ V ↦ (𝑥 ∈ 𝒫 (Atoms‘𝑘) ↦ ∩ {𝑦 ∈ (PSubSp‘𝑘) ∣ 𝑥 ⊆ 𝑦}))
cpolN 40272	class ⊥𝑃
df-polarityN 40273	⊢ ⊥𝑃 = (𝑙 ∈ V ↦ (𝑚 ∈ 𝒫 (Atoms‘𝑙) ↦ ((Atoms‘𝑙) ∩ ∩ 𝑝 ∈ 𝑚 ((pmap‘𝑙)‘((oc‘𝑙)‘𝑝)))))
cpscN 40304	class PSubCl
df-psubclN 40305	⊢ PSubCl = (𝑘 ∈ V ↦ {𝑠 ∣ (𝑠 ⊆ (Atoms‘𝑘) ∧ ((⊥𝑃‘𝑘)‘((⊥𝑃‘𝑘)‘𝑠)) = 𝑠)})
clh 40354	class LHyp
claut 40355	class LAut
cwpointsN 40356	class WAtoms
cpautN 40357	class PAut
df-lhyp 40358	⊢ LHyp = (𝑘 ∈ V ↦ {𝑥 ∈ (Base‘𝑘) ∣ 𝑥( ⋖ ‘𝑘)(1.‘𝑘)})
df-laut 40359	⊢ 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 40360	⊢ WAtoms = (𝑘 ∈ V ↦ (𝑑 ∈ (Atoms‘𝑘) ↦ ((Atoms‘𝑘) ∖ ((⊥𝑃‘𝑘)‘{𝑑}))))
df-pautN 40361	⊢ 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 40470	class LDil
cltrn 40471	class LTrn
cdilN 40472	class Dil
ctrnN 40473	class Trn
df-ldil 40474	⊢ LDil = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ {𝑓 ∈ (LAut‘𝑘) ∣ ∀𝑥 ∈ (Base‘𝑘)(𝑥(le‘𝑘)𝑤 → (𝑓‘𝑥) = 𝑥)}))
df-ltrn 40475	⊢ 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 40476	⊢ Dil = (𝑘 ∈ V ↦ (𝑑 ∈ (Atoms‘𝑘) ↦ {𝑓 ∈ (PAut‘𝑘) ∣ ∀𝑥 ∈ (PSubSp‘𝑘)(𝑥 ⊆ ((WAtoms‘𝑘)‘𝑑) → (𝑓‘𝑥) = 𝑥)}))
df-trnN 40477	⊢ 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 40528	class trL
df-trl 40529	⊢ 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 41112	class TGrp
df-tgrp 41113	⊢ TGrp = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ {⟨(Base‘ndx), ((LTrn‘𝑘)‘𝑤)⟩, ⟨(+g‘ndx), (𝑓 ∈ ((LTrn‘𝑘)‘𝑤), 𝑔 ∈ ((LTrn‘𝑘)‘𝑤) ↦ (𝑓 ∘ 𝑔))⟩}))
ctendo 41122	class TEndo
cedring 41123	class EDRing
cedring-rN 41124	class EDRingR
df-tendo 41125	⊢ 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 41126	⊢ EDRingR = (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 41127	⊢ 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 41372	class DVecA
df-dveca 41373	⊢ 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 41398	class DIsoA
df-disoa 41399	⊢ 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 41448	class DVecH
df-dvech 41449	⊢ 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)) ↦ ⟨((1st ‘𝑓) ∘ (1st ‘𝑔)), (ℎ ∈ ((LTrn‘𝑘)‘𝑤) ↦ (((2nd ‘𝑓)‘ℎ) ∘ ((2nd ‘𝑔)‘ℎ)))⟩)⟩, ⟨(Scalar‘ndx), ((EDRing‘𝑘)‘𝑤)⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑠 ∈ ((TEndo‘𝑘)‘𝑤), 𝑓 ∈ (((LTrn‘𝑘)‘𝑤) × ((TEndo‘𝑘)‘𝑤)) ↦ ⟨(𝑠‘(1st ‘𝑓)), (𝑠 ∘ (2nd ‘𝑓))⟩)⟩})))
cocaN 41489	class ocA
df-docaN 41490	⊢ 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 41501	class vA
df-djaN 41502	⊢ 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 41508	class DIsoB
df-dib 41509	⊢ DIsoB = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑥 ∈ dom ((DIsoA‘𝑘)‘𝑤) ↦ ((((DIsoA‘𝑘)‘𝑤)‘𝑥) × {(𝑓 ∈ ((LTrn‘𝑘)‘𝑤) ↦ ( I ↾ (Base‘𝑘)))}))))
cdic 41542	class DIsoC
df-dic 41543	⊢ 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 41598	class DIsoH
df-dih 41599	⊢ 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 41717	class ocH
df-doch 41718	⊢ 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 41764	class joinH
df-djh 41765	⊢ 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 41850	class LPol
df-lpolN 41851	⊢ LPol = (𝑤 ∈ V ↦ {𝑜 ∈ ((LSubSp‘𝑤) ↑m 𝒫 (Base‘𝑤)) ∣ ((𝑜‘(Base‘𝑤)) = {(0g‘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 41956	class LCDual
df-lcdual 41957	⊢ 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 41994	class mapd
df-mapd 41995	⊢ 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 42126	class HVMap
df-hvmap 42127	⊢ HVMap = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑥 ∈ ((Base‘((DVecH‘𝑘)‘𝑤)) ∖ {(0g‘((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 42161	class HDMap1
chdma 42162	class HDMap
df-hdmap1 42163	⊢ 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((2nd ‘𝑥) = (0g‘𝑢), (0g‘𝑐), (℩ℎ ∈ 𝑑 ((𝑚‘(𝑛‘{(2nd ‘𝑥)})) = (𝑗‘{ℎ}) ∧ (𝑚‘(𝑛‘{((1st ‘(1st ‘𝑥))(-g‘𝑢)(2nd ‘𝑥))})) = (𝑗‘{((2nd ‘(1st ‘𝑥))(-g‘𝑐)ℎ)})))))}))
df-hdmap 42164	⊢ 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 42253	class HGMap
df-hgmap 42254	⊢ 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 42302	class HLHil
df-hlhil 42303	⊢ 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 42332	class CSRing
df-csring 42333	⊢ CSRing = {𝑓 ∈ SRing ∣ (mulGrp‘𝑓) ∈ CMnd}
cprimroots 42455	class PrimRoots
df-primroots 42456	⊢ PrimRoots = (𝑟 ∈ CMnd, 𝑘 ∈ ℕ0 ↦ ⦋(Base‘𝑟) / 𝑏⦌{𝑎 ∈ 𝑏 ∣ ((𝑘(.g‘𝑟)𝑎) = (0g‘𝑟) ∧ ∀𝑙 ∈ ℕ0 ((𝑙(.g‘𝑟)𝑎) = (0g‘𝑟) → 𝑘 ∥ 𝑙))})
ax-exfinfld 42566	⊢ ∀𝑝 ∈ ℙ ∀𝑛 ∈ ℕ ∃𝑘 ∈ Field ((♯‘(Base‘𝑘)) = (𝑝↑𝑛) ∧ (chr‘𝑘) = 𝑝)
cresub 42729	class −ℝ
df-resub 42730	⊢ −ℝ = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (℩𝑧 ∈ ℝ (𝑦 + 𝑧) = 𝑥))
crediv 42804	class /ℝ
df-rediv 42805	⊢ /ℝ = (𝑥 ∈ ℝ, 𝑦 ∈ (ℝ ∖ {0}) ↦ (℩𝑧 ∈ ℝ (𝑦 · 𝑧) = 𝑥))
cprjsp 42953	class ℙ𝕣𝕠𝕛
df-prjsp 42954	⊢ ℙ𝕣𝕠𝕛 = (𝑣 ∈ LVec ↦ ⦋((Base‘𝑣) ∖ {(0g‘𝑣)}) / 𝑏⦌(𝑏 / {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝑏 ∧ 𝑦 ∈ 𝑏) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑣))𝑥 = (𝑙( ·𝑠 ‘𝑣)𝑦))}))
cprjspn 42966	class ℙ𝕣𝕠𝕛n
df-prjspn 42967	⊢ ℙ𝕣𝕠𝕛n = (𝑛 ∈ ℕ0, 𝑘 ∈ DivRing ↦ (ℙ𝕣𝕠𝕛‘(𝑘 freeLMod (0...𝑛))))
cprjcrv 42981	class ℙ𝕣𝕠𝕛Crv
df-prjcrv 42982	⊢ ℙ𝕣𝕠𝕛Crv = (𝑛 ∈ ℕ0, 𝑘 ∈ Field ↦ (𝑓 ∈ ∪ ran ((0...𝑛) mHomP 𝑘) ↦ {𝑝 ∈ (𝑛ℙ𝕣𝕠𝕛n𝑘) ∣ ((((0...𝑛) eval 𝑘)‘𝑓) “ 𝑝) = {(0g‘𝑘)}}))
cnacs 43053	class NoeACS
df-nacs 43054	⊢ NoeACS = (𝑥 ∈ V ↦ {𝑐 ∈ (ACS‘𝑥) ∣ ∀𝑠 ∈ 𝑐 ∃𝑔 ∈ (𝒫 𝑥 ∩ Fin)𝑠 = ((mrCls‘𝑐)‘𝑔)})
cmzpcl 43072	class mzPolyCld
cmzp 43073	class mzPoly
df-mzpcl 43074	⊢ 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 43075	⊢ mzPoly = (𝑣 ∈ V ↦ ∩ (mzPolyCld‘𝑣))
cdioph 43106	class Dioph
df-dioph 43107	⊢ 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 43187	class ◻NN
cpell1qr 43188	class Pell1QR
cpell1234qr 43189	class Pell1234QR
cpell14qr 43190	class Pell14QR
cpellfund 43191	class PellFund
df-squarenn 43192	⊢ ◻NN = {𝑥 ∈ ℕ ∣ (√‘𝑥) ∈ ℚ}
df-pell1qr 43193	⊢ Pell1QR = (𝑥 ∈ (ℕ ∖ ◻NN) ↦ {𝑦 ∈ ℝ ∣ ∃𝑧 ∈ ℕ0 ∃𝑤 ∈ ℕ0 (𝑦 = (𝑧 + ((√‘𝑥) · 𝑤)) ∧ ((𝑧↑2) − (𝑥 · (𝑤↑2))) = 1)})
df-pell14qr 43194	⊢ Pell14QR = (𝑥 ∈ (ℕ ∖ ◻NN) ↦ {𝑦 ∈ ℝ ∣ ∃𝑧 ∈ ℕ0 ∃𝑤 ∈ ℤ (𝑦 = (𝑧 + ((√‘𝑥) · 𝑤)) ∧ ((𝑧↑2) − (𝑥 · (𝑤↑2))) = 1)})
df-pell1234qr 43195	⊢ Pell1234QR = (𝑥 ∈ (ℕ ∖ ◻NN) ↦ {𝑦 ∈ ℝ ∣ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ (𝑦 = (𝑧 + ((√‘𝑥) · 𝑤)) ∧ ((𝑧↑2) − (𝑥 · (𝑤↑2))) = 1)})
df-pellfund 43196	⊢ PellFund = (𝑥 ∈ (ℕ ∖ ◻NN) ↦ inf({𝑧 ∈ (Pell14QR‘𝑥) ∣ 1 < 𝑧}, ℝ, < ))
crmx 43251	class Xrm
crmy 43252	class Yrm
df-rmx 43253	⊢ Xrm = (𝑎 ∈ (ℤ≥‘2), 𝑛 ∈ ℤ ↦ (1st ‘(◡(𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st ‘𝑏) + ((√‘((𝑎↑2) − 1)) · (2nd ‘𝑏))))‘((𝑎 + (√‘((𝑎↑2) − 1)))↑𝑛))))
df-rmy 43254	⊢ Yrm = (𝑎 ∈ (ℤ≥‘2), 𝑛 ∈ ℤ ↦ (2nd ‘(◡(𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st ‘𝑏) + ((√‘((𝑎↑2) − 1)) · (2nd ‘𝑏))))‘((𝑎 + (√‘((𝑎↑2) − 1)))↑𝑛))))
clfig 43418	class LFinGen
df-lfig 43419	⊢ LFinGen = {𝑤 ∈ LMod ∣ (Base‘𝑤) ∈ ((LSpan‘𝑤) “ (𝒫 (Base‘𝑤) ∩ Fin))}
clnm 43426	class LNoeM
df-lnm 43427	⊢ LNoeM = {𝑤 ∈ LMod ∣ ∀𝑖 ∈ (LSubSp‘𝑤)(𝑤 ↾s 𝑖) ∈ LFinGen}
clnr 43460	class LNoeR
df-lnr 43461	⊢ LNoeR = {𝑎 ∈ Ring ∣ (ringLMod‘𝑎) ∈ LNoeM}
cldgis 43472	class ldgIdlSeq
df-ldgis 43473	⊢ ldgIdlSeq = (𝑟 ∈ V ↦ (𝑖 ∈ (LIdeal‘(Poly1‘𝑟)) ↦ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘 ∈ 𝑖 (((deg1‘𝑟)‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe1‘𝑘)‘𝑥))})))
cmnc 43482	class Monic
cplylt 43483	class Poly<
df-mnc 43484	⊢ Monic = (𝑠 ∈ 𝒫 ℂ ↦ {𝑝 ∈ (Poly‘𝑠) ∣ ((coeff‘𝑝)‘(deg‘𝑝)) = 1})
df-plylt 43485	⊢ Poly< = (𝑠 ∈ 𝒫 ℂ, 𝑥 ∈ ℕ0 ↦ {𝑝 ∈ (Poly‘𝑠) ∣ (𝑝 = 0𝑝 ∨ (deg‘𝑝) < 𝑥)})
cdgraa 43491	class degAA
cmpaa 43492	class minPolyAA
df-dgraa 43493	⊢ degAA = (𝑥 ∈ 𝔸 ↦ inf({𝑑 ∈ ℕ ∣ ∃𝑝 ∈ ((Poly‘ℚ) ∖ {0𝑝})((deg‘𝑝) = 𝑑 ∧ (𝑝‘𝑥) = 0)}, ℝ, < ))
df-mpaa 43494	⊢ minPolyAA = (𝑥 ∈ 𝔸 ↦ (℩𝑝 ∈ (Poly‘ℚ)((deg‘𝑝) = (degAA‘𝑥) ∧ (𝑝‘𝑥) = 0 ∧ ((coeff‘𝑝)‘(degAA‘𝑥)) = 1)))
citgo 43508	class IntgOver
cza 43509	class ℤ
df-itgo 43510	⊢ IntgOver = (𝑠 ∈ 𝒫 ℂ ↦ {𝑥 ∈ ℂ ∣ ∃𝑝 ∈ (Poly‘𝑠)((𝑝‘𝑥) = 0 ∧ ((coeff‘𝑝)‘(deg‘𝑝)) = 1)})
df-za 43511	⊢ ℤ = (IntgOver‘ℤ)
cmend 43522	class MEndo
df-mend 43523	⊢ 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 43548	class CytP
df-cytp 43549	⊢ CytP = (𝑛 ∈ ℕ ↦ ((mulGrp‘(Poly1‘ℂfld)) Σg (𝑟 ∈ (◡(od‘((mulGrp‘ℂfld) ↾s (ℂ ∖ {0}))) “ {𝑛}) ↦ ((var1‘ℂfld)(-g‘(Poly1‘ℂfld))((algSc‘(Poly1‘ℂfld))‘𝑟)))))
ctopsep 43557	class TopSep
ctoplnd 43558	class TopLnd
df-topsep 43559	⊢ TopSep = {𝑗 ∈ Top ∣ ∃𝑥 ∈ 𝒫 ∪ 𝑗(𝑥 ≼ ω ∧ ((cls‘𝑗)‘𝑥) = ∪ 𝑗)}
df-toplnd 43560	⊢ TopLnd = {𝑥 ∈ Top ∣ ∀𝑦 ∈ 𝒫 𝑥(∪ 𝑥 = ∪ 𝑦 → ∃𝑧 ∈ 𝒫 𝑥(𝑧 ≼ ω ∧ ∪ 𝑥 = ∪ 𝑧))}
crcl 44022	class r*
df-rcl 44023	⊢ r* = (𝑥 ∈ V ↦ ∩ {𝑧 ∣ (𝑥 ⊆ 𝑧 ∧ ( I ↾ (dom 𝑧 ∪ ran 𝑧)) ⊆ 𝑧)})
whe 44122	wff 𝑅 hereditary 𝐴
df-he 44123	⊢ (𝑅 hereditary 𝐴 ↔ (𝑅 “ 𝐴) ⊆ 𝐴)
ax-frege1 44140	⊢ (𝜑 → (𝜓 → 𝜑))
ax-frege2 44141	⊢ ((𝜑 → (𝜓 → 𝜒)) → ((𝜑 → 𝜓) → (𝜑 → 𝜒)))
ax-frege8 44159	⊢ ((𝜑 → (𝜓 → 𝜒)) → (𝜓 → (𝜑 → 𝜒)))
ax-frege28 44180	⊢ ((𝜑 → 𝜓) → (¬ 𝜓 → ¬ 𝜑))
ax-frege31 44184	⊢ (¬ ¬ 𝜑 → 𝜑)
ax-frege41 44195	⊢ (𝜑 → ¬ ¬ 𝜑)
ax-frege52a 44207	⊢ ((𝜑 ↔ 𝜓) → (if-(𝜑, 𝜃, 𝜒) → if-(𝜓, 𝜃, 𝜒)))
ax-frege54a 44212	⊢ (𝜑 ↔ 𝜑)
ax-frege58a 44225	⊢ ((𝜓 ∧ 𝜒) → if-(𝜑, 𝜓, 𝜒))
ax-frege52c 44238	⊢ (𝐴 = 𝐵 → ([𝐴 / 𝑥]𝜑 → [𝐵 / 𝑥]𝜑))
ax-frege54c 44242	⊢ 𝐴 = 𝐴
ax-frege58b 44251	⊢ (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑)
cmnring 44561	class MndRing
df-mnring 44562	⊢ MndRing = (𝑟 ∈ V, 𝑚 ∈ V ↦ ⦋(𝑟 freeLMod (Base‘𝑚)) / 𝑣⦌(𝑣 sSet ⟨(.r‘ndx), (𝑥 ∈ (Base‘𝑣), 𝑦 ∈ (Base‘𝑣) ↦ (𝑣 Σg (𝑎 ∈ (Base‘𝑚), 𝑏 ∈ (Base‘𝑚) ↦ (𝑖 ∈ (Base‘𝑚) ↦ if(𝑖 = (𝑎(+g‘𝑚)𝑏), ((𝑥‘𝑎)(.r‘𝑟)(𝑦‘𝑏)), (0g‘𝑟))))))⟩))
cscott 44585	class Scott 𝐴
df-scott 44586	⊢ Scott 𝐴 = {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)}
ccoll 44600	class (𝐹 Coll 𝐴)
df-coll 44601	⊢ (𝐹 Coll 𝐴) = ∪ 𝑥 ∈ 𝐴 Scott (𝐹 “ {𝑥})
cbcc 44686	class C𝑐
df-bcc 44687	⊢ C𝑐 = (𝑐 ∈ ℂ, 𝑘 ∈ ℕ0 ↦ ((𝑐 FallFac 𝑘) / (!‘𝑘)))
cplusr 44806	class +𝑟
cminusr 44807	class -𝑟
ctimesr 44808	class .𝑣
cptdfc 44809	class PtDf(𝐴, 𝐵)
crr3c 44810	class RR3
cline3 44811	class line3
df-addr 44812	⊢ +𝑟 = (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑣 ∈ ℝ ↦ ((𝑥‘𝑣) + (𝑦‘𝑣))))
df-subr 44813	⊢ -𝑟 = (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑣 ∈ ℝ ↦ ((𝑥‘𝑣) − (𝑦‘𝑣))))
df-mulv 44814	⊢ .𝑣 = (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑣 ∈ ℝ ↦ (𝑥 · (𝑦‘𝑣))))
df-ptdf 44825	⊢ PtDf(𝐴, 𝐵) = (𝑥 ∈ ℝ ↦ (((𝑥.𝑣(𝐵-𝑟𝐴)) +𝑣 𝐴) “ {1, 2, 3}))
df-rr3 44826	⊢ RR3 = (ℝ ↑m {1, 2, 3})
df-line3 44827	⊢ line3 = {𝑥 ∈ 𝒫 RR3 ∣ (2o ≼ 𝑥 ∧ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑧 ≠ 𝑦 → ran PtDf(𝑦, 𝑧) = 𝑥))}
wvd1 44919	wff (   𝜑   ▶   𝜓   )
df-vd1 44920	⊢ ((   𝜑   ▶   𝜓   ) ↔ (𝜑 → 𝜓))
wvd2 44927	wff (   𝜑   ,   𝜓   ▶   𝜒   )
df-vd2 44928	⊢ ((   𝜑   ,   𝜓   ▶   𝜒   ) ↔ ((𝜑 ∧ 𝜓) → 𝜒))
wvhc2 44930	wff (   𝜑   ,   𝜓   )
df-vhc2 44931	⊢ ((   𝜑   ,   𝜓   ) ↔ (𝜑 ∧ 𝜓))
wvd3 44937	wff (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜃   )
wvhc3 44938	wff (   𝜑   ,   𝜓   ,   𝜒   )
df-vhc3 44939	⊢ ((   𝜑   ,   𝜓   ,   𝜒   ) ↔ (𝜑 ∧ 𝜓 ∧ 𝜒))
df-vd3 44940	⊢ ((   𝜑   ,   𝜓   ,   𝜒   ▶   𝜃   ) ↔ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃))
wrelp 45292	wff 𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵)
df-relp 45293	⊢ (𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵) ↔ (𝐻:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐻‘𝑥)𝑆(𝐻‘𝑦))))
clsi 46103	class lim inf
df-liminf 46104	⊢ lim inf = (𝑥 ∈ V ↦ sup(ran (𝑘 ∈ ℝ ↦ inf(((𝑥 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )), ℝ*, < ))
clsxlim 46170	class ~~>*
df-xlim 46171	⊢ ~~>* = (⇝𝑡‘(ordTop‘ ≤ ))
csalg 46660	class SAlg
df-salg 46661	⊢ SAlg = {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (∪ 𝑥 ∖ 𝑦) ∈ 𝑥 ∧ ∀𝑦 ∈ 𝒫 𝑥(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑥))}
csalon 46662	class SalOn
df-salon 46663	⊢ SalOn = (𝑥 ∈ V ↦ {𝑠 ∈ SAlg ∣ ∪ 𝑠 = 𝑥})
csalgen 46664	class SalGen
df-salgen 46665	⊢ SalGen = (𝑥 ∈ V ↦ ∩ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑥 ∧ 𝑥 ⊆ 𝑠)})
csumge0 46714	class Σ^
df-sumge0 46715	⊢ Σ^ = (𝑥 ∈ V ↦ if(+∞ ∈ ran 𝑥, +∞, sup(ran (𝑦 ∈ (𝒫 dom 𝑥 ∩ Fin) ↦ Σ𝑤 ∈ 𝑦 (𝑥‘𝑤)), ℝ*, < )))
cmea 46801	class Meas
df-mea 46802	⊢ 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 46841	class OutMeas
df-ome 46842	⊢ 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 46843	class CaraGen
df-caragen 46844	⊢ CaraGen = (𝑜 ∈ OutMeas ↦ {𝑒 ∈ 𝒫 ∪ dom 𝑜 ∣ ∀𝑎 ∈ 𝒫 ∪ dom 𝑜((𝑜‘(𝑎 ∩ 𝑒)) +𝑒 (𝑜‘(𝑎 ∖ 𝑒))) = (𝑜‘𝑎)})
covoln 46888	class voln*
df-ovoln 46889	⊢ 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 46890	class voln
df-voln 46891	⊢ voln = (𝑥 ∈ Fin ↦ ((voln*‘𝑥) ↾ (CaraGen‘(voln*‘𝑥))))
csmblfn 47047	class SMblFn
df-smblfn 47048	⊢ SMblFn = (𝑠 ∈ SAlg ↦ {𝑓 ∈ (ℝ ↑pm ∪ 𝑠) ∣ ∀𝑎 ∈ ℝ (◡𝑓 “ (-∞(,)𝑎)) ∈ (𝑠 ↾t dom 𝑓)})
caiota 47437	class (℩'𝑥𝜑)
df-aiota 47439	⊢ (℩'𝑥𝜑) = ∩ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}
wdfat 47470	wff 𝐹 defAt 𝐴
cafv 47471	class (𝐹'''𝐴)
caov 47472	class ((𝐴𝐹𝐵))
df-dfat 47473	⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
df-afv 47474	⊢ (𝐹'''𝐴) = (℩'𝑥𝐴𝐹𝑥)
df-aov 47475	⊢ ((𝐴𝐹𝐵)) = (𝐹'''⟨𝐴, 𝐵⟩)
cafv2 47562	class (𝐹''''𝐴)
df-afv2 47563	⊢ (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ∪ ran 𝐹)
cnelbr 47625	class _∉
df-nelbr 47626	⊢ _∉ = {⟨𝑥, 𝑦⟩ ∣ ¬ 𝑥 ∈ 𝑦}
ciccp 47767	class RePart
df-iccp 47768	⊢ RePart = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ* ↑m (0...𝑚)) ∣ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1))})
wich 47799	wff [𝑥⇄𝑦]𝜑
df-ich 47800	⊢ ([𝑥⇄𝑦]𝜑 ↔ ∀𝑥∀𝑦([𝑥 / 𝑎][𝑦 / 𝑥][𝑎 / 𝑦]𝜑 ↔ 𝜑))
cspr 47831	class Pairs
df-spr 47832	⊢ Pairs = (𝑣 ∈ V ↦ {𝑝 ∣ ∃𝑎 ∈ 𝑣 ∃𝑏 ∈ 𝑣 𝑝 = {𝑎, 𝑏}})
cprpr 47866	class Pairsproper
df-prpr 47867	⊢ Pairsproper = (𝑣 ∈ V ↦ {𝑝 ∣ ∃𝑎 ∈ 𝑣 ∃𝑏 ∈ 𝑣 (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})})
cfmtno 47881	class FermatNo
df-fmtno 47882	⊢ FermatNo = (𝑛 ∈ ℕ0 ↦ ((2↑(2↑𝑛)) + 1))
ceven 47978	class Even
codd 47979	class Odd
df-even 47980	⊢ Even = {𝑧 ∈ ℤ ∣ (𝑧 / 2) ∈ ℤ}
df-odd 47981	⊢ Odd = {𝑧 ∈ ℤ ∣ ((𝑧 + 1) / 2) ∈ ℤ}
cfppr 48078	class FPPr
df-fppr 48079	⊢ FPPr = (𝑛 ∈ ℕ ↦ {𝑥 ∈ (ℤ≥‘4) ∣ (𝑥 ∉ ℙ ∧ 𝑥 ∥ ((𝑛↑(𝑥 − 1)) − 1))})
cgbe 48099	class GoldbachEven
cgbow 48100	class GoldbachOddW
cgbo 48101	class GoldbachOdd
df-gbe 48102	⊢ GoldbachEven = {𝑧 ∈ Even ∣ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑧 = (𝑝 + 𝑞))}
df-gbow 48103	⊢ GoldbachOddW = {𝑧 ∈ Odd ∣ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑧 = ((𝑝 + 𝑞) + 𝑟)}
df-gbo 48104	⊢ GoldbachOdd = {𝑧 ∈ Odd ∣ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ ((𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑟 ∈ Odd ) ∧ 𝑧 = ((𝑝 + 𝑞) + 𝑟))}
ax-bgbltosilva 48164	⊢ ((𝑁 ∈ Even ∧ 4 < 𝑁 ∧ 𝑁 ≤ (4 · (;10↑;18))) → 𝑁 ∈ GoldbachEven )
ax-tgoldbachgt 48165	⊢ 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 48168	⊢ ∃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 48172	class ClNeighbVtx
df-clnbgr 48173	⊢ ClNeighbVtx = (𝑔 ∈ V, 𝑣 ∈ (Vtx‘𝑔) ↦ ({𝑣} ∪ {𝑛 ∈ (Vtx‘𝑔) ∣ ∃𝑒 ∈ (Edg‘𝑔){𝑣, 𝑛} ⊆ 𝑒}))
cisubgr 48214	class ISubGr
df-isubgr 48215	⊢ ISubGr = (𝑔 ∈ V, 𝑣 ∈ 𝒫 (Vtx‘𝑔) ↦ ⟨𝑣, ⦋(iEdg‘𝑔) / 𝑒⦌(𝑒 ↾ {𝑥 ∈ dom 𝑒 ∣ (𝑒‘𝑥) ⊆ 𝑣})⟩)
cgrisom 48228	class GraphIsom
cgrim 48229	class GraphIso
cgric 48230	class ≃𝑔𝑟
df-grisom 48231	⊢ 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 48232	⊢ 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 48235	⊢ ≃𝑔𝑟 = (◡ GraphIso “ (V ∖ 1o))
cgrtri 48291	class GrTriangles
df-grtri 48292	⊢ 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 48305	class StarGr
df-stgr 48306	⊢ StarGr = (𝑛 ∈ ℕ0 ↦ {⟨(Base‘ndx), (0...𝑛)⟩, ⟨(.ef‘ndx), ( I ↾ {𝑒 ∈ 𝒫 (0...𝑛) ∣ ∃𝑥 ∈ (1...𝑛)𝑒 = {0, 𝑥}})⟩})
cgrlim 48330	class GraphLocIso
cgrlic 48331	class ≃𝑙𝑔𝑟
df-grlim 48332	⊢ 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 48335	⊢ ≃𝑙𝑔𝑟 = (◡ GraphLocIso “ (V ∖ 1o))
cgpg 48394	class gPetersenGr
df-gpg 48395	⊢ 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 48487	class UPWalks
df-upwlks 48488	⊢ 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 48537	class clLaw
casslaw 48538	class assLaw
ccomlaw 48539	class comLaw
df-cllaw 48540	⊢ clLaw = {⟨𝑜, 𝑚⟩ ∣ ∀𝑥 ∈ 𝑚 ∀𝑦 ∈ 𝑚 (𝑥𝑜𝑦) ∈ 𝑚}
df-comlaw 48541	⊢ comLaw = {⟨𝑜, 𝑚⟩ ∣ ∀𝑥 ∈ 𝑚 ∀𝑦 ∈ 𝑚 (𝑥𝑜𝑦) = (𝑦𝑜𝑥)}
df-asslaw 48542	⊢ assLaw = {⟨𝑜, 𝑚⟩ ∣ ∀𝑥 ∈ 𝑚 ∀𝑦 ∈ 𝑚 ∀𝑧 ∈ 𝑚 ((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧))}
cintop 48550	class intOp
cclintop 48551	class clIntOp
cassintop 48552	class assIntOp
df-intop 48553	⊢ intOp = (𝑚 ∈ V, 𝑛 ∈ V ↦ (𝑛 ↑m (𝑚 × 𝑚)))
df-clintop 48554	⊢ clIntOp = (𝑚 ∈ V ↦ (𝑚 intOp 𝑚))
df-assintop 48555	⊢ assIntOp = (𝑚 ∈ V ↦ {𝑜 ∈ ( clIntOp ‘𝑚) ∣ 𝑜 assLaw 𝑚})
cmgm2 48569	class MgmALT
ccmgm2 48570	class CMgmALT
csgrp2 48571	class SGrpALT
ccsgrp2 48572	class CSGrpALT
df-mgm2 48573	⊢ MgmALT = {𝑚 ∣ (+g‘𝑚) clLaw (Base‘𝑚)}
df-cmgm2 48574	⊢ CMgmALT = {𝑚 ∈ MgmALT ∣ (+g‘𝑚) comLaw (Base‘𝑚)}
df-sgrp2 48575	⊢ SGrpALT = {𝑔 ∈ MgmALT ∣ (+g‘𝑔) assLaw (Base‘𝑔)}
df-csgrp2 48576	⊢ CSGrpALT = {𝑔 ∈ SGrpALT ∣ (+g‘𝑔) comLaw (Base‘𝑔)}
crngcALTV 48617	class RngCatALTV
df-rngcALTV 48618	⊢ 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 ∈ ((2nd ‘𝑣) RngHom 𝑧), 𝑓 ∈ ((1st ‘𝑣) RngHom (2nd ‘𝑣)) ↦ (𝑔 ∘ 𝑓)))⟩})
cringcALTV 48641	class RingCatALTV
df-ringcALTV 48642	⊢ 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 ∈ ((2nd ‘𝑣) RingHom 𝑧), 𝑓 ∈ ((1st ‘𝑣) RingHom (2nd ‘𝑣)) ↦ (𝑔 ∘ 𝑓)))⟩})
cdmatalt 48750	class DMatALT
cscmatalt 48751	class ScMatALT
df-dmatalt 48752	⊢ DMatALT = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ ⦋(𝑛 Mat 𝑟) / 𝑎⦌(𝑎 ↾s {𝑚 ∈ (Base‘𝑎) ∣ ∀𝑖 ∈ 𝑛 ∀𝑗 ∈ 𝑛 (𝑖 ≠ 𝑗 → (𝑖𝑚𝑗) = (0g‘𝑟))}))
df-scmatalt 48753	⊢ 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, (0g‘𝑟))}))
clinc 48758	class linC
clinco 48759	class LinCo
df-linc 48760	⊢ linC = (𝑚 ∈ V ↦ (𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑m 𝑣), 𝑣 ∈ 𝒫 (Base‘𝑚) ↦ (𝑚 Σg (𝑥 ∈ 𝑣 ↦ ((𝑠‘𝑥)( ·𝑠 ‘𝑚)𝑥)))))
df-lco 48761	⊢ LinCo = (𝑚 ∈ V, 𝑣 ∈ 𝒫 (Base‘𝑚) ↦ {𝑐 ∈ (Base‘𝑚) ∣ ∃𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑m 𝑣)(𝑠 finSupp (0g‘(Scalar‘𝑚)) ∧ 𝑐 = (𝑠( linC ‘𝑚)𝑣))})
clininds 48794	class linIndS
clindeps 48795	class linDepS
df-lininds 48796	⊢ linIndS = {⟨𝑠, 𝑚⟩ ∣ (𝑠 ∈ 𝒫 (Base‘𝑚) ∧ ∀𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑m 𝑠)((𝑓 finSupp (0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)𝑠) = (0g‘𝑚)) → ∀𝑥 ∈ 𝑠 (𝑓‘𝑥) = (0g‘(Scalar‘𝑚))))}
df-lindeps 48798	⊢ linDepS = {⟨𝑠, 𝑚⟩ ∣ ¬ 𝑠 linIndS 𝑚}
cfdiv 48891	class /f
df-fdiv 48892	⊢ /f = (𝑓 ∈ V, 𝑔 ∈ V ↦ ((𝑓 ∘f / 𝑔) ↾ (𝑔 supp 0)))
cbigo 48901	class Ο
df-bigo 48902	⊢ Ο = (𝑔 ∈ (ℝ ↑pm ℝ) ↦ {𝑓 ∈ (ℝ ↑pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(𝑓‘𝑦) ≤ (𝑚 · (𝑔‘𝑦))})
cblen 48923	class #b
df-blen 48924	⊢ #b = (𝑛 ∈ V ↦ if(𝑛 = 0, 1, ((⌊‘(2 logb (abs‘𝑛))) + 1)))
cdig 48949	class digit
df-dig 48950	⊢ digit = (𝑏 ∈ ℕ ↦ (𝑘 ∈ ℤ, 𝑟 ∈ (0[,)+∞) ↦ ((⌊‘((𝑏↑-𝑘) · 𝑟)) mod 𝑏)))
cnaryf 48980	class -aryF
df-naryf 48981	⊢ -aryF = (𝑛 ∈ ℕ0, 𝑥 ∈ V ↦ (𝑥 ↑m (𝑥 ↑m (0..^𝑛))))
citco 49011	class IterComp
cack 49012	class Ack
df-itco 49013	⊢ IterComp = (𝑓 ∈ V ↦ seq0((𝑔 ∈ V, 𝑗 ∈ V ↦ (𝑓 ∘ 𝑔)), (𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, ( I ↾ dom 𝑓), 𝑓))))
df-ack 49014	⊢ Ack = seq0((𝑓 ∈ V, 𝑗 ∈ V ↦ (𝑛 ∈ ℕ0 ↦ (((IterComp‘𝑓)‘(𝑛 + 1))‘1))), (𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, (𝑛 ∈ ℕ0 ↦ (𝑛 + 1)), 𝑖)))
cline 49081	class LineM
csph 49082	class Sphere
df-line 49083	⊢ LineM = (𝑤 ∈ V ↦ (𝑥 ∈ (Base‘𝑤), 𝑦 ∈ ((Base‘𝑤) ∖ {𝑥}) ↦ {𝑝 ∈ (Base‘𝑤) ∣ ∃𝑡 ∈ (Base‘(Scalar‘𝑤))𝑝 = ((((1r‘(Scalar‘𝑤))(-g‘(Scalar‘𝑤))𝑡)( ·𝑠 ‘𝑤)𝑥)(+g‘𝑤)(𝑡( ·𝑠 ‘𝑤)𝑦))}))
df-sph 49084	⊢ Sphere = (𝑤 ∈ V ↦ (𝑥 ∈ (Base‘𝑤), 𝑟 ∈ (0[,]+∞) ↦ {𝑝 ∈ (Base‘𝑤) ∣ (𝑝(dist‘𝑤)𝑥) = 𝑟}))
coppf 49475	class oppFunc
df-oppf 49476	⊢ oppFunc = (𝑓 ∈ V, 𝑔 ∈ V ↦ if((Rel 𝑔 ∧ Rel dom 𝑔), ⟨𝑓, tpos 𝑔⟩, ∅))
cup 49526	class UP
df-up 49527	⊢ 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((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝑏 ∀𝑔 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥ℎ𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦))𝑚))}))
cswapf 49612	class swapF
df-swapf 49613	⊢ swapF = (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 49669	class ∘F
df-fuco 49670	⊢ ∘F = (𝑝 ∈ V, 𝑒 ∈ V ↦ ⦋(1st ‘𝑝) / 𝑐⦌⦋(2nd ‘𝑝) / 𝑑⦌⦋((𝑑 Func 𝑒) × (𝑐 Func 𝑑)) / 𝑤⦌⟨( ∘func ↾ 𝑤), (𝑢 ∈ 𝑤, 𝑣 ∈ 𝑤 ↦ ⦋(1st ‘(2nd ‘𝑢)) / 𝑓⦌⦋(1st ‘(1st ‘𝑢)) / 𝑘⦌⦋(2nd ‘(1st ‘𝑢)) / 𝑙⦌⦋(1st ‘(2nd ‘𝑣)) / 𝑚⦌⦋(1st ‘(1st ‘𝑣)) / 𝑟⦌(𝑏 ∈ ((1st ‘𝑢)(𝑑 Nat 𝑒)(1st ‘𝑣)), 𝑎 ∈ ((2nd ‘𝑢)(𝑐 Nat 𝑑)(2nd ‘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 49726	class −∘F
df-prcof 49727	⊢ −∘F = (𝑝 ∈ V, 𝑓 ∈ V ↦ ⦋(1st ‘𝑝) / 𝑑⦌⦋(2nd ‘𝑝) / 𝑒⦌⦋(𝑑 Func 𝑒) / 𝑏⦌⟨(𝑘 ∈ 𝑏 ↦ (𝑘 ∘func 𝑓)), (𝑘 ∈ 𝑏, 𝑙 ∈ 𝑏 ↦ (𝑎 ∈ (𝑘(𝑑 Nat 𝑒)𝑙) ↦ (𝑎 ∘ (1st ‘𝑓))))⟩)
cthinc 49770	class ThinCat
df-thinc 49771	⊢ ThinCat = {𝑐 ∈ Cat ∣ [(Base‘𝑐) / 𝑏][(Hom ‘𝑐) / ℎ]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∃*𝑓 𝑓 ∈ (𝑥ℎ𝑦)}
ctermc 49825	class TermCat
df-termc 49826	⊢ TermCat = {𝑐 ∈ ThinCat ∣ ∃𝑥(Base‘𝑐) = {𝑥}}
cprstc 49902	class ProsetToCat
df-prstc 49903	⊢ ProsetToCat = (𝑘 ∈ Proset ↦ ((𝑘 sSet ⟨(Hom ‘ndx), ((le‘𝑘) × {1o})⟩) sSet ⟨(comp‘ndx), ∅⟩))
cmndtc 49930	class MndToCat
df-mndtc 49931	⊢ MndToCat = (𝑚 ∈ Mnd ↦ {⟨(Base‘ndx), {𝑚}⟩, ⟨(Hom ‘ndx), {⟨𝑚, 𝑚, (Base‘𝑚)⟩}⟩, ⟨(comp‘ndx), {⟨⟨𝑚, 𝑚, 𝑚⟩, (+g‘𝑚)⟩}⟩})
clan 49958	class Lan
cran 49959	class Ran
df-lan 49960	⊢ Lan = (𝑝 ∈ (V × V), 𝑒 ∈ V ↦ ⦋(1st ‘𝑝) / 𝑐⦌⦋(2nd ‘𝑝) / 𝑑⦌(𝑓 ∈ (𝑐 Func 𝑑), 𝑥 ∈ (𝑐 Func 𝑒) ↦ ((⟨𝑑, 𝑒⟩ −∘F 𝑓)((𝑑 FuncCat 𝑒) UP (𝑐 FuncCat 𝑒))𝑥)))
df-ran 49961	⊢ Ran = (𝑝 ∈ (V × V), 𝑒 ∈ V ↦ ⦋(1st ‘𝑝) / 𝑐⦌⦋(2nd ‘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 49996	class Limit
ccmd 49997	class Colimit
df-lmd 49998	⊢ Limit = (𝑐 ∈ V, 𝑑 ∈ V ↦ (𝑓 ∈ (𝑑 Func 𝑐) ↦ (( oppFunc ‘(𝑐Δfunc𝑑))((oppCat‘𝑐) UP (oppCat‘(𝑑 FuncCat 𝑐)))𝑓)))
df-cmd 49999	⊢ Colimit = (𝑐 ∈ V, 𝑑 ∈ V ↦ (𝑓 ∈ (𝑑 Func 𝑐) ↦ ((𝑐Δfunc𝑑)(𝑐 UP (𝑑 FuncCat 𝑐))𝑓)))
csetrecs 50036	class setrecs(𝐹)
df-setrecs 50037	⊢ setrecs(𝐹) = ∪ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)}
cpg 50062	class Pg
df-pg 50063	⊢ Pg = setrecs((𝑥 ∈ V ↦ (𝒫 𝑥 × 𝒫 𝑥)))
cge-real 50073	class ≥
cgt 50074	class >
df-gte 50075	⊢ ≥ = ◡ ≤
df-gt 50076	⊢ > = ◡ <
csinh 50083	class sinh
ccosh 50084	class cosh
ctanh 50085	class tanh
df-sinh 50086	⊢ sinh = (𝑥 ∈ ℂ ↦ ((sin‘(i · 𝑥)) / i))
df-cosh 50087	⊢ cosh = (𝑥 ∈ ℂ ↦ (cos‘(i · 𝑥)))
df-tanh 50088	⊢ tanh = (𝑥 ∈ (◡cosh “ (ℂ ∖ {0})) ↦ ((tan‘(i · 𝑥)) / i))
csec 50094	class sec
ccsc 50095	class csc
ccot 50096	class cot
df-sec 50097	⊢ sec = (𝑥 ∈ {𝑦 ∈ ℂ ∣ (cos‘𝑦) ≠ 0} ↦ (1 / (cos‘𝑥)))
df-csc 50098	⊢ csc = (𝑥 ∈ {𝑦 ∈ ℂ ∣ (sin‘𝑦) ≠ 0} ↦ (1 / (sin‘𝑥)))
df-cot 50099	⊢ cot = (𝑥 ∈ {𝑦 ∈ ℂ ∣ (sin‘𝑦) ≠ 0} ↦ ((cos‘𝑥) / (sin‘𝑥)))
clog- 50118	class log_
df-logbALT 50119	⊢ log_ = (𝑏 ∈ (ℂ ∖ {0, 1}) ↦ (𝑥 ∈ (ℂ ∖ {0}) ↦ ((log‘𝑥) / (log‘𝑏))))
wreflexive 50120	wff 𝑅Reflexive𝐴
df-reflexive 50121	⊢ (𝑅Reflexive𝐴 ↔ (𝑅 ⊆ (𝐴 × 𝐴) ∧ ∀𝑥 ∈ 𝐴 𝑥𝑅𝑥))
wirreflexive 50122	wff 𝑅Irreflexive𝐴
df-irreflexive 50123	⊢ (𝑅Irreflexive𝐴 ↔ (𝑅 ⊆ (𝐴 × 𝐴) ∧ ∀𝑥 ∈ 𝐴 ¬ 𝑥𝑅𝑥))
walsi 50139	wff ∀!𝑥(𝜑 → 𝜓)
walsc 50140	wff ∀!𝑥 ∈ 𝐴𝜑
df-alsi 50141	⊢ (∀!𝑥(𝜑 → 𝜓) ↔ (∀𝑥(𝜑 → 𝜓) ∧ ∃𝑥𝜑))
df-alsc 50142	⊢ (∀!𝑥 ∈ 𝐴𝜑 ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑥 𝑥 ∈ 𝐴))