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List of Syntax, Axioms (ax-) and Definitions (df-)
RefExpression (see link for any distinct variable requirements)
wn 3wff ¬ 𝜑
wi 4wff (𝜑𝜓)
ax-mp 5𝜑    &   (𝜑𝜓)       𝜓
ax-1 6(𝜑 → (𝜓𝜑))
ax-2 7((𝜑 → (𝜓𝜒)) → ((𝜑𝜓) → (𝜑𝜒)))
ax-3 8((¬ 𝜑 → ¬ 𝜓) → (𝜓𝜑))
wb 191wff (𝜑𝜓)
df-bi 192 ¬ (((𝜑𝜓) → ¬ ((𝜑𝜓) → ¬ (𝜓𝜑))) → ¬ (¬ ((𝜑𝜓) → ¬ (𝜓𝜑)) → (𝜑𝜓)))
wo 377wff (𝜑𝜓)
wa 378wff (𝜑𝜓)
df-or 379((𝜑𝜓) ↔ (¬ 𝜑𝜓))
df-an 380((𝜑𝜓) ↔ ¬ (𝜑 → ¬ 𝜓))
wif 996wff if-(𝜑, 𝜓, 𝜒)
df-ifp 997(if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑𝜓) ∨ (¬ 𝜑𝜒)))
w3o 1020wff (𝜑𝜓𝜒)
w3a 1021wff (𝜑𝜓𝜒)
df-3or 1022((𝜑𝜓𝜒) ↔ ((𝜑𝜓) ∨ 𝜒))
df-3an 1023((𝜑𝜓𝜒) ↔ ((𝜑𝜓) ∧ 𝜒))
wnan 1428wff (𝜑𝜓)
df-nan 1429((𝜑𝜓) ↔ ¬ (𝜑𝜓))
wxo 1446wff (𝜑𝜓)
df-xor 1447((𝜑𝜓) ↔ ¬ (𝜑𝜓))
wal 1466wff 𝑥𝜑
cv 1467class 𝑥
wceq 1468wff 𝐴 = 𝐵
wtru 1469wff
df-tru 1471(⊤ ↔ (∀𝑥 𝑥 = 𝑥 → ∀𝑥 𝑥 = 𝑥))
wfal 1473wff
df-fal 1474(⊥ ↔ ¬ ⊤)
whad 1520wff hadd(𝜑, 𝜓, 𝜒)
df-had 1521(hadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑𝜓) ⊻ 𝜒))
wcad 1533wff cadd(𝜑, 𝜓, 𝜒)
df-cad 1534(cadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑𝜓) ∨ (𝜒 ∧ (𝜑𝜓))))
wex 1692wff 𝑥𝜑
df-ex 1693(∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑)
wnf 1696wff 𝑥𝜑
df-nf 1697(Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑))
ax-gen 1698𝜑       𝑥𝜑
ax-4 1711(∀𝑥(𝜑𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))
ax-5 1789(𝜑 → ∀𝑥𝜑)
wsb 1828wff [𝑦 / 𝑥]𝜑
df-sb 1829([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
ax-6 1836 ¬ ∀𝑥 ¬ 𝑥 = 𝑦
ax-7 1883(𝑥 = 𝑦 → (𝑥 = 𝑧𝑦 = 𝑧))
wcel 1937wff 𝐴𝐵
ax-8 1939(𝑥 = 𝑦 → (𝑥𝑧𝑦𝑧))
ax-9 1946(𝑥 = 𝑦 → (𝑧𝑥𝑧𝑦))
ax-10 1965(¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
ax-11 1970(∀𝑥𝑦𝜑 → ∀𝑦𝑥𝜑)
ax-12 1983(𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
ax-13 2137𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
weu 2353wff ∃!𝑥𝜑
wmo 2354wff ∃*𝑥𝜑
df-eu 2357(∃!𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
df-mo 2358(∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
ax-ext 2485(∀𝑧(𝑧𝑥𝑧𝑦) → 𝑥 = 𝑦)
cab 2491class {𝑥𝜑}
df-clab 2492(𝑥 ∈ {𝑦𝜑} ↔ [𝑥 / 𝑦]𝜑)
df-cleq 2498(∀𝑥(𝑥𝑦𝑥𝑧) → 𝑦 = 𝑧)       (𝐴 = 𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
df-clel 2501(𝐴𝐵 ↔ ∃𝑥(𝑥 = 𝐴𝑥𝐵))
wnfc 2633wff 𝑥𝐴
df-nfc 2635(𝑥𝐴 ↔ ∀𝑦𝑥 𝑦𝐴)
wne 2675wff 𝐴𝐵
wnel 2676wff 𝐴𝐵
df-ne 2677(𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
df-nel 2678(𝐴𝐵 ↔ ¬ 𝐴𝐵)
wral 2791wff 𝑥𝐴 𝜑
wrex 2792wff 𝑥𝐴 𝜑
wreu 2793wff ∃!𝑥𝐴 𝜑
wrmo 2794wff ∃*𝑥𝐴 𝜑
crab 2795class {𝑥𝐴𝜑}
df-ral 2796(∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
df-rex 2797(∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
df-reu 2798(∃!𝑥𝐴 𝜑 ↔ ∃!𝑥(𝑥𝐴𝜑))
df-rmo 2799(∃*𝑥𝐴 𝜑 ↔ ∃*𝑥(𝑥𝐴𝜑))
df-rab 2800{𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
cvv 3066class V
df-v 3068V = {𝑥𝑥 = 𝑥}
wcdeq 3274wff CondEq(𝑥 = 𝑦𝜑)
df-cdeq 3275(CondEq(𝑥 = 𝑦𝜑) ↔ (𝑥 = 𝑦𝜑))
wsbc 3291wff [𝐴 / 𝑥]𝜑
df-sbc 3292([𝐴 / 𝑥]𝜑𝐴 ∈ {𝑥𝜑})
csb 3385class 𝐴 / 𝑥𝐵
df-csb 3386𝐴 / 𝑥𝐵 = {𝑦[𝐴 / 𝑥]𝑦𝐵}
cdif 3423class (𝐴𝐵)
cun 3424class (𝐴𝐵)
cin 3425class (𝐴𝐵)
wss 3426wff 𝐴𝐵
wpss 3427wff 𝐴𝐵
df-dif 3429(𝐴𝐵) = {𝑥 ∣ (𝑥𝐴 ∧ ¬ 𝑥𝐵)}
df-un 3431(𝐴𝐵) = {𝑥 ∣ (𝑥𝐴𝑥𝐵)}
df-in 3433(𝐴𝐵) = {𝑥 ∣ (𝑥𝐴𝑥𝐵)}
df-ss 3440(𝐴𝐵 ↔ (𝐴𝐵) = 𝐴)
df-pss 3442(𝐴𝐵 ↔ (𝐴𝐵𝐴𝐵))
csymdif 3689class (𝐴𝐵)
df-symdif 3690(𝐴𝐵) = ((𝐴𝐵) ∪ (𝐵𝐴))
c0 3757class
df-nul 3758∅ = (V ∖ V)
cif 3908class if(𝜑, 𝐴, 𝐵)
df-if 3909if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥𝐴𝜑) ∨ (𝑥𝐵 ∧ ¬ 𝜑))}
cpw 3978class 𝒫 𝐴
df-pw 3980𝒫 𝐴 = {𝑥𝑥𝐴}
csn 3995class {𝐴}
df-sn 3996{𝐴} = {𝑥𝑥 = 𝐴}
cpr 3997class {𝐴, 𝐵}
df-pr 3998{𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
ctp 3999class {𝐴, 𝐵, 𝐶}
df-tp 4000{𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
cop 4001class 𝐴, 𝐵
df-op 4002𝐴, 𝐵⟩ = {𝑥 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})}
cotp 4003class 𝐴, 𝐵, 𝐶
df-ot 4004𝐴, 𝐵, 𝐶⟩ = ⟨⟨𝐴, 𝐵⟩, 𝐶
cuni 4228class 𝐴
df-uni 4229 𝐴 = {𝑥 ∣ ∃𝑦(𝑥𝑦𝑦𝐴)}
cint 4264class 𝐴
df-int 4265 𝐴 = {𝑥 ∣ ∀𝑦(𝑦𝐴𝑥𝑦)}
ciun 4307class 𝑥𝐴 𝐵
ciin 4308class 𝑥𝐴 𝐵
df-iun 4309 𝑥𝐴 𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵}
df-iin 4310 𝑥𝐴 𝐵 = {𝑦 ∣ ∀𝑥𝐴 𝑦𝐵}
wdisj 4404wff Disj 𝑥𝐴 𝐵
df-disj 4405(Disj 𝑥𝐴 𝐵 ↔ ∀𝑦∃*𝑥𝐴 𝑦𝐵)
wbr 4434wff 𝐴𝑅𝐵
df-br 4435(𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
copab 4492class {⟨𝑥, 𝑦⟩ ∣ 𝜑}
cmpt 4493class (𝑥𝐴𝐵)
df-opab 4494{⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
df-mpt 4495(𝑥𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
wtr 4530wff Tr 𝐴
df-tr 4531(Tr 𝐴 𝐴𝐴)
ax-rep 4548(∀𝑤𝑦𝑧(∀𝑦𝜑𝑧 = 𝑦) → ∃𝑦𝑧(𝑧𝑦 ↔ ∃𝑤(𝑤𝑥 ∧ ∀𝑦𝜑)))
ax-sep 4558𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑))
ax-nul 4567𝑥𝑦 ¬ 𝑦𝑥
ax-pow 4619𝑦𝑧(∀𝑤(𝑤𝑧𝑤𝑥) → 𝑧𝑦)
ax-pr 4680𝑧𝑤((𝑤 = 𝑥𝑤 = 𝑦) → 𝑤𝑧)
cep 4789class E
cid 4790class I
df-eprel 4791 E = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦}
df-id 4795 I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
wpo 4799wff 𝑅 Po 𝐴
wor 4800wff 𝑅 Or 𝐴
df-po 4801(𝑅 Po 𝐴 ↔ ∀𝑥𝐴𝑦𝐴𝑧𝐴𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
df-so 4802(𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
wfr 4836wff 𝑅 Fr 𝐴
wse 4837wff 𝑅 Se 𝐴
wwe 4838wff 𝑅 We 𝐴
df-fr 4839(𝑅 Fr 𝐴 ↔ ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦))
df-se 4840(𝑅 Se 𝐴 ↔ ∀𝑥𝐴 {𝑦𝐴𝑦𝑅𝑥} ∈ V)
df-we 4841(𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴𝑅 Or 𝐴))
cxp 4878class (𝐴 × 𝐵)
ccnv 4879class 𝐴
cdm 4880class dom 𝐴
crn 4881class ran 𝐴
cres 4882class (𝐴𝐵)
cima 4883class (𝐴𝐵)
ccom 4884class (𝐴𝐵)
wrel 4885wff Rel 𝐴
df-xp 4886(𝐴 × 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝐵)}
df-rel 4887(Rel 𝐴𝐴 ⊆ (V × V))
df-cnv 4888𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥}
df-co 4889(𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐵𝑧𝑧𝐴𝑦)}
df-dm 4890dom 𝐴 = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦}
df-rn 4891ran 𝐴 = dom 𝐴
df-res 4892(𝐴𝐵) = (𝐴 ∩ (𝐵 × V))
df-ima 4893(𝐴𝐵) = ran (𝐴𝐵)
cpred 5430class Pred(𝑅, 𝐴, 𝑋)
df-pred 5431Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (𝑅 “ {𝑋}))
word 5473wff Ord 𝐴
con0 5474class On
wlim 5475wff Lim 𝐴
csuc 5476class suc 𝐴
df-ord 5477(Ord 𝐴 ↔ (Tr 𝐴 ∧ E We 𝐴))
df-on 5478On = {𝑥 ∣ Ord 𝑥}
df-lim 5479(Lim 𝐴 ↔ (Ord 𝐴𝐴 ≠ ∅ ∧ 𝐴 = 𝐴))
df-suc 5480suc 𝐴 = (𝐴 ∪ {𝐴})
cio 5595class (℩𝑥𝜑)
df-iota 5597(℩𝑥𝜑) = {𝑦 ∣ {𝑥𝜑} = {𝑦}}
wfun 5627wff Fun 𝐴
wfn 5628wff 𝐴 Fn 𝐵
wf 5629wff 𝐹:𝐴𝐵
wf1 5630wff 𝐹:𝐴1-1𝐵
wfo 5631wff 𝐹:𝐴onto𝐵
wf1o 5632wff 𝐹:𝐴1-1-onto𝐵
cfv 5633class (𝐹𝐴)
wiso 5634wff 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵)
df-fun 5635(Fun 𝐴 ↔ (Rel 𝐴 ∧ (𝐴𝐴) ⊆ I ))
df-fn 5636(𝐴 Fn 𝐵 ↔ (Fun 𝐴 ∧ dom 𝐴 = 𝐵))
df-f 5637(𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵))
df-f1 5638(𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ Fun 𝐹))
df-fo 5639(𝐹:𝐴onto𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵))
df-f1o 5640(𝐹:𝐴1-1-onto𝐵 ↔ (𝐹:𝐴1-1𝐵𝐹:𝐴onto𝐵))
df-fv 5641(𝐹𝐴) = (℩𝑥𝐴𝐹𝑥)
df-isom 5642(𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦))))
crio 6324class (𝑥𝐴 𝜑)
df-riota 6325(𝑥𝐴 𝜑) = (℩𝑥(𝑥𝐴𝜑))
co 6363class (𝐴𝐹𝐵)
coprab 6364class {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
cmpt2 6365class (𝑥𝐴, 𝑦𝐵𝐶)
df-ov 6366(𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
df-oprab 6367{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)}
df-mpt2 6368(𝑥𝐴, 𝑦𝐵𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
cof 6605class 𝑓 𝑅
cofr 6606class 𝑟 𝑅
df-of 6607𝑓 𝑅 = (𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓𝑥)𝑅(𝑔𝑥))))
df-ofr 6608𝑟 𝑅 = {⟨𝑓, 𝑔⟩ ∣ ∀𝑥 ∈ (dom 𝑓 ∩ dom 𝑔)(𝑓𝑥)𝑅(𝑔𝑥)}
crpss 6646class []
df-rpss 6647 [] = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦}
ax-un 6659𝑦𝑧(∃𝑤(𝑧𝑤𝑤𝑥) → 𝑧𝑦)
com 6769class ω
df-om 6770ω = {𝑥 ∈ On ∣ ∀𝑦(Lim 𝑦𝑥𝑦)}
c1st 6868class 1st
c2nd 6869class 2nd
df-1st 68701st = (𝑥 ∈ V ↦ dom {𝑥})
df-2nd 68712nd = (𝑥 ∈ V ↦ ran {𝑥})
csupp 6993class supp
df-supp 6994 supp = (𝑥 ∈ V, 𝑧 ∈ V ↦ {𝑖 ∈ dom 𝑥 ∣ (𝑥 “ {𝑖}) ≠ {𝑧}})
ctpos 7049class tpos 𝐹
df-tpos 7050tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}))
ccur 7089class curry 𝐴
cunc 7090class uncurry 𝐴
df-cur 7091curry 𝐹 = (𝑥 ∈ dom dom 𝐹 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦𝐹𝑧})
df-unc 7092uncurry 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(𝐹𝑥)𝑧}
cund 7096class Undef
df-undef 7097Undef = (𝑠 ∈ V ↦ 𝒫 𝑠)
cwrecs 7104class wrecs(𝑅, 𝐴, 𝐹)
df-wrecs 7105wrecs(𝑅, 𝐴, 𝐹) = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
wsmo 7141wff Smo 𝐴
df-smo 7142(Smo 𝐴 ↔ (𝐴:dom 𝐴⟶On ∧ Ord dom 𝐴 ∧ ∀𝑥 ∈ dom 𝐴𝑦 ∈ dom 𝐴(𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦))))
crecs 7166class recs(𝐹)
df-recs 7167recs(𝐹) = wrecs( E , On, 𝐹)
crdg 7204class rec(𝐹, 𝐼)
df-rdg 7205rec(𝐹, 𝐼) = recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔))))))
cseqom 7241class seq𝜔(𝐹, 𝐼)
df-seqom 7242seq𝜔(𝐹, 𝐼) = (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) “ ω)
c1o 7252class 1𝑜
c2o 7253class 2𝑜
c3o 7254class 3𝑜
c4o 7255class 4𝑜
coa 7256class +𝑜
comu 7257class ·𝑜
coe 7258class 𝑜
df-1o 72591𝑜 = suc ∅
df-2o 72602𝑜 = suc 1𝑜
df-3o 72613𝑜 = suc 2𝑜
df-4o 72624𝑜 = suc 3𝑜
df-oadd 7263 +𝑜 = (𝑥 ∈ On, 𝑦 ∈ On ↦ (rec((𝑧 ∈ V ↦ suc 𝑧), 𝑥)‘𝑦))
df-omul 7264 ·𝑜 = (𝑥 ∈ On, 𝑦 ∈ On ↦ (rec((𝑧 ∈ V ↦ (𝑧 +𝑜 𝑥)), ∅)‘𝑦))
df-oexp 7265𝑜 = (𝑥 ∈ On, 𝑦 ∈ On ↦ if(𝑥 = ∅, (1𝑜𝑦), (rec((𝑧 ∈ V ↦ (𝑧 ·𝑜 𝑥)), 1𝑜)‘𝑦)))
wer 7437wff 𝑅 Er 𝐴
cec 7438class [𝐴]𝑅
cqs 7439class (𝐴 / 𝑅)
df-er 7440(𝑅 Er 𝐴 ↔ (Rel 𝑅 ∧ dom 𝑅 = 𝐴 ∧ (𝑅 ∪ (𝑅𝑅)) ⊆ 𝑅))
df-ec 7442[𝐴]𝑅 = (𝑅 “ {𝐴})
df-qs 7446(𝐴 / 𝑅) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = [𝑥]𝑅}
cmap 7555class 𝑚
cpm 7556class pm
df-map 7557𝑚 = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓𝑓:𝑦𝑥})
df-pm 7558pm = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓 ∈ 𝒫 (𝑦 × 𝑥) ∣ Fun 𝑓})
cixp 7605class X𝑥𝐴 𝐵
df-ixp 7606X𝑥𝐴 𝐵 = {𝑓 ∣ (𝑓 Fn {𝑥𝑥𝐴} ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵)}
cen 7649class
cdom 7650class
csdm 7651class
cfn 7652class Fin
df-en 7653 ≈ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥1-1-onto𝑦}
df-dom 7654 ≼ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥1-1𝑦}
df-sdom 7655 ≺ = ( ≼ ∖ ≈ )
df-fin 7656Fin = {𝑥 ∣ ∃𝑦 ∈ ω 𝑥𝑦}
cfsupp 7968class finSupp
df-fsupp 7969 finSupp = {⟨𝑟, 𝑧⟩ ∣ (Fun 𝑟 ∧ (𝑟 supp 𝑧) ∈ Fin)}
cfi 8009class fi
df-fi 8010fi = (𝑥 ∈ V ↦ {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = 𝑦})
csup 8039class sup(𝐴, 𝐵, 𝑅)
cinf 8040class inf(𝐴, 𝐵, 𝑅)
df-sup 8041sup(𝐴, 𝐵, 𝑅) = {𝑥𝐵 ∣ (∀𝑦𝐴 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐵 (𝑦𝑅𝑥 → ∃𝑧𝐴 𝑦𝑅𝑧))}
df-inf 8042inf(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, 𝑅)
coi 8107class OrdIso(𝑅, 𝐴)
df-oi 8108OrdIso(𝑅, 𝐴) = if((𝑅 We 𝐴𝑅 Se 𝐴), (recs(( ∈ V ↦ (𝑣 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤}∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣))) ↾ {𝑥 ∈ On ∣ ∃𝑡𝐴𝑧 ∈ (recs(( ∈ V ↦ (𝑣 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤}∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣))) “ 𝑥)𝑧𝑅𝑡}), ∅)
char 8154class har
cwdom 8155class *
df-har 8156har = (𝑥 ∈ V ↦ {𝑦 ∈ On ∣ 𝑦𝑥})
df-wdom 8157* = {⟨𝑥, 𝑦⟩ ∣ (𝑥 = ∅ ∨ ∃𝑧 𝑧:𝑦onto𝑥)}
ax-reg 8190(∃𝑦 𝑦𝑥 → ∃𝑦(𝑦𝑥 ∧ ∀𝑧(𝑧𝑦 → ¬ 𝑧𝑥)))
ax-inf 8228𝑦(𝑥𝑦 ∧ ∀𝑧(𝑧𝑦 → ∃𝑤(𝑧𝑤𝑤𝑦)))
ax-inf2 8231𝑥(∃𝑦(𝑦𝑥 ∧ ∀𝑧 ¬ 𝑧𝑦) ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑧𝑥 ∧ ∀𝑤(𝑤𝑧 ↔ (𝑤𝑦𝑤 = 𝑦)))))
ccnf 8251class CNF
df-cnf 8252 CNF = (𝑥 ∈ On, 𝑦 ∈ On ↦ (𝑓 ∈ {𝑔 ∈ (𝑥𝑚 𝑦) ∣ 𝑔 finSupp ∅} ↦ OrdIso( E , (𝑓 supp ∅)) / (seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝑥𝑜 (𝑘)) ·𝑜 (𝑓‘(𝑘))) +𝑜 𝑧)), ∅)‘dom )))
ctc 8305class TC
df-tc 8306TC = (𝑥 ∈ V ↦ {𝑦 ∣ (𝑥𝑦 ∧ Tr 𝑦)})
cr1 8318class 𝑅1
crnk 8319class rank
df-r1 8320𝑅1 = rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅)
df-rank 8321rank = (𝑥 ∈ V ↦ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)})
ccrd 8454class card
cale 8455class
ccf 8456class cf
wacn 8457class AC 𝐴
df-card 8458card = (𝑥 ∈ V ↦ {𝑦 ∈ On ∣ 𝑦𝑥})
df-aleph 8459ℵ = rec(har, ω)
df-cf 8460cf = (𝑥 ∈ On ↦ {𝑦 ∣ ∃𝑧(𝑦 = (card‘𝑧) ∧ (𝑧𝑥 ∧ ∀𝑣𝑥𝑢𝑧 𝑣𝑢))})
df-acn 8461AC 𝐴 = {𝑥 ∣ (𝐴 ∈ V ∧ ∀𝑓 ∈ ((𝒫 𝑥 ∖ {∅}) ↑𝑚 𝐴)∃𝑔𝑦𝐴 (𝑔𝑦) ∈ (𝑓𝑦))}
wac 8631wff CHOICE
df-ac 8632(CHOICE ↔ ∀𝑥𝑓(𝑓𝑥𝑓 Fn dom 𝑥))
ccda 8682class +𝑐
df-cda 8683 +𝑐 = (𝑥 ∈ V, 𝑦 ∈ V ↦ ((𝑥 × {∅}) ∪ (𝑦 × {1𝑜})))
cfin1a 8793class FinIa
cfin2 8794class FinII
cfin4 8795class FinIV
cfin3 8796class FinIII
cfin5 8797class FinV
cfin6 8798class FinVI
cfin7 8799class FinVII
df-fin1a 8800FinIa = {𝑥 ∣ ∀𝑦 ∈ 𝒫 𝑥(𝑦 ∈ Fin ∨ (𝑥𝑦) ∈ Fin)}
df-fin2 8801FinII = {𝑥 ∣ ∀𝑦 ∈ 𝒫 𝒫 𝑥((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦)}
df-fin4 8802FinIV = {𝑥 ∣ ¬ ∃𝑦(𝑦𝑥𝑦𝑥)}
df-fin3 8803FinIII = {𝑥 ∣ 𝒫 𝑥 ∈ FinIV}
df-fin5 8804FinV = {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 ≺ (𝑥 +𝑐 𝑥))}
df-fin6 8805FinVI = {𝑥 ∣ (𝑥 ≺ 2𝑜𝑥 ≺ (𝑥 × 𝑥))}
df-fin7 8806FinVII = {𝑥 ∣ ¬ ∃𝑦 ∈ (On ∖ ω)𝑥𝑦}
ax-cc 8950(𝑥 ≈ ω → ∃𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧))
ax-dc 8961((∃𝑦𝑧 𝑦𝑥𝑧 ∧ ran 𝑥 ⊆ dom 𝑥) → ∃𝑓𝑛 ∈ ω (𝑓𝑛)𝑥(𝑓‘suc 𝑛))
ax-ac 8974𝑦𝑧𝑤((𝑧𝑤𝑤𝑥) → ∃𝑣𝑢(∃𝑡((𝑢𝑤𝑤𝑡) ∧ (𝑢𝑡𝑡𝑦)) ↔ 𝑢 = 𝑣))
ax-ac2 8978𝑦𝑧𝑣𝑢((𝑦𝑥 ∧ (𝑧𝑦 → ((𝑣𝑥 ∧ ¬ 𝑦 = 𝑣) ∧ 𝑧𝑣))) ∨ (¬ 𝑦𝑥 ∧ (𝑧𝑥 → ((𝑣𝑧𝑣𝑦) ∧ ((𝑢𝑧𝑢𝑦) → 𝑢 = 𝑣)))))
cgch 9130class GCH
df-gch 9131GCH = (Fin ∪ {𝑥 ∣ ∀𝑦 ¬ (𝑥𝑦𝑦 ≺ 𝒫 𝑥)})
cwina 9192class Inaccw
cina 9193class Inacc
df-wina 9194Inaccw = {𝑥 ∣ (𝑥 ≠ ∅ ∧ (cf‘𝑥) = 𝑥 ∧ ∀𝑦𝑥𝑧𝑥 𝑦𝑧)}
df-ina 9195Inacc = {𝑥 ∣ (𝑥 ≠ ∅ ∧ (cf‘𝑥) = 𝑥 ∧ ∀𝑦𝑥 𝒫 𝑦𝑥)}
cwun 9210class WUni
cwunm 9211class wUniCl
df-wun 9212WUni = {𝑢 ∣ (Tr 𝑢𝑢 ≠ ∅ ∧ ∀𝑥𝑢 ( 𝑥𝑢 ∧ 𝒫 𝑥𝑢 ∧ ∀𝑦𝑢 {𝑥, 𝑦} ∈ 𝑢))}
df-wunc 9213wUniCl = (𝑥 ∈ V ↦ {𝑢 ∈ WUni ∣ 𝑥𝑢})
ctsk 9258class Tarski
df-tsk 9259Tarski = {𝑦 ∣ (∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ ∃𝑤𝑦 𝒫 𝑧𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧𝑦𝑧𝑦))}
cgru 9300class Univ
df-gru 9301Univ = {𝑢 ∣ (Tr 𝑢 ∧ ∀𝑥𝑢 (𝒫 𝑥𝑢 ∧ ∀𝑦𝑢 {𝑥, 𝑦} ∈ 𝑢 ∧ ∀𝑦 ∈ (𝑢𝑚 𝑥) ran 𝑦𝑢))}
ax-groth 9333𝑦(𝑥𝑦 ∧ ∀𝑧𝑦 (∀𝑤(𝑤𝑧𝑤𝑦) ∧ ∃𝑤𝑦𝑣(𝑣𝑧𝑣𝑤)) ∧ ∀𝑧(𝑧𝑦 → (𝑧𝑦𝑧𝑦)))
ctskm 9347class tarskiMap
df-tskm 9348tarskiMap = (𝑥 ∈ V ↦ {𝑦 ∈ Tarski ∣ 𝑥𝑦})
cnpi 9354class N
cpli 9355class +N
cmi 9356class ·N
clti 9357class <N
cplpq 9358class +pQ
cmpq 9359class ·pQ
cltpq 9360class <pQ
ceq 9361class ~Q
cnq 9362class Q
c1q 9363class 1Q
cerq 9364class [Q]
cplq 9365class +Q
cmq 9366class ·Q
crq 9367class *Q
cltq 9368class <Q
cnp 9369class P
c1p 9370class 1P
cpp 9371class +P
cmp 9372class ·P
cltp 9373class <P
cer 9374class ~R
cnr 9375class R
c0r 9376class 0R
c1r 9377class 1R
cm1r 9378class -1R
cplr 9379class +R
cmr 9380class ·R
cltr 9381class <R
df-ni 9382N = (ω ∖ {∅})
df-pli 9383 +N = ( +𝑜 ↾ (N × N))
df-mi 9384 ·N = ( ·𝑜 ↾ (N × N))
df-lti 9385 <N = ( E ∩ (N × N))
df-plpq 9418 +pQ = (𝑥 ∈ (N × N), 𝑦 ∈ (N × N) ↦ ⟨(((1st𝑥) ·N (2nd𝑦)) +N ((1st𝑦) ·N (2nd𝑥))), ((2nd𝑥) ·N (2nd𝑦))⟩)
df-mpq 9419 ·pQ = (𝑥 ∈ (N × N), 𝑦 ∈ (N × N) ↦ ⟨((1st𝑥) ·N (1st𝑦)), ((2nd𝑥) ·N (2nd𝑦))⟩)
df-ltpq 9420 <pQ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ((1st𝑥) ·N (2nd𝑦)) <N ((1st𝑦) ·N (2nd𝑥)))}
df-enq 9421 ~Q = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·N 𝑢) = (𝑤 ·N 𝑣)))}
df-nq 9422Q = {𝑥 ∈ (N × N) ∣ ∀𝑦 ∈ (N × N)(𝑥 ~Q 𝑦 → ¬ (2nd𝑦) <N (2nd𝑥))}
df-erq 9423[Q] = ( ~Q ∩ ((N × N) × Q))
df-plq 9424 +Q = (([Q] ∘ +pQ ) ↾ (Q × Q))
df-mq 9425 ·Q = (([Q] ∘ ·pQ ) ↾ (Q × Q))
df-1nq 94261Q = ⟨1𝑜, 1𝑜
df-rq 9427*Q = ( ·Q “ {1Q})
df-ltnq 9428 <Q = ( <pQ ∩ (Q × Q))
df-np 9491P = {𝑥 ∣ ((∅ ⊊ 𝑥𝑥Q) ∧ ∀𝑦𝑥 (∀𝑧(𝑧 <Q 𝑦𝑧𝑥) ∧ ∃𝑧𝑥 𝑦 <Q 𝑧))}
df-1p 94921P = {𝑥𝑥 <Q 1Q}
df-plp 9493 +P = (𝑥P, 𝑦P ↦ {𝑤 ∣ ∃𝑣𝑥𝑢𝑦 𝑤 = (𝑣 +Q 𝑢)})
df-mp 9494 ·P = (𝑥P, 𝑦P ↦ {𝑤 ∣ ∃𝑣𝑥𝑢𝑦 𝑤 = (𝑣 ·Q 𝑢)})
df-ltp 9495<P = {⟨𝑥, 𝑦⟩ ∣ ((𝑥P𝑦P) ∧ 𝑥𝑦)}
df-enr 9565 ~R = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (P × P) ∧ 𝑦 ∈ (P × P)) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 +P 𝑢) = (𝑤 +P 𝑣)))}
df-nr 9566R = ((P × P) / ~R )
df-plr 9567 +R = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥R𝑦R) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~R𝑦 = [⟨𝑢, 𝑓⟩] ~R ) ∧ 𝑧 = [⟨(𝑤 +P 𝑢), (𝑣 +P 𝑓)⟩] ~R ))}
df-mr 9568 ·R = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥R𝑦R) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~R𝑦 = [⟨𝑢, 𝑓⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑓)), ((𝑤 ·P 𝑓) +P (𝑣 ·P 𝑢))⟩] ~R ))}
df-ltr 9569 <R = {⟨𝑥, 𝑦⟩ ∣ ((𝑥R𝑦R) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = [⟨𝑧, 𝑤⟩] ~R𝑦 = [⟨𝑣, 𝑢⟩] ~R ) ∧ (𝑧 +P 𝑢)<P (𝑤 +P 𝑣)))}
df-0r 95700R = [⟨1P, 1P⟩] ~R
df-1r 95711R = [⟨(1P +P 1P), 1P⟩] ~R
df-m1r 9572-1R = [⟨1P, (1P +P 1P)⟩] ~R
cc 9622class
cr 9623class
cc0 9624class 0
c1 9625class 1
ci 9626class i
caddc 9627class +
cltrr 9628class <
cmul 9629class ·
df-c 9630ℂ = (R × R)
df-0 96310 = ⟨0R, 0R
df-1 96321 = ⟨1R, 0R
df-i 9633i = ⟨0R, 1R
df-r 9634ℝ = (R × {0R})
df-add 9635 + = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨(𝑤 +R 𝑢), (𝑣 +R 𝑓)⟩))}
df-mul 9636 · = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) ∧ ∃𝑤𝑣𝑢𝑓((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 9637 < = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ∃𝑧𝑤((𝑥 = ⟨𝑧, 0R⟩ ∧ 𝑦 = ⟨𝑤, 0R⟩) ∧ 𝑧 <R 𝑤))}
ax-cnex 9680ℂ ∈ V
ax-resscn 9681ℝ ⊆ ℂ
ax-1cn 96821 ∈ ℂ
ax-icn 9683i ∈ ℂ
ax-addcl 9684((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) ∈ ℂ)
ax-addrcl 9685((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + 𝐵) ∈ ℝ)
ax-mulcl 9686((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) ∈ ℂ)
ax-mulrcl 9687((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 · 𝐵) ∈ ℝ)
ax-mulcom 9688((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) = (𝐵 · 𝐴))
ax-addass 9689((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶)))
ax-mulass 9690((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)))
ax-distr 9691((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶)))
ax-i2m1 9692((i · i) + 1) = 0
ax-1ne0 96931 ≠ 0
ax-1rid 9694(𝐴 ∈ ℝ → (𝐴 · 1) = 𝐴)
ax-rnegex 9695(𝐴 ∈ ℝ → ∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0)
ax-rrecex 9696((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → ∃𝑥 ∈ ℝ (𝐴 · 𝑥) = 1)
ax-cnre 9697(𝐴 ∈ ℂ → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝐴 = (𝑥 + (i · 𝑦)))
ax-pre-lttri 9698((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ ¬ (𝐴 = 𝐵𝐵 < 𝐴)))
ax-pre-lttrn 9699((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 < 𝐵𝐵 < 𝐶) → 𝐴 < 𝐶))
ax-pre-ltadd 9700((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐵 → (𝐶 + 𝐴) < (𝐶 + 𝐵)))
ax-pre-mulgt0 9701((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((0 < 𝐴 ∧ 0 < 𝐵) → 0 < (𝐴 · 𝐵)))
ax-pre-sup 9702((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑦 < 𝑥) → ∃𝑥 ∈ ℝ (∀𝑦𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧)))
ax-addf 9703 + :(ℂ × ℂ)⟶ℂ
ax-mulf 9704 · :(ℂ × ℂ)⟶ℂ
cpnf 9757class +∞
cmnf 9758class -∞
cxr 9759class *
clt 9760class <
cle 9761class
df-pnf 9762+∞ = 𝒫
df-mnf 9763-∞ = 𝒫 +∞
df-xr 9764* = (ℝ ∪ {+∞, -∞})
df-ltxr 9765 < = ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)} ∪ (((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ)))
df-le 9766 ≤ = ((ℝ* × ℝ*) ∖ < )
cmin 9947class
cneg 9948class -𝐴
df-sub 9949 − = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥))
df-neg 9950-𝐴 = (0 − 𝐴)
cdiv 10358class /
df-div 10359 / = (𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥))
cn 10698class
df-nn 10699ℕ = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) “ ω)
c2 10748class 2
c3 10749class 3
c4 10750class 4
c5 10751class 5
c6 10752class 6
c7 10753class 7
c8 10754class 8
c9 10755class 9
c10 10756class 10
df-2 107572 = (1 + 1)
df-3 107583 = (2 + 1)
df-4 107594 = (3 + 1)
df-5 107605 = (4 + 1)
df-6 107616 = (5 + 1)
df-7 107627 = (6 + 1)
df-8 107638 = (7 + 1)
df-9 107649 = (8 + 1)
df-10 1076510 = (9 + 1)
cn0 10960class 0
df-n0 109610 = (ℕ ∪ {0})
cz 11028class
df-z 11029ℤ = {𝑛 ∈ ℝ ∣ (𝑛 = 0 ∨ 𝑛 ∈ ℕ ∨ -𝑛 ∈ ℕ)}
cdc 11141class 𝐴𝐵
df-dec 11142𝐴𝐵 = ((10 · 𝐴) + 𝐵)
cuz 11249class
df-uz 11250 = (𝑗 ∈ ℤ ↦ {𝑘 ∈ ℤ ∣ 𝑗𝑘})
cq 11355class
df-q 11356ℚ = ( / “ (ℤ × ℕ))
crp 11393class +
df-rp 11394+ = {𝑥 ∈ ℝ ∣ 0 < 𝑥}
cxne 11497class -𝑒𝐴
cxad 11498class +𝑒
cxmu 11499class ·e
df-xneg 11500-𝑒𝐴 = if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴))
df-xadd 11501 +𝑒 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ if(𝑥 = +∞, if(𝑦 = -∞, 0, +∞), if(𝑥 = -∞, if(𝑦 = +∞, 0, -∞), if(𝑦 = +∞, +∞, if(𝑦 = -∞, -∞, (𝑥 + 𝑦))))))
df-xmul 11502 ·e = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ if((𝑥 = 0 ∨ 𝑦 = 0), 0, if((((0 < 𝑦𝑥 = +∞) ∨ (𝑦 < 0 ∧ 𝑥 = -∞)) ∨ ((0 < 𝑥𝑦 = +∞) ∨ (𝑥 < 0 ∧ 𝑦 = -∞))), +∞, if((((0 < 𝑦𝑥 = -∞) ∨ (𝑦 < 0 ∧ 𝑥 = +∞)) ∨ ((0 < 𝑥𝑦 = -∞) ∨ (𝑥 < 0 ∧ 𝑦 = +∞))), -∞, (𝑥 · 𝑦)))))
cioo 11730class (,)
cioc 11731class (,]
cico 11732class [,)
cicc 11733class [,]
df-ioo 11734(,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧𝑧 < 𝑦)})
df-ioc 11735(,] = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧𝑧𝑦)})
df-ico 11736[,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)})
df-icc 11737[,] = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧𝑦)})
cfz 11880class ...
df-fz 11881... = (𝑚 ∈ ℤ, 𝑛 ∈ ℤ ↦ {𝑘 ∈ ℤ ∣ (𝑚𝑘𝑘𝑛)})
cfzo 12016class ..^
df-fzo 12017..^ = (𝑚 ∈ ℤ, 𝑛 ∈ ℤ ↦ (𝑚...(𝑛 − 1)))
cfl 12134class
cceil 12135class
df-fl 12136⌊ = (𝑥 ∈ ℝ ↦ (𝑦 ∈ ℤ (𝑦𝑥𝑥 < (𝑦 + 1))))
df-ceil 12137⌈ = (𝑥 ∈ ℝ ↦ -(⌊‘-𝑥))
cmo 12204class mod
df-mod 12205 mod = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ+ ↦ (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦)))))
cseq 12327class seq𝑀( + , 𝐹)
df-seq 12328seq𝑀( + , 𝐹) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩) “ ω)
cexp 12386class
df-exp 12387↑ = (𝑥 ∈ ℂ, 𝑦 ∈ ℤ ↦ if(𝑦 = 0, 1, if(0 < 𝑦, (seq1( · , (ℕ × {𝑥}))‘𝑦), (1 / (seq1( · , (ℕ × {𝑥}))‘-𝑦)))))
cfa 12573class !
df-fac 12574! = ({⟨0, 1⟩} ∪ seq1( · , I ))
cbc 12601class C
df-bc 12602C = (𝑛 ∈ ℕ0, 𝑘 ∈ ℤ ↦ if(𝑘 ∈ (0...𝑛), ((!‘𝑛) / ((!‘(𝑛𝑘)) · (!‘𝑘))), 0))
chash 12629class #
df-hash 12630# = (((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω) ∘ card) ∪ ((V ∖ Fin) × {+∞}))
cword 12785class Word 𝑆
clsw 12786class lastS
cconcat 12787class ++
cs1 12788class ⟨“𝐴”⟩
csubstr 12789class substr
csplice 12790class splice
creverse 12791class reverse
creps 12792class repeatS
df-word 12793Word 𝑆 = {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆}
df-lsw 12794 lastS = (𝑤 ∈ V ↦ (𝑤‘((#‘𝑤) − 1)))
df-concat 12795 ++ = (𝑠 ∈ V, 𝑡 ∈ V ↦ (𝑥 ∈ (0..^((#‘𝑠) + (#‘𝑡))) ↦ if(𝑥 ∈ (0..^(#‘𝑠)), (𝑠𝑥), (𝑡‘(𝑥 − (#‘𝑠))))))
df-s1 12796⟨“𝐴”⟩ = {⟨0, ( I ‘𝐴)⟩}
df-substr 12797 substr = (𝑠 ∈ V, 𝑏 ∈ (ℤ × ℤ) ↦ if(((1st𝑏)..^(2nd𝑏)) ⊆ dom 𝑠, (𝑥 ∈ (0..^((2nd𝑏) − (1st𝑏))) ↦ (𝑠‘(𝑥 + (1st𝑏)))), ∅))
df-splice 12798 splice = (𝑠 ∈ V, 𝑏 ∈ V ↦ (((𝑠 substr ⟨0, (1st ‘(1st𝑏))⟩) ++ (2nd𝑏)) ++ (𝑠 substr ⟨(2nd ‘(1st𝑏)), (#‘𝑠)⟩)))
df-reverse 12799reverse = (𝑠 ∈ V ↦ (𝑥 ∈ (0..^(#‘𝑠)) ↦ (𝑠‘(((#‘𝑠) − 1) − 𝑥))))
df-reps 12800 repeatS = (𝑠 ∈ V, 𝑛 ∈ ℕ0 ↦ (𝑥 ∈ (0..^𝑛) ↦ 𝑠))
ccsh 13023class cyclShift
df-csh 13024 cyclShift = (𝑤 ∈ {𝑓 ∣ ∃𝑙 ∈ ℕ0 𝑓 Fn (0..^𝑙)}, 𝑛 ∈ ℤ ↦ if(𝑤 = ∅, ∅, ((𝑤 substr ⟨(𝑛 mod (#‘𝑤)), (#‘𝑤)⟩) ++ (𝑤 substr ⟨0, (𝑛 mod (#‘𝑤))⟩))))
cs2 13075class ⟨“𝐴𝐵”⟩
cs3 13076class ⟨“𝐴𝐵𝐶”⟩
cs4 13077class ⟨“𝐴𝐵𝐶𝐷”⟩
cs5 13078class ⟨“𝐴𝐵𝐶𝐷𝐸”⟩
cs6 13079class ⟨“𝐴𝐵𝐶𝐷𝐸𝐹”⟩
cs7 13080class ⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”⟩
cs8 13081class ⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺𝐻”⟩
df-s2 13082⟨“𝐴𝐵”⟩ = (⟨“𝐴”⟩ ++ ⟨“𝐵”⟩)
df-s3 13083⟨“𝐴𝐵𝐶”⟩ = (⟨“𝐴𝐵”⟩ ++ ⟨“𝐶”⟩)
df-s4 13084⟨“𝐴𝐵𝐶𝐷”⟩ = (⟨“𝐴𝐵𝐶”⟩ ++ ⟨“𝐷”⟩)
df-s5 13085⟨“𝐴𝐵𝐶𝐷𝐸”⟩ = (⟨“𝐴𝐵𝐶𝐷”⟩ ++ ⟨“𝐸”⟩)
df-s6 13086⟨“𝐴𝐵𝐶𝐷𝐸𝐹”⟩ = (⟨“𝐴𝐵𝐶𝐷𝐸”⟩ ++ ⟨“𝐹”⟩)
df-s7 13087⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”⟩ = (⟨“𝐴𝐵𝐶𝐷𝐸𝐹”⟩ ++ ⟨“𝐺”⟩)
df-s8 13088⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺𝐻”⟩ = (⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”⟩ ++ ⟨“𝐻”⟩)
ctcl 13207class t+
crtcl 13208class t*
df-trcl 13209t+ = (𝑥 ∈ V ↦ {𝑧 ∣ (𝑥𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)})
df-rtrcl 13210t* = (𝑥 ∈ V ↦ {𝑧 ∣ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧𝑥𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)})
crelexp 13243class 𝑟
df-relexp 13244𝑟 = (𝑟 ∈ V, 𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛)))
crtrcl 13278class t*rec
df-rtrclrec 13279t*rec = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))
cshi 13289class shift
df-shft 13290 shift = (𝑓 ∈ V, 𝑥 ∈ ℂ ↦ {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ ℂ ∧ (𝑦𝑥)𝑓𝑧)})
csgn 13309class sgn
df-sgn 13310sgn = (𝑥 ∈ ℝ* ↦ if(𝑥 = 0, 0, if(𝑥 < 0, -1, 1)))
ccj 13319class
cre 13320class
cim 13321class
df-cj 13322∗ = (𝑥 ∈ ℂ ↦ (𝑦 ∈ ℂ ((𝑥 + 𝑦) ∈ ℝ ∧ (i · (𝑥𝑦)) ∈ ℝ)))
df-re 13323ℜ = (𝑥 ∈ ℂ ↦ ((𝑥 + (∗‘𝑥)) / 2))
df-im 13324ℑ = (𝑥 ∈ ℂ ↦ (ℜ‘(𝑥 / i)))
csqrt 13456class
cabs 13457class abs
df-sqrt 13458√ = (𝑥 ∈ ℂ ↦ (𝑦 ∈ ℂ ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+)))
df-abs 13459abs = (𝑥 ∈ ℂ ↦ (√‘(𝑥 · (∗‘𝑥))))
clsp 13684class lim sup
clspold 13685class lim sup
df-limsup 13686lim sup = (𝑥 ∈ V ↦ inf(ran (𝑘 ∈ ℝ ↦ sup(((𝑥 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )), ℝ*, < ))
df-limsupOLD 13687lim sup = (𝑥 ∈ V ↦ sup(ran (𝑘 ∈ ℝ ↦ sup(((𝑥 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )), ℝ*, < ))
cli 13708class
crli 13709class 𝑟
co1 13710class 𝑂(1)
clo1 13711class ≤𝑂(1)
df-clim 13712 ⇝ = {⟨𝑓, 𝑦⟩ ∣ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)((𝑓𝑘) ∈ ℂ ∧ (abs‘((𝑓𝑘) − 𝑦)) < 𝑥))}
df-rlim 13713𝑟 = {⟨𝑓, 𝑥⟩ ∣ ((𝑓 ∈ (ℂ ↑pm ℝ) ∧ 𝑥 ∈ ℂ) ∧ ∀𝑦 ∈ ℝ+𝑧 ∈ ℝ ∀𝑤 ∈ dom 𝑓(𝑧𝑤 → (abs‘((𝑓𝑤) − 𝑥)) < 𝑦))}
df-o1 13714𝑂(1) = {𝑓 ∈ (ℂ ↑pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(abs‘(𝑓𝑦)) ≤ 𝑚}
df-lo1 13715≤𝑂(1) = {𝑓 ∈ (ℝ ↑pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(𝑓𝑦) ≤ 𝑚}
csu 13912class Σ𝑘𝐴 𝐵
df-sum 13913Σ𝑘𝐴 𝐵 = (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚))))
cprod 14119class 𝑘𝐴 𝐵
df-prod 14120𝑘𝐴 𝐵 = (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚))))
cfallfac 14217class FallFac
crisefac 14218class RiseFac
df-risefac 14219 RiseFac = (𝑥 ∈ ℂ, 𝑛 ∈ ℕ0 ↦ ∏𝑘 ∈ (0...(𝑛 − 1))(𝑥 + 𝑘))
df-fallfac 14220 FallFac = (𝑥 ∈ ℂ, 𝑛 ∈ ℕ0 ↦ ∏𝑘 ∈ (0...(𝑛 − 1))(𝑥𝑘))
cbp 14259class BernPoly
df-bpoly 14260 BernPoly = (𝑚 ∈ ℕ0, 𝑥 ∈ ℂ ↦ (wrecs( < , ℕ0, (𝑔 ∈ V ↦ (#‘dom 𝑔) / 𝑛((𝑥𝑛) − Σ𝑘 ∈ dom 𝑔((𝑛C𝑘) · ((𝑔𝑘) / ((𝑛𝑘) + 1))))))‘𝑚))
ce 14274class exp
ceu 14275class e
csin 14276class sin
ccos 14277class cos
ctan 14278class tan
cpi 14279class π
cpiold 14280class π
df-ef 14281exp = (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ ℕ0 ((𝑥𝑘) / (!‘𝑘)))
df-e 14282e = (exp‘1)
df-sin 14283sin = (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) − (exp‘(-i · 𝑥))) / (2 · i)))
df-cos 14284cos = (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥))) / 2))
df-tan 14285tan = (𝑥 ∈ (cos “ (ℂ ∖ {0})) ↦ ((sin‘𝑥) / (cos‘𝑥)))
df-pi 14286π = inf((ℝ+ ∩ (sin “ {0})), ℝ, < )
df-piOLD 14287π = sup((ℝ+ ∩ (sin “ {0})), ℝ, < )
cdvds 14465class
df-dvds 14466 ∥ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ ∃𝑛 ∈ ℤ (𝑛 · 𝑥) = 𝑦)}
cbits 14554class bits
csad 14555class sadd
csmu 14556class smul
df-bits 14557bits = (𝑛 ∈ ℤ ↦ {𝑚 ∈ ℕ0 ∣ ¬ 2 ∥ (⌊‘(𝑛 / (2↑𝑚)))})
df-sad 14587 sadd = (𝑥 ∈ 𝒫 ℕ0, 𝑦 ∈ 𝒫 ℕ0 ↦ {𝑘 ∈ ℕ0 ∣ hadd(𝑘𝑥, 𝑘𝑦, ∅ ∈ (seq0((𝑐 ∈ 2𝑜, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚𝑥, 𝑚𝑦, ∅ ∈ 𝑐), 1𝑜, ∅)), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑘))})
df-smu 14612 smul = (𝑥 ∈ 𝒫 ℕ0, 𝑦 ∈ 𝒫 ℕ0 ↦ {𝑘 ∈ ℕ0𝑘 ∈ (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝑥 ∧ (𝑛𝑚) ∈ 𝑦)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘(𝑘 + 1))})
cgcd 14630class gcd
df-gcd 14631 gcd = (𝑥 ∈ ℤ, 𝑦 ∈ ℤ ↦ if((𝑥 = 0 ∧ 𝑦 = 0), 0, sup({𝑛 ∈ ℤ ∣ (𝑛𝑥𝑛𝑦)}, ℝ, < )))
clcm 14709class lcm
clcmf 14710class lcm
clcmold 14711class lcm
clcmfold 14712class lcm
df-lcm 14713 lcm = (𝑥 ∈ ℤ, 𝑦 ∈ ℤ ↦ if((𝑥 = 0 ∨ 𝑦 = 0), 0, inf({𝑛 ∈ ℕ ∣ (𝑥𝑛𝑦𝑛)}, ℝ, < )))
df-lcmOLD 14714 lcm = (𝑥 ∈ ℤ, 𝑦 ∈ ℤ ↦ if((𝑥 = 0 ∨ 𝑦 = 0), 0, sup({𝑛 ∈ ℕ ∣ (𝑥𝑛𝑦𝑛)}, ℝ, < )))
df-lcmf 14715lcm = (𝑧 ∈ 𝒫 ℤ ↦ if(0 ∈ 𝑧, 0, inf({𝑛 ∈ ℕ ∣ ∀𝑚𝑧 𝑚𝑛}, ℝ, < )))
df-lcmfOLD 14716lcm = (𝑧 ∈ 𝒫 ℤ ↦ if(0 ∈ 𝑧, 0, sup({𝑛 ∈ ℕ ∣ ∀𝑚𝑧 𝑚𝑛}, ℝ, < )))
cprime 14784class
df-prm 14785ℙ = {𝑝 ∈ ℕ ∣ {𝑛 ∈ ℕ ∣ 𝑛𝑝} ≈ 2𝑜}
cnumer 14844class numer
cdenom 14845class denom
df-numer 14846numer = (𝑦 ∈ ℚ ↦ (1st ‘(𝑥 ∈ (ℤ × ℕ)(((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝑦 = ((1st𝑥) / (2nd𝑥))))))
df-denom 14847denom = (𝑦 ∈ ℚ ↦ (2nd ‘(𝑥 ∈ (ℤ × ℕ)(((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝑦 = ((1st𝑥) / (2nd𝑥))))))
codz 14871class od
codzold 14872class od
cphi 14873class ϕ
df-odz 14874od = (𝑛 ∈ ℕ ↦ (𝑥 ∈ {𝑥 ∈ ℤ ∣ (𝑥 gcd 𝑛) = 1} ↦ inf({𝑚 ∈ ℕ ∣ 𝑛 ∥ ((𝑥𝑚) − 1)}, ℝ, < )))
df-odzOLD 14875od = (𝑛 ∈ ℕ ↦ (𝑥 ∈ {𝑥 ∈ ℤ ∣ (𝑥 gcd 𝑛) = 1} ↦ sup({𝑚 ∈ ℕ ∣ 𝑛 ∥ ((𝑥𝑚) − 1)}, ℝ, < )))
df-phi 14876ϕ = (𝑛 ∈ ℕ ↦ (#‘{𝑥 ∈ (1...𝑛) ∣ (𝑥 gcd 𝑛) = 1}))
cpc 14948class pCnt
df-pc 14949 pCnt = (𝑝 ∈ ℙ, 𝑟 ∈ ℚ ↦ if(𝑟 = 0, +∞, (℩𝑧𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑟 = (𝑥 / 𝑦) ∧ 𝑧 = (sup({𝑛 ∈ ℕ0 ∣ (𝑝𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑝𝑛) ∥ 𝑦}, ℝ, < ))))))
cgz 15035class ℤ[i]
df-gz 15036ℤ[i] = {𝑥 ∈ ℂ ∣ ((ℜ‘𝑥) ∈ ℤ ∧ (ℑ‘𝑥) ∈ ℤ)}
cvdwa 15077class AP
cvdwm 15078class MonoAP
cvdwp 15079class PolyAP
df-vdwap 15080AP = (𝑘 ∈ ℕ0 ↦ (𝑎 ∈ ℕ, 𝑑 ∈ ℕ ↦ ran (𝑚 ∈ (0...(𝑘 − 1)) ↦ (𝑎 + (𝑚 · 𝑑)))))
df-vdwmc 15081 MonoAP = {⟨𝑘, 𝑓⟩ ∣ ∃𝑐(ran (AP‘𝑘) ∩ 𝒫 (𝑓 “ {𝑐})) ≠ ∅}
df-vdwpc 15082 PolyAP = {⟨⟨𝑚, 𝑘⟩, 𝑓⟩ ∣ ∃𝑎 ∈ ℕ ∃𝑑 ∈ (ℕ ↑𝑚 (1...𝑚))(∀𝑖 ∈ (1...𝑚)((𝑎 + (𝑑𝑖))(AP‘𝑘)(𝑑𝑖)) ⊆ (𝑓 “ {(𝑓‘(𝑎 + (𝑑𝑖)))}) ∧ (#‘ran (𝑖 ∈ (1...𝑚) ↦ (𝑓‘(𝑎 + (𝑑𝑖))))) = 𝑚)}
cram 15111class Ramsey
cramold 15112class Ramsey
df-ram 15114 Ramsey = (𝑚 ∈ ℕ0, 𝑟 ∈ V ↦ inf({𝑛 ∈ ℕ0 ∣ ∀𝑠(𝑛 ≤ (#‘𝑠) → ∀𝑓 ∈ (dom 𝑟𝑚 {𝑦 ∈ 𝒫 𝑠 ∣ (#‘𝑦) = 𝑚})∃𝑐 ∈ dom 𝑟𝑥 ∈ 𝒫 𝑠((𝑟𝑐) ≤ (#‘𝑥) ∧ ∀𝑦 ∈ 𝒫 𝑥((#‘𝑦) = 𝑚 → (𝑓𝑦) = 𝑐)))}, ℝ*, < ))
df-ramOLD 15115 Ramsey = (𝑚 ∈ ℕ0, 𝑟 ∈ V ↦ sup({𝑛 ∈ ℕ0 ∣ ∀𝑠(𝑛 ≤ (#‘𝑠) → ∀𝑓 ∈ (dom 𝑟𝑚 {𝑦 ∈ 𝒫 𝑠 ∣ (#‘𝑦) = 𝑚})∃𝑐 ∈ dom 𝑟𝑥 ∈ 𝒫 𝑠((𝑟𝑐) ≤ (#‘𝑥) ∧ ∀𝑦 ∈ 𝒫 𝑥((#‘𝑦) = 𝑚 → (𝑓𝑦) = 𝑐)))}, ℝ*, < ))
cprmo 15151class #p
df-prmo 15152#p = (𝑛 ∈ ℕ0 ↦ ∏𝑘 ∈ (1...𝑛)if(𝑘 ∈ ℙ, 𝑘, 1))
cstr 15278class Struct
cnx 15279class ndx
csts 15280class sSet
cslot 15281class Slot 𝐴
cbs 15282class Base
cress 15283class s
df-struct 15284 Struct = {⟨𝑓, 𝑥⟩ ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))}
df-ndx 15285ndx = ( I ↾ ℕ)
df-slot 15286Slot 𝐴 = (𝑥 ∈ V ↦ (𝑥𝐴))
df-base 15287Base = Slot 1
df-sets 15288 sSet = (𝑠 ∈ V, 𝑒 ∈ V ↦ ((𝑠 ↾ (V ∖ dom {𝑒})) ∪ {𝑒}))
df-ress 15289s = (𝑤 ∈ V, 𝑥 ∈ V ↦ if((Base‘𝑤) ⊆ 𝑥, 𝑤, (𝑤 sSet ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩)))
cplusg 15355class +g
cmulr 15356class .r
cstv 15357class *𝑟
csca 15358class Scalar
cvsca 15359class ·𝑠
cip 15360class ·𝑖
cts 15361class TopSet
cple 15362class le
coc 15363class oc
cds 15364class dist
cunif 15365class UnifSet
chom 15366class Hom
cco 15367class comp
df-plusg 15368+g = Slot 2
df-mulr 15369.r = Slot 3
df-starv 15370*𝑟 = Slot 4
df-sca 15371Scalar = Slot 5
df-vsca 15372 ·𝑠 = Slot 6
df-ip 15373·𝑖 = Slot 8
df-tset 15374TopSet = Slot 9
df-ple 15375le = Slot 10
df-ocomp 15376oc = Slot 11
df-ds 15377dist = Slot 12
df-unif 15378UnifSet = Slot 13
df-hom 15379Hom = Slot 14
df-cco 15380comp = Slot 15
crest 15484class t
ctopn 15485class TopOpen
df-rest 15486t = (𝑗 ∈ V, 𝑥 ∈ V ↦ ran (𝑦𝑗 ↦ (𝑦𝑥)))
df-topn 15487TopOpen = (𝑤 ∈ V ↦ ((TopSet‘𝑤) ↾t (Base‘𝑤)))
ctg 15501class topGen
cpt 15502class t
c0g 15503class 0g
cgsu 15504class Σg
df-0g 155050g = (𝑔 ∈ V ↦ (℩𝑒(𝑒 ∈ (Base‘𝑔) ∧ ∀𝑥 ∈ (Base‘𝑔)((𝑒(+g𝑔)𝑥) = 𝑥 ∧ (𝑥(+g𝑔)𝑒) = 𝑥))))
df-gsum 15506 Σg = (𝑤 ∈ V, 𝑓 ∈ V ↦ {𝑥 ∈ (Base‘𝑤) ∣ ∀𝑦 ∈ (Base‘𝑤)((𝑥(+g𝑤)𝑦) = 𝑦 ∧ (𝑦(+g𝑤)𝑥) = 𝑦)} / 𝑜if(ran 𝑓𝑜, (0g𝑤), if(dom 𝑓 ∈ ran ..., (℩𝑥𝑚𝑛 ∈ (ℤ𝑚)(dom 𝑓 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g𝑤), 𝑓)‘𝑛))), (℩𝑥𝑔[(𝑓 “ (V ∖ 𝑜)) / 𝑦](𝑔:(1...(#‘𝑦))–1-1-onto𝑦𝑥 = (seq1((+g𝑤), (𝑓𝑔))‘(#‘𝑦)))))))
df-topgen 15507topGen = (𝑥 ∈ V ↦ {𝑦𝑦 (𝑥 ∩ 𝒫 𝑦)})
df-pt 15508t = (𝑓 ∈ V ↦ (topGen‘{𝑥 ∣ ∃𝑔((𝑔 Fn dom 𝑓 ∧ ∀𝑦 ∈ dom 𝑓(𝑔𝑦) ∈ (𝑓𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (dom 𝑓𝑧)(𝑔𝑦) = (𝑓𝑦)) ∧ 𝑥 = X𝑦 ∈ dom 𝑓(𝑔𝑦))}))
cprds 15509class Xs
cpws 15510class s
df-prds 15511Xs = (𝑠 ∈ V, 𝑟 ∈ V ↦ X𝑥 ∈ dom 𝑟(Base‘(𝑟𝑥)) / 𝑣(𝑓𝑣, 𝑔𝑣X𝑥 ∈ dom 𝑟((𝑓𝑥)(Hom ‘(𝑟𝑥))(𝑔𝑥))) / (({⟨(Base‘ndx), 𝑣⟩, ⟨(+g‘ndx), (𝑓𝑣, 𝑔𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓𝑥)(+g‘(𝑟𝑥))(𝑔𝑥))))⟩, ⟨(.r‘ndx), (𝑓𝑣, 𝑔𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓𝑥)(.r‘(𝑟𝑥))(𝑔𝑥))))⟩} ∪ {⟨(Scalar‘ndx), 𝑠⟩, ⟨( ·𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑠), 𝑔𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ (𝑓( ·𝑠 ‘(𝑟𝑥))(𝑔𝑥))))⟩, ⟨(·𝑖‘ndx), (𝑓𝑣, 𝑔𝑣 ↦ (𝑠 Σg (𝑥 ∈ dom 𝑟 ↦ ((𝑓𝑥)(·𝑖‘(𝑟𝑥))(𝑔𝑥)))))⟩}) ∪ ({⟨(TopSet‘ndx), (∏t‘(TopOpen ∘ 𝑟))⟩, ⟨(le‘ndx), {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝑣 ∧ ∀𝑥 ∈ dom 𝑟(𝑓𝑥)(le‘(𝑟𝑥))(𝑔𝑥))}⟩, ⟨(dist‘ndx), (𝑓𝑣, 𝑔𝑣 ↦ sup((ran (𝑥 ∈ dom 𝑟 ↦ ((𝑓𝑥)(dist‘(𝑟𝑥))(𝑔𝑥))) ∪ {0}), ℝ*, < ))⟩} ∪ {⟨(Hom ‘ndx), ⟩, ⟨(comp‘ndx), (𝑎 ∈ (𝑣 × 𝑣), 𝑐𝑣 ↦ (𝑑 ∈ (𝑐(2nd𝑎)), 𝑒 ∈ (𝑎) ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑑𝑥)(⟨((1st𝑎)‘𝑥), ((2nd𝑎)‘𝑥)⟩(comp‘(𝑟𝑥))(𝑐𝑥))(𝑒𝑥)))))⟩})))
df-pws 15513s = (𝑟 ∈ V, 𝑖 ∈ V ↦ ((Scalar‘𝑟)Xs(𝑖 × {𝑟})))
cordt 15562class ordTop
cxrs 15563class *𝑠
df-ordt 15564ordTop = (𝑟 ∈ V ↦ (topGen‘(fi‘({dom 𝑟} ∪ ran ((𝑥 ∈ dom 𝑟 ↦ {𝑦 ∈ dom 𝑟 ∣ ¬ 𝑦𝑟𝑥}) ∪ (𝑥 ∈ dom 𝑟 ↦ {𝑦 ∈ dom 𝑟 ∣ ¬ 𝑥𝑟𝑦}))))))
df-xrs 15565*𝑠 = ({⟨(Base‘ndx), ℝ*⟩, ⟨(+g‘ndx), +𝑒 ⟩, ⟨(.r‘ndx), ·e ⟩} ∪ {⟨(TopSet‘ndx), (ordTop‘ ≤ )⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ if(𝑥𝑦, (𝑦 +𝑒 -𝑒𝑥), (𝑥 +𝑒 -𝑒𝑦)))⟩})
cqtop 15566class qTop
cimas 15567class s
cimasold 15568class s
cqus 15569class /s
cxps 15570class ×s
df-qtop 15571 qTop = (𝑗 ∈ V, 𝑓 ∈ V ↦ {𝑠 ∈ 𝒫 (𝑓 𝑗) ∣ ((𝑓𝑠) ∩ 𝑗) ∈ 𝑗})
df-imas 15572s = (𝑓 ∈ V, 𝑟 ∈ V ↦ (Base‘𝑟) / 𝑣(({⟨(Base‘ndx), ran 𝑓⟩, ⟨(+g‘ndx), 𝑝𝑣 𝑞𝑣 {⟨⟨(𝑓𝑝), (𝑓𝑞)⟩, (𝑓‘(𝑝(+g𝑟)𝑞))⟩}⟩, ⟨(.r‘ndx), 𝑝𝑣 𝑞𝑣 {⟨⟨(𝑓𝑝), (𝑓𝑞)⟩, (𝑓‘(𝑝(.r𝑟)𝑞))⟩}⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑟)⟩, ⟨( ·𝑠 ‘ndx), 𝑞𝑣 (𝑝 ∈ (Base‘(Scalar‘𝑟)), 𝑥 ∈ {(𝑓𝑞)} ↦ (𝑓‘(𝑝( ·𝑠𝑟)𝑞)))⟩, ⟨(·𝑖‘ndx), 𝑝𝑣 𝑞𝑣 {⟨⟨(𝑓𝑝), (𝑓𝑞)⟩, (𝑝(·𝑖𝑟)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 ∈ { ∈ ((𝑣 × 𝑣) ↑𝑚 (1...𝑛)) ∣ ((𝑓‘(1st ‘(‘1))) = 𝑥 ∧ (𝑓‘(2nd ‘(𝑛))) = 𝑦 ∧ ∀𝑖 ∈ (1...(𝑛 − 1))(𝑓‘(2nd ‘(𝑖))) = (𝑓‘(1st ‘(‘(𝑖 + 1)))))} ↦ (ℝ*𝑠 Σg ((dist‘𝑟) ∘ 𝑔))), ℝ*, < ))⟩}))
df-imasOLD 15573s = (𝑓 ∈ V, 𝑟 ∈ V ↦ (Base‘𝑟) / 𝑣(({⟨(Base‘ndx), ran 𝑓⟩, ⟨(+g‘ndx), 𝑝𝑣 𝑞𝑣 {⟨⟨(𝑓𝑝), (𝑓𝑞)⟩, (𝑓‘(𝑝(+g𝑟)𝑞))⟩}⟩, ⟨(.r‘ndx), 𝑝𝑣 𝑞𝑣 {⟨⟨(𝑓𝑝), (𝑓𝑞)⟩, (𝑓‘(𝑝(.r𝑟)𝑞))⟩}⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑟)⟩, ⟨( ·𝑠 ‘ndx), 𝑞𝑣 (𝑝 ∈ (Base‘(Scalar‘𝑟)), 𝑥 ∈ {(𝑓𝑞)} ↦ (𝑓‘(𝑝( ·𝑠𝑟)𝑞)))⟩, ⟨(·𝑖‘ndx), 𝑝𝑣 𝑞𝑣 {⟨⟨(𝑓𝑝), (𝑓𝑞)⟩, (𝑝(·𝑖𝑟)𝑞)⟩}⟩}) ∪ {⟨(TopSet‘ndx), ((TopOpen‘𝑟) qTop 𝑓)⟩, ⟨(le‘ndx), ((𝑓 ∘ (le‘𝑟)) ∘ 𝑓)⟩, ⟨(dist‘ndx), (𝑥 ∈ ran 𝑓, 𝑦 ∈ ran 𝑓 ↦ sup( 𝑛 ∈ ℕ ran (𝑔 ∈ { ∈ ((𝑣 × 𝑣) ↑𝑚 (1...𝑛)) ∣ ((𝑓‘(1st ‘(‘1))) = 𝑥 ∧ (𝑓‘(2nd ‘(𝑛))) = 𝑦 ∧ ∀𝑖 ∈ (1...(𝑛 − 1))(𝑓‘(2nd ‘(𝑖))) = (𝑓‘(1st ‘(‘(𝑖 + 1)))))} ↦ (ℝ*𝑠 Σg ((dist‘𝑟) ∘ 𝑔))), ℝ*, < ))⟩}))
df-qus 15574 /s = (𝑟 ∈ V, 𝑒 ∈ V ↦ ((𝑥 ∈ (Base‘𝑟) ↦ [𝑥]𝑒) “s 𝑟))
df-xps 15575 ×s = (𝑟 ∈ V, 𝑠 ∈ V ↦ ((𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Base‘𝑠) ↦ ({𝑥} +𝑐 {𝑦})) “s ((Scalar‘𝑟)Xs({𝑟} +𝑐 {𝑠}))))
cmre 15653class Moore
cmrc 15654class mrCls
cmri 15655class mrInd
cacs 15656class ACS
df-mre 15657Moore = (𝑥 ∈ V ↦ {𝑐 ∈ 𝒫 𝒫 𝑥 ∣ (𝑥𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → 𝑠𝑐))})
df-mrc 15658mrCls = (𝑐 ran Moore ↦ (𝑥 ∈ 𝒫 𝑐 {𝑠𝑐𝑥𝑠}))
df-mri 15659mrInd = (𝑐 ran Moore ↦ {𝑠 ∈ 𝒫 𝑐 ∣ ∀𝑥𝑠 ¬ 𝑥 ∈ ((mrCls‘𝑐)‘(𝑠 ∖ {𝑥}))})
df-acs 15660ACS = (𝑥 ∈ V ↦ {𝑐 ∈ (Moore‘𝑥) ∣ ∃𝑓(𝑓:𝒫 𝑥⟶𝒫 𝑥 ∧ ∀𝑠 ∈ 𝒫 𝑥(𝑠𝑐 (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠))})
ccat 15736class Cat
ccid 15737class Id
chomf 15738class Homf
ccomf 15739class compf
df-cat 15740Cat = {𝑐[(Base‘𝑐) / 𝑏][(Hom ‘𝑐) / ][(comp‘𝑐) / 𝑜]𝑥𝑏 (∃𝑔 ∈ (𝑥𝑥)∀𝑦𝑏 (∀𝑓 ∈ (𝑦𝑥)(𝑔(⟨𝑦, 𝑥𝑜𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝑦)(𝑓(⟨𝑥, 𝑥𝑜𝑦)𝑔) = 𝑓) ∧ ∀𝑦𝑏𝑧𝑏𝑓 ∈ (𝑥𝑦)∀𝑔 ∈ (𝑦𝑧)((𝑔(⟨𝑥, 𝑦𝑜𝑧)𝑓) ∈ (𝑥𝑧) ∧ ∀𝑤𝑏𝑘 ∈ (𝑧𝑤)((𝑘(⟨𝑦, 𝑧𝑜𝑤)𝑔)(⟨𝑥, 𝑦𝑜𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧𝑜𝑤)(𝑔(⟨𝑥, 𝑦𝑜𝑧)𝑓))))}
df-cid 15741Id = (𝑐 ∈ Cat ↦ (Base‘𝑐) / 𝑏(Hom ‘𝑐) / (comp‘𝑐) / 𝑜(𝑥𝑏 ↦ (𝑔 ∈ (𝑥𝑥)∀𝑦𝑏 (∀𝑓 ∈ (𝑦𝑥)(𝑔(⟨𝑦, 𝑥𝑜𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝑦)(𝑓(⟨𝑥, 𝑥𝑜𝑦)𝑔) = 𝑓))))
df-homf 15742Homf = (𝑐 ∈ V ↦ (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ (𝑥(Hom ‘𝑐)𝑦)))
df-comf 15743compf = (𝑐 ∈ V ↦ (𝑥 ∈ ((Base‘𝑐) × (Base‘𝑐)), 𝑦 ∈ (Base‘𝑐) ↦ (𝑔 ∈ ((2nd𝑥)(Hom ‘𝑐)𝑦), 𝑓 ∈ ((Hom ‘𝑐)‘𝑥) ↦ (𝑔(𝑥(comp‘𝑐)𝑦)𝑓))))
coppc 15782class oppCat
df-oppc 15783oppCat = (𝑓 ∈ V ↦ ((𝑓 sSet ⟨(Hom ‘ndx), tpos (Hom ‘𝑓)⟩) sSet ⟨(comp‘ndx), (𝑢 ∈ ((Base‘𝑓) × (Base‘𝑓)), 𝑧 ∈ (Base‘𝑓) ↦ tpos (⟨𝑧, (2nd𝑢)⟩(comp‘𝑓)(1st𝑢)))⟩))
cmon 15799class Mono
cepi 15800class Epi
df-mon 15801Mono = (𝑐 ∈ Cat ↦ (Base‘𝑐) / 𝑏(Hom ‘𝑐) / (𝑥𝑏, 𝑦𝑏 ↦ {𝑓 ∈ (𝑥𝑦) ∣ ∀𝑧𝑏 Fun (𝑔 ∈ (𝑧𝑥) ↦ (𝑓(⟨𝑧, 𝑥⟩(comp‘𝑐)𝑦)𝑔))}))
df-epi 15802Epi = (𝑐 ∈ Cat ↦ tpos (Mono‘(oppCat‘𝑐)))
csect 15815class Sect
cinv 15816class Inv
ciso 15817class Iso
df-sect 15818Sect = (𝑐 ∈ Cat ↦ (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ {⟨𝑓, 𝑔⟩ ∣ [(Hom ‘𝑐) / ]((𝑓 ∈ (𝑥𝑦) ∧ 𝑔 ∈ (𝑦𝑥)) ∧ (𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑥)𝑓) = ((Id‘𝑐)‘𝑥))}))
df-inv 15819Inv = (𝑐 ∈ Cat ↦ (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ ((𝑥(Sect‘𝑐)𝑦) ∩ (𝑦(Sect‘𝑐)𝑥))))
df-iso 15820Iso = (𝑐 ∈ Cat ↦ ((𝑥 ∈ V ↦ dom 𝑥) ∘ (Inv‘𝑐)))
ccic 15866class 𝑐
df-cic 15867𝑐 = (𝑐 ∈ Cat ↦ ((Iso‘𝑐) supp ∅))
cssc 15878class cat
cresc 15879class cat
csubc 15880class Subcat
df-ssc 15881cat = {⟨, 𝑗⟩ ∣ ∃𝑡(𝑗 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝑗𝑥))}
df-resc 15882cat = (𝑐 ∈ V, ∈ V ↦ ((𝑐s dom dom ) sSet ⟨(Hom ‘ndx), ⟩))
df-subc 15883Subcat = (𝑐 ∈ Cat ↦ { ∣ (cat (Homf𝑐) ∧ [dom dom / 𝑠]𝑥𝑠 (((Id‘𝑐)‘𝑥) ∈ (𝑥𝑥) ∧ ∀𝑦𝑠𝑧𝑠𝑓 ∈ (𝑥𝑦)∀𝑔 ∈ (𝑦𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑧)𝑓) ∈ (𝑥𝑧)))})
cfunc 15925class Func
cidfu 15926class idfunc
ccofu 15927class func
cresf 15928class f
df-func 15929 Func = (𝑡 ∈ Cat, 𝑢 ∈ Cat ↦ {⟨𝑓, 𝑔⟩ ∣ [(Base‘𝑡) / 𝑏](𝑓:𝑏⟶(Base‘𝑢) ∧ 𝑔X𝑧 ∈ (𝑏 × 𝑏)(((𝑓‘(1st𝑧))(Hom ‘𝑢)(𝑓‘(2nd𝑧))) ↑𝑚 ((Hom ‘𝑡)‘𝑧)) ∧ ∀𝑥𝑏 (((𝑥𝑔𝑥)‘((Id‘𝑡)‘𝑥)) = ((Id‘𝑢)‘(𝑓𝑥)) ∧ ∀𝑦𝑏𝑧𝑏𝑚 ∈ (𝑥(Hom ‘𝑡)𝑦)∀𝑛 ∈ (𝑦(Hom ‘𝑡)𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦⟩(comp‘𝑡)𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓𝑥), (𝑓𝑦)⟩(comp‘𝑢)(𝑓𝑧))((𝑥𝑔𝑦)‘𝑚))))})
df-idfu 15930idfunc = (𝑡 ∈ Cat ↦ (Base‘𝑡) / 𝑏⟨( I ↾ 𝑏), (𝑧 ∈ (𝑏 × 𝑏) ↦ ( I ↾ ((Hom ‘𝑡)‘𝑧)))⟩)
df-cofu 15931func = (𝑔 ∈ V, 𝑓 ∈ V ↦ ⟨((1st𝑔) ∘ (1st𝑓)), (𝑥 ∈ dom dom (2nd𝑓), 𝑦 ∈ dom dom (2nd𝑓) ↦ ((((1st𝑓)‘𝑥)(2nd𝑔)((1st𝑓)‘𝑦)) ∘ (𝑥(2nd𝑓)𝑦)))⟩)
df-resf 15932f = (𝑓 ∈ V, ∈ V ↦ ⟨((1st𝑓) ↾ dom dom ), (𝑥 ∈ dom ↦ (((2nd𝑓)‘𝑥) ↾ (𝑥)))⟩)
cful 15973class Full
cfth 15974class Faith
df-full 15975 Full = (𝑐 ∈ Cat, 𝑑 ∈ Cat ↦ {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝑐 Func 𝑑)𝑔 ∧ ∀𝑥 ∈ (Base‘𝑐)∀𝑦 ∈ (Base‘𝑐)ran (𝑥𝑔𝑦) = ((𝑓𝑥)(Hom ‘𝑑)(𝑓𝑦)))})
df-fth 15976 Faith = (𝑐 ∈ Cat, 𝑑 ∈ Cat ↦ {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝑐 Func 𝑑)𝑔 ∧ ∀𝑥 ∈ (Base‘𝑐)∀𝑦 ∈ (Base‘𝑐)Fun (𝑥𝑔𝑦))})
cnat 16012class Nat
cfuc 16013class FuncCat
df-nat 16014 Nat = (𝑡 ∈ Cat, 𝑢 ∈ Cat ↦ (𝑓 ∈ (𝑡 Func 𝑢), 𝑔 ∈ (𝑡 Func 𝑢) ↦ (1st𝑓) / 𝑟(1st𝑔) / 𝑠{𝑎X𝑥 ∈ (Base‘𝑡)((𝑟𝑥)(Hom ‘𝑢)(𝑠𝑥)) ∣ ∀𝑥 ∈ (Base‘𝑡)∀𝑦 ∈ (Base‘𝑡)∀ ∈ (𝑥(Hom ‘𝑡)𝑦)((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩(comp‘𝑢)(𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘)) = (((𝑥(2nd𝑔)𝑦)‘)(⟨(𝑟𝑥), (𝑠𝑥)⟩(comp‘𝑢)(𝑠𝑦))(𝑎𝑥))}))
df-fuc 16015 FuncCat = (𝑡 ∈ Cat, 𝑢 ∈ Cat ↦ {⟨(Base‘ndx), (𝑡 Func 𝑢)⟩, ⟨(Hom ‘ndx), (𝑡 Nat 𝑢)⟩, ⟨(comp‘ndx), (𝑣 ∈ ((𝑡 Func 𝑢) × (𝑡 Func 𝑢)), ∈ (𝑡 Func 𝑢) ↦ (1st𝑣) / 𝑓(2nd𝑣) / 𝑔(𝑏 ∈ (𝑔(𝑡 Nat 𝑢)), 𝑎 ∈ (𝑓(𝑡 Nat 𝑢)𝑔) ↦ (𝑥 ∈ (Base‘𝑡) ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝑢)((1st)‘𝑥))(𝑎𝑥)))))⟩})
cinito 16049class InitO
ctermo 16050class TermO
czeroo 16051class ZeroO
df-inito 16052InitO = (𝑐 ∈ Cat ↦ {𝑎 ∈ (Base‘𝑐) ∣ ∀𝑏 ∈ (Base‘𝑐)∃! ∈ (𝑎(Hom ‘𝑐)𝑏)})
df-termo 16053TermO = (𝑐 ∈ Cat ↦ {𝑎 ∈ (Base‘𝑐) ∣ ∀𝑏 ∈ (Base‘𝑐)∃! ∈ (𝑏(Hom ‘𝑐)𝑎)})
df-zeroo 16054ZeroO = (𝑐 ∈ Cat ↦ ((InitO‘𝑐) ∩ (TermO‘𝑐)))
cdoma 16081class doma
ccoda 16082class coda
carw 16083class Arrow
choma 16084class Homa
df-doma 16085doma = (1st ∘ 1st )
df-coda 16086coda = (2nd ∘ 1st )
df-homa 16087Homa = (𝑐 ∈ Cat ↦ (𝑥 ∈ ((Base‘𝑐) × (Base‘𝑐)) ↦ ({𝑥} × ((Hom ‘𝑐)‘𝑥))))
df-arw 16088Arrow = (𝑐 ∈ Cat ↦ ran (Homa𝑐))
cida 16114class Ida
ccoa 16115class compa
df-ida 16116Ida = (𝑐 ∈ Cat ↦ (𝑥 ∈ (Base‘𝑐) ↦ ⟨𝑥, 𝑥, ((Id‘𝑐)‘𝑥)⟩))
df-coa 16117compa = (𝑐 ∈ Cat ↦ (𝑔 ∈ (Arrow‘𝑐), 𝑓 ∈ { ∈ (Arrow‘𝑐) ∣ (coda) = (doma𝑔)} ↦ ⟨(doma𝑓), (coda𝑔), ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩(comp‘𝑐)(coda𝑔))(2nd𝑓))⟩))
csetc 16136class SetCat
df-setc 16137SetCat = (𝑢 ∈ V ↦ {⟨(Base‘ndx), 𝑢⟩, ⟨(Hom ‘ndx), (𝑥𝑢, 𝑦𝑢 ↦ (𝑦𝑚 𝑥))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝑢 × 𝑢), 𝑧𝑢 ↦ (𝑔 ∈ (𝑧𝑚 (2nd𝑣)), 𝑓 ∈ ((2nd𝑣) ↑𝑚 (1st𝑣)) ↦ (𝑔𝑓)))⟩})
ccatc 16155class CatCat
df-catc 16156CatCat = (𝑢 ∈ V ↦ (𝑢 ∩ Cat) / 𝑏{⟨(Base‘ndx), 𝑏⟩, ⟨(Hom ‘ndx), (𝑥𝑏, 𝑦𝑏 ↦ (𝑥 Func 𝑦))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝑏 × 𝑏), 𝑧𝑏 ↦ (𝑔 ∈ ((2nd𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔func 𝑓)))⟩})
cestrc 16173class ExtStrCat
df-estrc 16174ExtStrCat = (𝑢 ∈ V ↦ {⟨(Base‘ndx), 𝑢⟩, ⟨(Hom ‘ndx), (𝑥𝑢, 𝑦𝑢 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝑢 × 𝑢), 𝑧𝑢 ↦ (𝑔 ∈ ((Base‘𝑧) ↑𝑚 (Base‘(2nd𝑣))), 𝑓 ∈ ((Base‘(2nd𝑣)) ↑𝑚 (Base‘(1st𝑣))) ↦ (𝑔𝑓)))⟩})
cxpc 16219class ×c
c1stf 16220class 1stF
c2ndf 16221class 2ndF
cprf 16222class ⟨,⟩F
df-xpc 16223 ×c = (𝑟 ∈ V, 𝑠 ∈ V ↦ ((Base‘𝑟) × (Base‘𝑠)) / 𝑏(𝑢𝑏, 𝑣𝑏 ↦ (((1st𝑢)(Hom ‘𝑟)(1st𝑣)) × ((2nd𝑢)(Hom ‘𝑠)(2nd𝑣)))) / {⟨(Base‘ndx), 𝑏⟩, ⟨(Hom ‘ndx), ⟩, ⟨(comp‘ndx), (𝑥 ∈ (𝑏 × 𝑏), 𝑦𝑏 ↦ (𝑔 ∈ ((2nd𝑥)𝑦), 𝑓 ∈ (𝑥) ↦ ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝑟)(1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝑠)(2nd𝑦))(2nd𝑓))⟩))⟩})
df-1stf 16224 1stF = (𝑟 ∈ Cat, 𝑠 ∈ Cat ↦ ((Base‘𝑟) × (Base‘𝑠)) / 𝑏⟨(1st𝑏), (𝑥𝑏, 𝑦𝑏 ↦ (1st ↾ (𝑥(Hom ‘(𝑟 ×c 𝑠))𝑦)))⟩)
df-2ndf 16225 2ndF = (𝑟 ∈ Cat, 𝑠 ∈ Cat ↦ ((Base‘𝑟) × (Base‘𝑠)) / 𝑏⟨(2nd𝑏), (𝑥𝑏, 𝑦𝑏 ↦ (2nd ↾ (𝑥(Hom ‘(𝑟 ×c 𝑠))𝑦)))⟩)
df-prf 16226 ⟨,⟩F = (𝑓 ∈ V, 𝑔 ∈ V ↦ dom (1st𝑓) / 𝑏⟨(𝑥𝑏 ↦ ⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩), (𝑥𝑏, 𝑦𝑏 ↦ ( ∈ dom (𝑥(2nd𝑓)𝑦) ↦ ⟨((𝑥(2nd𝑓)𝑦)‘), ((𝑥(2nd𝑔)𝑦)‘)⟩))⟩)
cevlf 16260class evalF
ccurf 16261class curryF
cuncf 16262class uncurryF
cdiag 16263class Δfunc
df-evlf 16264 evalF = (𝑐 ∈ Cat, 𝑑 ∈ Cat ↦ ⟨(𝑓 ∈ (𝑐 Func 𝑑), 𝑥 ∈ (Base‘𝑐) ↦ ((1st𝑓)‘𝑥)), (𝑥 ∈ ((𝑐 Func 𝑑) × (Base‘𝑐)), 𝑦 ∈ ((𝑐 Func 𝑑) × (Base‘𝑐)) ↦ (1st𝑥) / 𝑚(1st𝑦) / 𝑛(𝑎 ∈ (𝑚(𝑐 Nat 𝑑)𝑛), 𝑔 ∈ ((2nd𝑥)(Hom ‘𝑐)(2nd𝑦)) ↦ ((𝑎‘(2nd𝑦))(⟨((1st𝑚)‘(2nd𝑥)), ((1st𝑚)‘(2nd𝑦))⟩(comp‘𝑑)((1st𝑛)‘(2nd𝑦)))(((2nd𝑥)(2nd𝑚)(2nd𝑦))‘𝑔))))⟩)
df-curf 16265 curryF = (𝑒 ∈ V, 𝑓 ∈ V ↦ (1st𝑒) / 𝑐(2nd𝑒) / 𝑑⟨(𝑥 ∈ (Base‘𝑐) ↦ ⟨(𝑦 ∈ (Base‘𝑑) ↦ (𝑥(1st𝑓)𝑦)), (𝑦 ∈ (Base‘𝑑), 𝑧 ∈ (Base‘𝑑) ↦ (𝑔 ∈ (𝑦(Hom ‘𝑑)𝑧) ↦ (((Id‘𝑐)‘𝑥)(⟨𝑥, 𝑦⟩(2nd𝑓)⟨𝑥, 𝑧⟩)𝑔)))⟩), (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ (𝑔 ∈ (𝑥(Hom ‘𝑐)𝑦) ↦ (𝑧 ∈ (Base‘𝑑) ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd𝑓)⟨𝑦, 𝑧⟩)((Id‘𝑑)‘𝑧)))))⟩)
df-uncf 16266 uncurryF = (𝑐 ∈ V, 𝑓 ∈ V ↦ (((𝑐‘1) evalF (𝑐‘2)) ∘func ((𝑓func ((𝑐‘0) 1stF (𝑐‘1))) ⟨,⟩F ((𝑐‘0) 2ndF (𝑐‘1)))))
df-diag 16267Δfunc = (𝑐 ∈ Cat, 𝑑 ∈ Cat ↦ (⟨𝑐, 𝑑⟩ curryF (𝑐 1stF 𝑑)))
chof 16299class HomF
cyon 16300class Yon
df-hof 16301HomF = (𝑐 ∈ Cat ↦ ⟨(Homf𝑐), (Base‘𝑐) / 𝑏(𝑥 ∈ (𝑏 × 𝑏), 𝑦 ∈ (𝑏 × 𝑏) ↦ (𝑓 ∈ ((1st𝑦)(Hom ‘𝑐)(1st𝑥)), 𝑔 ∈ ((2nd𝑥)(Hom ‘𝑐)(2nd𝑦)) ↦ ( ∈ ((Hom ‘𝑐)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝑐)(2nd𝑦)))(⟨(1st𝑦), (1st𝑥)⟩(comp‘𝑐)(2nd𝑦))𝑓))))⟩)
df-yon 16302Yon = (𝑐 ∈ Cat ↦ (⟨𝑐, (oppCat‘𝑐)⟩ curryF (HomF‘(oppCat‘𝑐))))
cpreset 16337class Preset
cdrs 16338class Dirset
df-preset 16339 Preset = {𝑓[(Base‘𝑓) / 𝑏][(le‘𝑓) / 𝑟]𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧))}
df-drs 16340Dirset = {𝑓 ∈ Preset ∣ [(Base‘𝑓) / 𝑏][(le‘𝑓) / 𝑟](𝑏 ≠ ∅ ∧ ∀𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑟𝑧𝑦𝑟𝑧))}
cpo 16351class Poset
cplt 16352class lt
club 16353class lub
cglb 16354class glb
cjn 16355class join
cmee 16356class meet
df-poset 16357Poset = {𝑓 ∣ ∃𝑏𝑟(𝑏 = (Base‘𝑓) ∧ 𝑟 = (le‘𝑓) ∧ ∀𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦𝑦𝑟𝑥) → 𝑥 = 𝑦) ∧ ((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧)))}
df-plt 16369lt = (𝑝 ∈ V ↦ ((le‘𝑝) ∖ I ))
df-lub 16385lub = (𝑝 ∈ V ↦ ((𝑠 ∈ 𝒫 (Base‘𝑝) ↦ (𝑥 ∈ (Base‘𝑝)(∀𝑦𝑠 𝑦(le‘𝑝)𝑥 ∧ ∀𝑧 ∈ (Base‘𝑝)(∀𝑦𝑠 𝑦(le‘𝑝)𝑧𝑥(le‘𝑝)𝑧)))) ↾ {𝑠 ∣ ∃!𝑥 ∈ (Base‘𝑝)(∀𝑦𝑠 𝑦(le‘𝑝)𝑥 ∧ ∀𝑧 ∈ (Base‘𝑝)(∀𝑦𝑠 𝑦(le‘𝑝)𝑧𝑥(le‘𝑝)𝑧))}))
df-glb 16386glb = (𝑝 ∈ V ↦ ((𝑠 ∈ 𝒫 (Base‘𝑝) ↦ (𝑥 ∈ (Base‘𝑝)(∀𝑦𝑠 𝑥(le‘𝑝)𝑦 ∧ ∀𝑧 ∈ (Base‘𝑝)(∀𝑦𝑠 𝑧(le‘𝑝)𝑦𝑧(le‘𝑝)𝑥)))) ↾ {𝑠 ∣ ∃!𝑥 ∈ (Base‘𝑝)(∀𝑦𝑠 𝑥(le‘𝑝)𝑦 ∧ ∀𝑧 ∈ (Base‘𝑝)(∀𝑦𝑠 𝑧(le‘𝑝)𝑦𝑧(le‘𝑝)𝑥))}))
df-join 16387join = (𝑝 ∈ V ↦ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦} (lub‘𝑝)𝑧})
df-meet 16388meet = (𝑝 ∈ V ↦ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦} (glb‘𝑝)𝑧})
ctos 16444class Toset
df-toset 16445Toset = {𝑓 ∈ Poset ∣ [(Base‘𝑓) / 𝑏][(le‘𝑓) / 𝑟]𝑥𝑏𝑦𝑏 (𝑥𝑟𝑦𝑦𝑟𝑥)}
cp0 16448class 0.
cp1 16449class 1.
df-p0 164500. = (𝑝 ∈ V ↦ ((glb‘𝑝)‘(Base‘𝑝)))
df-p1 164511. = (𝑝 ∈ V ↦ ((lub‘𝑝)‘(Base‘𝑝)))
clat 16456class Lat
df-lat 16457Lat = {𝑝 ∈ Poset ∣ (dom (join‘𝑝) = ((Base‘𝑝) × (Base‘𝑝)) ∧ dom (meet‘𝑝) = ((Base‘𝑝) × (Base‘𝑝)))}
ccla 16518class CLat
df-clat 16519CLat = {𝑝 ∈ Poset ∣ (dom (lub‘𝑝) = 𝒫 (Base‘𝑝) ∧ dom (glb‘𝑝) = 𝒫 (Base‘𝑝))}
codu 16539class ODual
df-odu 16540ODual = (𝑤 ∈ V ↦ (𝑤 sSet ⟨(le‘ndx), (le‘𝑤)⟩))
cipo 16562class toInc
df-ipo 16563toInc = (𝑓 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑓𝑥𝑦)} / 𝑜({⟨(Base‘ndx), 𝑓⟩, ⟨(TopSet‘ndx), (ordTop‘𝑜)⟩} ∪ {⟨(le‘ndx), 𝑜⟩, ⟨(oc‘ndx), (𝑥𝑓 {𝑦𝑓 ∣ (𝑦𝑥) = ∅})⟩}))
cdlat 16602class DLat
df-dlat 16603DLat = {𝑘 ∈ Lat ∣ [(Base‘𝑘) / 𝑏][(join‘𝑘) / 𝑗][(meet‘𝑘) / 𝑚]𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑚(𝑦𝑗𝑧)) = ((𝑥𝑚𝑦)𝑗(𝑥𝑚𝑧))}
cps 16609class PosetRel
ctsr 16610class TosetRel
df-ps 16611PosetRel = {𝑟 ∣ (Rel 𝑟 ∧ (𝑟𝑟) ⊆ 𝑟 ∧ (𝑟𝑟) = ( I ↾ 𝑟))}
df-tsr 16612 TosetRel = {𝑟 ∈ PosetRel ∣ (dom 𝑟 × dom 𝑟) ⊆ (𝑟𝑟)}
cdir 16639class DirRel
ctail 16640class tail
df-dir 16641DirRel = {𝑟 ∣ ((Rel 𝑟 ∧ ( I ↾ 𝑟) ⊆ 𝑟) ∧ ((𝑟𝑟) ⊆ 𝑟 ∧ ( 𝑟 × 𝑟) ⊆ (𝑟𝑟)))}
df-tail 16642tail = (𝑟 ∈ DirRel ↦ (𝑥 𝑟 ↦ (𝑟 “ {𝑥})))
cplusf 16650class +𝑓
cmgm 16651class Mgm
df-plusf 16652+𝑓 = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(+g𝑔)𝑦)))
df-mgm 16653Mgm = {𝑔[(Base‘𝑔) / 𝑏][(+g𝑔) / 𝑜]𝑥𝑏𝑦𝑏 (𝑥𝑜𝑦) ∈ 𝑏}
csgrp 16691class SGrp
df-sgrp 16692SGrp = {𝑔 ∈ Mgm ∣ [(Base‘𝑔) / 𝑏][(+g𝑔) / 𝑜]𝑥𝑏𝑦𝑏𝑧𝑏 ((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧))}
cmnd 16700class Mnd
cmndOLD 16701class MndOLD
df-mnd 16702Mnd = {𝑔 ∈ SGrp ∣ [(Base‘𝑔) / 𝑏][(+g𝑔) / 𝑝]𝑒𝑏𝑥𝑏 ((𝑒𝑝𝑥) = 𝑥 ∧ (𝑥𝑝𝑒) = 𝑥)}
cmhm 16740class MndHom
csubmnd 16741class SubMnd
df-mhm 16742 MndHom = (𝑠 ∈ Mnd, 𝑡 ∈ Mnd ↦ {𝑓 ∈ ((Base‘𝑡) ↑𝑚 (Base‘𝑠)) ∣ (∀𝑥 ∈ (Base‘𝑠)∀𝑦 ∈ (Base‘𝑠)(𝑓‘(𝑥(+g𝑠)𝑦)) = ((𝑓𝑥)(+g𝑡)(𝑓𝑦)) ∧ (𝑓‘(0g𝑠)) = (0g𝑡))})
df-submnd 16743SubMnd = (𝑠 ∈ Mnd ↦ {𝑡 ∈ 𝒫 (Base‘𝑠) ∣ ((0g𝑠) ∈ 𝑡 ∧ ∀𝑥𝑡𝑦𝑡 (𝑥(+g𝑠)𝑦) ∈ 𝑡)})
cfrmd 16791class freeMnd
cvrmd 16792class varFMnd
df-frmd 16793freeMnd = (𝑖 ∈ V ↦ {⟨(Base‘ndx), Word 𝑖⟩, ⟨(+g‘ndx), ( ++ ↾ (Word 𝑖 × Word 𝑖))⟩})
df-vrmd 16794varFMnd = (𝑖 ∈ V ↦ (𝑗𝑖 ↦ ⟨“𝑗”⟩))
cgrp 16829class Grp
cminusg 16830class invg
csg 16831class -g
cmg 16832class .g
df-grp 16833Grp = {𝑔 ∈ Mnd ∣ ∀𝑎 ∈ (Base‘𝑔)∃𝑚 ∈ (Base‘𝑔)(𝑚(+g𝑔)𝑎) = (0g𝑔)}
df-minusg 16834invg = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔) ↦ (𝑤 ∈ (Base‘𝑔)(𝑤(+g𝑔)𝑥) = (0g𝑔))))
df-sbg 16835-g = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(+g𝑔)((invg𝑔)‘𝑦))))
df-mulg 16836.g = (𝑔 ∈ V ↦ (𝑛 ∈ ℤ, 𝑥 ∈ (Base‘𝑔) ↦ if(𝑛 = 0, (0g𝑔), seq1((+g𝑔), (ℕ × {𝑥})) / 𝑠if(0 < 𝑛, (𝑠𝑛), ((invg𝑔)‘(𝑠‘-𝑛))))))
csubg 16971class SubGrp
cnsg 16972class NrmSGrp
cqg 16973class ~QG
df-subg 16974SubGrp = (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤s 𝑠) ∈ Grp})
df-nsg 16975NrmSGrp = (𝑤 ∈ Grp ↦ {𝑠 ∈ (SubGrp‘𝑤) ∣ [(Base‘𝑤) / 𝑏][(+g𝑤) / 𝑝]𝑥𝑏𝑦𝑏 ((𝑥𝑝𝑦) ∈ 𝑠 ↔ (𝑦𝑝𝑥) ∈ 𝑠)})
df-eqg 16976 ~QG = (𝑟 ∈ V, 𝑖 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (Base‘𝑟) ∧ (((invg𝑟)‘𝑥)(+g𝑟)𝑦) ∈ 𝑖)})
cghm 17040class GrpHom
df-ghm 17041 GrpHom = (𝑠 ∈ Grp, 𝑡 ∈ Grp ↦ {𝑔[(Base‘𝑠) / 𝑤](𝑔:𝑤⟶(Base‘𝑡) ∧ ∀𝑥𝑤𝑦𝑤 (𝑔‘(𝑥(+g𝑠)𝑦)) = ((𝑔𝑥)(+g𝑡)(𝑔𝑦)))})
cgim 17081class GrpIso
cgic 17082class 𝑔
df-gim 17083 GrpIso = (𝑠 ∈ Grp, 𝑡 ∈ Grp ↦ {𝑔 ∈ (𝑠 GrpHom 𝑡) ∣ 𝑔:(Base‘𝑠)–1-1-onto→(Base‘𝑡)})
df-gic 17084𝑔 = ( GrpIso “ (V ∖ 1𝑜))
cga 17104class GrpAct
df-ga 17105 GrpAct = (𝑔 ∈ Grp, 𝑠 ∈ V ↦ (Base‘𝑔) / 𝑏{𝑚 ∈ (𝑠𝑚 (𝑏 × 𝑠)) ∣ ∀𝑥𝑠 (((0g𝑔)𝑚𝑥) = 𝑥 ∧ ∀𝑦𝑏𝑧𝑏 ((𝑦(+g𝑔)𝑧)𝑚𝑥) = (𝑦𝑚(𝑧𝑚𝑥)))})
ccntz 17130class Cntz
ccntr 17131class Cntr
df-cntz 17132Cntz = (𝑚 ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘𝑚) ↦ {𝑥 ∈ (Base‘𝑚) ∣ ∀𝑦𝑠 (𝑥(+g𝑚)𝑦) = (𝑦(+g𝑚)𝑥)}))
df-cntr 17133Cntr = (𝑚 ∈ V ↦ ((Cntz‘𝑚)‘(Base‘𝑚)))
coppg 17157class oppg
df-oppg 17158oppg = (𝑤 ∈ V ↦ (𝑤 sSet ⟨(+g‘ndx), tpos (+g𝑤)⟩))
csymg 17179class SymGrp
df-symg 17180SymGrp = (𝑥 ∈ V ↦ {:𝑥1-1-onto𝑥} / 𝑏{⟨(Base‘ndx), 𝑏⟩, ⟨(+g‘ndx), (𝑓𝑏, 𝑔𝑏 ↦ (𝑓𝑔))⟩, ⟨(TopSet‘ndx), (∏t‘(𝑥 × {𝒫 𝑥}))⟩})
cpmtr 17243class pmTrsp
df-pmtr 17244pmTrsp = (𝑑 ∈ V ↦ (𝑝 ∈ {𝑦 ∈ 𝒫 𝑑𝑦 ≈ 2𝑜} ↦ (𝑧𝑑 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))))
cpsgn 17291class pmSgn
cevpm 17292class pmEven
df-psgn 17293pmSgn = (𝑑 ∈ V ↦ (𝑥 ∈ {𝑝 ∈ (Base‘(SymGrp‘𝑑)) ∣ dom (𝑝 ∖ I ) ∈ Fin} ↦ (℩𝑠𝑤 ∈ Word ran (pmTrsp‘𝑑)(𝑥 = ((SymGrp‘𝑑) Σg 𝑤) ∧ 𝑠 = (-1↑(#‘𝑤))))))
df-evpm 17294pmEven = (𝑑 ∈ V ↦ ((pmSgn‘𝑑) “ {1}))
cod 17326class od
codold 17327class od
cgex 17328class gEx
cgexold 17329class gEx
cpgp 17330class pGrp
cpgpold 17331class pGrp
cslw 17332class pSyl
df-od 17333od = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔) ↦ {𝑛 ∈ ℕ ∣ (𝑛(.g𝑔)𝑥) = (0g𝑔)} / 𝑖if(𝑖 = ∅, 0, inf(𝑖, ℝ, < ))))
df-odOLD 17334od = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔) ↦ {𝑛 ∈ ℕ ∣ (𝑛(.g𝑔)𝑥) = (0g𝑔)} / 𝑖if(𝑖 = ∅, 0, sup(𝑖, ℝ, < ))))
df-gex 17335gEx = (𝑔 ∈ V ↦ {𝑛 ∈ ℕ ∣ ∀𝑥 ∈ (Base‘𝑔)(𝑛(.g𝑔)𝑥) = (0g𝑔)} / 𝑖if(𝑖 = ∅, 0, inf(𝑖, ℝ, < )))
df-gexOLD 17336gEx = (𝑔 ∈ V ↦ {𝑛 ∈ ℕ ∣ ∀𝑥 ∈ (Base‘𝑔)(𝑛(.g𝑔)𝑥) = (0g𝑔)} / 𝑖if(𝑖 = ∅, 0, sup(𝑖, ℝ, < )))
df-pgp 17337 pGrp = {⟨𝑝, 𝑔⟩ ∣ ((𝑝 ∈ ℙ ∧ 𝑔 ∈ Grp) ∧ ∀𝑥 ∈ (Base‘𝑔)∃𝑛 ∈ ℕ0 ((od‘𝑔)‘𝑥) = (𝑝𝑛))}
df-pgpOLD 17338 pGrp = {⟨𝑝, 𝑔⟩ ∣ ((𝑝 ∈ ℙ ∧ 𝑔 ∈ Grp) ∧ ∀𝑥 ∈ (Base‘𝑔)∃𝑛 ∈ ℕ0 ((od‘𝑔)‘𝑥) = (𝑝𝑛))}
df-slw 17339 pSyl = (𝑝 ∈ ℙ, 𝑔 ∈ Grp ↦ { ∈ (SubGrp‘𝑔) ∣ ∀𝑘 ∈ (SubGrp‘𝑔)((𝑘𝑝 pGrp (𝑔s 𝑘)) ↔ = 𝑘)})
clsm 17447class LSSum
cpj1 17448class proj1
df-lsm 17449LSSum = (𝑤 ∈ V ↦ (𝑡 ∈ 𝒫 (Base‘𝑤), 𝑢 ∈ 𝒫 (Base‘𝑤) ↦ ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝑤)𝑦))))
df-pj1 17450proj1 = (𝑤 ∈ V ↦ (𝑡 ∈ 𝒫 (Base‘𝑤), 𝑢 ∈ 𝒫 (Base‘𝑤) ↦ (𝑧 ∈ (𝑡(LSSum‘𝑤)𝑢) ↦ (𝑥𝑡𝑦𝑢 𝑧 = (𝑥(+g𝑤)𝑦)))))
cefg 17517class ~FG
cfrgp 17518class freeGrp
cvrgp 17519class varFGrp
df-efg 17520 ~FG = (𝑖 ∈ V ↦ {𝑟 ∣ (𝑟 Er Word (𝑖 × 2𝑜) ∧ ∀𝑥 ∈ Word (𝑖 × 2𝑜)∀𝑛 ∈ (0...(#‘𝑥))∀𝑦𝑖𝑧 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜𝑧)⟩”⟩⟩))})
df-frgp 17521freeGrp = (𝑖 ∈ V ↦ ((freeMnd‘(𝑖 × 2𝑜)) /s ( ~FG𝑖)))
df-vrgp 17522varFGrp = (𝑖 ∈ V ↦ (𝑗𝑖 ↦ [⟨“⟨𝑗, ∅⟩”⟩]( ~FG𝑖)))
ccmn 17591class CMnd
cabl 17592class Abel
df-cmn 17593CMnd = {𝑔 ∈ Mnd ∣ ∀𝑎 ∈ (Base‘𝑔)∀𝑏 ∈ (Base‘𝑔)(𝑎(+g𝑔)𝑏) = (𝑏(+g𝑔)𝑎)}
df-abl 17594Abel = (Grp ∩ CMnd)
ccyg 17673class CycGrp
df-cyg 17674CycGrp = {𝑔 ∈ Grp ∣ ∃𝑥 ∈ (Base‘𝑔)ran (𝑛 ∈ ℤ ↦ (𝑛(.g𝑔)𝑥)) = (Base‘𝑔)}
cdprd 17786class DProd
cdpj 17787class dProj
df-dprd 17788 DProd = (𝑔 ∈ Grp, 𝑠 ∈ { ∣ (:dom ⟶(SubGrp‘𝑔) ∧ ∀𝑥 ∈ dom (∀𝑦 ∈ (dom ∖ {𝑥})(𝑥) ⊆ ((Cntz‘𝑔)‘(𝑦)) ∧ ((𝑥) ∩ ((mrCls‘(SubGrp‘𝑔))‘ ( “ (dom ∖ {𝑥})))) = {(0g𝑔)}))} ↦ ran (𝑓 ∈ {X𝑥 ∈ dom 𝑠(𝑠𝑥) ∣ finSupp (0g𝑔)} ↦ (𝑔 Σg 𝑓)))
df-dpj 17789dProj = (𝑔 ∈ Grp, 𝑠 ∈ (dom DProd “ {𝑔}) ↦ (𝑖 ∈ dom 𝑠 ↦ ((𝑠𝑖)(proj1𝑔)(𝑔 DProd (𝑠 ↾ (dom 𝑠 ∖ {𝑖}))))))
cmgp 17883class mulGrp
df-mgp 17884mulGrp = (𝑤 ∈ V ↦ (𝑤 sSet ⟨(+g‘ndx), (.r𝑤)⟩))
cur 17895class 1r
df-ur 178961r = (0g ∘ mulGrp)
csrg 17899class SRing
df-srg 17900SRing = {𝑓 ∈ CMnd ∣ ((mulGrp‘𝑓) ∈ Mnd ∧ [(Base‘𝑓) / 𝑟][(+g𝑓) / 𝑝][(.r𝑓) / 𝑡][(0g𝑓) / 𝑛]𝑥𝑟 (∀𝑦𝑟𝑧𝑟 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))) ∧ ((𝑛𝑡𝑥) = 𝑛 ∧ (𝑥𝑡𝑛) = 𝑛)))}
crg 17940class Ring
ccrg 17941class CRing
df-ring 17942Ring = {𝑓 ∈ Grp ∣ ((mulGrp‘𝑓) ∈ Mnd ∧ [(Base‘𝑓) / 𝑟][(+g𝑓) / 𝑝][(.r𝑓) / 𝑡]𝑥𝑟𝑦𝑟𝑧𝑟 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))))}
df-cring 17943CRing = {𝑓 ∈ Ring ∣ (mulGrp‘𝑓) ∈ CMnd}
coppr 18010class oppr
df-oppr 18011oppr = (𝑓 ∈ V ↦ (𝑓 sSet ⟨(.r‘ndx), tpos (.r𝑓)⟩))
cdsr 18026class r
cui 18027class Unit
cir 18028class Irred
df-dvdsr 18029r = (𝑤 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑤) ∧ ∃𝑧 ∈ (Base‘𝑤)(𝑧(.r𝑤)𝑥) = 𝑦)})
df-unit 18030Unit = (𝑤 ∈ V ↦ (((∥r𝑤) ∩ (∥r‘(oppr𝑤))) “ {(1r𝑤)}))
df-irred 18031Irred = (𝑤 ∈ V ↦ ((Base‘𝑤) ∖ (Unit‘𝑤)) / 𝑏{𝑧𝑏 ∣ ∀𝑥𝑏𝑦𝑏 (𝑥(.r𝑤)𝑦) ≠ 𝑧})
cinvr 18059class invr
df-invr 18060invr = (𝑟 ∈ V ↦ (invg‘((mulGrp‘𝑟) ↾s (Unit‘𝑟))))
cdvr 18070class /r
df-dvr 18071/r = (𝑟 ∈ V ↦ (𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Unit‘𝑟) ↦ (𝑥(.r𝑟)((invr𝑟)‘𝑦))))
crh 18100class RingHom
crs 18101class RingIso
cric 18102class 𝑟
df-rnghom 18103 RingHom = (𝑟 ∈ Ring, 𝑠 ∈ Ring ↦ (Base‘𝑟) / 𝑣(Base‘𝑠) / 𝑤{𝑓 ∈ (𝑤𝑚 𝑣) ∣ ((𝑓‘(1r𝑟)) = (1r𝑠) ∧ ∀𝑥𝑣𝑦𝑣 ((𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦))))})
df-rngiso 18104 RingIso = (𝑟 ∈ V, 𝑠 ∈ V ↦ {𝑓 ∈ (𝑟 RingHom 𝑠) ∣ 𝑓 ∈ (𝑠 RingHom 𝑟)})
df-ric 18106𝑟 = ( RingIso “ (V ∖ 1𝑜))
cdr 18135class DivRing
cfield 18136class Field
df-drng 18137DivRing = {𝑟 ∈ Ring ∣ (Unit‘𝑟) = ((Base‘𝑟) ∖ {(0g𝑟)})}
df-field 18138Field = (DivRing ∩ CRing)
csubrg 18164class SubRing
crgspn 18165class RingSpan
df-subrg 18166SubRing = (𝑤 ∈ Ring ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ ((𝑤s 𝑠) ∈ Ring ∧ (1r𝑤) ∈ 𝑠)})
df-rgspn 18167RingSpan = (𝑤 ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘𝑤) ↦ {𝑡 ∈ (SubRing‘𝑤) ∣ 𝑠𝑡}))
cabv 18204class AbsVal
df-abv 18205AbsVal = (𝑟 ∈ Ring ↦ {𝑓 ∈ ((0[,)+∞) ↑𝑚 (Base‘𝑟)) ∣ ∀𝑥 ∈ (Base‘𝑟)(((𝑓𝑥) = 0 ↔ 𝑥 = (0g𝑟)) ∧ ∀𝑦 ∈ (Base‘𝑟)((𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥(+g𝑟)𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦))))})
cstf 18231class *rf
csr 18232class *-Ring
df-staf 18233*rf = (𝑓 ∈ V ↦ (𝑥 ∈ (Base‘𝑓) ↦ ((*𝑟𝑓)‘𝑥)))
df-srng 18234*-Ring = {𝑓[(*rf𝑓) / 𝑖](𝑖 ∈ (𝑓 RingHom (oppr𝑓)) ∧ 𝑖 = 𝑖)}
clmod 18251class LMod
cscaf 18252class ·sf
df-lmod 18253LMod = {𝑔 ∈ Grp ∣ [(Base‘𝑔) / 𝑣][(+g𝑔) / 𝑎][(Scalar‘𝑔) / 𝑓][( ·𝑠𝑔) / 𝑠][(Base‘𝑓) / 𝑘][(+g𝑓) / 𝑝][(.r𝑓) / 𝑡](𝑓 ∈ Ring ∧ ∀𝑞𝑘𝑟𝑘𝑥𝑣𝑤𝑣 (((𝑟𝑠𝑤) ∈ 𝑣 ∧ (𝑟𝑠(𝑤𝑎𝑥)) = ((𝑟𝑠𝑤)𝑎(𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤)𝑎(𝑟𝑠𝑤))) ∧ (((𝑞𝑡𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ((1r𝑓)𝑠𝑤) = 𝑤)))}
df-scaf 18254 ·sf = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘(Scalar‘𝑔)), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥( ·𝑠𝑔)𝑦)))
clss 18315class LSubSp
df-lss 18316LSubSp = (𝑤 ∈ V ↦ {𝑠 ∈ (𝒫 (Base‘𝑤) ∖ {∅}) ∣ ∀𝑥 ∈ (Base‘(Scalar‘𝑤))∀𝑎𝑠𝑏𝑠 ((𝑥( ·𝑠𝑤)𝑎)(+g𝑤)𝑏) ∈ 𝑠})
clspn 18354class LSpan
df-lsp 18355LSpan = (𝑤 ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘𝑤) ↦ {𝑡 ∈ (LSubSp‘𝑤) ∣ 𝑠𝑡}))
clmhm 18402class LMHom
clmim 18403class LMIso
clmic 18404class 𝑚
df-lmhm 18405 LMHom = (𝑠 ∈ LMod, 𝑡 ∈ LMod ↦ {𝑓 ∈ (𝑠 GrpHom 𝑡) ∣ [(Scalar‘𝑠) / 𝑤]((Scalar‘𝑡) = 𝑤 ∧ ∀𝑥 ∈ (Base‘𝑤)∀𝑦 ∈ (Base‘𝑠)(𝑓‘(𝑥( ·𝑠𝑠)𝑦)) = (𝑥( ·𝑠𝑡)(𝑓𝑦)))})
df-lmim 18406 LMIso = (𝑠 ∈ LMod, 𝑡 ∈ LMod ↦ {𝑔 ∈ (𝑠 LMHom 𝑡) ∣ 𝑔:(Base‘𝑠)–1-1-onto→(Base‘𝑡)})
df-lmic 18407𝑚 = ( LMIso “ (V ∖ 1𝑜))
clbs 18457class LBasis
df-lbs 18458LBasis = (𝑤 ∈ V ↦ {𝑏 ∈ 𝒫 (Base‘𝑤) ∣ [(LSpan‘𝑤) / 𝑛][(Scalar‘𝑤) / 𝑠]((𝑛𝑏) = (Base‘𝑤) ∧ ∀𝑥𝑏𝑦 ∈ ((Base‘𝑠) ∖ {(0g𝑠)}) ¬ (𝑦( ·𝑠𝑤)𝑥) ∈ (𝑛‘(𝑏 ∖ {𝑥})))})
clvec 18485class LVec
df-lvec 18486LVec = {𝑓 ∈ LMod ∣ (Scalar‘𝑓) ∈ DivRing}
csra 18551class subringAlg
crglmod 18552class ringLMod
clidl 18553class LIdeal
crsp 18554class RSpan
df-sra 18555subringAlg = (𝑤 ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘𝑤) ↦ (((𝑤 sSet ⟨(Scalar‘ndx), (𝑤s 𝑠)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r𝑤)⟩) sSet ⟨(·𝑖‘ndx), (.r𝑤)⟩)))
df-rgmod 18556ringLMod = (𝑤 ∈ V ↦ ((subringAlg ‘𝑤)‘(Base‘𝑤)))
df-lidl 18557LIdeal = (LSubSp ∘ ringLMod)
df-rsp 18558RSpan = (LSpan ∘ ringLMod)
c2idl 18614class 2Ideal
df-2idl 186152Ideal = (𝑟 ∈ V ↦ ((LIdeal‘𝑟) ∩ (LIdeal‘(oppr𝑟))))
clpidl 18624class LPIdeal
clpir 18625class LPIR
df-lpidl 18626LPIdeal = (𝑤 ∈ Ring ↦ 𝑔 ∈ (Base‘𝑤){((RSpan‘𝑤)‘{𝑔})})
df-lpir 18627LPIR = {𝑤 ∈ Ring ∣ (LIdeal‘𝑤) = (LPIdeal‘𝑤)}
cnzr 18640class NzRing
df-nzr 18641NzRing = {𝑟 ∈ Ring ∣ (1r𝑟) ≠ (0g𝑟)}
crlreg 18662class RLReg
cdomn 18663class Domn
cidom 18664class IDomn
cpid 18665class PID
df-rlreg 18666RLReg = (𝑟 ∈ V ↦ {𝑥 ∈ (Base‘𝑟) ∣ ∀𝑦 ∈ (Base‘𝑟)((𝑥(.r𝑟)𝑦) = (0g𝑟) → 𝑦 = (0g𝑟))})
df-domn 18667Domn = {𝑟 ∈ NzRing ∣ [(Base‘𝑟) / 𝑏][(0g𝑟) / 𝑧]𝑥𝑏𝑦𝑏 ((𝑥(.r𝑟)𝑦) = 𝑧 → (𝑥 = 𝑧𝑦 = 𝑧))}
df-idom 18668IDomn = (CRing ∩ Domn)
df-pid 18669PID = (IDomn ∩ LPIR)
casa 18692class AssAlg
casp 18693class AlgSpan
cascl 18694class algSc
df-assa 18695AssAlg = {𝑤 ∈ (LMod ∩ Ring) ∣ [(Scalar‘𝑤) / 𝑓](𝑓 ∈ CRing ∧ ∀𝑟 ∈ (Base‘𝑓)∀𝑥 ∈ (Base‘𝑤)∀𝑦 ∈ (Base‘𝑤)[( ·𝑠𝑤) / 𝑠][(.r𝑤) / 𝑡](((𝑟𝑠𝑥)𝑡𝑦) = (𝑟𝑠(𝑥𝑡𝑦)) ∧ (𝑥𝑡(𝑟𝑠𝑦)) = (𝑟𝑠(𝑥𝑡𝑦))))}
df-asp 18696AlgSpan = (𝑤 ∈ AssAlg ↦ (𝑠 ∈ 𝒫 (Base‘𝑤) ↦ {𝑡 ∈ ((SubRing‘𝑤) ∩ (LSubSp‘𝑤)) ∣ 𝑠𝑡}))
df-ascl 18697algSc = (𝑤 ∈ V ↦ (𝑥 ∈ (Base‘(Scalar‘𝑤)) ↦ (𝑥( ·𝑠𝑤)(1r𝑤))))
cmps 18734class mPwSer
cmvr 18735class mVar
cmpl 18736class mPoly
cltb 18737class <bag
copws 18738class ordPwSer
df-psr 18739 mPwSer = (𝑖 ∈ V, 𝑟 ∈ V ↦ { ∈ (ℕ0𝑚 𝑖) ∣ ( “ ℕ) ∈ Fin} / 𝑑((Base‘𝑟) ↑𝑚 𝑑) / 𝑏({⟨(Base‘ndx), 𝑏⟩, ⟨(+g‘ndx), ( ∘𝑓 (+g𝑟) ↾ (𝑏 × 𝑏))⟩, ⟨(.r‘ndx), (𝑓𝑏, 𝑔𝑏 ↦ (𝑘𝑑 ↦ (𝑟 Σg (𝑥 ∈ {𝑦𝑑𝑦𝑟𝑘} ↦ ((𝑓𝑥)(.r𝑟)(𝑔‘(𝑘𝑓𝑥)))))))⟩} ∪ {⟨(Scalar‘ndx), 𝑟⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑟), 𝑓𝑏 ↦ ((𝑑 × {𝑥}) ∘𝑓 (.r𝑟)𝑓))⟩, ⟨(TopSet‘ndx), (∏t‘(𝑑 × {(TopOpen‘𝑟)}))⟩}))
df-mvr 18740 mVar = (𝑖 ∈ V, 𝑟 ∈ V ↦ (𝑥𝑖 ↦ (𝑓 ∈ { ∈ (ℕ0𝑚 𝑖) ∣ ( “ ℕ) ∈ Fin} ↦ if(𝑓 = (𝑦𝑖 ↦ if(𝑦 = 𝑥, 1, 0)), (1r𝑟), (0g𝑟)))))
df-mpl 18741 mPoly = (𝑖 ∈ V, 𝑟 ∈ V ↦ (𝑖 mPwSer 𝑟) / 𝑤(𝑤s {𝑓 ∈ (Base‘𝑤) ∣ 𝑓 finSupp (0g𝑟)}))
df-ltbag 18742 <bag = (𝑟 ∈ V, 𝑖 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ { ∈ (ℕ0𝑚 𝑖) ∣ ( “ ℕ) ∈ Fin} ∧ ∃𝑧𝑖 ((𝑥𝑧) < (𝑦𝑧) ∧ ∀𝑤𝑖 (𝑧𝑟𝑤 → (𝑥𝑤) = (𝑦𝑤))))})
df-opsr 18743 ordPwSer = (𝑖 ∈ V, 𝑠 ∈ V ↦ (𝑟 ∈ 𝒫 (𝑖 × 𝑖) ↦ (𝑖 mPwSer 𝑠) / 𝑝(𝑝 sSet ⟨(le‘ndx), {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (Base‘𝑝) ∧ ([{ ∈ (ℕ0𝑚 𝑖) ∣ ( “ ℕ) ∈ Fin} / 𝑑]𝑧𝑑 ((𝑥𝑧)(lt‘𝑠)(𝑦𝑧) ∧ ∀𝑤𝑑 (𝑤(𝑟 <bag 𝑖)𝑧 → (𝑥𝑤) = (𝑦𝑤))) ∨ 𝑥 = 𝑦))}⟩)))
ces 18886class evalSub
cevl 18887class eval
df-evls 18888 evalSub = (𝑖 ∈ V, 𝑠 ∈ CRing ↦ (Base‘𝑠) / 𝑏(𝑟 ∈ (SubRing‘𝑠) ↦ (𝑖 mPoly (𝑠s 𝑟)) / 𝑤(𝑓 ∈ (𝑤 RingHom (𝑠s (𝑏𝑚 𝑖)))((𝑓 ∘ (algSc‘𝑤)) = (𝑥𝑟 ↦ ((𝑏𝑚 𝑖) × {𝑥})) ∧ (𝑓 ∘ (𝑖 mVar (𝑠s 𝑟))) = (𝑥𝑖 ↦ (𝑔 ∈ (𝑏𝑚 𝑖) ↦ (𝑔𝑥)))))))
df-evl 18889 eval = (𝑖 ∈ V, 𝑟 ∈ V ↦ ((𝑖 evalSub 𝑟)‘(Base‘𝑟)))
cmhp 18919class mHomP
cpsd 18920class mPSDer
cslv 18921class selectVars
cai 18922class AlgInd
df-mhp 18923 mHomP = (𝑖 ∈ V, 𝑟 ∈ V ↦ (𝑛 ∈ ℕ0 ↦ {𝑓 ∈ (Base‘(𝑖 mPoly 𝑟)) ∣ (𝑓 supp (0g𝑟)) ⊆ {𝑔 ∈ { ∈ (ℕ0𝑚 𝑖) ∣ ( “ ℕ) ∈ Fin} ∣ Σ𝑗 ∈ ℕ0 (𝑔𝑗) = 𝑛}}))
df-psd 18924 mPSDer = (𝑖 ∈ V, 𝑟 ∈ V ↦ (𝑥𝑖 ↦ (𝑓 ∈ (Base‘(𝑖 mPwSer 𝑟)) ↦ (𝑘 ∈ { ∈ (ℕ0𝑚 𝑖) ∣ ( “ ℕ) ∈ Fin} ↦ (((𝑘𝑥) + 1)(.g𝑟)(𝑓‘(𝑘𝑓 + (𝑦𝑖 ↦ if(𝑦 = 𝑥, 1, 0)))))))))
df-selv 18925 selectVars = (𝑖 ∈ V, 𝑟 ∈ V ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑓 ∈ (𝑖 mPoly 𝑟) ↦ ((𝑖𝑗) mPoly 𝑟) / 𝑠(𝑥 ∈ (Scalar‘𝑠) ↦ (𝑥( ·𝑠𝑠)(1r𝑠))) / 𝑐((((𝑖 evalSub 𝑠)‘(𝑐s 𝑟))‘(𝑐𝑓))‘(𝑥𝑖 ↦ if(𝑥𝑗, ((𝑗 mVar ((𝑖𝑗) mPoly 𝑟))‘𝑥), (𝑐 ∘ (((𝑖𝑗) mVar 𝑟)‘𝑥))))))))
df-algind 18926 AlgInd = (𝑤 ∈ V, 𝑘 ∈ 𝒫 (Base‘𝑤) ↦ {𝑣 ∈ 𝒫 (Base‘𝑤) ∣ Fun (𝑓 ∈ (Base‘(𝑣 mPoly (𝑤s 𝑘))) ↦ ((((𝑣 evalSub 𝑤)‘𝑘)‘𝑓)‘( I ↾ 𝑣)))})
cps1 18927class PwSer1
cv1 18928class var1
cpl1 18929class Poly1
cco1 18930class coe1
ctp1 18931class toPoly1
df-psr1 18932PwSer1 = (𝑟 ∈ V ↦ ((1𝑜 ordPwSer 𝑟)‘∅))
df-vr1 18933var1 = (𝑟 ∈ V ↦ ((1𝑜 mVar 𝑟)‘∅))
df-ply1 18934Poly1 = (𝑟 ∈ V ↦ ((PwSer1𝑟) ↾s (Base‘(1𝑜 mPoly 𝑟))))
df-coe1 18935coe1 = (𝑓 ∈ V ↦ (𝑛 ∈ ℕ0 ↦ (𝑓‘(1𝑜 × {𝑛}))))
df-toply1 18936toPoly1 = (𝑓 ∈ V ↦ (𝑛 ∈ (ℕ0𝑚 1𝑜) ↦ (𝑓‘(𝑛‘∅))))
ces1 19060class evalSub1
ce1 19061class eval1
df-evls1 19062 evalSub1 = (𝑠 ∈ V, 𝑟 ∈ 𝒫 (Base‘𝑠) ↦ (Base‘𝑠) / 𝑏((𝑥 ∈ (𝑏𝑚 (𝑏𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝑏 ↦ (1𝑜 × {𝑦})))) ∘ ((1𝑜 evalSub 𝑠)‘𝑟)))
df-evl1 19063eval1 = (𝑟 ∈ V ↦ (Base‘𝑟) / 𝑏((𝑥 ∈ (𝑏𝑚 (𝑏𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝑏 ↦ (1𝑜 × {𝑦})))) ∘ (1𝑜 eval 𝑟)))
cpsmet 19112class PsMet
cxmt 19113class ∞Met
cme 19114class Met
cbl 19115class ball
cfbas 19116class fBas
cfg 19117class filGen
cmopn 19118class MetOpen
cmetu 19119class metUnif
df-psmet 19120PsMet = (𝑥 ∈ V ↦ {𝑑 ∈ (ℝ*𝑚 (𝑥 × 𝑥)) ∣ ∀𝑦𝑥 ((𝑦𝑑𝑦) = 0 ∧ ∀𝑧𝑥𝑤𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) +𝑒 (𝑤𝑑𝑧)))})
df-xmet 19121∞Met = (𝑥 ∈ V ↦ {𝑑 ∈ (ℝ*𝑚 (𝑥 × 𝑥)) ∣ ∀𝑦𝑥𝑧𝑥 (((𝑦𝑑𝑧) = 0 ↔ 𝑦 = 𝑧) ∧ ∀𝑤𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) +𝑒 (𝑤𝑑𝑧)))})
df-met 19122Met = (𝑥 ∈ V ↦ {𝑑 ∈ (ℝ ↑𝑚 (𝑥 × 𝑥)) ∣ ∀𝑦𝑥𝑧𝑥 (((𝑦𝑑𝑧) = 0 ↔ 𝑦 = 𝑧) ∧ ∀𝑤𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) + (𝑤𝑑𝑧)))})
df-bl 19123ball = (𝑑 ∈ V ↦ (𝑥 ∈ dom dom 𝑑, 𝑧 ∈ ℝ* ↦ {𝑦 ∈ dom dom 𝑑 ∣ (𝑥𝑑𝑦) < 𝑧}))
df-mopn 19124MetOpen = (𝑑 ran ∞Met ↦ (topGen‘ran (ball‘𝑑)))
df-fbas 19125fBas = (𝑤 ∈ V ↦ {𝑥 ∈ 𝒫 𝒫 𝑤 ∣ (𝑥 ≠ ∅ ∧ ∅ ∉ 𝑥 ∧ ∀𝑦𝑥𝑧𝑥 (𝑥 ∩ 𝒫 (𝑦𝑧)) ≠ ∅)})
df-fg 19126filGen = (𝑤 ∈ V, 𝑥 ∈ (fBas‘𝑤) ↦ {𝑦 ∈ 𝒫 𝑤 ∣ (𝑥 ∩ 𝒫 𝑦) ≠ ∅})
df-metu 19127metUnif = (𝑑 ran PsMet ↦ ((dom dom 𝑑 × dom dom 𝑑)filGenran (𝑎 ∈ ℝ+ ↦ (𝑑 “ (0[,)𝑎)))))
ccnfld 19128class fld
df-cnfld 19129fld = (({⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), · ⟩} ∪ {⟨(*𝑟‘ndx), ∗⟩}) ∪ ({⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∪ {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩}))
zring 19197class ring
df-zring 19198ring = (ℂflds ℤ)
czrh 19229class ℤRHom
czlm 19230class ℤMod
cchr 19231class chr
czn 19232class ℤ/n
df-zrh 19233ℤRHom = (𝑟 ∈ V ↦ (ℤring RingHom 𝑟))
df-zlm 19234ℤMod = (𝑔 ∈ V ↦ ((𝑔 sSet ⟨(Scalar‘ndx), ℤring⟩) sSet ⟨( ·𝑠 ‘ndx), (.g𝑔)⟩))
df-chr 19235chr = (𝑔 ∈ V ↦ ((od‘𝑔)‘(1r𝑔)))
df-zn 19236ℤ/nℤ = (𝑛 ∈ ℕ0ring / 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 19330class fld
df-refld 19331fld = (ℂflds ℝ)
cphl 19349class PreHil
cipf 19350class ·if
df-phl 19351PreHil = {𝑔 ∈ LVec ∣ [(Base‘𝑔) / 𝑣][(·𝑖𝑔) / ][(Scalar‘𝑔) / 𝑓](𝑓 ∈ *-Ring ∧ ∀𝑥𝑣 ((𝑦𝑣 ↦ (𝑦𝑥)) ∈ (𝑔 LMHom (ringLMod‘𝑓)) ∧ ((𝑥𝑥) = (0g𝑓) → 𝑥 = (0g𝑔)) ∧ ∀𝑦𝑣 ((*𝑟𝑓)‘(𝑥𝑦)) = (𝑦𝑥)))}
df-ipf 19352·if = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(·𝑖𝑔)𝑦)))
cocv 19381class ocv
ccss 19382class CSubSp
cthl 19383class toHL
df-ocv 19384ocv = ( ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘) ↦ {𝑥 ∈ (Base‘) ∣ ∀𝑦𝑠 (𝑥(·𝑖)𝑦) = (0g‘(Scalar‘))}))
df-css 19385CSubSp = ( ∈ V ↦ {𝑠𝑠 = ((ocv‘)‘((ocv‘)‘𝑠))})
df-thl 19386toHL = ( ∈ V ↦ ((toInc‘(CSubSp‘)) sSet ⟨(oc‘ndx), (ocv‘)⟩))
cpj 19421class proj
chs 19422class Hil
cobs 19423class OBasis
df-pj 19424proj = ( ∈ V ↦ ((𝑥 ∈ (LSubSp‘) ↦ (𝑥(proj1)((ocv‘)‘𝑥))) ∩ (V × ((Base‘) ↑𝑚 (Base‘)))))
df-hil 19425Hil = { ∈ PreHil ∣ dom (proj‘) = (CSubSp‘)}
df-obs 19426OBasis = ( ∈ PreHil ↦ {𝑏 ∈ 𝒫 (Base‘) ∣ (∀𝑥𝑏𝑦𝑏 (𝑥(·𝑖)𝑦) = if(𝑥 = 𝑦, (1r‘(Scalar‘)), (0g‘(Scalar‘))) ∧ ((ocv‘)‘𝑏) = {(0g)})})
cdsmm 19452class m
df-dsmm 19453m = (𝑠 ∈ V, 𝑟 ∈ V ↦ ((𝑠Xs𝑟) ↾s {𝑓X𝑥 ∈ dom 𝑟(Base‘(𝑟𝑥)) ∣ {𝑥 ∈ dom 𝑟 ∣ (𝑓𝑥) ≠ (0g‘(𝑟𝑥))} ∈ Fin}))
cfrlm 19467class freeLMod
df-frlm 19468 freeLMod = (𝑟 ∈ V, 𝑖 ∈ V ↦ (𝑟m (𝑖 × {(ringLMod‘𝑟)})))
cuvc 19498class unitVec
df-uvc 19499 unitVec = (𝑟 ∈ V, 𝑖 ∈ V ↦ (𝑗𝑖 ↦ (𝑘𝑖 ↦ if(𝑘 = 𝑗, (1r𝑟), (0g𝑟)))))
clindf 19520class LIndF
clinds 19521class LIndS
df-lindf 19522 LIndF = {⟨𝑓, 𝑤⟩ ∣ (𝑓:dom 𝑓⟶(Base‘𝑤) ∧ [(Scalar‘𝑤) / 𝑠]𝑥 ∈ dom 𝑓𝑘 ∈ ((Base‘𝑠) ∖ {(0g𝑠)}) ¬ (𝑘( ·𝑠𝑤)(𝑓𝑥)) ∈ ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥}))))}
df-linds 19523LIndS = (𝑤 ∈ V ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ ( I ↾ 𝑠) LIndF 𝑤})
cmmul 19566class maMul
df-mamu 19567 maMul = (𝑟 ∈ V, 𝑜 ∈ V ↦ (1st ‘(1st𝑜)) / 𝑚(2nd ‘(1st𝑜)) / 𝑛(2nd𝑜) / 𝑝(𝑥 ∈ ((Base‘𝑟) ↑𝑚 (𝑚 × 𝑛)), 𝑦 ∈ ((Base‘𝑟) ↑𝑚 (𝑛 × 𝑝)) ↦ (𝑖𝑚, 𝑘𝑝 ↦ (𝑟 Σg (𝑗𝑛 ↦ ((𝑖𝑥𝑗)(.r𝑟)(𝑗𝑦𝑘)))))))
cmat 19590class Mat
df-mat 19591 Mat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ ((𝑟 freeLMod (𝑛 × 𝑛)) sSet ⟨(.r‘ndx), (𝑟 maMul ⟨𝑛, 𝑛, 𝑛⟩)⟩))
cdmat 19671class DMat
cscmat 19672class ScMat
df-dmat 19673 DMat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ {𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ∣ ∀𝑖𝑛𝑗𝑛 (𝑖𝑗 → (𝑖𝑚𝑗) = (0g𝑟))})
df-scmat 19674 ScMat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ (𝑛 Mat 𝑟) / 𝑎{𝑚 ∈ (Base‘𝑎) ∣ ∃𝑐 ∈ (Base‘𝑟)𝑚 = (𝑐( ·𝑠𝑎)(1r𝑎))})
cmvmul 19723class maVecMul
df-mvmul 19724 maVecMul = (𝑟 ∈ V, 𝑜 ∈ V ↦ (1st𝑜) / 𝑚(2nd𝑜) / 𝑛(𝑥 ∈ ((Base‘𝑟) ↑𝑚 (𝑚 × 𝑛)), 𝑦 ∈ ((Base‘𝑟) ↑𝑚 𝑛) ↦ (𝑖𝑚 ↦ (𝑟 Σg (𝑗𝑛 ↦ ((𝑖𝑥𝑗)(.r𝑟)(𝑦𝑗)))))))
cmarrep 19739class matRRep
cmatrepV 19740class matRepV
df-marrep 19741 matRRep = (𝑛 ∈ V, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)), 𝑠 ∈ (Base‘𝑟) ↦ (𝑘𝑛, 𝑙𝑛 ↦ (𝑖𝑛, 𝑗𝑛 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑠, (0g𝑟)), (𝑖𝑚𝑗))))))
df-marepv 19742 matRepV = (𝑛 ∈ V, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)), 𝑣 ∈ ((Base‘𝑟) ↑𝑚 𝑛) ↦ (𝑘𝑛 ↦ (𝑖𝑛, 𝑗𝑛 ↦ if(𝑗 = 𝑘, (𝑣𝑖), (𝑖𝑚𝑗))))))
csubma 19759class subMat
df-subma 19760 subMat = (𝑛 ∈ V, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑘𝑛, 𝑙𝑛 ↦ (𝑖 ∈ (𝑛 ∖ {𝑘}), 𝑗 ∈ (𝑛 ∖ {𝑙}) ↦ (𝑖𝑚𝑗)))))
cmdat 19767class maDet
df-mdet 19768 maDet = (𝑛 ∈ V, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑟 Σg (𝑝 ∈ (Base‘(SymGrp‘𝑛)) ↦ ((((ℤRHom‘𝑟) ∘ (pmSgn‘𝑛))‘𝑝)(.r𝑟)((mulGrp‘𝑟) Σg (𝑥𝑛 ↦ ((𝑝𝑥)𝑚𝑥))))))))
cmadu 19815class maAdju
cminmar1 19816class minMatR1
df-madu 19817 maAdju = (𝑛 ∈ V, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑖𝑛, 𝑗𝑛 ↦ ((𝑛 maDet 𝑟)‘(𝑘𝑛, 𝑙𝑛 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, (1r𝑟), (0g𝑟)), (𝑘𝑚𝑙)))))))
df-minmar1 19818 minMatR1 = (𝑛 ∈ V, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑘𝑛, 𝑙𝑛 ↦ (𝑖𝑛, 𝑗𝑛 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, (1r𝑟), (0g𝑟)), (𝑖𝑚𝑗))))))
ccpmat 19885class ConstPolyMat
cmat2pmat 19886class matToPolyMat
ccpmat2mat 19887class cPolyMatToMat
df-cpmat 19888 ConstPolyMat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ {𝑚 ∈ (Base‘(𝑛 Mat (Poly1𝑟))) ∣ ∀𝑖𝑛𝑗𝑛𝑘 ∈ ℕ ((coe1‘(𝑖𝑚𝑗))‘𝑘) = (0g𝑟)})
df-mat2pmat 19889 matToPolyMat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑥𝑛, 𝑦𝑛 ↦ ((algSc‘(Poly1𝑟))‘(𝑥𝑚𝑦)))))
df-cpmat2mat 19890 cPolyMatToMat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ (𝑚 ∈ (𝑛 ConstPolyMat 𝑟) ↦ (𝑥𝑛, 𝑦𝑛 ↦ ((coe1‘(𝑥𝑚𝑦))‘0))))
cdecpmat 19944class decompPMat
df-decpmat 19945 decompPMat = (𝑚 ∈ V, 𝑘 ∈ ℕ0 ↦ (𝑖 ∈ dom dom 𝑚, 𝑗 ∈ dom dom 𝑚 ↦ ((coe1‘(𝑖𝑚𝑗))‘𝑘)))
cpm2mp 19974class pMatToMatPoly
df-pm2mp 19975 pMatToMatPoly = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat (Poly1𝑟))) ↦ (𝑛 Mat 𝑟) / 𝑎(Poly1𝑎) / 𝑞(𝑞 Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘)( ·𝑠𝑞)(𝑘(.g‘(mulGrp‘𝑞))(var1𝑎)))))))
cchpmat 20008class CharPlyMat
df-chpmat 20009 CharPlyMat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ ((𝑛 maDet (Poly1𝑟))‘(((var1𝑟)( ·𝑠 ‘(𝑛 Mat (Poly1𝑟)))(1r‘(𝑛 Mat (Poly1𝑟))))(-g‘(𝑛 Mat (Poly1𝑟)))((𝑛 matToPolyMat 𝑟)‘𝑚)))))
ctop 20075class Top
ctopon 20076class TopOn
ctps 20077class TopSp
ctb 20078class TopBases
df-top 20079Top = {𝑥 ∣ (∀𝑦 ∈ 𝒫 𝑥 𝑦𝑥 ∧ ∀𝑦𝑥𝑧𝑥 (𝑦𝑧) ∈ 𝑥)}
df-bases 20080TopBases = {𝑥 ∣ ∀𝑦𝑥𝑧𝑥 (𝑦𝑧) ⊆ (𝑥 ∩ 𝒫 (𝑦𝑧))}
df-topon 20081TopOn = (𝑏 ∈ V ↦ {𝑗 ∈ Top ∣ 𝑏 = 𝑗})
df-topsp 20082TopSp = {𝑓 ∣ (TopOpen‘𝑓) ∈ (TopOn‘(Base‘𝑓))}
ccld 20188class Clsd
cnt 20189class int
ccl 20190class cls
df-cld 20191Clsd = (𝑗 ∈ Top ↦ {𝑥 ∈ 𝒫 𝑗 ∣ ( 𝑗𝑥) ∈ 𝑗})
df-ntr 20192int = (𝑗 ∈ Top ↦ (𝑥 ∈ 𝒫 𝑗 (𝑗 ∩ 𝒫 𝑥)))
df-cls 20193cls = (𝑗 ∈ Top ↦ (𝑥 ∈ 𝒫 𝑗 {𝑦 ∈ (Clsd‘𝑗) ∣ 𝑥𝑦}))
cnei 20270class nei
df-nei 20271nei = (𝑗 ∈ Top ↦ (𝑥 ∈ 𝒫 𝑗 ↦ {𝑦 ∈ 𝒫 𝑗 ∣ ∃𝑔𝑗 (𝑥𝑔𝑔𝑦)}))
clp 20307class limPt
cperf 20308class Perf
df-lp 20309limPt = (𝑗 ∈ Top ↦ (𝑥 ∈ 𝒫 𝑗 ↦ {𝑦𝑦 ∈ ((cls‘𝑗)‘(𝑥 ∖ {𝑦}))}))
df-perf 20310Perf = {𝑗 ∈ Top ∣ ((limPt‘𝑗)‘ 𝑗) = 𝑗}
ccn 20397class Cn
ccnp 20398class CnP
clm 20399class 𝑡
df-cn 20400 Cn = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ {𝑓 ∈ ( 𝑘𝑚 𝑗) ∣ ∀𝑦𝑘 (𝑓𝑦) ∈ 𝑗})
df-cnp 20401 CnP = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ (𝑥 𝑗 ↦ {𝑓 ∈ ( 𝑘𝑚 𝑗) ∣ ∀𝑦𝑘 ((𝑓𝑥) ∈ 𝑦 → ∃𝑔𝑗 (𝑥𝑔 ∧ (𝑓𝑔) ⊆ 𝑦))}))
df-lm 20402𝑡 = (𝑗 ∈ Top ↦ {⟨𝑓, 𝑥⟩ ∣ (𝑓 ∈ ( 𝑗pm ℂ) ∧ 𝑥 𝑗 ∧ ∀𝑢𝑗 (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢))})
ct0 20479class Kol2
ct1 20480class Fre
cha 20481class Haus
creg 20482class Reg
cnrm 20483class Nrm
ccnrm 20484class CNrm
cpnrm 20485class PNrm
df-t0 20486Kol2 = {𝑗 ∈ Top ∣ ∀𝑥 𝑗𝑦 𝑗(∀𝑜𝑗 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦)}
df-t1 20487Fre = {𝑥 ∈ Top ∣ ∀𝑎 𝑥{𝑎} ∈ (Clsd‘𝑥)}
df-haus 20488Haus = {𝑗 ∈ Top ∣ ∀𝑥 𝑗𝑦 𝑗(𝑥𝑦 → ∃𝑛𝑗𝑚𝑗 (𝑥𝑛𝑦𝑚 ∧ (𝑛𝑚) = ∅))}
df-reg 20489Reg = {𝑗 ∈ Top ∣ ∀𝑥𝑗𝑦𝑥𝑧𝑗 (𝑦𝑧 ∧ ((cls‘𝑗)‘𝑧) ⊆ 𝑥)}
df-nrm 20490Nrm = {𝑗 ∈ Top ∣ ∀𝑥𝑗𝑦 ∈ ((Clsd‘𝑗) ∩ 𝒫 𝑥)∃𝑧𝑗 (𝑦𝑧 ∧ ((cls‘𝑗)‘𝑧) ⊆ 𝑥)}
df-cnrm 20491CNrm = {𝑗 ∈ Top ∣ ∀𝑥 ∈ 𝒫 𝑗(𝑗t 𝑥) ∈ Nrm}
df-pnrm 20492PNrm = {𝑗 ∈ Nrm ∣ (Clsd‘𝑗) ⊆ ran (𝑓 ∈ (𝑗𝑚 ℕ) ↦ ran 𝑓)}
ccmp 20558class Comp
df-cmp 20559Comp = {𝑥 ∈ Top ∣ ∀𝑦 ∈ 𝒫 𝑥( 𝑥 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) 𝑥 = 𝑧)}
ccon 20583class Con
df-con 20584Con = {𝑗 ∈ Top ∣ (𝑗 ∩ (Clsd‘𝑗)) = {∅, 𝑗}}
c1stc 20609class 1st𝜔
c2ndc 20610class 2nd𝜔
df-1stc 206111st𝜔 = {𝑗 ∈ Top ∣ ∀𝑥 𝑗𝑦 ∈ 𝒫 𝑗(𝑦 ≼ ω ∧ ∀𝑧𝑗 (𝑥𝑧𝑥 (𝑦 ∩ 𝒫 𝑧)))}
df-2ndc 206122nd𝜔 = {𝑗 ∣ ∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝑗)}
clly 20636class Locally 𝐴
cnlly 20637class 𝑛-Locally 𝐴
df-lly 20638Locally 𝐴 = {𝑗 ∈ Top ∣ ∀𝑥𝑗𝑦𝑥𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝐴)}
df-nlly 20639𝑛-Locally 𝐴 = {𝑗 ∈ Top ∣ ∀𝑥𝑗𝑦𝑥𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗t 𝑢) ∈ 𝐴}
cref 20674class Ref
cptfin 20675class PtFin
clocfin 20676class LocFin
df-ref 20677Ref = {⟨𝑥, 𝑦⟩ ∣ ( 𝑦 = 𝑥 ∧ ∀𝑧𝑥𝑤𝑦 𝑧𝑤)}
df-ptfin 20678PtFin = {𝑥 ∣ ∀𝑦 𝑥{𝑧𝑥𝑦𝑧} ∈ Fin}
df-locfin 20679LocFin = (𝑥 ∈ Top ↦ {𝑦 ∣ ( 𝑥 = 𝑦 ∧ ∀𝑝 𝑥𝑛𝑥 (𝑝𝑛 ∧ {𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))})
ckgen 20705class 𝑘Gen
df-kgen 20706𝑘Gen = (𝑗 ∈ Top ↦ {𝑥 ∈ 𝒫 𝑗 ∣ ∀𝑘 ∈ 𝒫 𝑗((𝑗t 𝑘) ∈ Comp → (𝑥𝑘) ∈ (𝑗t 𝑘))})
ctx 20732class ×t
cxko 20733class ^ko
df-tx 20734 ×t = (𝑟 ∈ V, 𝑠 ∈ V ↦ (topGen‘ran (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦))))
df-xko 20735 ^ko = (𝑠 ∈ Top, 𝑟 ∈ Top ↦ (topGen‘(fi‘ran (𝑘 ∈ {𝑥 ∈ 𝒫 𝑟 ∣ (𝑟t 𝑥) ∈ Comp}, 𝑣𝑠 ↦ {𝑓 ∈ (𝑟 Cn 𝑠) ∣ (𝑓𝑘) ⊆ 𝑣}))))
ckq 20865class KQ
df-kq 20866KQ = (𝑗 ∈ Top ↦ (𝑗 qTop (𝑥 𝑗 ↦ {𝑦𝑗𝑥𝑦})))
chmeo 20925class Homeo
chmph 20926class
df-hmeo 20927Homeo = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ {𝑓 ∈ (𝑗 Cn 𝑘) ∣ 𝑓 ∈ (𝑘 Cn 𝑗)})
df-hmph 20928 ≃ = (Homeo “ (V ∖ 1𝑜))
cfil 21018class Fil
df-fil 21019Fil = (𝑧 ∈ V ↦ {𝑓 ∈ (fBas‘𝑧) ∣ ∀𝑥 ∈ 𝒫 𝑧((𝑓 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥𝑓)})
cufil 21072class UFil
cufl 21073class UFL
df-ufil 21074UFil = (𝑔 ∈ V ↦ {𝑓 ∈ (Fil‘𝑔) ∣ ∀𝑥 ∈ 𝒫 𝑔(𝑥𝑓 ∨ (𝑔𝑥) ∈ 𝑓)})
df-ufl 21075UFL = {𝑥 ∣ ∀𝑓 ∈ (Fil‘𝑥)∃𝑔 ∈ (UFil‘𝑥)𝑓𝑔}
cfm 21106class FilMap
cflim 21107class fLim
cflf 21108class fLimf
cfcls 21109class fClus
cfcf 21110class fClusf
df-fm 21111 FilMap = (𝑥 ∈ V, 𝑓 ∈ V ↦ (𝑦 ∈ (fBas‘dom 𝑓) ↦ (𝑥filGenran (𝑡𝑦 ↦ (𝑓𝑡)))))
df-flim 21112 fLim = (𝑗 ∈ Top, 𝑓 ran Fil ↦ {𝑥 𝑗 ∣ (((nei‘𝑗)‘{𝑥}) ⊆ 𝑓𝑓 ⊆ 𝒫 𝑗)})
df-flf 21113 fLimf = (𝑥 ∈ Top, 𝑦 ran Fil ↦ (𝑓 ∈ ( 𝑥𝑚 𝑦) ↦ (𝑥 fLim (( 𝑥 FilMap 𝑓)‘𝑦))))
df-fcls 21114 fClus = (𝑗 ∈ Top, 𝑓 ran Fil ↦ if( 𝑗 = 𝑓, 𝑥𝑓 ((cls‘𝑗)‘𝑥), ∅))
df-fcf 21115 fClusf = (𝑗 ∈ Top, 𝑓 ran Fil ↦ (𝑔 ∈ ( 𝑗𝑚 𝑓) ↦ (𝑗 fClus (( 𝑗 FilMap 𝑔)‘𝑓))))
ccnext 21232class CnExt
df-cnext 21233CnExt = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ (𝑓 ∈ ( 𝑘pm 𝑗) ↦ 𝑥 ∈ ((cls‘𝑗)‘dom 𝑓)({𝑥} × ((𝑘 fLimf (((nei‘𝑗)‘{𝑥}) ↾t dom 𝑓))‘𝑓))))
ctmd 21243class TopMnd
ctgp 21244class TopGrp
df-tmd 21245TopMnd = {𝑓 ∈ (Mnd ∩ TopSp) ∣ [(TopOpen‘𝑓) / 𝑗](+𝑓𝑓) ∈ ((𝑗 ×t 𝑗) Cn 𝑗)}
df-tgp 21246TopGrp = {𝑓 ∈ (Grp ∩ TopMnd) ∣ [(TopOpen‘𝑓) / 𝑗](invg𝑓) ∈ (𝑗 Cn 𝑗)}
ctsu 21298class tsums
df-tsms 21299 tsums = (𝑤 ∈ V, 𝑓 ∈ V ↦ (𝒫 dom 𝑓 ∩ Fin) / 𝑠(((TopOpen‘𝑤) fLimf (𝑠filGenran (𝑧𝑠 ↦ {𝑦𝑠𝑧𝑦})))‘(𝑦𝑠 ↦ (𝑤 Σg (𝑓𝑦)))))
ctrg 21328class TopRing
ctdrg 21329class TopDRing
ctlm 21330class TopMod
ctvc 21331class TopVec
df-trg 21332TopRing = {𝑟 ∈ (TopGrp ∩ Ring) ∣ (mulGrp‘𝑟) ∈ TopMnd}
df-tdrg 21333TopDRing = {𝑟 ∈ (TopRing ∩ DivRing) ∣ ((mulGrp‘𝑟) ↾s (Unit‘𝑟)) ∈ TopGrp}
df-tlm 21334TopMod = {𝑤 ∈ (TopMnd ∩ LMod) ∣ ((Scalar‘𝑤) ∈ TopRing ∧ ( ·sf𝑤) ∈ (((TopOpen‘(Scalar‘𝑤)) ×t (TopOpen‘𝑤)) Cn (TopOpen‘𝑤)))}
df-tvc 21335TopVec = {𝑤 ∈ TopMod ∣ (Scalar‘𝑤) ∈ TopDRing}
cust 21372class UnifOn
df-ust 21373UnifOn = (𝑥 ∈ V ↦ {𝑢 ∣ (𝑢 ⊆ 𝒫 (𝑥 × 𝑥) ∧ (𝑥 × 𝑥) ∈ 𝑢 ∧ ∀𝑣𝑢 (∀𝑤 ∈ 𝒫 (𝑥 × 𝑥)(𝑣𝑤𝑤𝑢) ∧ ∀𝑤𝑢 (𝑣𝑤) ∈ 𝑢 ∧ (( I ↾ 𝑥) ⊆ 𝑣𝑣𝑢 ∧ ∃𝑤𝑢 (𝑤𝑤) ⊆ 𝑣)))})
cutop 21403class unifTop
df-utop 21404unifTop = (𝑢 ran UnifOn ↦ {𝑎 ∈ 𝒫 dom 𝑢 ∣ ∀𝑥𝑎𝑣𝑢 (𝑣 “ {𝑥}) ⊆ 𝑎})
cuss 21426class UnifSt
cusp 21427class UnifSp
ctus 21428class toUnifSp
df-uss 21429UnifSt = (𝑓 ∈ V ↦ ((UnifSet‘𝑓) ↾t ((Base‘𝑓) × (Base‘𝑓))))
df-usp 21430UnifSp = {𝑓 ∣ ((UnifSt‘𝑓) ∈ (UnifOn‘(Base‘𝑓)) ∧ (TopOpen‘𝑓) = (unifTop‘(UnifSt‘𝑓)))}
df-tus 21431toUnifSp = (𝑢 ran UnifOn ↦ ({⟨(Base‘ndx), dom 𝑢⟩, ⟨(UnifSet‘ndx), 𝑢⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑢)⟩))
cucn 21448class Cnu
df-ucn 21449 Cnu = (𝑢 ran UnifOn, 𝑣 ran UnifOn ↦ {𝑓 ∈ (dom 𝑣𝑚 dom 𝑢) ∣ ∀𝑠𝑣𝑟𝑢𝑥 ∈ dom 𝑢𝑦 ∈ dom 𝑢(𝑥𝑟𝑦 → (𝑓𝑥)𝑠(𝑓𝑦))})
ccfilu 21459class CauFilu
df-cfilu 21460CauFilu = (𝑢 ran UnifOn ↦ {𝑓 ∈ (fBas‘dom 𝑢) ∣ ∀𝑣𝑢𝑎𝑓 (𝑎 × 𝑎) ⊆ 𝑣})
ccusp 21470class CUnifSp
df-cusp 21471CUnifSp = {𝑤 ∈ UnifSp ∣ ∀𝑐 ∈ (Fil‘(Base‘𝑤))(𝑐 ∈ (CauFilu‘(UnifSt‘𝑤)) → ((TopOpen‘𝑤) fLim 𝑐) ≠ ∅)}
cxme 21490class ∞MetSp
cmt 21491class MetSp
ctmt 21492class toMetSp
df-xms 21493∞MetSp = {𝑓 ∈ TopSp ∣ (TopOpen‘𝑓) = (MetOpen‘((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))))}
df-ms 21494MetSp = {𝑓 ∈ ∞MetSp ∣ ((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))) ∈ (Met‘(Base‘𝑓))}
df-tms 21495toMetSp = (𝑑 ran ∞Met ↦ ({⟨(Base‘ndx), dom dom 𝑑⟩, ⟨(dist‘ndx), 𝑑⟩} sSet ⟨(TopSet‘ndx), (MetOpen‘𝑑)⟩))
cnm 21749class norm
cngp 21750class NrmGrp
ctng 21751class toNrmGrp
cnrg 21752class NrmRing
cnlm 21753class NrmMod
cnvc 21754class NrmVec
df-nm 21755norm = (𝑤 ∈ V ↦ (𝑥 ∈ (Base‘𝑤) ↦ (𝑥(dist‘𝑤)(0g𝑤))))
df-ngp 21756NrmGrp = {𝑔 ∈ (Grp ∩ MetSp) ∣ ((norm‘𝑔) ∘ (-g𝑔)) ⊆ (dist‘𝑔)}
df-tng 21757 toNrmGrp = (𝑔 ∈ V, 𝑓 ∈ V ↦ ((𝑔 sSet ⟨(dist‘ndx), (𝑓 ∘ (-g𝑔))⟩) sSet ⟨(TopSet‘ndx), (MetOpen‘(𝑓 ∘ (-g𝑔)))⟩))
df-nrg 21758NrmRing = {𝑤 ∈ NrmGrp ∣ (norm‘𝑤) ∈ (AbsVal‘𝑤)}
df-nlm 21759NrmMod = {𝑤 ∈ (NrmGrp ∩ LMod) ∣ [(Scalar‘𝑤) / 𝑓](𝑓 ∈ NrmRing ∧ ∀𝑥 ∈ (Base‘𝑓)∀𝑦 ∈ (Base‘𝑤)((norm‘𝑤)‘(𝑥( ·𝑠𝑤)𝑦)) = (((norm‘𝑓)‘𝑥) · ((norm‘𝑤)‘𝑦)))}
df-nvc 21760NrmVec = (NrmMod ∩ LVec)
cnmo 21864class normOp
cnmoold 21865class normOp
cnghm 21866class NGHom
cnghmold 21867class NGHom
cnmhm 21868class NMHom
df-nmo 21869 normOp = (𝑠 ∈ NrmGrp, 𝑡 ∈ NrmGrp ↦ (𝑓 ∈ (𝑠 GrpHom 𝑡) ↦ inf({𝑟 ∈ (0[,)+∞) ∣ ∀𝑥 ∈ (Base‘𝑠)((norm‘𝑡)‘(𝑓𝑥)) ≤ (𝑟 · ((norm‘𝑠)‘𝑥))}, ℝ*, < )))
df-nmoOLD 21870 normOp = (𝑠 ∈ NrmGrp, 𝑡 ∈ NrmGrp ↦ (𝑓 ∈ (𝑠 GrpHom 𝑡) ↦ sup({𝑟 ∈ (0[,)+∞) ∣ ∀𝑥 ∈ (Base‘𝑠)((norm‘𝑡)‘(𝑓𝑥)) ≤ (𝑟 · ((norm‘𝑠)‘𝑥))}, ℝ*, < )))
df-nghm 21871 NGHom = (𝑠 ∈ NrmGrp, 𝑡 ∈ NrmGrp ↦ ((𝑠 normOp 𝑡) “ ℝ))
df-nghmOLD 21872 NGHom = (𝑠 ∈ NrmGrp, 𝑡 ∈ NrmGrp ↦ ((𝑠 normOp 𝑡) “ ℝ))
df-nmhm 21873 NMHom = (𝑠 ∈ NrmMod, 𝑡 ∈ NrmMod ↦ ((𝑠 LMHom 𝑡) ∩ (𝑠 NGHom 𝑡)))
cii 22065class II
ccncf 22066class cn
df-ii 22067II = (MetOpen‘((abs ∘ − ) ↾ ((0[,]1) × (0[,]1))))
df-cncf 22068cn→ = (𝑎 ∈ 𝒫 ℂ, 𝑏 ∈ 𝒫 ℂ ↦ {𝑓 ∈ (𝑏𝑚 𝑎) ∣ ∀𝑥𝑎𝑒 ∈ ℝ+𝑑 ∈ ℝ+𝑦𝑎 ((abs‘(𝑥𝑦)) < 𝑑 → (abs‘((𝑓𝑥) − (𝑓𝑦))) < 𝑒)})
chtpy 22156class Htpy
cphtpy 22157class PHtpy
cphtpc 22158class ph
df-htpy 22159 Htpy = (𝑥 ∈ Top, 𝑦 ∈ Top ↦ (𝑓 ∈ (𝑥 Cn 𝑦), 𝑔 ∈ (𝑥 Cn 𝑦) ↦ { ∈ ((𝑥 ×t II) Cn 𝑦) ∣ ∀𝑠 𝑥((𝑠0) = (𝑓𝑠) ∧ (𝑠1) = (𝑔𝑠))}))
df-phtpy 22160PHtpy = (𝑥 ∈ Top ↦ (𝑓 ∈ (II Cn 𝑥), 𝑔 ∈ (II Cn 𝑥) ↦ { ∈ (𝑓(II Htpy 𝑥)𝑔) ∣ ∀𝑠 ∈ (0[,]1)((0𝑠) = (𝑓‘0) ∧ (1𝑠) = (𝑓‘1))}))
df-phtpc 22181ph = (𝑥 ∈ Top ↦ {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ (II Cn 𝑥) ∧ (𝑓(PHtpy‘𝑥)𝑔) ≠ ∅)})
cpco 22190class *𝑝
comi 22191class Ω1
comn 22192class Ω𝑛
cpi1 22193class π1
cpin 22194class πn
df-pco 22195*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 22196 Ω1 = (𝑗 ∈ Top, 𝑦 𝑗 ↦ {⟨(Base‘ndx), {𝑓 ∈ (II Cn 𝑗) ∣ ((𝑓‘0) = 𝑦 ∧ (𝑓‘1) = 𝑦)}⟩, ⟨(+g‘ndx), (*𝑝𝑗)⟩, ⟨(TopSet‘ndx), (𝑗 ^ko II)⟩})
df-omn 22197 Ω𝑛 = (𝑗 ∈ Top, 𝑦 𝑗 ↦ seq0(((𝑥 ∈ V, 𝑝 ∈ V ↦ ⟨((TopOpen‘(1st𝑥)) Ω1 (2nd𝑥)), ((0[,]1) × {(2nd𝑥)})⟩) ∘ 1st ), ⟨{⟨(Base‘ndx), 𝑗⟩, ⟨(TopSet‘ndx), 𝑗⟩}, 𝑦⟩))
df-pi1 22198 π1 = (𝑗 ∈ Top, 𝑦 𝑗 ↦ ((𝑗 Ω1 𝑦) /s ( ≃ph𝑗)))
df-pin 22199 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 22252class ℂMod
df-clm 22253ℂMod = {𝑤 ∈ LMod ∣ [(Scalar‘𝑤) / 𝑓][(Base‘𝑓) / 𝑘](𝑓 = (ℂflds 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))}
ccvs 22294class ℂVec
df-cvs 22295ℂVec = (ℂMod ∩ LVec)
ccph 22303class ℂPreHil
ctch 22304class toℂHil
df-cph 22305ℂPreHil = {𝑤 ∈ (PreHil ∩ NrmMod) ∣ [(Scalar‘𝑤) / 𝑓][(Base‘𝑓) / 𝑘](𝑓 = (ℂflds 𝑘) ∧ (√ “ (𝑘 ∩ (0[,)+∞))) ⊆ 𝑘 ∧ (norm‘𝑤) = (𝑥 ∈ (Base‘𝑤) ↦ (√‘(𝑥(·𝑖𝑤)𝑥))))}
df-tch 22306toℂHil = (𝑤 ∈ V ↦ (𝑤 toNrmGrp (𝑥 ∈ (Base‘𝑤) ↦ (√‘(𝑥(·𝑖𝑤)𝑥)))))
ccfil 22381class CauFil
cca 22382class Cau
cms 22383class CMet
df-cfil 22384CauFil = (𝑑 ran ∞Met ↦ {𝑓 ∈ (Fil‘dom dom 𝑑) ∣ ∀𝑥 ∈ ℝ+𝑦𝑓 (𝑑 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)})
df-cau 22385Cau = (𝑑 ran ∞Met ↦ {𝑓 ∈ (dom dom 𝑑pm ℂ) ∣ ∀𝑥 ∈ ℝ+𝑗 ∈ ℤ (𝑓 ↾ (ℤ𝑗)):(ℤ𝑗)⟶((𝑓𝑗)(ball‘𝑑)𝑥)})
df-cmet 22386CMet = (𝑥 ∈ V ↦ {𝑑 ∈ (Met‘𝑥) ∣ ∀𝑓 ∈ (CauFil‘𝑑)((MetOpen‘𝑑) fLim 𝑓) ≠ ∅})
ccms 22459class CMetSp
cbn 22460class Ban
chl 22461class ℂHil
df-cms 22462CMetSp = {𝑤 ∈ MetSp ∣ [(Base‘𝑤) / 𝑏]((dist‘𝑤) ↾ (𝑏 × 𝑏)) ∈ (CMet‘𝑏)}
df-bn 22463Ban = {𝑤 ∈ (NrmVec ∩ CMetSp) ∣ (Scalar‘𝑤) ∈ CMetSp}
df-hl 22464ℂHil = (Ban ∩ ℂPreHil)
crrx 22501class ℝ^
cehl 22502class 𝔼hil
df-rrx 22503ℝ^ = (𝑖 ∈ V ↦ (toℂHil‘(ℝfld freeLMod 𝑖)))
df-ehl 22504𝔼hil = (𝑛 ∈ ℕ0 ↦ (ℝ^‘(1...𝑛)))
covol 22572class vol*
covolold 22573class vol*
cvol 22574class vol
df-ovol 22575vol* = (𝑥 ∈ 𝒫 ℝ ↦ inf({𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ)(𝑥 ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))}, ℝ*, < ))
df-ovolOLD 22576vol* = (𝑥 ∈ 𝒫 ℝ ↦ sup({𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ)(𝑥 ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))}, ℝ*, < ))
df-vol 22577vol = (vol* ↾ {𝑥 ∣ ∀𝑦 ∈ (vol* “ ℝ)(vol*‘𝑦) = ((vol*‘(𝑦𝑥)) + (vol*‘(𝑦𝑥)))})
cmbf 22733class MblFn
citg1 22734class 1
citg2 22735class 2
cibl 22736class 𝐿1
citg 22737class 𝐴𝐵 d𝑥
df-mbf 22738MblFn = {𝑓 ∈ (ℂ ↑pm ℝ) ∣ ∀𝑥 ∈ ran (,)(((ℜ ∘ 𝑓) “ 𝑥) ∈ dom vol ∧ ((ℑ ∘ 𝑓) “ 𝑥) ∈ dom vol)}
df-itg1 227391 = (𝑓 ∈ {𝑔 ∈ MblFn ∣ (𝑔:ℝ⟶ℝ ∧ ran 𝑔 ∈ Fin ∧ (vol‘(𝑔 “ (ℝ ∖ {0}))) ∈ ℝ)} ↦ Σ𝑥 ∈ (ran 𝑓 ∖ {0})(𝑥 · (vol‘(𝑓 “ {𝑥}))))
df-itg2 227402 = (𝑓 ∈ ((0[,]+∞) ↑𝑚 ℝ) ↦ sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔𝑟𝑓𝑥 = (∫1𝑔))}, ℝ*, < ))
df-ibl 22741𝐿1 = {𝑓 ∈ MblFn ∣ ∀𝑘 ∈ (0...3)(∫2‘(𝑥 ∈ ℝ ↦ (ℜ‘((𝑓𝑥) / (i↑𝑘))) / 𝑦if((𝑥 ∈ dom 𝑓 ∧ 0 ≤ 𝑦), 𝑦, 0))) ∈ ℝ}
df-itg 22742𝐴𝐵 d𝑥 = Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ (ℜ‘(𝐵 / (i↑𝑘))) / 𝑦if((𝑥𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0))))
c0p 22788class 0𝑝
df-0p 227890𝑝 = (ℂ × {0})
cdit 22962class ⨜[𝐴𝐵]𝐶 d𝑥
df-ditg 22963⨜[𝐴𝐵]𝐶 d𝑥 = if(𝐴𝐵, ∫(𝐴(,)𝐵)𝐶 d𝑥, -∫(𝐵(,)𝐴)𝐶 d𝑥)
climc 22978class lim
cdv 22979class D
cdvn 22980class D𝑛
ccpn 22981class Cn
df-limc 22982 lim = (𝑓 ∈ (ℂ ↑pm ℂ), 𝑥 ∈ ℂ ↦ {𝑦[(TopOpen‘ℂfld) / 𝑗](𝑧 ∈ (dom 𝑓 ∪ {𝑥}) ↦ if(𝑧 = 𝑥, 𝑦, (𝑓𝑧))) ∈ (((𝑗t (dom 𝑓 ∪ {𝑥})) CnP 𝑗)‘𝑥)})
df-dv 22983 D = (𝑠 ∈ 𝒫 ℂ, 𝑓 ∈ (ℂ ↑pm 𝑠) ↦ 𝑥 ∈ ((int‘((TopOpen‘ℂfld) ↾t 𝑠))‘dom 𝑓)({𝑥} × ((𝑧 ∈ (dom 𝑓 ∖ {𝑥}) ↦ (((𝑓𝑧) − (𝑓𝑥)) / (𝑧𝑥))) lim 𝑥)))
df-dvn 22984 D𝑛 = (𝑠 ∈ 𝒫 ℂ, 𝑓 ∈ (ℂ ↑pm 𝑠) ↦ seq0(((𝑥 ∈ V ↦ (𝑠 D 𝑥)) ∘ 1st ), (ℕ0 × {𝑓})))
df-cpn 22985Cn = (𝑠 ∈ 𝒫 ℂ ↦ (𝑥 ∈ ℕ0 ↦ {𝑓 ∈ (ℂ ↑pm 𝑠) ∣ ((𝑠 D𝑛 𝑓)‘𝑥) ∈ (dom 𝑓cn→ℂ)}))
cmdg 23163class mDeg
cdg1 23164class deg1
df-mdeg 23165 mDeg = (𝑖 ∈ V, 𝑟 ∈ V ↦ (𝑓 ∈ (Base‘(𝑖 mPoly 𝑟)) ↦ sup(ran ( ∈ (𝑓 supp (0g𝑟)) ↦ (ℂfld Σg )), ℝ*, < )))
df-deg1 23166 deg1 = (𝑟 ∈ V ↦ (1𝑜 mDeg 𝑟))
cmn1 23235class Monic1p
cuc1p 23236class Unic1p
cq1p 23237class quot1p
cr1p 23238class rem1p
cig1p 23239class idlGen1p
cig1pold 23240class idlGen1p
df-mon1 23241Monic1p = (𝑟 ∈ V ↦ {𝑓 ∈ (Base‘(Poly1𝑟)) ∣ (𝑓 ≠ (0g‘(Poly1𝑟)) ∧ ((coe1𝑓)‘(( deg1𝑟)‘𝑓)) = (1r𝑟))})
df-uc1p 23242Unic1p = (𝑟 ∈ V ↦ {𝑓 ∈ (Base‘(Poly1𝑟)) ∣ (𝑓 ≠ (0g‘(Poly1𝑟)) ∧ ((coe1𝑓)‘(( deg1𝑟)‘𝑓)) ∈ (Unit‘𝑟))})
df-q1p 23243quot1p = (𝑟 ∈ V ↦ (Poly1𝑟) / 𝑝(Base‘𝑝) / 𝑏(𝑓𝑏, 𝑔𝑏 ↦ (𝑞𝑏 (( deg1𝑟)‘(𝑓(-g𝑝)(𝑞(.r𝑝)𝑔))) < (( deg1𝑟)‘𝑔))))
df-r1p 23244rem1p = (𝑟 ∈ V ↦ (Base‘(Poly1𝑟)) / 𝑏(𝑓𝑏, 𝑔𝑏 ↦ (𝑓(-g‘(Poly1𝑟))((𝑓(quot1p𝑟)𝑔)(.r‘(Poly1𝑟))𝑔))))
df-ig1p 23245idlGen1p = (𝑟 ∈ V ↦ (𝑖 ∈ (LIdeal‘(Poly1𝑟)) ↦ if(𝑖 = {(0g‘(Poly1𝑟))}, (0g‘(Poly1𝑟)), (𝑔 ∈ (𝑖 ∩ (Monic1p𝑟))(( deg1𝑟)‘𝑔) = inf((( deg1𝑟) “ (𝑖 ∖ {(0g‘(Poly1𝑟))})), ℝ, < )))))
df-ig1pOLD 23246idlGen1p = (𝑟 ∈ V ↦ (𝑖 ∈ (LIdeal‘(Poly1𝑟)) ↦ if(𝑖 = {(0g‘(Poly1𝑟))}, (0g‘(Poly1𝑟)), (𝑔 ∈ (𝑖 ∩ (Monic1p𝑟))(( deg1𝑟)‘𝑔) = sup((( deg1𝑟) “ (𝑖 ∖ {(0g‘(Poly1𝑟))})), ℝ, < )))))
cply 23299class Poly
cidp 23300class Xp
ccoe 23301class coeff
cdgr 23302class deg
df-ply 23303Poly = (𝑥 ∈ 𝒫 ℂ ↦ {𝑓 ∣ ∃𝑛 ∈ ℕ0𝑎 ∈ ((𝑥 ∪ {0}) ↑𝑚0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))})
df-idp 23304Xp = ( I ↾ ℂ)
df-coe 23305coeff = (𝑓 ∈ (Poly‘ℂ) ↦ (𝑎 ∈ (ℂ ↑𝑚0)∃𝑛 ∈ ℕ0 ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))))))
df-dgr 23306deg = (𝑓 ∈ (Poly‘ℂ) ↦ sup(((coeff‘𝑓) “ (ℂ ∖ {0})), ℕ0, < ))
cquot 23404class quot
df-quot 23405 quot = (𝑓 ∈ (Poly‘ℂ), 𝑔 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ↦ (𝑞 ∈ (Poly‘ℂ)[(𝑓𝑓 − (𝑔𝑓 · 𝑞)) / 𝑟](𝑟 = 0𝑝 ∨ (deg‘𝑟) < (deg‘𝑔))))
caa 23428class 𝔸
df-aa 23429𝔸 = 𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑓 “ {0})
ctayl 23469class Tayl
cana 23470class Ana
df-tayl 23471 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)) · ((𝑥𝑎)↑𝑘)))))))
df-ana 23472Ana = (𝑠 ∈ {ℝ, ℂ} ↦ {𝑓 ∈ (ℂ ↑pm 𝑠) ∣ ∀𝑥 ∈ dom 𝑓 𝑥 ∈ ((int‘((TopOpen‘ℂfld) ↾t 𝑠))‘dom (𝑓 ∩ (+∞(𝑠 Tayl 𝑓)𝑥)))})
culm 23492class 𝑢
df-ulm 23493𝑢 = (𝑠 ∈ V ↦ {⟨𝑓, 𝑦⟩ ∣ ∃𝑛 ∈ ℤ (𝑓:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑠) ∧ 𝑦:𝑠⟶ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑠 (abs‘(((𝑓𝑘)‘𝑧) − (𝑦𝑧))) < 𝑥)})
clog 23665class log
ccxp 23666class 𝑐
df-log 23667log = (exp ↾ (ℑ “ (-π(,]π)))
df-cxp 23668𝑐 = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ if(𝑥 = 0, if(𝑦 = 0, 1, 0), (exp‘(𝑦 · (log‘𝑥)))))
clogb 23862class logb
df-logb 23863 logb = (𝑥 ∈ (ℂ ∖ {0, 1}), 𝑦 ∈ (ℂ ∖ {0}) ↦ ((log‘𝑦) / (log‘𝑥)))
casin 23949class arcsin
cacos 23950class arccos
catan 23951class arctan
df-asin 23952arcsin = (𝑥 ∈ ℂ ↦ (-i · (log‘((i · 𝑥) + (√‘(1 − (𝑥↑2)))))))
df-acos 23953arccos = (𝑥 ∈ ℂ ↦ ((π / 2) − (arcsin‘𝑥)))
df-atan 23954arctan = (𝑥 ∈ (ℂ ∖ {-i, i}) ↦ ((i / 2) · ((log‘(1 − (i · 𝑥))) − (log‘(1 + (i · 𝑥))))))
carea 24042class area
df-area 24043area = (𝑠 ∈ {𝑡 ∈ 𝒫 (ℝ × ℝ) ∣ (∀𝑥 ∈ ℝ (𝑡 “ {𝑥}) ∈ (vol “ ℝ) ∧ (𝑥 ∈ ℝ ↦ (vol‘(𝑡 “ {𝑥}))) ∈ 𝐿1)} ↦ ∫ℝ(vol‘(𝑠 “ {𝑥})) d𝑥)
cem 24078class γ
df-em 24079γ = Σ𝑘 ∈ ℕ ((1 / 𝑘) − (log‘(1 + (1 / 𝑘))))
czeta 24099class ζ
df-zeta 24100ζ = (𝑓 ∈ ((ℂ ∖ {1})–cn→ℂ)∀𝑠 ∈ (ℂ ∖ {1})((1 − (2↑𝑐(1 − 𝑠))) · (𝑓𝑠)) = Σ𝑛 ∈ ℕ0𝑘 ∈ (0...𝑛)(((-1↑𝑘) · (𝑛C𝑘)) · ((𝑘 + 1)↑𝑐𝑠)) / (2↑(𝑛 + 1))))
clgam 24102class log Γ
cgam 24103class Γ
cigam 24104class 1/Γ
df-lgam 24105log Γ = (𝑧 ∈ (ℂ ∖ (ℤ ∖ ℕ)) ↦ (Σ𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))) − (log‘𝑧)))
df-gam 24106Γ = (exp ∘ log Γ)
df-igam 241071/Γ = (𝑥 ∈ ℂ ↦ if(𝑥 ∈ (ℤ ∖ ℕ), 0, (1 / (Γ‘𝑥))))
ccht 24178class θ
cvma 24179class Λ
cchp 24180class ψ
cppi 24181class π
cmu 24182class μ
csgm 24183class σ
df-cht 24184θ = (𝑥 ∈ ℝ ↦ Σ𝑝 ∈ ((0[,]𝑥) ∩ ℙ)(log‘𝑝))
df-vma 24185Λ = (𝑥 ∈ ℕ ↦ {𝑝 ∈ ℙ ∣ 𝑝𝑥} / 𝑠if((#‘𝑠) = 1, (log‘ 𝑠), 0))
df-chp 24186ψ = (𝑥 ∈ ℝ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))(Λ‘𝑛))
df-ppi 24187π = (𝑥 ∈ ℝ ↦ (#‘((0[,]𝑥) ∩ ℙ)))
df-mu 24188μ = (𝑥 ∈ ℕ ↦ if(∃𝑝 ∈ ℙ (𝑝↑2) ∥ 𝑥, 0, (-1↑(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑥}))))
df-sgm 24189 σ = (𝑥 ∈ ℂ, 𝑛 ∈ ℕ ↦ Σ𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝𝑛} (𝑘𝑐𝑥))
cdchr 24321class DChr
df-dchr 24322DChr = (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 24383class /L
df-lgs 24384 /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)))))
cstrkg 24639class TarskiG
cstrkgc 24640class TarskiGC
cstrkgb 24641class TarskiGB
cstrkgcb 24642class TarskiGCB
cstrkgld 24643class DimTarskiG
cstrkge 24644class TarskiGE
citv 24645class Itv
clng 24646class LineG
df-itv 24647Itv = Slot 16
df-lng 24648LineG = Slot 17
df-trkgc 24657TarskiGC = {𝑓[(Base‘𝑓) / 𝑝][(dist‘𝑓) / 𝑑](∀𝑥𝑝𝑦𝑝 (𝑥𝑑𝑦) = (𝑦𝑑𝑥) ∧ ∀𝑥𝑝𝑦𝑝𝑧𝑝 ((𝑥𝑑𝑦) = (𝑧𝑑𝑧) → 𝑥 = 𝑦))}
df-trkgb 24658TarskiGB = {𝑓[(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](∀𝑥𝑝𝑦𝑝 (𝑦 ∈ (𝑥𝑖𝑥) → 𝑥 = 𝑦) ∧ ∀𝑥𝑝𝑦𝑝𝑧𝑝𝑢𝑝𝑣𝑝 ((𝑢 ∈ (𝑥𝑖𝑧) ∧ 𝑣 ∈ (𝑦𝑖𝑧)) → ∃𝑎𝑝 (𝑎 ∈ (𝑢𝑖𝑦) ∧ 𝑎 ∈ (𝑣𝑖𝑥))) ∧ ∀𝑠 ∈ 𝒫 𝑝𝑡 ∈ 𝒫 𝑝(∃𝑎𝑝𝑥𝑠𝑦𝑡 𝑥 ∈ (𝑎𝑖𝑦) → ∃𝑏𝑝𝑥𝑠𝑦𝑡 𝑏 ∈ (𝑥𝑖𝑦)))}
df-trkgcb 24659TarskiGCB = {𝑓[(Base‘𝑓) / 𝑝][(dist‘𝑓) / 𝑑][(Itv‘𝑓) / 𝑖](∀𝑥𝑝𝑦𝑝𝑧𝑝𝑢𝑝𝑎𝑝𝑏𝑝𝑐𝑝𝑣𝑝 (((𝑥𝑦𝑦 ∈ (𝑥𝑖𝑧) ∧ 𝑏 ∈ (𝑎𝑖𝑐)) ∧ (((𝑥𝑑𝑦) = (𝑎𝑑𝑏) ∧ (𝑦𝑑𝑧) = (𝑏𝑑𝑐)) ∧ ((𝑥𝑑𝑢) = (𝑎𝑑𝑣) ∧ (𝑦𝑑𝑢) = (𝑏𝑑𝑣)))) → (𝑧𝑑𝑢) = (𝑐𝑑𝑣)) ∧ ∀𝑥𝑝𝑦𝑝𝑎𝑝𝑏𝑝𝑧𝑝 (𝑦 ∈ (𝑥𝑖𝑧) ∧ (𝑦𝑑𝑧) = (𝑎𝑑𝑏)))}
df-trkge 24660TarskiGE = {𝑓[(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖]𝑥𝑝𝑦𝑝𝑧𝑝𝑢𝑝𝑣𝑝 ((𝑢 ∈ (𝑥𝑖𝑣) ∧ 𝑢 ∈ (𝑦𝑖𝑧) ∧ 𝑥𝑢) → ∃𝑎𝑝𝑏𝑝 (𝑦 ∈ (𝑥𝑖𝑎) ∧ 𝑧 ∈ (𝑥𝑖𝑏) ∧ 𝑣 ∈ (𝑎𝑖𝑏)))}
df-trkgld 24661DimTarskiG≥ = {⟨𝑔, 𝑛⟩ ∣ [(Base‘𝑔) / 𝑝][(dist‘𝑔) / 𝑑][(Itv‘𝑔) / 𝑖]𝑓(𝑓:(1..^𝑛)–1-1𝑝 ∧ ∃𝑥𝑝𝑦𝑝𝑧𝑝 (∀𝑗 ∈ (2..^𝑛)(((𝑓‘1)𝑑𝑥) = ((𝑓𝑗)𝑑𝑥) ∧ ((𝑓‘1)𝑑𝑦) = ((𝑓𝑗)𝑑𝑦) ∧ ((𝑓‘1)𝑑𝑧) = ((𝑓𝑗)𝑑𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))))}
df-trkg 24662TarskiG = ((TarskiGC ∩ TarskiGB) ∩ (TarskiGCB ∩ {𝑓[(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})}))
ccgrg 24716class cgrG
df-cgrg 24717cgrG = (𝑔 ∈ V ↦ {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ ((Base‘𝑔) ↑pm ℝ) ∧ 𝑏 ∈ ((Base‘𝑔) ↑pm ℝ)) ∧ (dom 𝑎 = dom 𝑏 ∧ ∀𝑖 ∈ dom 𝑎𝑗 ∈ dom 𝑎((𝑎𝑖)(dist‘𝑔)(𝑎𝑗)) = ((𝑏𝑖)(dist‘𝑔)(𝑏𝑗))))})
cismt 24738class Ismt
df-ismt 24739Ismt = (𝑔 ∈ V, ∈ V ↦ {𝑓 ∣ (𝑓:(Base‘𝑔)–1-1-onto→(Base‘) ∧ ∀𝑎 ∈ (Base‘𝑔)∀𝑏 ∈ (Base‘𝑔)((𝑓𝑎)(dist‘)(𝑓𝑏)) = (𝑎(dist‘𝑔)𝑏))})
cleg 24788class ≤G
df-leg 24789≤G = (𝑔 ∈ V ↦ {⟨𝑒, 𝑓⟩ ∣ [(Base‘𝑔) / 𝑝][(dist‘𝑔) / 𝑑][(Itv‘𝑔) / 𝑖]𝑥𝑝𝑦𝑝 (𝑓 = (𝑥𝑑𝑦) ∧ ∃𝑧𝑝 (𝑧 ∈ (𝑥𝑖𝑦) ∧ 𝑒 = (𝑥𝑑𝑧)))})
chlg 24806class hlG
df-hlg 24807hlG = (𝑔 ∈ V ↦ (𝑐 ∈ (Base‘𝑔) ↦ {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (Base‘𝑔) ∧ 𝑏 ∈ (Base‘𝑔)) ∧ (𝑎𝑐𝑏𝑐 ∧ (𝑎 ∈ (𝑐(Itv‘𝑔)𝑏) ∨ 𝑏 ∈ (𝑐(Itv‘𝑔)𝑎))))}))
cmir 24858class pInvG
df-mir 24859pInvG = (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 24899class ∟G
df-rag 24900∟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 24901class ⟂G
df-perpg 24902⟂G = (𝑔 ∈ V ↦ {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ ran (LineG‘𝑔) ∧ 𝑏 ∈ ran (LineG‘𝑔)) ∧ ∃𝑥 ∈ (𝑎𝑏)∀𝑢𝑎𝑣𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝑔))})
chpg 24960class hpG
df-hpg 24961hpG = (𝑔 ∈ V ↦ (𝑑 ∈ ran (LineG‘𝑔) ↦ {⟨𝑎, 𝑏⟩ ∣ [(Base‘𝑔) / 𝑝][(Itv‘𝑔) / 𝑖]𝑐𝑝 (((𝑎 ∈ (𝑝𝑑) ∧ 𝑐 ∈ (𝑝𝑑)) ∧ ∃𝑡𝑑 𝑡 ∈ (𝑎𝑖𝑐)) ∧ ((𝑏 ∈ (𝑝𝑑) ∧ 𝑐 ∈ (𝑝𝑑)) ∧ ∃𝑡𝑑 𝑡 ∈ (𝑏𝑖𝑐)))}))
cmid 24975class midG
clmi 24976class lInvG
df-mid 24977midG = (𝑔 ∈ V ↦ (𝑎 ∈ (Base‘𝑔), 𝑏 ∈ (Base‘𝑔) ↦ (𝑚 ∈ (Base‘𝑔)𝑏 = (((pInvG‘𝑔)‘𝑚)‘𝑎))))
df-lmi 24978lInvG = (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 25010class cgrA
df-cgra 25011cgrA = (𝑔 ∈ V ↦ {⟨𝑎, 𝑏⟩ ∣ [(Base‘𝑔) / 𝑝][(hlG‘𝑔) / 𝑘]((𝑎 ∈ (𝑝𝑚 (0..^3)) ∧ 𝑏 ∈ (𝑝𝑚 (0..^3))) ∧ ∃𝑥𝑝𝑦𝑝 (𝑎(cgrG‘𝑔)⟨“𝑥(𝑏‘1)𝑦”⟩ ∧ 𝑥(𝑘‘(𝑏‘1))(𝑏‘0) ∧ 𝑦(𝑘‘(𝑏‘1))(𝑏‘2)))})
cinag 25037class inA
cleag 25038class
df-inag 25039inA = (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 25043 = (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 25056class eqltrG
df-eqlg 25057eqltrG = (𝑔 ∈ V ↦ {𝑥 ∈ ((Base‘𝑔) ↑𝑚 (0..^3)) ∣ 𝑥(cgrG‘𝑔)⟨“(𝑥‘1)(𝑥‘2)(𝑥‘0)”⟩})
cttg 25064class toTG
df-ttg 25065toTG = (𝑤 ∈ V ↦ (𝑥 ∈ (Base‘𝑤), 𝑦 ∈ (Base‘𝑤) ↦ {𝑧 ∈ (Base‘𝑤) ∣ ∃𝑘 ∈ (0[,]1)(𝑧(-g𝑤)𝑥) = (𝑘( ·𝑠𝑤)(𝑦(-g𝑤)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 25079class 𝔼
cbtwn 25080class Btwn
ccgr 25081class Cgr
df-ee 25082𝔼 = (𝑛 ∈ ℕ ↦ (ℝ ↑𝑚 (1...𝑛)))
df-btwn 25083 Btwn = {⟨⟨𝑥, 𝑧⟩, 𝑦⟩ ∣ ∃𝑛 ∈ ℕ ((𝑥 ∈ (𝔼‘𝑛) ∧ 𝑧 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛)) ∧ ∃𝑡 ∈ (0[,]1)∀𝑖 ∈ (1...𝑛)(𝑦𝑖) = (((1 − 𝑡) · (𝑥𝑖)) + (𝑡 · (𝑧𝑖))))}
df-cgr 25084Cgr = {⟨𝑥, 𝑦⟩ ∣ ∃𝑛 ∈ ℕ ((𝑥 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑦 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛))) ∧ Σ𝑖 ∈ (1...𝑛)((((1st𝑥)‘𝑖) − ((2nd𝑥)‘𝑖))↑2) = Σ𝑖 ∈ (1...𝑛)((((1st𝑦)‘𝑖) − ((2nd𝑦)‘𝑖))↑2))}
ceeng 25168class EEG
df-eeng 25169EEG = (𝑛 ∈ ℕ ↦ ({⟨(Base‘ndx), (𝔼‘𝑛)⟩, ⟨(dist‘ndx), (𝑥 ∈ (𝔼‘𝑛), 𝑦 ∈ (𝔼‘𝑛) ↦ Σ𝑖 ∈ (1...𝑛)(((𝑥𝑖) − (𝑦𝑖))↑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⟩)})⟩}))
cuhg 25178class UHGrph
cushg 25179class USHGrph
df-uhgra 25180 UHGrph = {⟨𝑣, 𝑒⟩ ∣ 𝑒:dom 𝑒⟶(𝒫 𝑣 ∖ {∅})}
df-ushgra 25181 USHGrph = {⟨𝑣, 𝑒⟩ ∣ 𝑒:dom 𝑒1-1→(𝒫 𝑣 ∖ {∅})}
cumg 25200class UMGrph
df-umgra 25201 UMGrph = {⟨𝑣, 𝑒⟩ ∣ 𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}}
cuslg 25217class USLGrph
cusg 25218class USGrph
cedg 25219class Edges
df-uslgra 25220 USLGrph = {⟨𝑣, 𝑒⟩ ∣ 𝑒:dom 𝑒1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}}
df-usgra 25221 USGrph = {⟨𝑣, 𝑒⟩ ∣ 𝑒:dom 𝑒1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) = 2}}
df-edg 25224Edges = (𝑔 ∈ V ↦ ran (2nd𝑔))
cnbgra 25306class Neighbors
ccusgra 25307class ComplUSGrph
cuvtx 25308class UnivVertex
df-nbgra 25309 Neighbors = (𝑔 ∈ V, 𝑘 ∈ (1st𝑔) ↦ {𝑛 ∈ (1st𝑔) ∣ {𝑘, 𝑛} ∈ ran (2nd𝑔)})
df-cusgra 25310 ComplUSGrph = {⟨𝑣, 𝑒⟩ ∣ (𝑣 USGrph 𝑒 ∧ ∀𝑘𝑣𝑛 ∈ (𝑣 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝑒)}
df-uvtx 25311 UnivVertex = (𝑣 ∈ V, 𝑒 ∈ V ↦ {𝑛𝑣 ∣ ∀𝑘 ∈ (𝑣 ∖ {𝑛}){𝑘, 𝑛} ∈ ran 𝑒})
cwalk 25387class Walks
ctrail 25388class Trails
cpath 25389class Paths
cspath 25390class SPaths
cwlkon 25391class WalkOn
ctrlon 25392class TrailOn
cpthon 25393class PathOn
cspthon 25394class SPathOn
ccrct 25395class Circuits
ccycl 25396class Cycles
df-wlk 25397 Walks = (𝑣 ∈ V, 𝑒 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom 𝑒𝑝:(0...(#‘𝑓))⟶𝑣 ∧ ∀𝑘 ∈ (0..^(#‘𝑓))(𝑒‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))})})
df-trail 25398 Trails = (𝑣 ∈ V, 𝑒 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑣 Walks 𝑒)𝑝 ∧ Fun 𝑓)})
df-pth 25399 Paths = (𝑣 ∈ V, 𝑒 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑣 Trails 𝑒)𝑝 ∧ Fun (𝑝 ↾ (1..^(#‘𝑓))) ∧ ((𝑝 “ {0, (#‘𝑓)}) ∩ (𝑝 “ (1..^(#‘𝑓)))) = ∅)})
df-spth 25400 SPaths = (𝑣 ∈ V, 𝑒 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑣 Trails 𝑒)𝑝 ∧ Fun 𝑝)})
df-crct 25401 Circuits = (𝑣 ∈ V, 𝑒 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑣 Trails 𝑒)𝑝 ∧ (𝑝‘0) = (𝑝‘(#‘𝑓)))})
df-cycl 25402 Cycles = (𝑣 ∈ V, 𝑒 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑣 Paths 𝑒)𝑝 ∧ (𝑝‘0) = (𝑝‘(#‘𝑓)))})
df-wlkon 25403 WalkOn = (𝑣 ∈ V, 𝑒 ∈ V ↦ (𝑎𝑣, 𝑏𝑣 ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑣 Walks 𝑒)𝑝 ∧ (𝑝‘0) = 𝑎 ∧ (𝑝‘(#‘𝑓)) = 𝑏)}))
df-trlon 25404 TrailOn = (𝑣 ∈ V, 𝑒 ∈ V ↦ (𝑎𝑣, 𝑏𝑣 ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑎(𝑣 WalkOn 𝑒)𝑏)𝑝𝑓(𝑣 Trails 𝑒)𝑝)}))
df-pthon 25405 PathOn = (𝑣 ∈ V, 𝑒 ∈ V ↦ (𝑎𝑣, 𝑏𝑣 ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑎(𝑣 WalkOn 𝑒)𝑏)𝑝𝑓(𝑣 Paths 𝑒)𝑝)}))
df-spthon 25406 SPathOn = (𝑣 ∈ V, 𝑒 ∈ V ↦ (𝑎𝑣, 𝑏𝑣 ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑎(𝑣 WalkOn 𝑒)𝑏)𝑝𝑓(𝑣 SPaths 𝑒)𝑝)}))
cconngra 25558class ConnGrph
df-conngra 25559 ConnGrph = {⟨𝑣, 𝑒⟩ ∣ ∀𝑘𝑣𝑛𝑣𝑓𝑝 𝑓(𝑘(𝑣 PathOn 𝑒)𝑛)𝑝}
cwwlk 25566class WWalks
cwwlkn 25567class WWalksN
df-wwlk 25568 WWalks = (𝑣 ∈ V, 𝑒 ∈ V ↦ {𝑤 ∈ Word 𝑣 ∣ (𝑤 ≠ ∅ ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝑒)})
df-wwlkn 25569 WWalksN = (𝑣 ∈ V, 𝑒 ∈ V ↦ (𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑣 WWalks 𝑒) ∣ (#‘𝑤) = (𝑛 + 1)}))
cclwlk 25636class ClWalks
cclwwlk 25637class ClWWalks
cclwwlkn 25638class ClWWalksN
df-clwlk 25639 ClWalks = (𝑣 ∈ V, 𝑒 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑣 Walks 𝑒)𝑝 ∧ (𝑝‘0) = (𝑝‘(#‘𝑓)))})
df-clwwlk 25640 ClWWalks = (𝑣 ∈ V, 𝑒 ∈ V ↦ {𝑤 ∈ Word 𝑣 ∣ (∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝑒 ∧ {( lastS ‘𝑤), (𝑤‘0)} ∈ ran 𝑒)})
df-clwwlkn 25641 ClWWalksN = (𝑣 ∈ V, 𝑒 ∈ V ↦ (𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑣 ClWWalks 𝑒) ∣ (#‘𝑤) = 𝑛}))
c2wlkot 25742class 2WalksOt
c2wlkonot 25743class 2WalksOnOt
c2spthot 25744class 2SPathsOt
c2pthonot 25745class 2SPathOnOt
df-2wlkonot 25746 2WalksOnOt = (𝑣 ∈ V, 𝑒 ∈ V ↦ (𝑎𝑣, 𝑏𝑣 ↦ {𝑡 ∈ ((𝑣 × 𝑣) × 𝑣) ∣ ∃𝑓𝑝(𝑓(𝑎(𝑣 WalkOn 𝑒)𝑏)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑡)) = 𝑎 ∧ (2nd ‘(1st𝑡)) = (𝑝‘1) ∧ (2nd𝑡) = 𝑏))}))
df-2wlksot 25747 2WalksOt = (𝑣 ∈ V, 𝑒 ∈ V ↦ {𝑡 ∈ ((𝑣 × 𝑣) × 𝑣) ∣ ∃𝑎𝑣𝑏𝑣 𝑡 ∈ (𝑎(𝑣 2WalksOnOt 𝑒)𝑏)})
df-2spthonot 25748 2SPathOnOt = (𝑣 ∈ V, 𝑒 ∈ V ↦ (𝑎𝑣, 𝑏𝑣 ↦ {𝑡 ∈ ((𝑣 × 𝑣) × 𝑣) ∣ ∃𝑓𝑝(𝑓(𝑎(𝑣 SPathOn 𝑒)𝑏)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑡)) = 𝑎 ∧ (2nd ‘(1st𝑡)) = (𝑝‘1) ∧ (2nd𝑡) = 𝑏))}))
df-2spthsot 25749 2SPathsOt = (𝑣 ∈ V, 𝑒 ∈ V ↦ {𝑡 ∈ ((𝑣 × 𝑣) × 𝑣) ∣ ∃𝑎𝑣𝑏𝑣 𝑡 ∈ (𝑎(𝑣 2SPathOnOt 𝑒)𝑏)})
cvdg 25781class VDeg
df-vdgr 25782 VDeg = (𝑣 ∈ V, 𝑒 ∈ V ↦ (𝑢𝑣 ↦ ((#‘{𝑥 ∈ dom 𝑒𝑢 ∈ (𝑒𝑥)}) +𝑒 (#‘{𝑥 ∈ dom 𝑒 ∣ (𝑒𝑥) = {𝑢}}))))
crgra 25810class RegGrph
crusgra 25811class RegUSGrph
df-rgra 25812 RegGrph = {⟨⟨𝑣, 𝑒⟩, 𝑘⟩ ∣ (𝑘 ∈ ℕ0 ∧ ∀𝑝𝑣 ((𝑣 VDeg 𝑒)‘𝑝) = 𝑘)}
df-rusgra 25813 RegUSGrph = {⟨⟨𝑣, 𝑒⟩, 𝑘⟩ ∣ (𝑣 USGrph 𝑒 ∧ ⟨𝑣, 𝑒⟩ RegGrph 𝑘)}
ceup 25850class EulPaths
df-eupa 25851 EulPaths = (𝑣 ∈ V, 𝑒 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑣 UMGrph 𝑒 ∧ ∃𝑛 ∈ ℕ0 (𝑓:(1...𝑛)–1-1-onto→dom 𝑒𝑝:(0...𝑛)⟶𝑣 ∧ ∀𝑘 ∈ (1...𝑛)(𝑒‘(𝑓𝑘)) = {(𝑝‘(𝑘 − 1)), (𝑝𝑘)}))})
cfrgra 25876class FriendGrph
df-frgra 25877 FriendGrph = {⟨𝑣, 𝑒⟩ ∣ (𝑣 USGrph 𝑒 ∧ ∀𝑘𝑣𝑙 ∈ (𝑣 ∖ {𝑘})∃!𝑥𝑣 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝑒)}
cplig 26068class Plig
df-plig 26069Plig = {𝑥 ∣ (∀𝑎 𝑥𝑏 𝑥(𝑎𝑏 → ∃!𝑙𝑥 (𝑎𝑙𝑏𝑙)) ∧ ∀𝑙𝑥𝑎 𝑥𝑏 𝑥(𝑎𝑏𝑎𝑙𝑏𝑙) ∧ ∃𝑎 𝑥𝑏 𝑥𝑐 𝑥𝑙𝑥 ¬ (𝑎𝑙𝑏𝑙𝑐𝑙))}
crpm 26073class RPrime
df-rprm 26074RPrime = (𝑤 ∈ V ↦ (Base‘𝑤) / 𝑏{𝑝 ∈ (𝑏 ∖ ((Unit‘𝑤) ∪ {(0g𝑤)})) ∣ ∀𝑥𝑏𝑦𝑏 [(∥r𝑤) / 𝑑](𝑝𝑑(𝑥(.r𝑤)𝑦) → (𝑝𝑑𝑥𝑝𝑑𝑦))})
cgr 26077class GrpOp
cgi 26078class GId
cgn 26079class inv
cgs 26080class /𝑔
cgx 26081class 𝑔
df-grpo 26082GrpOp = {𝑔 ∣ ∃𝑡(𝑔:(𝑡 × 𝑡)⟶𝑡 ∧ ∀𝑥𝑡𝑦𝑡𝑧𝑡 ((𝑥𝑔𝑦)𝑔𝑧) = (𝑥𝑔(𝑦𝑔𝑧)) ∧ ∃𝑢𝑡𝑥𝑡 ((𝑢𝑔𝑥) = 𝑥 ∧ ∃𝑦𝑡 (𝑦𝑔𝑥) = 𝑢))}
df-gid 26083GId = (𝑔 ∈ V ↦ (𝑢 ∈ ran 𝑔𝑥 ∈ ran 𝑔((𝑢𝑔𝑥) = 𝑥 ∧ (𝑥𝑔𝑢) = 𝑥)))
df-ginv 26084inv = (𝑔 ∈ GrpOp ↦ (𝑥 ∈ ran 𝑔 ↦ (𝑧 ∈ ran 𝑔(𝑧𝑔𝑥) = (GId‘𝑔))))
df-gdiv 26085 /𝑔 = (𝑔 ∈ GrpOp ↦ (𝑥 ∈ ran 𝑔, 𝑦 ∈ ran 𝑔 ↦ (𝑥𝑔((inv‘𝑔)‘𝑦))))
df-gx 26086𝑔 = (𝑔 ∈ GrpOp ↦ (𝑥 ∈ ran 𝑔, 𝑦 ∈ ℤ ↦ if(𝑦 = 0, (GId‘𝑔), if(0 < 𝑦, (seq1(𝑔, (ℕ × {𝑥}))‘𝑦), ((inv‘𝑔)‘(seq1(𝑔, (ℕ × {𝑥}))‘-𝑦))))))
cablo 26172class AbelOp
df-ablo 26173AbelOp = {𝑔 ∈ GrpOp ∣ ∀𝑥 ∈ ran 𝑔𝑦 ∈ ran 𝑔(𝑥𝑔𝑦) = (𝑦𝑔𝑥)}
csubgo 26192class SubGrpOp
df-subgo 26193SubGrpOp = (𝑔 ∈ GrpOp ↦ (GrpOp ∩ 𝒫 𝑔))
cass 26203class Ass
df-ass 26204Ass = {𝑔 ∣ ∀𝑥 ∈ dom dom 𝑔𝑦 ∈ dom dom 𝑔𝑧 ∈ dom dom 𝑔((𝑥𝑔𝑦)𝑔𝑧) = (𝑥𝑔(𝑦𝑔𝑧))}
cexid 26205class ExId
df-exid 26206 ExId = {𝑔 ∣ ∃𝑥 ∈ dom dom 𝑔𝑦 ∈ dom dom 𝑔((𝑥𝑔𝑦) = 𝑦 ∧ (𝑦𝑔𝑥) = 𝑦)}
cmagm 26209class Magma
df-mgmOLD 26210Magma = {𝑔 ∣ ∃𝑡 𝑔:(𝑡 × 𝑡)⟶𝑡}
csem 26221class SemiGrp
df-sgrOLD 26222SemiGrp = (Magma ∩ Ass)
cmndo 26228class MndOp
df-mndo 26229MndOp = (SemiGrp ∩ ExId )
cghomOLD 26248class GrpOpHom
df-ghomOLD 26249 GrpOpHom = (𝑔 ∈ GrpOp, ∈ GrpOp ↦ {𝑓 ∣ (𝑓:ran 𝑔⟶ran ∧ ∀𝑥 ∈ ran 𝑔𝑦 ∈ ran 𝑔((𝑓𝑥)(𝑓𝑦)) = (𝑓‘(𝑥𝑔𝑦)))})
cgisoOLD 26250class GrpOpIso
df-gisoOLD 26251 GrpOpIso = (𝑔 ∈ GrpOp, ∈ GrpOp ↦ {𝑓 ∈ (𝑔 GrpOpHom ) ∣ 𝑓:ran 𝑔1-1-onto→ran })
crngo 26266class RingOps
df-rngo 26267RingOps = {⟨𝑔, ⟩ ∣ ((𝑔 ∈ AbelOp ∧ :(ran 𝑔 × ran 𝑔)⟶ran 𝑔) ∧ (∀𝑥 ∈ ran 𝑔𝑦 ∈ ran 𝑔𝑧 ∈ ran 𝑔(((𝑥𝑦)𝑧) = (𝑥(𝑦𝑧)) ∧ (𝑥(𝑦𝑔𝑧)) = ((𝑥𝑦)𝑔(𝑥𝑧)) ∧ ((𝑥𝑔𝑦)𝑧) = ((𝑥𝑧)𝑔(𝑦𝑧))) ∧ ∃𝑥 ∈ ran 𝑔𝑦 ∈ ran 𝑔((𝑥𝑦) = 𝑦 ∧ (𝑦𝑥) = 𝑦)))}
cdrng 26296class DivRingOps
df-drngo 26297DivRingOps = {⟨𝑔, ⟩ ∣ (⟨𝑔, ⟩ ∈ RingOps ∧ ( ↾ ((ran 𝑔 ∖ {(GId‘𝑔)}) × (ran 𝑔 ∖ {(GId‘𝑔)}))) ∈ GrpOp)}
csfld 26299class *-Fld
df-sfld 26300*-Fld = {⟨𝑟, 𝑛⟩ ∣ (𝑟 ∈ DivRingOps ∧ 𝑛:ran (1st𝑟)⟶ran (1st𝑟) ∧ ∀𝑥 ∈ dom 𝑛𝑦 ∈ dom 𝑛((𝑛‘(𝑥(1st𝑟)𝑦)) = ((𝑛𝑥)(1st𝑟)(𝑛𝑦)) ∧ (𝑛‘(𝑥(2nd𝑟)𝑦)) = ((𝑛𝑦)(2nd𝑟)(𝑛𝑥)) ∧ (𝑛‘(𝑛𝑥)) = 𝑥))}
ccm2 26301class Com2
df-com2 26302Com2 = {⟨𝑔, ⟩ ∣ ∀𝑎 ∈ ran 𝑔𝑏 ∈ ran 𝑔(𝑎𝑏) = (𝑏𝑎)}
cfld 26304class Fld
df-fld 26305Fld = (DivRingOps ∩ Com2)
cvc 26327class CVecOLD
df-vc 26328CVecOLD = {⟨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 26366class NrmCVec
cpv 26367class +𝑣
cba 26368class BaseSet
cns 26369class ·𝑠OLD
cn0v 26370class 0vec
cnsb 26371class 𝑣
cnmcv 26372class normCV
cims 26373class IndMet
df-nv 26374NrmCVec = {⟨⟨𝑔, 𝑠⟩, 𝑛⟩ ∣ (⟨𝑔, 𝑠⟩ ∈ CVecOLD𝑛:ran 𝑔⟶ℝ ∧ ∀𝑥 ∈ ran 𝑔(((𝑛𝑥) = 0 → 𝑥 = (GId‘𝑔)) ∧ ∀𝑦 ∈ ℂ (𝑛‘(𝑦𝑠𝑥)) = ((abs‘𝑦) · (𝑛𝑥)) ∧ ∀𝑦 ∈ ran 𝑔(𝑛‘(𝑥𝑔𝑦)) ≤ ((𝑛𝑥) + (𝑛𝑦))))}
df-va 26377 +𝑣 = (1st ∘ 1st )
df-ba 26378BaseSet = (𝑥 ∈ V ↦ ran ( +𝑣𝑥))
df-sm 26379 ·𝑠OLD = (2nd ∘ 1st )
df-0v 263800vec = (GId ∘ +𝑣 )
df-vs 26381𝑣 = ( /𝑔 ∘ +𝑣 )
df-nmcv 26382normCV = 2nd
df-ims 26383IndMet = (𝑢 ∈ NrmCVec ↦ ((normCV𝑢) ∘ ( −𝑣𝑢)))
cdip 26499class ·𝑖OLD
df-dip 26500·𝑖OLD = (𝑢 ∈ NrmCVec ↦ (𝑥 ∈ (BaseSet‘𝑢), 𝑦 ∈ (BaseSet‘𝑢) ↦ (Σ𝑘 ∈ (1...4)((i↑𝑘) · (((normCV𝑢)‘(𝑥( +𝑣𝑢)((i↑𝑘)( ·𝑠OLD𝑢)𝑦)))↑2)) / 4)))
css 26523class SubSp
df-ssp 26524SubSp = (𝑢 ∈ NrmCVec ↦ {𝑤 ∈ NrmCVec ∣ (( +𝑣𝑤) ⊆ ( +𝑣𝑢) ∧ ( ·𝑠OLD𝑤) ⊆ ( ·𝑠OLD𝑢) ∧ (normCV𝑤) ⊆ (normCV𝑢))})
clno 26544class LnOp
cnmoo 26545class normOpOLD
cblo 26546class BLnOp
c0o 26547class 0op
df-lno 26548 LnOp = (𝑢 ∈ NrmCVec, 𝑤 ∈ NrmCVec ↦ {𝑡 ∈ ((BaseSet‘𝑤) ↑𝑚 (BaseSet‘𝑢)) ∣ ∀𝑥 ∈ ℂ ∀𝑦 ∈ (BaseSet‘𝑢)∀𝑧 ∈ (BaseSet‘𝑢)(𝑡‘((𝑥( ·𝑠OLD𝑢)𝑦)( +𝑣𝑢)𝑧)) = ((𝑥( ·𝑠OLD𝑤)(𝑡𝑦))( +𝑣𝑤)(𝑡𝑧))})
df-nmoo 26549 normOpOLD = (𝑢 ∈ NrmCVec, 𝑤 ∈ NrmCVec ↦ (𝑡 ∈ ((BaseSet‘𝑤) ↑𝑚 (BaseSet‘𝑢)) ↦ sup({𝑥 ∣ ∃𝑧 ∈ (BaseSet‘𝑢)(((normCV𝑢)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV𝑤)‘(𝑡𝑧)))}, ℝ*, < )))
df-blo 26550 BLnOp = (𝑢 ∈ NrmCVec, 𝑤 ∈ NrmCVec ↦ {𝑡 ∈ (𝑢 LnOp 𝑤) ∣ ((𝑢 normOpOLD 𝑤)‘𝑡) < +∞})
df-0o 26551 0op = (𝑢 ∈ NrmCVec, 𝑤 ∈ NrmCVec ↦ ((BaseSet‘𝑢) × {(0vec𝑤)}))
caj 26552class adj
chmo 26553class HmOp
df-aj 26554adj = (𝑢 ∈ NrmCVec, 𝑤 ∈ NrmCVec ↦ {⟨𝑡, 𝑠⟩ ∣ (𝑡:(BaseSet‘𝑢)⟶(BaseSet‘𝑤) ∧ 𝑠:(BaseSet‘𝑤)⟶(BaseSet‘𝑢) ∧ ∀𝑥 ∈ (BaseSet‘𝑢)∀𝑦 ∈ (BaseSet‘𝑤)((𝑡𝑥)(·𝑖OLD𝑤)𝑦) = (𝑥(·𝑖OLD𝑢)(𝑠𝑦)))})
df-hmo 26555HmOp = (𝑢 ∈ NrmCVec ↦ {𝑡 ∈ dom (𝑢adj𝑢) ∣ ((𝑢adj𝑢)‘𝑡) = 𝑡})
ccphlo 26616class CPreHilOLD
df-ph 26617CPreHilOLD = (NrmCVec ∩ {⟨⟨𝑔, 𝑠⟩, 𝑛⟩ ∣ ∀𝑥 ∈ ran 𝑔𝑦 ∈ ran 𝑔(((𝑛‘(𝑥𝑔𝑦))↑2) + ((𝑛‘(𝑥𝑔(-1𝑠𝑦)))↑2)) = (2 · (((𝑛𝑥)↑2) + ((𝑛𝑦)↑2)))})
ccbn 26667class CBan
df-cbn 26668CBan = {𝑢 ∈ NrmCVec ∣ (IndMet‘𝑢) ∈ (CMet‘(BaseSet‘𝑢))}
chlo 26700class CHilOLD
df-hlo 26701CHilOLD = (CBan ∩ CPreHilOLD)
The list of syntax, axioms (ax-) and definitions (df-) for the Hilbert Space Explorer starts here
chil 26735class
cva 26736class +
csm 26737class ·
csp 26738class ·ih
cno 26739class norm
c0v 26740class 0
cmv 26741class
ccau 26742class Cauchy
chli 26743class 𝑣
csh 26744class S
cch 26745class C
cort 26746class
cph 26747class +
cspn 26748class span
chj 26749class
chsup 26750class
c0h 26751class 0
ccm 26752class 𝐶
cpjh 26753class proj
chos 26754class +op
chot 26755class ·op
chod 26756class op
chfs 26757class +fn
chft 26758class ·fn
ch0o 26759class 0hop
chio 26760class Iop
cnop 26761class normop
ccop 26762class ConOp
clo 26763class LinOp
cbo 26764class BndLinOp
cuo 26765class UniOp
cho 26766class HrmOp
cnmf 26767class normfn
cnl 26768class null
ccnfn 26769class ConFn
clf 26770class LinFn
cado 26771class adj
cbr 26772class bra
ck 26773class ketbra
cleo 26774class op
cei 26775class eigvec
cel 26776class eigval
cspc 26777class Lambda
cst 26778class States
chst 26779class CHStates
ccv 26780class
cat 26781class HAtoms
cmd 26782class 𝑀
cdmd 26783class 𝑀*
df-hnorm 26784norm = (𝑥 ∈ dom dom ·ih ↦ (√‘(𝑥 ·ih 𝑥)))
df-hba 26785 ℋ = (BaseSet‘⟨⟨ + , · ⟩, norm⟩)
df-h0v 267860 = (0vec‘⟨⟨ + , · ⟩, norm⟩)
df-hvsub 26787 = (𝑥 ∈ ℋ, 𝑦 ∈ ℋ ↦ (𝑥 + (-1 · 𝑦)))
df-hlim 26788𝑣 = {⟨𝑓, 𝑤⟩ ∣ ((𝑓:ℕ⟶ ℋ ∧ 𝑤 ∈ ℋ) ∧ ∀𝑥 ∈ ℝ+𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ𝑦)(norm‘((𝑓𝑧) − 𝑤)) < 𝑥)}
df-hcau 26789Cauchy = {𝑓 ∈ ( ℋ ↑𝑚 ℕ) ∣ ∀𝑥 ∈ ℝ+𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ𝑦)(norm‘((𝑓𝑦) − (𝑓𝑧))) < 𝑥}
ax-hilex 26815 ℋ ∈ V
ax-hfvadd 26816 + :( ℋ × ℋ)⟶ ℋ
ax-hvcom 26817((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 + 𝐵) = (𝐵 + 𝐴))
ax-hvass 26818((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶)))
ax-hv0cl 268190 ∈ ℋ
ax-hvaddid 26820(𝐴 ∈ ℋ → (𝐴 + 0) = 𝐴)
ax-hfvmul 26821 · :(ℂ × ℋ)⟶ ℋ
ax-hvmulid 26822(𝐴 ∈ ℋ → (1 · 𝐴) = 𝐴)
ax-hvmulass 26823((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)))
ax-hvdistr1 26824((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶)))
ax-hvdistr2 26825((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝐴 + 𝐵) · 𝐶) = ((𝐴 · 𝐶) + (𝐵 · 𝐶)))
ax-hvmul0 26826(𝐴 ∈ ℋ → (0 · 𝐴) = 0)
ax-hfi 26895 ·ih :( ℋ × ℋ)⟶ℂ
ax-his1 26898((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ih 𝐵) = (∗‘(𝐵 ·ih 𝐴)))
ax-his2 26899((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 + 𝐵) ·ih 𝐶) = ((𝐴 ·ih 𝐶) + (𝐵 ·ih 𝐶)))
ax-his3 26900((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 · 𝐵) ·ih 𝐶) = (𝐴 · (𝐵 ·ih 𝐶)))
ax-his4 26901((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → 0 < (𝐴 ·ih 𝐴))
ax-hcompl 27018(𝐹 ∈ Cauchy → ∃𝑥 ∈ ℋ 𝐹𝑣 𝑥)
df-sh 27023 S = { ∈ 𝒫 ℋ ∣ (0 ∧ ( + “ ( × )) ⊆ ∧ ( · “ (ℂ × )) ⊆ )}
df-ch 27037 C = {S ∣ ( ⇝𝑣 “ (𝑚 ℕ)) ⊆ }
df-oc 27068⊥ = (𝑥 ∈ 𝒫 ℋ ↦ {𝑦 ∈ ℋ ∣ ∀𝑧𝑥 (𝑦 ·ih 𝑧) = 0})
df-ch0 270690 = {0}
df-shs 27124 + = (𝑥S , 𝑦S ↦ ( + “ (𝑥 × 𝑦)))
df-span 27125span = (𝑥 ∈ 𝒫 ℋ ↦ {𝑦S𝑥𝑦})
df-chj 27126 = (𝑥 ∈ 𝒫 ℋ, 𝑦 ∈ 𝒫 ℋ ↦ (⊥‘(⊥‘(𝑥𝑦))))
df-chsup 27127 = (𝑥 ∈ 𝒫 𝒫 ℋ ↦ (⊥‘(⊥‘ 𝑥)))
df-pjh 27211proj = (C ↦ (𝑥 ∈ ℋ ↦ (𝑧𝑦 ∈ (⊥‘)𝑥 = (𝑧 + 𝑦))))
df-cm 27399 𝐶 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥C𝑦C ) ∧ 𝑥 = ((𝑥𝑦) ∨ (𝑥 ∩ (⊥‘𝑦))))}
df-hosum 27546 +op = (𝑓 ∈ ( ℋ ↑𝑚 ℋ), 𝑔 ∈ ( ℋ ↑𝑚 ℋ) ↦ (𝑥 ∈ ℋ ↦ ((𝑓𝑥) + (𝑔𝑥))))
df-homul 27547 ·op = (𝑓 ∈ ℂ, 𝑔 ∈ ( ℋ ↑𝑚 ℋ) ↦ (𝑥 ∈ ℋ ↦ (𝑓 · (𝑔𝑥))))
df-hodif 27548op = (𝑓 ∈ ( ℋ ↑𝑚 ℋ), 𝑔 ∈ ( ℋ ↑𝑚 ℋ) ↦ (𝑥 ∈ ℋ ↦ ((𝑓𝑥) − (𝑔𝑥))))
df-hfsum 27549 +fn = (𝑓 ∈ (ℂ ↑𝑚 ℋ), 𝑔 ∈ (ℂ ↑𝑚 ℋ) ↦ (𝑥 ∈ ℋ ↦ ((𝑓𝑥) + (𝑔𝑥))))
df-hfmul 27550 ·fn = (𝑓 ∈ ℂ, 𝑔 ∈ (ℂ ↑𝑚 ℋ) ↦ (𝑥 ∈ ℋ ↦ (𝑓 · (𝑔𝑥))))
df-h0op 27564 0hop = (proj‘0)
df-iop 27565 Iop = (proj‘ ℋ)
df-nmop 27655normop = (𝑡 ∈ ( ℋ ↑𝑚 ℋ) ↦ sup({𝑥 ∣ ∃𝑧 ∈ ℋ ((norm𝑧) ≤ 1 ∧ 𝑥 = (norm‘(𝑡𝑧)))}, ℝ*, < ))
df-cnop 27656ConOp = {𝑡 ∈ ( ℋ ↑𝑚 ℋ) ∣ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤 ∈ ℋ ((norm‘(𝑤 𝑥)) < 𝑧 → (norm‘((𝑡𝑤) − (𝑡𝑥))) < 𝑦)}
df-lnop 27657LinOp = {𝑡 ∈ ( ℋ ↑𝑚 ℋ) ∣ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ (𝑡‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · (𝑡𝑦)) + (𝑡𝑧))}
df-bdop 27658BndLinOp = {𝑡 ∈ LinOp ∣ (normop𝑡) < +∞}
df-unop 27659UniOp = {𝑡 ∣ (𝑡: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑡𝑥) ·ih (𝑡𝑦)) = (𝑥 ·ih 𝑦))}
df-hmop 27660HrmOp = {𝑡 ∈ ( ℋ ↑𝑚 ℋ) ∣ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑡𝑦)) = ((𝑡𝑥) ·ih 𝑦)}
df-nmfn 27661normfn = (𝑡 ∈ (ℂ ↑𝑚 ℋ) ↦ sup({𝑥 ∣ ∃𝑧 ∈ ℋ ((norm𝑧) ≤ 1 ∧ 𝑥 = (abs‘(𝑡𝑧)))}, ℝ*, < ))
df-nlfn 27662null = (𝑡 ∈ (ℂ ↑𝑚 ℋ) ↦ (𝑡 “ {0}))
df-cnfn 27663ConFn = {𝑡 ∈ (ℂ ↑𝑚 ℋ) ∣ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℝ+𝑧 ∈ ℝ+