MPE Home Metamath Proof Explorer This is the Unicode version.
Change to GIF version
List of Syntax, Axioms (ax-) and Definitions (df-)
RefExpression (see link for any distinct variable requirements)
wn 3wff ¬ 𝜑
wi 4wff (𝜑 → 𝜓)
ax-mp 5𝜑    &   (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 209wff (𝜑 ↔ 𝜓)
df-bi 210 ¬ (((𝜑 ↔ 𝜓) → ¬ ((𝜑 → 𝜓) → ¬ (𝜓 → 𝜑))) → ¬ (¬ ((𝜑 → 𝜓) → ¬ (𝜓 → 𝜑)) → (𝜑 ↔ 𝜓)))
wa 400wff (𝜑 ∧ 𝜓)
df-an 401((𝜑 ∧ 𝜓) ↔ ¬ (𝜑 → ¬ 𝜓))
wo 860wff (𝜑 ∨ 𝜓)
df-or 861((𝜑 ∨ 𝜓) ↔ (¬ 𝜑 → 𝜓))
wif 1076wff if-(𝜑, 𝜓, 𝜒)
df-ifp 1077(if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒)))
w3o 1100wff (𝜑 ∨ 𝜓 ∨ 𝜒)
w3a 1101wff (𝜑 ∧ 𝜓 ∧ 𝜒)
df-3or 1102((𝜑 ∨ 𝜓 ∨ 𝜒) ↔ ((𝜑 ∨ 𝜓) ∨ 𝜒))
df-3an 1103((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ((𝜑 ∧ 𝜓) ∧ 𝜒))
wnan 1519wff (𝜑 ⊼ 𝜓)
df-nan 1520((𝜑 ⊼ 𝜓) ↔ ¬ (𝜑 ∧ 𝜓))
wxo 1539wff (𝜑 ⊻ 𝜓)
df-xor 1540((𝜑 ⊻ 𝜓) ↔ ¬ (𝜑 ↔ 𝜓))
wnor 1556wff (𝜑 ⊽ 𝜓)
df-nor 1557((𝜑 ⊽ 𝜓) ↔ ¬ (𝜑 ∨ 𝜓))
wal 1566wff ∀𝑥𝜑
cv 1567class 𝑥
wceq 1568wff 𝐴 = 𝐵
wtru 1569wff ⊤
df-tru 1571(⊤ ↔ (∀𝑥 𝑥 = 𝑥 → ∀𝑥 𝑥 = 𝑥))
wfal 1580wff ⊥
df-fal 1581(⊥ ↔ ¬ ⊤)
whad 1621wff hadd(𝜑, 𝜓, 𝜒)
df-had 1622(hadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ⊻ 𝜓) ⊻ 𝜒))
wcad 1634wff cadd(𝜑, 𝜓, 𝜒)
df-cad 1635(cadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ∧ 𝜓) ∨ (𝜒 ∧ (𝜑 ⊻ 𝜓))))
wex 1807wff ∃𝑥𝜑
df-ex 1808(∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑)
wnf 1811wff Ⅎ𝑥𝜑
df-nf 1812(Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
ax-gen 1823𝜑    ⇒   ∀𝑥𝜑
ax-4 1837(∀𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))
ax-5 1938(𝜑 → ∀𝑥𝜑)
ax-6 1995 ¬ ∀𝑥 ¬ 𝑥 = 𝑦
ax-7 2036(𝑥 = 𝑦 → (𝑥 = 𝑧 → 𝑦 = 𝑧))
wsb 2098wff [𝑦 / 𝑥]𝜑
df-sb 2099([𝑡 / 𝑥]𝜑 ↔ (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) ∧ ∀𝑧(𝑧 = 𝑡 → ∀𝑥(𝑥 = 𝑧 → 𝜑))))
wcel 2150wff 𝐴 ∈ 𝐵
ax-8 2152(𝑥 = 𝑦 → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧))
ax-9 2160(𝑥 = 𝑦 → (𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦))
ax-10 2183(¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
ax-11 2199(∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑)
ax-12 2220(𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
ax-13 2411(¬ 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
wmo 2572wff ∃*𝑥𝜑
df-mo 2574(∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦) ↔ ∃𝑧∀𝑥(𝜑 → 𝑥 = 𝑧))    ⇒   (∃*𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))
weu 2603wff ∃!𝑥𝜑
df-eu 2604(∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃*𝑥𝜑))
ax-ext 2742(∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → 𝑥 = 𝑦)
cab 2748class {𝑥 ∣ 𝜑}
df-clab 2749(𝑥 ∈ {𝑦 ∣ 𝜑} ↔ [𝑥 / 𝑦]𝜑)
df-cleq 2762(𝑦 = 𝑧 ↔ ∀𝑢(𝑢 ∈ 𝑦 ↔ 𝑢 ∈ 𝑧))    &   (𝑡 = 𝑡 ↔ ∀𝑣(𝑣 ∈ 𝑡 ↔ 𝑣 ∈ 𝑡))    ⇒   (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
df-clel 2845(𝑦 ∈ 𝑧 ↔ ∃𝑢(𝑢 = 𝑦 ∧ 𝑢 ∈ 𝑧))    &   (𝑡 ∈ 𝑡 ↔ ∃𝑣(𝑣 = 𝑡 ∧ 𝑣 ∈ 𝑡))    ⇒   (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵))
wnfc 2917wff Ⅎ𝑥𝐴
df-nfc 2919(Ⅎ𝑥𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴)
wne 2965wff 𝐴 ≠ 𝐵
df-ne 2966(𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵)
wnel 3071wff 𝐴 ∉ 𝐵
df-nel 3072(𝐴 ∉ 𝐵 ↔ ¬ 𝐴 ∈ 𝐵)
wral 3086wff ∀𝑥 ∈ 𝐴 𝜑
df-ral 3087(∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
wrex 3096wff ∃𝑥 ∈ 𝐴 𝜑
df-rex 3097(∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
wreu 3374wff ∃!𝑥 ∈ 𝐴 𝜑
wrmo 3375wff ∃*𝑥 ∈ 𝐴 𝜑
df-rmo 3376(∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
df-reu 3377(∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
crab 3423class {𝑥 ∈ 𝐴 ∣ 𝜑}
df-rab 3424{𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
cvv 3462class V
df-v 3464V = {𝑥 ∣ 𝑥 = 𝑥}
wcdeq 3734wff CondEq(𝑥 = 𝑦 → 𝜑)
df-cdeq 3735(CondEq(𝑥 = 𝑦 → 𝜑) ↔ (𝑥 = 𝑦 → 𝜑))
wsbc 3752wff [𝐴 / 𝑥]𝜑
df-sbc 3753([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑})
csb 3861class ⦋𝐴 / 𝑥⦌𝐵
df-csb 3862⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵}
cdif 3910class (𝐴 ∖ 𝐵)
cun 3911class (𝐴 ∪ 𝐵)
cin 3912class (𝐴 ∩ 𝐵)
wss 3913wff 𝐴 ⊆ 𝐵
wpss 3914wff 𝐴 ⊊ 𝐵
df-dif 3916(𝐴 ∖ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)}
df-un 3918(𝐴 ∪ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)}
df-in 3920(𝐴 ∩ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)}
df-ss 3930(𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
df-pss 3933(𝐴 ⊊ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ 𝐵))
csymdif 4213class (𝐴 △ 𝐵)
df-symdif 4214(𝐴 △ 𝐵) = ((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐴))
c0 4294class ∅
df-nul 4295∅ = (V ∖ V)
cif 4492class if(𝜑, 𝐴, 𝐵)
df-if 4493if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑))}
cpw 4567class 𝒫 𝐴
df-pw 4569𝒫 𝐴 = {𝑥 ∣ 𝑥 ⊆ 𝐴}
csn 4594class {𝐴}
df-sn 4595{𝐴} = {𝑥 ∣ 𝑥 = 𝐴}
cpr 4596class {𝐴, 𝐵}
df-pr 4597{𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
ctp 4598class {𝐴, 𝐵, 𝐶}
df-tp 4599{𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
cop 4600class ⟨𝐴, 𝐵⟩
df-op 4601⟨𝐴, 𝐵⟩ = {𝑥 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})}
cotp 4602class ⟨𝐴, 𝐵, 𝐶⟩
df-ot 4603⟨𝐴, 𝐵, 𝐶⟩ = ⟨⟨𝐴, 𝐵⟩, 𝐶⟩
cuni 4877class ∪ 𝐴
df-uni 4878∪ 𝐴 = {𝑥 ∣ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)}
cint 4917class ∩ 𝐴
df-int 4918∩ 𝐴 = {𝑥 ∣ ∀𝑦(𝑦 ∈ 𝐴 → 𝑥 ∈ 𝑦)}
ciun 4961class ∪ 𝑥 ∈ 𝐴 𝐵
ciin 4962class ∩ 𝑥 ∈ 𝐴 𝐵
df-iun 4963∪ 𝑥 ∈ 𝐴 𝐵 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵}
df-iin 4964∩ 𝑥 ∈ 𝐴 𝐵 = {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵}
wdisj 5081wff Disj 𝑥 ∈ 𝐴 𝐵
df-disj 5082(Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
wbr 5114wff 𝐴𝑅𝐵
df-br 5115(𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
copab 5178class {⟨𝑥, 𝑦⟩ ∣ 𝜑}
df-opab 5179{⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
cmpt 5197class (𝑥 ∈ 𝐴 ↦ 𝐵)
df-mpt 5198(𝑥 ∈ 𝐴 ↦ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
wtr 5223wff Tr 𝐴
df-tr 5224(Tr 𝐴 ↔ ∪ 𝐴 ⊆ 𝐴)
ax-rep 5243(∀𝑤∃𝑦∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦) → ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)))
ax-sep 5262∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))
ax-nul 5274∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥
ax-pow 5340∃𝑦∀𝑧(∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦)
ax-pr 5408∃𝑧∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧)
cid 5559class I
df-id 5560 I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
cep 5564class E
df-eprel 5565 E = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ 𝑦}
wpo 5571wff 𝑅 Po 𝐴
wor 5572wff 𝑅 Or 𝐴
df-po 5573(𝑅 Po 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
df-so 5574(𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)))
wfr 5615wff 𝑅 Fr 𝐴
wse 5616wff 𝑅 Se 𝐴
wwe 5617wff 𝑅 We 𝐴
df-fr 5618(𝑅 Fr 𝐴 ↔ ∀𝑥((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦))
df-se 5619(𝑅 Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V)
df-we 5620(𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴 ∧ 𝑅 Or 𝐴))
cxp 5663class (𝐴 × 𝐵)
ccnv 5664class ◡𝐴
cdm 5665class dom 𝐴
crn 5666class ran 𝐴
cres 5667class (𝐴 ↾ 𝐵)
cima 5668class (𝐴 “ 𝐵)
ccom 5669class (𝐴 ∘ 𝐵)
wrel 5670wff Rel 𝐴
df-xp 5671(𝐴 × 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)}
df-rel 5672(Rel 𝐴 ↔ 𝐴 ⊆ (V × V))
df-cnv 5673◡𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥}
df-co 5674(𝐴 ∘ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)}
df-dm 5675dom 𝐴 = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦}
df-rn 5676ran 𝐴 = dom ◡𝐴
df-res 5677(𝐴 ↾ 𝐵) = (𝐴 ∩ (𝐵 × V))
df-ima 5678(𝐴 “ 𝐵) = ran (𝐴 ↾ 𝐵)
cpred 6305class Pred(𝑅, 𝐴, 𝑋)
df-pred 6306Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋}))
word 6363wff Ord 𝐴
con0 6364class On
wlim 6365wff Lim 𝐴
csuc 6366class suc 𝐴
df-ord 6367(Ord 𝐴 ↔ (Tr 𝐴 ∧ E We 𝐴))
df-on 6368On = {𝑥 ∣ Ord 𝑥}
df-lim 6369(Lim 𝐴 ↔ (Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴))
df-suc 6370suc 𝐴 = (𝐴 ∪ {𝐴})
cio 6494class (℩𝑥𝜑)
df-iota 6496(℩𝑥𝜑) = ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}
wfun 6534wff Fun 𝐴
wfn 6535wff 𝐴 Fn 𝐵
wf 6536wff 𝐹:𝐴⟶𝐵
wf1 6537wff 𝐹:𝐴–1-1→𝐵
wfo 6538wff 𝐹:𝐴–onto→𝐵
wf1o 6539wff 𝐹:𝐴–1-1-onto→𝐵
cfv 6540class (𝐹‘𝐴)
wiso 6541wff 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵)
df-fun 6542(Fun 𝐴 ↔ (Rel 𝐴 ∧ (𝐴 ∘ ◡𝐴) ⊆ I ))
df-fn 6543(𝐴 Fn 𝐵 ↔ (Fun 𝐴 ∧ dom 𝐴 = 𝐵))
df-f 6544(𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵))
df-f1 6545(𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ Fun ◡𝐹))
df-fo 6546(𝐹:𝐴–onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵))
df-f1o 6547(𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹:𝐴–1-1→𝐵 ∧ 𝐹:𝐴–onto→𝐵))
df-fv 6548(𝐹‘𝐴) = (℩𝑥𝐴𝐹𝑥)
df-isom 6549(𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐻:𝐴–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦))))
crio 7370class (℩𝑥 ∈ 𝐴 𝜑)
df-riota 7371(℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
co 7414class (𝐴𝐹𝐵)
coprab 7415class {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
cmpo 7416class (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
df-ov 7417(𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
df-oprab 7418{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)}
df-mpo 7419(𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)}
cof 7676class ∘f 𝑅
cofr 7677class ∘r 𝑅
df-of 7678 ∘f 𝑅 = (𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥))))
df-ofr 7679 ∘r 𝑅 = {⟨𝑓, 𝑔⟩ ∣ ∀𝑥 ∈ (dom 𝑓 ∩ dom 𝑔)(𝑓‘𝑥)𝑅(𝑔‘𝑥)}
crpss 7723class [⊊]
df-rpss 7724 [⊊] = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ⊊ 𝑦}
ax-un 7736∃𝑦∀𝑧(∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦)
com 7865class ω
df-om 7866ω = {𝑥 ∈ On ∣ ∀𝑦(Lim 𝑦 → 𝑥 ∈ 𝑦)}
c1st 7987class 1st
c2nd 7988class 2nd
df-1st 79891st = (𝑥 ∈ V ↦ ∪ dom {𝑥})
df-2nd 79902nd = (𝑥 ∈ V ↦ ∪ ran {𝑥})
csupp 8159class supp
df-supp 8160 supp = (𝑥 ∈ V, 𝑧 ∈ V ↦ {𝑖 ∈ dom 𝑥 ∣ (𝑥 “ {𝑖}) ≠ {𝑧}})
ctpos 8224class tpos 𝐹
df-tpos 8225tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}))
ccur 8264class curry 𝐴
cunc 8265class uncurry 𝐴
df-cur 8266curry 𝐹 = (𝑥 ∈ dom dom 𝐹 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩𝐹𝑧})
df-unc 8267uncurry 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(𝐹‘𝑥)𝑧}
cund 8271class Undef
df-undef 8272Undef = (𝑠 ∈ V ↦ 𝒫 ∪ 𝑠)
cfrecs 8280class frecs(𝑅, 𝐴, 𝐹)
df-frecs 8281frecs(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 8311class wrecs(𝑅, 𝐴, 𝐹)
df-wrecs 8312wrecs(𝑅, 𝐴, 𝐹) = frecs(𝑅, 𝐴, (𝐹 ∘ 2nd ))
wsmo 8335wff Smo 𝐴
df-smo 8336(Smo 𝐴 ↔ (𝐴:dom 𝐴⟶On ∧ Ord dom 𝐴 ∧ ∀𝑥 ∈ dom 𝐴∀𝑦 ∈ dom 𝐴(𝑥 ∈ 𝑦 → (𝐴‘𝑥) ∈ (𝐴‘𝑦))))
crecs 8360class recs(𝐹)
df-recs 8361recs(𝐹) = wrecs( E , On, 𝐹)
crdg 8399class rec(𝐹, 𝐼)
df-rdg 8400rec(𝐹, 𝐼) = recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔))))))
cseqom 8437class seqω(𝐹, 𝐼)
df-seqom 8438seqω(𝐹, 𝐼) = (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) “ ω)
c1o 8449class 1o
c2o 8450class 2o
c3o 8451class 3o
c4o 8452class 4o
coa 8453class +o
comu 8454class ·o
coe 8455class ↑o
df-1o 84561o = suc ∅
df-2o 84572o = suc 1o
df-3o 84583o = suc 2o
df-4o 84594o = suc 3o
df-oadd 8460 +o = (𝑥 ∈ On, 𝑦 ∈ On ↦ (rec((𝑧 ∈ V ↦ suc 𝑧), 𝑥)‘𝑦))
df-omul 8461 ·o = (𝑥 ∈ On, 𝑦 ∈ On ↦ (rec((𝑧 ∈ V ↦ (𝑧 +o 𝑥)), ∅)‘𝑦))
df-oexp 8462 ↑o = (𝑥 ∈ On, 𝑦 ∈ On ↦ if(𝑥 = ∅, (1o ∖ 𝑦), (rec((𝑧 ∈ V ↦ (𝑧 ·o 𝑥)), 1o)‘𝑦)))
cnadd 8654class +no
df-nadd 8655 +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 8694wff 𝑅 Er 𝐴
cec 8695class [𝐴]𝑅
cqs 8696class (𝐴 / 𝑅)
df-er 8697(𝑅 Er 𝐴 ↔ (Rel 𝑅 ∧ dom 𝑅 = 𝐴 ∧ (◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅))
df-ec 8699[𝐴]𝑅 = (𝑅 “ {𝐴})
df-qs 8703(𝐴 / 𝑅) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅}
cmap 8827class ↑m
cpm 8828class ↑pm
df-map 8829 ↑m = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓 ∣ 𝑓:𝑦⟶𝑥})
df-pm 8830 ↑pm = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓 ∈ 𝒫 (𝑦 × 𝑥) ∣ Fun 𝑓})
cixp 8898class X𝑥 ∈ 𝐴 𝐵
df-ixp 8899X𝑥 ∈ 𝐴 𝐵 = {𝑓 ∣ (𝑓 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵)}
cen 8943class ≈
cdom 8944class ≼
csdm 8945class ≺
cfn 8946class Fin
df-en 8947 ≈ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥–1-1-onto→𝑦}
df-dom 8948 ≼ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥–1-1→𝑦}
df-sdom 8949 ≺ = ( ≼ ∖ ≈ )
df-fin 8950Fin = {𝑥 ∣ ∃𝑦 ∈ ω 𝑥 ≈ 𝑦}
cfsupp 9324class finSupp
df-fsupp 9325 finSupp = {⟨𝑟, 𝑧⟩ ∣ (Fun 𝑟 ∧ (𝑟 supp 𝑧) ∈ Fin)}
cfi 9373class fi
df-fi 9374fi = (𝑥 ∈ V ↦ {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦})
csup 9403class sup(𝐴, 𝐵, 𝑅)
cinf 9404class inf(𝐴, 𝐵, 𝑅)
df-sup 9405sup(𝐴, 𝐵, 𝑅) = ∪ {𝑥 ∈ 𝐵 ∣ (∀𝑦 ∈ 𝐴 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐵 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐴 𝑦𝑅𝑧))}
df-inf 9406inf(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, ◡𝑅)
coi 9474class OrdIso(𝑅, 𝐴)
df-oi 9475OrdIso(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 9521class har
df-har 9522har = (𝑥 ∈ V ↦ {𝑦 ∈ On ∣ 𝑦 ≼ 𝑥})
cwdom 9529class ≼*
df-wdom 9530 ≼* = {⟨𝑥, 𝑦⟩ ∣ (𝑥 = ∅ ∨ ∃𝑧 𝑧:𝑦–onto→𝑥)}
ax-reg 9557(∃𝑦 𝑦 ∈ 𝑥 → ∃𝑦(𝑦 ∈ 𝑥 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑥)))
ax-inf 9610∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦)))
ax-inf2 9613∃𝑥(∃𝑦(𝑦 ∈ 𝑥 ∧ ∀𝑧 ¬ 𝑧 ∈ 𝑦) ∧ ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑧 ∈ 𝑥 ∧ ∀𝑤(𝑤 ∈ 𝑧 ↔ (𝑤 ∈ 𝑦 ∨ 𝑤 = 𝑦)))))
ccnf 9633class CNF
df-cnf 9634 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 9679class t++𝑅
df-ttrcl 9680t++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 9706class TC
df-tc 9707TC = (𝑥 ∈ V ↦ ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ Tr 𝑦)})
cr1 9737class 𝑅1
crnk 9738class rank
df-r1 9739𝑅1 = rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅)
df-rank 9740rank = (𝑥 ∈ V ↦ ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)})
cscott 9860class Scott 𝐴
df-scott 9861Scott 𝐴 = {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)}
cdju 9887class (𝐴 ⊔ 𝐵)
cinl 9888class inl
cinr 9889class inr
df-dju 9890(𝐴 ⊔ 𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵))
df-inl 9891inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩)
df-inr 9892inr = (𝑥 ∈ V ↦ ⟨1o, 𝑥⟩)
ccrd 9924class card
cale 9925class ℵ
ccf 9926class cf
wacn 9927class AC 𝐴
df-card 9928card = (𝑥 ∈ V ↦ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥})
df-aleph 9929ℵ = rec(har, ω)
df-cf 9930cf = (𝑥 ∈ On ↦ ∩ {𝑦 ∣ ∃𝑧(𝑦 = (card‘𝑧) ∧ (𝑧 ⊆ 𝑥 ∧ ∀𝑣 ∈ 𝑥 ∃𝑢 ∈ 𝑧 𝑣 ⊆ 𝑢))})
df-acn 9931AC 𝐴 = {𝑥 ∣ (𝐴 ∈ V ∧ ∀𝑓 ∈ ((𝒫 𝑥 ∖ {∅}) ↑m 𝐴)∃𝑔∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝑓‘𝑦))}
wac 10102wff CHOICE
df-ac 10103(CHOICE ↔ ∀𝑥∃𝑓(𝑓 ⊆ 𝑥 ∧ 𝑓 Fn dom 𝑥))
cfin1a 10265class FinIa
cfin2 10266class FinII
cfin4 10267class FinIV
cfin3 10268class FinIII
cfin5 10269class FinV
cfin6 10270class FinVI
cfin7 10271class FinVII
df-fin1a 10272FinIa = {𝑥 ∣ ∀𝑦 ∈ 𝒫 𝑥(𝑦 ∈ Fin ∨ (𝑥 ∖ 𝑦) ∈ Fin)}
df-fin2 10273FinII = {𝑥 ∣ ∀𝑦 ∈ 𝒫 𝒫 𝑥((𝑦 ≠ ∅ ∧ [⊊] Or 𝑦) → ∪ 𝑦 ∈ 𝑦)}
df-fin4 10274FinIV = {𝑥 ∣ ¬ ∃𝑦(𝑦 ⊊ 𝑥 ∧ 𝑦 ≈ 𝑥)}
df-fin3 10275FinIII = {𝑥 ∣ 𝒫 𝑥 ∈ FinIV}
df-fin5 10276FinV = {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 ≺ (𝑥 ⊔ 𝑥))}
df-fin6 10277FinVI = {𝑥 ∣ (𝑥 ≺ 2o ∨ 𝑥 ≺ (𝑥 × 𝑥))}
df-fin7 10278FinVII = {𝑥 ∣ ¬ ∃𝑦 ∈ (On ∖ ω)𝑥 ≈ 𝑦}
ax-cc 10422(𝑥 ≈ ω → ∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧))
ax-dc 10433((∃𝑦∃𝑧 𝑦𝑥𝑧 ∧ ran 𝑥 ⊆ dom 𝑥) → ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛))
ax-ac 10446∃𝑦∀𝑧∀𝑤((𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → ∃𝑣∀𝑢(∃𝑡((𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑡) ∧ (𝑢 ∈ 𝑡 ∧ 𝑡 ∈ 𝑦)) ↔ 𝑢 = 𝑣))
ax-ac2 10450∃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 10608class GCH
df-gch 10609GCH = (Fin ∪ {𝑥 ∣ ∀𝑦 ¬ (𝑥 ≺ 𝑦 ∧ 𝑦 ≺ 𝒫 𝑥)})
cwina 10670class Inaccw
cina 10671class Inacc
df-wina 10672Inaccw = {𝑥 ∣ (𝑥 ≠ ∅ ∧ (cf‘𝑥) = 𝑥 ∧ ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝑥 𝑦 ≺ 𝑧)}
df-ina 10673Inacc = {𝑥 ∣ (𝑥 ≠ ∅ ∧ (cf‘𝑥) = 𝑥 ∧ ∀𝑦 ∈ 𝑥 𝒫 𝑦 ≺ 𝑥)}
cwun 10688class WUni
cwunm 10689class wUniCl
df-wun 10690WUni = {𝑢 ∣ (Tr 𝑢 ∧ 𝑢 ≠ ∅ ∧ ∀𝑥 ∈ 𝑢 (∪ 𝑥 ∈ 𝑢 ∧ 𝒫 𝑥 ∈ 𝑢 ∧ ∀𝑦 ∈ 𝑢 {𝑥, 𝑦} ∈ 𝑢))}
df-wunc 10691wUniCl = (𝑥 ∈ V ↦ ∩ {𝑢 ∈ WUni ∣ 𝑥 ⊆ 𝑢})
ctsk 10736class Tarski
df-tsk 10737Tarski = {𝑦 ∣ (∀𝑧 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑦 ∧ ∃𝑤 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧 ≈ 𝑦 ∨ 𝑧 ∈ 𝑦))}
cgru 10778class Univ
df-gru 10779Univ = {𝑢 ∣ (Tr 𝑢 ∧ ∀𝑥 ∈ 𝑢 (𝒫 𝑥 ∈ 𝑢 ∧ ∀𝑦 ∈ 𝑢 {𝑥, 𝑦} ∈ 𝑢 ∧ ∀𝑦 ∈ (𝑢 ↑m 𝑥)∪ ran 𝑦 ∈ 𝑢))}
ax-groth 10811∃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 10825class tarskiMap
df-tskm 10826tarskiMap = (𝑥 ∈ V ↦ ∩ {𝑦 ∈ Tarski ∣ 𝑥 ∈ 𝑦})
cnpi 10832class N
cpli 10833class +N
cmi 10834class ·N
clti 10835class <N
cplpq 10836class +pQ
cmpq 10837class ·pQ
cltpq 10838class <pQ
ceq 10839class ~Q
cnq 10840class Q
c1q 10841class 1Q
cerq 10842class [Q]
cplq 10843class +Q
cmq 10844class ·Q
crq 10845class *Q
cltq 10846class <Q
cnp 10847class P
c1p 10848class 1P
cpp 10849class +P
cmp 10850class ·P
cltp 10851class <P
cer 10852class ~R
cnr 10853class R
c0r 10854class 0R
c1r 10855class 1R
cm1r 10856class -1R
cplr 10857class +R
cmr 10858class ·R
cltr 10859class <R
df-ni 10860N = (ω ∖ {∅})
df-pli 10861 +N = ( +o ↾ (N × N))
df-mi 10862 ·N = ( ·o ↾ (N × N))
df-lti 10863 <N = ( E ∩ (N × N))
df-plpq 10896 +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 10897 ·pQ = (𝑥 ∈ (N × N), 𝑦 ∈ (N × N) ↦ ⟨((1st ‘𝑥) ·N (1st ‘𝑦)), ((2nd ‘𝑥) ·N (2nd ‘𝑦))⟩)
df-ltpq 10898 <pQ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ((1st ‘𝑥) ·N (2nd ‘𝑦)) <N ((1st ‘𝑦) ·N (2nd ‘𝑥)))}
df-enq 10899 ~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 10900Q = {𝑥 ∈ (N × N) ∣ ∀𝑦 ∈ (N × N)(𝑥 ~Q 𝑦 → ¬ (2nd ‘𝑦) <N (2nd ‘𝑥))}
df-erq 10901[Q] = ( ~Q ∩ ((N × N) × Q))
df-plq 10902 +Q = (([Q] ∘ +pQ ) ↾ (Q × Q))
df-mq 10903 ·Q = (([Q] ∘ ·pQ ) ↾ (Q × Q))
df-1nq 109041Q = ⟨1o, 1o⟩
df-rq 10905*Q = (◡ ·Q “ {1Q})
df-ltnq 10906 <Q = ( <pQ ∩ (Q × Q))
df-np 10969P = {𝑥 ∣ ((∅ ⊊ 𝑥 ∧ 𝑥 ⊊ Q) ∧ ∀𝑦 ∈ 𝑥 (∀𝑧(𝑧 <Q 𝑦 → 𝑧 ∈ 𝑥) ∧ ∃𝑧 ∈ 𝑥 𝑦 <Q 𝑧))}
df-1p 109701P = {𝑥 ∣ 𝑥 <Q 1Q}
df-plp 10971 +P = (𝑥 ∈ P, 𝑦 ∈ P ↦ {𝑤 ∣ ∃𝑣 ∈ 𝑥 ∃𝑢 ∈ 𝑦 𝑤 = (𝑣 +Q 𝑢)})
df-mp 10972 ·P = (𝑥 ∈ P, 𝑦 ∈ P ↦ {𝑤 ∣ ∃𝑣 ∈ 𝑥 ∃𝑢 ∈ 𝑦 𝑤 = (𝑣 ·Q 𝑢)})
df-ltp 10973<P = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ 𝑥 ⊊ 𝑦)}
df-enr 11043 ~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 11044R = ((P × P) / ~R )
df-plr 11045 +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 11046 ·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 11047 <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 110480R = [⟨1P, 1P⟩] ~R
df-1r 110491R = [⟨(1P +P 1P), 1P⟩] ~R
df-m1r 11050-1R = [⟨1P, (1P +P 1P)⟩] ~R
cc 11101class ℂ
cr 11102class ℝ
cc0 11103class 0
c1 11104class 1
ci 11105class i
caddc 11106class +
cltrr 11107class <ℝ
cmul 11108class ·
df-c 11109ℂ = (R × R)
df-0 111100 = ⟨0R, 0R⟩
df-1 111111 = ⟨1R, 0R⟩
df-i 11112i = ⟨0R, 1R⟩
df-r 11113ℝ = (R × {0R})
df-add 11114 + = {⟨⟨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 11115 · = {⟨⟨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 11116 <ℝ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ∃𝑧∃𝑤((𝑥 = ⟨𝑧, 0R⟩ ∧ 𝑦 = ⟨𝑤, 0R⟩) ∧ 𝑧 <R 𝑤))}
ax-cnex 11159ℂ ∈ V
ax-resscn 11160ℝ ⊆ ℂ
ax-1cn 111611 ∈ ℂ
ax-icn 11162i ∈ ℂ
ax-addcl 11163((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) ∈ ℂ)
ax-addrcl 11164((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + 𝐵) ∈ ℝ)
ax-mulcl 11165((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) ∈ ℂ)
ax-mulrcl 11166((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 · 𝐵) ∈ ℝ)
ax-mulcom 11167((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) = (𝐵 · 𝐴))
ax-addass 11168((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶)))
ax-mulass 11169((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)))
ax-distr 11170((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶)))
ax-i2m1 11171((i · i) + 1) = 0
ax-1ne0 111721 ≠ 0
ax-1rid 11173(𝐴 ∈ ℝ → (𝐴 · 1) = 𝐴)
ax-rnegex 11174(𝐴 ∈ ℝ → ∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0)
ax-rrecex 11175((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → ∃𝑥 ∈ ℝ (𝐴 · 𝑥) = 1)
ax-cnre 11176(𝐴 ∈ ℂ → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝐴 = (𝑥 + (i · 𝑦)))
ax-pre-lttri 11177((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 <ℝ 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 <ℝ 𝐴)))
ax-pre-lttrn 11178((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 <ℝ 𝐵 ∧ 𝐵 <ℝ 𝐶) → 𝐴 <ℝ 𝐶))
ax-pre-ltadd 11179((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 <ℝ 𝐵 → (𝐶 + 𝐴) <ℝ (𝐶 + 𝐵)))
ax-pre-mulgt0 11180((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((0 <ℝ 𝐴 ∧ 0 <ℝ 𝐵) → 0 <ℝ (𝐴 · 𝐵)))
ax-pre-sup 11181((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 11182 + :(ℂ × ℂ)⟶ℂ
ax-mulf 11183 · :(ℂ × ℂ)⟶ℂ
cpnf 11243class +∞
cmnf 11244class -∞
cxr 11245class ℝ*
clt 11246class <
cle 11247class ≤
df-pnf 11248+∞ = 𝒫 ∪ ℂ
df-mnf 11249-∞ = 𝒫 +∞
df-xr 11250ℝ* = (ℝ ∪ {+∞, -∞})
df-ltxr 11251 < = ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)} ∪ (((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ)))
df-le 11252 ≤ = ((ℝ* × ℝ*) ∖ ◡ < )
cmin 11444class −
cneg 11445class -𝐴
df-sub 11446 − = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥))
df-neg 11447-𝐴 = (0 − 𝐴)
cdiv 11874class /
df-div 11875 / = (𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥))
cind 12221class 𝟭
df-ind 12222𝟭 = (𝑜 ∈ V ↦ (𝑎 ∈ 𝒫 𝑜 ↦ (𝑥 ∈ 𝑜 ↦ if(𝑥 ∈ 𝑎, 1, 0))))
cn 12236class ℕ
df-nn 12237ℕ = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) “ ω)
c2 12298class 2
c3 12299class 3
c4 12300class 4
c5 12301class 5
c6 12302class 6
c7 12303class 7
c8 12304class 8
c9 12305class 9
df-2 123062 = (1 + 1)
df-3 123073 = (2 + 1)
df-4 123084 = (3 + 1)
df-5 123095 = (4 + 1)
df-6 123106 = (5 + 1)
df-7 123117 = (6 + 1)
df-8 123128 = (7 + 1)
df-9 123139 = (8 + 1)
cn0 12507class ℕ0
df-n0 12508ℕ0 = (ℕ ∪ {0})
cxnn0 12580class ℕ0*
df-xnn0 12581ℕ0* = (ℕ0 ∪ {+∞})
cz 12594class ℤ
df-z 12595ℤ = {𝑛 ∈ ℝ ∣ (𝑛 = 0 ∨ 𝑛 ∈ ℕ ∨ -𝑛 ∈ ℕ)}
cdc 12714class 𝐴𝐵
df-dec 12715𝐴𝐵 = (((9 + 1) · 𝐴) + 𝐵)
cuz 12865class ℤ≥
df-uz 12866ℤ≥ = (𝑗 ∈ ℤ ↦ {𝑘 ∈ ℤ ∣ 𝑗 ≤ 𝑘})
cq 12975class ℚ
df-q 12976ℚ = ( / “ (ℤ × ℕ))
crp 13019class ℝ+
df-rp 13020ℝ+ = {𝑥 ∈ ℝ ∣ 0 < 𝑥}
cxne 13137class -𝑒𝐴
cxad 13138class +𝑒
cxmu 13139class ·e
df-xneg 13140-𝑒𝐴 = if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴))
df-xadd 13141 +𝑒 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ if(𝑥 = +∞, if(𝑦 = -∞, 0, +∞), if(𝑥 = -∞, if(𝑦 = +∞, 0, -∞), if(𝑦 = +∞, +∞, if(𝑦 = -∞, -∞, (𝑥 + 𝑦))))))
df-xmul 13142 ·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 13375class (,)
cioc 13376class (,]
cico 13377class [,)
cicc 13378class [,]
df-ioo 13379(,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)})
df-ioc 13380(,] = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 ≤ 𝑦)})
df-ico 13381[,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)})
df-icc 13382[,] = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)})
cfz 13538class ...
df-fz 13539... = (𝑚 ∈ ℤ, 𝑛 ∈ ℤ ↦ {𝑘 ∈ ℤ ∣ (𝑚 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛)})
cfzo 13685class ..^
df-fzo 13686..^ = (𝑚 ∈ ℤ, 𝑛 ∈ ℤ ↦ (𝑚...(𝑛 − 1)))
cfl 13826class ⌊
cceil 13827class ⌈
df-fl 13828⌊ = (𝑥 ∈ ℝ ↦ (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))))
df-ceil 13829⌈ = (𝑥 ∈ ℝ ↦ -(⌊‘-𝑥))
cmo 13905class mod
df-mod 13906 mod = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ+ ↦ (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦)))))
cseq 14040class seq𝑀( + , 𝐹)
df-seq 14041seq𝑀( + , 𝐹) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) “ ω)
cexp 14100class ↑
df-exp 14101↑ = (𝑥 ∈ ℂ, 𝑦 ∈ ℤ ↦ if(𝑦 = 0, 1, if(0 < 𝑦, (seq1( · , (ℕ × {𝑥}))‘𝑦), (1 / (seq1( · , (ℕ × {𝑥}))‘-𝑦)))))
cfa 14312class !
df-fac 14313! = ({⟨0, 1⟩} ∪ seq1( · , I ))
cbc 14341class C
df-bc 14342C = (𝑛 ∈ ℕ0, 𝑘 ∈ ℤ ↦ if(𝑘 ∈ (0...𝑛), ((!‘𝑛) / ((!‘(𝑛 − 𝑘)) · (!‘𝑘))), 0))
chash 14369class ♯
df-hash 14370♯ = (((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω) ∘ card) ∪ ((V ∖ Fin) × {+∞}))
cword 14553class Word 𝑆
df-word 14554Word 𝑆 = {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆}
clsw 14602class lastS
df-lsw 14603lastS = (𝑤 ∈ V ↦ (𝑤‘((♯‘𝑤) − 1)))
cconcat 14610class ++
df-concat 14611 ++ = (𝑠 ∈ V, 𝑡 ∈ V ↦ (𝑥 ∈ (0..^((♯‘𝑠) + (♯‘𝑡))) ↦ if(𝑥 ∈ (0..^(♯‘𝑠)), (𝑠‘𝑥), (𝑡‘(𝑥 − (♯‘𝑠))))))
cs1 14636class ⟨“𝐴”⟩
df-s1 14637⟨“𝐴”⟩ = {⟨0, ( I ‘𝐴)⟩}
csubstr 14681class substr
df-substr 14682 substr = (𝑠 ∈ V, 𝑏 ∈ (ℤ × ℤ) ↦ if(((1st ‘𝑏)..^(2nd ‘𝑏)) ⊆ dom 𝑠, (𝑥 ∈ (0..^((2nd ‘𝑏) − (1st ‘𝑏))) ↦ (𝑠‘(𝑥 + (1st ‘𝑏)))), ∅))
cpfx 14711class prefix
df-pfx 14712 prefix = (𝑠 ∈ V, 𝑙 ∈ ℕ0 ↦ (𝑠 substr ⟨0, 𝑙⟩))
csplice 14789class splice
df-splice 14790 splice = (𝑠 ∈ V, 𝑏 ∈ V ↦ (((𝑠 prefix (1st ‘(1st ‘𝑏))) ++ (2nd ‘𝑏)) ++ (𝑠 substr ⟨(2nd ‘(1st ‘𝑏)), (♯‘𝑠)⟩)))
creverse 14798class reverse
df-reverse 14799reverse = (𝑠 ∈ V ↦ (𝑥 ∈ (0..^(♯‘𝑠)) ↦ (𝑠‘(((♯‘𝑠) − 1) − 𝑥))))
creps 14808class repeatS
df-reps 14809 repeatS = (𝑠 ∈ V, 𝑛 ∈ ℕ0 ↦ (𝑥 ∈ (0..^𝑛) ↦ 𝑠))
ccsh 14828class cyclShift
df-csh 14829 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 14881class ⟨“𝐴𝐵”⟩
cs3 14882class ⟨“𝐴𝐵𝐶”⟩
cs4 14883class ⟨“𝐴𝐵𝐶𝐷”⟩
cs5 14884class ⟨“𝐴𝐵𝐶𝐷𝐸”⟩
cs6 14885class ⟨“𝐴𝐵𝐶𝐷𝐸𝐹”⟩
cs7 14886class ⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”⟩
cs8 14887class ⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺𝐻”⟩
df-s2 14888⟨“𝐴𝐵”⟩ = (⟨“𝐴”⟩ ++ ⟨“𝐵”⟩)
df-s3 14889⟨“𝐴𝐵𝐶”⟩ = (⟨“𝐴𝐵”⟩ ++ ⟨“𝐶”⟩)
df-s4 14890⟨“𝐴𝐵𝐶𝐷”⟩ = (⟨“𝐴𝐵𝐶”⟩ ++ ⟨“𝐷”⟩)
df-s5 14891⟨“𝐴𝐵𝐶𝐷𝐸”⟩ = (⟨“𝐴𝐵𝐶𝐷”⟩ ++ ⟨“𝐸”⟩)
df-s6 14892⟨“𝐴𝐵𝐶𝐷𝐸𝐹”⟩ = (⟨“𝐴𝐵𝐶𝐷𝐸”⟩ ++ ⟨“𝐹”⟩)
df-s7 14893⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”⟩ = (⟨“𝐴𝐵𝐶𝐷𝐸𝐹”⟩ ++ ⟨“𝐺”⟩)
df-s8 14894⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺𝐻”⟩ = (⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”⟩ ++ ⟨“𝐻”⟩)
ctcl 15025class t+
crtcl 15026class t*
df-trcl 15027t+ = (𝑥 ∈ V ↦ ∩ {𝑧 ∣ (𝑥 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)})
df-rtrcl 15028t* = (𝑥 ∈ V ↦ ∩ {𝑧 ∣ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧 ∧ 𝑥 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)})
crelexp 15059class ↑𝑟
df-relexp 15060↑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 15095class t*rec
df-rtrclrec 15096t*rec = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))
cshi 15106class shift
df-shft 15107 shift = (𝑓 ∈ V, 𝑥 ∈ ℂ ↦ {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ ℂ ∧ (𝑦 − 𝑥)𝑓𝑧)})
csgn 15126class sgn
df-sgn 15127sgn = (𝑥 ∈ ℝ* ↦ if(𝑥 = 0, 0, if(𝑥 < 0, -1, 1)))
ccj 15150class ∗
cre 15151class ℜ
cim 15152class ℑ
df-cj 15153∗ = (𝑥 ∈ ℂ ↦ (℩𝑦 ∈ ℂ ((𝑥 + 𝑦) ∈ ℝ ∧ (i · (𝑥 − 𝑦)) ∈ ℝ)))
df-re 15154ℜ = (𝑥 ∈ ℂ ↦ ((𝑥 + (∗‘𝑥)) / 2))
df-im 15155ℑ = (𝑥 ∈ ℂ ↦ (ℜ‘(𝑥 / i)))
csqrt 15287class √
cabs 15288class abs
df-sqrt 15289√ = (𝑥 ∈ ℂ ↦ (℩𝑦 ∈ ℂ ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉ ℝ+)))
df-abs 15290abs = (𝑥 ∈ ℂ ↦ (√‘(𝑥 · (∗‘𝑥))))
clsp 15524class lim sup
df-limsup 15525lim sup = (𝑥 ∈ V ↦ inf(ran (𝑘 ∈ ℝ ↦ sup(((𝑥 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )), ℝ*, < ))
cli 15538class ⇝
crli 15539class ⇝𝑟
co1 15540class 𝑂(1)
clo1 15541class ≤𝑂(1)
df-clim 15542 ⇝ = {⟨𝑓, 𝑦⟩ ∣ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝑓‘𝑘) ∈ ℂ ∧ (abs‘((𝑓‘𝑘) − 𝑦)) < 𝑥))}
df-rlim 15543 ⇝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 15544𝑂(1) = {𝑓 ∈ (ℂ ↑pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(abs‘(𝑓‘𝑦)) ≤ 𝑚}
df-lo1 15545≤𝑂(1) = {𝑓 ∈ (ℝ ↑pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(𝑓‘𝑦) ≤ 𝑚}
csu 15740class Σ𝑘 ∈ 𝐴 𝐵
df-sum 15741Σ𝑘 ∈ 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 15960class ∏𝑘 ∈ 𝐴 𝐵
df-prod 15961∏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 16061class FallFac
crisefac 16062class RiseFac
df-risefac 16063 RiseFac = (𝑥 ∈ ℂ, 𝑛 ∈ ℕ0 ↦ ∏𝑘 ∈ (0...(𝑛 − 1))(𝑥 + 𝑘))
df-fallfac 16064 FallFac = (𝑥 ∈ ℂ, 𝑛 ∈ ℕ0 ↦ ∏𝑘 ∈ (0...(𝑛 − 1))(𝑥 − 𝑘))
cbp 16103class BernPoly
df-bpoly 16104 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 16118class exp
ceu 16119class e
csin 16120class sin
ccos 16121class cos
ctan 16122class tan
cpi 16123class π
df-ef 16124exp = (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ ℕ0 ((𝑥↑𝑘) / (!‘𝑘)))
df-e 16125e = (exp‘1)
df-sin 16126sin = (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) − (exp‘(-i · 𝑥))) / (2 · i)))
df-cos 16127cos = (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥))) / 2))
df-tan 16128tan = (𝑥 ∈ (◡cos “ (ℂ ∖ {0})) ↦ ((sin‘𝑥) / (cos‘𝑥)))
df-pi 16129π = inf((ℝ+ ∩ (◡sin “ {0})), ℝ, < )
ctau 16261class τ
df-tau 16262τ = inf((ℝ+ ∩ (◡cos “ {1})), ℝ, < )
cdvds 16313class ∥
df-dvds 16314 ∥ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ ∃𝑛 ∈ ℤ (𝑛 · 𝑥) = 𝑦)}
cbits 16480class bits
csad 16481class sadd
csmu 16482class smul
df-bits 16483bits = (𝑛 ∈ ℤ ↦ {𝑚 ∈ ℕ0 ∣ ¬ 2 ∥ (⌊‘(𝑛 / (2↑𝑚)))})
df-sad 16512 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 16537 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 16555class gcd
df-gcd 16556 gcd = (𝑥 ∈ ℤ, 𝑦 ∈ ℤ ↦ if((𝑥 = 0 ∧ 𝑦 = 0), 0, sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑥 ∧ 𝑛 ∥ 𝑦)}, ℝ, < )))
clcm 16649class lcm
clcmf 16650class lcm
df-lcm 16651 lcm = (𝑥 ∈ ℤ, 𝑦 ∈ ℤ ↦ if((𝑥 = 0 ∨ 𝑦 = 0), 0, inf({𝑛 ∈ ℕ ∣ (𝑥 ∥ 𝑛 ∧ 𝑦 ∥ 𝑛)}, ℝ, < )))
df-lcmf 16652lcm = (𝑧 ∈ 𝒫 ℤ ↦ if(0 ∈ 𝑧, 0, inf({𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑧 𝑚 ∥ 𝑛}, ℝ, < )))
cprime 16732class ℙ
df-prm 16733ℙ = {𝑝 ∈ ℕ ∣ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑝} ≈ 2o}
cnumer 16795class numer
cdenom 16796class denom
df-numer 16797numer = (𝑦 ∈ ℚ ↦ (1st ‘(℩𝑥 ∈ (ℤ × ℕ)(((1st ‘𝑥) gcd (2nd ‘𝑥)) = 1 ∧ 𝑦 = ((1st ‘𝑥) / (2nd ‘𝑥))))))
df-denom 16798denom = (𝑦 ∈ ℚ ↦ (2nd ‘(℩𝑥 ∈ (ℤ × ℕ)(((1st ‘𝑥) gcd (2nd ‘𝑥)) = 1 ∧ 𝑦 = ((1st ‘𝑥) / (2nd ‘𝑥))))))
codz 16825class odℤ
cphi 16826class ϕ
df-odz 16827odℤ = (𝑛 ∈ ℕ ↦ (𝑥 ∈ {𝑥 ∈ ℤ ∣ (𝑥 gcd 𝑛) = 1} ↦ inf({𝑚 ∈ ℕ ∣ 𝑛 ∥ ((𝑥↑𝑚) − 1)}, ℝ, < )))
df-phi 16828ϕ = (𝑛 ∈ ℕ ↦ (♯‘{𝑥 ∈ (1...𝑛) ∣ (𝑥 gcd 𝑛) = 1}))
cpc 16899class pCnt
df-pc 16900 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 16992class ℤ[i]
df-gz 16993ℤ[i] = {𝑥 ∈ ℂ ∣ ((ℜ‘𝑥) ∈ ℤ ∧ (ℑ‘𝑥) ∈ ℤ)}
cvdwa 17028class AP
cvdwm 17029class MonoAP
cvdwp 17030class PolyAP
df-vdwap 17031AP = (𝑘 ∈ ℕ0 ↦ (𝑎 ∈ ℕ, 𝑑 ∈ ℕ ↦ ran (𝑚 ∈ (0...(𝑘 − 1)) ↦ (𝑎 + (𝑚 · 𝑑)))))
df-vdwmc 17032 MonoAP = {⟨𝑘, 𝑓⟩ ∣ ∃𝑐(ran (AP‘𝑘) ∩ 𝒫 (◡𝑓 “ {𝑐})) ≠ ∅}
df-vdwpc 17033 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 17062class Ramsey
df-ram 17064 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 17094class #p
df-prmo 17095#p = (𝑛 ∈ ℕ0 ↦ ∏𝑘 ∈ (1...𝑛)if(𝑘 ∈ ℙ, 𝑘, 1))
cstr 17209class Struct
df-struct 17210 Struct = {⟨𝑓, 𝑥⟩ ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))}
csts 17226class sSet
df-sets 17227 sSet = (𝑠 ∈ V, 𝑒 ∈ V ↦ ((𝑠 ↾ (V ∖ dom {𝑒})) ∪ {𝑒}))
cslot 17244class Slot 𝐴
df-slot 17245Slot 𝐴 = (𝑥 ∈ V ↦ (𝑥‘𝐴))
cnx 17256class ndx
df-ndx 17257ndx = ( I ↾ ℕ)
cbs 17272class Base
df-base 17273Base = Slot 1
cress 17293class ↾s
df-ress 17294 ↾s = (𝑤 ∈ V, 𝑥 ∈ V ↦ if((Base‘𝑤) ⊆ 𝑥, 𝑤, (𝑤 sSet ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩)))
cplusg 17313class +g
cmulr 17314class .r
cstv 17315class *𝑟
csca 17316class Scalar
cvsca 17317class ·𝑠
cip 17318class ·𝑖
cts 17319class TopSet
cple 17320class le
coc 17321class oc
cds 17322class dist
cunif 17323class UnifSet
chom 17324class Hom
cco 17325class comp
df-plusg 17326+g = Slot 2
df-mulr 17327.r = Slot 3
df-starv 17328*𝑟 = Slot 4
df-sca 17329Scalar = Slot 5
df-vsca 17330 ·𝑠 = Slot 6
df-ip 17331·𝑖 = Slot 8
df-tset 17332TopSet = Slot 9
df-ple 17333le = Slot 10
df-ocomp 17334oc = Slot 11
df-ds 17335dist = Slot 12
df-unif 17336UnifSet = Slot 13
df-hom 17337Hom = Slot 14
df-cco 17338comp = Slot 15
crest 17476class ↾t
ctopn 17477class TopOpen
df-rest 17478 ↾t = (𝑗 ∈ V, 𝑥 ∈ V ↦ ran (𝑦 ∈ 𝑗 ↦ (𝑦 ∩ 𝑥)))
df-topn 17479TopOpen = (𝑤 ∈ V ↦ ((TopSet‘𝑤) ↾t (Base‘𝑤)))
ctg 17493class topGen
cpt 17494class ∏t
c0g 17495class 0g
cgsu 17496class Σg
df-0g 174970g = (𝑔 ∈ V ↦ (℩𝑒(𝑒 ∈ (Base‘𝑔) ∧ ∀𝑥 ∈ (Base‘𝑔)((𝑒(+g‘𝑔)𝑥) = 𝑥 ∧ (𝑥(+g‘𝑔)𝑒) = 𝑥))))
df-gsum 17498 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 17499topGen = (𝑥 ∈ V ↦ {𝑦 ∣ 𝑦 ⊆ ∪ (𝑥 ∩ 𝒫 𝑦)})
df-pt 17500∏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 17501class Xs
cpws 17502class ↑s
df-prds 17503Xs = (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 17505 ↑s = (𝑟 ∈ V, 𝑖 ∈ V ↦ ((Scalar‘𝑟)Xs(𝑖 × {𝑟})))
cordt 17556class ordTop
cxrs 17557class ℝ*𝑠
df-ordt 17558ordTop = (𝑟 ∈ V ↦ (topGen‘(fi‘({dom 𝑟} ∪ ran ((𝑥 ∈ dom 𝑟 ↦ {𝑦 ∈ dom 𝑟 ∣ ¬ 𝑦𝑟𝑥}) ∪ (𝑥 ∈ dom 𝑟 ↦ {𝑦 ∈ dom 𝑟 ∣ ¬ 𝑥𝑟𝑦}))))))
df-xrs 17559ℝ*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 17560class qTop
cimas 17561class “s
cqus 17562class /s
cxps 17563class ×s
df-qtop 17564 qTop = (𝑗 ∈ V, 𝑓 ∈ V ↦ {𝑠 ∈ 𝒫 (𝑓 “ ∪ 𝑗) ∣ ((◡𝑓 “ 𝑠) ∩ ∪ 𝑗) ∈ 𝑗})
df-imas 17565 “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 17566 /s = (𝑟 ∈ V, 𝑒 ∈ V ↦ ((𝑥 ∈ (Base‘𝑟) ↦ [𝑥]𝑒) “s 𝑟))
df-xps 17567 ×s = (𝑟 ∈ V, 𝑠 ∈ V ↦ (◡(𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Base‘𝑠) ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}) “s ((Scalar‘𝑟)Xs{⟨∅, 𝑟⟩, ⟨1o, 𝑠⟩})))
cmre 17637class Moore
cmrc 17638class mrCls
cmri 17639class mrInd
cacs 17640class ACS
df-mre 17641Moore = (𝑥 ∈ V ↦ {𝑐 ∈ 𝒫 𝒫 𝑥 ∣ (𝑥 ∈ 𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝑐))})
df-mrc 17642mrCls = (𝑐 ∈ ∪ ran Moore ↦ (𝑥 ∈ 𝒫 ∪ 𝑐 ↦ ∩ {𝑠 ∈ 𝑐 ∣ 𝑥 ⊆ 𝑠}))
df-mri 17643mrInd = (𝑐 ∈ ∪ ran Moore ↦ {𝑠 ∈ 𝒫 ∪ 𝑐 ∣ ∀𝑥 ∈ 𝑠 ¬ 𝑥 ∈ ((mrCls‘𝑐)‘(𝑠 ∖ {𝑥}))})
df-acs 17644ACS = (𝑥 ∈ V ↦ {𝑐 ∈ (Moore‘𝑥) ∣ ∃𝑓(𝑓:𝒫 𝑥⟶𝒫 𝑥 ∧ ∀𝑠 ∈ 𝒫 𝑥(𝑠 ∈ 𝑐 ↔ ∪ (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠))})
ccat 17723class Cat
ccid 17724class Id
chomf 17725class Homf
ccomf 17726class compf
df-cat 17727Cat = {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 17728Id = (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 17729Homf = (𝑐 ∈ V ↦ (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ (𝑥(Hom ‘𝑐)𝑦)))
df-comf 17730compf = (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 17770class oppCat
df-oppc 17771oppCat = (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 17788class Mono
cepi 17789class Epi
df-mon 17790Mono = (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 17791Epi = (𝑐 ∈ Cat ↦ tpos (Mono‘(oppCat‘𝑐)))
csect 17804class Sect
cinv 17805class Inv
ciso 17806class Iso
df-sect 17807Sect = (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 17808Inv = (𝑐 ∈ Cat ↦ (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ ((𝑥(Sect‘𝑐)𝑦) ∩ ◡(𝑦(Sect‘𝑐)𝑥))))
df-iso 17809Iso = (𝑐 ∈ Cat ↦ ((𝑥 ∈ V ↦ dom 𝑥) ∘ (Inv‘𝑐)))
ccic 17855class ≃𝑐
df-cic 17856 ≃𝑐 = (𝑐 ∈ Cat ↦ ((Iso‘𝑐) supp ∅))
cssc 17867class ⊆cat
cresc 17868class ↾cat
csubc 17869class Subcat
df-ssc 17870 ⊆cat = {⟨ℎ, 𝑗⟩ ∣ ∃𝑡(𝑗 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝑗‘𝑥))}
df-resc 17871 ↾cat = (𝑐 ∈ V, ℎ ∈ V ↦ ((𝑐 ↾s dom dom ℎ) sSet ⟨(Hom ‘ndx), ℎ⟩))
df-subc 17872Subcat = (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 17914class Func
cidfu 17915class idfunc
ccofu 17916class ∘func
cresf 17917class ↾f
df-func 17918 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 17919idfunc = (𝑡 ∈ Cat ↦ ⦋(Base‘𝑡) / 𝑏⦌⟨( I ↾ 𝑏), (𝑧 ∈ (𝑏 × 𝑏) ↦ ( I ↾ ((Hom ‘𝑡)‘𝑧)))⟩)
df-cofu 17920 ∘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 17921 ↾f = (𝑓 ∈ V, ℎ ∈ V ↦ ⟨((1st ‘𝑓) ↾ dom dom ℎ), (𝑥 ∈ dom ℎ ↦ (((2nd ‘𝑓)‘𝑥) ↾ (ℎ‘𝑥)))⟩)
cful 17964class Full
cfth 17965class Faith
df-full 17966 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 17967 Faith = (𝑐 ∈ Cat, 𝑑 ∈ Cat ↦ {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝑐 Func 𝑑)𝑔 ∧ ∀𝑥 ∈ (Base‘𝑐)∀𝑦 ∈ (Base‘𝑐)Fun ◡(𝑥𝑔𝑦))})
cnat 18004class Nat
cfuc 18005class FuncCat
df-nat 18006 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 18007 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 18041class InitO
ctermo 18042class TermO
czeroo 18043class ZeroO
df-inito 18044InitO = (𝑐 ∈ Cat ↦ {𝑎 ∈ (Base‘𝑐) ∣ ∀𝑏 ∈ (Base‘𝑐)∃!ℎ ℎ ∈ (𝑎(Hom ‘𝑐)𝑏)})
df-termo 18045TermO = (𝑐 ∈ Cat ↦ {𝑎 ∈ (Base‘𝑐) ∣ ∀𝑏 ∈ (Base‘𝑐)∃!ℎ ℎ ∈ (𝑏(Hom ‘𝑐)𝑎)})
df-zeroo 18046ZeroO = (𝑐 ∈ Cat ↦ ((InitO‘𝑐) ∩ (TermO‘𝑐)))
cdoma 18080class doma
ccoda 18081class coda
carw 18082class Arrow
choma 18083class Homa
df-doma 18084doma = (1st ∘ 1st )
df-coda 18085coda = (2nd ∘ 1st )
df-homa 18086Homa = (𝑐 ∈ Cat ↦ (𝑥 ∈ ((Base‘𝑐) × (Base‘𝑐)) ↦ ({𝑥} × ((Hom ‘𝑐)‘𝑥))))
df-arw 18087Arrow = (𝑐 ∈ Cat ↦ ∪ ran (Homa‘𝑐))
cida 18113class Ida
ccoa 18114class compa
df-ida 18115Ida = (𝑐 ∈ Cat ↦ (𝑥 ∈ (Base‘𝑐) ↦ ⟨𝑥, 𝑥, ((Id‘𝑐)‘𝑥)⟩))
df-coa 18116compa = (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 18135class SetCat
df-setc 18136SetCat = (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 18158class CatCat
df-catc 18159CatCat = (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 18181class ExtStrCat
df-estrc 18182ExtStrCat = (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 18227class ×c
c1stf 18228class 1stF
c2ndf 18229class 2ndF
cprf 18230class ⟨,⟩F
df-xpc 18231 ×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 18232 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 18233 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 18234 ⟨,⟩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 18268class evalF
ccurf 18269class curryF
cuncf 18270class uncurryF
cdiag 18271class Δfunc
df-evlf 18272 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 18273 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 18274 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 18275Δfunc = (𝑐 ∈ Cat, 𝑑 ∈ Cat ↦ (⟨𝑐, 𝑑⟩ curryF (𝑐 1stF 𝑑)))
chof 18307class HomF
cyon 18308class Yon
df-hof 18309HomF = (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 18310Yon = (𝑐 ∈ Cat ↦ (⟨𝑐, (oppCat‘𝑐)⟩ curryF (HomF‘(oppCat‘𝑐))))
codu 18345class ODual
df-odu 18346ODual = (𝑤 ∈ V ↦ (𝑤 sSet ⟨(le‘ndx), ◡(le‘𝑤)⟩))
cproset 18351class Proset
cdrs 18352class Dirset
df-proset 18353 Proset = {𝑓 ∣ [(Base‘𝑓) / 𝑏][(le‘𝑓) / 𝑟]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) → 𝑥𝑟𝑧))}
df-drs 18354Dirset = {𝑓 ∈ Proset ∣ [(Base‘𝑓) / 𝑏][(le‘𝑓) / 𝑟](𝑏 ≠ ∅ ∧ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∃𝑧 ∈ 𝑏 (𝑥𝑟𝑧 ∧ 𝑦𝑟𝑧))}
cpo 18366class Poset
cplt 18367class lt
club 18368class lub
cglb 18369class glb
cjn 18370class join
cmee 18371class meet
df-poset 18372Poset = {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 18387lt = (𝑝 ∈ V ↦ ((le‘𝑝) ∖ I ))
df-lub 18403lub = (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 18404glb = (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 18405join = (𝑝 ∈ V ↦ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦} (lub‘𝑝)𝑧})
df-meet 18406meet = (𝑝 ∈ V ↦ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦} (glb‘𝑝)𝑧})
ctos 18473class Toset
df-toset 18474Toset = {𝑓 ∈ Poset ∣ [(Base‘𝑓) / 𝑏][(le‘𝑓) / 𝑟]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥𝑟𝑦 ∨ 𝑦𝑟𝑥)}
cp0 18480class 0.
cp1 18481class 1.
df-p0 184820. = (𝑝 ∈ V ↦ ((glb‘𝑝)‘(Base‘𝑝)))
df-p1 184831. = (𝑝 ∈ V ↦ ((lub‘𝑝)‘(Base‘𝑝)))
clat 18490class Lat
df-lat 18491Lat = {𝑝 ∈ Poset ∣ (dom (join‘𝑝) = ((Base‘𝑝) × (Base‘𝑝)) ∧ dom (meet‘𝑝) = ((Base‘𝑝) × (Base‘𝑝)))}
ccla 18557class CLat
df-clat 18558CLat = {𝑝 ∈ Poset ∣ (dom (lub‘𝑝) = 𝒫 (Base‘𝑝) ∧ dom (glb‘𝑝) = 𝒫 (Base‘𝑝))}
cdlat 18579class DLat
df-dlat 18580DLat = {𝑘 ∈ Lat ∣ [(Base‘𝑘) / 𝑏][(join‘𝑘) / 𝑗][(meet‘𝑘) / 𝑚]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 (𝑥𝑚(𝑦𝑗𝑧)) = ((𝑥𝑚𝑦)𝑗(𝑥𝑚𝑧))}
cipo 18586class toInc
df-ipo 18587toInc = (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 18623class PosetRel
ctsr 18624class TosetRel
df-ps 18625PosetRel = {𝑟 ∣ (Rel 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟 ∧ (𝑟 ∩ ◡𝑟) = ( I ↾ ∪ ∪ 𝑟))}
df-tsr 18626 TosetRel = {𝑟 ∈ PosetRel ∣ (dom 𝑟 × dom 𝑟) ⊆ (𝑟 ∪ ◡𝑟)}
cdir 18653class DirRel
ctail 18654class tail
df-dir 18655DirRel = {𝑟 ∣ ((Rel 𝑟 ∧ ( I ↾ ∪ ∪ 𝑟) ⊆ 𝑟) ∧ ((𝑟 ∘ 𝑟) ⊆ 𝑟 ∧ (∪ ∪ 𝑟 × ∪ ∪ 𝑟) ⊆ (◡𝑟 ∘ 𝑟)))}
df-tail 18656tail = (𝑟 ∈ DirRel ↦ (𝑥 ∈ ∪ ∪ 𝑟 ↦ (𝑟 “ {𝑥})))
cchn 18664class ( < Chain 𝐴)
df-chn 18665( < Chain 𝐴) = {𝑐 ∈ Word 𝐴 ∣ ∀𝑛 ∈ (dom 𝑐 ∖ {0})(𝑐‘(𝑛 − 1)) < (𝑐‘𝑛)}
cplusf 18698class +𝑓
cmgm 18699class Mgm
df-plusf 18700+𝑓 = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(+g‘𝑔)𝑦)))
df-mgm 18701Mgm = {𝑔 ∣ [(Base‘𝑔) / 𝑏][(+g‘𝑔) / 𝑜]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥𝑜𝑦) ∈ 𝑏}
cmgmhm 18751class MgmHom
csubmgm 18752class SubMgm
df-mgmhm 18753 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 18754SubMgm = (𝑠 ∈ Mgm ↦ {𝑡 ∈ 𝒫 (Base‘𝑠) ∣ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (𝑥(+g‘𝑠)𝑦) ∈ 𝑡})
csgrp 18779class Smgrp
df-sgrp 18780Smgrp = {𝑔 ∈ Mgm ∣ [(Base‘𝑔) / 𝑏][(+g‘𝑔) / 𝑜]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧))}
cmnd 18795class Mnd
df-mnd 18796Mnd = {𝑔 ∈ Smgrp ∣ [(Base‘𝑔) / 𝑏][(+g‘𝑔) / 𝑝]∃𝑒 ∈ 𝑏 ∀𝑥 ∈ 𝑏 ((𝑒𝑝𝑥) = 𝑥 ∧ (𝑥𝑝𝑒) = 𝑥)}
cmhm 18842class MndHom
csubmnd 18843class SubMnd
df-mhm 18844 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 18845SubMnd = (𝑠 ∈ Mnd ↦ {𝑡 ∈ 𝒫 (Base‘𝑠) ∣ ((0g‘𝑠) ∈ 𝑡 ∧ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (𝑥(+g‘𝑠)𝑦) ∈ 𝑡)})
cfrmd 18909class freeMnd
cvrmd 18910class varFMnd
df-frmd 18911freeMnd = (𝑖 ∈ V ↦ {⟨(Base‘ndx), Word 𝑖⟩, ⟨(+g‘ndx), ( ++ ↾ (Word 𝑖 × Word 𝑖))⟩})
df-vrmd 18912varFMnd = (𝑖 ∈ V ↦ (𝑗 ∈ 𝑖 ↦ ⟨“𝑗”⟩))
cefmnd 18930class EndoFMnd
df-efmnd 18931EndoFMnd = (𝑥 ∈ V ↦ ⦋(𝑥 ↑m 𝑥) / 𝑏⦌{⟨(Base‘ndx), 𝑏⟩, ⟨(+g‘ndx), (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (𝑓 ∘ 𝑔))⟩, ⟨(TopSet‘ndx), (∏t‘(𝑥 × {𝒫 𝑥}))⟩})
cgrp 19003class Grp
cminusg 19004class invg
csg 19005class -g
df-grp 19006Grp = {𝑔 ∈ Mnd ∣ ∀𝑎 ∈ (Base‘𝑔)∃𝑚 ∈ (Base‘𝑔)(𝑚(+g‘𝑔)𝑎) = (0g‘𝑔)}
df-minusg 19007invg = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔) ↦ (℩𝑤 ∈ (Base‘𝑔)(𝑤(+g‘𝑔)𝑥) = (0g‘𝑔))))
df-sbg 19008-g = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(+g‘𝑔)((invg‘𝑔)‘𝑦))))
cmg 19136class .g
df-mulg 19137.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 19189class SubGrp
cnsg 19190class NrmSGrp
cqg 19191class ~QG
df-subg 19192SubGrp = (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp})
df-nsg 19193NrmSGrp = (𝑤 ∈ Grp ↦ {𝑠 ∈ (SubGrp‘𝑤) ∣ [(Base‘𝑤) / 𝑏][(+g‘𝑤) / 𝑝]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ((𝑥𝑝𝑦) ∈ 𝑠 ↔ (𝑦𝑝𝑥) ∈ 𝑠)})
df-eqg 19194 ~QG = (𝑟 ∈ V, 𝑖 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (Base‘𝑟) ∧ (((invg‘𝑟)‘𝑥)(+g‘𝑟)𝑦) ∈ 𝑖)})
cghm 19286class GrpHom
df-ghm 19287 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 19330class GrpIso
cgic 19331class ≃𝑔
df-gim 19332 GrpIso = (𝑠 ∈ Grp, 𝑡 ∈ Grp ↦ {𝑔 ∈ (𝑠 GrpHom 𝑡) ∣ 𝑔:(Base‘𝑠)–1-1-onto→(Base‘𝑡)})
df-gic 19333 ≃𝑔 = (◡ GrpIso “ (V ∖ 1o))
cga 19362class GrpAct
df-ga 19363 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 19388class Cntz
ccntr 19389class Cntr
df-cntz 19390Cntz = (𝑚 ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘𝑚) ↦ {𝑥 ∈ (Base‘𝑚) ∣ ∀𝑦 ∈ 𝑠 (𝑥(+g‘𝑚)𝑦) = (𝑦(+g‘𝑚)𝑥)}))
df-cntr 19391Cntr = (𝑚 ∈ V ↦ ((Cntz‘𝑚)‘(Base‘𝑚)))
coppg 19418class oppg
df-oppg 19419oppg = (𝑤 ∈ V ↦ (𝑤 sSet ⟨(+g‘ndx), tpos (+g‘𝑤)⟩))
csymg 19442class SymGrp
df-symg 19443SymGrp = (𝑥 ∈ V ↦ ((EndoFMnd‘𝑥) ↾s {ℎ ∣ ℎ:𝑥–1-1-onto→𝑥}))
cpmtr 19514class pmTrsp
df-pmtr 19515pmTrsp = (𝑑 ∈ V ↦ (𝑝 ∈ {𝑦 ∈ 𝒫 𝑑 ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ 𝑑 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))))
cpsgn 19562class pmSgn
cevpm 19563class pmEven
df-psgn 19564pmSgn = (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 19565pmEven = (𝑑 ∈ V ↦ (◡(pmSgn‘𝑑) “ {1}))
cod 19597class od
cgex 19598class gEx
cpgp 19599class pGrp
cslw 19600class pSyl
df-od 19601od = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔) ↦ ⦋{𝑛 ∈ ℕ ∣ (𝑛(.g‘𝑔)𝑥) = (0g‘𝑔)} / 𝑖⦌if(𝑖 = ∅, 0, inf(𝑖, ℝ, < ))))
df-gex 19602gEx = (𝑔 ∈ V ↦ ⦋{𝑛 ∈ ℕ ∣ ∀𝑥 ∈ (Base‘𝑔)(𝑛(.g‘𝑔)𝑥) = (0g‘𝑔)} / 𝑖⦌if(𝑖 = ∅, 0, inf(𝑖, ℝ, < )))
df-pgp 19603 pGrp = {⟨𝑝, 𝑔⟩ ∣ ((𝑝 ∈ ℙ ∧ 𝑔 ∈ Grp) ∧ ∀𝑥 ∈ (Base‘𝑔)∃𝑛 ∈ ℕ0 ((od‘𝑔)‘𝑥) = (𝑝↑𝑛))}
df-slw 19604 pSyl = (𝑝 ∈ ℙ, 𝑔 ∈ Grp ↦ {ℎ ∈ (SubGrp‘𝑔) ∣ ∀𝑘 ∈ (SubGrp‘𝑔)((ℎ ⊆ 𝑘 ∧ 𝑝 pGrp (𝑔 ↾s 𝑘)) ↔ ℎ = 𝑘)})
clsm 19707class LSSum
cpj1 19708class proj1
df-lsm 19709LSSum = (𝑤 ∈ V ↦ (𝑡 ∈ 𝒫 (Base‘𝑤), 𝑢 ∈ 𝒫 (Base‘𝑤) ↦ ran (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥(+g‘𝑤)𝑦))))
df-pj1 19710proj1 = (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 19779class ~FG
cfrgp 19780class freeGrp
cvrgp 19781class varFGrp
df-efg 19782 ~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 19783freeGrp = (𝑖 ∈ V ↦ ((freeMnd‘(𝑖 × 2o)) /s ( ~FG ‘𝑖)))
df-vrgp 19784varFGrp = (𝑖 ∈ V ↦ (𝑗 ∈ 𝑖 ↦ [⟨“⟨𝑗, ∅⟩”⟩]( ~FG ‘𝑖)))
ccmn 19853class CMnd
cabl 19854class Abel
df-cmn 19855CMnd = {𝑔 ∈ Mnd ∣ ∀𝑎 ∈ (Base‘𝑔)∀𝑏 ∈ (Base‘𝑔)(𝑎(+g‘𝑔)𝑏) = (𝑏(+g‘𝑔)𝑎)}
df-abl 19856Abel = (Grp ∩ CMnd)
ccyg 19950class CycGrp
df-cyg 19951CycGrp = {𝑔 ∈ Grp ∣ ∃𝑥 ∈ (Base‘𝑔)ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝑔)𝑥)) = (Base‘𝑔)}
cdprd 20068class DProd
cdpj 20069class dProj
df-dprd 20070 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 20071dProj = (𝑔 ∈ Grp, 𝑠 ∈ (dom DProd “ {𝑔}) ↦ (𝑖 ∈ dom 𝑠 ↦ ((𝑠‘𝑖)(proj1‘𝑔)(𝑔 DProd (𝑠 ↾ (dom 𝑠 ∖ {𝑖}))))))
csimpg 20165class SimpGrp
df-simpg 20166SimpGrp = {𝑔 ∈ Grp ∣ (NrmSGrp‘𝑔) ≈ 2o}
comnd 20192class oMnd
cogrp 20193class oGrp
df-omnd 20194oMnd = {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 20195oGrp = (Grp ∩ oMnd)
cmgp 20219class mulGrp
df-mgp 20220mulGrp = (𝑤 ∈ V ↦ (𝑤 sSet ⟨(+g‘ndx), (.r‘𝑤)⟩))
crng 20233class Rng
df-rng 20234Rng = {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 20266class 1r
df-ur 202671r = (0g ∘ mulGrp)
csrg 20271class SRing
df-srg 20272SRing = {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 20318class Ring
ccrg 20319class CRing
df-ring 20320Ring = {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 20321CRing = {𝑓 ∈ Ring ∣ (mulGrp‘𝑓) ∈ CMnd}
coppr 20421class oppr
df-oppr 20422oppr = (𝑓 ∈ V ↦ (𝑓 sSet ⟨(.r‘ndx), tpos (.r‘𝑓)⟩))
cdsr 20439class ∥r
cui 20440class Unit
cir 20441class Irred
df-dvdsr 20442∥r = (𝑤 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑤) ∧ ∃𝑧 ∈ (Base‘𝑤)(𝑧(.r‘𝑤)𝑥) = 𝑦)})
df-unit 20443Unit = (𝑤 ∈ V ↦ (◡((∥r‘𝑤) ∩ (∥r‘(oppr‘𝑤))) “ {(1r‘𝑤)}))
df-irred 20444Irred = (𝑤 ∈ V ↦ ⦋((Base‘𝑤) ∖ (Unit‘𝑤)) / 𝑏⦌{𝑧 ∈ 𝑏 ∣ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥(.r‘𝑤)𝑦) ≠ 𝑧})
cinvr 20472class invr
df-invr 20473invr = (𝑟 ∈ V ↦ (invg‘((mulGrp‘𝑟) ↾s (Unit‘𝑟))))
cdvr 20485class /r
df-dvr 20486/r = (𝑟 ∈ V ↦ (𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Unit‘𝑟) ↦ (𝑥(.r‘𝑟)((invr‘𝑟)‘𝑦))))
crpm 20517class RPrime
df-rprm 20518RPrime = (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 20519class RngHom
crngim 20520class RngIso
df-rnghm 20521 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 20522 RngIso = (𝑟 ∈ V, 𝑠 ∈ V ↦ {𝑓 ∈ (𝑟 RngHom 𝑠) ∣ ◡𝑓 ∈ (𝑠 RngHom 𝑟)})
crh 20554class RingHom
crs 20555class RingIso
cric 20556class ≃𝑟
df-rhm 20557 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 20558 RingIso = (𝑟 ∈ V, 𝑠 ∈ V ↦ {𝑓 ∈ (𝑟 RingHom 𝑠) ∣ ◡𝑓 ∈ (𝑠 RingHom 𝑟)})
df-ric 20560 ≃𝑟 = (◡ RingIso “ (V ∖ 1o))
cnzr 20598class NzRing
df-nzr 20599NzRing = {𝑟 ∈ Ring ∣ (1r‘𝑟) ≠ (0g‘𝑟)}
clring 20626class LRing
df-lring 20627LRing = {𝑟 ∈ NzRing ∣ ∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)((𝑥(+g‘𝑟)𝑦) = (1r‘𝑟) → (𝑥 ∈ (Unit‘𝑟) ∨ 𝑦 ∈ (Unit‘𝑟)))}
csubrng 20633class SubRng
df-subrng 20634SubRng = (𝑤 ∈ Rng ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Rng})
csubrg 20657class SubRing
df-subrg 20658SubRing = (𝑤 ∈ Ring ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ ((𝑤 ↾s 𝑠) ∈ Ring ∧ (1r‘𝑤) ∈ 𝑠)})
crgspn 20698class RingSpan
df-rgspn 20699RingSpan = (𝑤 ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘𝑤) ↦ ∩ {𝑡 ∈ (SubRing‘𝑤) ∣ 𝑠 ⊆ 𝑡}))
crngc 20704class RngCat
df-rngc 20705RngCat = (𝑢 ∈ V ↦ ((ExtStrCat‘𝑢) ↾cat ( RngHom ↾ ((𝑢 ∩ Rng) × (𝑢 ∩ Rng)))))
cringc 20733class RingCat
df-ringc 20734RingCat = (𝑢 ∈ V ↦ ((ExtStrCat‘𝑢) ↾cat ( RingHom ↾ ((𝑢 ∩ Ring) × (𝑢 ∩ Ring)))))
crlreg 20779class RLReg
cdomn 20780class Domn
cidom 20781class IDomn
df-rlreg 20782RLReg = (𝑟 ∈ V ↦ {𝑥 ∈ (Base‘𝑟) ∣ ∀𝑦 ∈ (Base‘𝑟)((𝑥(.r‘𝑟)𝑦) = (0g‘𝑟) → 𝑦 = (0g‘𝑟))})
df-domn 20783Domn = {𝑟 ∈ NzRing ∣ [(Base‘𝑟) / 𝑏][(0g‘𝑟) / 𝑧]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ((𝑥(.r‘𝑟)𝑦) = 𝑧 → (𝑥 = 𝑧 ∨ 𝑦 = 𝑧))}
df-idom 20784IDomn = (CRing ∩ Domn)
cdr 20816class DivRing
cfield 20817class Field
df-drng 20818DivRing = {𝑟 ∈ Ring ∣ (Unit‘𝑟) = ((Base‘𝑟) ∖ {(0g‘𝑟)})}
df-field 20819Field = (DivRing ∩ CRing)
csdrg 20872class SubDRing
df-sdrg 20873SubDRing = (𝑤 ∈ DivRing ↦ {𝑠 ∈ (SubRing‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ DivRing})
cabv 20894class AbsVal
df-abv 20895AbsVal = (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 20923class *rf
csr 20924class *-Ring
df-staf 20925*rf = (𝑓 ∈ V ↦ (𝑥 ∈ (Base‘𝑓) ↦ ((*𝑟‘𝑓)‘𝑥)))
df-srng 20926*-Ring = {𝑓 ∣ [(*rf‘𝑓) / 𝑖](𝑖 ∈ (𝑓 RingHom (oppr‘𝑓)) ∧ 𝑖 = ◡𝑖)}
corng 20943class oRing
cofld 20944class oField
df-orng 20945oRing = {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 20946oField = (Field ∩ oRing)
clmod 20964class LMod
cscaf 20965class ·sf
df-lmod 20966LMod = {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 20967 ·sf = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘(Scalar‘𝑔)), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥( ·𝑠 ‘𝑔)𝑦)))
clss 21035class LSubSp
df-lss 21036LSubSp = (𝑤 ∈ V ↦ {𝑠 ∈ (𝒫 (Base‘𝑤) ∖ {∅}) ∣ ∀𝑥 ∈ (Base‘(Scalar‘𝑤))∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑥( ·𝑠 ‘𝑤)𝑎)(+g‘𝑤)𝑏) ∈ 𝑠})
clspn 21075class LSpan
df-lsp 21076LSpan = (𝑤 ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘𝑤) ↦ ∩ {𝑡 ∈ (LSubSp‘𝑤) ∣ 𝑠 ⊆ 𝑡}))
clmhm 21123class LMHom
clmim 21124class LMIso
clmic 21125class ≃𝑚
df-lmhm 21126 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 21127 LMIso = (𝑠 ∈ LMod, 𝑡 ∈ LMod ↦ {𝑔 ∈ (𝑠 LMHom 𝑡) ∣ 𝑔:(Base‘𝑠)–1-1-onto→(Base‘𝑡)})
df-lmic 21128 ≃𝑚 = (◡ LMIso “ (V ∖ 1o))
clbs 21178class LBasis
df-lbs 21179LBasis = (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 21206class LVec
df-lvec 21207LVec = {𝑓 ∈ LMod ∣ (Scalar‘𝑓) ∈ DivRing}
csra 21275class subringAlg
crglmod 21276class ringLMod
df-sra 21277subringAlg = (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 21278ringLMod = (𝑤 ∈ V ↦ ((subringAlg ‘𝑤)‘(Base‘𝑤)))
clidl 21313class LIdeal
crsp 21314class RSpan
df-lidl 21315LIdeal = (LSubSp ∘ ringLMod)
df-rsp 21316RSpan = (LSpan ∘ ringLMod)
c2idl 21371class 2Ideal
df-2idl 213722Ideal = (𝑟 ∈ V ↦ ((LIdeal‘𝑟) ∩ (LIdeal‘(oppr‘𝑟))))
cprmidl 21443class PrmIdeal
df-prmidl 21444PrmIdeal = (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)))})
clpidl 21471class LPIdeal
clpir 21472class LPIR
df-lpidl 21473LPIdeal = (𝑤 ∈ Ring ↦ ∪ 𝑔 ∈ (Base‘𝑤){((RSpan‘𝑤)‘{𝑔})})
df-lpir 21474LPIR = {𝑤 ∈ Ring ∣ (LIdeal‘𝑤) = (LPIdeal‘𝑤)}
cpid 21487class PID
df-pid 21488PID = (IDomn ∩ LPIR)
cpsmet 21489class PsMet
cxmet 21490class ∞Met
cmet 21491class Met
cbl 21492class ball
cfbas 21493class fBas
cfg 21494class filGen
cmopn 21495class MetOpen
cmetu 21496class metUnif
df-psmet 21497PsMet = (𝑥 ∈ V ↦ {𝑑 ∈ (ℝ* ↑m (𝑥 × 𝑥)) ∣ ∀𝑦 ∈ 𝑥 ((𝑦𝑑𝑦) = 0 ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) +𝑒 (𝑤𝑑𝑧)))})
df-xmet 21498∞Met = (𝑥 ∈ V ↦ {𝑑 ∈ (ℝ* ↑m (𝑥 × 𝑥)) ∣ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (((𝑦𝑑𝑧) = 0 ↔ 𝑦 = 𝑧) ∧ ∀𝑤 ∈ 𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) +𝑒 (𝑤𝑑𝑧)))})
df-met 21499Met = (𝑥 ∈ V ↦ {𝑑 ∈ (ℝ ↑m (𝑥 × 𝑥)) ∣ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (((𝑦𝑑𝑧) = 0 ↔ 𝑦 = 𝑧) ∧ ∀𝑤 ∈ 𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) + (𝑤𝑑𝑧)))})
df-bl 21500ball = (𝑑 ∈ V ↦ (𝑥 ∈ dom dom 𝑑, 𝑧 ∈ ℝ* ↦ {𝑦 ∈ dom dom 𝑑 ∣ (𝑥𝑑𝑦) < 𝑧}))
df-mopn 21501MetOpen = (𝑑 ∈ ∪ ran ∞Met ↦ (topGen‘ran (ball‘𝑑)))
df-fbas 21502fBas = (𝑤 ∈ V ↦ {𝑥 ∈ 𝒫 𝒫 𝑤 ∣ (𝑥 ≠ ∅ ∧ ∅ ∉ 𝑥 ∧ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑥 ∩ 𝒫 (𝑦 ∩ 𝑧)) ≠ ∅)})
df-fg 21503filGen = (𝑤 ∈ V, 𝑥 ∈ (fBas‘𝑤) ↦ {𝑦 ∈ 𝒫 𝑤 ∣ (𝑥 ∩ 𝒫 𝑦) ≠ ∅})
df-metu 21504metUnif = (𝑑 ∈ ∪ ran PsMet ↦ ((dom dom 𝑑 × dom dom 𝑑)filGenran (𝑎 ∈ ℝ+ ↦ (◡𝑑 “ (0[,)𝑎)))))
ccnfld 21505class ℂfld
df-cnfld 21506ℂfld = (({⟨(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 21579class ℤring
df-zring 21580ℤring = (ℂfld ↾s ℤ)
czrh 21632class ℤRHom
czlm 21633class ℤMod
cchr 21634class chr
czn 21635class ℤ/nℤ
df-zrh 21636ℤRHom = (𝑟 ∈ V ↦ ∪ (ℤring RingHom 𝑟))
df-zlm 21637ℤMod = (𝑔 ∈ V ↦ ((𝑔 sSet ⟨(Scalar‘ndx), ℤring⟩) sSet ⟨( ·𝑠 ‘ndx), (.g‘𝑔)⟩))
df-chr 21638chr = (𝑔 ∈ V ↦ ((od‘𝑔)‘(1r‘𝑔)))
df-zn 21639ℤ/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 21737class ℝfld
df-refld 21738ℝfld = (ℂfld ↾s ℝ)
cphl 21757class PreHil
cipf 21758class ·if
df-phl 21759PreHil = {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 21760·if = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(·𝑖‘𝑔)𝑦)))
cocv 21793class ocv
ccss 21794class ClSubSp
cthl 21795class toHL
df-ocv 21796ocv = (ℎ ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘ℎ) ↦ {𝑥 ∈ (Base‘ℎ) ∣ ∀𝑦 ∈ 𝑠 (𝑥(·𝑖‘ℎ)𝑦) = (0g‘(Scalar‘ℎ))}))
df-css 21797ClSubSp = (ℎ ∈ V ↦ {𝑠 ∣ 𝑠 = ((ocv‘ℎ)‘((ocv‘ℎ)‘𝑠))})
df-thl 21798toHL = (ℎ ∈ V ↦ ((toInc‘(ClSubSp‘ℎ)) sSet ⟨(oc‘ndx), (ocv‘ℎ)⟩))
cpj 21833class proj
chil 21834class Hil
cobs 21835class OBasis
df-pj 21836proj = (ℎ ∈ V ↦ ((𝑥 ∈ (LSubSp‘ℎ) ↦ (𝑥(proj1‘ℎ)((ocv‘ℎ)‘𝑥))) ∩ (V × ((Base‘ℎ) ↑m (Base‘ℎ)))))
df-hil 21837Hil = {ℎ ∈ PreHil ∣ dom (proj‘ℎ) = (ClSubSp‘ℎ)}
df-obs 21838OBasis = (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 21864class ⊕m
df-dsmm 21865 ⊕m = (𝑠 ∈ V, 𝑟 ∈ V ↦ ((𝑠Xs𝑟) ↾s {𝑓 ∈ X𝑥 ∈ dom 𝑟(Base‘(𝑟‘𝑥)) ∣ {𝑥 ∈ dom 𝑟 ∣ (𝑓‘𝑥) ≠ (0g‘(𝑟‘𝑥))} ∈ Fin}))
cfrlm 21879class freeLMod
df-frlm 21880 freeLMod = (𝑟 ∈ V, 𝑖 ∈ V ↦ (𝑟 ⊕m (𝑖 × {(ringLMod‘𝑟)})))
cuvc 21915class unitVec
df-uvc 21916 unitVec = (𝑟 ∈ V, 𝑖 ∈ V ↦ (𝑗 ∈ 𝑖 ↦ (𝑘 ∈ 𝑖 ↦ if(𝑘 = 𝑗, (1r‘𝑟), (0g‘𝑟)))))
clindf 21937class LIndF
clinds 21938class LIndS
df-lindf 21939 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 21940LIndS = (𝑤 ∈ V ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ ( I ↾ 𝑠) LIndF 𝑤})
casa 21983class AssAlg
casp 21984class AlgSpan
cascl 21985class algSc
df-assa 21986AssAlg = {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 21987AlgSpan = (𝑤 ∈ AssAlg ↦ (𝑠 ∈ 𝒫 (Base‘𝑤) ↦ ∩ {𝑡 ∈ ((SubRing‘𝑤) ∩ (LSubSp‘𝑤)) ∣ 𝑠 ⊆ 𝑡}))
df-ascl 21988algSc = (𝑤 ∈ V ↦ (𝑥 ∈ (Base‘(Scalar‘𝑤)) ↦ (𝑥( ·𝑠 ‘𝑤)(1r‘𝑤))))
cmps 22037class mPwSer
cmvr 22038class mVar
cmpl 22039class mPoly
cltb 22040class <bag
copws 22041class ordPwSer
df-psr 22042 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 22043 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 22044 mPoly = (𝑖 ∈ V, 𝑟 ∈ V ↦ ⦋(𝑖 mPwSer 𝑟) / 𝑤⦌(𝑤 ↾s {𝑓 ∈ (Base‘𝑤) ∣ 𝑓 finSupp (0g‘𝑟)}))
df-ltbag 22045 <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 22046 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 22206class evalSub
cevl 22207class eval
df-evls 22208 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 22209 eval = (𝑖 ∈ V, 𝑟 ∈ V ↦ ((𝑖 evalSub 𝑟)‘(Base‘𝑟)))
cslv 22250class selectVars
df-selv 22251 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))))))))
cmhp 22279class mHomP
cpsd 22280class mPSDer
cai 22281class AlgInd
df-mhp 22282 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 22302 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 22317 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 22318class PwSer1
cv1 22319class var1
cpl1 22320class Poly1
cco1 22321class coe1
ctp1 22322class toPoly1
df-psr1 22323PwSer1 = (𝑟 ∈ V ↦ ((1o ordPwSer 𝑟)‘∅))
df-vr1 22324var1 = (𝑟 ∈ V ↦ ((1o mVar 𝑟)‘∅))
df-ply1 22325Poly1 = (𝑟 ∈ V ↦ ((PwSer1‘𝑟) ↾s (Base‘(1o mPoly 𝑟))))
df-coe1 22326coe1 = (𝑓 ∈ V ↦ (𝑛 ∈ ℕ0 ↦ (𝑓‘(1o × {𝑛}))))
df-toply1 22327toPoly1 = (𝑓 ∈ V ↦ (𝑛 ∈ (ℕ0 ↑m 1o) ↦ (𝑓‘(𝑛‘∅))))
ces1 22456class evalSub1
ce1 22457class eval1
df-evls1 22458 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 22459eval1 = (𝑟 ∈ V ↦ ⦋(Base‘𝑟) / 𝑏⦌((𝑥 ∈ (𝑏 ↑m (𝑏 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝑏 ↦ (1o × {𝑦})))) ∘ (1o eval 𝑟)))
cmmul 22530class maMul
df-mamu 22531 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 22547class Mat
df-mat 22548 Mat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ ((𝑟 freeLMod (𝑛 × 𝑛)) sSet ⟨(.r‘ndx), (𝑟 maMul ⟨𝑛, 𝑛, 𝑛⟩)⟩))
cdmat 22628class DMat
cscmat 22629class ScMat
df-dmat 22630 DMat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ {𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ∣ ∀𝑖 ∈ 𝑛 ∀𝑗 ∈ 𝑛 (𝑖 ≠ 𝑗 → (𝑖𝑚𝑗) = (0g‘𝑟))})
df-scmat 22631 ScMat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ ⦋(𝑛 Mat 𝑟) / 𝑎⦌{𝑚 ∈ (Base‘𝑎) ∣ ∃𝑐 ∈ (Base‘𝑟)𝑚 = (𝑐( ·𝑠 ‘𝑎)(1r‘𝑎))})
cmvmul 22680class maVecMul
df-mvmul 22681 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 22696class matRRep
cmatrepV 22697class matRepV
df-marrep 22698 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 22699 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 22716class subMat
df-subma 22717 subMat = (𝑛 ∈ V, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑘 ∈ 𝑛, 𝑙 ∈ 𝑛 ↦ (𝑖 ∈ (𝑛 ∖ {𝑘}), 𝑗 ∈ (𝑛 ∖ {𝑙}) ↦ (𝑖𝑚𝑗)))))
cmdat 22724class maDet
df-mdet 22725 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 22772class maAdju
cminmar1 22773class minMatR1
df-madu 22774 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 22775 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 22843class ConstPolyMat
cmat2pmat 22844class matToPolyMat
ccpmat2mat 22845class cPolyMatToMat
df-cpmat 22846 ConstPolyMat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ {𝑚 ∈ (Base‘(𝑛 Mat (Poly1‘𝑟))) ∣ ∀𝑖 ∈ 𝑛 ∀𝑗 ∈ 𝑛 ∀𝑘 ∈ ℕ ((coe1‘(𝑖𝑚𝑗))‘𝑘) = (0g‘𝑟)})
df-mat2pmat 22847 matToPolyMat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑥 ∈ 𝑛, 𝑦 ∈ 𝑛 ↦ ((algSc‘(Poly1‘𝑟))‘(𝑥𝑚𝑦)))))
df-cpmat2mat 22848 cPolyMatToMat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ (𝑚 ∈ (𝑛 ConstPolyMat 𝑟) ↦ (𝑥 ∈ 𝑛, 𝑦 ∈ 𝑛 ↦ ((coe1‘(𝑥𝑚𝑦))‘0))))
cdecpmat 22902class decompPMat
df-decpmat 22903 decompPMat = (𝑚 ∈ V, 𝑘 ∈ ℕ0 ↦ (𝑖 ∈ dom dom 𝑚, 𝑗 ∈ dom dom 𝑚 ↦ ((coe1‘(𝑖𝑚𝑗))‘𝑘)))
cpm2mp 22932class pMatToMatPoly
df-pm2mp 22933 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 22966class CharPlyMat
df-chpmat 22967 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 23033class Top
df-top 23034Top = {𝑥 ∣ (∀𝑦 ∈ 𝒫 𝑥∪ 𝑦 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑦 ∩ 𝑧) ∈ 𝑥)}
ctopon 23050class TopOn
df-topon 23051TopOn = (𝑏 ∈ V ↦ {𝑗 ∈ Top ∣ 𝑏 = ∪ 𝑗})
ctps 23072class TopSp
df-topsp 23073TopSp = {𝑓 ∣ (TopOpen‘𝑓) ∈ (TopOn‘(Base‘𝑓))}
ctb 23085class TopBases
df-bases 23086TopBases = {𝑥 ∣ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑦 ∩ 𝑧) ⊆ ∪ (𝑥 ∩ 𝒫 (𝑦 ∩ 𝑧))}
ccld 23156class Clsd
cnt 23157class int
ccl 23158class cls
df-cld 23159Clsd = (𝑗 ∈ Top ↦ {𝑥 ∈ 𝒫 ∪ 𝑗 ∣ (∪ 𝑗 ∖ 𝑥) ∈ 𝑗})
df-ntr 23160int = (𝑗 ∈ Top ↦ (𝑥 ∈ 𝒫 ∪ 𝑗 ↦ ∪ (𝑗 ∩ 𝒫 𝑥)))
df-cls 23161cls = (𝑗 ∈ Top ↦ (𝑥 ∈ 𝒫 ∪ 𝑗 ↦ ∩ {𝑦 ∈ (Clsd‘𝑗) ∣ 𝑥 ⊆ 𝑦}))
cnei 23237class nei
df-nei 23238nei = (𝑗 ∈ Top ↦ (𝑥 ∈ 𝒫 ∪ 𝑗 ↦ {𝑦 ∈ 𝒫 ∪ 𝑗 ∣ ∃𝑔 ∈ 𝑗 (𝑥 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑦)}))
clp 23274class limPt
cperf 23275class Perf
df-lp 23276limPt = (𝑗 ∈ Top ↦ (𝑥 ∈ 𝒫 ∪ 𝑗 ↦ {𝑦 ∣ 𝑦 ∈ ((cls‘𝑗)‘(𝑥 ∖ {𝑦}))}))
df-perf 23277Perf = {𝑗 ∈ Top ∣ ((limPt‘𝑗)‘∪ 𝑗) = ∪ 𝑗}
ccn 23364class Cn
ccnp 23365class CnP
clm 23366class ⇝𝑡
df-cn 23367 Cn = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ {𝑓 ∈ (∪ 𝑘 ↑m ∪ 𝑗) ∣ ∀𝑦 ∈ 𝑘 (◡𝑓 “ 𝑦) ∈ 𝑗})
df-cnp 23368 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 23369⇝𝑡 = (𝑗 ∈ Top ↦ {⟨𝑓, 𝑥⟩ ∣ (𝑓 ∈ (∪ 𝑗 ↑pm ℂ) ∧ 𝑥 ∈ ∪ 𝑗 ∧ ∀𝑢 ∈ 𝑗 (𝑥 ∈ 𝑢 → ∃𝑦 ∈ ran ℤ≥(𝑓 ↾ 𝑦):𝑦⟶𝑢))})
ct0 23446class Kol2
ct1 23447class Fre
cha 23448class Haus
creg 23449class Reg
cnrm 23450class Nrm
ccnrm 23451class CNrm
cpnrm 23452class PNrm
df-t0 23453Kol2 = {𝑗 ∈ Top ∣ ∀𝑥 ∈ ∪ 𝑗∀𝑦 ∈ ∪ 𝑗(∀𝑜 ∈ 𝑗 (𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜) → 𝑥 = 𝑦)}
df-t1 23454Fre = {𝑥 ∈ Top ∣ ∀𝑎 ∈ ∪ 𝑥{𝑎} ∈ (Clsd‘𝑥)}
df-haus 23455Haus = {𝑗 ∈ Top ∣ ∀𝑥 ∈ ∪ 𝑗∀𝑦 ∈ ∪ 𝑗(𝑥 ≠ 𝑦 → ∃𝑛 ∈ 𝑗 ∃𝑚 ∈ 𝑗 (𝑥 ∈ 𝑛 ∧ 𝑦 ∈ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅))}
df-reg 23456Reg = {𝑗 ∈ Top ∣ ∀𝑥 ∈ 𝑗 ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝑗 (𝑦 ∈ 𝑧 ∧ ((cls‘𝑗)‘𝑧) ⊆ 𝑥)}
df-nrm 23457Nrm = {𝑗 ∈ Top ∣ ∀𝑥 ∈ 𝑗 ∀𝑦 ∈ ((Clsd‘𝑗) ∩ 𝒫 𝑥)∃𝑧 ∈ 𝑗 (𝑦 ⊆ 𝑧 ∧ ((cls‘𝑗)‘𝑧) ⊆ 𝑥)}
df-cnrm 23458CNrm = {𝑗 ∈ Top ∣ ∀𝑥 ∈ 𝒫 ∪ 𝑗(𝑗 ↾t 𝑥) ∈ Nrm}
df-pnrm 23459PNrm = {𝑗 ∈ Nrm ∣ (Clsd‘𝑗) ⊆ ran (𝑓 ∈ (𝑗 ↑m ℕ) ↦ ∩ ran 𝑓)}
ccmp 23526class Comp
df-cmp 23527Comp = {𝑥 ∈ Top ∣ ∀𝑦 ∈ 𝒫 𝑥(∪ 𝑥 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝑥 = ∪ 𝑧)}
cconn 23551class Conn
df-conn 23552Conn = {𝑗 ∈ Top ∣ (𝑗 ∩ (Clsd‘𝑗)) = {∅, ∪ 𝑗}}
c1stc 23577class 1stω
c2ndc 23578class 2ndω
df-1stc 235791stω = {𝑗 ∈ Top ∣ ∀𝑥 ∈ ∪ 𝑗∃𝑦 ∈ 𝒫 𝑗(𝑦 ≼ ω ∧ ∀𝑧 ∈ 𝑗 (𝑥 ∈ 𝑧 → 𝑥 ∈ ∪ (𝑦 ∩ 𝒫 𝑧)))}
df-2ndc 235802ndω = {𝑗 ∣ ∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝑗)}
clly 23604class Locally 𝐴
cnlly 23605class 𝑛-Locally 𝐴
df-lly 23606Locally 𝐴 = {𝑗 ∈ Top ∣ ∀𝑥 ∈ 𝑗 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝐴)}
df-nlly 23607𝑛-Locally 𝐴 = {𝑗 ∈ Top ∣ ∀𝑥 ∈ 𝑗 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗 ↾t 𝑢) ∈ 𝐴}
cref 23642class Ref
cptfin 23643class PtFin
clocfin 23644class LocFin
df-ref 23645Ref = {⟨𝑥, 𝑦⟩ ∣ (∪ 𝑦 = ∪ 𝑥 ∧ ∀𝑧 ∈ 𝑥 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)}
df-ptfin 23646PtFin = {𝑥 ∣ ∀𝑦 ∈ ∪ 𝑥{𝑧 ∈ 𝑥 ∣ 𝑦 ∈ 𝑧} ∈ Fin}
df-locfin 23647LocFin = (𝑥 ∈ Top ↦ {𝑦 ∣ (∪ 𝑥 = ∪ 𝑦 ∧ ∀𝑝 ∈ ∪ 𝑥∃𝑛 ∈ 𝑥 (𝑝 ∈ 𝑛 ∧ {𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))})
ckgen 23673class 𝑘Gen
df-kgen 23674𝑘Gen = (𝑗 ∈ Top ↦ {𝑥 ∈ 𝒫 ∪ 𝑗 ∣ ∀𝑘 ∈ 𝒫 ∪ 𝑗((𝑗 ↾t 𝑘) ∈ Comp → (𝑥 ∩ 𝑘) ∈ (𝑗 ↾t 𝑘))})
ctx 23700class ×t
cxko 23701class ↑ko
df-tx 23702 ×t = (𝑟 ∈ V, 𝑠 ∈ V ↦ (topGen‘ran (𝑥 ∈ 𝑟, 𝑦 ∈ 𝑠 ↦ (𝑥 × 𝑦))))
df-xko 23703 ↑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 23833class KQ
df-kq 23834KQ = (𝑗 ∈ Top ↦ (𝑗 qTop (𝑥 ∈ ∪ 𝑗 ↦ {𝑦 ∈ 𝑗 ∣ 𝑥 ∈ 𝑦})))
chmeo 23893class Homeo
chmph 23894class ≃
df-hmeo 23895Homeo = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ {𝑓 ∈ (𝑗 Cn 𝑘) ∣ ◡𝑓 ∈ (𝑘 Cn 𝑗)})
df-hmph 23896 ≃ = (◡Homeo “ (V ∖ 1o))
cfil 23985class Fil
df-fil 23986Fil = (𝑧 ∈ V ↦ {𝑓 ∈ (fBas‘𝑧) ∣ ∀𝑥 ∈ 𝒫 𝑧((𝑓 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥 ∈ 𝑓)})
cufil 24039class UFil
cufl 24040class UFL
df-ufil 24041UFil = (𝑔 ∈ V ↦ {𝑓 ∈ (Fil‘𝑔) ∣ ∀𝑥 ∈ 𝒫 𝑔(𝑥 ∈ 𝑓 ∨ (𝑔 ∖ 𝑥) ∈ 𝑓)})
df-ufl 24042UFL = {𝑥 ∣ ∀𝑓 ∈ (Fil‘𝑥)∃𝑔 ∈ (UFil‘𝑥)𝑓 ⊆ 𝑔}
cfm 24073class FilMap
cflim 24074class fLim
cflf 24075class fLimf
cfcls 24076class fClus
cfcf 24077class fClusf
df-fm 24078 FilMap = (𝑥 ∈ V, 𝑓 ∈ V ↦ (𝑦 ∈ (fBas‘dom 𝑓) ↦ (𝑥filGenran (𝑡 ∈ 𝑦 ↦ (𝑓 “ 𝑡)))))
df-flim 24079 fLim = (𝑗 ∈ Top, 𝑓 ∈ ∪ ran Fil ↦ {𝑥 ∈ ∪ 𝑗 ∣ (((nei‘𝑗)‘{𝑥}) ⊆ 𝑓 ∧ 𝑓 ⊆ 𝒫 ∪ 𝑗)})
df-flf 24080 fLimf = (𝑥 ∈ Top, 𝑦 ∈ ∪ ran Fil ↦ (𝑓 ∈ (∪ 𝑥 ↑m ∪ 𝑦) ↦ (𝑥 fLim ((∪ 𝑥 FilMap 𝑓)‘𝑦))))
df-fcls 24081 fClus = (𝑗 ∈ Top, 𝑓 ∈ ∪ ran Fil ↦ if(∪ 𝑗 = ∪ 𝑓, ∩ 𝑥 ∈ 𝑓 ((cls‘𝑗)‘𝑥), ∅))
df-fcf 24082 fClusf = (𝑗 ∈ Top, 𝑓 ∈ ∪ ran Fil ↦ (𝑔 ∈ (∪ 𝑗 ↑m ∪ 𝑓) ↦ (𝑗 fClus ((∪ 𝑗 FilMap 𝑔)‘𝑓))))
ccnext 24199class CnExt
df-cnext 24200CnExt = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ (𝑓 ∈ (∪ 𝑘 ↑pm ∪ 𝑗) ↦ ∪ 𝑥 ∈ ((cls‘𝑗)‘dom 𝑓)({𝑥} × ((𝑘 fLimf (((nei‘𝑗)‘{𝑥}) ↾t dom 𝑓))‘𝑓))))
ctmd 24210class TopMnd
ctgp 24211class TopGrp
df-tmd 24212TopMnd = {𝑓 ∈ (Mnd ∩ TopSp) ∣ [(TopOpen‘𝑓) / 𝑗](+𝑓‘𝑓) ∈ ((𝑗 ×t 𝑗) Cn 𝑗)}
df-tgp 24213TopGrp = {𝑓 ∈ (Grp ∩ TopMnd) ∣ [(TopOpen‘𝑓) / 𝑗](invg‘𝑓) ∈ (𝑗 Cn 𝑗)}
ctsu 24266class tsums
df-tsms 24267 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 24296class TopRing
ctdrg 24297class TopDRing
ctlm 24298class TopMod
ctvc 24299class TopVec
df-trg 24300TopRing = {𝑟 ∈ (TopGrp ∩ Ring) ∣ (mulGrp‘𝑟) ∈ TopMnd}
df-tdrg 24301TopDRing = {𝑟 ∈ (TopRing ∩ DivRing) ∣ ((mulGrp‘𝑟) ↾s (Unit‘𝑟)) ∈ TopGrp}
df-tlm 24302TopMod = {𝑤 ∈ (TopMnd ∩ LMod) ∣ ((Scalar‘𝑤) ∈ TopRing ∧ ( ·sf ‘𝑤) ∈ (((TopOpen‘(Scalar‘𝑤)) ×t (TopOpen‘𝑤)) Cn (TopOpen‘𝑤)))}
df-tvc 24303TopVec = {𝑤 ∈ TopMod ∣ (Scalar‘𝑤) ∈ TopDRing}
cust 24340class UnifOn
df-ust 24341UnifOn = (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 24370class unifTop
df-utop 24371unifTop = (𝑢 ∈ ∪ ran UnifOn ↦ {𝑎 ∈ 𝒫 dom ∪ 𝑢 ∣ ∀𝑥 ∈ 𝑎 ∃𝑣 ∈ 𝑢 (𝑣 “ {𝑥}) ⊆ 𝑎})
cuss 24393class UnifSt
cusp 24394class UnifSp
ctus 24395class toUnifSp
df-uss 24396UnifSt = (𝑓 ∈ V ↦ ((UnifSet‘𝑓) ↾t ((Base‘𝑓) × (Base‘𝑓))))
df-usp 24397UnifSp = {𝑓 ∣ ((UnifSt‘𝑓) ∈ (UnifOn‘(Base‘𝑓)) ∧ (TopOpen‘𝑓) = (unifTop‘(UnifSt‘𝑓)))}
df-tus 24398toUnifSp = (𝑢 ∈ ∪ ran UnifOn ↦ ({⟨(Base‘ndx), dom ∪ 𝑢⟩, ⟨(UnifSet‘ndx), 𝑢⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑢)⟩))
cucn 24414class Cnu
df-ucn 24415 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 24425class CauFilu
df-cfilu 24426CauFilu = (𝑢 ∈ ∪ ran UnifOn ↦ {𝑓 ∈ (fBas‘dom ∪ 𝑢) ∣ ∀𝑣 ∈ 𝑢 ∃𝑎 ∈ 𝑓 (𝑎 × 𝑎) ⊆ 𝑣})
ccusp 24436class CUnifSp
df-cusp 24437CUnifSp = {𝑤 ∈ UnifSp ∣ ∀𝑐 ∈ (Fil‘(Base‘𝑤))(𝑐 ∈ (CauFilu‘(UnifSt‘𝑤)) → ((TopOpen‘𝑤) fLim 𝑐) ≠ ∅)}
cxms 24457class ∞MetSp
cms 24458class MetSp
ctms 24459class toMetSp
df-xms 24460∞MetSp = {𝑓 ∈ TopSp ∣ (TopOpen‘𝑓) = (MetOpen‘((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))))}
df-ms 24461MetSp = {𝑓 ∈ ∞MetSp ∣ ((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))) ∈ (Met‘(Base‘𝑓))}
df-tms 24462toMetSp = (𝑑 ∈ ∪ ran ∞Met ↦ ({⟨(Base‘ndx), dom dom 𝑑⟩, ⟨(dist‘ndx), 𝑑⟩} sSet ⟨(TopSet‘ndx), (MetOpen‘𝑑)⟩))
cnm 24716class norm
cngp 24717class NrmGrp
ctng 24718class toNrmGrp
cnrg 24719class NrmRing
cnlm 24720class NrmMod
cnvc 24721class NrmVec
df-nm 24722norm = (𝑤 ∈ V ↦ (𝑥 ∈ (Base‘𝑤) ↦ (𝑥(dist‘𝑤)(0g‘𝑤))))
df-ngp 24723NrmGrp = {𝑔 ∈ (Grp ∩ MetSp) ∣ ((norm‘𝑔) ∘ (-g‘𝑔)) ⊆ (dist‘𝑔)}
df-tng 24724 toNrmGrp = (𝑔 ∈ V, 𝑓 ∈ V ↦ ((𝑔 sSet ⟨(dist‘ndx), (𝑓 ∘ (-g‘𝑔))⟩) sSet ⟨(TopSet‘ndx), (MetOpen‘(𝑓 ∘ (-g‘𝑔)))⟩))
df-nrg 24725NrmRing = {𝑤 ∈ NrmGrp ∣ (norm‘𝑤) ∈ (AbsVal‘𝑤)}
df-nlm 24726NrmMod = {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 24727NrmVec = (NrmMod ∩ LVec)
cnmo 24845class normOp
cnghm 24846class NGHom
cnmhm 24847class NMHom
df-nmo 24848 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 24849 NGHom = (𝑠 ∈ NrmGrp, 𝑡 ∈ NrmGrp ↦ (◡(𝑠 normOp 𝑡) “ ℝ))
df-nmhm 24850 NMHom = (𝑠 ∈ NrmMod, 𝑡 ∈ NrmMod ↦ ((𝑠 LMHom 𝑡) ∩ (𝑠 NGHom 𝑡)))
cii 25017class II
ccncf 25018class –cn→
df-ii 25019II = (MetOpen‘((abs ∘ − ) ↾ ((0[,]1) × (0[,]1))))
df-cncf 25020–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 25109class Htpy
cphtpy 25110class PHtpy
cphtpc 25111class ≃ph
df-htpy 25112 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 25113PHtpy = (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 25134 ≃ph = (𝑥 ∈ Top ↦ {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ (II Cn 𝑥) ∧ (𝑓(PHtpy‘𝑥)𝑔) ≠ ∅)})
cpco 25142class *𝑝
comi 25143class Ω1
comn 25144class Ω𝑛
cpi1 25145class π1
cpin 25146class πn
df-pco 25147*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 25148 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 25149 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 25150 π1 = (𝑗 ∈ Top, 𝑦 ∈ ∪ 𝑗 ↦ ((𝑗 Ω1 𝑦) /s ( ≃ph‘𝑗)))
df-pin 25151 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 25204class ℂMod
df-clm 25205ℂMod = {𝑤 ∈ LMod ∣ [(Scalar‘𝑤) / 𝑓][(Base‘𝑓) / 𝑘](𝑓 = (ℂfld ↾s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))}
ccvs 25265class ℂVec
df-cvs 25266ℂVec = (ℂMod ∩ LVec)
ccph 25308class ℂPreHil
ctcph 25309class toℂPreHil
df-cph 25310ℂPreHil = {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 25311toℂPreHil = (𝑤 ∈ V ↦ (𝑤 toNrmGrp (𝑥 ∈ (Base‘𝑤) ↦ (√‘(𝑥(·𝑖‘𝑤)𝑥)))))
ccfil 25394class CauFil
ccau 25395class Cau
ccmet 25396class CMet
df-cfil 25397CauFil = (𝑑 ∈ ∪ ran ∞Met ↦ {𝑓 ∈ (Fil‘dom dom 𝑑) ∣ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ 𝑓 (𝑑 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)})
df-cau 25398Cau = (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 25399CMet = (𝑥 ∈ V ↦ {𝑑 ∈ (Met‘𝑥) ∣ ∀𝑓 ∈ (CauFil‘𝑑)((MetOpen‘𝑑) fLim 𝑓) ≠ ∅})
ccms 25474class CMetSp
cbn 25475class Ban
chl 25476class ℂHil
df-cms 25477CMetSp = {𝑤 ∈ MetSp ∣ [(Base‘𝑤) / 𝑏]((dist‘𝑤) ↾ (𝑏 × 𝑏)) ∈ (CMet‘𝑏)}
df-bn 25478Ban = {𝑤 ∈ (NrmVec ∩ CMetSp) ∣ (Scalar‘𝑤) ∈ CMetSp}
df-hl 25479ℂHil = (Ban ∩ ℂPreHil)
crrx 25525class ℝ^
cehl 25526class 𝔼hil
df-rrx 25527ℝ^ = (𝑖 ∈ V ↦ (toℂPreHil‘(ℝfld freeLMod 𝑖)))
df-ehl 25528𝔼hil = (𝑛 ∈ ℕ0 ↦ (ℝ^‘(1...𝑛)))
covol 25604class vol*
cvol 25605class vol
df-ovol 25606vol* = (𝑥 ∈ 𝒫 ℝ ↦ inf({𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝑥 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))}, ℝ*, < ))
df-vol 25607vol = (vol* ↾ {𝑥 ∣ ∀𝑦 ∈ (◡vol* “ ℝ)(vol*‘𝑦) = ((vol*‘(𝑦 ∩ 𝑥)) + (vol*‘(𝑦 ∖ 𝑥)))})
cmbf 25756class MblFn
citg1 25757class ∫1
citg2 25758class ∫2
cibl 25759class 𝐿1
citg 25760class ∫𝐴𝐵 d𝑥
df-mbf 25761MblFn = {𝑓 ∈ (ℂ ↑pm ℝ) ∣ ∀𝑥 ∈ ran (,)((◡(ℜ ∘ 𝑓) “ 𝑥) ∈ dom vol ∧ (◡(ℑ ∘ 𝑓) “ 𝑥) ∈ dom vol)}
df-itg1 25762∫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 25763∫2 = (𝑓 ∈ ((0[,]+∞) ↑m ℝ) ↦ sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔 ∘r ≤ 𝑓 ∧ 𝑥 = (∫1‘𝑔))}, ℝ*, < ))
df-ibl 25764𝐿1 = {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 25765∫𝐴𝐵 d𝑥 = Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐵 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0))))
c0p 25811class 0𝑝
df-0p 258120𝑝 = (ℂ × {0})
cdit 25988class ⨜[𝐴 → 𝐵]𝐶 d𝑥
df-ditg 25989⨜[𝐴 → 𝐵]𝐶 d𝑥 = if(𝐴 ≤ 𝐵, ∫(𝐴(,)𝐵)𝐶 d𝑥, -∫(𝐵(,)𝐴)𝐶 d𝑥)
climc 26004class limℂ
cdv 26005class D
cdvn 26006class D𝑛
ccpn 26007class 𝓑C𝑛
df-limc 26008 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 26009 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 26010 D𝑛 = (𝑠 ∈ 𝒫 ℂ, 𝑓 ∈ (ℂ ↑pm 𝑠) ↦ seq0(((𝑥 ∈ V ↦ (𝑠 D 𝑥)) ∘ 1st ), (ℕ0 × {𝑓})))
df-cpn 26011𝓑C𝑛 = (𝑠 ∈ 𝒫 ℂ ↦ (𝑥 ∈ ℕ0 ↦ {𝑓 ∈ (ℂ ↑pm 𝑠) ∣ ((𝑠 D𝑛 𝑓)‘𝑥) ∈ (dom 𝑓–cn→ℂ)}))
cmdg 26193class mDeg
cdg1 26194class deg1
df-mdeg 26195 mDeg = (𝑖 ∈ V, 𝑟 ∈ V ↦ (𝑓 ∈ (Base‘(𝑖 mPoly 𝑟)) ↦ sup(ran (ℎ ∈ (𝑓 supp (0g‘𝑟)) ↦ (ℂfld Σg ℎ)), ℝ*, < )))
df-deg1 26196deg1 = (𝑟 ∈ V ↦ (1o mDeg 𝑟))
cmn1 26266class Monic1p
cuc1p 26267class Unic1p
cq1p 26268class quot1p
cr1p 26269class rem1p
cig1p 26270class idlGen1p
df-mon1 26271Monic1p = (𝑟 ∈ V ↦ {𝑓 ∈ (Base‘(Poly1‘𝑟)) ∣ (𝑓 ≠ (0g‘(Poly1‘𝑟)) ∧ ((coe1‘𝑓)‘((deg1‘𝑟)‘𝑓)) = (1r‘𝑟))})
df-uc1p 26272Unic1p = (𝑟 ∈ V ↦ {𝑓 ∈ (Base‘(Poly1‘𝑟)) ∣ (𝑓 ≠ (0g‘(Poly1‘𝑟)) ∧ ((coe1‘𝑓)‘((deg1‘𝑟)‘𝑓)) ∈ (Unit‘𝑟))})
df-q1p 26273quot1p = (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 26274rem1p = (𝑟 ∈ V ↦ ⦋(Base‘(Poly1‘𝑟)) / 𝑏⦌(𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (𝑓(-g‘(Poly1‘𝑟))((𝑓(quot1p‘𝑟)𝑔)(.r‘(Poly1‘𝑟))𝑔))))
df-ig1p 26275idlGen1p = (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 26324class Poly
cidp 26325class Xp
ccoe 26326class coeff
cdgr 26327class deg
df-ply 26328Poly = (𝑥 ∈ 𝒫 ℂ ↦ {𝑓 ∣ ∃𝑛 ∈ ℕ0 ∃𝑎 ∈ ((𝑥 ∪ {0}) ↑m ℕ0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))})
df-idp 26329Xp = ( I ↾ ℂ)
df-coe 26330coeff = (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 26331deg = (𝑓 ∈ (Poly‘ℂ) ↦ sup((◡(coeff‘𝑓) “ (ℂ ∖ {0})), ℕ0, < ))
cquot 26434class quot
df-quot 26435 quot = (𝑓 ∈ (Poly‘ℂ), 𝑔 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ↦ (℩𝑞 ∈ (Poly‘ℂ)[(𝑓 ∘f − (𝑔 ∘f · 𝑞)) / 𝑟](𝑟 = 0𝑝 ∨ (deg‘𝑟) < (deg‘𝑔))))
caa 26458class 𝔸
df-aa 26459𝔸 = ∪ 𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝})(◡𝑓 “ {0})
ctayl 26496class Tayl
cana 26497class Ana
df-tayl 26498 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 26499Ana = (𝑠 ∈ {ℝ, ℂ} ↦ {𝑓 ∈ (ℂ ↑pm 𝑠) ∣ ∀𝑥 ∈ dom 𝑓 𝑥 ∈ ((int‘((TopOpen‘ℂfld) ↾t 𝑠))‘dom (𝑓 ∩ (+∞(𝑠 Tayl 𝑓)𝑥)))})
culm 26519class ⇝𝑢
df-ulm 26520⇝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 26699class log
ccxp 26700class ↑𝑐
df-log 26701log = ◡(exp ↾ (◡ℑ “ (-π(,]π)))
df-cxp 26702↑𝑐 = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ if(𝑥 = 0, if(𝑦 = 0, 1, 0), (exp‘(𝑦 · (log‘𝑥)))))
clogb 26909class logb
df-logb 26910 logb = (𝑥 ∈ (ℂ ∖ {0, 1}), 𝑦 ∈ (ℂ ∖ {0}) ↦ ((log‘𝑦) / (log‘𝑥)))
casin 27007class arcsin
cacos 27008class arccos
catan 27009class arctan
df-asin 27010arcsin = (𝑥 ∈ ℂ ↦ (-i · (log‘((i · 𝑥) + (√‘(1 − (𝑥↑2)))))))
df-acos 27011arccos = (𝑥 ∈ ℂ ↦ ((π / 2) − (arcsin‘𝑥)))
df-atan 27012arctan = (𝑥 ∈ (ℂ ∖ {-i, i}) ↦ ((i / 2) · ((log‘(1 − (i · 𝑥))) − (log‘(1 + (i · 𝑥))))))
carea 27100class area
df-area 27101area = (𝑠 ∈ {𝑡 ∈ 𝒫 (ℝ × ℝ) ∣ (∀𝑥 ∈ ℝ (𝑡 “ {𝑥}) ∈ (◡vol “ ℝ) ∧ (𝑥 ∈ ℝ ↦ (vol‘(𝑡 “ {𝑥}))) ∈ 𝐿1)} ↦ ∫ℝ(vol‘(𝑠 “ {𝑥})) d𝑥)
cem 27136class γ
df-em 27137γ = Σ𝑘 ∈ ℕ ((1 / 𝑘) − (log‘(1 + (1 / 𝑘))))
czeta 27157class ζ
df-zeta 27158ζ = (℩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 27160class log Γ
cgam 27161class Γ
cigam 27162class 1/Γ
df-lgam 27163log Γ = (𝑧 ∈ (ℂ ∖ (ℤ ∖ ℕ)) ↦ (Σ𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))) − (log‘𝑧)))
df-gam 27164Γ = (exp ∘ log Γ)
df-igam 271651/Γ = (𝑥 ∈ ℂ ↦ if(𝑥 ∈ (ℤ ∖ ℕ), 0, (1 / (Γ‘𝑥))))
ccht 27235class θ
cvma 27236class Λ
cchp 27237class ψ
cppi 27238class π
cmu 27239class μ
csgm 27240class σ
df-cht 27241θ = (𝑥 ∈ ℝ ↦ Σ𝑝 ∈ ((0[,]𝑥) ∩ ℙ)(log‘𝑝))
df-vma 27242Λ = (𝑥 ∈ ℕ ↦ ⦋{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥} / 𝑠⦌if((♯‘𝑠) = 1, (log‘∪ 𝑠), 0))
df-chp 27243ψ = (𝑥 ∈ ℝ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))(Λ‘𝑛))
df-ppi 27244π = (𝑥 ∈ ℝ ↦ (♯‘((0[,]𝑥) ∩ ℙ)))
df-mu 27245μ = (𝑥 ∈ ℕ ↦ if(∃𝑝 ∈ ℙ (𝑝↑2) ∥ 𝑥, 0, (-1↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥}))))
df-sgm 27246 σ = (𝑥 ∈ ℂ, 𝑛 ∈ ℕ ↦ Σ𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝑛} (𝑘↑𝑐𝑥))
cdchr 27376class DChr
df-dchr 27377DChr = (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 27438class /L
df-lgs 27439 /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 27784class No
clts 27785class <s
cbday 27786class bday
df-no 27787 No = {𝑓 ∣ ∃𝑎 ∈ On 𝑓:𝑎⟶{1o, 2o}}
df-lts 27788 <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 27789 bday = (𝑥 ∈ No ↦ dom 𝑥)
cles 27888class ≤s
df-les 27889 ≤s = (( No × No ) ∖ ◡ <s )
cslts 27930class <<s
df-slts 27931 <<s = {⟨𝑎, 𝑏⟩ ∣ (𝑎 ⊆ No ∧ 𝑏 ⊆ No ∧ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 𝑥 <s 𝑦)}
ccuts 27932class |s
df-cuts 27933 |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 27978class 0s
c1s 27979class 1s
df-0s 27980 0s = (∅ |s ∅)
df-1s 27981 1s = ({ 0s } |s ∅)
cmade 27995class M
cold 27996class O
cnew 27997class N
cleft 27998class L
cright 27999class R
df-made 28000 M = recs((𝑓 ∈ V ↦ ( |s “ (𝒫 ∪ ran 𝑓 × 𝒫 ∪ ran 𝑓))))
df-old 28001 O = (𝑥 ∈ On ↦ ∪ ( M “ 𝑥))
df-new 28002 N = (𝑥 ∈ On ↦ (( M ‘𝑥) ∖ ( O ‘𝑥)))
df-left 28003 L = (𝑥 ∈ No ↦ {𝑦 ∈ ( O ‘( bday ‘𝑥)) ∣ 𝑦 <s 𝑥})
df-right 28004 R = (𝑥 ∈ No ↦ {𝑦 ∈ ( O ‘( bday ‘𝑥)) ∣ 𝑥 <s 𝑦})
cnorec 28110class norec (𝐹)
df-norec 28111 norec (𝐹) = frecs({⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}, No , 𝐹)
cnorec2 28121class norec2 (𝐹)
df-norec2 28122 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 28132class +s
df-adds 28133 +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 28192class -us
csubs 28193class -s
df-negs 28194 -us = norec ((𝑥 ∈ V, 𝑛 ∈ V ↦ ((𝑛 “ ( R ‘𝑥)) |s (𝑛 “ ( L ‘𝑥)))))
df-subs 28195 -s = (𝑥 ∈ No , 𝑦 ∈ No ↦ (𝑥 +s ( -us ‘𝑦)))
cmuls 28279class ·s
df-muls 28280 ·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 28360class /su
df-divs 28361 /su = (𝑥 ∈ No , 𝑦 ∈ ( No ∖ { 0s }) ↦ (℩𝑧 ∈ No (𝑦 ·s 𝑧) = 𝑥))
cabss 28410class abss
df-abss 28411abss = (𝑥 ∈ No ↦ if( 0s ≤s 𝑥, 𝑥, ( -us ‘𝑥)))
cons 28424class Ons
df-ons 28425Ons = {𝑥 ∈ No ∣ ( R ‘𝑥) = ∅}
cseqs 28456class seqs𝑀( + , 𝐹)
df-seqs 28457seqs𝑀( + , 𝐹) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑦 + (𝐹‘(𝑥 +s 1s )))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) “ ω)
cn0s 28485class ℕ0s
cnns 28486class ℕs
df-n0s 28487ℕ0s = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 0s ) “ ω)
df-nns 28488ℕs = (ℕ0s ∖ { 0s })
czs 28551class ℤs
df-zs 28552ℤs = ( -s “ (ℕs × ℕs))
c2s 28583class 2s
df-2s 285842s = ({ 1s } |s ∅)
cexps 28585class ↑s
df-exps 28586↑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 28587class ℤs[1/2]
df-z12s 28588ℤs[1/2] = {𝑥 ∣ ∃𝑦 ∈ ℤs ∃𝑧 ∈ ℕ0s 𝑥 = (𝑦 /su (2s↑s𝑧))}
creno 28662class ℝs
df-reno 28663ℝs = {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 28676class TarskiG
cstrkgc 28677class TarskiGC
cstrkgb 28678class TarskiGB
cstrkgcb 28679class TarskiGCB
cstrkgld 28680class DimTarskiG≥
cstrkge 28681class TarskiGE
citv 28682class Itv
clng 28683class LineG
df-itv 28684Itv = Slot 16
df-lng 28685LineG = Slot 17
df-trkgc 28697TarskiGC = {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 28698TarskiGB = {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 28699TarskiGCB = {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 28700TarskiGE = {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 28701DimTarskiG≥ = {⟨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 28702TarskiG = ((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 28759class cgrG
df-cgrg 28760cgrG = (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 28781class Ismt
df-ismt 28782Ismt = (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 28831class ≤G
df-leg 28832≤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 28849class hlG
df-hlg 28850hlG = (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 28909class pInvG
df-mir 28910pInvG = (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 28952class ∟G
df-rag 28953∟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 28954class ⟂G
df-perpg 28955⟂G = (𝑔 ∈ V ↦ {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ ran (LineG‘𝑔) ∧ 𝑏 ∈ ran (LineG‘𝑔)) ∧ ∃𝑥 ∈ (𝑎 ∩ 𝑏)∀𝑢 ∈ 𝑎 ∀𝑣 ∈ 𝑏 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝑔))})
chpg 29018class hpG
df-hpg 29019hpG = (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)))}))
cplng 29033class hlG
df-plng 29034hlG = (𝑔 ∈ V ↦ (𝑎 ∈ ran (LineG‘𝑔), 𝑟 ∈ ((Base‘𝑔) ∖ 𝑎) ↦ {𝑥 ∈ (Base‘𝑔) ∣ (𝑥 ∈ 𝑎 ∨ 𝑥((hpG‘𝑔)‘𝑎)𝑟 ∨ ∃𝑡 ∈ 𝑎 𝑡 ∈ (𝑥(Itv‘𝑔)𝑟))}))
cmid 29059class midG
clmi 29060class lInvG
df-mid 29061midG = (𝑔 ∈ V ↦ (𝑎 ∈ (Base‘𝑔), 𝑏 ∈ (Base‘𝑔) ↦ (℩𝑚 ∈ (Base‘𝑔)𝑏 = (((pInvG‘𝑔)‘𝑚)‘𝑎))))
df-lmi 29062lInvG = (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 29095class cgrA
df-cgra 29096cgrA = (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 29129class inA
cleag 29130class ≤∠
df-inag 29131inA = (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 29140≤∠ = (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 29159class eqltrG
df-eqlg 29160eqltrG = (𝑔 ∈ V ↦ {𝑥 ∈ ((Base‘𝑔) ↑m (0..^3)) ∣ 𝑥(cgrG‘𝑔)⟨“(𝑥‘1)(𝑥‘2)(𝑥‘0)”⟩})
cprlng 29163class parlnG
df-prlng 29164parlnG = (𝑔 ∈ V ↦ {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ ran (LineG‘𝑔) ∧ 𝑏 ∈ ran (LineG‘𝑔)) ∧ (𝑎 = 𝑏 ∨ (∃ℎ ∈ ran (hlG‘𝑔)(𝑎 ⊆ ℎ ∧ 𝑏 ⊆ ℎ) ∧ (𝑎 ∩ 𝑏) = ∅)))})
cttg 29192class toTG
df-ttg 29193toTG = (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 29207class 𝔼
cbtwn 29208class Btwn
ccgr 29209class Cgr
df-ee 29210𝔼 = (𝑛 ∈ ℕ ↦ (ℝ ↑m (1...𝑛)))
df-btwn 29211 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 29212Cgr = {⟨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 29297class EEG
df-eeng 29298EEG = (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 29308class .ef
df-edgf 29309.ef = Slot 18
cvtx 29316class Vtx
ciedg 29317class iEdg
df-vtx 29318Vtx = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔)))
df-iedg 29319iEdg = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (2nd ‘𝑔), (.ef‘𝑔)))
cedg 29367class Edg
df-edg 29368Edg = (𝑔 ∈ V ↦ ran (iEdg‘𝑔))
cuhgr 29376class UHGraph
cushgr 29377class USHGraph
df-uhgr 29378UHGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶(𝒫 𝑣 ∖ {∅})}
df-ushgr 29379USHGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→(𝒫 𝑣 ∖ {∅})}
cupgr 29400class UPGraph
cumgr 29401class UMGraph
df-upgr 29402UPGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}}
df-umgr 29403UMGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2}}
cuspgr 29468class USPGraph
cusgr 29469class USGraph
df-uspgr 29470USPGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}}
df-usgr 29471USGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2}}
csubgr 29587class SubGraph
df-subgr 29588 SubGraph = {⟨𝑠, 𝑔⟩ ∣ ((Vtx‘𝑠) ⊆ (Vtx‘𝑔) ∧ (iEdg‘𝑠) = ((iEdg‘𝑔) ↾ dom (iEdg‘𝑠)) ∧ (Edg‘𝑠) ⊆ 𝒫 (Vtx‘𝑠))}
cfusgr 29636class FinUSGraph
df-fusgr 29637FinUSGraph = {𝑔 ∈ USGraph ∣ (Vtx‘𝑔) ∈ Fin}
cnbgr 29652class NeighbVtx
df-nbgr 29653 NeighbVtx = (𝑔 ∈ V, 𝑣 ∈ (Vtx‘𝑔) ↦ {𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣}) ∣ ∃𝑒 ∈ (Edg‘𝑔){𝑣, 𝑛} ⊆ 𝑒})
cuvtx 29705class UnivVtx
df-uvtx 29706UnivVtx = (𝑔 ∈ V ↦ {𝑣 ∈ (Vtx‘𝑔) ∣ ∀𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣})𝑛 ∈ (𝑔 NeighbVtx 𝑣)})
ccplgr 29729class ComplGraph
ccusgr 29730class ComplUSGraph
df-cplgr 29731ComplGraph = {𝑔 ∣ (UnivVtx‘𝑔) = (Vtx‘𝑔)}
df-cusgr 29732ComplUSGraph = (USGraph ∩ ComplGraph)
cvtxdg 29785class VtxDeg
df-vtxdg 29786VtxDeg = (𝑔 ∈ V ↦ ⦋(Vtx‘𝑔) / 𝑣⦌⦋(iEdg‘𝑔) / 𝑒⦌(𝑢 ∈ 𝑣 ↦ ((♯‘{𝑥 ∈ dom 𝑒 ∣ 𝑢 ∈ (𝑒‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom 𝑒 ∣ (𝑒‘𝑥) = {𝑢}}))))
crgr 29875class RegGraph
crusgr 29876class RegUSGraph
df-rgr 29877 RegGraph = {⟨𝑔, 𝑘⟩ ∣ (𝑘 ∈ ℕ0* ∧ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 𝑘)}
df-rusgr 29878 RegUSGraph = {⟨𝑔, 𝑘⟩ ∣ (𝑔 ∈ USGraph ∧ 𝑔 RegGraph 𝑘)}
cewlks 29915class EdgWalks
cwlks 29916class Walks
cwlkson 29917class WalksOn
df-ewlks 29918 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 29919Walks = (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 29920WalksOn = (𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝑔)𝑝 ∧ (𝑝‘0) = 𝑎 ∧ (𝑝‘(♯‘𝑓)) = 𝑏)}))
ctrls 30008class Trails
ctrlson 30009class TrailsOn
df-trls 30010Trails = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝑔)𝑝 ∧ Fun ◡𝑓)})
df-trlson 30011TrailsOn = (𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑎(WalksOn‘𝑔)𝑏)𝑝 ∧ 𝑓(Trails‘𝑔)𝑝)}))
cpths 30029class Paths
cspths 30030class SPaths
cpthson 30031class PathsOn
cspthson 30032class SPathsOn
df-pths 30033Paths = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝑔)𝑝 ∧ Fun ◡(𝑝 ↾ (1..^(♯‘𝑓))) ∧ ((𝑝 “ {0, (♯‘𝑓)}) ∩ (𝑝 “ (1..^(♯‘𝑓)))) = ∅)})
df-spths 30034SPaths = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝑔)𝑝 ∧ Fun ◡𝑝)})
df-pthson 30035PathsOn = (𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑎(TrailsOn‘𝑔)𝑏)𝑝 ∧ 𝑓(Paths‘𝑔)𝑝)}))
df-spthson 30036SPathsOn = (𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑎(TrailsOn‘𝑔)𝑏)𝑝 ∧ 𝑓(SPaths‘𝑔)𝑝)}))
cclwlks 30089class ClWalks
df-clwlks 30090ClWalks = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝑔)𝑝 ∧ (𝑝‘0) = (𝑝‘(♯‘𝑓)))})
ccrcts 30103class Circuits
ccycls 30104class Cycles
df-crcts 30105Circuits = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝑔)𝑝 ∧ (𝑝‘0) = (𝑝‘(♯‘𝑓)))})
df-cycls 30106Cycles = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Paths‘𝑔)𝑝 ∧ (𝑝‘0) = (𝑝‘(♯‘𝑓)))})
cwwlks 30144class WWalks
cwwlksn 30145class WWalksN
cwwlksnon 30146class WWalksNOn
cwwspthsn 30147class WSPathsN
cwwspthsnon 30148class WSPathsNOn
df-wwlks 30149WWalks = (𝑔 ∈ V ↦ {𝑤 ∈ Word (Vtx‘𝑔) ∣ (𝑤 ≠ ∅ ∧ ∀𝑖 ∈ (0..^((♯‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝑔))})
df-wwlksn 30150 WWalksN = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ {𝑤 ∈ (WWalks‘𝑔) ∣ (♯‘𝑤) = (𝑛 + 1)})
df-wwlksnon 30151 WWalksNOn = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {𝑤 ∈ (𝑛 WWalksN 𝑔) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤‘𝑛) = 𝑏)}))
df-wspthsn 30152 WSPathsN = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ {𝑤 ∈ (𝑛 WWalksN 𝑔) ∣ ∃𝑓 𝑓(SPaths‘𝑔)𝑤})
df-wspthsnon 30153 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 30302class ClWWalks
df-clwwlk 30303ClWWalks = (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 30345class ClWWalksN
df-clwwlkn 30346 ClWWalksN = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ {𝑤 ∈ (ClWWalks‘𝑔) ∣ (♯‘𝑤) = 𝑛})
cclwwlknon 30408class ClWWalksNOn
df-clwwlknon 30409ClWWalksNOn = (𝑔 ∈ V ↦ (𝑣 ∈ (Vtx‘𝑔), 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝑔) ∣ (𝑤‘0) = 𝑣}))
cconngr 30507class ConnGraph
df-conngr 30508ConnGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣]∀𝑘 ∈ 𝑣 ∀𝑛 ∈ 𝑣 ∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝}
ceupth 30518class EulerPaths
df-eupth 30519EulerPaths = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝑔)𝑝 ∧ 𝑓:(0..^(♯‘𝑓))–onto→dom (iEdg‘𝑔))})
cfrgr 30579class FriendGraph
df-frgr 30580 FriendGraph = {𝑔 ∈ USGraph ∣ [(Vtx‘𝑔) / 𝑣][(Edg‘𝑔) / 𝑒]∀𝑘 ∈ 𝑣 ∀𝑙 ∈ (𝑣 ∖ {𝑘})∃!𝑥 ∈ 𝑣 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝑒}
ax-flt 30793((𝑁 ∈ (ℤ≥‘3) ∧ (𝑋 ∈ ℕ ∧ 𝑌 ∈ ℕ ∧ 𝑍 ∈ ℕ)) → ((𝑋↑𝑁) + (𝑌↑𝑁)) ≠ (𝑍↑𝑁))
cplig 30796class Plig
df-plig 30797Plig = {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 30811class GrpOp
cgi 30812class GId
cgn 30813class inv
cgs 30814class /𝑔
df-grpo 30815GrpOp = {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 30816GId = (𝑔 ∈ V ↦ (℩𝑢 ∈ ran 𝑔∀𝑥 ∈ ran 𝑔((𝑢𝑔𝑥) = 𝑥 ∧ (𝑥𝑔𝑢) = 𝑥)))
df-ginv 30817inv = (𝑔 ∈ GrpOp ↦ (𝑥 ∈ ran 𝑔 ↦ (℩𝑧 ∈ ran 𝑔(𝑧𝑔𝑥) = (GId‘𝑔))))
df-gdiv 30818 /𝑔 = (𝑔 ∈ GrpOp ↦ (𝑥 ∈ ran 𝑔, 𝑦 ∈ ran 𝑔 ↦ (𝑥𝑔((inv‘𝑔)‘𝑦))))
cablo 30866class AbelOp
df-ablo 30867AbelOp = {𝑔 ∈ GrpOp ∣ ∀𝑥 ∈ ran 𝑔∀𝑦 ∈ ran 𝑔(𝑥𝑔𝑦) = (𝑦𝑔𝑥)}
cvc 30880class CVecOLD
df-vc 30881CVecOLD = {⟨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 30906class NrmCVec
cpv 30907class +𝑣
cba 30908class BaseSet
cns 30909class ·𝑠OLD
cn0v 30910class 0vec
cnsb 30911class −𝑣
cnmcv 30912class normCV
cims 30913class IndMet
df-nv 30914NrmCVec = {⟨⟨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 30917 +𝑣 = (1st ∘ 1st )
df-ba 30918BaseSet = (𝑥 ∈ V ↦ ran ( +𝑣 ‘𝑥))
df-sm 30919 ·𝑠OLD = (2nd ∘ 1st )
df-0v 309200vec = (GId ∘ +𝑣 )
df-vs 30921 −𝑣 = ( /𝑔 ∘ +𝑣 )
df-nmcv 30922normCV = 2nd
df-ims 30923IndMet = (𝑢 ∈ NrmCVec ↦ ((normCV‘𝑢) ∘ ( −𝑣 ‘𝑢)))
cdip 31022class ·𝑖OLD
df-dip 31023·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 31043class SubSp
df-ssp 31044SubSp = (𝑢 ∈ NrmCVec ↦ {𝑤 ∈ NrmCVec ∣ (( +𝑣 ‘𝑤) ⊆ ( +𝑣 ‘𝑢) ∧ ( ·𝑠OLD ‘𝑤) ⊆ ( ·𝑠OLD ‘𝑢) ∧ (normCV‘𝑤) ⊆ (normCV‘𝑢))})
clno 31062class LnOp
cnmoo 31063class normOpOLD
cblo 31064class BLnOp
c0o 31065class 0op
df-lno 31066 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 31067 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 31068 BLnOp = (𝑢 ∈ NrmCVec, 𝑤 ∈ NrmCVec ↦ {𝑡 ∈ (𝑢 LnOp 𝑤) ∣ ((𝑢 normOpOLD 𝑤)‘𝑡) < +∞})
df-0o 31069 0op = (𝑢 ∈ NrmCVec, 𝑤 ∈ NrmCVec ↦ ((BaseSet‘𝑢) × {(0vec‘𝑤)}))
caj 31070class adj
chmo 31071class HmOp
df-aj 31072adj = (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 31073HmOp = (𝑢 ∈ NrmCVec ↦ {𝑡 ∈ dom (𝑢adj𝑢) ∣ ((𝑢adj𝑢)‘𝑡) = 𝑡})
ccphlo 31134class CPreHilOLD
df-ph 31135CPreHilOLD = (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 31184class CBan
df-cbn 31185CBan = {𝑢 ∈ NrmCVec ∣ (IndMet‘𝑢) ∈ (CMet‘(BaseSet‘𝑢))}
chlo 31207class CHilOLD
df-hlo 31208CHilOLD = (CBan ∩ CPreHilOLD)
The list of syntax, axioms (ax-) and definitions (df-) for the Hilbert Space Explorer starts here
chba 31241class ℋ
cva 31242class +ℎ
csm 31243class ·ℎ
csp 31244class ·ih
cno 31245class normℎ
c0v 31246class 0ℎ
cmv 31247class −ℎ
ccauold 31248class Cauchy
chli 31249class ⇝𝑣
csh 31250class Sℋ
cch 31251class Cℋ
cort 31252class ⊥
cph 31253class +ℋ
cspn 31254class span
chj 31255class ∨ℋ
chsup 31256class ∨ℋ
c0h 31257class 0ℋ
ccm 31258class 𝐶ℋ
cpjh 31259class projℎ
chos 31260class +op
chot 31261class ·op
chod 31262class −op
chfs 31263class +fn
chft 31264class ·fn
ch0o 31265class 0hop
chio 31266class Iop
cnop 31267class normop
ccop 31268class ContOp
clo 31269class LinOp
cbo 31270class BndLinOp
cuo 31271class UniOp
cho 31272class HrmOp
cnmf 31273class normfn
cnl 31274class null
ccnfn 31275class ContFn
clf 31276class LinFn
cado 31277class adjℎ
cbr 31278class bra
ck 31279class ketbra
cleo 31280class ≤op
cei 31281class eigvec
cel 31282class eigval
cspc 31283class Lambda
cst 31284class States
chst 31285class CHStates
ccv 31286class ⋖ℋ
cat 31287class HAtoms
cmd 31288class 𝑀ℋ
cdmd 31289class 𝑀ℋ*
df-hnorm 31290normℎ = (𝑥 ∈ dom dom ·ih ↦ (√‘(𝑥 ·ih 𝑥)))
df-hba 31291 ℋ = (BaseSet‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩)
df-h0v 312920ℎ = (0vec‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩)
df-hvsub 31293 −ℎ = (𝑥 ∈ ℋ, 𝑦 ∈ ℋ ↦ (𝑥 +ℎ (-1 ·ℎ 𝑦)))
df-hlim 31294 ⇝𝑣 = {⟨𝑓, 𝑤⟩ ∣ ((𝑓:ℕ⟶ ℋ ∧ 𝑤 ∈ ℋ) ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ≥‘𝑦)(normℎ‘((𝑓‘𝑧) −ℎ 𝑤)) < 𝑥)}
df-hcau 31295Cauchy = {𝑓 ∈ ( ℋ ↑m ℕ) ∣ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ≥‘𝑦)(normℎ‘((𝑓‘𝑦) −ℎ (𝑓‘𝑧))) < 𝑥}
ax-hilex 31321 ℋ ∈ V
ax-hfvadd 31322 +ℎ :( ℋ × ℋ)⟶ ℋ
ax-hvcom 31323((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 +ℎ 𝐵) = (𝐵 +ℎ 𝐴))
ax-hvass 31324((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 +ℎ 𝐵) +ℎ 𝐶) = (𝐴 +ℎ (𝐵 +ℎ 𝐶)))
ax-hv0cl 313250ℎ ∈ ℋ
ax-hvaddid 31326(𝐴 ∈ ℋ → (𝐴 +ℎ 0ℎ) = 𝐴)
ax-hfvmul 31327 ·ℎ :(ℂ × ℋ)⟶ ℋ
ax-hvmulid 31328(𝐴 ∈ ℋ → (1 ·ℎ 𝐴) = 𝐴)
ax-hvmulass 31329((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝐴 · 𝐵) ·ℎ 𝐶) = (𝐴 ·ℎ (𝐵 ·ℎ 𝐶)))
ax-hvdistr1 31330((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 ·ℎ (𝐵 +ℎ 𝐶)) = ((𝐴 ·ℎ 𝐵) +ℎ (𝐴 ·ℎ 𝐶)))
ax-hvdistr2 31331((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝐴 + 𝐵) ·ℎ 𝐶) = ((𝐴 ·ℎ 𝐶) +ℎ (𝐵 ·ℎ 𝐶)))
ax-hvmul0 31332(𝐴 ∈ ℋ → (0 ·ℎ 𝐴) = 0ℎ)
ax-hfi 31401 ·ih :( ℋ × ℋ)⟶ℂ
ax-his1 31404((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ih 𝐵) = (∗‘(𝐵 ·ih 𝐴)))
ax-his2 31405((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 +ℎ 𝐵) ·ih 𝐶) = ((𝐴 ·ih 𝐶) + (𝐵 ·ih 𝐶)))
ax-his3 31406((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 ·ℎ 𝐵) ·ih 𝐶) = (𝐴 · (𝐵 ·ih 𝐶)))
ax-his4 31407((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → 0 < (𝐴 ·ih 𝐴))
ax-hcompl 31524(𝐹 ∈ Cauchy → ∃𝑥 ∈ ℋ 𝐹 ⇝𝑣 𝑥)
df-sh 31529 Sℋ = {ℎ ∈ 𝒫 ℋ ∣ (0ℎ ∈ ℎ ∧ ( +ℎ “ (ℎ × ℎ)) ⊆ ℎ ∧ ( ·ℎ “ (ℂ × ℎ)) ⊆ ℎ)}
df-ch 31543 Cℋ = {ℎ ∈ Sℋ ∣ ( ⇝𝑣 “ (ℎ ↑m ℕ)) ⊆ ℎ}
df-oc 31574⊥ = (𝑥 ∈ 𝒫 ℋ ↦ {𝑦 ∈ ℋ ∣ ∀𝑧 ∈ 𝑥 (𝑦 ·ih 𝑧) = 0})
df-ch0 315750ℋ = {0ℎ}
df-shs 31630 +ℋ = (𝑥 ∈ Sℋ , 𝑦 ∈ Sℋ ↦ ( +ℎ “ (𝑥 × 𝑦)))
df-span 31631span = (𝑥 ∈ 𝒫 ℋ ↦ ∩ {𝑦 ∈ Sℋ ∣ 𝑥 ⊆ 𝑦})
df-chj 31632 ∨ℋ = (𝑥 ∈ 𝒫 ℋ, 𝑦 ∈ 𝒫 ℋ ↦ (⊥‘(⊥‘(𝑥 ∪ 𝑦))))
df-chsup 31633 ∨ℋ = (𝑥 ∈ 𝒫 𝒫 ℋ ↦ (⊥‘(⊥‘∪ 𝑥)))
df-pjh 31717projℎ = (ℎ ∈ Cℋ ↦ (𝑥 ∈ ℋ ↦ (℩𝑧 ∈ ℎ ∃𝑦 ∈ (⊥‘ℎ)𝑥 = (𝑧 +ℎ 𝑦))))
df-cm 31905 𝐶ℋ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ Cℋ ∧ 𝑦 ∈ Cℋ ) ∧ 𝑥 = ((𝑥 ∩ 𝑦) ∨ℋ (𝑥 ∩ (⊥‘𝑦))))}
df-hosum 32052 +op = (𝑓 ∈ ( ℋ ↑m ℋ), 𝑔 ∈ ( ℋ ↑m ℋ) ↦ (𝑥 ∈ ℋ ↦ ((𝑓‘𝑥) +ℎ (𝑔‘𝑥))))
df-homul 32053 ·op = (𝑓 ∈ ℂ, 𝑔 ∈ ( ℋ ↑m ℋ) ↦ (𝑥 ∈ ℋ ↦ (𝑓 ·ℎ (𝑔‘𝑥))))
df-hodif 32054 −op = (𝑓 ∈ ( ℋ ↑m ℋ), 𝑔 ∈ ( ℋ ↑m ℋ) ↦ (𝑥 ∈ ℋ ↦ ((𝑓‘𝑥) −ℎ (𝑔‘𝑥))))
df-hfsum 32055 +fn = (𝑓 ∈ (ℂ ↑m ℋ), 𝑔 ∈ (ℂ ↑m ℋ) ↦ (𝑥 ∈ ℋ ↦ ((𝑓‘𝑥) + (𝑔‘𝑥))))
df-hfmul 32056 ·fn = (𝑓 ∈ ℂ, 𝑔 ∈ (ℂ ↑m ℋ) ↦ (𝑥 ∈ ℋ ↦ (𝑓 · (𝑔‘𝑥))))
df-h0op 32070 0hop = (projℎ‘0ℋ)
df-iop 32071 Iop = (projℎ‘ ℋ)
df-nmop 32161normop = (𝑡 ∈ ( ℋ ↑m ℋ) ↦ sup({𝑥 ∣ ∃𝑧 ∈ ℋ ((normℎ‘𝑧) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑡‘𝑧)))}, ℝ*, < ))
df-cnop 32162ContOp = {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 32163LinOp = {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 32164BndLinOp = {𝑡 ∈ LinOp ∣ (normop‘𝑡) < +∞}
df-unop 32165UniOp = {𝑡 ∣ (𝑡: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑡‘𝑥) ·ih (𝑡‘𝑦)) = (𝑥 ·ih 𝑦))}
df-hmop 32166HrmOp = {𝑡 ∈ ( ℋ ↑m ℋ) ∣ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑡‘𝑦)) = ((𝑡‘𝑥) ·ih 𝑦)}
df-nmfn 32167normfn = (𝑡 ∈ (ℂ ↑m ℋ) ↦ sup({𝑥 ∣ ∃𝑧 ∈ ℋ ((normℎ‘𝑧) ≤ 1 ∧ 𝑥 = (abs‘(𝑡‘𝑧)))}, ℝ*, < ))
df-nlfn 32168null = (𝑡 ∈ (ℂ ↑m ℋ) ↦ (◡𝑡 “ {0}))
df-cnfn 32169ContFn = {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 32170LinFn = {𝑡 ∈ (ℂ ↑m ℋ) ∣ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ (𝑡‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑥 · (𝑡‘𝑦)) + (𝑡‘𝑧))}
df-adjh 32171adjℎ = {⟨𝑡, 𝑢⟩ ∣ (𝑡: ℋ⟶ ℋ ∧ 𝑢: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑡‘𝑥) ·ih 𝑦) = (𝑥 ·ih (𝑢‘𝑦)))}
df-bra 32172bra = (𝑥 ∈ ℋ ↦ (𝑦 ∈ ℋ ↦ (𝑦 ·ih 𝑥)))
df-kb 32173 ketbra = (𝑥 ∈ ℋ, 𝑦 ∈ ℋ ↦ (𝑧 ∈ ℋ ↦ ((𝑧 ·ih 𝑦) ·ℎ 𝑥)))
df-leop 32174 ≤op = {⟨𝑡, 𝑢⟩ ∣ ((𝑢 −op 𝑡) ∈ HrmOp ∧ ∀𝑥 ∈ ℋ 0 ≤ (((𝑢 −op 𝑡)‘𝑥) ·ih 𝑥))}
df-eigvec 32175eigvec = (𝑡 ∈ ( ℋ ↑m ℋ) ↦ {𝑥 ∈ ( ℋ ∖ 0ℋ) ∣ ∃𝑧 ∈ ℂ (𝑡‘𝑥) = (𝑧 ·ℎ 𝑥)})
df-eigval 32176eigval = (𝑡 ∈ ( ℋ ↑m ℋ) ↦ (𝑥 ∈ (eigvec‘𝑡) ↦ (((𝑡‘𝑥) ·ih 𝑥) / ((normℎ‘𝑥)↑2))))
df-spec 32177Lambda = (𝑡 ∈ ( ℋ ↑m ℋ) ↦ {𝑥 ∈ ℂ ∣ ¬ (𝑡 −op (𝑥 ·op ( I ↾ ℋ))): ℋ–1-1→ ℋ})
df-st 32533States = {𝑓 ∈ ((0[,]1) ↑m Cℋ ) ∣ ((𝑓‘ ℋ) = 1 ∧ ∀𝑥 ∈ Cℋ ∀𝑦 ∈ Cℋ (𝑥 ⊆ (⊥‘𝑦) → (𝑓‘(𝑥 ∨ℋ 𝑦)) = ((𝑓‘𝑥) + (𝑓‘𝑦))))}
df-hst 32534CHStates = {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 32601 ⋖ℋ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ Cℋ ∧ 𝑦 ∈ Cℋ ) ∧ (𝑥 ⊊ 𝑦 ∧ ¬ ∃𝑧 ∈ Cℋ (𝑥 ⊊ 𝑧 ∧ 𝑧 ⊊ 𝑦)))}
df-md 32602 𝑀ℋ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ Cℋ ∧ 𝑦 ∈ Cℋ ) ∧ ∀𝑧 ∈ Cℋ (𝑧 ⊆ 𝑦 → ((𝑧 ∨ℋ 𝑥) ∩ 𝑦) = (𝑧 ∨ℋ (𝑥 ∩ 𝑦))))}
df-dmd 32603 𝑀ℋ* = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ Cℋ ∧ 𝑦 ∈ Cℋ ) ∧ ∀𝑧 ∈ Cℋ (𝑦 ⊆ 𝑧 → ((𝑧 ∩ 𝑥) ∨ℋ 𝑦) = (𝑧 ∩ (𝑥 ∨ℋ 𝑦))))}
df-at 32660HAtoms = {𝑥 ∈ Cℋ ∣ 0ℋ ⋖ℋ 𝑥}
The list of syntax, axioms (ax-) and definitions (df-) for the User Mathboxes starts here
w2reu 32794wff ∃!𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵𝜑
df-2reu 32795(∃!𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵𝜑 ↔ (∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ ∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑))
cdp2 33160class 𝐴𝐵
df-dp2 33161𝐴𝐵 = (𝐴 + (𝐵 / 10))
cdp 33177class .
df-dp 33178. = (𝑥 ∈ ℕ0, 𝑦 ∈ ℝ ↦ 𝑥𝑦)
cxdiv 33206class /𝑒
df-xdiv 33207 /𝑒 = (𝑥 ∈ ℝ*, 𝑦 ∈ (ℝ ∖ {0}) ↦ (℩𝑧 ∈ ℝ* (𝑦 ·e 𝑧) = 𝑥))
cmnt 33268class Monot
cmgc 33269class MGalConn
df-mnt 33270Monot = (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 33271MGalConn = (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 33294ℝfld = (Scalar‘ℝ*𝑠)
ax-xrsvsca 33295 ·e = ( ·𝑠 ‘ℝ*𝑠)
ctocyc 33396class toCyc
df-tocyc 33397toCyc = (𝑑 ∈ V ↦ (𝑤 ∈ {𝑢 ∈ Word 𝑑 ∣ 𝑢:dom 𝑢–1-1→𝑑} ↦ (( I ↾ (𝑑 ∖ ran 𝑤)) ∪ ((𝑤 cyclShift 1) ∘ ◡𝑤))))
csgns 33448class sgns
df-sgns 33449sgns = (𝑟 ∈ V ↦ (𝑥 ∈ (Base‘𝑟) ↦ if(𝑥 = (0g‘𝑟), 0, if((0g‘𝑟)(lt‘𝑟)𝑥, 1, -1))))
cfxp 33453class FixPts
df-fxp 33454FixPts = (𝑏 ∈ V, 𝑎 ∈ V ↦ {𝑥 ∈ 𝑏 ∣ ∀𝑝 ∈ dom dom 𝑎(𝑝𝑎𝑥) = 𝑥})
cinftm 33466class ⋘
carchi 33467class Archi
df-inftm 33468⋘ = (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 33469Archi = {𝑤 ∣ (⋘‘𝑤) = ∅}
cslmd 33490class SLMod
df-slmd 33491SLMod = {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 33543class ~RL
crloc 33544class RLocal
df-erl 33545 ~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 33546 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 33581class EuclF
df-euf 33582EuclF = Slot 21
cedom 33585class EDomn
df-edom 33586EDomn = {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 33593class Frac
df-frac 33594 Frac = (𝑟 ∈ V ↦ (𝑟 RLocal (RLReg‘𝑟)))
cfldgen 33601class fldGen
df-fldgen 33602 fldGen = (𝑓 ∈ V, 𝑠 ∈ V ↦ ∩ {𝑎 ∈ (SubDRing‘𝑓) ∣ 𝑠 ⊆ 𝑎})
cresv 33616class ↾v
df-resv 33617 ↾v = (𝑤 ∈ V, 𝑥 ∈ V ↦ if((Base‘(Scalar‘𝑤)) ⊆ 𝑥, 𝑤, (𝑤 sSet ⟨(Scalar‘ndx), ((Scalar‘𝑤) ↾s 𝑥)⟩)))
cmxidl 33712class MaxIdeal
df-mxidl 33713MaxIdeal = (𝑟 ∈ Ring ↦ {𝑖 ∈ (LIdeal‘𝑟) ∣ (𝑖 ≠ (Base‘𝑟) ∧ ∀𝑗 ∈ (LIdeal‘𝑟)(𝑖 ⊆ 𝑗 → (𝑗 = 𝑖 ∨ 𝑗 = (Base‘𝑟))))})
cidlsrg 33760class IDLsrg
df-idlsrg 33761IDLsrg = (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 33798class UFD
df-ufd 33799UFD = {𝑟 ∈ IDomn ∣ ∀𝑖 ∈ ((PrmIdeal‘𝑟) ∖ {{(0g‘𝑟)}})(𝑖 ∩ (RPrime‘𝑟)) ≠ ∅}
cextv 33889class extendVars
df-extv 33890extendVars = (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 33915class SymPoly
cesply 33916class eSymPoly
df-sply 33917SymPoly = (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 33918eSymPoly = (𝑖 ∈ V, 𝑟 ∈ V ↦ (𝑘 ∈ ℕ0 ↦ ((ℤRHom‘𝑟) ∘ ((𝟭‘{ℎ ∈ (ℕ0 ↑m 𝑖) ∣ ℎ finSupp 0})‘((𝟭‘𝑖) “ {𝑐 ∈ 𝒫 𝑖 ∣ (♯‘𝑐) = 𝑘})))))
cldim 33959class dim
df-dim 33960dim = (𝑓 ∈ V ↦ ∪ (♯ “ (LBasis‘𝑓)))
cfldext 33998class /FldExt
cfinext 33999class /FinExt
cextdg 34000class [:]
df-fldext 34001/FldExt = {⟨𝑒, 𝑓⟩ ∣ ((𝑒 ∈ Field ∧ 𝑓 ∈ Field) ∧ (𝑓 = (𝑒 ↾s (Base‘𝑓)) ∧ (Base‘𝑓) ∈ (SubRing‘𝑒)))}
df-extdg 34002[:] = (𝑒 ∈ V, 𝑓 ∈ (/FldExt “ {𝑒}) ↦ (dim‘((subringAlg ‘𝑒)‘(Base‘𝑓))))
df-finext 34003/FinExt = {⟨𝑒, 𝑓⟩ ∣ (𝑒/FldExt𝑓 ∧ (𝑒[:]𝑓) ∈ ℕ0)}
cirng 34043class IntgRing
df-irng 34044 IntgRing = (𝑟 ∈ V, 𝑠 ∈ V ↦ ∪ 𝑓 ∈ (Monic1p‘(𝑟 ↾s 𝑠))(◡((𝑟 evalSub1 𝑠)‘𝑓) “ {(0g‘𝑟)}))
calgext 34055class /AlgExt
df-algext 34056/AlgExt = {⟨𝑒, 𝑓⟩ ∣ (𝑒/FldExt𝑓 ∧ (𝑒 IntgRing (Base‘𝑓)) = (Base‘𝑒))}
cminply 34059class minPoly
df-minply 34060 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 34089class Constr
df-constr 34090Constr = ∪ (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 34153class subMat1
df-smat 34154subMat1 = (𝑚 ∈ V ↦ (𝑘 ∈ ℕ, 𝑙 ∈ ℕ ↦ (𝑚 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝑘, 𝑖, (𝑖 + 1)), if(𝑗 < 𝑙, 𝑗, (𝑗 + 1))⟩))))
clmat 34171class litMat
df-lmat 34172litMat = (𝑚 ∈ V ↦ (𝑖 ∈ (1...(♯‘𝑚)), 𝑗 ∈ (1...(♯‘(𝑚‘0))) ↦ ((𝑚‘(𝑖 − 1))‘(𝑗 − 1))))
ccref 34202class CovHasRef𝐴
df-cref 34203CovHasRef𝐴 = {𝑗 ∈ Top ∣ ∀𝑦 ∈ 𝒫 𝑗(∪ 𝑗 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑗 ∩ 𝐴)𝑧Ref𝑦)}
cldlf 34212class Ldlf
df-ldlf 34213Ldlf = CovHasRef{𝑥 ∣ 𝑥 ≼ ω}
cpcmp 34215class Paracomp
df-pcmp 34216Paracomp = {𝑗 ∣ 𝑗 ∈ CovHasRef(LocFin‘𝑗)}
crspec 34222class Spec
df-rspec 34223Spec = (𝑟 ∈ Ring ↦ ((IDLsrg‘𝑟) ↾s (PrmIdeal‘𝑟)))
cmetid 34246class ~Met
cpstm 34247class pstoMet
df-metid 34248~Met = (𝑑 ∈ ∪ ran PsMet ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ dom dom 𝑑 ∧ 𝑦 ∈ dom dom 𝑑) ∧ (𝑥𝑑𝑦) = 0)})
df-pstm 34249pstoMet = (𝑑 ∈ ∪ ran PsMet ↦ (𝑎 ∈ (dom dom 𝑑 / (~Met‘𝑑)), 𝑏 ∈ (dom dom 𝑑 / (~Met‘𝑑)) ↦ ∪ {𝑧 ∣ ∃𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝑧 = (𝑥𝑑𝑦)}))
chcmp 34316class HCmp
df-hcmp 34317HCmp = {⟨𝑢, 𝑤⟩ ∣ ((𝑢 ∈ ∪ ran UnifOn ∧ 𝑤 ∈ CUnifSp) ∧ ((UnifSt‘𝑤) ↾t dom ∪ 𝑢) = 𝑢 ∧ ((cls‘(TopOpen‘𝑤))‘dom ∪ 𝑢) = (Base‘𝑤))}
cqqh 34330class ℚHom
df-qqh 34331ℚHom = (𝑟 ∈ V ↦ ran (𝑥 ∈ ℤ, 𝑦 ∈ (◡(ℤRHom‘𝑟) “ (Unit‘𝑟)) ↦ ⟨(𝑥 / 𝑦), (((ℤRHom‘𝑟)‘𝑥)(/r‘𝑟)((ℤRHom‘𝑟)‘𝑦))⟩))
crrh 34353class ℝHom
crrext 34354class ℝExt
df-rrh 34355ℝHom = (𝑟 ∈ V ↦ (((topGen‘ran (,))CnExt(TopOpen‘𝑟))‘(ℚHom‘𝑟)))
df-rrext 34359 ℝExt = {𝑟 ∈ (NrmRing ∩ DivRing) ∣ (((ℤMod‘𝑟) ∈ NrmMod ∧ (chr‘𝑟) = 0) ∧ (𝑟 ∈ CUnifSp ∧ (UnifSt‘𝑟) = (metUnif‘((dist‘𝑟) ↾ ((Base‘𝑟) × (Base‘𝑟))))))}
cxrh 34376class ℝ*Hom
df-xrh 34377ℝ*Hom = (𝑟 ∈ V ↦ (𝑥 ∈ ℝ* ↦ if(𝑥 ∈ ℝ, ((ℝHom‘𝑟)‘𝑥), if(𝑥 = +∞, ((lub‘𝑟)‘((ℝHom‘𝑟) “ ℝ)), ((glb‘𝑟)‘((ℝHom‘𝑟) “ ℝ))))))
cmntop 34382class ManTop
df-mntop 34383ManTop = {⟨𝑛, 𝑗⟩ ∣ (𝑛 ∈ ℕ0 ∧ (𝑗 ∈ 2ndω ∧ 𝑗 ∈ Haus ∧ 𝑗 ∈ Locally [(TopOpen‘(𝔼hil‘𝑛))] ≃ ))}
cesum 34387class Σ*𝑘 ∈ 𝐴𝐵
df-esum 34388Σ*𝑘 ∈ 𝐴𝐵 = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵))
cofc 34455class ∘f/c 𝑅
df-ofc 34456 ∘f/c 𝑅 = (𝑓 ∈ V, 𝑐 ∈ V ↦ (𝑥 ∈ dom 𝑓 ↦ ((𝑓‘𝑥)𝑅𝑐)))
csiga 34468class sigAlgebra
df-siga 34469sigAlgebra = (𝑜 ∈ V ↦ {𝑠 ∣ (𝑠 ⊆ 𝒫 𝑜 ∧ (𝑜 ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑜 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑠)))})
csigagen 34498class sigaGen
df-sigagen 34499sigaGen = (𝑥 ∈ V ↦ ∩ {𝑠 ∈ (sigAlgebra‘∪ 𝑥) ∣ 𝑥 ⊆ 𝑠})
cbrsiga 34541class 𝔅ℝ
df-brsiga 34542𝔅ℝ = (sigaGen‘(topGen‘ran (,)))
csx 34548class ×s
df-sx 34549 ×s = (𝑠 ∈ V, 𝑡 ∈ V ↦ (sigaGen‘ran (𝑥 ∈ 𝑠, 𝑦 ∈ 𝑡 ↦ (𝑥 × 𝑦))))
cmeas 34555class measures
df-meas 34556measures = (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 34592class δ
df-dde 34593δ = (𝑎 ∈ 𝒫 ℝ ↦ if(0 ∈ 𝑎, 1, 0))
cae 34597class a.e.
cfae 34598class ~ a.e.
df-ae 34599a.e. = {⟨𝑎, 𝑚⟩ ∣ (𝑚‘(∪ dom 𝑚 ∖ 𝑎)) = 0}
df-fae 34605~ 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 34609class MblFnM
df-mbfm 34610MblFnM = (𝑠 ∈ ∪ ran sigAlgebra, 𝑡 ∈ ∪ ran sigAlgebra ↦ {𝑓 ∈ (∪ 𝑡 ↑m ∪ 𝑠) ∣ ∀𝑥 ∈ 𝑡 (◡𝑓 “ 𝑥) ∈ 𝑠})
coms 34651class toOMeas
df-oms 34652toOMeas = (𝑟 ∈ V ↦ (𝑎 ∈ 𝒫 ∪ dom 𝑟 ↦ inf(ran (𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑟 ∣ (𝑎 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω)} ↦ Σ*𝑦 ∈ 𝑥(𝑟‘𝑦)), (0[,]+∞), < )))
ccarsg 34661class toCaraSiga
df-carsg 34662toCaraSiga = (𝑚 ∈ V ↦ {𝑎 ∈ 𝒫 ∪ dom 𝑚 ∣ ∀𝑒 ∈ 𝒫 ∪ dom 𝑚((𝑚‘(𝑒 ∩ 𝑎)) +𝑒 (𝑚‘(𝑒 ∖ 𝑎))) = (𝑚‘𝑒)})
citgm 34687class itgm
csitm 34688class sitm
csitg 34689class sitg
df-sitg 34690sitg = (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 34691sitm = (𝑤 ∈ V, 𝑚 ∈ ∪ ran measures ↦ (𝑓 ∈ dom (𝑤sitg𝑚), 𝑔 ∈ dom (𝑤sitg𝑚) ↦ (((ℝ*𝑠 ↾s (0[,]+∞))sitg𝑚)‘(𝑓 ∘f (dist‘𝑤)𝑔))))
df-itgm 34713itgm = (𝑤 ∈ V, 𝑚 ∈ ∪ ran measures ↦ (((metUnif‘(𝑤sitm𝑚))CnExt(UnifSt‘𝑤))‘(𝑤sitg𝑚)))
csseq 34743class seqstr
df-sseq 34744seqstr = (𝑚 ∈ V, 𝑓 ∈ V ↦ (𝑚 ∪ (lastS ∘ seq(♯‘𝑚)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝑓‘𝑥)”⟩)), (ℕ0 × {(𝑚 ++ ⟨“(𝑓‘𝑚)”⟩)})))))
cfib 34756class Fibci
df-fib 34757Fibci = (⟨“01”⟩seqstr(𝑤 ∈ (Word ℕ0 ∩ (◡♯ “ (ℤ≥‘2))) ↦ ((𝑤‘((♯‘𝑤) − 2)) + (𝑤‘((♯‘𝑤) − 1)))))
cprb 34767class Prob
df-prob 34768Prob = {𝑝 ∈ ∪ ran measures ∣ (𝑝‘∪ dom 𝑝) = 1}
ccprob 34791class cprob
df-cndprob 34792cprob = (𝑝 ∈ Prob ↦ (𝑎 ∈ dom 𝑝, 𝑏 ∈ dom 𝑝 ↦ ((𝑝‘(𝑎 ∩ 𝑏)) / (𝑝‘𝑏))))
crrv 34800class rRndVar
df-rrv 34801rRndVar = (𝑝 ∈ Prob ↦ (dom 𝑝MblFnM𝔅ℝ))
corvc 34816class ∘RV/𝑐𝑅
df-orvc 34817∘RV/𝑐𝑅 = (𝑥 ∈ {𝑥 ∣ Fun 𝑥}, 𝑎 ∈ V ↦ (◡𝑥 “ {𝑦 ∣ 𝑦𝑅𝑎}))
crepr 34965class repr
df-repr 34966repr = (𝑠 ∈ ℕ0 ↦ (𝑏 ∈ 𝒫 ℕ, 𝑚 ∈ ℤ ↦ {𝑐 ∈ (𝑏 ↑m (0..^𝑠)) ∣ Σ𝑎 ∈ (0..^𝑠)(𝑐‘𝑎) = 𝑚}))
cvts 34992class vts
df-vts 34993vts = (𝑙 ∈ (ℂ ↑m ℕ), 𝑛 ∈ ℕ0 ↦ (𝑥 ∈ ℂ ↦ Σ𝑎 ∈ (1...𝑛)((𝑙‘𝑎) · (exp‘((i · (2 · π)) · (𝑎 · 𝑥))))))
ax-hgt749 35001∀𝑛 ∈ {𝑧 ∈ ℤ ∣ ¬ 2 ∥ 𝑧} ((10↑27) ≤ 𝑛 → ∃ℎ ∈ ((0[,)+∞) ↑m ℕ)∃𝑘 ∈ ((0[,)+∞) ↑m ℕ)(∀𝑚 ∈ ℕ (𝑘‘𝑚) ≤ (1.079955) ∧ ∀𝑚 ∈ ℕ (ℎ‘𝑚) ≤ (1.414) ∧ ((0.00042248) · (𝑛↑2)) ≤ ∫(0(,)1)(((((Λ ∘f · ℎ)vts𝑛)‘𝑥) · ((((Λ ∘f · 𝑘)vts𝑛)‘𝑥)↑2)) · (exp‘((i · (2 · π)) · (-𝑛 · 𝑥)))) d𝑥))
ax-ros335 35002∀𝑥 ∈ ℝ+ (ψ‘𝑥) < ((1.03883) · 𝑥)
ax-ros336 35003∀𝑥 ∈ ℝ+ ((ψ‘𝑥) − (θ‘𝑥)) < ((1.4262) · (√‘𝑥))
cstrkg2d 35021class TarskiG2D
df-trkg2d 35022TarskiG2D = {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 35029class AFS
df-afs 35030AFS = (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 35034class leftpad
df-lpad 35035 leftpad = (𝑐 ∈ V, 𝑤 ∈ V ↦ (𝑙 ∈ ℕ0 ↦ (((0..^(𝑙 − (♯‘𝑤))) × {𝑐}) ++ 𝑤)))
w-bnj17 35045wff (𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃)
df-bnj17 35046((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ ((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃))
c-bnj14 35047class pred(𝑋, 𝐴, 𝑅)
df-bnj14 35048 pred(𝑋, 𝐴, 𝑅) = {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋}
w-bnj13 35049wff 𝑅 Se 𝐴
df-bnj13 35050(𝑅 Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 pred(𝑥, 𝐴, 𝑅) ∈ V)
w-bnj15 35051wff 𝑅 FrSe 𝐴
df-bnj15 35052(𝑅 FrSe 𝐴 ↔ (𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴))
c-bnj18 35053class trCl(𝑋, 𝐴, 𝑅)
df-bnj18 35054 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 35055wff TrFo(𝐵, 𝐴, 𝑅)
df-bnj19 35056( TrFo(𝐵, 𝐴, 𝑅) ↔ ∀𝑥 ∈ 𝐵 pred(𝑥, 𝐴, 𝑅) ⊆ 𝐵)
ax-regs 35497(∃𝑥𝜑 → ∃𝑦(∀𝑥(𝑥 = 𝑦 → 𝜑) ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ ∀𝑥(𝑥 = 𝑧 → 𝜑))))
ckard 35520class kard
df-kard 35521kard = (𝑥 ∈ V ↦ Scott {𝑦 ∣ 𝑦 ≈ 𝑥})
cacycgr 35592class AcyclicGraph
df-acycgr 35593AcyclicGraph = {𝑔 ∣ ¬ ∃𝑓∃𝑝(𝑓(Cycles‘𝑔)𝑝 ∧ 𝑓 ≠ ∅)}
ax-7d 35609(∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑)
ax-8d 35610(𝑥 = 𝑦 → (𝑥 = 𝑧 → 𝑦 = 𝑧))
ax-9d1 35611 ¬ ∀𝑥 ¬ 𝑥 = 𝑥
ax-9d2 35612 ¬ ∀𝑥 ¬ 𝑥 = 𝑦
ax-10d 35613(∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
ax-11d 35614(𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
cretr 35667class Retr
df-retr 35668 Retr = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ {𝑟 ∈ (𝑗 Cn 𝑘) ∣ ∃𝑠 ∈ (𝑘 Cn 𝑗)((𝑟 ∘ 𝑠)(𝑗 Htpy 𝑗)( I ↾ ∪ 𝑗)) ≠ ∅})
cpconn 35669class PConn
csconn 35670class SConn
df-pconn 35671PConn = {𝑗 ∈ Top ∣ ∀𝑥 ∈ ∪ 𝑗∀𝑦 ∈ ∪ 𝑗∃𝑓 ∈ (II Cn 𝑗)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)}
df-sconn 35672SConn = {𝑗 ∈ PConn ∣ ∀𝑓 ∈ (II Cn 𝑗)((𝑓‘0) = (𝑓‘1) → 𝑓( ≃ph‘𝑗)((0[,]1) × {(𝑓‘0)}))}
ccvm 35705class CovMap
df-cvm 35706 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 35783class ∈𝑔
cgna 35784class ⊼𝑔
cgol 35785class ∀𝑔𝑁𝑈
csat 35786class Sat
cfmla 35787class Fmla
csate 35788class Sat∈
cprv 35789class ⊧
df-goel 35790∈𝑔 = (𝑥 ∈ (ω × ω) ↦ ⟨∅, 𝑥⟩)
df-gona 35791⊼𝑔 = (𝑥 ∈ (V × V) ↦ ⟨1o, 𝑥⟩)
df-goal 35792∀𝑔𝑁𝑈 = ⟨2o, ⟨𝑁, 𝑈⟩⟩
df-sat 35793 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 35794 Sat∈ = (𝑚 ∈ V, 𝑢 ∈ V ↦ (((𝑚 Sat ( E ∩ (𝑚 × 𝑚)))‘ω)‘𝑢))
df-fmla 35795Fmla = (𝑛 ∈ suc ω ↦ dom ((∅ Sat ∅)‘𝑛))
df-prv 35796⊧ = {⟨𝑚, 𝑢⟩ ∣ (𝑚 Sat∈ 𝑢) = (𝑚 ↑m ω)}
cgon 35882class ¬𝑔𝑈
cgoa 35883class ∧𝑔
cgoi 35884class →𝑔
cgoo 35885class ∨𝑔
cgob 35886class ↔𝑔
cgoq 35887class =𝑔
cgox 35888class ∃𝑔𝑁𝑈
df-gonot 35889¬𝑔𝑈 = (𝑈⊼𝑔𝑈)
df-goan 35890∧𝑔 = (𝑢 ∈ V, 𝑣 ∈ V ↦ ¬𝑔(𝑢⊼𝑔𝑣))
df-goim 35891 →𝑔 = (𝑢 ∈ V, 𝑣 ∈ V ↦ (𝑢⊼𝑔¬𝑔𝑣))
df-goor 35892 ∨𝑔 = (𝑢 ∈ V, 𝑣 ∈ V ↦ (¬𝑔𝑢 →𝑔 𝑣))
df-gobi 35893 ↔𝑔 = (𝑢 ∈ V, 𝑣 ∈ V ↦ ((𝑢 →𝑔 𝑣)∧𝑔(𝑣 →𝑔 𝑢)))
df-goeq 35894=𝑔 = (𝑢 ∈ ω, 𝑣 ∈ ω ↦ ⦋suc (𝑢 ∪ 𝑣) / 𝑤⦌∀𝑔𝑤((𝑤∈𝑔𝑢) ↔𝑔 (𝑤∈𝑔𝑣)))
df-goex 35895∃𝑔𝑁𝑈 = ¬𝑔∀𝑔𝑁¬𝑔𝑈
cgze 35896class AxExt
cgzr 35897class AxRep
cgzp 35898class AxPow
cgzu 35899class AxUn
cgzg 35900class AxReg
cgzi 35901class AxInf
cgzf 35902class ZF
df-gzext 35903AxExt = (∀𝑔2o((2o∈𝑔∅) ↔𝑔 (2o∈𝑔1o)) →𝑔 (∅=𝑔1o))
df-gzrep 35904AxRep = (𝑢 ∈ (Fmla‘ω) ↦ (∀𝑔3o∃𝑔1o∀𝑔2o(∀𝑔1o𝑢 →𝑔 (2o=𝑔1o)) →𝑔 ∀𝑔1o∀𝑔2o((2o∈𝑔1o) ↔𝑔 ∃𝑔3o((3o∈𝑔∅)∧𝑔∀𝑔1o𝑢))))
df-gzpow 35905AxPow = ∃𝑔1o∀𝑔2o(∀𝑔1o((1o∈𝑔2o) ↔𝑔 (1o∈𝑔∅)) →𝑔 (2o∈𝑔1o))
df-gzun 35906AxUn = ∃𝑔1o∀𝑔2o(∃𝑔1o((2o∈𝑔1o)∧𝑔(1o∈𝑔∅)) →𝑔 (2o∈𝑔1o))
df-gzreg 35907AxReg = (∃𝑔1o(1o∈𝑔∅) →𝑔 ∃𝑔1o((1o∈𝑔∅)∧𝑔∀𝑔2o((2o∈𝑔1o) →𝑔 ¬𝑔(2o∈𝑔∅))))
df-gzinf 35908AxInf = ∃𝑔1o((∅∈𝑔1o)∧𝑔∀𝑔2o((2o∈𝑔1o) →𝑔 ∃𝑔∅((2o∈𝑔∅)∧𝑔(∅∈𝑔1o))))
df-gzf 35909ZF = {𝑚 ∣ ((Tr 𝑚 ∧ 𝑚⊧AxExt ∧ 𝑚⊧AxPow) ∧ (𝑚⊧AxUn ∧ 𝑚⊧AxReg ∧ 𝑚⊧AxInf) ∧ ∀𝑢 ∈ (Fmla‘ω)𝑚⊧(AxRep‘𝑢))}
cmcn 35910class mCN
cmvar 35911class mVR
cmty 35912class mType
cmvt 35913class mVT
cmtc 35914class mTC
cmax 35915class mAx
cmrex 35916class mREx
cmex 35917class mEx
cmdv 35918class mDV
cmvrs 35919class mVars
cmrsub 35920class mRSubst
cmsub 35921class mSubst
cmvh 35922class mVH
cmpst 35923class mPreSt
cmsr 35924class mStRed
cmsta 35925class mStat
cmfs 35926class mFS
cmcls 35927class mCls
cmpps 35928class mPPSt
cmthm 35929class mThm
df-mcn 35930mCN = Slot 1
df-mvar 35931mVR = Slot 2
df-mty 35932mType = Slot 3
df-mtc 35933mTC = Slot 4
df-mmax 35934mAx = Slot 5
df-mvt 35935mVT = (𝑡 ∈ V ↦ ran (mType‘𝑡))
df-mrex 35936mREx = (𝑡 ∈ V ↦ Word ((mCN‘𝑡) ∪ (mVR‘𝑡)))
df-mex 35937mEx = (𝑡 ∈ V ↦ ((mTC‘𝑡) × (mREx‘𝑡)))
df-mdv 35938mDV = (𝑡 ∈ V ↦ (((mVR‘𝑡) × (mVR‘𝑡)) ∖ I ))
df-mvrs 35939mVars = (𝑡 ∈ V ↦ (𝑒 ∈ (mEx‘𝑡) ↦ (ran (2nd ‘𝑒) ∩ (mVR‘𝑡))))
df-mrsub 35940mRSubst = (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 35941mSubst = (𝑡 ∈ V ↦ (𝑓 ∈ ((mREx‘𝑡) ↑pm (mVR‘𝑡)) ↦ (𝑒 ∈ (mEx‘𝑡) ↦ ⟨(1st ‘𝑒), (((mRSubst‘𝑡)‘𝑓)‘(2nd ‘𝑒))⟩)))
df-mvh 35942mVH = (𝑡 ∈ V ↦ (𝑣 ∈ (mVR‘𝑡) ↦ ⟨((mType‘𝑡)‘𝑣), ⟨“𝑣”⟩⟩))
df-mpst 35943mPreSt = (𝑡 ∈ V ↦ (({𝑑 ∈ 𝒫 (mDV‘𝑡) ∣ ◡𝑑 = 𝑑} × (𝒫 (mEx‘𝑡) ∩ Fin)) × (mEx‘𝑡)))
df-msr 35944mStRed = (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 35945mStat = (𝑡 ∈ V ↦ ran (mStRed‘𝑡))
df-mfs 35946mFS = {𝑡 ∣ ((((mCN‘𝑡) ∩ (mVR‘𝑡)) = ∅ ∧ (mType‘𝑡):(mVR‘𝑡)⟶(mTC‘𝑡)) ∧ ((mAx‘𝑡) ⊆ (mStat‘𝑡) ∧ ∀𝑣 ∈ (mVT‘𝑡) ¬ (◡(mType‘𝑡) “ {𝑣}) ∈ Fin))}
df-mcls 35947mCls = (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 35948mPPSt = (𝑡 ∈ V ↦ {⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∣ (⟨𝑑, ℎ, 𝑎⟩ ∈ (mPreSt‘𝑡) ∧ 𝑎 ∈ (𝑑(mCls‘𝑡)ℎ))})
df-mthm 35949mThm = (𝑡 ∈ V ↦ (◡(mStRed‘𝑡) “ ((mStRed‘𝑡) “ (mPPSt‘𝑡))))
cm0s 36035class m0St
cmsa 36036class mSA
cmwgfs 36037class mWGFS
cmsy 36038class mSyn
cmesy 36039class mESyn
cmgfs 36040class mGFS
cmtree 36041class mTree
cmst 36042class mST
cmsax 36043class mSAX
cmufs 36044class mUFS
df-m0s 36045m0St = (𝑎 ∈ V ↦ ⟨∅, ∅, 𝑎⟩)
df-msa 36046mSA = (𝑡 ∈ V ↦ {𝑎 ∈ (mEx‘𝑡) ∣ ((m0St‘𝑎) ∈ (mAx‘𝑡) ∧ (1st ‘𝑎) ∈ (mVT‘𝑡) ∧ Fun (◡(2nd ‘𝑎) ↾ (mVR‘𝑡)))})
df-mwgfs 36047mWGFS = {𝑡 ∈ mFS ∣ ∀𝑑∀ℎ∀𝑎((⟨𝑑, ℎ, 𝑎⟩ ∈ (mAx‘𝑡) ∧ (1st ‘𝑎) ∈ (mVT‘𝑡)) → ∃𝑠 ∈ ran (mSubst‘𝑡)𝑎 ∈ (𝑠 “ (mSA‘𝑡)))}
df-msyn 36048mSyn = Slot 6
df-mesyn 36049mESyn = (𝑡 ∈ V ↦ (𝑐 ∈ (mTC‘𝑡), 𝑒 ∈ (mREx‘𝑡) ↦ (((mSyn‘𝑡)‘𝑐)m0St𝑒)))
df-mgfs 36050mGFS = {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 36051mTree = (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 36052mST = (𝑡 ∈ V ↦ ((∅(mTree‘𝑡)∅) ↾ ((mEx‘𝑡) ↾ (mVT‘𝑡))))
df-msax 36053mSAX = (𝑡 ∈ V ↦ (𝑝 ∈ (mSA‘𝑡) ↦ ((mVH‘𝑡) “ ((mVars‘𝑡)‘𝑝))))
df-mufs 36054mUFS = {𝑡 ∈ mGFS ∣ Fun (mST‘𝑡)}
cmuv 36055class mUV
cmvl 36056class mVL
cmvsb 36057class mVSubst
cmfsh 36058class mFresh
cmfr 36059class mFRel
cmevl 36060class mEval
cmdl 36061class mMdl
cusyn 36062class mUSyn
cgmdl 36063class mGMdl
cmitp 36064class mItp
cmfitp 36065class mFromItp
df-muv 36066mUV = Slot 7
df-mfsh 36067mFresh = Slot 19
df-mevl 36068mEval = Slot 20
df-mvl 36069mVL = (𝑡 ∈ V ↦ X𝑣 ∈ (mVR‘𝑡)((mUV‘𝑡) “ {((mType‘𝑡)‘𝑣)}))
df-mvsb 36070mVSubst = (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 36071mFRel = (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 36072mMdl = {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 36073mUSyn = (𝑡 ∈ V ↦ (𝑣 ∈ (mUV‘𝑡) ↦ ⟨((mSyn‘𝑡)‘(1st ‘𝑣)), (2nd ‘𝑣)⟩))
df-gmdl 36074mGMdl = {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 36075mItp = (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 36076mFromItp = (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 36077class cplMetSp
chlb 36078class HomLimB
chlim 36079class HomLim
cpfl 36080class polyFld
csf1 36081class splitFld1
csf 36082class splitFld
cpsl 36083class polySplitLim
df-cplmet 36084 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 36085 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 36086 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 36087 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 36094 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 36095 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 36096 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 36097class ZRing
cgf 36098class GF
cgfo 36099class GF∞
ceqp 36100class ~Qp
crqp 36101class /Qp
cqp 36102class Qp
czp 36103class Zp
cqpa 36104class _Qp
ccp 36105class Cp
df-zrng 36106ZRing = (𝑟 ∈ V ↦ (𝑟 IntgRing ran (ℤRHom‘𝑟)))
df-gf 36107 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 36108GF∞ = (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 36109~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 36110/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 36111Qp = (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 36112Zp = (ZRing ∘ Qp)
df-qpa 36113_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 36114Cp = ( cplMetSp ∘ _Qp)
ccloneop 36145class CloneOp
df-cloneop 36146CloneOp = (𝑎 ∈ V ↦ {𝑥 ∣ ∃𝑛 ∈ (ω ∖ 1o)𝑥 ∈ (𝑎 ↑m (𝑎 ↑m 𝑛))})
cprj 36147class prj
df-prj 36148prj = (𝑎 ∈ V ↦ (𝑛 ∈ (ω ∖ 1o), 𝑖 ∈ 𝑛 ↦ (𝑥 ∈ (𝑎 ↑m 𝑛) ↦ (𝑥‘𝑖))))
csuppos 36149class suppos
df-suppos 36150suppos = (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 36258class wsuc(𝑅, 𝐴, 𝑋)
cwlim 36259class WLim(𝑅, 𝐴)
df-wsuc 36260wsuc(𝑅, 𝐴, 𝑋) = inf(Pred(◡𝑅, 𝐴, 𝑋), 𝐴, 𝑅)
df-wlim 36261WLim(𝑅, 𝐴) = {𝑥 ∈ 𝐴 ∣ (𝑥 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑥 = sup(Pred(𝑅, 𝐴, 𝑥), 𝐴, 𝑅))}
ctxp 36278class (𝐴 ⊗ 𝐵)
cpprod 36279class pprod(𝑅, 𝑆)
csset 36280class SSet
ctrans 36281class Trans
cbigcup 36282class Bigcup
cfix 36283class Fix 𝐴
climits 36284class Limits
cfuns 36285class Funs
csingle 36286class Singleton
csingles 36287class Singletons
cimage 36288class Image𝐴
ccart 36289class Cart
cimg 36290class Img
cdomain 36291class Domain
crange 36292class Range
capply 36293class Apply
ccup 36294class Cup
ccap 36295class Cap
csuccf 36296class Succ
cfunpart 36297class Funpart𝐹
cfullfn 36298class FullFun𝐹
crestrict 36299class Restrict
cub 36300class UB𝑅
clb 36301class LB𝑅
df-txp 36302(𝐴 ⊗ 𝐵) = ((◡(1st ↾ (V × V)) ∘ 𝐴) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐵))
df-pprod 36303pprod(𝐴, 𝐵) = ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V))))
df-sset 36304 SSet = ((V × V) ∖ ran ( E ⊗ (V ∖ E )))
df-trans 36305 Trans = (V ∖ ran (( E ∘ E ) ∖ E ))
df-bigcup 36306 Bigcup = ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ E ) ⊗ V)))
df-fix 36307 Fix 𝐴 = dom (𝐴 ∩ I )
df-limits 36308 Limits = ((On ∩ Fix Bigcup ) ∖ {∅})
df-funs 36309 Funs = (𝒫 (V × V) ∖ Fix ( E ∘ ((1st ⊗ ((V ∖ I ) ∘ 2nd )) ∘ ◡ E )))
df-singleton 36310Singleton = ((V × V) ∖ ran ((V ⊗ E ) △ ( I ⊗ V)))
df-singles 36311 Singletons = ran Singleton
df-image 36312Image𝐴 = ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ ◡𝐴) ⊗ V)))
df-cart 36313Cart = (((V × V) × V) ∖ ran ((V ⊗ E ) △ (pprod( E , E ) ⊗ V)))
df-img 36314Img = (Image((2nd ∘ 1st ) ↾ (1st ↾ (V × V))) ∘ Cart)
df-domain 36315Domain = Image(1st ↾ (V × V))
df-range 36316Range = Image(2nd ↾ (V × V))
df-cup 36317Cup = (((V × V) × V) ∖ ran ((V ⊗ E ) △ (((◡1st ∘ E ) ∪ (◡2nd ∘ E )) ⊗ V)))
df-cap 36318Cap = (((V × V) × V) ∖ ran ((V ⊗ E ) △ (((◡1st ∘ E ) ∩ (◡2nd ∘ E )) ⊗ V)))
df-restrict 36319Restrict = (Cap ∘ (1st ⊗ (Cart ∘ (2nd ⊗ (Range ∘ 1st )))))
df-succf 36320Succ = (Cup ∘ ( I ⊗ Singleton))
df-apply 36321Apply = (( Bigcup ∘ Bigcup ) ∘ (((V × V) ∖ ran ((V ⊗ E ) △ (( E ↾ Singletons ) ⊗ V))) ∘ ((Singleton ∘ Img) ∘ pprod( I , Singleton))))
df-funpart 36322Funpart𝐹 = (𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))
df-fullfun 36323FullFun𝐹 = (Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))
df-ub 36324UB𝑅 = ((V × V) ∖ ((V ∖ 𝑅) ∘ ◡ E ))
df-lb 36325LB𝑅 = UB◡𝑅
caltop 36406class ⟪𝐴, 𝐵⟫
caltxp 36407class (𝐴 ×× 𝐵)
df-altop 36408⟪𝐴, 𝐵⟫ = {{𝐴}, {𝐴, {𝐵}}}
df-altxp 36409(𝐴 ×× 𝐵) = {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = ⟪𝑥, 𝑦⟫}
cofs 36432class OuterFiveSeg
df-ofs 36433 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 36479class TransportTo
df-transport 36480TransportTo = {⟨⟨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 36485class InnerFiveSeg
ccgr3 36486class Cgr3
ccolin 36487class Colinear
cfs 36488class FiveSeg
df-colinear 36489 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 36490 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 36491Cgr3 = {⟨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 36492 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 36556class Seg≤
df-segle 36557 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 36569class OutsideOf
df-outsideof 36570OutsideOf = ( Colinear ∖ Btwn )
cline2 36584class Line
cray 36585class Ray
clines2 36586class LinesEE
df-line2 36587Line = {⟨⟨𝑎, 𝑏⟩, 𝑙⟩ ∣ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎 ≠ 𝑏) ∧ 𝑙 = [⟨𝑎, 𝑏⟩]◡ Colinear )}
df-ray 36588Ray = {⟨⟨𝑝, 𝑎⟩, 𝑟⟩ ∣ ∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑎) ∧ 𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩})}
df-lines2 36589LinesEE = ran Line
cfwddif 36608class △
df-fwddif 36609 △ = (𝑓 ∈ (ℂ ↑pm ℂ) ↦ (𝑥 ∈ {𝑦 ∈ dom 𝑓 ∣ (𝑦 + 1) ∈ dom 𝑓} ↦ ((𝑓‘(𝑥 + 1)) − (𝑓‘𝑥))))
cfwddifn 36610class △n
df-fwddifn 36611 △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 36622class Hf
df-hf 36623 Hf = ∪ (𝑅1 “ ω)
cnmul 36637class ·no
df-nmul 36638 ·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), (𝑝 ∈ V, 𝑚 ∈ V ↦ ⦋(1st ‘𝑝) / 𝑎⦌⦋(2nd ‘𝑝) / 𝑏⦌∩ {𝑧 ∈ On ∣ ∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑏 ((𝑐𝑚𝑏) +no (𝑎𝑚𝑑)) ∈ (𝑧 +no (𝑐𝑚𝑑))}))
cfne 36795class Fne
df-fne 36796Fne = {⟨𝑥, 𝑦⟩ ∣ (∪ 𝑥 = ∪ 𝑦 ∧ ∀𝑧 ∈ 𝑥 𝑧 ⊆ ∪ (𝑦 ∩ 𝒫 𝑧))}
w3nand 36856wff (𝜑 ⊼ 𝜓 ⊼ 𝜒)
df-3nand 36857((𝜑 ⊼ 𝜓 ⊼ 𝜒) ↔ (𝜑 → (𝜓 → ¬ 𝜒)))
cgcdOLD 36918class gcdOLD (𝐴, 𝐵)
df-gcdOLD 36919 gcdOLD (𝐴, 𝐵) = sup({𝑥 ∈ ℕ ∣ ((𝐴 / 𝑥) ∈ ℕ ∧ (𝐵 / 𝑥) ∈ ℕ)}, ℕ, < )
ax-tco 36931∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦)))
cttc 36945class TC+ 𝐴
df-ttc 36946TC+ 𝐴 = ∪ 𝑥 ∈ 𝐴 ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω)
cprvb 37138wff Prv 𝜑
ax-prv1 37139𝜑    ⇒   Prv 𝜑
ax-prv2 37140(Prv (𝜑 → 𝜓) → (Prv 𝜑 → Prv 𝜓))
ax-prv3 37141(Prv 𝜑 → Prv Prv 𝜑)
wmoo 37218wff ∃**𝑥𝜑
df-bj-mo 37219(∃**𝑥𝜑 ↔ ∀𝑧∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))
wnnf 37299wff Ⅎ'𝑥𝜑
df-bj-nnf 37300(Ⅎ'𝑥𝜑 ↔ ((∃𝑥𝜑 → 𝜑) ∧ (𝜑 → ∀𝑥𝜑)))
bj-cgab 37517class {𝐴 ∣ 𝑥 ∣ 𝜑}
df-bj-gab 37518{𝐴 ∣ 𝑥 ∣ 𝜑} = {𝑦 ∣ ∃𝑥(𝐴 = 𝑦 ∧ 𝜑)}
wrnf 37525wff Ⅎ𝑥 ∈ 𝐴𝜑
df-bj-rnf 37526(Ⅎ𝑥 ∈ 𝐴𝜑 ↔ (∃𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 𝜑))
bj-csngl 37549class sngl 𝐴
df-bj-sngl 37550sngl 𝐴 = {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = {𝑦}}
bj-ctag 37558class tag 𝐴
df-bj-tag 37559tag 𝐴 = (sngl 𝐴 ∪ {∅})
bj-cproj 37574class (𝐴 Proj 𝐵)
df-bj-proj 37575(𝐴 Proj 𝐵) = {𝑥 ∣ {𝑥} ∈ (𝐵 “ {𝐴})}
bj-c1upl 37581class ⦅𝐴⦆
df-bj-1upl 37582⦅𝐴⦆ = ({∅} × tag 𝐴)
bj-cpr1 37584class pr1 𝐴
df-bj-pr1 37585pr1 𝐴 = (∅ Proj 𝐴)
bj-c2uple 37594class ⦅𝐴, 𝐵⦆
df-bj-2upl 37595⦅𝐴, 𝐵⦆ = (⦅𝐴⦆ ∪ ({1o} × tag 𝐵))
bj-cpr2 37598class pr2 𝐴
df-bj-pr2 37599pr2 𝐴 = (1o Proj 𝐴)
ax-bj-sn 37617∀𝑥∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ 𝑧 = 𝑥)
ax-bj-bun 37621∀𝑥∀𝑦∃𝑧∀𝑡(𝑡 ∈ 𝑧 ↔ (𝑡 ∈ 𝑥 ∨ 𝑡 ∈ 𝑦))
ax-bj-adj 37626∀𝑥∀𝑦∃𝑧∀𝑡(𝑡 ∈ 𝑧 ↔ (𝑡 ∈ 𝑥 ∨ 𝑡 = 𝑦))
celwise 37669class elwise
df-elwise 37670elwise = (𝑜 ∈ V ↦ (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑧 ∣ ∃𝑢 ∈ 𝑥 ∃𝑣 ∈ 𝑦 𝑧 = (𝑢𝑜𝑣)}))
cmoore 37693class Moore
df-bj-moore 37694Moore = {𝑥 ∣ ∀𝑦 ∈ 𝒫 𝑥(∪ 𝑥 ∩ ∩ 𝑦) ∈ 𝑥}
cmpt3 37710class (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵, 𝑧 ∈ 𝐶 ↦ 𝐷)
df-bj-mpt3 37711(𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵, 𝑧 ∈ 𝐶 ↦ 𝐷) = {⟨𝑠, 𝑡⟩ ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 (𝑠 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝑡 = 𝐷)}
csethom 37712class Set⟶
df-bj-sethom 37713 Set⟶ = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓 ∣ 𝑓:𝑥⟶𝑦})
ctophom 37714class Top⟶
df-bj-tophom 37715 Top⟶ = (𝑥 ∈ TopSp, 𝑦 ∈ TopSp ↦ {𝑓 ∈ ((Base‘𝑥) Set⟶ (Base‘𝑦)) ∣ ∀𝑢 ∈ (TopOpen‘𝑦)(◡𝑓 “ 𝑢) ∈ (TopOpen‘𝑥)})
cmgmhom 37716class Mgm⟶
df-bj-mgmhom 37717 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 37718class TopMgm⟶
df-bj-topmgmhom 37719 TopMgm⟶ = (𝑥 ∈ TopMnd, 𝑦 ∈ TopMnd ↦ ((𝑥 Top⟶ 𝑦) ∩ (𝑥 Mgm⟶ 𝑦)))
ccur- 37720class curry_
df-bj-cur 37721curry_ = (𝑥 ∈ V, 𝑦 ∈ V, 𝑧 ∈ V ↦ (𝑓 ∈ ((𝑥 × 𝑦) Set⟶ 𝑧) ↦ (𝑎 ∈ 𝑥 ↦ (𝑏 ∈ 𝑦 ↦ (𝑓‘⟨𝑎, 𝑏⟩)))))
cunc- 37722class uncurry_
df-bj-unc 37723uncurry_ = (𝑥 ∈ V, 𝑦 ∈ V, 𝑧 ∈ V ↦ (𝑓 ∈ (𝑥 Set⟶ (𝑦 Set⟶ 𝑧)) ↦ (𝑎 ∈ 𝑥, 𝑏 ∈ 𝑦 ↦ ((𝑓‘𝑎)‘𝑏))))
cstrset 37724class [𝐵 / 𝐴]struct𝑆
df-strset 37725[𝐵 / 𝐴]struct𝑆 = ((𝑆 ↾ (V ∖ {(𝐴‘ndx)})) ∪ {⟨(𝐴‘ndx), 𝐵⟩})
cdiag2 37764class Id
df-bj-diag 37765Id = (𝑥 ∈ V ↦ ( I ↾ 𝑥))
cimdir 37770class 𝒫*
df-imdir 37771𝒫* = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑟 ∈ 𝒫 (𝑎 × 𝑏) ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝑎 ∧ 𝑦 ⊆ 𝑏) ∧ (𝑟 “ 𝑥) = 𝑦)}))
ciminv 37783class 𝒫*
df-iminv 37784𝒫* = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑟 ∈ 𝒫 (𝑎 × 𝑏) ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝑎 ∧ 𝑦 ⊆ 𝑏) ∧ 𝑥 = (◡𝑟 “ 𝑦))}))
cfractemp 37788class {R
df-bj-fractemp 37789{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 37790class +∞eiτ
df-bj-inftyexpitau 37791+∞eiτ = (𝑥 ∈ ℝ ↦ ⟨({R‘(1st ‘𝑥)), {R}⟩)
cccinftyN 37792class ℂ∞N
df-bj-ccinftyN 37793ℂ∞N = ran +∞eiτ
chalf 37795class 1/2
df-bj-onehalf 377961/2 = (℩𝑥 ∈ R (𝑥 +R 𝑥) = 1R)
cinftyexpi 37798class +∞ei
df-bj-inftyexpi 37799+∞ei = (𝑥 ∈ (-π(,]π) ↦ ⟨𝑥, ℂ⟩)
cccinfty 37803class ℂ∞
df-bj-ccinfty 37804ℂ∞ = ran +∞ei
cccbar 37807class ℂ̅
df-bj-ccbar 37808ℂ̅ = (ℂ ∪ ℂ∞)
cpinfty 37811class +∞
df-bj-pinfty 37812+∞ = (+∞ei‘0)
cminfty 37815class -∞
df-bj-minfty 37816-∞ = (+∞ei‘π)
crrbar 37820class ℝ̅
df-bj-rrbar 37821ℝ̅ = (ℝ ∪ {-∞, +∞})
cinfty 37822class ∞
df-bj-infty 37823∞ = 𝒫 ∪ ℂ
ccchat 37824class ℂ̂
df-bj-cchat 37825ℂ̂ = (ℂ ∪ {∞})
crrhat 37826class ℝ̂
df-bj-rrhat 37827ℝ̂ = (ℝ ∪ {∞})
caddcc 37829class +ℂ̅
df-bj-addc 37830 +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 37831class -ℂ̅
df-bj-oppc 37832-ℂ̅ = (𝑥 ∈ (ℂ̅ ∪ ℂ̂) ↦ if(𝑥 = ∞, ∞, if(𝑥 ∈ ℂ, (℩𝑦 ∈ ℂ (𝑥 +ℂ̅ 𝑦) = 0), (+∞eiτ‘(𝑥 +ℂ̅ ⟨1/2, 0R⟩)))))
cltxr 37833class <ℝ̅
df-bj-lt 37834<ℝ̅ = ({𝑥 ∈ (ℝ̅ × ℝ̅) ∣ ∃𝑦∃𝑧(((1st ‘𝑥) = ⟨𝑦, 0R⟩ ∧ (2nd ‘𝑥) = ⟨𝑧, 0R⟩) ∧ 𝑦 <R 𝑧)} ∪ ((({-∞} × ℝ) ∪ (ℝ × {+∞})) ∪ ({-∞} × {+∞})))
carg 37835class Arg
df-bj-arg 37836Arg = (𝑥 ∈ (ℂ̅ ∖ {0}) ↦ if(𝑥 ∈ ℂ, (ℑ‘(log‘𝑥)), if(𝑥<ℝ̅0, π, (((1st ‘𝑥) / (2 · π)) − π))))
cmulc 37837class ·ℂ̅
df-bj-mulc 37838 ·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 37839class -1ℂ̅
df-bj-invc 37840-1ℂ̅ = (𝑥 ∈ (ℂ̅ ∪ ℂ̂) ↦ if(𝑥 = 0, ∞, if(𝑥 ∈ ℂ, (℩𝑦 ∈ ℂ (𝑥 ·ℂ̅ 𝑦) = 1), 0)))
ciomnn 37841class iω↪ℕ
df-bj-iomnn 37842iω↪ℕ = ((𝑛 ∈ ω ↦ ⟨[⟨{𝑟 ∈ Q ∣ 𝑟 <Q ⟨suc 𝑛, 1o⟩}, 1P⟩] ~R , 0R⟩) ∪ {⟨ω, +∞⟩})
cnnbar 37852class ℕ̅
df-bj-nnbar 37853ℕ̅ = (ℕ0 ∪ {+∞})
czzbar 37854class ℤ̅
df-bj-zzbar 37855ℤ̅ = (ℤ ∪ {-∞, +∞})
czzhat 37856class ℤ̂
df-bj-zzhat 37857ℤ̂ = (ℤ ∪ {∞})
cdivc 37858class ∥ℂ
df-bj-divc 37859 ∥ℂ = {⟨𝑥, 𝑦⟩ ∣ (⟨𝑥, 𝑦⟩ ∈ ((ℂ̅ × ℂ̅) ∪ (ℂ̂ × ℂ̂)) ∧ ∃𝑛 ∈ (ℤ̅ ∪ ℤ̂)(𝑛 ·ℂ̅ 𝑥) = 𝑦)}
cfinsum 37875class FinSum
df-bj-finsum 37876 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 37884class ℝ-Vec
df-bj-rvec 37885ℝ-Vec = (LMod ∩ (◡Scalar “ {ℝfld}))
cend 37905class End
df-bj-end 37906End = (𝑐 ∈ Cat ↦ (𝑥 ∈ (Base‘𝑐) ↦ {⟨(Base‘ndx), (𝑥(Hom ‘𝑐)𝑥)⟩, ⟨(+g‘ndx), (⟨𝑥, 𝑥⟩(comp‘𝑐)𝑥)⟩}))
cfinxp 37977class (𝑈↑↑𝑁)
df-finxp 37978(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 38013((𝜑 → 𝜓) → ((𝜓 → 𝜒) → (𝜑 → 𝜒)))
ax-luk2 38014((¬ 𝜑 → 𝜑) → 𝜑)
ax-luk3 38015(𝜑 → (¬ 𝜑 → 𝜓))
ax-wl-13v 38087(¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
ax-wl-cleq 38096(𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
ax-wl-clel 38097(𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵))
ctotbnd 38365class TotBnd
cbnd 38366class Bnd
df-totbnd 38367TotBnd = (𝑥 ∈ V ↦ {𝑚 ∈ (Met‘𝑥) ∣ ∀𝑑 ∈ ℝ+ ∃𝑣 ∈ Fin (∪ 𝑣 = 𝑥 ∧ ∀𝑏 ∈ 𝑣 ∃𝑦 ∈ 𝑥 𝑏 = (𝑦(ball‘𝑚)𝑑))})
df-bnd 38378Bnd = (𝑥 ∈ V ↦ {𝑚 ∈ (Met‘𝑥) ∣ ∀𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ 𝑥 = (𝑦(ball‘𝑚)𝑟)})
cismty 38397class Ismty
df-ismty 38398 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 38424class ℝn
df-rrn 38425ℝn = (𝑖 ∈ Fin ↦ (𝑥 ∈ (ℝ ↑m 𝑖), 𝑦 ∈ (ℝ ↑m 𝑖) ↦ (√‘Σ𝑘 ∈ 𝑖 (((𝑥‘𝑘) − (𝑦‘𝑘))↑2))))
cass 38441class Ass
df-ass 38442Ass = {𝑔 ∣ ∀𝑥 ∈ dom dom 𝑔∀𝑦 ∈ dom dom 𝑔∀𝑧 ∈ dom dom 𝑔((𝑥𝑔𝑦)𝑔𝑧) = (𝑥𝑔(𝑦𝑔𝑧))}
cexid 38443class ExId
df-exid 38444 ExId = {𝑔 ∣ ∃𝑥 ∈ dom dom 𝑔∀𝑦 ∈ dom dom 𝑔((𝑥𝑔𝑦) = 𝑦 ∧ (𝑦𝑔𝑥) = 𝑦)}
cmagm 38447class Magma
df-mgmOLD 38448Magma = {𝑔 ∣ ∃𝑡 𝑔:(𝑡 × 𝑡)⟶𝑡}
csem 38459class SemiGrp
df-sgrOLD 38460SemiGrp = (Magma ∩ Ass)
cmndo 38465class MndOp
df-mndo 38466MndOp = (SemiGrp ∩ ExId )
cghomOLD 38482class GrpOpHom
df-ghomOLD 38483 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 38493class RingOps
df-rngo 38494RingOps = {⟨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 38547class DivRingOps
df-drngo 38548DivRingOps = {⟨𝑔, ℎ⟩ ∣ (⟨𝑔, ℎ⟩ ∈ RingOps ∧ (ℎ ↾ ((ran 𝑔 ∖ {(GId‘𝑔)}) × (ran 𝑔 ∖ {(GId‘𝑔)}))) ∈ GrpOp)}
crngohom 38559class RingOpsHom
crngoiso 38560class RingOpsIso
crisc 38561class ≃𝑟
df-rngohom 38562 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 38575 RingOpsIso = (𝑟 ∈ RingOps, 𝑠 ∈ RingOps ↦ {𝑓 ∈ (𝑟 RingOpsHom 𝑠) ∣ 𝑓:ran (1st ‘𝑟)–1-1-onto→ran (1st ‘𝑠)})
df-risc 38582 ≃𝑟 = {⟨𝑟, 𝑠⟩ ∣ ((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑟 RingOpsIso 𝑠))}
ccm2 38588class Com2
df-com2 38589Com2 = {⟨𝑔, ℎ⟩ ∣ ∀𝑎 ∈ ran 𝑔∀𝑏 ∈ ran 𝑔(𝑎ℎ𝑏) = (𝑏ℎ𝑎)}
cfld 38590class Fld
df-fld 38591Fld = (DivRingOps ∩ Com2)
ccring 38592class CRingOps
df-crngo 38593CRingOps = (RingOps ∩ Com2)
cidl 38606class Idl
cpridl 38607class PrIdl
cmaxidl 38608class MaxIdl
df-idl 38609Idl = (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 38610PrIdl = (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 38611MaxIdl = (𝑟 ∈ RingOps ↦ {𝑖 ∈ (Idl‘𝑟) ∣ (𝑖 ≠ ran (1st ‘𝑟) ∧ ∀𝑗 ∈ (Idl‘𝑟)(𝑖 ⊆ 𝑗 → (𝑗 = 𝑖 ∨ 𝑗 = ran (1st ‘𝑟))))})
cprrng 38645class PrRing
cdmn 38646class Dmn
df-prrngo 38647PrRing = {𝑟 ∈ RingOps ∣ {(GId‘(1st ‘𝑟))} ∈ (PrIdl‘𝑟)}
df-dmn 38648Dmn = (PrRing ∩ Com2)
cigen 38658class IdlGen
df-igen 38659 IdlGen = (𝑟 ∈ RingOps, 𝑠 ∈ 𝒫 ran (1st ‘𝑟) ↦ ∩ {𝑗 ∈ (Idl‘𝑟) ∣ 𝑠 ⊆ 𝑗})
cxrn 38773class (𝐴 ⋉ 𝐵)
cqmap 38774class QMap 𝑅
cadjliftmap 38775class (𝑅 AdjLiftMap 𝐴)
cblockliftmap 38776class (𝑅 BlockLiftMap 𝐴)
csucmap 38777class SucMap
csuccl 38778class Suc
cpre 38779class pre 𝑁
cblockliftfix 38780class BlockLiftFix
cshiftstable 38781class (𝑆 ShiftStable 𝐹)
ccoss 38782class ≀ 𝑅
ccoels 38783class ∼ 𝐴
crels 38784class Rels
cssr 38785class S
crefs 38786class Refs
crefrels 38787class RefRels
wrefrel 38788wff RefRel 𝑅
ccnvrefs 38789class CnvRefs
ccnvrefrels 38790class CnvRefRels
wcnvrefrel 38791wff CnvRefRel 𝑅
csyms 38792class Syms
csymrels 38793class SymRels
wsymrel 38794wff SymRel 𝑅
ctrs 38795class Trs
ctrrels 38796class TrRels
wtrrel 38797wff TrRel 𝑅
ceqvrels 38798class EqvRels
weqvrel 38799wff EqvRel 𝑅
ccoeleqvrels 38800class CoElEqvRels
wcoeleqvrel 38801wff CoElEqvRel 𝐴
credunds 38802class Redunds
wredund 38803wff 𝐴 Redund ⟨𝐵, 𝐶⟩
wredundp 38804wff redund (𝜑, 𝜓, 𝜒)
cdmqss 38805class DomainQss
wdmqs 38806wff 𝑅 DomainQs 𝐴
cers 38807class Ers
werALTV 38808wff 𝑅 ErALTV 𝐴
cpeters 38809class PetErs
cpet2ers 38810class Pet2Ers
ccomembers 38811class CoMembErs
wcomember 38812wff CoMembEr 𝐴
cfunss 38813class Funss
cfunsALTV 38814class FunsALTV
wfunALTV 38815wff FunALTV 𝐹
cdisjss 38816class Disjss
cdisjs 38817class Disjs
wdisjALTV 38818wff Disj 𝑅
celdisjs 38819class ElDisjs
weldisj 38820wff ElDisj 𝐴
wantisymrel 38821wff AntisymRel 𝑅
cparts 38822class Parts
wpart 38823wff 𝑅 Part 𝐴
cmembparts 38824class MembParts
wmembpart 38825wff MembPart 𝐴
cpetparts 38826class PetParts
cpet2parts 38827class Pet2Parts
df-xrn 38979(𝐴 ⋉ 𝐵) = ((◡(1st ↾ (V × V)) ∘ 𝐴) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐵))
df-rels 39039 Rels = 𝒫 (V × V)
df-qmap 39045 QMap 𝑅 = (𝑥 ∈ dom 𝑅 ↦ [𝑥]𝑅)
df-adjliftmap 39054(𝑅 AdjLiftMap 𝐴) = QMap ((𝑅 ∪ ◡ E ) ↾ 𝐴)
df-blockliftmap 39058(𝑅 BlockLiftMap 𝐴) = QMap (𝑅 ⋉ (◡ E ↾ 𝐴))
df-sucmap 39061 SucMap = {⟨𝑚, 𝑛⟩ ∣ suc 𝑚 = 𝑛}
df-succl 39068 Suc = ran SucMap
df-pre 39074 pre 𝑁 = (℩𝑚𝑚 ∈ Pred( SucMap , dom SucMap , 𝑁))
df-blockliftfix 39080 BlockLiftFix = {⟨𝑟, 𝑎⟩ ∣ (dom (𝑟 ⋉ (◡ E ↾ 𝑎)) / (𝑟 ⋉ (◡ E ↾ 𝑎))) = 𝑎}
df-shiftstable 39081(𝑆 ShiftStable 𝐹) = ((𝑆 ∘ 𝐹) ∩ 𝐹)
df-coss 39100 ≀ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)}
df-coels 39101 ∼ 𝐴 = ≀ (◡ E ↾ 𝐴)
df-ssr 39177 S = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ⊆ 𝑦}
df-refs 39189 Refs = {𝑥 ∣ ( I ∩ (dom 𝑥 × ran 𝑥)) S (𝑥 ∩ (dom 𝑥 × ran 𝑥))}
df-refrels 39190 RefRels = ( Refs ∩ Rels )
df-refrel 39191( RefRel 𝑅 ↔ (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅))
df-cnvrefs 39204 CnvRefs = {𝑥 ∣ ( I ∩ (dom 𝑥 × ran 𝑥))◡ S (𝑥 ∩ (dom 𝑥 × ran 𝑥))}
df-cnvrefrels 39205 CnvRefRels = ( CnvRefs ∩ Rels )
df-cnvrefrel 39206( CnvRefRel 𝑅 ↔ ((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ ( I ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅))
df-syms 39221 Syms = {𝑥 ∣ ◡(𝑥 ∩ (dom 𝑥 × ran 𝑥)) S (𝑥 ∩ (dom 𝑥 × ran 𝑥))}
df-symrels 39222 SymRels = ( Syms ∩ Rels )
df-symrel 39223( SymRel 𝑅 ↔ (◡(𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅))
df-trs 39255 Trs = {𝑥 ∣ ((𝑥 ∩ (dom 𝑥 × ran 𝑥)) ∘ (𝑥 ∩ (dom 𝑥 × ran 𝑥))) S (𝑥 ∩ (dom 𝑥 × ran 𝑥))}
df-trrels 39256 TrRels = ( Trs ∩ Rels )
df-trrel 39257( TrRel 𝑅 ↔ (((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∘ (𝑅 ∩ (dom 𝑅 × ran 𝑅))) ⊆ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅))
df-eqvrels 39267 EqvRels = (( RefRels ∩ SymRels ) ∩ TrRels )
df-eqvrel 39268( EqvRel 𝑅 ↔ ( RefRel 𝑅 ∧ SymRel 𝑅 ∧ TrRel 𝑅))
df-coeleqvrels 39269 CoElEqvRels = {𝑎 ∣ ≀ (◡ E ↾ 𝑎) ∈ EqvRels }
df-coeleqvrel 39270( CoElEqvRel 𝐴 ↔ EqvRel ≀ (◡ E ↾ 𝐴))
df-redunds 39306 Redunds = ◡{⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ (𝑥 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑧) = (𝑦 ∩ 𝑧))}
df-redund 39307(𝐴 Redund ⟨𝐵, 𝐶⟩ ↔ (𝐴 ⊆ 𝐵 ∧ (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐶)))
df-redundp 39308( redund (𝜑, 𝜓, 𝜒) ↔ ((𝜑 → 𝜓) ∧ ((𝜑 ∧ 𝜒) ↔ (𝜓 ∧ 𝜒))))
df-dmqss 39321 DomainQss = {⟨𝑥, 𝑦⟩ ∣ (dom 𝑥 / 𝑥) = 𝑦}
df-dmqs 39322(𝑅 DomainQs 𝐴 ↔ (dom 𝑅 / 𝑅) = 𝐴)
df-ers 39347 Ers = ( DomainQss ↾ EqvRels )
df-erALTV 39348(𝑅 ErALTV 𝐴 ↔ ( EqvRel 𝑅 ∧ 𝑅 DomainQs 𝐴))
df-comembers 39349 CoMembErs = {𝑎 ∣ ≀ (◡ E ↾ 𝑎) Ers 𝑎}
df-comember 39350( CoMembEr 𝐴 ↔ ≀ (◡ E ↾ 𝐴) ErALTV 𝐴)
df-funss 39364 Funss = {𝑥 ∣ ≀ 𝑥 ∈ CnvRefRels }
df-funsALTV 39365 FunsALTV = ( Funss ∩ Rels )
df-funALTV 39366( FunALTV 𝐹 ↔ ( CnvRefRel ≀ 𝐹 ∧ Rel 𝐹))
df-disjss 39387 Disjss = {𝑥 ∣ ≀ ◡𝑥 ∈ CnvRefRels }
df-disjs 39388 Disjs = ( Disjss ∩ Rels )
df-disjALTV 39389( Disj 𝑅 ↔ ( CnvRefRel ≀ ◡𝑅 ∧ Rel 𝑅))
df-eldisjs 39390 ElDisjs = {𝑎 ∣ (◡ E ↾ 𝑎) ∈ Disjs }
df-eldisj 39391( ElDisj 𝐴 ↔ Disj (◡ E ↾ 𝐴))
df-antisymrel 39462( AntisymRel 𝑅 ↔ ( CnvRefRel (𝑅 ∩ ◡𝑅) ∧ Rel 𝑅))
df-parts 39467 Parts = ( DomainQss ↾ Disjs )
df-part 39468(𝑅 Part 𝐴 ↔ ( Disj 𝑅 ∧ 𝑅 DomainQs 𝐴))
df-membparts 39469 MembParts = {𝑎 ∣ (◡ E ↾ 𝑎) Parts 𝑎}
df-membpart 39470( MembPart 𝐴 ↔ (◡ E ↾ 𝐴) Part 𝐴)
df-petparts 39567 PetParts = {⟨𝑟, 𝑛⟩ ∣ ((𝑟 ∈ Rels ∧ 𝑛 ∈ MembParts ) ∧ (𝑟 ⋉ (◡ E ↾ 𝑛)) Parts 𝑛)}
df-peters 39568 PetErs = {⟨𝑟, 𝑛⟩ ∣ ((𝑟 ∈ Rels ∧ 𝑛 ∈ CoMembErs ) ∧ ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) Ers 𝑛)}
df-pet2parts 39569 Pet2Parts = ( SucMap ShiftStable PetParts )
df-pet2ers 39570 Pet2Ers = ( SucMap ShiftStable PetErs )
wprt 39595wff Prt 𝐴
df-prt 39596(Prt 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅))
ax-c5 39607(∀𝑥𝜑 → 𝜑)
ax-c4 39608(∀𝑥(∀𝑥𝜑 → 𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))
ax-c7 39609(¬ ∀𝑥 ¬ ∀𝑥𝜑 → 𝜑)
ax-c10 39610(∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → 𝜑)
ax-c11 39611(∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑))
ax-c11n 39612(∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
ax-c15 39613(¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))))
ax-c9 39614(¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))
ax-c14 39615(¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥 ∈ 𝑦 → ∀𝑧 𝑥 ∈ 𝑦)))
ax-c16 39616(∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))
ax-riotaBAD 39677(℩𝑥 ∈ 𝐴 𝜑) = if(∃!𝑥 ∈ 𝐴 𝜑, (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)), (Undef‘{𝑥 ∣ 𝑥 ∈ 𝐴}))
clsa 39698class LSAtoms
clsh 39699class LSHyp
df-lsatoms 39700LSAtoms = (𝑤 ∈ V ↦ ran (𝑣 ∈ ((Base‘𝑤) ∖ {(0g‘𝑤)}) ↦ ((LSpan‘𝑤)‘{𝑣})))
df-lshyp 39701LSHyp = (𝑤 ∈ V ↦ {𝑠 ∈ (LSubSp‘𝑤) ∣ (𝑠 ≠ (Base‘𝑤) ∧ ∃𝑣 ∈ (Base‘𝑤)((LSpan‘𝑤)‘(𝑠 ∪ {𝑣})) = (Base‘𝑤))})
clcv 39742class ⋖L
df-lcv 39743 ⋖L = (𝑤 ∈ V ↦ {⟨𝑡, 𝑢⟩ ∣ ((𝑡 ∈ (LSubSp‘𝑤) ∧ 𝑢 ∈ (LSubSp‘𝑤)) ∧ (𝑡 ⊊ 𝑢 ∧ ¬ ∃𝑠 ∈ (LSubSp‘𝑤)(𝑡 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑢)))})
clfn 39781class LFnl
df-lfl 39782LFnl = (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 39809class LKer
df-lkr 39810LKer = (𝑤 ∈ V ↦ (𝑓 ∈ (LFnl‘𝑤) ↦ (◡𝑓 “ {(0g‘(Scalar‘𝑤))})))
cld 39847class LDual
df-ldual 39848LDual = (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 39896class OP
ccmtN 39897class cm
col 39898class OL
coml 39899class OML
df-oposet 39900OP = {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 39901cm = (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 39902OL = (Lat ∩ OP)
df-oml 39903OML = {𝑙 ∈ OL ∣ ∀𝑎 ∈ (Base‘𝑙)∀𝑏 ∈ (Base‘𝑙)(𝑎(le‘𝑙)𝑏 → 𝑏 = (𝑎(join‘𝑙)(𝑏(meet‘𝑙)((oc‘𝑙)‘𝑎))))}
ccvr 39986class ⋖
catm 39987class Atoms
cal 39988class AtLat
clc 39989class CvLat
df-covers 39990 ⋖ = (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 39991Atoms = (𝑝 ∈ V ↦ {𝑎 ∈ (Base‘𝑝) ∣ (0.‘𝑝)( ⋖ ‘𝑝)𝑎})
df-atl 40022AtLat = {𝑘 ∈ Lat ∣ ((Base‘𝑘) ∈ dom (glb‘𝑘) ∧ ∀𝑥 ∈ (Base‘𝑘)(𝑥 ≠ (0.‘𝑘) → ∃𝑝 ∈ (Atoms‘𝑘)𝑝(le‘𝑘)𝑥))}
df-cvlat 40046CvLat = {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 40074class HL
df-hlat 40075HL = {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 40215class LLines
clpl 40216class LPlanes
clvol 40217class LVols
clines 40218class Lines
cpointsN 40219class Points
cpsubsp 40220class PSubSp
cpmap 40221class pmap
df-llines 40222LLines = (𝑘 ∈ V ↦ {𝑥 ∈ (Base‘𝑘) ∣ ∃𝑝 ∈ (Atoms‘𝑘)𝑝( ⋖ ‘𝑘)𝑥})
df-lplanes 40223LPlanes = (𝑘 ∈ V ↦ {𝑥 ∈ (Base‘𝑘) ∣ ∃𝑝 ∈ (LLines‘𝑘)𝑝( ⋖ ‘𝑘)𝑥})
df-lvols 40224LVols = (𝑘 ∈ V ↦ {𝑥 ∈ (Base‘𝑘) ∣ ∃𝑝 ∈ (LPlanes‘𝑘)𝑝( ⋖ ‘𝑘)𝑥})
df-lines 40225Lines = (𝑘 ∈ V ↦ {𝑠 ∣ ∃𝑞 ∈ (Atoms‘𝑘)∃𝑟 ∈ (Atoms‘𝑘)(𝑞 ≠ 𝑟 ∧ 𝑠 = {𝑝 ∈ (Atoms‘𝑘) ∣ 𝑝(le‘𝑘)(𝑞(join‘𝑘)𝑟)})})
df-pointsN 40226Points = (𝑘 ∈ V ↦ {𝑞 ∣ ∃𝑝 ∈ (Atoms‘𝑘)𝑞 = {𝑝}})
df-psubsp 40227PSubSp = (𝑘 ∈ V ↦ {𝑠 ∣ (𝑠 ⊆ (Atoms‘𝑘) ∧ ∀𝑝 ∈ 𝑠 ∀𝑞 ∈ 𝑠 ∀𝑟 ∈ (Atoms‘𝑘)(𝑟(le‘𝑘)(𝑝(join‘𝑘)𝑞) → 𝑟 ∈ 𝑠))})
df-pmap 40228pmap = (𝑘 ∈ V ↦ (𝑎 ∈ (Base‘𝑘) ↦ {𝑝 ∈ (Atoms‘𝑘) ∣ 𝑝(le‘𝑘)𝑎}))
cpadd 40519class +𝑃
df-padd 40520+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 40611class PCl
df-pclN 40612PCl = (𝑘 ∈ V ↦ (𝑥 ∈ 𝒫 (Atoms‘𝑘) ↦ ∩ {𝑦 ∈ (PSubSp‘𝑘) ∣ 𝑥 ⊆ 𝑦}))
cpolN 40626class ⊥𝑃
df-polarityN 40627⊥𝑃 = (𝑙 ∈ V ↦ (𝑚 ∈ 𝒫 (Atoms‘𝑙) ↦ ((Atoms‘𝑙) ∩ ∩ 𝑝 ∈ 𝑚 ((pmap‘𝑙)‘((oc‘𝑙)‘𝑝)))))
cpscN 40658class PSubCl
df-psubclN 40659PSubCl = (𝑘 ∈ V ↦ {𝑠 ∣ (𝑠 ⊆ (Atoms‘𝑘) ∧ ((⊥𝑃‘𝑘)‘((⊥𝑃‘𝑘)‘𝑠)) = 𝑠)})
clh 40708class LHyp
claut 40709class LAut
cwpointsN 40710class WAtoms
cpautN 40711class PAut
df-lhyp 40712LHyp = (𝑘 ∈ V ↦ {𝑥 ∈ (Base‘𝑘) ∣ 𝑥( ⋖ ‘𝑘)(1.‘𝑘)})
df-laut 40713LAut = (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 40714WAtoms = (𝑘 ∈ V ↦ (𝑑 ∈ (Atoms‘𝑘) ↦ ((Atoms‘𝑘) ∖ ((⊥𝑃‘𝑘)‘{𝑑}))))
df-pautN 40715PAut = (𝑘 ∈ V ↦ {𝑓 ∣ (𝑓:(PSubSp‘𝑘)–1-1-onto→(PSubSp‘𝑘) ∧ ∀𝑥 ∈ (PSubSp‘𝑘)∀𝑦 ∈ (PSubSp‘𝑘)(𝑥 ⊆ 𝑦 ↔ (𝑓‘𝑥) ⊆ (𝑓‘𝑦)))})
cldil 40824class LDil
cltrn 40825class LTrn
cdilN 40826class Dil
ctrnN 40827class Trn
df-ldil 40828LDil = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ {𝑓 ∈ (LAut‘𝑘) ∣ ∀𝑥 ∈ (Base‘𝑘)(𝑥(le‘𝑘)𝑤 → (𝑓‘𝑥) = 𝑥)}))
df-ltrn 40829LTrn = (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 40830Dil = (𝑘 ∈ V ↦ (𝑑 ∈ (Atoms‘𝑘) ↦ {𝑓 ∈ (PAut‘𝑘) ∣ ∀𝑥 ∈ (PSubSp‘𝑘)(𝑥 ⊆ ((WAtoms‘𝑘)‘𝑑) → (𝑓‘𝑥) = 𝑥)}))
df-trnN 40831Trn = (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 40882class trL
df-trl 40883trL = (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 41466class TGrp
df-tgrp 41467TGrp = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ {⟨(Base‘ndx), ((LTrn‘𝑘)‘𝑤)⟩, ⟨(+g‘ndx), (𝑓 ∈ ((LTrn‘𝑘)‘𝑤), 𝑔 ∈ ((LTrn‘𝑘)‘𝑤) ↦ (𝑓 ∘ 𝑔))⟩}))
ctendo 41476class TEndo
cedring 41477class EDRing
cedring-rN 41478class EDRingR
df-tendo 41479TEndo = (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 41480EDRingR = (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 41481EDRing = (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 41726class DVecA
df-dveca 41727DVecA = (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 41752class DIsoA
df-disoa 41753DIsoA = (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 41802class DVecH
df-dvech 41803DVecH = (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 41843class ocA
df-docaN 41844ocA = (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 41855class vA
df-djaN 41856vA = (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 41862class DIsoB
df-dib 41863DIsoB = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑥 ∈ dom ((DIsoA‘𝑘)‘𝑤) ↦ ((((DIsoA‘𝑘)‘𝑤)‘𝑥) × {(𝑓 ∈ ((LTrn‘𝑘)‘𝑤) ↦ ( I ↾ (Base‘𝑘)))}))))
cdic 41896class DIsoC
df-dic 41897DIsoC = (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 41952class DIsoH
df-dih 41953DIsoH = (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 42071class ocH
df-doch 42072ocH = (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 42118class joinH
df-djh 42119joinH = (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 42204class LPol
df-lpolN 42205LPol = (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 42310class LCDual
df-lcdual 42311LCDual = (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 42348class mapd
df-mapd 42349mapd = (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 42480class HVMap
df-hvmap 42481HVMap = (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 42515class HDMap1
chdma 42516class HDMap
df-hdmap1 42517HDMap1 = (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 42518HDMap = (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 42607class HGMap
df-hgmap 42608HGMap = (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 42656class HLHil
df-hlhil 42657HLHil = (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 42686class CSRing
df-csring 42687 CSRing = {𝑓 ∈ SRing ∣ (mulGrp‘𝑓) ∈ CMnd}
cprimroots 42808class PrimRoots
df-primroots 42809 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 42919∀𝑝 ∈ ℙ ∀𝑛 ∈ ℕ ∃𝑘 ∈ Field ((♯‘(Base‘𝑘)) = (𝑝↑𝑛) ∧ (chr‘𝑘) = 𝑝)
cresub 43076class −ℝ
df-resub 43077 −ℝ = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (℩𝑧 ∈ ℝ (𝑦 + 𝑧) = 𝑥))
crediv 43151class /ℝ
df-rediv 43152 /ℝ = (𝑥 ∈ ℝ, 𝑦 ∈ (ℝ ∖ {0}) ↦ (℩𝑧 ∈ ℝ (𝑦 · 𝑧) = 𝑥))
cprjsp 43285class ℙ𝕣𝕠𝕛
df-prjsp 43286ℙ𝕣𝕠𝕛 = (𝑣 ∈ LVec ↦ ⦋((Base‘𝑣) ∖ {(0g‘𝑣)}) / 𝑏⦌(𝑏 / {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝑏 ∧ 𝑦 ∈ 𝑏) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑣))𝑥 = (𝑙( ·𝑠 ‘𝑣)𝑦))}))
cprjspn 43298class ℙ𝕣𝕠𝕛n
df-prjspn 43299ℙ𝕣𝕠𝕛n = (𝑛 ∈ ℕ0, 𝑘 ∈ DivRing ↦ (ℙ𝕣𝕠𝕛‘(𝑘 freeLMod (0...𝑛))))
cprjcrv 43313class ℙ𝕣𝕠𝕛Crv
df-prjcrv 43314ℙ𝕣𝕠𝕛Crv = (𝑛 ∈ ℕ0, 𝑘 ∈ Field ↦ (𝑓 ∈ ∪ ran ((0...𝑛) mHomP 𝑘) ↦ {𝑝 ∈ (𝑛ℙ𝕣𝕠𝕛n𝑘) ∣ ((((0...𝑛) eval 𝑘)‘𝑓) “ 𝑝) = {(0g‘𝑘)}}))
cnacs 43385class NoeACS
df-nacs 43386NoeACS = (𝑥 ∈ V ↦ {𝑐 ∈ (ACS‘𝑥) ∣ ∀𝑠 ∈ 𝑐 ∃𝑔 ∈ (𝒫 𝑥 ∩ Fin)𝑠 = ((mrCls‘𝑐)‘𝑔)})
cmzpcl 43404class mzPolyCld
cmzp 43405class mzPoly
df-mzpcl 43406mzPolyCld = (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 43407mzPoly = (𝑣 ∈ V ↦ ∩ (mzPolyCld‘𝑣))
cdioph 43438class Dioph
df-dioph 43439Dioph = (𝑛 ∈ ℕ0 ↦ ran (𝑘 ∈ (ℤ≥‘𝑛), 𝑝 ∈ (mzPoly‘(1...𝑘)) ↦ {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑛)) ∧ (𝑝‘𝑢) = 0)}))
csquarenn 43515class ◻NN
cpell1qr 43516class Pell1QR
cpell1234qr 43517class Pell1234QR
cpell14qr 43518class Pell14QR
cpellfund 43519class PellFund
df-squarenn 43520◻NN = {𝑥 ∈ ℕ ∣ (√‘𝑥) ∈ ℚ}
df-pell1qr 43521Pell1QR = (𝑥 ∈ (ℕ ∖ ◻NN) ↦ {𝑦 ∈ ℝ ∣ ∃𝑧 ∈ ℕ0 ∃𝑤 ∈ ℕ0 (𝑦 = (𝑧 + ((√‘𝑥) · 𝑤)) ∧ ((𝑧↑2) − (𝑥 · (𝑤↑2))) = 1)})
df-pell14qr 43522Pell14QR = (𝑥 ∈ (ℕ ∖ ◻NN) ↦ {𝑦 ∈ ℝ ∣ ∃𝑧 ∈ ℕ0 ∃𝑤 ∈ ℤ (𝑦 = (𝑧 + ((√‘𝑥) · 𝑤)) ∧ ((𝑧↑2) − (𝑥 · (𝑤↑2))) = 1)})
df-pell1234qr 43523Pell1234QR = (𝑥 ∈ (ℕ ∖ ◻NN) ↦ {𝑦 ∈ ℝ ∣ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ (𝑦 = (𝑧 + ((√‘𝑥) · 𝑤)) ∧ ((𝑧↑2) − (𝑥 · (𝑤↑2))) = 1)})
df-pellfund 43524PellFund = (𝑥 ∈ (ℕ ∖ ◻NN) ↦ inf({𝑧 ∈ (Pell14QR‘𝑥) ∣ 1 < 𝑧}, ℝ, < ))
crmx 43579class Xrm
crmy 43580class Yrm
df-rmx 43581 Xrm = (𝑎 ∈ (ℤ≥‘2), 𝑛 ∈ ℤ ↦ (1st ‘(◡(𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st ‘𝑏) + ((√‘((𝑎↑2) − 1)) · (2nd ‘𝑏))))‘((𝑎 + (√‘((𝑎↑2) − 1)))↑𝑛))))
df-rmy 43582 Yrm = (𝑎 ∈ (ℤ≥‘2), 𝑛 ∈ ℤ ↦ (2nd ‘(◡(𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st ‘𝑏) + ((√‘((𝑎↑2) − 1)) · (2nd ‘𝑏))))‘((𝑎 + (√‘((𝑎↑2) − 1)))↑𝑛))))
clfig 43746class LFinGen
df-lfig 43747LFinGen = {𝑤 ∈ LMod ∣ (Base‘𝑤) ∈ ((LSpan‘𝑤) “ (𝒫 (Base‘𝑤) ∩ Fin))}
clnm 43754class LNoeM
df-lnm 43755LNoeM = {𝑤 ∈ LMod ∣ ∀𝑖 ∈ (LSubSp‘𝑤)(𝑤 ↾s 𝑖) ∈ LFinGen}
clnr 43788class LNoeR
df-lnr 43789LNoeR = {𝑎 ∈ Ring ∣ (ringLMod‘𝑎) ∈ LNoeM}
cldgis 43800class ldgIdlSeq
df-ldgis 43801ldgIdlSeq = (𝑟 ∈ V ↦ (𝑖 ∈ (LIdeal‘(Poly1‘𝑟)) ↦ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘 ∈ 𝑖 (((deg1‘𝑟)‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe1‘𝑘)‘𝑥))})))
cmnc 43810class Monic
cplylt 43811class Poly<
df-mnc 43812 Monic = (𝑠 ∈ 𝒫 ℂ ↦ {𝑝 ∈ (Poly‘𝑠) ∣ ((coeff‘𝑝)‘(deg‘𝑝)) = 1})
df-plylt 43813 Poly< = (𝑠 ∈ 𝒫 ℂ, 𝑥 ∈ ℕ0 ↦ {𝑝 ∈ (Poly‘𝑠) ∣ (𝑝 = 0𝑝 ∨ (deg‘𝑝) < 𝑥)})
cdgraa 43819class degAA
cmpaa 43820class minPolyAA
df-dgraa 43821degAA = (𝑥 ∈ 𝔸 ↦ inf({𝑑 ∈ ℕ ∣ ∃𝑝 ∈ ((Poly‘ℚ) ∖ {0𝑝})((deg‘𝑝) = 𝑑 ∧ (𝑝‘𝑥) = 0)}, ℝ, < ))
df-mpaa 43822minPolyAA = (𝑥 ∈ 𝔸 ↦ (℩𝑝 ∈ (Poly‘ℚ)((deg‘𝑝) = (degAA‘𝑥) ∧ (𝑝‘𝑥) = 0 ∧ ((coeff‘𝑝)‘(degAA‘𝑥)) = 1)))
citgo 43836class IntgOver
cza 43837class ℤ
df-itgo 43838IntgOver = (𝑠 ∈ 𝒫 ℂ ↦ {𝑥 ∈ ℂ ∣ ∃𝑝 ∈ (Poly‘𝑠)((𝑝‘𝑥) = 0 ∧ ((coeff‘𝑝)‘(deg‘𝑝)) = 1)})
df-za 43839ℤ = (IntgOver‘ℤ)
cmend 43850class MEndo
df-mend 43851MEndo = (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 43876class CytP
df-cytp 43877CytP = (𝑛 ∈ ℕ ↦ ((mulGrp‘(Poly1‘ℂfld)) Σg (𝑟 ∈ (◡(od‘((mulGrp‘ℂfld) ↾s (ℂ ∖ {0}))) “ {𝑛}) ↦ ((var1‘ℂfld)(-g‘(Poly1‘ℂfld))((algSc‘(Poly1‘ℂfld))‘𝑟)))))
ctopsep 43885class TopSep
ctoplnd 43886class TopLnd
df-topsep 43887TopSep = {𝑗 ∈ Top ∣ ∃𝑥 ∈ 𝒫 ∪ 𝑗(𝑥 ≼ ω ∧ ((cls‘𝑗)‘𝑥) = ∪ 𝑗)}
df-toplnd 43888TopLnd = {𝑥 ∈ Top ∣ ∀𝑦 ∈ 𝒫 𝑥(∪ 𝑥 = ∪ 𝑦 → ∃𝑧 ∈ 𝒫 𝑥(𝑧 ≼ ω ∧ ∪ 𝑥 = ∪ 𝑧))}
crcl 44350class r*
df-rcl 44351r* = (𝑥 ∈ V ↦ ∩ {𝑧 ∣ (𝑥 ⊆ 𝑧 ∧ ( I ↾ (dom 𝑧 ∪ ran 𝑧)) ⊆ 𝑧)})
whe 44450wff 𝑅 hereditary 𝐴
df-he 44451(𝑅 hereditary 𝐴 ↔ (𝑅 “ 𝐴) ⊆ 𝐴)
ax-frege1 44468(𝜑 → (𝜓 → 𝜑))
ax-frege2 44469((𝜑 → (𝜓 → 𝜒)) → ((𝜑 → 𝜓) → (𝜑 → 𝜒)))
ax-frege8 44487((𝜑 → (𝜓 → 𝜒)) → (𝜓 → (𝜑 → 𝜒)))
ax-frege28 44508((𝜑 → 𝜓) → (¬ 𝜓 → ¬ 𝜑))
ax-frege31 44512(¬ ¬ 𝜑 → 𝜑)
ax-frege41 44523(𝜑 → ¬ ¬ 𝜑)
ax-frege52a 44535((𝜑 ↔ 𝜓) → (if-(𝜑, 𝜃, 𝜒) → if-(𝜓, 𝜃, 𝜒)))
ax-frege54a 44540(𝜑 ↔ 𝜑)
ax-frege58a 44553((𝜓 ∧ 𝜒) → if-(𝜑, 𝜓, 𝜒))
ax-frege52c 44566(𝐴 = 𝐵 → ([𝐴 / 𝑥]𝜑 → [𝐵 / 𝑥]𝜑))
ax-frege54c 44570𝐴 = 𝐴
ax-frege58b 44579(∀𝑥𝜑 → [𝑦 / 𝑥]𝜑)
cmnring 44887class MndRing
df-mnring 44888 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))))))⟩))
ccoll 44912class (𝐹 Coll 𝐴)
df-coll 44913(𝐹 Coll 𝐴) = ∪ 𝑥 ∈ 𝐴 Scott (𝐹 “ {𝑥})
cbcc 44998class C𝑐
df-bcc 44999C𝑐 = (𝑐 ∈ ℂ, 𝑘 ∈ ℕ0 ↦ ((𝑐 FallFac 𝑘) / (!‘𝑘)))
cplusr 45117class +𝑟
cminusr 45118class -𝑟
ctimesr 45119class .𝑣
cptdfc 45120class PtDf(𝐴, 𝐵)
crr3c 45121class RR3
cline3 45122class line3
df-addr 45123+𝑟 = (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑣 ∈ ℝ ↦ ((𝑥‘𝑣) + (𝑦‘𝑣))))
df-subr 45124-𝑟 = (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑣 ∈ ℝ ↦ ((𝑥‘𝑣) − (𝑦‘𝑣))))
df-mulv 45125.𝑣 = (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑣 ∈ ℝ ↦ (𝑥 · (𝑦‘𝑣))))
df-ptdf 45136PtDf(𝐴, 𝐵) = (𝑥 ∈ ℝ ↦ (((𝑥.𝑣(𝐵-𝑟𝐴)) +𝑣 𝐴) “ {1, 2, 3}))
df-rr3 45137RR3 = (ℝ ↑m {1, 2, 3})
df-line3 45138line3 = {𝑥 ∈ 𝒫 RR3 ∣ (2o ≼ 𝑥 ∧ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑧 ≠ 𝑦 → ran PtDf(𝑦, 𝑧) = 𝑥))}
wvd1 45230wff (   𝜑   ▶   𝜓   )
df-vd1 45231((   𝜑   ▶   𝜓   ) ↔ (𝜑 → 𝜓))
wvd2 45238wff (   𝜑   ,   𝜓   ▶   𝜒   )
df-vd2 45239((   𝜑   ,   𝜓   ▶   𝜒   ) ↔ ((𝜑 ∧ 𝜓) → 𝜒))
wvhc2 45241wff (   𝜑   ,   𝜓   )
df-vhc2 45242((   𝜑   ,   𝜓   ) ↔ (𝜑 ∧ 𝜓))
wvd3 45248wff (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜃   )
wvhc3 45249wff (   𝜑   ,   𝜓   ,   𝜒   )
df-vhc3 45250((   𝜑   ,   𝜓   ,   𝜒   ) ↔ (𝜑 ∧ 𝜓 ∧ 𝜒))
df-vd3 45251((   𝜑   ,   𝜓   ,   𝜒   ▶   𝜃   ) ↔ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃))
wrelp 45603wff 𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵)
df-relp 45604(𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵) ↔ (𝐻:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐻‘𝑥)𝑆(𝐻‘𝑦))))
clsi 46417class lim inf
df-liminf 46418lim inf = (𝑥 ∈ V ↦ sup(ran (𝑘 ∈ ℝ ↦ inf(((𝑥 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )), ℝ*, < ))
clsxlim 46484class ~~>*
df-xlim 46485~~>* = (⇝𝑡‘(ordTop‘ ≤ ))
csalg 46974class SAlg
df-salg 46975SAlg = {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (∪ 𝑥 ∖ 𝑦) ∈ 𝑥 ∧ ∀𝑦 ∈ 𝒫 𝑥(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑥))}
csalon 46976class SalOn
df-salon 46977SalOn = (𝑥 ∈ V ↦ {𝑠 ∈ SAlg ∣ ∪ 𝑠 = 𝑥})
csalgen 46978class SalGen
df-salgen 46979SalGen = (𝑥 ∈ V ↦ ∩ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑥 ∧ 𝑥 ⊆ 𝑠)})
csumge0 47028class Σ^
df-sumge0 47029Σ^ = (𝑥 ∈ V ↦ if(+∞ ∈ ran 𝑥, +∞, sup(ran (𝑦 ∈ (𝒫 dom 𝑥 ∩ Fin) ↦ Σ𝑤 ∈ 𝑦 (𝑥‘𝑤)), ℝ*, < )))
cmea 47115class Meas
df-mea 47116Meas = {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 47155class OutMeas
df-ome 47156OutMeas = {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 47157class CaraGen
df-caragen 47158CaraGen = (𝑜 ∈ OutMeas ↦ {𝑒 ∈ 𝒫 ∪ dom 𝑜 ∣ ∀𝑎 ∈ 𝒫 ∪ dom 𝑜((𝑜‘(𝑎 ∩ 𝑒)) +𝑒 (𝑜‘(𝑎 ∖ 𝑒))) = (𝑜‘𝑎)})
covoln 47202class voln*
df-ovoln 47203voln* = (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 47204class voln
df-voln 47205voln = (𝑥 ∈ Fin ↦ ((voln*‘𝑥) ↾ (CaraGen‘(voln*‘𝑥))))
csmblfn 47361class SMblFn
df-smblfn 47362SMblFn = (𝑠 ∈ SAlg ↦ {𝑓 ∈ (ℝ ↑pm ∪ 𝑠) ∣ ∀𝑎 ∈ ℝ (◡𝑓 “ (-∞(,)𝑎)) ∈ (𝑠 ↾t dom 𝑓)})
caiota 47769class (℩'𝑥𝜑)
df-aiota 47771(℩'𝑥𝜑) = ∩ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}
wdfat 47802wff 𝐹 defAt 𝐴
cafv 47803class (𝐹'''𝐴)
caov 47804class ((𝐴𝐹𝐵))
df-dfat 47805(𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
df-afv 47806(𝐹'''𝐴) = (℩'𝑥𝐴𝐹𝑥)
df-aov 47807 ((𝐴𝐹𝐵)) = (𝐹'''⟨𝐴, 𝐵⟩)
cafv2 47894class (𝐹''''𝐴)
df-afv2 47895(𝐹''''𝐴) = if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ∪ ran 𝐹)
cnelbr 47957class _∉
df-nelbr 47958 _∉ = {⟨𝑥, 𝑦⟩ ∣ ¬ 𝑥 ∈ 𝑦}
ciccp 48111class RePart
df-iccp 48112RePart = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ* ↑m (0...𝑚)) ∣ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1))})
wich 48143wff [𝑥⇄𝑦]𝜑
df-ich 48144([𝑥⇄𝑦]𝜑 ↔ ∀𝑥∀𝑦([𝑥 / 𝑎][𝑦 / 𝑥][𝑎 / 𝑦]𝜑 ↔ 𝜑))
cspr 48175class Pairs
df-spr 48176Pairs = (𝑣 ∈ V ↦ {𝑝 ∣ ∃𝑎 ∈ 𝑣 ∃𝑏 ∈ 𝑣 𝑝 = {𝑎, 𝑏}})
cprpr 48210class Pairsproper
df-prpr 48211Pairsproper = (𝑣 ∈ V ↦ {𝑝 ∣ ∃𝑎 ∈ 𝑣 ∃𝑏 ∈ 𝑣 (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})})
cfmtno 48228class FermatNo
df-fmtno 48229FermatNo = (𝑛 ∈ ℕ0 ↦ ((2↑(2↑𝑛)) + 1))
ceven 48338class Even
codd 48339class Odd
df-even 48340 Even = {𝑧 ∈ ℤ ∣ (𝑧 / 2) ∈ ℤ}
df-odd 48341 Odd = {𝑧 ∈ ℤ ∣ ((𝑧 + 1) / 2) ∈ ℤ}
cfppr 48438class FPPr
df-fppr 48439 FPPr = (𝑛 ∈ ℕ ↦ {𝑥 ∈ (ℤ≥‘4) ∣ (𝑥 ∉ ℙ ∧ 𝑥 ∥ ((𝑛↑(𝑥 − 1)) − 1))})
cgbe 48459class GoldbachEven
cgbow 48460class GoldbachOddW
cgbo 48461class GoldbachOdd
df-gbe 48462 GoldbachEven = {𝑧 ∈ Even ∣ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑧 = (𝑝 + 𝑞))}
df-gbow 48463 GoldbachOddW = {𝑧 ∈ Odd ∣ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑧 = ((𝑝 + 𝑞) + 𝑟)}
df-gbo 48464 GoldbachOdd = {𝑧 ∈ Odd ∣ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ ((𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑟 ∈ Odd ) ∧ 𝑧 = ((𝑝 + 𝑞) + 𝑟))}
ax-bgbltosilva 48524((𝑁 ∈ Even ∧ 4 < 𝑁 ∧ 𝑁 ≤ (4 · (10↑18))) → 𝑁 ∈ GoldbachEven )
ax-tgoldbachgt 48525𝑂 = {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 48528∃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 48532class ClNeighbVtx
df-clnbgr 48533 ClNeighbVtx = (𝑔 ∈ V, 𝑣 ∈ (Vtx‘𝑔) ↦ ({𝑣} ∪ {𝑛 ∈ (Vtx‘𝑔) ∣ ∃𝑒 ∈ (Edg‘𝑔){𝑣, 𝑛} ⊆ 𝑒}))
cisubgr 48574class ISubGr
df-isubgr 48575 ISubGr = (𝑔 ∈ V, 𝑣 ∈ 𝒫 (Vtx‘𝑔) ↦ ⟨𝑣, ⦋(iEdg‘𝑔) / 𝑒⦌(𝑒 ↾ {𝑥 ∈ dom 𝑒 ∣ (𝑒‘𝑥) ⊆ 𝑣})⟩)
cgrisom 48588class GraphIsom
cgrim 48589class GraphIso
cgric 48590class ≃𝑔𝑟
df-grisom 48591 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 48592 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 48595 ≃𝑔𝑟 = (◡ GraphIso “ (V ∖ 1o))
cgrtri 48651class GrTriangles
df-grtri 48652GrTriangles = (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 48665class StarGr
df-stgr 48666StarGr = (𝑛 ∈ ℕ0 ↦ {⟨(Base‘ndx), (0...𝑛)⟩, ⟨(.ef‘ndx), ( I ↾ {𝑒 ∈ 𝒫 (0...𝑛) ∣ ∃𝑥 ∈ (1...𝑛)𝑒 = {0, 𝑥}})⟩})
cgrlim 48690class GraphLocIso
cgrlic 48691class ≃𝑙𝑔𝑟
df-grlim 48692 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 48695 ≃𝑙𝑔𝑟 = (◡ GraphLocIso “ (V ∖ 1o))
cgpg 48754class gPetersenGr
df-gpg 48755 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 48847class UPWalks
df-upwlks 48848UPWalks = (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 48897class clLaw
casslaw 48898class assLaw
ccomlaw 48899class comLaw
df-cllaw 48900 clLaw = {⟨𝑜, 𝑚⟩ ∣ ∀𝑥 ∈ 𝑚 ∀𝑦 ∈ 𝑚 (𝑥𝑜𝑦) ∈ 𝑚}
df-comlaw 48901 comLaw = {⟨𝑜, 𝑚⟩ ∣ ∀𝑥 ∈ 𝑚 ∀𝑦 ∈ 𝑚 (𝑥𝑜𝑦) = (𝑦𝑜𝑥)}
df-asslaw 48902 assLaw = {⟨𝑜, 𝑚⟩ ∣ ∀𝑥 ∈ 𝑚 ∀𝑦 ∈ 𝑚 ∀𝑧 ∈ 𝑚 ((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧))}
cintop 48910class intOp
cclintop 48911class clIntOp
cassintop 48912class assIntOp
df-intop 48913 intOp = (𝑚 ∈ V, 𝑛 ∈ V ↦ (𝑛 ↑m (𝑚 × 𝑚)))
df-clintop 48914 clIntOp = (𝑚 ∈ V ↦ (𝑚 intOp 𝑚))
df-assintop 48915 assIntOp = (𝑚 ∈ V ↦ {𝑜 ∈ ( clIntOp ‘𝑚) ∣ 𝑜 assLaw 𝑚})
cmgm2 48929class MgmALT
ccmgm2 48930class CMgmALT
csgrp2 48931class SGrpALT
ccsgrp2 48932class CSGrpALT
df-mgm2 48933MgmALT = {𝑚 ∣ (+g‘𝑚) clLaw (Base‘𝑚)}
df-cmgm2 48934CMgmALT = {𝑚 ∈ MgmALT ∣ (+g‘𝑚) comLaw (Base‘𝑚)}
df-sgrp2 48935SGrpALT = {𝑔 ∈ MgmALT ∣ (+g‘𝑔) assLaw (Base‘𝑔)}
df-csgrp2 48936CSGrpALT = {𝑔 ∈ SGrpALT ∣ (+g‘𝑔) comLaw (Base‘𝑔)}
crngcALTV 48977class RngCatALTV
df-rngcALTV 48978RngCatALTV = (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 49001class RingCatALTV
df-ringcALTV 49002RingCatALTV = (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)))⟩})
cprmrng 49048class PrmRing
df-prmring 49049PrmRing = {𝑟 ∈ Ring ∣ {(0g‘𝑟)} ∈ (PrmIdeal‘𝑟)}
cdmatalt 49125class DMatALT
cscmatalt 49126class ScMatALT
df-dmatalt 49127 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 49128 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 49133class linC
clinco 49134class LinCo
df-linc 49135 linC = (𝑚 ∈ V ↦ (𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑m 𝑣), 𝑣 ∈ 𝒫 (Base‘𝑚) ↦ (𝑚 Σg (𝑥 ∈ 𝑣 ↦ ((𝑠‘𝑥)( ·𝑠 ‘𝑚)𝑥)))))
df-lco 49136 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 49169class linIndS
clindeps 49170class linDepS
df-lininds 49171 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 49173 linDepS = {⟨𝑠, 𝑚⟩ ∣ ¬ 𝑠 linIndS 𝑚}
cfdiv 49266class /f
df-fdiv 49267 /f = (𝑓 ∈ V, 𝑔 ∈ V ↦ ((𝑓 ∘f / 𝑔) ↾ (𝑔 supp 0)))
cbigo 49276class Ο
df-bigo 49277Ο = (𝑔 ∈ (ℝ ↑pm ℝ) ↦ {𝑓 ∈ (ℝ ↑pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(𝑓‘𝑦) ≤ (𝑚 · (𝑔‘𝑦))})
cblen 49298class #b
df-blen 49299#b = (𝑛 ∈ V ↦ if(𝑛 = 0, 1, ((⌊‘(2 logb (abs‘𝑛))) + 1)))
cdig 49324class digit
df-dig 49325digit = (𝑏 ∈ ℕ ↦ (𝑘 ∈ ℤ, 𝑟 ∈ (0[,)+∞) ↦ ((⌊‘((𝑏↑-𝑘) · 𝑟)) mod 𝑏)))
cnaryf 49355class -aryF
df-naryf 49356-aryF = (𝑛 ∈ ℕ0, 𝑥 ∈ V ↦ (𝑥 ↑m (𝑥 ↑m (0..^𝑛))))
citco 49386class IterComp
cack 49387class Ack
df-itco 49388IterComp = (𝑓 ∈ V ↦ seq0((𝑔 ∈ V, 𝑗 ∈ V ↦ (𝑓 ∘ 𝑔)), (𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, ( I ↾ dom 𝑓), 𝑓))))
df-ack 49389Ack = seq0((𝑓 ∈ V, 𝑗 ∈ V ↦ (𝑛 ∈ ℕ0 ↦ (((IterComp‘𝑓)‘(𝑛 + 1))‘1))), (𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, (𝑛 ∈ ℕ0 ↦ (𝑛 + 1)), 𝑖)))
cline 49456class LineM
csph 49457class Sphere
df-line 49458LineM = (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 49459Sphere = (𝑤 ∈ V ↦ (𝑥 ∈ (Base‘𝑤), 𝑟 ∈ (0[,]+∞) ↦ {𝑝 ∈ (Base‘𝑤) ∣ (𝑝(dist‘𝑤)𝑥) = 𝑟}))
coppf 49849class oppFunc
df-oppf 49850 oppFunc = (𝑓 ∈ V, 𝑔 ∈ V ↦ if((Rel 𝑔 ∧ Rel dom 𝑔), ⟨𝑓, tpos 𝑔⟩, ∅))
cup 49900class UP
df-up 49901 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 49986class swapF
df-swapf 49987 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 50043class ∘F
df-fuco 50044 ∘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 50100class −∘F
df-prcof 50101 −∘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 50144class ThinCat
df-thinc 50145ThinCat = {𝑐 ∈ Cat ∣ [(Base‘𝑐) / 𝑏][(Hom ‘𝑐) / ℎ]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∃*𝑓 𝑓 ∈ (𝑥ℎ𝑦)}
ctermc 50199class TermCat
df-termc 50200TermCat = {𝑐 ∈ ThinCat ∣ ∃𝑥(Base‘𝑐) = {𝑥}}
cprstc 50276class ProsetToCat
df-prstc 50277ProsetToCat = (𝑘 ∈ Proset ↦ ((𝑘 sSet ⟨(Hom ‘ndx), ((le‘𝑘) × {1o})⟩) sSet ⟨(comp‘ndx), ∅⟩))
cmndtc 50304class MndToCat
df-mndtc 50305MndToCat = (𝑚 ∈ Mnd ↦ {⟨(Base‘ndx), {𝑚}⟩, ⟨(Hom ‘ndx), {⟨𝑚, 𝑚, (Base‘𝑚)⟩}⟩, ⟨(comp‘ndx), {⟨⟨𝑚, 𝑚, 𝑚⟩, (+g‘𝑚)⟩}⟩})
clan 50332class Lan
cran 50333class Ran
df-lan 50334 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 50335 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 50370class Limit
ccmd 50371class Colimit
df-lmd 50372 Limit = (𝑐 ∈ V, 𝑑 ∈ V ↦ (𝑓 ∈ (𝑑 Func 𝑐) ↦ (( oppFunc ‘(𝑐Δfunc𝑑))((oppCat‘𝑐) UP (oppCat‘(𝑑 FuncCat 𝑐)))𝑓)))
df-cmd 50373 Colimit = (𝑐 ∈ V, 𝑑 ∈ V ↦ (𝑓 ∈ (𝑑 Func 𝑐) ↦ ((𝑐Δfunc𝑑)(𝑐 UP (𝑑 FuncCat 𝑐))𝑓)))
csetrecs 50410class setrecs(𝐹)
df-setrecs 50411setrecs(𝐹) = ∪ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)}
cpg 50436class Pg
df-pg 50437Pg = setrecs((𝑥 ∈ V ↦ (𝒫 𝑥 × 𝒫 𝑥)))
cge-real 50447class ≥
cgt 50448class >
df-gte 50449 ≥ = ◡ ≤
df-gt 50450 > = ◡ <
csinh 50457class sinh
ccosh 50458class cosh
ctanh 50459class tanh
df-sinh 50460sinh = (𝑥 ∈ ℂ ↦ ((sin‘(i · 𝑥)) / i))
df-cosh 50461cosh = (𝑥 ∈ ℂ ↦ (cos‘(i · 𝑥)))
df-tanh 50462tanh = (𝑥 ∈ (◡cosh “ (ℂ ∖ {0})) ↦ ((tan‘(i · 𝑥)) / i))
csec 50468class sec
ccsc 50469class csc
ccot 50470class cot
df-sec 50471sec = (𝑥 ∈ {𝑦 ∈ ℂ ∣ (cos‘𝑦) ≠ 0} ↦ (1 / (cos‘𝑥)))
df-csc 50472csc = (𝑥 ∈ {𝑦 ∈ ℂ ∣ (sin‘𝑦) ≠ 0} ↦ (1 / (sin‘𝑥)))
df-cot 50473cot = (𝑥 ∈ {𝑦 ∈ ℂ ∣ (sin‘𝑦) ≠ 0} ↦ ((cos‘𝑥) / (sin‘𝑥)))
clog- 50492class log_
df-logbALT 50493log_ = (𝑏 ∈ (ℂ ∖ {0, 1}) ↦ (𝑥 ∈ (ℂ ∖ {0}) ↦ ((log‘𝑥) / (log‘𝑏))))
wreflexive 50494wff 𝑅Reflexive𝐴
df-reflexive 50495(𝑅Reflexive𝐴 ↔ (𝑅 ⊆ (𝐴 × 𝐴) ∧ ∀𝑥 ∈ 𝐴 𝑥𝑅𝑥))
wirreflexive 50496wff 𝑅Irreflexive𝐴
df-irreflexive 50497(𝑅Irreflexive𝐴 ↔ (𝑅 ⊆ (𝐴 × 𝐴) ∧ ∀𝑥 ∈ 𝐴 ¬ 𝑥𝑅𝑥))
wals 50513wff ∀∃𝑥(𝜑 → 𝜓)
wrals 50514wff ∀∃𝑥 ∈ 𝐴(𝜑 → 𝜓)
df-als 50515(∀∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥(𝜑 → 𝜓) ∧ ∃𝑥𝜑))
df-rals 50516(∀∃𝑥 ∈ 𝐴(𝜑 → 𝜓) ↔ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ∧ ∃𝑥 ∈ 𝐴 𝜑))
  Copyright terms: Public domain W3C validator