List of Syntax, Axioms (ax-) and
Definitions (df-)
Ref | Expression (see link for any distinct variable requirements)
|
wn 3 | wff ¬ 𝜑 |
wi 4 | wff (𝜑 → 𝜓) |
ax-mp 5 | ⊢ 𝜑
& ⊢ (𝜑 → 𝜓) ⇒ ⊢ 𝜓 |
ax-1 6 | ⊢ (𝜑 → (𝜓 → 𝜑)) |
ax-2 7 | ⊢ ((𝜑 → (𝜓 → 𝜒)) → ((𝜑 → 𝜓) → (𝜑 → 𝜒))) |
ax-3 8 | ⊢ ((¬ 𝜑 → ¬ 𝜓) → (𝜓 → 𝜑)) |
wb 198 | wff (𝜑 ↔ 𝜓) |
df-bi 199 | ⊢ ¬ (((𝜑 ↔ 𝜓) → ¬ ((𝜑 → 𝜓) → ¬ (𝜓 → 𝜑))) → ¬ (¬ ((𝜑 → 𝜓) → ¬ (𝜓 → 𝜑)) → (𝜑 ↔ 𝜓))) |
wa 387 | wff (𝜑 ∧ 𝜓) |
df-an 388 | ⊢ ((𝜑 ∧ 𝜓) ↔ ¬ (𝜑 → ¬ 𝜓)) |
wo 833 | wff (𝜑 ∨ 𝜓) |
df-or 834 | ⊢ ((𝜑 ∨ 𝜓) ↔ (¬ 𝜑 → 𝜓)) |
wif 1043 | wff if-(𝜑, 𝜓, 𝜒) |
df-ifp 1044 | ⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒))) |
w3o 1067 | wff (𝜑 ∨ 𝜓 ∨ 𝜒) |
w3a 1068 | wff (𝜑 ∧ 𝜓 ∧ 𝜒) |
df-3or 1069 | ⊢ ((𝜑 ∨ 𝜓 ∨ 𝜒) ↔ ((𝜑 ∨ 𝜓) ∨ 𝜒)) |
df-3an 1070 | ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ((𝜑 ∧ 𝜓) ∧ 𝜒)) |
wnan 1463 | wff (𝜑 ⊼ 𝜓) |
df-nan 1464 | ⊢ ((𝜑 ⊼ 𝜓) ↔ ¬ (𝜑 ∧ 𝜓)) |
wxo 1488 | wff (𝜑 ⊻ 𝜓) |
df-xor 1489 | ⊢ ((𝜑 ⊻ 𝜓) ↔ ¬ (𝜑 ↔ 𝜓)) |
wal 1505 | wff ∀𝑥𝜑 |
cv 1506 | class 𝑥 |
wceq 1507 | wff 𝐴 = 𝐵 |
wtru 1508 | wff ⊤ |
df-tru 1510 | ⊢ (⊤ ↔ (∀𝑥 𝑥 = 𝑥 → ∀𝑥 𝑥 = 𝑥)) |
wfal 1519 | wff ⊥ |
df-fal 1520 | ⊢ (⊥ ↔ ¬
⊤) |
whad 1556 | wff hadd(𝜑, 𝜓, 𝜒) |
df-had 1557 | ⊢ (hadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ⊻ 𝜓) ⊻ 𝜒)) |
wcad 1569 | wff cadd(𝜑, 𝜓, 𝜒) |
df-cad 1570 | ⊢ (cadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ∧ 𝜓) ∨ (𝜒 ∧ (𝜑 ⊻ 𝜓)))) |
wex 1742 | wff ∃𝑥𝜑 |
df-ex 1743 | ⊢ (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑) |
wnf 1746 | wff Ⅎ𝑥𝜑 |
df-nf 1747 | ⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑)) |
ax-gen 1758 | ⊢ 𝜑 ⇒ ⊢ ∀𝑥𝜑 |
ax-4 1772 | ⊢ (∀𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓)) |
ax-5 1869 | ⊢ (𝜑 → ∀𝑥𝜑) |
ax-6 1928 | ⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦 |
ax-7 1964 | ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 → 𝑦 = 𝑧)) |
wsb 2013 | wff [𝑦 / 𝑥]𝜑 |
df-sb 2014 | ⊢ ([𝑡 / 𝑥]𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) |
wcel 2048 | wff 𝐴 ∈ 𝐵 |
ax-8 2050 | ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧)) |
ax-9 2057 | ⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦)) |
ax-10 2077 | ⊢ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑) |
ax-11 2091 | ⊢ (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑) |
ax-12 2104 | ⊢ (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) |
ax-13 2299 | ⊢ (¬ 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧)) |
wmo 2542 | wff ∃*𝑥𝜑 |
df-mo 2544 | ⊢ (∃*𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)) |
weu 2579 | wff ∃!𝑥𝜑 |
df-eu 2580 | ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃*𝑥𝜑)) |
ax-ext 2745 | ⊢ (∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → 𝑥 = 𝑦) |
cab 2753 | class {𝑥 ∣ 𝜑} |
df-clab 2754 | ⊢ (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ [𝑥 / 𝑦]𝜑) |
df-cleq 2765 | ⊢ (𝑦 = 𝑧 ↔ ∀𝑢(𝑢 ∈ 𝑦 ↔ 𝑢 ∈ 𝑧))
& ⊢ (𝑡 = 𝑡 ↔ ∀𝑣(𝑣 ∈ 𝑡 ↔ 𝑣 ∈ 𝑡)) ⇒ ⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) |
df-clel 2840 | ⊢ (𝑦 ∈ 𝑧 ↔ ∃𝑢(𝑢 = 𝑦 ∧ 𝑢 ∈ 𝑧))
& ⊢ (𝑡 ∈ 𝑡 ↔ ∃𝑣(𝑣 = 𝑡 ∧ 𝑣 ∈ 𝑡)) ⇒ ⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵)) |
wnfc 2910 | wff Ⅎ𝑥𝐴 |
df-nfc 2912 | ⊢ (Ⅎ𝑥𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴) |
wne 2961 | wff 𝐴 ≠ 𝐵 |
df-ne 2962 | ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵) |
wnel 3067 | wff 𝐴 ∉ 𝐵 |
df-nel 3068 | ⊢ (𝐴 ∉ 𝐵 ↔ ¬ 𝐴 ∈ 𝐵) |
wral 3082 | wff ∀𝑥 ∈ 𝐴 𝜑 |
wrex 3083 | wff ∃𝑥 ∈ 𝐴 𝜑 |
wreu 3084 | wff ∃!𝑥 ∈ 𝐴 𝜑 |
wrmo 3085 | wff ∃*𝑥 ∈ 𝐴 𝜑 |
crab 3086 | class {𝑥 ∈ 𝐴 ∣ 𝜑} |
df-ral 3087 | ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)) |
df-rex 3088 | ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) |
df-reu 3089 | ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) |
df-rmo 3090 | ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) |
df-rab 3091 | ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} |
cvv 3409 | class V |
df-v 3411 | ⊢ V = {𝑥 ∣ 𝑥 = 𝑥} |
wcdeq 3659 | wff CondEq(𝑥 = 𝑦 → 𝜑) |
df-cdeq 3660 | ⊢ (CondEq(𝑥 = 𝑦 → 𝜑) ↔ (𝑥 = 𝑦 → 𝜑)) |
wsbc 3677 | wff [𝐴 / 𝑥]𝜑 |
df-sbc 3678 | ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑}) |
csb 3782 | class ⦋𝐴 / 𝑥⦌𝐵 |
df-csb 3783 | ⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} |
cdif 3822 | class (𝐴 ∖ 𝐵) |
cun 3823 | class (𝐴 ∪ 𝐵) |
cin 3824 | class (𝐴 ∩ 𝐵) |
wss 3825 | wff 𝐴 ⊆ 𝐵 |
wpss 3826 | wff 𝐴 ⊊ 𝐵 |
df-dif 3828 | ⊢ (𝐴 ∖ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)} |
df-un 3830 | ⊢ (𝐴 ∪ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)} |
df-in 3832 | ⊢ (𝐴 ∩ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)} |
df-ss 3839 | ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ 𝐵) = 𝐴) |
df-pss 3841 | ⊢ (𝐴 ⊊ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ 𝐵)) |
csymdif 4100 | class (𝐴 △ 𝐵) |
df-symdif 4101 | ⊢ (𝐴 △ 𝐵) = ((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐴)) |
c0 4173 | class ∅ |
df-nul 4174 | ⊢ ∅ = (V ∖ V) |
cif 4344 | class if(𝜑, 𝐴, 𝐵) |
df-if 4345 | ⊢ if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑))} |
cpw 4416 | class 𝒫 𝐴 |
df-pw 4418 | ⊢ 𝒫 𝐴 = {𝑥 ∣ 𝑥 ⊆ 𝐴} |
csn 4435 | class {𝐴} |
df-sn 4436 | ⊢ {𝐴} = {𝑥 ∣ 𝑥 = 𝐴} |
cpr 4437 | class {𝐴, 𝐵} |
df-pr 4438 | ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵}) |
ctp 4439 | class {𝐴, 𝐵, 𝐶} |
df-tp 4440 | ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶}) |
cop 4441 | class 〈𝐴, 𝐵〉 |
df-op 4442 | ⊢ 〈𝐴, 𝐵〉 = {𝑥 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})} |
cotp 4443 | class 〈𝐴, 𝐵, 𝐶〉 |
df-ot 4444 | ⊢ 〈𝐴, 𝐵, 𝐶〉 = 〈〈𝐴, 𝐵〉, 𝐶〉 |
cuni 4706 | class ∪
𝐴 |
df-uni 4707 | ⊢ ∪ 𝐴 = {𝑥 ∣ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)} |
cint 4743 | class ∩
𝐴 |
df-int 4744 | ⊢ ∩ 𝐴 = {𝑥 ∣ ∀𝑦(𝑦 ∈ 𝐴 → 𝑥 ∈ 𝑦)} |
ciun 4786 | class ∪ 𝑥 ∈ 𝐴 𝐵 |
ciin 4787 | class ∩ 𝑥 ∈ 𝐴 𝐵 |
df-iun 4788 | ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵} |
df-iin 4789 | ⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵} |
wdisj 4891 | wff Disj 𝑥 ∈ 𝐴 𝐵 |
df-disj 4892 | ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵) |
wbr 4923 | wff 𝐴𝑅𝐵 |
df-br 4924 | ⊢ (𝐴𝑅𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝑅) |
copab 4985 | class {〈𝑥, 𝑦〉 ∣ 𝜑} |
df-opab 4986 | ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑)} |
cmpt 5002 | class (𝑥 ∈ 𝐴 ↦ 𝐵) |
df-mpt 5003 | ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} |
wtr 5024 | wff Tr 𝐴 |
df-tr 5025 | ⊢ (Tr 𝐴 ↔ ∪ 𝐴 ⊆ 𝐴) |
ax-rep 5043 | ⊢ (∀𝑤∃𝑦∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦) → ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))) |
ax-sep 5054 | ⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)) |
ax-nul 5061 | ⊢ ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥 |
ax-pow 5113 | ⊢ ∃𝑦∀𝑧(∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦) |
ax-pr 5180 | ⊢ ∃𝑧∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧) |
cid 5304 | class I |
df-id 5305 | ⊢ I = {〈𝑥, 𝑦〉 ∣ 𝑥 = 𝑦} |
cep 5309 | class E |
df-eprel 5310 | ⊢ E = {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝑦} |
wpo 5317 | wff 𝑅 Po 𝐴 |
wor 5318 | wff 𝑅 Or 𝐴 |
df-po 5319 | ⊢ (𝑅 Po 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))) |
df-so 5320 | ⊢ (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥))) |
wfr 5356 | wff 𝑅 Fr 𝐴 |
wse 5357 | wff 𝑅 Se 𝐴 |
wwe 5358 | wff 𝑅 We 𝐴 |
df-fr 5359 | ⊢ (𝑅 Fr 𝐴 ↔ ∀𝑥((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦)) |
df-se 5360 | ⊢ (𝑅 Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V) |
df-we 5361 | ⊢ (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴 ∧ 𝑅 Or 𝐴)) |
cxp 5398 | class (𝐴 × 𝐵) |
ccnv 5399 | class ◡𝐴 |
cdm 5400 | class dom 𝐴 |
crn 5401 | class ran 𝐴 |
cres 5402 | class (𝐴 ↾ 𝐵) |
cima 5403 | class (𝐴 “ 𝐵) |
ccom 5404 | class (𝐴 ∘ 𝐵) |
wrel 5405 | wff Rel 𝐴 |
df-xp 5406 | ⊢ (𝐴 × 𝐵) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)} |
df-rel 5407 | ⊢ (Rel 𝐴 ↔ 𝐴 ⊆ (V × V)) |
df-cnv 5408 | ⊢ ◡𝐴 = {〈𝑥, 𝑦〉 ∣ 𝑦𝐴𝑥} |
df-co 5409 | ⊢ (𝐴 ∘ 𝐵) = {〈𝑥, 𝑦〉 ∣ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)} |
df-dm 5410 | ⊢ dom 𝐴 = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦} |
df-rn 5411 | ⊢ ran 𝐴 = dom ◡𝐴 |
df-res 5412 | ⊢ (𝐴 ↾ 𝐵) = (𝐴 ∩ (𝐵 × V)) |
df-ima 5413 | ⊢ (𝐴 “ 𝐵) = ran (𝐴 ↾ 𝐵) |
cpred 5979 | class Pred(𝑅, 𝐴, 𝑋) |
df-pred 5980 | ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋})) |
word 6022 | wff Ord 𝐴 |
con0 6023 | class On |
wlim 6024 | wff Lim 𝐴 |
csuc 6025 | class suc 𝐴 |
df-ord 6026 | ⊢ (Ord 𝐴 ↔ (Tr 𝐴 ∧ E We 𝐴)) |
df-on 6027 | ⊢ On = {𝑥 ∣ Ord 𝑥} |
df-lim 6028 | ⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴)) |
df-suc 6029 | ⊢ suc 𝐴 = (𝐴 ∪ {𝐴}) |
cio 6144 | class (℩𝑥𝜑) |
df-iota 6146 | ⊢ (℩𝑥𝜑) = ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} |
wfun 6176 | wff Fun 𝐴 |
wfn 6177 | wff 𝐴 Fn 𝐵 |
wf 6178 | wff 𝐹:𝐴⟶𝐵 |
wf1 6179 | wff 𝐹:𝐴–1-1→𝐵 |
wfo 6180 | wff 𝐹:𝐴–onto→𝐵 |
wf1o 6181 | wff 𝐹:𝐴–1-1-onto→𝐵 |
cfv 6182 | class (𝐹‘𝐴) |
wiso 6183 | wff 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) |
df-fun 6184 | ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ (𝐴 ∘ ◡𝐴) ⊆ I )) |
df-fn 6185 | ⊢ (𝐴 Fn 𝐵 ↔ (Fun 𝐴 ∧ dom 𝐴 = 𝐵)) |
df-f 6186 | ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵)) |
df-f1 6187 | ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ Fun ◡𝐹)) |
df-fo 6188 | ⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵)) |
df-f1o 6189 | ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹:𝐴–1-1→𝐵 ∧ 𝐹:𝐴–onto→𝐵)) |
df-fv 6190 | ⊢ (𝐹‘𝐴) = (℩𝑥𝐴𝐹𝑥) |
df-isom 6191 | ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐻:𝐴–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)))) |
crio 6930 | class (℩𝑥 ∈ 𝐴 𝜑) |
df-riota 6931 | ⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) |
co 6970 | class (𝐴𝐹𝐵) |
coprab 6971 | class {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} |
cmpo 6972 | class (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) |
df-ov 6973 | ⊢ (𝐴𝐹𝐵) = (𝐹‘〈𝐴, 𝐵〉) |
df-oprab 6974 | ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜑)} |
df-mpo 6975 | ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)} |
cof 7219 | class ∘𝑓
𝑅 |
cofr 7220 | class ∘𝑟
𝑅 |
df-of 7221 | ⊢ ∘𝑓
𝑅 = (𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥)))) |
df-ofr 7222 | ⊢ ∘𝑟 𝑅 = {〈𝑓, 𝑔〉 ∣ ∀𝑥 ∈ (dom 𝑓 ∩ dom 𝑔)(𝑓‘𝑥)𝑅(𝑔‘𝑥)} |
crpss 7260 | class
[⊊] |
df-rpss 7261 | ⊢ [⊊] = {〈𝑥, 𝑦〉 ∣ 𝑥 ⊊ 𝑦} |
ax-un 7273 | ⊢ ∃𝑦∀𝑧(∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦) |
com 7390 | class ω |
df-om 7391 | ⊢ ω = {𝑥 ∈ On ∣ ∀𝑦(Lim 𝑦 → 𝑥 ∈ 𝑦)} |
c1st 7492 | class 1st |
c2nd 7493 | class 2nd |
df-1st 7494 | ⊢ 1st = (𝑥 ∈ V ↦ ∪ dom {𝑥}) |
df-2nd 7495 | ⊢ 2nd = (𝑥 ∈ V ↦ ∪ ran {𝑥}) |
csupp 7626 | class supp |
df-supp 7627 | ⊢ supp = (𝑥 ∈ V, 𝑧 ∈ V ↦ {𝑖 ∈ dom 𝑥 ∣ (𝑥 “ {𝑖}) ≠ {𝑧}}) |
ctpos 7687 | class tpos 𝐹 |
df-tpos 7688 | ⊢ tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})) |
ccur 7727 | class curry 𝐴 |
cunc 7728 | class uncurry 𝐴 |
df-cur 7729 | ⊢ curry 𝐹 = (𝑥 ∈ dom dom 𝐹 ↦ {〈𝑦, 𝑧〉 ∣ 〈𝑥, 𝑦〉𝐹𝑧}) |
df-unc 7730 | ⊢ uncurry 𝐹 = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝑦(𝐹‘𝑥)𝑧} |
cund 7734 | class Undef |
df-undef 7735 | ⊢ Undef = (𝑠 ∈ V ↦ 𝒫 ∪ 𝑠) |
cwrecs 7742 | class wrecs(𝑅, 𝐴, 𝐹) |
df-wrecs 7743 | ⊢ wrecs(𝑅, 𝐴, 𝐹) = ∪ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} |
wsmo 7779 | wff Smo 𝐴 |
df-smo 7780 | ⊢ (Smo 𝐴 ↔ (𝐴:dom 𝐴⟶On ∧ Ord dom 𝐴 ∧ ∀𝑥 ∈ dom 𝐴∀𝑦 ∈ dom 𝐴(𝑥 ∈ 𝑦 → (𝐴‘𝑥) ∈ (𝐴‘𝑦)))) |
crecs 7804 | class recs(𝐹) |
df-recs 7805 | ⊢ recs(𝐹) = wrecs( E , On, 𝐹) |
crdg 7842 | class rec(𝐹, 𝐼) |
df-rdg 7843 | ⊢ rec(𝐹, 𝐼) = recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔)))))) |
cseqom 7879 | class seq𝜔(𝐹, 𝐼) |
df-seqom 7880 | ⊢ seq𝜔(𝐹, 𝐼) = (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ 〈suc 𝑖, (𝑖𝐹𝑣)〉), 〈∅, ( I ‘𝐼)〉) “
ω) |
c1o 7890 | class 1o |
c2o 7891 | class 2o |
c3o 7892 | class 3o |
c4o 7893 | class 4o |
coa 7894 | class +o |
comu 7895 | class
·o |
coe 7896 | class
↑o |
df-1o 7897 | ⊢ 1o = suc
∅ |
df-2o 7898 | ⊢ 2o = suc
1o |
df-3o 7899 | ⊢ 3o = suc
2o |
df-4o 7900 | ⊢ 4o = suc
3o |
df-oadd 7901 | ⊢ +o = (𝑥 ∈ On, 𝑦 ∈ On ↦ (rec((𝑧 ∈ V ↦ suc 𝑧), 𝑥)‘𝑦)) |
df-omul 7902 | ⊢ ·o = (𝑥 ∈ On, 𝑦 ∈ On ↦ (rec((𝑧 ∈ V ↦ (𝑧 +o 𝑥)), ∅)‘𝑦)) |
df-oexp 7903 | ⊢ ↑o = (𝑥 ∈ On, 𝑦 ∈ On ↦ if(𝑥 = ∅, (1o ∖ 𝑦), (rec((𝑧 ∈ V ↦ (𝑧 ·o 𝑥)), 1o)‘𝑦))) |
wer 8078 | wff 𝑅 Er 𝐴 |
cec 8079 | class [𝐴]𝑅 |
cqs 8080 | class (𝐴 / 𝑅) |
df-er 8081 | ⊢ (𝑅 Er 𝐴 ↔ (Rel 𝑅 ∧ dom 𝑅 = 𝐴 ∧ (◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅)) |
df-ec 8083 | ⊢ [𝐴]𝑅 = (𝑅 “ {𝐴}) |
df-qs 8087 | ⊢ (𝐴 / 𝑅) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅} |
cmap 8198 | class
↑𝑚 |
cpm 8199 | class
↑pm |
df-map 8200 | ⊢ ↑𝑚 = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓 ∣ 𝑓:𝑦⟶𝑥}) |
df-pm 8201 | ⊢ ↑pm =
(𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓 ∈ 𝒫 (𝑦 × 𝑥) ∣ Fun 𝑓}) |
cixp 8251 | class X𝑥 ∈ 𝐴 𝐵 |
df-ixp 8252 | ⊢ X𝑥 ∈ 𝐴 𝐵 = {𝑓 ∣ (𝑓 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵)} |
cen 8295 | class ≈ |
cdom 8296 | class ≼ |
csdm 8297 | class ≺ |
cfn 8298 | class Fin |
df-en 8299 | ⊢ ≈ = {〈𝑥, 𝑦〉 ∣ ∃𝑓 𝑓:𝑥–1-1-onto→𝑦} |
df-dom 8300 | ⊢ ≼ = {〈𝑥, 𝑦〉 ∣ ∃𝑓 𝑓:𝑥–1-1→𝑦} |
df-sdom 8301 | ⊢ ≺ = ( ≼ ∖ ≈
) |
df-fin 8302 | ⊢ Fin = {𝑥 ∣ ∃𝑦 ∈ ω 𝑥 ≈ 𝑦} |
cfsupp 8620 | class finSupp |
df-fsupp 8621 | ⊢ finSupp = {〈𝑟, 𝑧〉 ∣ (Fun 𝑟 ∧ (𝑟 supp 𝑧) ∈ Fin)} |
cfi 8661 | class fi |
df-fi 8662 | ⊢ fi = (𝑥 ∈ V ↦ {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦}) |
csup 8691 | class sup(𝐴, 𝐵, 𝑅) |
cinf 8692 | class inf(𝐴, 𝐵, 𝑅) |
df-sup 8693 | ⊢ sup(𝐴, 𝐵, 𝑅) = ∪ {𝑥 ∈ 𝐵 ∣ (∀𝑦 ∈ 𝐴 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐵 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐴 𝑦𝑅𝑧))} |
df-inf 8694 | ⊢ inf(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, ◡𝑅) |
coi 8760 | class OrdIso(𝑅, 𝐴) |
df-oi 8761 | ⊢ OrdIso(𝑅, 𝐴) = if((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴), (recs((ℎ ∈ V ↦ (℩𝑣 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤}∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣))) ↾ {𝑥 ∈ On ∣ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (recs((ℎ ∈ V ↦ (℩𝑣 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤}∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣))) “ 𝑥)𝑧𝑅𝑡}), ∅) |
char 8807 | class har |
cwdom 8808 | class
≼* |
df-har 8809 | ⊢ har = (𝑥 ∈ V ↦ {𝑦 ∈ On ∣ 𝑦 ≼ 𝑥}) |
df-wdom 8810 | ⊢ ≼* = {〈𝑥, 𝑦〉 ∣ (𝑥 = ∅ ∨ ∃𝑧 𝑧:𝑦–onto→𝑥)} |
ax-reg 8843 | ⊢ (∃𝑦 𝑦 ∈ 𝑥 → ∃𝑦(𝑦 ∈ 𝑥 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑥))) |
ax-inf 8887 | ⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦))) |
ax-inf2 8890 | ⊢ ∃𝑥(∃𝑦(𝑦 ∈ 𝑥 ∧ ∀𝑧 ¬ 𝑧 ∈ 𝑦) ∧ ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑧 ∈ 𝑥 ∧ ∀𝑤(𝑤 ∈ 𝑧 ↔ (𝑤 ∈ 𝑦 ∨ 𝑤 = 𝑦))))) |
ccnf 8910 | class CNF |
df-cnf 8911 | ⊢ 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))) |
ctc 8964 | class TC |
df-tc 8965 | ⊢ TC = (𝑥 ∈ V ↦ ∩ {𝑦
∣ (𝑥 ⊆ 𝑦 ∧ Tr 𝑦)}) |
cr1 8977 | class
𝑅1 |
crnk 8978 | class rank |
df-r1 8979 | ⊢ 𝑅1 =
rec((𝑥 ∈ V ↦
𝒫 𝑥),
∅) |
df-rank 8980 | ⊢ rank = (𝑥 ∈ V ↦ ∩ {𝑦
∈ On ∣ 𝑥 ∈
(𝑅1‘suc 𝑦)}) |
cdju 9113 | class (𝐴 ⊔ 𝐵) |
cinl 9114 | class inl |
cinr 9115 | class inr |
df-dju 9116 | ⊢ (𝐴 ⊔ 𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵)) |
df-inl 9117 | ⊢ inl = (𝑥 ∈ V ↦ 〈∅, 𝑥〉) |
df-inr 9118 | ⊢ inr = (𝑥 ∈ V ↦ 〈1o, 𝑥〉) |
ccrd 9150 | class card |
cale 9151 | class ℵ |
ccf 9152 | class cf |
wacn 9153 | class AC 𝐴 |
df-card 9154 | ⊢ card = (𝑥 ∈ V ↦ ∩ {𝑦
∈ On ∣ 𝑦 ≈
𝑥}) |
df-aleph 9155 | ⊢ ℵ = rec(har, ω) |
df-cf 9156 | ⊢ cf = (𝑥 ∈ On ↦ ∩ {𝑦
∣ ∃𝑧(𝑦 = (card‘𝑧) ∧ (𝑧 ⊆ 𝑥 ∧ ∀𝑣 ∈ 𝑥 ∃𝑢 ∈ 𝑧 𝑣 ⊆ 𝑢))}) |
df-acn 9157 | ⊢ AC 𝐴 = {𝑥 ∣ (𝐴 ∈ V ∧ ∀𝑓 ∈ ((𝒫 𝑥 ∖ {∅})
↑𝑚 𝐴)∃𝑔∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝑓‘𝑦))} |
wac 9327 | wff
CHOICE |
df-ac 9328 | ⊢ (CHOICE ↔
∀𝑥∃𝑓(𝑓 ⊆ 𝑥 ∧ 𝑓 Fn dom 𝑥)) |
ccda 9379 | class
+𝑐 |
df-cda 9380 | ⊢ +𝑐 = (𝑥 ∈ V, 𝑦 ∈ V ↦ ((𝑥 × {∅}) ∪ (𝑦 × {1o}))) |
cfin1a 9490 | class FinIa |
cfin2 9491 | class
FinII |
cfin4 9492 | class
FinIV |
cfin3 9493 | class
FinIII |
cfin5 9494 | class FinV |
cfin6 9495 | class
FinVI |
cfin7 9496 | class
FinVII |
df-fin1a 9497 | ⊢ FinIa = {𝑥 ∣ ∀𝑦 ∈ 𝒫 𝑥(𝑦 ∈ Fin ∨ (𝑥 ∖ 𝑦) ∈ Fin)} |
df-fin2 9498 | ⊢ FinII = {𝑥 ∣ ∀𝑦 ∈ 𝒫 𝒫 𝑥((𝑦 ≠ ∅ ∧ [⊊] Or
𝑦) → ∪ 𝑦
∈ 𝑦)} |
df-fin4 9499 | ⊢ FinIV = {𝑥 ∣ ¬ ∃𝑦(𝑦 ⊊ 𝑥 ∧ 𝑦 ≈ 𝑥)} |
df-fin3 9500 | ⊢ FinIII = {𝑥 ∣ 𝒫 𝑥 ∈ FinIV} |
df-fin5 9501 | ⊢ FinV = {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 ≺ (𝑥 ⊔ 𝑥))} |
df-fin6 9502 | ⊢ FinVI = {𝑥 ∣ (𝑥 ≺ 2o ∨ 𝑥 ≺ (𝑥 × 𝑥))} |
df-fin7 9503 | ⊢ FinVII = {𝑥 ∣ ¬ ∃𝑦 ∈ (On ∖ ω)𝑥 ≈ 𝑦} |
ax-cc 9647 | ⊢ (𝑥 ≈ ω → ∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)) |
ax-dc 9658 | ⊢ ((∃𝑦∃𝑧 𝑦𝑥𝑧 ∧ ran 𝑥 ⊆ dom 𝑥) → ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛)) |
ax-ac 9671 | ⊢ ∃𝑦∀𝑧∀𝑤((𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → ∃𝑣∀𝑢(∃𝑡((𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑡) ∧ (𝑢 ∈ 𝑡 ∧ 𝑡 ∈ 𝑦)) ↔ 𝑢 = 𝑣)) |
ax-ac2 9675 | ⊢ ∃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 9832 | class GCH |
df-gch 9833 | ⊢ GCH = (Fin ∪ {𝑥 ∣ ∀𝑦 ¬ (𝑥 ≺ 𝑦 ∧ 𝑦 ≺ 𝒫 𝑥)}) |
cwina 9894 | class
Inaccw |
cina 9895 | class Inacc |
df-wina 9896 | ⊢ Inaccw = {𝑥 ∣ (𝑥 ≠ ∅ ∧ (cf‘𝑥) = 𝑥 ∧ ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝑥 𝑦 ≺ 𝑧)} |
df-ina 9897 | ⊢ Inacc = {𝑥 ∣ (𝑥 ≠ ∅ ∧ (cf‘𝑥) = 𝑥 ∧ ∀𝑦 ∈ 𝑥 𝒫 𝑦 ≺ 𝑥)} |
cwun 9912 | class WUni |
cwunm 9913 | class wUniCl |
df-wun 9914 | ⊢ WUni = {𝑢 ∣ (Tr 𝑢 ∧ 𝑢 ≠ ∅ ∧ ∀𝑥 ∈ 𝑢 (∪ 𝑥 ∈ 𝑢 ∧ 𝒫 𝑥 ∈ 𝑢 ∧ ∀𝑦 ∈ 𝑢 {𝑥, 𝑦} ∈ 𝑢))} |
df-wunc 9915 | ⊢ wUniCl = (𝑥 ∈ V ↦ ∩ {𝑢
∈ WUni ∣ 𝑥
⊆ 𝑢}) |
ctsk 9960 | class Tarski |
df-tsk 9961 | ⊢ Tarski = {𝑦 ∣ (∀𝑧 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑦 ∧ ∃𝑤 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧 ≈ 𝑦 ∨ 𝑧 ∈ 𝑦))} |
cgru 10002 | class Univ |
df-gru 10003 | ⊢ Univ = {𝑢 ∣ (Tr 𝑢 ∧ ∀𝑥 ∈ 𝑢 (𝒫 𝑥 ∈ 𝑢 ∧ ∀𝑦 ∈ 𝑢 {𝑥, 𝑦} ∈ 𝑢 ∧ ∀𝑦 ∈ (𝑢 ↑𝑚 𝑥)∪
ran 𝑦 ∈ 𝑢))} |
ax-groth 10035 | ⊢ ∃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 10049 | class tarskiMap |
df-tskm 10050 | ⊢ tarskiMap = (𝑥 ∈ V ↦ ∩ {𝑦
∈ Tarski ∣ 𝑥
∈ 𝑦}) |
cnpi 10056 | class N |
cpli 10057 | class
+N |
cmi 10058 | class
·N |
clti 10059 | class
<N |
cplpq 10060 | class
+pQ |
cmpq 10061 | class
·pQ |
cltpq 10062 | class
<pQ |
ceq 10063 | class
~Q |
cnq 10064 | class Q |
c1q 10065 | class
1Q |
cerq 10066 | class
[Q] |
cplq 10067 | class
+Q |
cmq 10068 | class
·Q |
crq 10069 | class
*Q |
cltq 10070 | class
<Q |
cnp 10071 | class P |
c1p 10072 | class
1P |
cpp 10073 | class
+P |
cmp 10074 | class
·P |
cltp 10075 | class
<P |
cer 10076 | class
~R |
cnr 10077 | class R |
c0r 10078 | class
0R |
c1r 10079 | class
1R |
cm1r 10080 | class
-1R |
cplr 10081 | class
+R |
cmr 10082 | class
·R |
cltr 10083 | class
<R |
df-ni 10084 | ⊢ N = (ω
∖ {∅}) |
df-pli 10085 | ⊢ +N = (
+o ↾ (N ×
N)) |
df-mi 10086 | ⊢
·N = ( ·o ↾
(N × N)) |
df-lti 10087 | ⊢ <N = ( E ∩
(N × N)) |
df-plpq 10120 | ⊢ +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 10121 | ⊢ ·pQ = (𝑥 ∈ (N ×
N), 𝑦 ∈
(N × N) ↦ 〈((1st
‘𝑥)
·N (1st ‘𝑦)), ((2nd ‘𝑥)
·N (2nd ‘𝑦))〉) |
df-ltpq 10122 | ⊢ <pQ =
{〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ (N ×
N) ∧ 𝑦
∈ (N × N)) ∧ ((1st
‘𝑥)
·N (2nd ‘𝑦)) <N
((1st ‘𝑦)
·N (2nd ‘𝑥)))} |
df-enq 10123 | ⊢ ~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 10124 | ⊢ Q = {𝑥 ∈ (N ×
N) ∣ ∀𝑦 ∈ (N ×
N)(𝑥
~Q 𝑦 → ¬ (2nd ‘𝑦) <N
(2nd ‘𝑥))} |
df-erq 10125 | ⊢ [Q] = (
~Q ∩ ((N × N)
× Q)) |
df-plq 10126 | ⊢ +Q =
(([Q] ∘ +pQ ) ↾
(Q × Q)) |
df-mq 10127 | ⊢
·Q = (([Q] ∘
·pQ ) ↾ (Q ×
Q)) |
df-1nq 10128 | ⊢ 1Q =
〈1o, 1o〉 |
df-rq 10129 | ⊢ *Q =
(◡ ·Q
“ {1Q}) |
df-ltnq 10130 | ⊢ <Q = (
<pQ ∩ (Q ×
Q)) |
df-np 10193 | ⊢ P = {𝑥 ∣ ((∅ ⊊
𝑥 ∧ 𝑥 ⊊ Q) ∧
∀𝑦 ∈ 𝑥 (∀𝑧(𝑧 <Q 𝑦 → 𝑧 ∈ 𝑥) ∧ ∃𝑧 ∈ 𝑥 𝑦 <Q 𝑧))} |
df-1p 10194 | ⊢ 1P =
{𝑥 ∣ 𝑥 <Q
1Q} |
df-plp 10195 | ⊢ +P = (𝑥 ∈ P, 𝑦 ∈ P ↦
{𝑤 ∣ ∃𝑣 ∈ 𝑥 ∃𝑢 ∈ 𝑦 𝑤 = (𝑣 +Q 𝑢)}) |
df-mp 10196 | ⊢
·P = (𝑥 ∈ P, 𝑦 ∈ P ↦ {𝑤 ∣ ∃𝑣 ∈ 𝑥 ∃𝑢 ∈ 𝑦 𝑤 = (𝑣 ·Q 𝑢)}) |
df-ltp 10197 | ⊢ <P = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧
𝑥 ⊊ 𝑦)} |
df-enr 10267 | ⊢ ~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 10268 | ⊢ R =
((P × P) / ~R
) |
df-plr 10269 | ⊢ +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 10270 | ⊢
·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 10271 | ⊢ <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 10272 | ⊢ 0R =
[〈1P, 1P〉]
~R |
df-1r 10273 | ⊢ 1R =
[〈(1P +P
1P), 1P〉]
~R |
df-m1r 10274 | ⊢ -1R =
[〈1P, (1P
+P 1P)〉]
~R |
cc 10325 | class ℂ |
cr 10326 | class ℝ |
cc0 10327 | class 0 |
c1 10328 | class 1 |
ci 10329 | class i |
caddc 10330 | class + |
cltrr 10331 | class
<ℝ |
cmul 10332 | class · |
df-c 10333 | ⊢ ℂ = (R
× R) |
df-0 10334 | ⊢ 0 =
〈0R,
0R〉 |
df-1 10335 | ⊢ 1 =
〈1R,
0R〉 |
df-i 10336 | ⊢ i =
〈0R,
1R〉 |
df-r 10337 | ⊢ ℝ = (R
× {0R}) |
df-add 10338 | ⊢ + = {〈〈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 10339 | ⊢ · = {〈〈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 10340 | ⊢ <ℝ =
{〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧
∃𝑧∃𝑤((𝑥 = 〈𝑧, 0R〉 ∧
𝑦 = 〈𝑤,
0R〉) ∧ 𝑧 <R 𝑤))} |
ax-cnex 10383 | ⊢ ℂ ∈ V |
ax-resscn 10384 | ⊢ ℝ ⊆ ℂ |
ax-1cn 10385 | ⊢ 1 ∈ ℂ |
ax-icn 10386 | ⊢ i ∈ ℂ |
ax-addcl 10387 | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) ∈ ℂ) |
ax-addrcl 10388 | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + 𝐵) ∈ ℝ) |
ax-mulcl 10389 | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) ∈ ℂ) |
ax-mulrcl 10390 | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 · 𝐵) ∈ ℝ) |
ax-mulcom 10391 | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) = (𝐵 · 𝐴)) |
ax-addass 10392 | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶))) |
ax-mulass 10393 | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶))) |
ax-distr 10394 | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶))) |
ax-i2m1 10395 | ⊢ ((i · i) + 1) = 0 |
ax-1ne0 10396 | ⊢ 1 ≠ 0 |
ax-1rid 10397 | ⊢ (𝐴 ∈ ℝ → (𝐴 · 1) = 𝐴) |
ax-rnegex 10398 | ⊢ (𝐴 ∈ ℝ → ∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0) |
ax-rrecex 10399 | ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → ∃𝑥 ∈ ℝ (𝐴 · 𝑥) = 1) |
ax-cnre 10400 | ⊢ (𝐴 ∈ ℂ → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝐴 = (𝑥 + (i · 𝑦))) |
ax-pre-lttri 10401 | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 <ℝ 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 <ℝ 𝐴))) |
ax-pre-lttrn 10402 | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 <ℝ 𝐵 ∧ 𝐵 <ℝ 𝐶) → 𝐴 <ℝ 𝐶)) |
ax-pre-ltadd 10403 | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 <ℝ 𝐵 → (𝐶 + 𝐴) <ℝ (𝐶 + 𝐵))) |
ax-pre-mulgt0 10404 | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((0
<ℝ 𝐴
∧ 0 <ℝ 𝐵) → 0 <ℝ (𝐴 · 𝐵))) |
ax-pre-sup 10405 | ⊢ ((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 10406 | ⊢ + :(ℂ ×
ℂ)⟶ℂ |
ax-mulf 10407 | ⊢ · :(ℂ ×
ℂ)⟶ℂ |
cpnf 10463 | class +∞ |
cmnf 10464 | class -∞ |
cxr 10465 | class
ℝ* |
clt 10466 | class < |
cle 10467 | class ≤ |
df-pnf 10468 | ⊢ +∞ = 𝒫 ∪ ℂ |
df-mnf 10469 | ⊢ -∞ = 𝒫
+∞ |
df-xr 10470 | ⊢ ℝ* = (ℝ
∪ {+∞, -∞}) |
df-ltxr 10471 | ⊢ < = ({〈𝑥, 𝑦〉 ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)} ∪ (((ℝ ∪ {-∞}) ×
{+∞}) ∪ ({-∞} × ℝ))) |
df-le 10472 | ⊢ ≤ = ((ℝ*
× ℝ*) ∖ ◡
< ) |
cmin 10662 | class − |
cneg 10663 | class -𝐴 |
df-sub 10664 | ⊢ − = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥)) |
df-neg 10665 | ⊢ -𝐴 = (0 − 𝐴) |
cdiv 11090 | class / |
df-div 11091 | ⊢ / = (𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦
(℩𝑧 ∈
ℂ (𝑦 · 𝑧) = 𝑥)) |
cn 11431 | class ℕ |
df-nn 11432 | ⊢ ℕ = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) “
ω) |
c2 11488 | class 2 |
c3 11489 | class 3 |
c4 11490 | class 4 |
c5 11491 | class 5 |
c6 11492 | class 6 |
c7 11493 | class 7 |
c8 11494 | class 8 |
c9 11495 | class 9 |
df-2 11496 | ⊢ 2 = (1 + 1) |
df-3 11497 | ⊢ 3 = (2 + 1) |
df-4 11498 | ⊢ 4 = (3 + 1) |
df-5 11499 | ⊢ 5 = (4 + 1) |
df-6 11500 | ⊢ 6 = (5 + 1) |
df-7 11501 | ⊢ 7 = (6 + 1) |
df-8 11502 | ⊢ 8 = (7 + 1) |
df-9 11503 | ⊢ 9 = (8 + 1) |
cn0 11700 | class
ℕ0 |
df-n0 11701 | ⊢ ℕ0 = (ℕ
∪ {0}) |
cxnn0 11772 | class
ℕ0* |
df-xnn0 11773 | ⊢ ℕ0* =
(ℕ0 ∪ {+∞}) |
cz 11786 | class ℤ |
df-z 11787 | ⊢ ℤ = {𝑛 ∈ ℝ ∣ (𝑛 = 0 ∨ 𝑛 ∈ ℕ ∨ -𝑛 ∈ ℕ)} |
cdc 11904 | class ;𝐴𝐵 |
df-dec 11905 | ⊢ ;𝐴𝐵 = (((9 + 1) · 𝐴) + 𝐵) |
cuz 12051 | class
ℤ≥ |
df-uz 12052 | ⊢ ℤ≥ = (𝑗 ∈ ℤ ↦ {𝑘 ∈ ℤ ∣ 𝑗 ≤ 𝑘}) |
cq 12155 | class ℚ |
df-q 12156 | ⊢ ℚ = ( / “ (ℤ
× ℕ)) |
crp 12197 | class
ℝ+ |
df-rp 12198 | ⊢ ℝ+ = {𝑥 ∈ ℝ ∣ 0 <
𝑥} |
cxne 12314 | class -𝑒𝐴 |
cxad 12315 | class
+𝑒 |
cxmu 12316 | class
·e |
df-xneg 12317 | ⊢ -𝑒𝐴 = if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴)) |
df-xadd 12318 | ⊢ +𝑒 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ*
↦ if(𝑥 = +∞,
if(𝑦 = -∞, 0,
+∞), if(𝑥 = -∞,
if(𝑦 = +∞, 0,
-∞), if(𝑦 = +∞,
+∞, if(𝑦 = -∞,
-∞, (𝑥 + 𝑦)))))) |
df-xmul 12319 | ⊢ ·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 12547 | class (,) |
cioc 12548 | class (,] |
cico 12549 | class [,) |
cicc 12550 | class [,] |
df-ioo 12551 | ⊢ (,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ*
↦ {𝑧 ∈
ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)}) |
df-ioc 12552 | ⊢ (,] = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ*
↦ {𝑧 ∈
ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 ≤ 𝑦)}) |
df-ico 12553 | ⊢ [,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ*
↦ {𝑧 ∈
ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}) |
df-icc 12554 | ⊢ [,] = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ*
↦ {𝑧 ∈
ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)}) |
cfz 12701 | class ... |
df-fz 12702 | ⊢ ... = (𝑚 ∈ ℤ, 𝑛 ∈ ℤ ↦ {𝑘 ∈ ℤ ∣ (𝑚 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛)}) |
cfzo 12842 | class ..^ |
df-fzo 12843 | ⊢ ..^ = (𝑚 ∈ ℤ, 𝑛 ∈ ℤ ↦ (𝑚...(𝑛 − 1))) |
cfl 12968 | class ⌊ |
cceil 12969 | class ⌈ |
df-fl 12970 | ⊢ ⌊ = (𝑥 ∈ ℝ ↦ (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1)))) |
df-ceil 12971 | ⊢ ⌈ = (𝑥 ∈ ℝ ↦
-(⌊‘-𝑥)) |
cmo 13045 | class mod |
df-mod 13046 | ⊢ mod = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ+ ↦ (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦))))) |
cseq 13177 | class seq𝑀( + , 𝐹) |
df-seq 13178 | ⊢ seq𝑀( + , 𝐹) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉) “ ω) |
cexp 13237 | class ↑ |
df-exp 13238 | ⊢ ↑ = (𝑥 ∈ ℂ, 𝑦 ∈ ℤ ↦ if(𝑦 = 0, 1, if(0 < 𝑦, (seq1( · , (ℕ × {𝑥}))‘𝑦), (1 / (seq1( · , (ℕ ×
{𝑥}))‘-𝑦))))) |
cfa 13441 | class ! |
df-fac 13442 | ⊢ ! = ({〈0, 1〉} ∪ seq1( ·
, I )) |
cbc 13470 | class C |
df-bc 13471 | ⊢ C = (𝑛 ∈ ℕ0, 𝑘 ∈ ℤ ↦ if(𝑘 ∈ (0...𝑛), ((!‘𝑛) / ((!‘(𝑛 − 𝑘)) · (!‘𝑘))), 0)) |
chash 13498 | class ♯ |
df-hash 13499 | ⊢ ♯ = (((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω) ∘ card)
∪ ((V ∖ Fin) × {+∞})) |
cword 13662 | class Word 𝑆 |
df-word 13663 | ⊢ Word 𝑆 = {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆} |
clsw 13715 | class lastS |
df-lsw 13716 | ⊢ lastS = (𝑤 ∈ V ↦ (𝑤‘((♯‘𝑤) − 1))) |
cconcat 13723 | class ++ |
df-concat 13724 | ⊢ ++ = (𝑠 ∈ V, 𝑡 ∈ V ↦ (𝑥 ∈ (0..^((♯‘𝑠) + (♯‘𝑡))) ↦ if(𝑥 ∈
(0..^(♯‘𝑠)),
(𝑠‘𝑥), (𝑡‘(𝑥 − (♯‘𝑠)))))) |
cs1 13748 | class 〈“𝐴”〉 |
df-s1 13749 | ⊢ 〈“𝐴”〉 = {〈0, ( I ‘𝐴)〉} |
csubstr 13793 | class substr |
df-substr 13794 | ⊢ substr = (𝑠 ∈ V, 𝑏 ∈ (ℤ × ℤ) ↦
if(((1st ‘𝑏)..^(2nd ‘𝑏)) ⊆ dom 𝑠, (𝑥 ∈ (0..^((2nd ‘𝑏) − (1st
‘𝑏))) ↦ (𝑠‘(𝑥 + (1st ‘𝑏)))), ∅)) |
cpfx 13842 | class prefix |
df-pfx 13843 | ⊢ prefix = (𝑠 ∈ V, 𝑙 ∈ ℕ0 ↦ (𝑠 substr 〈0, 𝑙〉)) |
csplice 13948 | class splice |
cspliceOLD 13949 | class spliceOLD |
df-splice 13950 | ⊢ splice = (𝑠 ∈ V, 𝑏 ∈ V ↦ (((𝑠 prefix (1st
‘(1st ‘𝑏))) ++ (2nd ‘𝑏)) ++ (𝑠 substr 〈(2nd
‘(1st ‘𝑏)), (♯‘𝑠)〉))) |
df-spliceOLD 13951 | ⊢ spliceOLD = (𝑠 ∈ V, 𝑏 ∈ V ↦ (((𝑠 substr 〈0, (1st
‘(1st ‘𝑏))〉) ++ (2nd ‘𝑏)) ++ (𝑠 substr 〈(2nd
‘(1st ‘𝑏)), (♯‘𝑠)〉))) |
creverse 13967 | class reverse |
df-reverse 13968 | ⊢ reverse = (𝑠 ∈ V ↦ (𝑥 ∈ (0..^(♯‘𝑠)) ↦ (𝑠‘(((♯‘𝑠) − 1) − 𝑥)))) |
creps 13977 | class repeatS |
df-reps 13978 | ⊢ repeatS = (𝑠 ∈ V, 𝑛 ∈ ℕ0 ↦ (𝑥 ∈ (0..^𝑛) ↦ 𝑠)) |
ccsh 13997 | class cyclShift |
ccshOLD 13998 | class cyclShiftOLD |
df-csh 13999 | ⊢ cyclShift = (𝑤 ∈ {𝑓 ∣ ∃𝑙 ∈ ℕ0 𝑓 Fn (0..^𝑙)}, 𝑛 ∈ ℤ ↦ if(𝑤 = ∅, ∅, ((𝑤 substr 〈(𝑛 mod (♯‘𝑤)), (♯‘𝑤)〉) ++ (𝑤 prefix (𝑛 mod (♯‘𝑤)))))) |
df-cshOLD 14000 | ⊢ cyclShiftOLD = (𝑤 ∈ {𝑓 ∣ ∃𝑙 ∈ ℕ0 𝑓 Fn (0..^𝑙)}, 𝑛 ∈ ℤ ↦ if(𝑤 = ∅, ∅, ((𝑤 substr 〈(𝑛 mod (♯‘𝑤)), (♯‘𝑤)〉) ++ (𝑤 substr 〈0, (𝑛 mod (♯‘𝑤))〉)))) |
cs2 14055 | class 〈“𝐴𝐵”〉 |
cs3 14056 | class 〈“𝐴𝐵𝐶”〉 |
cs4 14057 | class 〈“𝐴𝐵𝐶𝐷”〉 |
cs5 14058 | class 〈“𝐴𝐵𝐶𝐷𝐸”〉 |
cs6 14059 | class 〈“𝐴𝐵𝐶𝐷𝐸𝐹”〉 |
cs7 14060 | class 〈“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”〉 |
cs8 14061 | class 〈“𝐴𝐵𝐶𝐷𝐸𝐹𝐺𝐻”〉 |
df-s2 14062 | ⊢ 〈“𝐴𝐵”〉 = (〈“𝐴”〉 ++
〈“𝐵”〉) |
df-s3 14063 | ⊢ 〈“𝐴𝐵𝐶”〉 = (〈“𝐴𝐵”〉 ++ 〈“𝐶”〉) |
df-s4 14064 | ⊢ 〈“𝐴𝐵𝐶𝐷”〉 = (〈“𝐴𝐵𝐶”〉 ++ 〈“𝐷”〉) |
df-s5 14065 | ⊢ 〈“𝐴𝐵𝐶𝐷𝐸”〉 = (〈“𝐴𝐵𝐶𝐷”〉 ++ 〈“𝐸”〉) |
df-s6 14066 | ⊢ 〈“𝐴𝐵𝐶𝐷𝐸𝐹”〉 = (〈“𝐴𝐵𝐶𝐷𝐸”〉 ++ 〈“𝐹”〉) |
df-s7 14067 | ⊢ 〈“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”〉 = (〈“𝐴𝐵𝐶𝐷𝐸𝐹”〉 ++ 〈“𝐺”〉) |
df-s8 14068 | ⊢ 〈“𝐴𝐵𝐶𝐷𝐸𝐹𝐺𝐻”〉 = (〈“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”〉 ++ 〈“𝐻”〉) |
ctcl 14196 | class t+ |
crtcl 14197 | class t* |
df-trcl 14198 | ⊢ t+ = (𝑥 ∈ V ↦ ∩ {𝑧
∣ (𝑥 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)}) |
df-rtrcl 14199 | ⊢ t* = (𝑥 ∈ V ↦ ∩ {𝑧
∣ (( I ↾ (dom 𝑥
∪ ran 𝑥)) ⊆ 𝑧 ∧ 𝑥 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)}) |
crelexp 14230 | class
↑𝑟 |
df-relexp 14231 | ⊢ ↑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 14265 | class t*rec |
df-rtrclrec 14266 | ⊢ t*rec = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛)) |
cshi 14276 | class shift |
df-shft 14277 | ⊢ shift = (𝑓 ∈ V, 𝑥 ∈ ℂ ↦ {〈𝑦, 𝑧〉 ∣ (𝑦 ∈ ℂ ∧ (𝑦 − 𝑥)𝑓𝑧)}) |
csgn 14296 | class sgn |
df-sgn 14297 | ⊢ sgn = (𝑥 ∈ ℝ* ↦ if(𝑥 = 0, 0, if(𝑥 < 0, -1, 1))) |
ccj 14306 | class ∗ |
cre 14307 | class ℜ |
cim 14308 | class ℑ |
df-cj 14309 | ⊢ ∗ = (𝑥 ∈ ℂ ↦ (℩𝑦 ∈ ℂ ((𝑥 + 𝑦) ∈ ℝ ∧ (i · (𝑥 − 𝑦)) ∈ ℝ))) |
df-re 14310 | ⊢ ℜ = (𝑥 ∈ ℂ ↦ ((𝑥 + (∗‘𝑥)) / 2)) |
df-im 14311 | ⊢ ℑ = (𝑥 ∈ ℂ ↦ (ℜ‘(𝑥 / i))) |
csqrt 14443 | class √ |
cabs 14444 | class abs |
df-sqrt 14445 | ⊢ √ = (𝑥 ∈ ℂ ↦ (℩𝑦 ∈ ℂ ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉
ℝ+))) |
df-abs 14446 | ⊢ abs = (𝑥 ∈ ℂ ↦ (√‘(𝑥 · (∗‘𝑥)))) |
clsp 14678 | class lim sup |
df-limsup 14679 | ⊢ lim sup = (𝑥 ∈ V ↦ inf(ran (𝑘 ∈ ℝ ↦ sup(((𝑥 “ (𝑘[,)+∞)) ∩ ℝ*),
ℝ*, < )), ℝ*, < )) |
cli 14692 | class ⇝ |
crli 14693 | class
⇝𝑟 |
co1 14694 | class 𝑂(1) |
clo1 14695 | class
≤𝑂(1) |
df-clim 14696 | ⊢ ⇝ = {〈𝑓, 𝑦〉 ∣ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+
∃𝑗 ∈ ℤ
∀𝑘 ∈
(ℤ≥‘𝑗)((𝑓‘𝑘) ∈ ℂ ∧ (abs‘((𝑓‘𝑘) − 𝑦)) < 𝑥))} |
df-rlim 14697 | ⊢ ⇝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 14698 | ⊢ 𝑂(1) = {𝑓 ∈ (ℂ
↑pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(abs‘(𝑓‘𝑦)) ≤ 𝑚} |
df-lo1 14699 | ⊢ ≤𝑂(1) = {𝑓 ∈ (ℝ ↑pm
ℝ) ∣ ∃𝑥
∈ ℝ ∃𝑚
∈ ℝ ∀𝑦
∈ (dom 𝑓 ∩ (𝑥[,)+∞))(𝑓‘𝑦) ≤ 𝑚} |
csu 14893 | class Σ𝑘 ∈ 𝐴 𝐵 |
df-sum 14894 | ⊢ Σ𝑘 ∈ 𝐴 𝐵 = (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)))) |
cprod 15109 | class ∏𝑘 ∈ 𝐴 𝐵 |
df-prod 15110 | ⊢ ∏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 15208 | class FallFac |
crisefac 15209 | class RiseFac |
df-risefac 15210 | ⊢ RiseFac = (𝑥 ∈ ℂ, 𝑛 ∈ ℕ0 ↦
∏𝑘 ∈ (0...(𝑛 − 1))(𝑥 + 𝑘)) |
df-fallfac 15211 | ⊢ FallFac = (𝑥 ∈ ℂ, 𝑛 ∈ ℕ0 ↦
∏𝑘 ∈ (0...(𝑛 − 1))(𝑥 − 𝑘)) |
cbp 15250 | class BernPoly |
df-bpoly 15251 | ⊢ 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 15265 | class exp |
ceu 15266 | class e |
csin 15267 | class sin |
ccos 15268 | class cos |
ctan 15269 | class tan |
cpi 15270 | class π |
df-ef 15271 | ⊢ exp = (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ ℕ0
((𝑥↑𝑘) / (!‘𝑘))) |
df-e 15272 | ⊢ e =
(exp‘1) |
df-sin 15273 | ⊢ sin = (𝑥 ∈ ℂ ↦ (((exp‘(i
· 𝑥)) −
(exp‘(-i · 𝑥))) / (2 · i))) |
df-cos 15274 | ⊢ cos = (𝑥 ∈ ℂ ↦ (((exp‘(i
· 𝑥)) +
(exp‘(-i · 𝑥))) / 2)) |
df-tan 15275 | ⊢ tan = (𝑥 ∈ (◡cos “ (ℂ ∖ {0})) ↦
((sin‘𝑥) /
(cos‘𝑥))) |
df-pi 15276 | ⊢ π = inf((ℝ+
∩ (◡sin “ {0})), ℝ,
< ) |
ctau 15405 | class τ |
df-tau 15406 | ⊢ τ = inf((ℝ+ ∩
(◡cos “ {1})), ℝ, <
) |
cdvds 15457 | class ∥ |
df-dvds 15458 | ⊢ ∥ = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ ∃𝑛 ∈ ℤ (𝑛 · 𝑥) = 𝑦)} |
cbits 15618 | class bits |
csad 15619 | class sadd |
csmu 15620 | class smul |
df-bits 15621 | ⊢ bits = (𝑛 ∈ ℤ ↦ {𝑚 ∈ ℕ0 ∣ ¬ 2
∥ (⌊‘(𝑛 /
(2↑𝑚)))}) |
df-sad 15650 | ⊢ 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 15675 | ⊢ 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 15693 | class gcd |
df-gcd 15694 | ⊢ gcd = (𝑥 ∈ ℤ, 𝑦 ∈ ℤ ↦ if((𝑥 = 0 ∧ 𝑦 = 0), 0, sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑥 ∧ 𝑛 ∥ 𝑦)}, ℝ, < ))) |
clcm 15778 | class lcm |
clcmf 15779 | class lcm |
df-lcm 15780 | ⊢ lcm = (𝑥 ∈ ℤ, 𝑦 ∈ ℤ ↦ if((𝑥 = 0 ∨ 𝑦 = 0), 0, inf({𝑛 ∈ ℕ ∣ (𝑥 ∥ 𝑛 ∧ 𝑦 ∥ 𝑛)}, ℝ, < ))) |
df-lcmf 15781 | ⊢ lcm = (𝑧 ∈ 𝒫 ℤ ↦ if(0 ∈
𝑧, 0, inf({𝑛 ∈ ℕ ∣
∀𝑚 ∈ 𝑧 𝑚 ∥ 𝑛}, ℝ, < ))) |
cprime 15861 | class ℙ |
df-prm 15862 | ⊢ ℙ = {𝑝 ∈ ℕ ∣ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑝} ≈ 2o} |
cnumer 15919 | class numer |
cdenom 15920 | class denom |
df-numer 15921 | ⊢ numer = (𝑦 ∈ ℚ ↦ (1st
‘(℩𝑥
∈ (ℤ × ℕ)(((1st ‘𝑥) gcd (2nd ‘𝑥)) = 1 ∧ 𝑦 = ((1st ‘𝑥) / (2nd ‘𝑥)))))) |
df-denom 15922 | ⊢ denom = (𝑦 ∈ ℚ ↦ (2nd
‘(℩𝑥
∈ (ℤ × ℕ)(((1st ‘𝑥) gcd (2nd ‘𝑥)) = 1 ∧ 𝑦 = ((1st ‘𝑥) / (2nd ‘𝑥)))))) |
codz 15946 | class
odℤ |
cphi 15947 | class ϕ |
df-odz 15948 | ⊢ odℤ = (𝑛 ∈ ℕ ↦ (𝑥 ∈ {𝑥 ∈ ℤ ∣ (𝑥 gcd 𝑛) = 1} ↦ inf({𝑚 ∈ ℕ ∣ 𝑛 ∥ ((𝑥↑𝑚) − 1)}, ℝ, <
))) |
df-phi 15949 | ⊢ ϕ = (𝑛 ∈ ℕ ↦ (♯‘{𝑥 ∈ (1...𝑛) ∣ (𝑥 gcd 𝑛) = 1})) |
cpc 16019 | class pCnt |
df-pc 16020 | ⊢ 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 16111 | class ℤ[i] |
df-gz 16112 | ⊢ ℤ[i] = {𝑥 ∈ ℂ ∣ ((ℜ‘𝑥) ∈ ℤ ∧
(ℑ‘𝑥) ∈
ℤ)} |
cvdwa 16147 | class AP |
cvdwm 16148 | class MonoAP |
cvdwp 16149 | class PolyAP |
df-vdwap 16150 | ⊢ AP = (𝑘 ∈ ℕ0 ↦ (𝑎 ∈ ℕ, 𝑑 ∈ ℕ ↦ ran
(𝑚 ∈ (0...(𝑘 − 1)) ↦ (𝑎 + (𝑚 · 𝑑))))) |
df-vdwmc 16151 | ⊢ MonoAP = {〈𝑘, 𝑓〉 ∣ ∃𝑐(ran (AP‘𝑘) ∩ 𝒫 (◡𝑓 “ {𝑐})) ≠ ∅} |
df-vdwpc 16152 | ⊢ 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 16181 | class Ramsey |
df-ram 16183 | ⊢ 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 16213 | class #p |
df-prmo 16214 | ⊢ #p = (𝑛 ∈ ℕ0 ↦
∏𝑘 ∈ (1...𝑛)if(𝑘 ∈ ℙ, 𝑘, 1)) |
cstr 16325 | class Struct |
cnx 16326 | class ndx |
csts 16327 | class sSet |
cslot 16328 | class Slot 𝐴 |
cbs 16329 | class Base |
cress 16330 | class
↾s |
df-struct 16331 | ⊢ Struct = {〈𝑓, 𝑥〉 ∣ (𝑥 ∈ ( ≤ ∩ (ℕ ×
ℕ)) ∧ Fun (𝑓
∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))} |
df-ndx 16332 | ⊢ ndx = ( I ↾ ℕ) |
df-slot 16333 | ⊢ Slot 𝐴 = (𝑥 ∈ V ↦ (𝑥‘𝐴)) |
df-base 16335 | ⊢ Base = Slot 1 |
df-sets 16336 | ⊢ sSet = (𝑠 ∈ V, 𝑒 ∈ V ↦ ((𝑠 ↾ (V ∖ dom {𝑒})) ∪ {𝑒})) |
df-ress 16337 | ⊢ ↾s = (𝑤 ∈ V, 𝑥 ∈ V ↦ if((Base‘𝑤) ⊆ 𝑥, 𝑤, (𝑤 sSet 〈(Base‘ndx), (𝑥 ∩ (Base‘𝑤))〉))) |
cplusg 16411 | class +g |
cmulr 16412 | class .r |
cstv 16413 | class
*𝑟 |
csca 16414 | class Scalar |
cvsca 16415 | class
·𝑠 |
cip 16416 | class
·𝑖 |
cts 16417 | class TopSet |
cple 16418 | class le |
coc 16419 | class oc |
cds 16420 | class dist |
cunif 16421 | class UnifSet |
chom 16422 | class Hom |
cco 16423 | class comp |
df-plusg 16424 | ⊢ +g = Slot 2 |
df-mulr 16425 | ⊢ .r = Slot 3 |
df-starv 16426 | ⊢ *𝑟 = Slot
4 |
df-sca 16427 | ⊢ Scalar = Slot 5 |
df-vsca 16428 | ⊢ ·𝑠 = Slot
6 |
df-ip 16429 | ⊢
·𝑖 = Slot 8 |
df-tset 16430 | ⊢ TopSet = Slot 9 |
df-ple 16431 | ⊢ le = Slot ;10 |
df-ocomp 16432 | ⊢ oc = Slot ;11 |
df-ds 16433 | ⊢ dist = Slot ;12 |
df-unif 16434 | ⊢ UnifSet = Slot ;13 |
df-hom 16435 | ⊢ Hom = Slot ;14 |
df-cco 16436 | ⊢ comp = Slot ;15 |
crest 16540 | class
↾t |
ctopn 16541 | class TopOpen |
df-rest 16542 | ⊢ ↾t = (𝑗 ∈ V, 𝑥 ∈ V ↦ ran (𝑦 ∈ 𝑗 ↦ (𝑦 ∩ 𝑥))) |
df-topn 16543 | ⊢ TopOpen = (𝑤 ∈ V ↦ ((TopSet‘𝑤) ↾t
(Base‘𝑤))) |
ctg 16557 | class topGen |
cpt 16558 | class
∏t |
c0g 16559 | class 0g |
cgsu 16560 | class
Σg |
df-0g 16561 | ⊢ 0g = (𝑔 ∈ V ↦ (℩𝑒(𝑒 ∈ (Base‘𝑔) ∧ ∀𝑥 ∈ (Base‘𝑔)((𝑒(+g‘𝑔)𝑥) = 𝑥 ∧ (𝑥(+g‘𝑔)𝑒) = 𝑥)))) |
df-gsum 16562 | ⊢ 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 16563 | ⊢ topGen = (𝑥 ∈ V ↦ {𝑦 ∣ 𝑦 ⊆ ∪ (𝑥 ∩ 𝒫 𝑦)}) |
df-pt 16564 | ⊢ ∏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 16565 | class Xs |
cpws 16566 | class
↑s |
df-prds 16567 | ⊢ Xs = (𝑠 ∈ V, 𝑟 ∈ V ↦ ⦋X𝑥 ∈
dom 𝑟(Base‘(𝑟‘𝑥)) / 𝑣⦌⦋(𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ X𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥))) / ℎ⦌(({〈(Base‘ndx),
𝑣〉,
〈(+g‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(+g‘(𝑟‘𝑥))(𝑔‘𝑥))))〉, 〈(.r‘ndx),
(𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(.r‘(𝑟‘𝑥))(𝑔‘𝑥))))〉} ∪ {〈(Scalar‘ndx),
𝑠〉, 〈(
·𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑠), 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ (𝑓( ·𝑠
‘(𝑟‘𝑥))(𝑔‘𝑥))))〉,
〈(·𝑖‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑠 Σg (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(·𝑖‘(𝑟‘𝑥))(𝑔‘𝑥)))))〉}) ∪ ({〈(TopSet‘ndx),
(∏t‘(TopOpen ∘ 𝑟))〉, 〈(le‘ndx), {〈𝑓, 𝑔〉 ∣ ({𝑓, 𝑔} ⊆ 𝑣 ∧ ∀𝑥 ∈ dom 𝑟(𝑓‘𝑥)(le‘(𝑟‘𝑥))(𝑔‘𝑥))}〉, 〈(dist‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ sup((ran (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(dist‘(𝑟‘𝑥))(𝑔‘𝑥))) ∪ {0}), ℝ*, <
))〉} ∪ {〈(Hom ‘ndx), ℎ〉, 〈(comp‘ndx), (𝑎 ∈ (𝑣 × 𝑣), 𝑐 ∈ 𝑣 ↦ (𝑑 ∈ (𝑐ℎ(2nd ‘𝑎)), 𝑒 ∈ (ℎ‘𝑎) ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑑‘𝑥)(〈((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)〉(comp‘(𝑟‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥)))))〉}))) |
df-pws 16569 | ⊢ ↑s = (𝑟 ∈ V, 𝑖 ∈ V ↦ ((Scalar‘𝑟)Xs(𝑖 × {𝑟}))) |
cordt 16618 | class ordTop |
cxrs 16619 | class
ℝ*𝑠 |
df-ordt 16620 | ⊢ ordTop = (𝑟 ∈ V ↦
(topGen‘(fi‘({dom 𝑟} ∪ ran ((𝑥 ∈ dom 𝑟 ↦ {𝑦 ∈ dom 𝑟 ∣ ¬ 𝑦𝑟𝑥}) ∪ (𝑥 ∈ dom 𝑟 ↦ {𝑦 ∈ dom 𝑟 ∣ ¬ 𝑥𝑟𝑦})))))) |
df-xrs 16621 | ⊢ ℝ*𝑠 =
({〈(Base‘ndx), ℝ*〉,
〈(+g‘ndx), +𝑒 〉,
〈(.r‘ndx), ·e 〉} ∪
{〈(TopSet‘ndx), (ordTop‘ ≤ )〉, 〈(le‘ndx),
≤ 〉, 〈(dist‘ndx), (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ*
↦ if(𝑥 ≤ 𝑦, (𝑦 +𝑒
-𝑒𝑥),
(𝑥 +𝑒
-𝑒𝑦)))〉}) |
cqtop 16622 | class qTop |
cimas 16623 | class
“s |
cqus 16624 | class
/s |
cxps 16625 | class
×s |
df-qtop 16626 | ⊢ qTop = (𝑗 ∈ V, 𝑓 ∈ V ↦ {𝑠 ∈ 𝒫 (𝑓 “ ∪ 𝑗) ∣ ((◡𝑓 “ 𝑠) ∩ ∪ 𝑗) ∈ 𝑗}) |
df-imas 16627 | ⊢ “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 16628 | ⊢ /s = (𝑟 ∈ V, 𝑒 ∈ V ↦ ((𝑥 ∈ (Base‘𝑟) ↦ [𝑥]𝑒) “s 𝑟)) |
df-xps 16629 | ⊢ ×s = (𝑟 ∈ V, 𝑠 ∈ V ↦ (◡(𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Base‘𝑠) ↦ ◡({𝑥} +𝑐 {𝑦})) “s
((Scalar‘𝑟)Xs◡({𝑟} +𝑐 {𝑠})))) |
cmre 16701 | class Moore |
cmrc 16702 | class mrCls |
cmri 16703 | class mrInd |
cacs 16704 | class ACS |
df-mre 16705 | ⊢ Moore = (𝑥 ∈ V ↦ {𝑐 ∈ 𝒫 𝒫 𝑥 ∣ (𝑥 ∈ 𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → ∩ 𝑠
∈ 𝑐))}) |
df-mrc 16706 | ⊢ mrCls = (𝑐 ∈ ∪ ran
Moore ↦ (𝑥 ∈
𝒫 ∪ 𝑐 ↦ ∩ {𝑠 ∈ 𝑐 ∣ 𝑥 ⊆ 𝑠})) |
df-mri 16707 | ⊢ mrInd = (𝑐 ∈ ∪ ran
Moore ↦ {𝑠 ∈
𝒫 ∪ 𝑐 ∣ ∀𝑥 ∈ 𝑠 ¬ 𝑥 ∈ ((mrCls‘𝑐)‘(𝑠 ∖ {𝑥}))}) |
df-acs 16708 | ⊢ ACS = (𝑥 ∈ V ↦ {𝑐 ∈ (Moore‘𝑥) ∣ ∃𝑓(𝑓:𝒫 𝑥⟶𝒫 𝑥 ∧ ∀𝑠 ∈ 𝒫 𝑥(𝑠 ∈ 𝑐 ↔ ∪ (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠))}) |
ccat 16783 | class Cat |
ccid 16784 | class Id |
chomf 16785 | class
Homf |
ccomf 16786 | class
compf |
df-cat 16787 | ⊢ Cat = {𝑐 ∣ [(Base‘𝑐) / 𝑏][(Hom ‘𝑐) / ℎ][(comp‘𝑐) / 𝑜]∀𝑥 ∈ 𝑏 (∃𝑔 ∈ (𝑥ℎ𝑥)∀𝑦 ∈ 𝑏 (∀𝑓 ∈ (𝑦ℎ𝑥)(𝑔(〈𝑦, 𝑥〉𝑜𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥ℎ𝑦)(𝑓(〈𝑥, 𝑥〉𝑜𝑦)𝑔) = 𝑓) ∧ ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ∀𝑓 ∈ (𝑥ℎ𝑦)∀𝑔 ∈ (𝑦ℎ𝑧)((𝑔(〈𝑥, 𝑦〉𝑜𝑧)𝑓) ∈ (𝑥ℎ𝑧) ∧ ∀𝑤 ∈ 𝑏 ∀𝑘 ∈ (𝑧ℎ𝑤)((𝑘(〈𝑦, 𝑧〉𝑜𝑤)𝑔)(〈𝑥, 𝑦〉𝑜𝑤)𝑓) = (𝑘(〈𝑥, 𝑧〉𝑜𝑤)(𝑔(〈𝑥, 𝑦〉𝑜𝑧)𝑓))))} |
df-cid 16788 | ⊢ Id = (𝑐 ∈ Cat ↦
⦋(Base‘𝑐) / 𝑏⦌⦋(Hom
‘𝑐) / ℎ⦌⦋(comp‘𝑐) / 𝑜⦌(𝑥 ∈ 𝑏 ↦ (℩𝑔 ∈ (𝑥ℎ𝑥)∀𝑦 ∈ 𝑏 (∀𝑓 ∈ (𝑦ℎ𝑥)(𝑔(〈𝑦, 𝑥〉𝑜𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥ℎ𝑦)(𝑓(〈𝑥, 𝑥〉𝑜𝑦)𝑔) = 𝑓)))) |
df-homf 16789 | ⊢ Homf = (𝑐 ∈ V ↦ (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ (𝑥(Hom ‘𝑐)𝑦))) |
df-comf 16790 | ⊢ compf = (𝑐 ∈ V ↦ (𝑥 ∈ ((Base‘𝑐) × (Base‘𝑐)), 𝑦 ∈ (Base‘𝑐) ↦ (𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝑐)𝑦), 𝑓 ∈ ((Hom ‘𝑐)‘𝑥) ↦ (𝑔(𝑥(comp‘𝑐)𝑦)𝑓)))) |
coppc 16829 | class oppCat |
df-oppc 16830 | ⊢ oppCat = (𝑓 ∈ V ↦ ((𝑓 sSet 〈(Hom ‘ndx), tpos (Hom
‘𝑓)〉) sSet
〈(comp‘ndx), (𝑢
∈ ((Base‘𝑓)
× (Base‘𝑓)),
𝑧 ∈ (Base‘𝑓) ↦ tpos (〈𝑧, (2nd ‘𝑢)〉(comp‘𝑓)(1st ‘𝑢)))〉)) |
cmon 16846 | class Mono |
cepi 16847 | class Epi |
df-mon 16848 | ⊢ Mono = (𝑐 ∈ Cat ↦
⦋(Base‘𝑐) / 𝑏⦌⦋(Hom
‘𝑐) / ℎ⦌(𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ {𝑓 ∈ (𝑥ℎ𝑦) ∣ ∀𝑧 ∈ 𝑏 Fun ◡(𝑔 ∈ (𝑧ℎ𝑥) ↦ (𝑓(〈𝑧, 𝑥〉(comp‘𝑐)𝑦)𝑔))})) |
df-epi 16849 | ⊢ Epi = (𝑐 ∈ Cat ↦ tpos
(Mono‘(oppCat‘𝑐))) |
csect 16862 | class Sect |
cinv 16863 | class Inv |
ciso 16864 | class Iso |
df-sect 16865 | ⊢ Sect = (𝑐 ∈ Cat ↦ (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ {〈𝑓, 𝑔〉 ∣ [(Hom ‘𝑐) / ℎ]((𝑓 ∈ (𝑥ℎ𝑦) ∧ 𝑔 ∈ (𝑦ℎ𝑥)) ∧ (𝑔(〈𝑥, 𝑦〉(comp‘𝑐)𝑥)𝑓) = ((Id‘𝑐)‘𝑥))})) |
df-inv 16866 | ⊢ Inv = (𝑐 ∈ Cat ↦ (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ ((𝑥(Sect‘𝑐)𝑦) ∩ ◡(𝑦(Sect‘𝑐)𝑥)))) |
df-iso 16867 | ⊢ Iso = (𝑐 ∈ Cat ↦ ((𝑥 ∈ V ↦ dom 𝑥) ∘ (Inv‘𝑐))) |
ccic 16913 | class
≃𝑐 |
df-cic 16914 | ⊢ ≃𝑐 = (𝑐 ∈ Cat ↦
((Iso‘𝑐) supp
∅)) |
cssc 16925 | class
⊆cat |
cresc 16926 | class
↾cat |
csubc 16927 | class Subcat |
df-ssc 16928 | ⊢ ⊆cat = {〈ℎ, 𝑗〉 ∣ ∃𝑡(𝑗 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝑗‘𝑥))} |
df-resc 16929 | ⊢ ↾cat = (𝑐 ∈ V, ℎ ∈ V ↦ ((𝑐 ↾s dom dom ℎ) sSet 〈(Hom ‘ndx),
ℎ〉)) |
df-subc 16930 | ⊢ Subcat = (𝑐 ∈ Cat ↦ {ℎ ∣ (ℎ ⊆cat (Homf
‘𝑐) ∧ [dom
dom ℎ / 𝑠]∀𝑥 ∈ 𝑠 (((Id‘𝑐)‘𝑥) ∈ (𝑥ℎ𝑥) ∧ ∀𝑦 ∈ 𝑠 ∀𝑧 ∈ 𝑠 ∀𝑓 ∈ (𝑥ℎ𝑦)∀𝑔 ∈ (𝑦ℎ𝑧)(𝑔(〈𝑥, 𝑦〉(comp‘𝑐)𝑧)𝑓) ∈ (𝑥ℎ𝑧)))}) |
cfunc 16972 | class Func |
cidfu 16973 | class
idfunc |
ccofu 16974 | class
∘func |
cresf 16975 | class
↾f |
df-func 16976 | ⊢ 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 16977 | ⊢ idfunc = (𝑡 ∈ Cat ↦
⦋(Base‘𝑡) / 𝑏⦌〈( I ↾ 𝑏), (𝑧 ∈ (𝑏 × 𝑏) ↦ ( I ↾ ((Hom ‘𝑡)‘𝑧)))〉) |
df-cofu 16978 | ⊢ ∘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 16979 | ⊢ ↾f = (𝑓 ∈ V, ℎ ∈ V ↦ 〈((1st
‘𝑓) ↾ dom dom
ℎ), (𝑥 ∈ dom ℎ ↦ (((2nd ‘𝑓)‘𝑥) ↾ (ℎ‘𝑥)))〉) |
cful 17020 | class Full |
cfth 17021 | class Faith |
df-full 17022 | ⊢ 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 17023 | ⊢ Faith = (𝑐 ∈ Cat, 𝑑 ∈ Cat ↦ {〈𝑓, 𝑔〉 ∣ (𝑓(𝑐 Func 𝑑)𝑔 ∧ ∀𝑥 ∈ (Base‘𝑐)∀𝑦 ∈ (Base‘𝑐)Fun ◡(𝑥𝑔𝑦))}) |
cnat 17059 | class Nat |
cfuc 17060 | class FuncCat |
df-nat 17061 | ⊢ 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 17062 | ⊢ 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 17096 | class InitO |
ctermo 17097 | class TermO |
czeroo 17098 | class ZeroO |
df-inito 17099 | ⊢ InitO = (𝑐 ∈ Cat ↦ {𝑎 ∈ (Base‘𝑐) ∣ ∀𝑏 ∈ (Base‘𝑐)∃!ℎ ℎ ∈ (𝑎(Hom ‘𝑐)𝑏)}) |
df-termo 17100 | ⊢ TermO = (𝑐 ∈ Cat ↦ {𝑎 ∈ (Base‘𝑐) ∣ ∀𝑏 ∈ (Base‘𝑐)∃!ℎ ℎ ∈ (𝑏(Hom ‘𝑐)𝑎)}) |
df-zeroo 17101 | ⊢ ZeroO = (𝑐 ∈ Cat ↦ ((InitO‘𝑐) ∩ (TermO‘𝑐))) |
cdoma 17128 | class
doma |
ccoda 17129 | class
coda |
carw 17130 | class Arrow |
choma 17131 | class
Homa |
df-doma 17132 | ⊢ doma = (1st
∘ 1st ) |
df-coda 17133 | ⊢ coda = (2nd
∘ 1st ) |
df-homa 17134 | ⊢ Homa = (𝑐 ∈ Cat ↦ (𝑥 ∈ ((Base‘𝑐) × (Base‘𝑐)) ↦ ({𝑥} × ((Hom ‘𝑐)‘𝑥)))) |
df-arw 17135 | ⊢ Arrow = (𝑐 ∈ Cat ↦ ∪ ran (Homa‘𝑐)) |
cida 17161 | class
Ida |
ccoa 17162 | class
compa |
df-ida 17163 | ⊢ Ida = (𝑐 ∈ Cat ↦ (𝑥 ∈ (Base‘𝑐) ↦ 〈𝑥, 𝑥, ((Id‘𝑐)‘𝑥)〉)) |
df-coa 17164 | ⊢ compa = (𝑐 ∈ Cat ↦ (𝑔 ∈ (Arrow‘𝑐), 𝑓 ∈ {ℎ ∈ (Arrow‘𝑐) ∣ (coda‘ℎ) =
(doma‘𝑔)} ↦
〈(doma‘𝑓), (coda‘𝑔), ((2nd ‘𝑔)(〈(doma‘𝑓),
(doma‘𝑔)〉(comp‘𝑐)(coda‘𝑔))(2nd ‘𝑓))〉)) |
csetc 17183 | class SetCat |
df-setc 17184 | ⊢ SetCat = (𝑢 ∈ V ↦ {〈(Base‘ndx),
𝑢〉, 〈(Hom
‘ndx), (𝑥 ∈
𝑢, 𝑦 ∈ 𝑢 ↦ (𝑦 ↑𝑚 𝑥))〉,
〈(comp‘ndx), (𝑣
∈ (𝑢 × 𝑢), 𝑧 ∈ 𝑢 ↦ (𝑔 ∈ (𝑧 ↑𝑚 (2nd
‘𝑣)), 𝑓 ∈ ((2nd
‘𝑣)
↑𝑚 (1st ‘𝑣)) ↦ (𝑔 ∘ 𝑓)))〉}) |
ccatc 17202 | class CatCat |
df-catc 17203 | ⊢ CatCat = (𝑢 ∈ V ↦ ⦋(𝑢 ∩ Cat) / 𝑏⦌{〈(Base‘ndx), 𝑏〉, 〈(Hom ‘ndx),
(𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ (𝑥 Func 𝑦))〉, 〈(comp‘ndx), (𝑣 ∈ (𝑏 × 𝑏), 𝑧 ∈ 𝑏 ↦ (𝑔 ∈ ((2nd ‘𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔 ∘func 𝑓)))〉}) |
cestrc 17220 | class ExtStrCat |
df-estrc 17221 | ⊢ ExtStrCat = (𝑢 ∈ V ↦ {〈(Base‘ndx),
𝑢〉, 〈(Hom
‘ndx), (𝑥 ∈
𝑢, 𝑦 ∈ 𝑢 ↦ ((Base‘𝑦) ↑𝑚
(Base‘𝑥)))〉,
〈(comp‘ndx), (𝑣
∈ (𝑢 × 𝑢), 𝑧 ∈ 𝑢 ↦ (𝑔 ∈ ((Base‘𝑧) ↑𝑚
(Base‘(2nd ‘𝑣))), 𝑓 ∈ ((Base‘(2nd
‘𝑣))
↑𝑚 (Base‘(1st ‘𝑣))) ↦ (𝑔 ∘ 𝑓)))〉}) |
cxpc 17266 | class
×c |
c1stf 17267 | class
1stF |
c2ndf 17268 | class
2ndF |
cprf 17269 | class
〈,〉F |
df-xpc 17270 | ⊢ ×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 17271 | ⊢ 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 17272 | ⊢ 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 17273 | ⊢ 〈,〉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 17307 | class
evalF |
ccurf 17308 | class
curryF |
cuncf 17309 | class
uncurryF |
cdiag 17310 | class
Δfunc |
df-evlf 17311 | ⊢ 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 17312 | ⊢ 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 17313 | ⊢ 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 17314 | ⊢ Δfunc = (𝑐 ∈ Cat, 𝑑 ∈ Cat ↦ (〈𝑐, 𝑑〉 curryF (𝑐
1stF 𝑑))) |
chof 17346 | class
HomF |
cyon 17347 | class Yon |
df-hof 17348 | ⊢ HomF = (𝑐 ∈ Cat ↦
〈(Homf ‘𝑐), ⦋(Base‘𝑐) / 𝑏⦌(𝑥 ∈ (𝑏 × 𝑏), 𝑦 ∈ (𝑏 × 𝑏) ↦ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝑐)(1st ‘𝑥)), 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝑐)(2nd ‘𝑦)) ↦ (ℎ ∈ ((Hom ‘𝑐)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝑐)(2nd ‘𝑦))ℎ)(〈(1st ‘𝑦), (1st ‘𝑥)〉(comp‘𝑐)(2nd ‘𝑦))𝑓))))〉) |
df-yon 17349 | ⊢ Yon = (𝑐 ∈ Cat ↦ (〈𝑐, (oppCat‘𝑐)〉 curryF
(HomF‘(oppCat‘𝑐)))) |
cproset 17384 | class Proset |
cdrs 17385 | class Dirset |
df-proset 17386 | ⊢ Proset = {𝑓 ∣ [(Base‘𝑓) / 𝑏][(le‘𝑓) / 𝑟]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) → 𝑥𝑟𝑧))} |
df-drs 17387 | ⊢ Dirset = {𝑓 ∈ Proset ∣
[(Base‘𝑓) /
𝑏][(le‘𝑓) / 𝑟](𝑏 ≠ ∅ ∧ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∃𝑧 ∈ 𝑏 (𝑥𝑟𝑧 ∧ 𝑦𝑟𝑧))} |
cpo 17398 | class Poset |
cplt 17399 | class lt |
club 17400 | class lub |
cglb 17401 | class glb |
cjn 17402 | class join |
cmee 17403 | class meet |
df-poset 17404 | ⊢ Poset = {𝑓 ∣ ∃𝑏∃𝑟(𝑏 = (Base‘𝑓) ∧ 𝑟 = (le‘𝑓) ∧ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑥) → 𝑥 = 𝑦) ∧ ((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) → 𝑥𝑟𝑧)))} |
df-plt 17416 | ⊢ lt = (𝑝 ∈ V ↦ ((le‘𝑝) ∖ I )) |
df-lub 17432 | ⊢ lub = (𝑝 ∈ V ↦ ((𝑠 ∈ 𝒫 (Base‘𝑝) ↦ (℩𝑥 ∈ (Base‘𝑝)(∀𝑦 ∈ 𝑠 𝑦(le‘𝑝)𝑥 ∧ ∀𝑧 ∈ (Base‘𝑝)(∀𝑦 ∈ 𝑠 𝑦(le‘𝑝)𝑧 → 𝑥(le‘𝑝)𝑧)))) ↾ {𝑠 ∣ ∃!𝑥 ∈ (Base‘𝑝)(∀𝑦 ∈ 𝑠 𝑦(le‘𝑝)𝑥 ∧ ∀𝑧 ∈ (Base‘𝑝)(∀𝑦 ∈ 𝑠 𝑦(le‘𝑝)𝑧 → 𝑥(le‘𝑝)𝑧))})) |
df-glb 17433 | ⊢ glb = (𝑝 ∈ V ↦ ((𝑠 ∈ 𝒫 (Base‘𝑝) ↦ (℩𝑥 ∈ (Base‘𝑝)(∀𝑦 ∈ 𝑠 𝑥(le‘𝑝)𝑦 ∧ ∀𝑧 ∈ (Base‘𝑝)(∀𝑦 ∈ 𝑠 𝑧(le‘𝑝)𝑦 → 𝑧(le‘𝑝)𝑥)))) ↾ {𝑠 ∣ ∃!𝑥 ∈ (Base‘𝑝)(∀𝑦 ∈ 𝑠 𝑥(le‘𝑝)𝑦 ∧ ∀𝑧 ∈ (Base‘𝑝)(∀𝑦 ∈ 𝑠 𝑧(le‘𝑝)𝑦 → 𝑧(le‘𝑝)𝑥))})) |
df-join 17434 | ⊢ join = (𝑝 ∈ V ↦ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ {𝑥, 𝑦} (lub‘𝑝)𝑧}) |
df-meet 17435 | ⊢ meet = (𝑝 ∈ V ↦ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ {𝑥, 𝑦} (glb‘𝑝)𝑧}) |
ctos 17491 | class Toset |
df-toset 17492 | ⊢ Toset = {𝑓 ∈ Poset ∣
[(Base‘𝑓) /
𝑏][(le‘𝑓) / 𝑟]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥𝑟𝑦 ∨ 𝑦𝑟𝑥)} |
cp0 17495 | class 0. |
cp1 17496 | class 1. |
df-p0 17497 | ⊢ 0. = (𝑝 ∈ V ↦ ((glb‘𝑝)‘(Base‘𝑝))) |
df-p1 17498 | ⊢ 1. = (𝑝 ∈ V ↦ ((lub‘𝑝)‘(Base‘𝑝))) |
clat 17503 | class Lat |
df-lat 17504 | ⊢ Lat = {𝑝 ∈ Poset ∣ (dom (join‘𝑝) = ((Base‘𝑝) × (Base‘𝑝)) ∧ dom (meet‘𝑝) = ((Base‘𝑝) × (Base‘𝑝)))} |
ccla 17565 | class CLat |
df-clat 17566 | ⊢ CLat = {𝑝 ∈ Poset ∣ (dom (lub‘𝑝) = 𝒫 (Base‘𝑝) ∧ dom (glb‘𝑝) = 𝒫 (Base‘𝑝))} |
codu 17586 | class ODual |
df-odu 17587 | ⊢ ODual = (𝑤 ∈ V ↦ (𝑤 sSet 〈(le‘ndx), ◡(le‘𝑤)〉)) |
cipo 17609 | class toInc |
df-ipo 17610 | ⊢ toInc = (𝑓 ∈ V ↦ ⦋{〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝑓 ∧ 𝑥 ⊆ 𝑦)} / 𝑜⦌({〈(Base‘ndx),
𝑓〉,
〈(TopSet‘ndx), (ordTop‘𝑜)〉} ∪ {〈(le‘ndx), 𝑜〉, 〈(oc‘ndx),
(𝑥 ∈ 𝑓 ↦ ∪ {𝑦
∈ 𝑓 ∣ (𝑦 ∩ 𝑥) = ∅})〉})) |
cdlat 17649 | class DLat |
df-dlat 17650 | ⊢ DLat = {𝑘 ∈ Lat ∣ [(Base‘𝑘) / 𝑏][(join‘𝑘) / 𝑗][(meet‘𝑘) / 𝑚]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 (𝑥𝑚(𝑦𝑗𝑧)) = ((𝑥𝑚𝑦)𝑗(𝑥𝑚𝑧))} |
cps 17656 | class PosetRel |
ctsr 17657 | class TosetRel |
df-ps 17658 | ⊢ PosetRel = {𝑟 ∣ (Rel 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟 ∧ (𝑟 ∩ ◡𝑟) = ( I ↾ ∪
∪ 𝑟))} |
df-tsr 17659 | ⊢ TosetRel = {𝑟 ∈ PosetRel ∣ (dom 𝑟 × dom 𝑟) ⊆ (𝑟 ∪ ◡𝑟)} |
cdir 17686 | class DirRel |
ctail 17687 | class tail |
df-dir 17688 | ⊢ DirRel = {𝑟 ∣ ((Rel 𝑟 ∧ ( I ↾ ∪ ∪ 𝑟) ⊆ 𝑟) ∧ ((𝑟 ∘ 𝑟) ⊆ 𝑟 ∧ (∪ ∪ 𝑟
× ∪ ∪ 𝑟) ⊆ (◡𝑟 ∘ 𝑟)))} |
df-tail 17689 | ⊢ tail = (𝑟 ∈ DirRel ↦ (𝑥 ∈ ∪ ∪ 𝑟
↦ (𝑟 “ {𝑥}))) |
cplusf 17697 | class
+𝑓 |
cmgm 17698 | class Mgm |
df-plusf 17699 | ⊢ +𝑓 = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(+g‘𝑔)𝑦))) |
df-mgm 17700 | ⊢ Mgm = {𝑔 ∣ [(Base‘𝑔) / 𝑏][(+g‘𝑔) / 𝑜]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥𝑜𝑦) ∈ 𝑏} |
csgrp 17741 | class SGrp |
df-sgrp 17742 | ⊢ SGrp = {𝑔 ∈ Mgm ∣ [(Base‘𝑔) / 𝑏][(+g‘𝑔) / 𝑜]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧))} |
cmnd 17752 | class Mnd |
df-mnd 17753 | ⊢ Mnd = {𝑔 ∈ SGrp ∣
[(Base‘𝑔) /
𝑏][(+g‘𝑔) / 𝑝]∃𝑒 ∈ 𝑏 ∀𝑥 ∈ 𝑏 ((𝑒𝑝𝑥) = 𝑥 ∧ (𝑥𝑝𝑒) = 𝑥)} |
cmhm 17791 | class MndHom |
csubmnd 17792 | class SubMnd |
df-mhm 17793 | ⊢ 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 17794 | ⊢ SubMnd = (𝑠 ∈ Mnd ↦ {𝑡 ∈ 𝒫 (Base‘𝑠) ∣
((0g‘𝑠)
∈ 𝑡 ∧
∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (𝑥(+g‘𝑠)𝑦) ∈ 𝑡)}) |
cfrmd 17843 | class freeMnd |
cvrmd 17844 | class
varFMnd |
df-frmd 17845 | ⊢ freeMnd = (𝑖 ∈ V ↦ {〈(Base‘ndx),
Word 𝑖〉,
〈(+g‘ndx), ( ++ ↾ (Word 𝑖 × Word 𝑖))〉}) |
df-vrmd 17846 | ⊢ varFMnd = (𝑖 ∈ V ↦ (𝑗 ∈ 𝑖 ↦ 〈“𝑗”〉)) |
cgrp 17881 | class Grp |
cminusg 17882 | class invg |
csg 17883 | class -g |
df-grp 17884 | ⊢ Grp = {𝑔 ∈ Mnd ∣ ∀𝑎 ∈ (Base‘𝑔)∃𝑚 ∈ (Base‘𝑔)(𝑚(+g‘𝑔)𝑎) = (0g‘𝑔)} |
df-minusg 17885 | ⊢ invg = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔) ↦ (℩𝑤 ∈ (Base‘𝑔)(𝑤(+g‘𝑔)𝑥) = (0g‘𝑔)))) |
df-sbg 17886 | ⊢ -g = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(+g‘𝑔)((invg‘𝑔)‘𝑦)))) |
cmg 18001 | class .g |
df-mulg 18002 | ⊢ .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 18047 | class SubGrp |
cnsg 18048 | class NrmSGrp |
cqg 18049 | class
~QG |
df-subg 18050 | ⊢ SubGrp = (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp}) |
df-nsg 18051 | ⊢ NrmSGrp = (𝑤 ∈ Grp ↦ {𝑠 ∈ (SubGrp‘𝑤) ∣ [(Base‘𝑤) / 𝑏][(+g‘𝑤) / 𝑝]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ((𝑥𝑝𝑦) ∈ 𝑠 ↔ (𝑦𝑝𝑥) ∈ 𝑠)}) |
df-eqg 18052 | ⊢ ~QG = (𝑟 ∈ V, 𝑖 ∈ V ↦ {〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ (Base‘𝑟) ∧ (((invg‘𝑟)‘𝑥)(+g‘𝑟)𝑦) ∈ 𝑖)}) |
cghm 18116 | class GrpHom |
df-ghm 18117 | ⊢ 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 18158 | class GrpIso |
cgic 18159 | class
≃𝑔 |
df-gim 18160 | ⊢ GrpIso = (𝑠 ∈ Grp, 𝑡 ∈ Grp ↦ {𝑔 ∈ (𝑠 GrpHom 𝑡) ∣ 𝑔:(Base‘𝑠)–1-1-onto→(Base‘𝑡)}) |
df-gic 18161 | ⊢ ≃𝑔 = (◡ GrpIso “ (V ∖
1o)) |
cga 18180 | class GrpAct |
df-ga 18181 | ⊢ 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 18206 | class Cntz |
ccntr 18207 | class Cntr |
df-cntz 18208 | ⊢ Cntz = (𝑚 ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘𝑚) ↦ {𝑥 ∈ (Base‘𝑚) ∣ ∀𝑦 ∈ 𝑠 (𝑥(+g‘𝑚)𝑦) = (𝑦(+g‘𝑚)𝑥)})) |
df-cntr 18209 | ⊢ Cntr = (𝑚 ∈ V ↦ ((Cntz‘𝑚)‘(Base‘𝑚))) |
coppg 18234 | class
oppg |
df-oppg 18235 | ⊢ oppg = (𝑤 ∈ V ↦ (𝑤 sSet 〈(+g‘ndx), tpos
(+g‘𝑤)〉)) |
csymg 18256 | class SymGrp |
df-symg 18257 | ⊢ SymGrp = (𝑥 ∈ V ↦ ⦋{ℎ ∣ ℎ:𝑥–1-1-onto→𝑥} / 𝑏⦌{〈(Base‘ndx), 𝑏〉,
〈(+g‘ndx), (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (𝑓 ∘ 𝑔))〉, 〈(TopSet‘ndx),
(∏t‘(𝑥 × {𝒫 𝑥}))〉}) |
cpmtr 18320 | class pmTrsp |
df-pmtr 18321 | ⊢ pmTrsp = (𝑑 ∈ V ↦ (𝑝 ∈ {𝑦 ∈ 𝒫 𝑑 ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ 𝑑 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧)))) |
cpsgn 18368 | class pmSgn |
cevpm 18369 | class pmEven |
df-psgn 18370 | ⊢ pmSgn = (𝑑 ∈ V ↦ (𝑥 ∈ {𝑝 ∈ (Base‘(SymGrp‘𝑑)) ∣ dom (𝑝 ∖ I ) ∈ Fin} ↦
(℩𝑠∃𝑤 ∈ Word ran
(pmTrsp‘𝑑)(𝑥 = ((SymGrp‘𝑑) Σg
𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤)))))) |
df-evpm 18371 | ⊢ pmEven = (𝑑 ∈ V ↦ (◡(pmSgn‘𝑑) “ {1})) |
cod 18404 | class od |
cgex 18405 | class gEx |
cpgp 18406 | class pGrp |
cslw 18407 | class pSyl |
df-od 18408 | ⊢ od = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔) ↦ ⦋{𝑛 ∈ ℕ ∣ (𝑛(.g‘𝑔)𝑥) = (0g‘𝑔)} / 𝑖⦌if(𝑖 = ∅, 0, inf(𝑖, ℝ, < )))) |
df-gex 18409 | ⊢ gEx = (𝑔 ∈ V ↦ ⦋{𝑛 ∈ ℕ ∣
∀𝑥 ∈
(Base‘𝑔)(𝑛(.g‘𝑔)𝑥) = (0g‘𝑔)} / 𝑖⦌if(𝑖 = ∅, 0, inf(𝑖, ℝ, < ))) |
df-pgp 18410 | ⊢ pGrp = {〈𝑝, 𝑔〉 ∣ ((𝑝 ∈ ℙ ∧ 𝑔 ∈ Grp) ∧ ∀𝑥 ∈ (Base‘𝑔)∃𝑛 ∈ ℕ0 ((od‘𝑔)‘𝑥) = (𝑝↑𝑛))} |
df-slw 18411 | ⊢ pSyl = (𝑝 ∈ ℙ, 𝑔 ∈ Grp ↦ {ℎ ∈ (SubGrp‘𝑔) ∣ ∀𝑘 ∈ (SubGrp‘𝑔)((ℎ ⊆ 𝑘 ∧ 𝑝 pGrp (𝑔 ↾s 𝑘)) ↔ ℎ = 𝑘)}) |
clsm 18510 | class LSSum |
cpj1 18511 | class
proj1 |
df-lsm 18512 | ⊢ LSSum = (𝑤 ∈ V ↦ (𝑡 ∈ 𝒫 (Base‘𝑤), 𝑢 ∈ 𝒫 (Base‘𝑤) ↦ ran (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥(+g‘𝑤)𝑦)))) |
df-pj1 18513 | ⊢ proj1 = (𝑤 ∈ V ↦ (𝑡 ∈ 𝒫 (Base‘𝑤), 𝑢 ∈ 𝒫 (Base‘𝑤) ↦ (𝑧 ∈ (𝑡(LSSum‘𝑤)𝑢) ↦ (℩𝑥 ∈ 𝑡 ∃𝑦 ∈ 𝑢 𝑧 = (𝑥(+g‘𝑤)𝑦))))) |
cefg 18580 | class
~FG |
cfrgp 18581 | class freeGrp |
cvrgp 18582 | class
varFGrp |
df-efg 18583 | ⊢ ~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 18584 | ⊢ freeGrp = (𝑖 ∈ V ↦ ((freeMnd‘(𝑖 × 2o))
/s ( ~FG ‘𝑖))) |
df-vrgp 18585 | ⊢ varFGrp = (𝑖 ∈ V ↦ (𝑗 ∈ 𝑖 ↦ [〈“〈𝑗, ∅〉”〉](
~FG ‘𝑖))) |
ccmn 18656 | class CMnd |
cabl 18657 | class Abel |
df-cmn 18658 | ⊢ CMnd = {𝑔 ∈ Mnd ∣ ∀𝑎 ∈ (Base‘𝑔)∀𝑏 ∈ (Base‘𝑔)(𝑎(+g‘𝑔)𝑏) = (𝑏(+g‘𝑔)𝑎)} |
df-abl 18659 | ⊢ Abel = (Grp ∩ CMnd) |
ccyg 18742 | class CycGrp |
df-cyg 18743 | ⊢ CycGrp = {𝑔 ∈ Grp ∣ ∃𝑥 ∈ (Base‘𝑔)ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝑔)𝑥)) = (Base‘𝑔)} |
cdprd 18855 | class DProd |
cdpj 18856 | class dProj |
df-dprd 18857 | ⊢ 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 18858 | ⊢ dProj = (𝑔 ∈ Grp, 𝑠 ∈ (dom DProd “ {𝑔}) ↦ (𝑖 ∈ dom 𝑠 ↦ ((𝑠‘𝑖)(proj1‘𝑔)(𝑔 DProd (𝑠 ↾ (dom 𝑠 ∖ {𝑖})))))) |
cmgp 18952 | class mulGrp |
df-mgp 18953 | ⊢ mulGrp = (𝑤 ∈ V ↦ (𝑤 sSet 〈(+g‘ndx),
(.r‘𝑤)〉)) |
cur 18964 | class 1r |
df-ur 18965 | ⊢ 1r = (0g
∘ mulGrp) |
csrg 18968 | class SRing |
df-srg 18969 | ⊢ SRing = {𝑓 ∈ CMnd ∣ ((mulGrp‘𝑓) ∈ Mnd ∧
[(Base‘𝑓) /
𝑟][(+g‘𝑓) / 𝑝][(.r‘𝑓) / 𝑡][(0g‘𝑓) / 𝑛]∀𝑥 ∈ 𝑟 (∀𝑦 ∈ 𝑟 ∀𝑧 ∈ 𝑟 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))) ∧ ((𝑛𝑡𝑥) = 𝑛 ∧ (𝑥𝑡𝑛) = 𝑛)))} |
crg 19010 | class Ring |
ccrg 19011 | class CRing |
df-ring 19012 | ⊢ Ring = {𝑓 ∈ Grp ∣ ((mulGrp‘𝑓) ∈ Mnd ∧
[(Base‘𝑓) /
𝑟][(+g‘𝑓) / 𝑝][(.r‘𝑓) / 𝑡]∀𝑥 ∈ 𝑟 ∀𝑦 ∈ 𝑟 ∀𝑧 ∈ 𝑟 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))))} |
df-cring 19013 | ⊢ CRing = {𝑓 ∈ Ring ∣ (mulGrp‘𝑓) ∈ CMnd} |
coppr 19085 | class
oppr |
df-oppr 19086 | ⊢ oppr = (𝑓 ∈ V ↦ (𝑓 sSet 〈(.r‘ndx), tpos
(.r‘𝑓)〉)) |
cdsr 19101 | class
∥r |
cui 19102 | class Unit |
cir 19103 | class Irred |
df-dvdsr 19104 | ⊢ ∥r = (𝑤 ∈ V ↦ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ (Base‘𝑤) ∧ ∃𝑧 ∈ (Base‘𝑤)(𝑧(.r‘𝑤)𝑥) = 𝑦)}) |
df-unit 19105 | ⊢ Unit = (𝑤 ∈ V ↦ (◡((∥r‘𝑤) ∩
(∥r‘(oppr‘𝑤))) “
{(1r‘𝑤)})) |
df-irred 19106 | ⊢ Irred = (𝑤 ∈ V ↦
⦋((Base‘𝑤) ∖ (Unit‘𝑤)) / 𝑏⦌{𝑧 ∈ 𝑏 ∣ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥(.r‘𝑤)𝑦) ≠ 𝑧}) |
cinvr 19134 | class invr |
df-invr 19135 | ⊢ invr = (𝑟 ∈ V ↦
(invg‘((mulGrp‘𝑟) ↾s (Unit‘𝑟)))) |
cdvr 19145 | class /r |
df-dvr 19146 | ⊢ /r = (𝑟 ∈ V ↦ (𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Unit‘𝑟) ↦ (𝑥(.r‘𝑟)((invr‘𝑟)‘𝑦)))) |
crpm 19175 | class RPrime |
df-rprm 19176 | ⊢ RPrime = (𝑤 ∈ V ↦
⦋(Base‘𝑤) / 𝑏⦌{𝑝 ∈ (𝑏 ∖ ((Unit‘𝑤) ∪ {(0g‘𝑤)})) ∣ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏
[(∥r‘𝑤) / 𝑑](𝑝𝑑(𝑥(.r‘𝑤)𝑦) → (𝑝𝑑𝑥 ∨ 𝑝𝑑𝑦))}) |
crh 19177 | class RingHom |
crs 19178 | class RingIso |
cric 19179 | class
≃𝑟 |
df-rnghom 19180 | ⊢ 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-rngiso 19181 | ⊢ RingIso = (𝑟 ∈ V, 𝑠 ∈ V ↦ {𝑓 ∈ (𝑟 RingHom 𝑠) ∣ ◡𝑓 ∈ (𝑠 RingHom 𝑟)}) |
df-ric 19183 | ⊢ ≃𝑟 = (◡ RingIso “ (V ∖
1o)) |
cdr 19215 | class DivRing |
cfield 19216 | class Field |
df-drng 19217 | ⊢ DivRing = {𝑟 ∈ Ring ∣ (Unit‘𝑟) = ((Base‘𝑟) ∖
{(0g‘𝑟)})} |
df-field 19218 | ⊢ Field = (DivRing ∩
CRing) |
csubrg 19244 | class SubRing |
crgspn 19245 | class RingSpan |
df-subrg 19246 | ⊢ SubRing = (𝑤 ∈ Ring ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ ((𝑤 ↾s 𝑠) ∈ Ring ∧
(1r‘𝑤)
∈ 𝑠)}) |
df-rgspn 19247 | ⊢ RingSpan = (𝑤 ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘𝑤) ↦ ∩ {𝑡
∈ (SubRing‘𝑤)
∣ 𝑠 ⊆ 𝑡})) |
csdrg 19284 | class SubDRing |
df-sdrg 19285 | ⊢ SubDRing = (𝑤 ∈ DivRing ↦ {𝑠 ∈ (SubRing‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ DivRing}) |
cabv 19299 | class AbsVal |
df-abv 19300 | ⊢ AbsVal = (𝑟 ∈ Ring ↦ {𝑓 ∈ ((0[,)+∞)
↑𝑚 (Base‘𝑟)) ∣ ∀𝑥 ∈ (Base‘𝑟)(((𝑓‘𝑥) = 0 ↔ 𝑥 = (0g‘𝑟)) ∧ ∀𝑦 ∈ (Base‘𝑟)((𝑓‘(𝑥(.r‘𝑟)𝑦)) = ((𝑓‘𝑥) · (𝑓‘𝑦)) ∧ (𝑓‘(𝑥(+g‘𝑟)𝑦)) ≤ ((𝑓‘𝑥) + (𝑓‘𝑦))))}) |
cstf 19326 | class
*rf |
csr 19327 | class *-Ring |
df-staf 19328 | ⊢ *rf = (𝑓 ∈ V ↦ (𝑥 ∈ (Base‘𝑓) ↦ ((*𝑟‘𝑓)‘𝑥))) |
df-srng 19329 | ⊢ *-Ring = {𝑓 ∣
[(*rf‘𝑓) / 𝑖](𝑖 ∈ (𝑓 RingHom (oppr‘𝑓)) ∧ 𝑖 = ◡𝑖)} |
clmod 19346 | class LMod |
cscaf 19347 | class
·sf |
df-lmod 19348 | ⊢ LMod = {𝑔 ∈ Grp ∣ [(Base‘𝑔) / 𝑣][(+g‘𝑔) / 𝑎][(Scalar‘𝑔) / 𝑓][(
·𝑠 ‘𝑔) / 𝑠][(Base‘𝑓) / 𝑘][(+g‘𝑓) / 𝑝][(.r‘𝑓) / 𝑡](𝑓 ∈ Ring ∧ ∀𝑞 ∈ 𝑘 ∀𝑟 ∈ 𝑘 ∀𝑥 ∈ 𝑣 ∀𝑤 ∈ 𝑣 (((𝑟𝑠𝑤) ∈ 𝑣 ∧ (𝑟𝑠(𝑤𝑎𝑥)) = ((𝑟𝑠𝑤)𝑎(𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤)𝑎(𝑟𝑠𝑤))) ∧ (((𝑞𝑡𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ((1r‘𝑓)𝑠𝑤) = 𝑤)))} |
df-scaf 19349 | ⊢ ·sf = (𝑔 ∈ V ↦ (𝑥 ∈
(Base‘(Scalar‘𝑔)), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥( ·𝑠
‘𝑔)𝑦))) |
clss 19415 | class LSubSp |
df-lss 19416 | ⊢ LSubSp = (𝑤 ∈ V ↦ {𝑠 ∈ (𝒫 (Base‘𝑤) ∖ {∅}) ∣
∀𝑥 ∈
(Base‘(Scalar‘𝑤))∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑥( ·𝑠
‘𝑤)𝑎)(+g‘𝑤)𝑏) ∈ 𝑠}) |
clspn 19455 | class LSpan |
df-lsp 19456 | ⊢ LSpan = (𝑤 ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘𝑤) ↦ ∩ {𝑡
∈ (LSubSp‘𝑤)
∣ 𝑠 ⊆ 𝑡})) |
clmhm 19503 | class LMHom |
clmim 19504 | class LMIso |
clmic 19505 | class
≃𝑚 |
df-lmhm 19506 | ⊢ 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 19507 | ⊢ LMIso = (𝑠 ∈ LMod, 𝑡 ∈ LMod ↦ {𝑔 ∈ (𝑠 LMHom 𝑡) ∣ 𝑔:(Base‘𝑠)–1-1-onto→(Base‘𝑡)}) |
df-lmic 19508 | ⊢ ≃𝑚 = (◡ LMIso “ (V ∖
1o)) |
clbs 19558 | class LBasis |
df-lbs 19559 | ⊢ LBasis = (𝑤 ∈ V ↦ {𝑏 ∈ 𝒫 (Base‘𝑤) ∣
[(LSpan‘𝑤) /
𝑛][(Scalar‘𝑤) / 𝑠]((𝑛‘𝑏) = (Base‘𝑤) ∧ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ ((Base‘𝑠) ∖ {(0g‘𝑠)}) ¬ (𝑦( ·𝑠
‘𝑤)𝑥) ∈ (𝑛‘(𝑏 ∖ {𝑥})))}) |
clvec 19586 | class LVec |
df-lvec 19587 | ⊢ LVec = {𝑓 ∈ LMod ∣ (Scalar‘𝑓) ∈
DivRing} |
csra 19652 | class subringAlg |
crglmod 19653 | class ringLMod |
clidl 19654 | class LIdeal |
crsp 19655 | class RSpan |
df-sra 19656 | ⊢ subringAlg = (𝑤 ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘𝑤) ↦ (((𝑤 sSet 〈(Scalar‘ndx), (𝑤 ↾s 𝑠)〉) sSet 〈(
·𝑠 ‘ndx), (.r‘𝑤)〉) sSet
〈(·𝑖‘ndx),
(.r‘𝑤)〉))) |
df-rgmod 19657 | ⊢ ringLMod = (𝑤 ∈ V ↦ ((subringAlg ‘𝑤)‘(Base‘𝑤))) |
df-lidl 19658 | ⊢ LIdeal = (LSubSp ∘
ringLMod) |
df-rsp 19659 | ⊢ RSpan = (LSpan ∘
ringLMod) |
c2idl 19715 | class 2Ideal |
df-2idl 19716 | ⊢ 2Ideal = (𝑟 ∈ V ↦ ((LIdeal‘𝑟) ∩
(LIdeal‘(oppr‘𝑟)))) |
clpidl 19725 | class LPIdeal |
clpir 19726 | class LPIR |
df-lpidl 19727 | ⊢ LPIdeal = (𝑤 ∈ Ring ↦ ∪ 𝑔 ∈ (Base‘𝑤){((RSpan‘𝑤)‘{𝑔})}) |
df-lpir 19728 | ⊢ LPIR = {𝑤 ∈ Ring ∣ (LIdeal‘𝑤) = (LPIdeal‘𝑤)} |
cnzr 19741 | class NzRing |
df-nzr 19742 | ⊢ NzRing = {𝑟 ∈ Ring ∣
(1r‘𝑟)
≠ (0g‘𝑟)} |
crlreg 19763 | class RLReg |
cdomn 19764 | class Domn |
cidom 19765 | class IDomn |
cpid 19766 | class PID |
df-rlreg 19767 | ⊢ RLReg = (𝑟 ∈ V ↦ {𝑥 ∈ (Base‘𝑟) ∣ ∀𝑦 ∈ (Base‘𝑟)((𝑥(.r‘𝑟)𝑦) = (0g‘𝑟) → 𝑦 = (0g‘𝑟))}) |
df-domn 19768 | ⊢ Domn = {𝑟 ∈ NzRing ∣
[(Base‘𝑟) /
𝑏][(0g‘𝑟) / 𝑧]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ((𝑥(.r‘𝑟)𝑦) = 𝑧 → (𝑥 = 𝑧 ∨ 𝑦 = 𝑧))} |
df-idom 19769 | ⊢ IDomn = (CRing ∩ Domn) |
df-pid 19770 | ⊢ PID = (IDomn ∩ LPIR) |
casa 19793 | class AssAlg |
casp 19794 | class AlgSpan |
cascl 19795 | class algSc |
df-assa 19796 | ⊢ AssAlg = {𝑤 ∈ (LMod ∩ Ring) ∣
[(Scalar‘𝑤) /
𝑓](𝑓 ∈ CRing ∧
∀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 19797 | ⊢ AlgSpan = (𝑤 ∈ AssAlg ↦ (𝑠 ∈ 𝒫 (Base‘𝑤) ↦ ∩ {𝑡
∈ ((SubRing‘𝑤)
∩ (LSubSp‘𝑤))
∣ 𝑠 ⊆ 𝑡})) |
df-ascl 19798 | ⊢ algSc = (𝑤 ∈ V ↦ (𝑥 ∈ (Base‘(Scalar‘𝑤)) ↦ (𝑥( ·𝑠
‘𝑤)(1r‘𝑤)))) |
cmps 19835 | class mPwSer |
cmvr 19836 | class mVar |
cmpl 19837 | class mPoly |
cltb 19838 | class
<bag |
copws 19839 | class ordPwSer |
df-psr 19840 | ⊢ 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 19841 | ⊢ 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 19842 | ⊢ mPoly = (𝑖 ∈ V, 𝑟 ∈ V ↦ ⦋(𝑖 mPwSer 𝑟) / 𝑤⦌(𝑤 ↾s {𝑓 ∈ (Base‘𝑤) ∣ 𝑓 finSupp (0g‘𝑟)})) |
df-ltbag 19843 | ⊢ <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 19844 | ⊢ 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 19987 | class evalSub |
cevl 19988 | class eval |
df-evls 19989 | ⊢ 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 19990 | ⊢ eval = (𝑖 ∈ V, 𝑟 ∈ V ↦ ((𝑖 evalSub 𝑟)‘(Base‘𝑟))) |
cmhp 20020 | class mHomP |
cpsd 20021 | class mPSDer |
cslv 20022 | class selectVars |
cai 20023 | class AlgInd |
df-mhp 20024 | ⊢ mHomP = (𝑖 ∈ V, 𝑟 ∈ V ↦ (𝑛 ∈ ℕ0 ↦ {𝑓 ∈ (Base‘(𝑖 mPoly 𝑟)) ∣ (𝑓 supp (0g‘𝑟)) ⊆ {𝑔 ∈ {ℎ ∈ (ℕ0
↑𝑚 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣
Σ𝑗 ∈
ℕ0 (𝑔‘𝑗) = 𝑛}})) |
df-psd 20025 | ⊢ 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-selv 20026 | ⊢ selectVars = (𝑖 ∈ V, 𝑟 ∈ V ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑓 ∈ (𝑖 mPoly 𝑟) ↦ ⦋((𝑖 ∖ 𝑗) mPoly 𝑟) / 𝑠⦌⦋(𝑥 ∈ (Scalar‘𝑠) ↦ (𝑥( ·𝑠
‘𝑠)(1r‘𝑠))) / 𝑐⦌((((𝑖 evalSub 𝑠)‘(𝑐 “s 𝑟))‘(𝑐 ∘ 𝑓))‘(𝑥 ∈ 𝑖 ↦ if(𝑥 ∈ 𝑗, ((𝑗 mVar ((𝑖 ∖ 𝑗) mPoly 𝑟))‘𝑥), (𝑐 ∘ (((𝑖 ∖ 𝑗) mVar 𝑟)‘𝑥)))))))) |
df-algind 20027 | ⊢ 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 20036 | class
PwSer1 |
cv1 20037 | class var1 |
cpl1 20038 | class
Poly1 |
cco1 20039 | class coe1 |
ctp1 20040 | class
toPoly1 |
df-psr1 20041 | ⊢ PwSer1 = (𝑟 ∈ V ↦ ((1o ordPwSer
𝑟)‘∅)) |
df-vr1 20042 | ⊢ var1 = (𝑟 ∈ V ↦ ((1o mVar 𝑟)‘∅)) |
df-ply1 20043 | ⊢ Poly1 = (𝑟 ∈ V ↦
((PwSer1‘𝑟) ↾s
(Base‘(1o mPoly 𝑟)))) |
df-coe1 20044 | ⊢ coe1 = (𝑓 ∈ V ↦ (𝑛 ∈ ℕ0 ↦ (𝑓‘(1o ×
{𝑛})))) |
df-toply1 20045 | ⊢ toPoly1 = (𝑓 ∈ V ↦ (𝑛 ∈ (ℕ0
↑𝑚 1o) ↦ (𝑓‘(𝑛‘∅)))) |
ces1 20169 | class
evalSub1 |
ce1 20170 | class
eval1 |
df-evls1 20171 | ⊢ 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 20172 | ⊢ eval1 = (𝑟 ∈ V ↦
⦋(Base‘𝑟) / 𝑏⦌((𝑥 ∈ (𝑏 ↑𝑚 (𝑏 ↑𝑚
1o)) ↦ (𝑥
∘ (𝑦 ∈ 𝑏 ↦ (1o ×
{𝑦})))) ∘
(1o eval 𝑟))) |
cpsmet 20221 | class PsMet |
cxmet 20222 | class ∞Met |
cmet 20223 | class Met |
cbl 20224 | class ball |
cfbas 20225 | class fBas |
cfg 20226 | class filGen |
cmopn 20227 | class MetOpen |
cmetu 20228 | class metUnif |
df-psmet 20229 | ⊢ PsMet = (𝑥 ∈ V ↦ {𝑑 ∈ (ℝ*
↑𝑚 (𝑥 × 𝑥)) ∣ ∀𝑦 ∈ 𝑥 ((𝑦𝑑𝑦) = 0 ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) +𝑒 (𝑤𝑑𝑧)))}) |
df-xmet 20230 | ⊢ ∞Met = (𝑥 ∈ V ↦ {𝑑 ∈ (ℝ*
↑𝑚 (𝑥 × 𝑥)) ∣ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (((𝑦𝑑𝑧) = 0 ↔ 𝑦 = 𝑧) ∧ ∀𝑤 ∈ 𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) +𝑒 (𝑤𝑑𝑧)))}) |
df-met 20231 | ⊢ Met = (𝑥 ∈ V ↦ {𝑑 ∈ (ℝ ↑𝑚
(𝑥 × 𝑥)) ∣ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (((𝑦𝑑𝑧) = 0 ↔ 𝑦 = 𝑧) ∧ ∀𝑤 ∈ 𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) + (𝑤𝑑𝑧)))}) |
df-bl 20232 | ⊢ ball = (𝑑 ∈ V ↦ (𝑥 ∈ dom dom 𝑑, 𝑧 ∈ ℝ* ↦ {𝑦 ∈ dom dom 𝑑 ∣ (𝑥𝑑𝑦) < 𝑧})) |
df-mopn 20233 | ⊢ MetOpen = (𝑑 ∈ ∪ ran
∞Met ↦ (topGen‘ran (ball‘𝑑))) |
df-fbas 20234 | ⊢ fBas = (𝑤 ∈ V ↦ {𝑥 ∈ 𝒫 𝒫 𝑤 ∣ (𝑥 ≠ ∅ ∧ ∅ ∉ 𝑥 ∧ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑥 ∩ 𝒫 (𝑦 ∩ 𝑧)) ≠ ∅)}) |
df-fg 20235 | ⊢ filGen = (𝑤 ∈ V, 𝑥 ∈ (fBas‘𝑤) ↦ {𝑦 ∈ 𝒫 𝑤 ∣ (𝑥 ∩ 𝒫 𝑦) ≠ ∅}) |
df-metu 20236 | ⊢ metUnif = (𝑑 ∈ ∪ ran
PsMet ↦ ((dom dom 𝑑
× dom dom 𝑑)filGenran
(𝑎 ∈
ℝ+ ↦ (◡𝑑 “ (0[,)𝑎))))) |
ccnfld 20237 | class
ℂfld |
df-cnfld 20238 | ⊢ ℂfld =
(({〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), +
〉, 〈(.r‘ndx), · 〉} ∪
{〈(*𝑟‘ndx), ∗〉}) ∪
({〈(TopSet‘ndx), (MetOpen‘(abs ∘ − ))〉,
〈(le‘ndx), ≤ 〉, 〈(dist‘ndx), (abs ∘ −
)〉} ∪ {〈(UnifSet‘ndx), (metUnif‘(abs ∘ −
))〉})) |
zring 20309 | class
ℤring |
df-zring 20310 | ⊢ ℤring =
(ℂfld ↾s ℤ) |
czrh 20339 | class ℤRHom |
czlm 20340 | class ℤMod |
cchr 20341 | class chr |
czn 20342 | class
ℤ/nℤ |
df-zrh 20343 | ⊢ ℤRHom = (𝑟 ∈ V ↦ ∪ (ℤring RingHom 𝑟)) |
df-zlm 20344 | ⊢ ℤMod = (𝑔 ∈ V ↦ ((𝑔 sSet 〈(Scalar‘ndx),
ℤring〉) sSet 〈( ·𝑠
‘ndx), (.g‘𝑔)〉)) |
df-chr 20345 | ⊢ chr = (𝑔 ∈ V ↦ ((od‘𝑔)‘(1r‘𝑔))) |
df-zn 20346 | ⊢ ℤ/nℤ = (𝑛 ∈ ℕ0
↦ ⦋ℤring / 𝑧⦌⦋(𝑧 /s (𝑧 ~QG
((RSpan‘𝑧)‘{𝑛}))) / 𝑠⦌(𝑠 sSet 〈(le‘ndx),
⦋((ℤRHom‘𝑠) ↾ if(𝑛 = 0, ℤ, (0..^𝑛))) / 𝑓⦌((𝑓 ∘ ≤ ) ∘ ◡𝑓)〉)) |
crefld 20440 | class
ℝfld |
df-refld 20441 | ⊢ ℝfld =
(ℂfld ↾s ℝ) |
cphl 20460 | class PreHil |
cipf 20461 | class
·if |
df-phl 20462 | ⊢ PreHil = {𝑔 ∈ LVec ∣
[(Base‘𝑔) /
𝑣][(·𝑖‘𝑔) / ℎ][(Scalar‘𝑔) / 𝑓](𝑓
∈ *-Ring ∧ ∀𝑥 ∈ 𝑣 ((𝑦 ∈ 𝑣
↦ (𝑦ℎ𝑥)) ∈
(𝑔 LMHom (ringLMod‘𝑓)) ∧ ((𝑥ℎ𝑥) = (0g‘𝑓) → 𝑥 =
(0g‘𝑔)) ∧ ∀𝑦 ∈ 𝑣 ((*𝑟‘𝑓)‘(𝑥ℎ𝑦)) = (𝑦ℎ𝑥)))} |
df-ipf 20463 | ⊢ ·if = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(·𝑖‘𝑔)𝑦))) |
cocv 20496 | class ocv |
ccss 20497 | class ClSubSp |
cthl 20498 | class toHL |
df-ocv 20499 | ⊢ ocv = (ℎ ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘ℎ) ↦ {𝑥 ∈ (Base‘ℎ) ∣ ∀𝑦 ∈ 𝑠 (𝑥(·𝑖‘ℎ)𝑦) = (0g‘(Scalar‘ℎ))})) |
df-css 20500 | ⊢ ClSubSp = (ℎ ∈ V ↦ {𝑠 ∣ 𝑠 = ((ocv‘ℎ)‘((ocv‘ℎ)‘𝑠))}) |
df-thl 20501 | ⊢ toHL = (ℎ ∈ V ↦
((toInc‘(ClSubSp‘ℎ)) sSet 〈(oc‘ndx), (ocv‘ℎ)〉)) |
cpj 20536 | class proj |
chil 20537 | class Hil |
cobs 20538 | class OBasis |
df-pj 20539 | ⊢ proj = (ℎ ∈ V ↦ ((𝑥 ∈ (LSubSp‘ℎ) ↦ (𝑥(proj1‘ℎ)((ocv‘ℎ)‘𝑥))) ∩ (V × ((Base‘ℎ) ↑𝑚
(Base‘ℎ))))) |
df-hil 20540 | ⊢ Hil = {ℎ ∈ PreHil ∣ dom (proj‘ℎ) = (ClSubSp‘ℎ)} |
df-obs 20541 | ⊢ OBasis = (ℎ ∈ PreHil ↦ {𝑏 ∈ 𝒫 (Base‘ℎ) ∣ (∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥(·𝑖‘ℎ)𝑦) = if(𝑥 = 𝑦, (1r‘(Scalar‘ℎ)),
(0g‘(Scalar‘ℎ))) ∧ ((ocv‘ℎ)‘𝑏) = {(0g‘ℎ)})}) |
cdsmm 20567 | class
⊕m |
df-dsmm 20568 | ⊢ ⊕m = (𝑠 ∈ V, 𝑟 ∈ V ↦ ((𝑠Xs𝑟) ↾s {𝑓 ∈ X𝑥 ∈ dom 𝑟(Base‘(𝑟‘𝑥)) ∣ {𝑥 ∈ dom 𝑟 ∣ (𝑓‘𝑥) ≠ (0g‘(𝑟‘𝑥))} ∈ Fin})) |
cfrlm 20582 | class freeLMod |
df-frlm 20583 | ⊢ freeLMod = (𝑟 ∈ V, 𝑖 ∈ V ↦ (𝑟 ⊕m (𝑖 × {(ringLMod‘𝑟)}))) |
cuvc 20618 | class unitVec |
df-uvc 20619 | ⊢ unitVec = (𝑟 ∈ V, 𝑖 ∈ V ↦ (𝑗 ∈ 𝑖 ↦ (𝑘 ∈ 𝑖 ↦ if(𝑘 = 𝑗, (1r‘𝑟), (0g‘𝑟))))) |
clindf 20640 | class LIndF |
clinds 20641 | class LIndS |
df-lindf 20642 | ⊢ 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 20643 | ⊢ LIndS = (𝑤 ∈ V ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ ( I ↾ 𝑠) LIndF 𝑤}) |
cmmul 20686 | class maMul |
df-mamu 20687 | ⊢ 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 20710 | class Mat |
df-mat 20711 | ⊢ Mat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ ((𝑟 freeLMod (𝑛 × 𝑛)) sSet 〈(.r‘ndx),
(𝑟 maMul 〈𝑛, 𝑛, 𝑛〉)〉)) |
cdmat 20791 | class DMat |
cscmat 20792 | class ScMat |
df-dmat 20793 | ⊢ DMat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ {𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ∣ ∀𝑖 ∈ 𝑛 ∀𝑗 ∈ 𝑛 (𝑖 ≠ 𝑗 → (𝑖𝑚𝑗) = (0g‘𝑟))}) |
df-scmat 20794 | ⊢ ScMat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ ⦋(𝑛 Mat 𝑟) / 𝑎⦌{𝑚 ∈ (Base‘𝑎) ∣ ∃𝑐 ∈ (Base‘𝑟)𝑚 = (𝑐( ·𝑠
‘𝑎)(1r‘𝑎))}) |
cmvmul 20843 | class maVecMul |
df-mvmul 20844 | ⊢ 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 20859 | class matRRep |
cmatrepV 20860 | class matRepV |
df-marrep 20861 | ⊢ 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 20862 | ⊢ 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 20879 | class subMat |
df-subma 20880 | ⊢ subMat = (𝑛 ∈ V, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑘 ∈ 𝑛, 𝑙 ∈ 𝑛 ↦ (𝑖 ∈ (𝑛 ∖ {𝑘}), 𝑗 ∈ (𝑛 ∖ {𝑙}) ↦ (𝑖𝑚𝑗))))) |
cmdat 20887 | class maDet |
df-mdet 20888 | ⊢ 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 20935 | class maAdju |
cminmar1 20936 | class minMatR1 |
df-madu 20937 | ⊢ 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 20938 | ⊢ 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 21005 | class ConstPolyMat |
cmat2pmat 21006 | class matToPolyMat |
ccpmat2mat 21007 | class cPolyMatToMat |
df-cpmat 21008 | ⊢ ConstPolyMat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ {𝑚 ∈ (Base‘(𝑛 Mat (Poly1‘𝑟))) ∣ ∀𝑖 ∈ 𝑛 ∀𝑗 ∈ 𝑛 ∀𝑘 ∈ ℕ
((coe1‘(𝑖𝑚𝑗))‘𝑘) = (0g‘𝑟)}) |
df-mat2pmat 21009 | ⊢ matToPolyMat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑥 ∈ 𝑛, 𝑦 ∈ 𝑛 ↦
((algSc‘(Poly1‘𝑟))‘(𝑥𝑚𝑦))))) |
df-cpmat2mat 21010 | ⊢ cPolyMatToMat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ (𝑚 ∈ (𝑛 ConstPolyMat 𝑟) ↦ (𝑥 ∈ 𝑛, 𝑦 ∈ 𝑛 ↦ ((coe1‘(𝑥𝑚𝑦))‘0)))) |
cdecpmat 21064 | class decompPMat |
df-decpmat 21065 | ⊢ decompPMat = (𝑚 ∈ V, 𝑘 ∈ ℕ0 ↦ (𝑖 ∈ dom dom 𝑚, 𝑗 ∈ dom dom 𝑚 ↦ ((coe1‘(𝑖𝑚𝑗))‘𝑘))) |
cpm2mp 21094 | class pMatToMatPoly |
df-pm2mp 21095 | ⊢ 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 21128 | class CharPlyMat |
df-chpmat 21129 | ⊢ 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 21195 | class Top |
df-top 21196 | ⊢ Top = {𝑥 ∣ (∀𝑦 ∈ 𝒫 𝑥∪ 𝑦 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑦 ∩ 𝑧) ∈ 𝑥)} |
ctopon 21212 | class TopOn |
df-topon 21213 | ⊢ TopOn = (𝑏 ∈ V ↦ {𝑗 ∈ Top ∣ 𝑏 = ∪ 𝑗}) |
ctps 21234 | class TopSp |
df-topsp 21235 | ⊢ TopSp = {𝑓 ∣ (TopOpen‘𝑓) ∈ (TopOn‘(Base‘𝑓))} |
ctb 21247 | class TopBases |
df-bases 21248 | ⊢ TopBases = {𝑥 ∣ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑦 ∩ 𝑧) ⊆ ∪ (𝑥 ∩ 𝒫 (𝑦 ∩ 𝑧))} |
ccld 21318 | class Clsd |
cnt 21319 | class int |
ccl 21320 | class cls |
df-cld 21321 | ⊢ Clsd = (𝑗 ∈ Top ↦ {𝑥 ∈ 𝒫 ∪ 𝑗
∣ (∪ 𝑗 ∖ 𝑥) ∈ 𝑗}) |
df-ntr 21322 | ⊢ int = (𝑗 ∈ Top ↦ (𝑥 ∈ 𝒫 ∪ 𝑗
↦ ∪ (𝑗 ∩ 𝒫 𝑥))) |
df-cls 21323 | ⊢ cls = (𝑗 ∈ Top ↦ (𝑥 ∈ 𝒫 ∪ 𝑗
↦ ∩ {𝑦 ∈ (Clsd‘𝑗) ∣ 𝑥 ⊆ 𝑦})) |
cnei 21399 | class nei |
df-nei 21400 | ⊢ nei = (𝑗 ∈ Top ↦ (𝑥 ∈ 𝒫 ∪ 𝑗
↦ {𝑦 ∈ 𝒫
∪ 𝑗 ∣ ∃𝑔 ∈ 𝑗 (𝑥 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑦)})) |
clp 21436 | class limPt |
cperf 21437 | class Perf |
df-lp 21438 | ⊢ limPt = (𝑗 ∈ Top ↦ (𝑥 ∈ 𝒫 ∪ 𝑗
↦ {𝑦 ∣ 𝑦 ∈ ((cls‘𝑗)‘(𝑥 ∖ {𝑦}))})) |
df-perf 21439 | ⊢ Perf = {𝑗 ∈ Top ∣ ((limPt‘𝑗)‘∪ 𝑗) =
∪ 𝑗} |
ccn 21526 | class Cn |
ccnp 21527 | class CnP |
clm 21528 | class
⇝𝑡 |
df-cn 21529 | ⊢ Cn = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ {𝑓 ∈ (∪ 𝑘 ↑𝑚
∪ 𝑗) ∣ ∀𝑦 ∈ 𝑘 (◡𝑓 “ 𝑦) ∈ 𝑗}) |
df-cnp 21530 | ⊢ 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 21531 | ⊢ ⇝𝑡 =
(𝑗 ∈ Top ↦
{〈𝑓, 𝑥〉 ∣ (𝑓 ∈ (∪ 𝑗
↑pm ℂ) ∧ 𝑥 ∈ ∪ 𝑗 ∧ ∀𝑢 ∈ 𝑗 (𝑥 ∈ 𝑢 → ∃𝑦 ∈ ran ℤ≥(𝑓 ↾ 𝑦):𝑦⟶𝑢))}) |
ct0 21608 | class Kol2 |
ct1 21609 | class Fre |
cha 21610 | class Haus |
creg 21611 | class Reg |
cnrm 21612 | class Nrm |
ccnrm 21613 | class CNrm |
cpnrm 21614 | class PNrm |
df-t0 21615 | ⊢ Kol2 = {𝑗 ∈ Top ∣ ∀𝑥 ∈ ∪ 𝑗∀𝑦 ∈ ∪ 𝑗(∀𝑜 ∈ 𝑗 (𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜) → 𝑥 = 𝑦)} |
df-t1 21616 | ⊢ Fre = {𝑥 ∈ Top ∣ ∀𝑎 ∈ ∪ 𝑥{𝑎} ∈ (Clsd‘𝑥)} |
df-haus 21617 | ⊢ Haus = {𝑗 ∈ Top ∣ ∀𝑥 ∈ ∪ 𝑗∀𝑦 ∈ ∪ 𝑗(𝑥 ≠ 𝑦 → ∃𝑛 ∈ 𝑗 ∃𝑚 ∈ 𝑗 (𝑥 ∈ 𝑛 ∧ 𝑦 ∈ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅))} |
df-reg 21618 | ⊢ Reg = {𝑗 ∈ Top ∣ ∀𝑥 ∈ 𝑗 ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝑗 (𝑦 ∈ 𝑧 ∧ ((cls‘𝑗)‘𝑧) ⊆ 𝑥)} |
df-nrm 21619 | ⊢ Nrm = {𝑗 ∈ Top ∣ ∀𝑥 ∈ 𝑗 ∀𝑦 ∈ ((Clsd‘𝑗) ∩ 𝒫 𝑥)∃𝑧 ∈ 𝑗 (𝑦 ⊆ 𝑧 ∧ ((cls‘𝑗)‘𝑧) ⊆ 𝑥)} |
df-cnrm 21620 | ⊢ CNrm = {𝑗 ∈ Top ∣ ∀𝑥 ∈ 𝒫 ∪ 𝑗(𝑗 ↾t 𝑥) ∈ Nrm} |
df-pnrm 21621 | ⊢ PNrm = {𝑗 ∈ Nrm ∣ (Clsd‘𝑗) ⊆ ran (𝑓 ∈ (𝑗 ↑𝑚 ℕ) ↦
∩ ran 𝑓)} |
ccmp 21688 | class Comp |
df-cmp 21689 | ⊢ Comp = {𝑥 ∈ Top ∣ ∀𝑦 ∈ 𝒫 𝑥(∪
𝑥 = ∪ 𝑦
→ ∃𝑧 ∈
(𝒫 𝑦 ∩
Fin)∪ 𝑥 = ∪ 𝑧)} |
cconn 21713 | class Conn |
df-conn 21714 | ⊢ Conn = {𝑗 ∈ Top ∣ (𝑗 ∩ (Clsd‘𝑗)) = {∅, ∪
𝑗}} |
c1stc 21739 | class
1st𝜔 |
c2ndc 21740 | class
2nd𝜔 |
df-1stc 21741 | ⊢ 1st𝜔 = {𝑗 ∈ Top ∣ ∀𝑥 ∈ ∪ 𝑗∃𝑦 ∈ 𝒫 𝑗(𝑦 ≼ ω ∧ ∀𝑧 ∈ 𝑗 (𝑥 ∈ 𝑧 → 𝑥 ∈ ∪ (𝑦 ∩ 𝒫 𝑧)))} |
df-2ndc 21742 | ⊢ 2nd𝜔 = {𝑗 ∣ ∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝑗)} |
clly 21766 | class Locally 𝐴 |
cnlly 21767 | class 𝑛-Locally 𝐴 |
df-lly 21768 | ⊢ Locally 𝐴 = {𝑗 ∈ Top ∣ ∀𝑥 ∈ 𝑗 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝐴)} |
df-nlly 21769 | ⊢ 𝑛-Locally 𝐴 = {𝑗 ∈ Top ∣ ∀𝑥 ∈ 𝑗 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗 ↾t 𝑢) ∈ 𝐴} |
cref 21804 | class Ref |
cptfin 21805 | class PtFin |
clocfin 21806 | class LocFin |
df-ref 21807 | ⊢ Ref = {〈𝑥, 𝑦〉 ∣ (∪
𝑦 = ∪ 𝑥
∧ ∀𝑧 ∈
𝑥 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)} |
df-ptfin 21808 | ⊢ PtFin = {𝑥 ∣ ∀𝑦 ∈ ∪ 𝑥{𝑧 ∈ 𝑥 ∣ 𝑦 ∈ 𝑧} ∈ Fin} |
df-locfin 21809 | ⊢ LocFin = (𝑥 ∈ Top ↦ {𝑦 ∣ (∪ 𝑥 = ∪
𝑦 ∧ ∀𝑝 ∈ ∪ 𝑥∃𝑛 ∈ 𝑥 (𝑝 ∈ 𝑛 ∧ {𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈
Fin))}) |
ckgen 21835 | class 𝑘Gen |
df-kgen 21836 | ⊢ 𝑘Gen = (𝑗 ∈ Top ↦ {𝑥 ∈ 𝒫 ∪ 𝑗
∣ ∀𝑘 ∈
𝒫 ∪ 𝑗((𝑗 ↾t 𝑘) ∈ Comp → (𝑥 ∩ 𝑘) ∈ (𝑗 ↾t 𝑘))}) |
ctx 21862 | class
×t |
cxko 21863 | class
^ko |
df-tx 21864 | ⊢ ×t = (𝑟 ∈ V, 𝑠 ∈ V ↦ (topGen‘ran (𝑥 ∈ 𝑟, 𝑦 ∈ 𝑠 ↦ (𝑥 × 𝑦)))) |
df-xko 21865 | ⊢ ^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 21995 | class KQ |
df-kq 21996 | ⊢ KQ = (𝑗 ∈ Top ↦ (𝑗 qTop (𝑥 ∈ ∪ 𝑗 ↦ {𝑦 ∈ 𝑗 ∣ 𝑥 ∈ 𝑦}))) |
chmeo 22055 | class Homeo |
chmph 22056 | class ≃ |
df-hmeo 22057 | ⊢ Homeo = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ {𝑓 ∈ (𝑗 Cn 𝑘) ∣ ◡𝑓 ∈ (𝑘 Cn 𝑗)}) |
df-hmph 22058 | ⊢ ≃ = (◡Homeo “ (V ∖
1o)) |
cfil 22147 | class Fil |
df-fil 22148 | ⊢ Fil = (𝑧 ∈ V ↦ {𝑓 ∈ (fBas‘𝑧) ∣ ∀𝑥 ∈ 𝒫 𝑧((𝑓 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥 ∈ 𝑓)}) |
cufil 22201 | class UFil |
cufl 22202 | class UFL |
df-ufil 22203 | ⊢ UFil = (𝑔 ∈ V ↦ {𝑓 ∈ (Fil‘𝑔) ∣ ∀𝑥 ∈ 𝒫 𝑔(𝑥 ∈ 𝑓 ∨ (𝑔 ∖ 𝑥) ∈ 𝑓)}) |
df-ufl 22204 | ⊢ UFL = {𝑥 ∣ ∀𝑓 ∈ (Fil‘𝑥)∃𝑔 ∈ (UFil‘𝑥)𝑓 ⊆ 𝑔} |
cfm 22235 | class FilMap |
cflim 22236 | class fLim |
cflf 22237 | class fLimf |
cfcls 22238 | class fClus |
cfcf 22239 | class fClusf |
df-fm 22240 | ⊢ FilMap = (𝑥 ∈ V, 𝑓 ∈ V ↦ (𝑦 ∈ (fBas‘dom 𝑓) ↦ (𝑥filGenran (𝑡 ∈ 𝑦 ↦ (𝑓 “ 𝑡))))) |
df-flim 22241 | ⊢ fLim = (𝑗 ∈ Top, 𝑓 ∈ ∪ ran Fil
↦ {𝑥 ∈ ∪ 𝑗
∣ (((nei‘𝑗)‘{𝑥}) ⊆ 𝑓 ∧ 𝑓 ⊆ 𝒫 ∪ 𝑗)}) |
df-flf 22242 | ⊢ fLimf = (𝑥 ∈ Top, 𝑦 ∈ ∪ ran Fil
↦ (𝑓 ∈ (∪ 𝑥
↑𝑚 ∪ 𝑦) ↦ (𝑥 fLim ((∪ 𝑥 FilMap 𝑓)‘𝑦)))) |
df-fcls 22243 | ⊢ fClus = (𝑗 ∈ Top, 𝑓 ∈ ∪ ran Fil
↦ if(∪ 𝑗 = ∪ 𝑓, ∩ 𝑥 ∈ 𝑓 ((cls‘𝑗)‘𝑥), ∅)) |
df-fcf 22244 | ⊢ fClusf = (𝑗 ∈ Top, 𝑓 ∈ ∪ ran Fil
↦ (𝑔 ∈ (∪ 𝑗
↑𝑚 ∪ 𝑓) ↦ (𝑗 fClus ((∪ 𝑗 FilMap 𝑔)‘𝑓)))) |
ccnext 22361 | class CnExt |
df-cnext 22362 | ⊢ CnExt = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ (𝑓 ∈ (∪ 𝑘 ↑pm
∪ 𝑗) ↦ ∪
𝑥 ∈ ((cls‘𝑗)‘dom 𝑓)({𝑥} × ((𝑘 fLimf (((nei‘𝑗)‘{𝑥}) ↾t dom 𝑓))‘𝑓)))) |
ctmd 22372 | class TopMnd |
ctgp 22373 | class TopGrp |
df-tmd 22374 | ⊢ TopMnd = {𝑓 ∈ (Mnd ∩ TopSp) ∣
[(TopOpen‘𝑓) /
𝑗](+𝑓‘𝑓) ∈ ((𝑗 ×t 𝑗) Cn 𝑗)} |
df-tgp 22375 | ⊢ TopGrp = {𝑓 ∈ (Grp ∩ TopMnd) ∣
[(TopOpen‘𝑓) /
𝑗](invg‘𝑓) ∈ (𝑗 Cn 𝑗)} |
ctsu 22427 | class tsums |
df-tsms 22428 | ⊢ 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 22457 | class TopRing |
ctdrg 22458 | class TopDRing |
ctlm 22459 | class TopMod |
ctvc 22460 | class TopVec |
df-trg 22461 | ⊢ TopRing = {𝑟 ∈ (TopGrp ∩ Ring) ∣
(mulGrp‘𝑟) ∈
TopMnd} |
df-tdrg 22462 | ⊢ TopDRing = {𝑟 ∈ (TopRing ∩ DivRing) ∣
((mulGrp‘𝑟)
↾s (Unit‘𝑟)) ∈ TopGrp} |
df-tlm 22463 | ⊢ TopMod = {𝑤 ∈ (TopMnd ∩ LMod) ∣
((Scalar‘𝑤) ∈
TopRing ∧ ( ·sf ‘𝑤) ∈ (((TopOpen‘(Scalar‘𝑤)) ×t
(TopOpen‘𝑤)) Cn
(TopOpen‘𝑤)))} |
df-tvc 22464 | ⊢ TopVec = {𝑤 ∈ TopMod ∣ (Scalar‘𝑤) ∈
TopDRing} |
cust 22501 | class UnifOn |
df-ust 22502 | ⊢ UnifOn = (𝑥 ∈ V ↦ {𝑢 ∣ (𝑢 ⊆ 𝒫 (𝑥 × 𝑥) ∧ (𝑥 × 𝑥) ∈ 𝑢 ∧ ∀𝑣 ∈ 𝑢 (∀𝑤 ∈ 𝒫 (𝑥 × 𝑥)(𝑣 ⊆ 𝑤 → 𝑤 ∈ 𝑢) ∧ ∀𝑤 ∈ 𝑢 (𝑣 ∩ 𝑤) ∈ 𝑢 ∧ (( I ↾ 𝑥) ⊆ 𝑣 ∧ ◡𝑣 ∈ 𝑢 ∧ ∃𝑤 ∈ 𝑢 (𝑤 ∘ 𝑤) ⊆ 𝑣)))}) |
cutop 22532 | class unifTop |
df-utop 22533 | ⊢ unifTop = (𝑢 ∈ ∪ ran
UnifOn ↦ {𝑎 ∈
𝒫 dom ∪ 𝑢 ∣ ∀𝑥 ∈ 𝑎 ∃𝑣 ∈ 𝑢 (𝑣 “ {𝑥}) ⊆ 𝑎}) |
cuss 22555 | class UnifSt |
cusp 22556 | class UnifSp |
ctus 22557 | class toUnifSp |
df-uss 22558 | ⊢ UnifSt = (𝑓 ∈ V ↦ ((UnifSet‘𝑓) ↾t
((Base‘𝑓) ×
(Base‘𝑓)))) |
df-usp 22559 | ⊢ UnifSp = {𝑓 ∣ ((UnifSt‘𝑓) ∈ (UnifOn‘(Base‘𝑓)) ∧ (TopOpen‘𝑓) =
(unifTop‘(UnifSt‘𝑓)))} |
df-tus 22560 | ⊢ toUnifSp = (𝑢 ∈ ∪ ran
UnifOn ↦ ({〈(Base‘ndx), dom ∪ 𝑢〉,
〈(UnifSet‘ndx), 𝑢〉} sSet 〈(TopSet‘ndx),
(unifTop‘𝑢)〉)) |
cucn 22577 | class Cnu |
df-ucn 22578 | ⊢ 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 22588 | class
CauFilu |
df-cfilu 22589 | ⊢ CauFilu = (𝑢 ∈ ∪ ran
UnifOn ↦ {𝑓 ∈
(fBas‘dom ∪ 𝑢) ∣ ∀𝑣 ∈ 𝑢 ∃𝑎 ∈ 𝑓 (𝑎 × 𝑎) ⊆ 𝑣}) |
ccusp 22599 | class CUnifSp |
df-cusp 22600 | ⊢ CUnifSp = {𝑤 ∈ UnifSp ∣ ∀𝑐 ∈
(Fil‘(Base‘𝑤))(𝑐 ∈
(CauFilu‘(UnifSt‘𝑤)) → ((TopOpen‘𝑤) fLim 𝑐) ≠ ∅)} |
cxms 22620 | class ∞MetSp |
cms 22621 | class MetSp |
ctms 22622 | class toMetSp |
df-xms 22623 | ⊢ ∞MetSp = {𝑓 ∈ TopSp ∣ (TopOpen‘𝑓) =
(MetOpen‘((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))))} |
df-ms 22624 | ⊢ MetSp = {𝑓 ∈ ∞MetSp ∣
((dist‘𝑓) ↾
((Base‘𝑓) ×
(Base‘𝑓))) ∈
(Met‘(Base‘𝑓))} |
df-tms 22625 | ⊢ toMetSp = (𝑑 ∈ ∪ ran
∞Met ↦ ({〈(Base‘ndx), dom dom 𝑑〉, 〈(dist‘ndx), 𝑑〉} sSet
〈(TopSet‘ndx), (MetOpen‘𝑑)〉)) |
cnm 22879 | class norm |
cngp 22880 | class NrmGrp |
ctng 22881 | class toNrmGrp |
cnrg 22882 | class NrmRing |
cnlm 22883 | class NrmMod |
cnvc 22884 | class NrmVec |
df-nm 22885 | ⊢ norm = (𝑤 ∈ V ↦ (𝑥 ∈ (Base‘𝑤) ↦ (𝑥(dist‘𝑤)(0g‘𝑤)))) |
df-ngp 22886 | ⊢ NrmGrp = {𝑔 ∈ (Grp ∩ MetSp) ∣
((norm‘𝑔) ∘
(-g‘𝑔))
⊆ (dist‘𝑔)} |
df-tng 22887 | ⊢ toNrmGrp = (𝑔 ∈ V, 𝑓 ∈ V ↦ ((𝑔 sSet 〈(dist‘ndx), (𝑓 ∘
(-g‘𝑔))〉) sSet 〈(TopSet‘ndx),
(MetOpen‘(𝑓 ∘
(-g‘𝑔)))〉)) |
df-nrg 22888 | ⊢ NrmRing = {𝑤 ∈ NrmGrp ∣ (norm‘𝑤) ∈ (AbsVal‘𝑤)} |
df-nlm 22889 | ⊢ NrmMod = {𝑤 ∈ (NrmGrp ∩ LMod) ∣
[(Scalar‘𝑤) /
𝑓](𝑓 ∈ NrmRing ∧
∀𝑥 ∈
(Base‘𝑓)∀𝑦 ∈ (Base‘𝑤)((norm‘𝑤)‘(𝑥( ·𝑠
‘𝑤)𝑦)) = (((norm‘𝑓)‘𝑥) · ((norm‘𝑤)‘𝑦)))} |
df-nvc 22890 | ⊢ NrmVec = (NrmMod ∩ LVec) |
cnmo 23007 | class normOp |
cnghm 23008 | class NGHom |
cnmhm 23009 | class NMHom |
df-nmo 23010 | ⊢ 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 23011 | ⊢ NGHom = (𝑠 ∈ NrmGrp, 𝑡 ∈ NrmGrp ↦ (◡(𝑠 normOp 𝑡) “ ℝ)) |
df-nmhm 23012 | ⊢ NMHom = (𝑠 ∈ NrmMod, 𝑡 ∈ NrmMod ↦ ((𝑠 LMHom 𝑡) ∩ (𝑠 NGHom 𝑡))) |
cii 23176 | class II |
ccncf 23177 | class –cn→ |
df-ii 23178 | ⊢ II = (MetOpen‘((abs
∘ − ) ↾ ((0[,]1) × (0[,]1)))) |
df-cncf 23179 | ⊢ –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 23264 | class Htpy |
cphtpy 23265 | class PHtpy |
cphtpc 23266 | class
≃ph |
df-htpy 23267 | ⊢ 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 23268 | ⊢ PHtpy = (𝑥 ∈ Top ↦ (𝑓 ∈ (II Cn 𝑥), 𝑔 ∈ (II Cn 𝑥) ↦ {ℎ ∈ (𝑓(II Htpy 𝑥)𝑔) ∣ ∀𝑠 ∈ (0[,]1)((0ℎ𝑠) = (𝑓‘0) ∧ (1ℎ𝑠) = (𝑓‘1))})) |
df-phtpc 23289 | ⊢ ≃ph = (𝑥 ∈ Top ↦ {〈𝑓, 𝑔〉 ∣ ({𝑓, 𝑔} ⊆ (II Cn 𝑥) ∧ (𝑓(PHtpy‘𝑥)𝑔) ≠ ∅)}) |
cpco 23297 | class
*𝑝 |
comi 23298 | class
Ω1 |
comn 23299 | class
Ω𝑛 |
cpi1 23300 | class
π1 |
cpin 23301 | class
πn |
df-pco 23302 | ⊢ *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 23303 | ⊢ 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 23304 | ⊢ 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 23305 | ⊢ π1 = (𝑗 ∈ Top, 𝑦 ∈ ∪ 𝑗 ↦ ((𝑗 Ω1 𝑦) /s (
≃ph‘𝑗))) |
df-pin 23306 | ⊢ 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 23359 | class ℂMod |
df-clm 23360 | ⊢ ℂMod = {𝑤 ∈ LMod ∣
[(Scalar‘𝑤) /
𝑓][(Base‘𝑓) / 𝑘](𝑓 = (ℂfld ↾s
𝑘) ∧ 𝑘 ∈
(SubRing‘ℂfld))} |
ccvs 23420 | class ℂVec |
df-cvs 23421 | ⊢ ℂVec = (ℂMod ∩
LVec) |
ccph 23463 | class ℂPreHil |
ctcph 23464 | class toℂPreHil |
df-cph 23465 | ⊢ ℂPreHil = {𝑤 ∈ (PreHil ∩ NrmMod) ∣
[(Scalar‘𝑤) /
𝑓][(Base‘𝑓) / 𝑘](𝑓 = (ℂfld ↾s
𝑘) ∧ (√ “
(𝑘 ∩ (0[,)+∞)))
⊆ 𝑘 ∧
(norm‘𝑤) = (𝑥 ∈ (Base‘𝑤) ↦ (√‘(𝑥(·𝑖‘𝑤)𝑥))))} |
df-tcph 23466 | ⊢ toℂPreHil = (𝑤 ∈ V ↦ (𝑤 toNrmGrp (𝑥 ∈ (Base‘𝑤) ↦ (√‘(𝑥(·𝑖‘𝑤)𝑥))))) |
ccfil 23548 | class CauFil |
ccau 23549 | class Cau |
ccmet 23550 | class CMet |
df-cfil 23551 | ⊢ CauFil = (𝑑 ∈ ∪ ran
∞Met ↦ {𝑓
∈ (Fil‘dom dom 𝑑) ∣ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ 𝑓 (𝑑 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)}) |
df-cau 23552 | ⊢ Cau = (𝑑 ∈ ∪ ran
∞Met ↦ {𝑓
∈ (dom dom 𝑑
↑pm ℂ) ∣ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℤ (𝑓 ↾
(ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶((𝑓‘𝑗)(ball‘𝑑)𝑥)}) |
df-cmet 23553 | ⊢ CMet = (𝑥 ∈ V ↦ {𝑑 ∈ (Met‘𝑥) ∣ ∀𝑓 ∈ (CauFil‘𝑑)((MetOpen‘𝑑) fLim 𝑓) ≠ ∅}) |
ccms 23628 | class CMetSp |
cbn 23629 | class Ban |
chl 23630 | class ℂHil |
df-cms 23631 | ⊢ CMetSp = {𝑤 ∈ MetSp ∣
[(Base‘𝑤) /
𝑏]((dist‘𝑤) ↾ (𝑏 × 𝑏)) ∈ (CMet‘𝑏)} |
df-bn 23632 | ⊢ Ban = {𝑤 ∈ (NrmVec ∩ CMetSp) ∣
(Scalar‘𝑤) ∈
CMetSp} |
df-hl 23633 | ⊢ ℂHil = (Ban ∩
ℂPreHil) |
crrx 23679 | class ℝ^ |
cehl 23680 | class
𝔼hil |
df-rrx 23681 | ⊢ ℝ^ = (𝑖 ∈ V ↦
(toℂPreHil‘(ℝfld freeLMod 𝑖))) |
df-ehl 23682 | ⊢ 𝔼hil = (𝑛 ∈ ℕ0 ↦
(ℝ^‘(1...𝑛))) |
covol 23756 | class vol* |
cvol 23757 | class vol |
df-ovol 23758 | ⊢ vol* = (𝑥 ∈ 𝒫 ℝ ↦ inf({𝑦 ∈ ℝ*
∣ ∃𝑓 ∈ ((
≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ)(𝑥 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − )
∘ 𝑓)),
ℝ*, < ))}, ℝ*, < )) |
df-vol 23759 | ⊢ vol = (vol* ↾ {𝑥 ∣ ∀𝑦 ∈ (◡vol* “ ℝ)(vol*‘𝑦) = ((vol*‘(𝑦 ∩ 𝑥)) + (vol*‘(𝑦 ∖ 𝑥)))}) |
cmbf 23908 | class MblFn |
citg1 23909 | class
∫1 |
citg2 23910 | class
∫2 |
cibl 23911 | class
𝐿1 |
citg 23912 | class ∫𝐴𝐵 d𝑥 |
df-mbf 23913 | ⊢ MblFn = {𝑓 ∈ (ℂ ↑pm
ℝ) ∣ ∀𝑥
∈ ran (,)((◡(ℜ ∘ 𝑓) “ 𝑥) ∈ dom vol ∧ (◡(ℑ ∘ 𝑓) “ 𝑥) ∈ dom vol)} |
df-itg1 23914 | ⊢ ∫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 23915 | ⊢ ∫2 = (𝑓 ∈ ((0[,]+∞)
↑𝑚 ℝ) ↦ sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔 ∘𝑟
≤ 𝑓 ∧ 𝑥 =
(∫1‘𝑔))}, ℝ*, <
)) |
df-ibl 23916 | ⊢ 𝐿1 = {𝑓 ∈ MblFn ∣ ∀𝑘 ∈
(0...3)(∫2‘(𝑥 ∈ ℝ ↦
⦋(ℜ‘((𝑓‘𝑥) / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ dom 𝑓 ∧ 0 ≤ 𝑦), 𝑦, 0))) ∈ ℝ} |
df-itg 23917 | ⊢ ∫𝐴𝐵 d𝑥 = Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦
⦋(ℜ‘(𝐵 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0)))) |
c0p 23963 | class
0𝑝 |
df-0p 23964 | ⊢ 0𝑝 = (ℂ
× {0}) |
cdit 24137 | class ⨜[𝐴 → 𝐵]𝐶 d𝑥 |
df-ditg 24138 | ⊢ ⨜[𝐴 → 𝐵]𝐶 d𝑥 = if(𝐴 ≤ 𝐵, ∫(𝐴(,)𝐵)𝐶 d𝑥, -∫(𝐵(,)𝐴)𝐶 d𝑥) |
climc 24153 | class
limℂ |
cdv 24154 | class D |
cdvn 24155 | class
D𝑛 |
ccpn 24156 | class
Cn |
df-limc 24157 | ⊢ 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 24158 | ⊢ 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 24159 | ⊢ D𝑛 = (𝑠 ∈ 𝒫 ℂ, 𝑓 ∈ (ℂ ↑pm
𝑠) ↦ seq0(((𝑥 ∈ V ↦ (𝑠 D 𝑥)) ∘ 1st ),
(ℕ0 × {𝑓}))) |
df-cpn 24160 | ⊢ Cn = (𝑠 ∈ 𝒫 ℂ ↦ (𝑥 ∈ ℕ0
↦ {𝑓 ∈ (ℂ
↑pm 𝑠) ∣ ((𝑠 D𝑛 𝑓)‘𝑥) ∈ (dom 𝑓–cn→ℂ)})) |
cmdg 24340 | class mDeg |
cdg1 24341 | class
deg1 |
df-mdeg 24342 | ⊢ mDeg = (𝑖 ∈ V, 𝑟 ∈ V ↦ (𝑓 ∈ (Base‘(𝑖 mPoly 𝑟)) ↦ sup(ran (ℎ ∈ (𝑓 supp (0g‘𝑟)) ↦ (ℂfld
Σg ℎ)), ℝ*, <
))) |
df-deg1 24343 | ⊢ deg1 = (𝑟 ∈ V ↦ (1o mDeg 𝑟)) |
cmn1 24412 | class
Monic1p |
cuc1p 24413 | class
Unic1p |
cq1p 24414 | class
quot1p |
cr1p 24415 | class
rem1p |
cig1p 24416 | class
idlGen1p |
df-mon1 24417 | ⊢ Monic1p = (𝑟 ∈ V ↦ {𝑓 ∈
(Base‘(Poly1‘𝑟)) ∣ (𝑓 ≠
(0g‘(Poly1‘𝑟)) ∧ ((coe1‘𝑓)‘(( deg1
‘𝑟)‘𝑓)) = (1r‘𝑟))}) |
df-uc1p 24418 | ⊢ Unic1p = (𝑟 ∈ V ↦ {𝑓 ∈
(Base‘(Poly1‘𝑟)) ∣ (𝑓 ≠
(0g‘(Poly1‘𝑟)) ∧ ((coe1‘𝑓)‘(( deg1
‘𝑟)‘𝑓)) ∈ (Unit‘𝑟))}) |
df-q1p 24419 | ⊢ quot1p = (𝑟 ∈ V ↦
⦋(Poly1‘𝑟) / 𝑝⦌⦋(Base‘𝑝) / 𝑏⦌(𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (℩𝑞 ∈ 𝑏 (( deg1 ‘𝑟)‘(𝑓(-g‘𝑝)(𝑞(.r‘𝑝)𝑔))) < (( deg1 ‘𝑟)‘𝑔)))) |
df-r1p 24420 | ⊢ rem1p = (𝑟 ∈ V ↦
⦋(Base‘(Poly1‘𝑟)) / 𝑏⦌(𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (𝑓(-g‘(Poly1‘𝑟))((𝑓(quot1p‘𝑟)𝑔)(.r‘(Poly1‘𝑟))𝑔)))) |
df-ig1p 24421 | ⊢ idlGen1p = (𝑟 ∈ V ↦ (𝑖 ∈
(LIdeal‘(Poly1‘𝑟)) ↦ if(𝑖 =
{(0g‘(Poly1‘𝑟))},
(0g‘(Poly1‘𝑟)), (℩𝑔 ∈ (𝑖 ∩ (Monic1p‘𝑟))(( deg1
‘𝑟)‘𝑔) = inf((( deg1
‘𝑟) “ (𝑖 ∖
{(0g‘(Poly1‘𝑟))})), ℝ, < ))))) |
cply 24467 | class Poly |
cidp 24468 | class
Xp |
ccoe 24469 | class coeff |
cdgr 24470 | class deg |
df-ply 24471 | ⊢ Poly = (𝑥 ∈ 𝒫 ℂ ↦ {𝑓 ∣ ∃𝑛 ∈ ℕ0
∃𝑎 ∈ ((𝑥 ∪ {0})
↑𝑚 ℕ0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))}) |
df-idp 24472 | ⊢ Xp = ( I ↾
ℂ) |
df-coe 24473 | ⊢ coeff = (𝑓 ∈ (Poly‘ℂ) ↦
(℩𝑎 ∈
(ℂ ↑𝑚 ℕ0)∃𝑛 ∈ ℕ0
((𝑎 “
(ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))))) |
df-dgr 24474 | ⊢ deg = (𝑓 ∈ (Poly‘ℂ) ↦
sup((◡(coeff‘𝑓) “ (ℂ ∖ {0})),
ℕ0, < )) |
cquot 24572 | class quot |
df-quot 24573 | ⊢ quot = (𝑓 ∈ (Poly‘ℂ), 𝑔 ∈ ((Poly‘ℂ)
∖ {0𝑝}) ↦ (℩𝑞 ∈ (Poly‘ℂ)[(𝑓 ∘𝑓
− (𝑔
∘𝑓 · 𝑞)) / 𝑟](𝑟 = 0𝑝 ∨
(deg‘𝑟) <
(deg‘𝑔)))) |
caa 24596 | class 𝔸 |
df-aa 24597 | ⊢ 𝔸 = ∪ 𝑓 ∈ ((Poly‘ℤ) ∖
{0𝑝})(◡𝑓 “ {0}) |
ctayl 24634 | class Tayl |
cana 24635 | class Ana |
df-tayl 24636 | ⊢ 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 24637 | ⊢ Ana = (𝑠 ∈ {ℝ, ℂ} ↦ {𝑓 ∈ (ℂ
↑pm 𝑠) ∣ ∀𝑥 ∈ dom 𝑓 𝑥 ∈
((int‘((TopOpen‘ℂfld) ↾t 𝑠))‘dom (𝑓 ∩ (+∞(𝑠 Tayl 𝑓)𝑥)))}) |
culm 24657 | class
⇝𝑢 |
df-ulm 24658 | ⊢ ⇝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 24829 | class log |
ccxp 24830 | class
↑𝑐 |
df-log 24831 | ⊢ log = ◡(exp ↾ (◡ℑ “
(-π(,]π))) |
df-cxp 24832 | ⊢ ↑𝑐 = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ if(𝑥 = 0, if(𝑦 = 0, 1, 0), (exp‘(𝑦 · (log‘𝑥))))) |
clogb 25033 | class
logb |
df-logb 25034 | ⊢ logb = (𝑥 ∈ (ℂ ∖ {0, 1}), 𝑦 ∈ (ℂ ∖ {0})
↦ ((log‘𝑦) /
(log‘𝑥))) |
casin 25131 | class arcsin |
cacos 25132 | class arccos |
catan 25133 | class arctan |
df-asin 25134 | ⊢ arcsin = (𝑥 ∈ ℂ ↦ (-i ·
(log‘((i · 𝑥)
+ (√‘(1 − (𝑥↑2))))))) |
df-acos 25135 | ⊢ arccos = (𝑥 ∈ ℂ ↦ ((π / 2) −
(arcsin‘𝑥))) |
df-atan 25136 | ⊢ arctan = (𝑥 ∈ (ℂ ∖ {-i, i}) ↦ ((i
/ 2) · ((log‘(1 − (i · 𝑥))) − (log‘(1 + (i · 𝑥)))))) |
carea 25225 | class area |
df-area 25226 | ⊢ area = (𝑠 ∈ {𝑡 ∈ 𝒫 (ℝ × ℝ)
∣ (∀𝑥 ∈
ℝ (𝑡 “ {𝑥}) ∈ (◡vol “ ℝ) ∧ (𝑥 ∈ ℝ ↦
(vol‘(𝑡 “
{𝑥}))) ∈
𝐿1)} ↦ ∫ℝ(vol‘(𝑠 “ {𝑥})) d𝑥) |
cem 25261 | class γ |
df-em 25262 | ⊢ γ = Σ𝑘 ∈ ℕ ((1 / 𝑘) − (log‘(1 + (1 /
𝑘)))) |
czeta 25282 | class ζ |
df-zeta 25283 | ⊢ ζ = (℩𝑓 ∈ ((ℂ ∖ {1})–cn→ℂ)∀𝑠 ∈ (ℂ ∖ {1})((1 −
(2↑𝑐(1 − 𝑠))) · (𝑓‘𝑠)) = Σ𝑛 ∈ ℕ0 (Σ𝑘 ∈ (0...𝑛)(((-1↑𝑘) · (𝑛C𝑘)) · ((𝑘 + 1)↑𝑐𝑠)) / (2↑(𝑛 + 1)))) |
clgam 25285 | class log Γ |
cgam 25286 | class Γ |
cigam 25287 | class 1/Γ |
df-lgam 25288 | ⊢ log Γ = (𝑧 ∈ (ℂ ∖ (ℤ ∖
ℕ)) ↦ (Σ𝑚
∈ ℕ ((𝑧 ·
(log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))) − (log‘𝑧))) |
df-gam 25289 | ⊢ Γ = (exp ∘ log
Γ) |
df-igam 25290 | ⊢ 1/Γ = (𝑥 ∈ ℂ ↦ if(𝑥 ∈ (ℤ ∖ ℕ), 0, (1 /
(Γ‘𝑥)))) |
ccht 25360 | class θ |
cvma 25361 | class Λ |
cchp 25362 | class ψ |
cppi 25363 | class π |
cmu 25364 | class μ |
csgm 25365 | class σ |
df-cht 25366 | ⊢ θ = (𝑥 ∈ ℝ ↦ Σ𝑝 ∈ ((0[,]𝑥) ∩ ℙ)(log‘𝑝)) |
df-vma 25367 | ⊢ Λ = (𝑥 ∈ ℕ ↦ ⦋{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥} / 𝑠⦌if((♯‘𝑠) = 1, (log‘∪ 𝑠),
0)) |
df-chp 25368 | ⊢ ψ = (𝑥 ∈ ℝ ↦ Σ𝑛 ∈
(1...(⌊‘𝑥))(Λ‘𝑛)) |
df-ppi 25369 | ⊢ π = (𝑥 ∈ ℝ ↦
(♯‘((0[,]𝑥)
∩ ℙ))) |
df-mu 25370 | ⊢ μ = (𝑥 ∈ ℕ ↦ if(∃𝑝 ∈ ℙ (𝑝↑2) ∥ 𝑥, 0,
(-1↑(♯‘{𝑝
∈ ℙ ∣ 𝑝
∥ 𝑥})))) |
df-sgm 25371 | ⊢ σ = (𝑥 ∈ ℂ, 𝑛 ∈ ℕ ↦ Σ𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝑛} (𝑘↑𝑐𝑥)) |
cdchr 25500 | class DChr |
df-dchr 25501 | ⊢ DChr = (𝑛 ∈ ℕ ↦
⦋(ℤ/nℤ‘𝑛) / 𝑧⦌⦋{𝑥 ∈ ((mulGrp‘𝑧) MndHom
(mulGrp‘ℂfld)) ∣ (((Base‘𝑧) ∖ (Unit‘𝑧)) × {0}) ⊆ 𝑥} / 𝑏⦌{〈(Base‘ndx), 𝑏〉,
〈(+g‘ndx), ( ∘𝑓 · ↾
(𝑏 × 𝑏))〉}) |
clgs 25562 | class
/L |
df-lgs 25563 | ⊢ /L = (𝑎 ∈ ℤ, 𝑛 ∈ ℤ ↦ if(𝑛 = 0, if((𝑎↑2) = 1, 1, 0), (if((𝑛 < 0 ∧ 𝑎 < 0), -1, 1) · (seq1( · ,
(𝑚 ∈ ℕ ↦
if(𝑚 ∈ ℙ,
(if(𝑚 = 2, if(2 ∥
𝑎, 0, if((𝑎 mod 8) ∈ {1, 7}, 1, -1)),
((((𝑎↑((𝑚 − 1) / 2)) + 1) mod 𝑚) − 1))↑(𝑚 pCnt 𝑛)), 1)))‘(abs‘𝑛))))) |
cstrkg 25908 | class TarskiG |
cstrkgc 25909 | class
TarskiGC |
cstrkgb 25910 | class
TarskiGB |
cstrkgcb 25911 | class
TarskiGCB |
cstrkgld 25912 | class
DimTarskiG≥ |
cstrkge 25913 | class
TarskiGE |
citv 25914 | class Itv |
clng 25915 | class LineG |
df-itv 25916 | ⊢ Itv = Slot ;16 |
df-lng 25917 | ⊢ LineG = Slot ;17 |
df-trkgc 25926 | ⊢ TarskiGC = {𝑓 ∣ [(Base‘𝑓) / 𝑝][(dist‘𝑓) / 𝑑](∀𝑥 ∈ 𝑝 ∀𝑦 ∈ 𝑝 (𝑥𝑑𝑦) = (𝑦𝑑𝑥) ∧ ∀𝑥 ∈ 𝑝 ∀𝑦 ∈ 𝑝 ∀𝑧 ∈ 𝑝 ((𝑥𝑑𝑦) = (𝑧𝑑𝑧) → 𝑥 = 𝑦))} |
df-trkgb 25927 | ⊢ TarskiGB = {𝑓 ∣ [(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](∀𝑥 ∈ 𝑝 ∀𝑦 ∈ 𝑝 (𝑦 ∈ (𝑥𝑖𝑥) → 𝑥 = 𝑦) ∧ ∀𝑥 ∈ 𝑝 ∀𝑦 ∈ 𝑝 ∀𝑧 ∈ 𝑝 ∀𝑢 ∈ 𝑝 ∀𝑣 ∈ 𝑝 ((𝑢 ∈ (𝑥𝑖𝑧) ∧ 𝑣 ∈ (𝑦𝑖𝑧)) → ∃𝑎 ∈ 𝑝 (𝑎 ∈ (𝑢𝑖𝑦) ∧ 𝑎 ∈ (𝑣𝑖𝑥))) ∧ ∀𝑠 ∈ 𝒫 𝑝∀𝑡 ∈ 𝒫 𝑝(∃𝑎 ∈ 𝑝 ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑥 ∈ (𝑎𝑖𝑦) → ∃𝑏 ∈ 𝑝 ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑏 ∈ (𝑥𝑖𝑦)))} |
df-trkgcb 25928 | ⊢ TarskiGCB = {𝑓 ∣ [(Base‘𝑓) / 𝑝][(dist‘𝑓) / 𝑑][(Itv‘𝑓) / 𝑖](∀𝑥 ∈ 𝑝 ∀𝑦 ∈ 𝑝 ∀𝑧 ∈ 𝑝 ∀𝑢 ∈ 𝑝 ∀𝑎 ∈ 𝑝 ∀𝑏 ∈ 𝑝 ∀𝑐 ∈ 𝑝 ∀𝑣 ∈ 𝑝 (((𝑥 ≠ 𝑦 ∧ 𝑦 ∈ (𝑥𝑖𝑧) ∧ 𝑏 ∈ (𝑎𝑖𝑐)) ∧ (((𝑥𝑑𝑦) = (𝑎𝑑𝑏) ∧ (𝑦𝑑𝑧) = (𝑏𝑑𝑐)) ∧ ((𝑥𝑑𝑢) = (𝑎𝑑𝑣) ∧ (𝑦𝑑𝑢) = (𝑏𝑑𝑣)))) → (𝑧𝑑𝑢) = (𝑐𝑑𝑣)) ∧ ∀𝑥 ∈ 𝑝 ∀𝑦 ∈ 𝑝 ∀𝑎 ∈ 𝑝 ∀𝑏 ∈ 𝑝 ∃𝑧 ∈ 𝑝 (𝑦 ∈ (𝑥𝑖𝑧) ∧ (𝑦𝑑𝑧) = (𝑎𝑑𝑏)))} |
df-trkge 25929 | ⊢ TarskiGE = {𝑓 ∣ [(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖]∀𝑥 ∈ 𝑝 ∀𝑦 ∈ 𝑝 ∀𝑧 ∈ 𝑝 ∀𝑢 ∈ 𝑝 ∀𝑣 ∈ 𝑝 ((𝑢 ∈ (𝑥𝑖𝑣) ∧ 𝑢 ∈ (𝑦𝑖𝑧) ∧ 𝑥 ≠ 𝑢) → ∃𝑎 ∈ 𝑝 ∃𝑏 ∈ 𝑝 (𝑦 ∈ (𝑥𝑖𝑎) ∧ 𝑧 ∈ (𝑥𝑖𝑏) ∧ 𝑣 ∈ (𝑎𝑖𝑏)))} |
df-trkgld 25930 | ⊢ DimTarskiG≥ = {〈𝑔, 𝑛〉 ∣ [(Base‘𝑔) / 𝑝][(dist‘𝑔) / 𝑑][(Itv‘𝑔) / 𝑖]∃𝑓(𝑓:(1..^𝑛)–1-1→𝑝 ∧ ∃𝑥 ∈ 𝑝 ∃𝑦 ∈ 𝑝 ∃𝑧 ∈ 𝑝 (∀𝑗 ∈ (2..^𝑛)(((𝑓‘1)𝑑𝑥) = ((𝑓‘𝑗)𝑑𝑥) ∧ ((𝑓‘1)𝑑𝑦) = ((𝑓‘𝑗)𝑑𝑦) ∧ ((𝑓‘1)𝑑𝑧) = ((𝑓‘𝑗)𝑑𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))))} |
df-trkg 25931 | ⊢ TarskiG = ((TarskiGC ∩
TarskiGB) ∩ (TarskiGCB ∩ {𝑓 ∣ [(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥 ∈ 𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})})) |
ccgrg 25988 | class cgrG |
df-cgrg 25989 | ⊢ cgrG = (𝑔 ∈ V ↦ {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ ((Base‘𝑔) ↑pm ℝ) ∧
𝑏 ∈ ((Base‘𝑔) ↑pm
ℝ)) ∧ (dom 𝑎 =
dom 𝑏 ∧ ∀𝑖 ∈ dom 𝑎∀𝑗 ∈ dom 𝑎((𝑎‘𝑖)(dist‘𝑔)(𝑎‘𝑗)) = ((𝑏‘𝑖)(dist‘𝑔)(𝑏‘𝑗))))}) |
cismt 26010 | class Ismt |
df-ismt 26011 | ⊢ Ismt = (𝑔 ∈ V, ℎ ∈ V ↦ {𝑓 ∣ (𝑓:(Base‘𝑔)–1-1-onto→(Base‘ℎ) ∧ ∀𝑎 ∈ (Base‘𝑔)∀𝑏 ∈ (Base‘𝑔)((𝑓‘𝑎)(dist‘ℎ)(𝑓‘𝑏)) = (𝑎(dist‘𝑔)𝑏))}) |
cleg 26060 | class ≤G |
df-leg 26061 | ⊢ ≤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 26078 | class hlG |
df-hlg 26079 | ⊢ hlG = (𝑔 ∈ V ↦ (𝑐 ∈ (Base‘𝑔) ↦ {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ (Base‘𝑔) ∧ 𝑏 ∈ (Base‘𝑔)) ∧ (𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐 ∧ (𝑎 ∈ (𝑐(Itv‘𝑔)𝑏) ∨ 𝑏 ∈ (𝑐(Itv‘𝑔)𝑎))))})) |
cmir 26130 | class pInvG |
df-mir 26131 | ⊢ pInvG = (𝑔 ∈ V ↦ (𝑚 ∈ (Base‘𝑔) ↦ (𝑎 ∈ (Base‘𝑔) ↦ (℩𝑏 ∈ (Base‘𝑔)((𝑚(dist‘𝑔)𝑏) = (𝑚(dist‘𝑔)𝑎) ∧ 𝑚 ∈ (𝑏(Itv‘𝑔)𝑎)))))) |
crag 26171 | class ∟G |
df-rag 26172 | ⊢ ∟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 26173 | class ⟂G |
df-perpg 26174 | ⊢ ⟂G = (𝑔 ∈ V ↦ {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ ran (LineG‘𝑔) ∧ 𝑏 ∈ ran (LineG‘𝑔)) ∧ ∃𝑥 ∈ (𝑎 ∩ 𝑏)∀𝑢 ∈ 𝑎 ∀𝑣 ∈ 𝑏 〈“𝑢𝑥𝑣”〉 ∈ (∟G‘𝑔))}) |
chpg 26235 | class hpG |
df-hpg 26236 | ⊢ hpG = (𝑔 ∈ V ↦ (𝑑 ∈ ran (LineG‘𝑔) ↦ {〈𝑎, 𝑏〉 ∣ [(Base‘𝑔) / 𝑝][(Itv‘𝑔) / 𝑖]∃𝑐 ∈ 𝑝 (((𝑎 ∈ (𝑝 ∖ 𝑑) ∧ 𝑐 ∈ (𝑝 ∖ 𝑑)) ∧ ∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑎𝑖𝑐)) ∧ ((𝑏 ∈ (𝑝 ∖ 𝑑) ∧ 𝑐 ∈ (𝑝 ∖ 𝑑)) ∧ ∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑏𝑖𝑐)))})) |
cmid 26250 | class midG |
clmi 26251 | class lInvG |
df-mid 26252 | ⊢ midG = (𝑔 ∈ V ↦ (𝑎 ∈ (Base‘𝑔), 𝑏 ∈ (Base‘𝑔) ↦ (℩𝑚 ∈ (Base‘𝑔)𝑏 = (((pInvG‘𝑔)‘𝑚)‘𝑎)))) |
df-lmi 26253 | ⊢ lInvG = (𝑔 ∈ V ↦ (𝑚 ∈ ran (LineG‘𝑔) ↦ (𝑎 ∈ (Base‘𝑔) ↦ (℩𝑏 ∈ (Base‘𝑔)((𝑎(midG‘𝑔)𝑏) ∈ 𝑚 ∧ (𝑚(⟂G‘𝑔)(𝑎(LineG‘𝑔)𝑏) ∨ 𝑎 = 𝑏)))))) |
ccgra 26285 | class cgrA |
df-cgra 26286 | ⊢ cgrA = (𝑔 ∈ V ↦ {〈𝑎, 𝑏〉 ∣ [(Base‘𝑔) / 𝑝][(hlG‘𝑔) / 𝑘]((𝑎 ∈ (𝑝 ↑𝑚 (0..^3)) ∧
𝑏 ∈ (𝑝 ↑𝑚
(0..^3))) ∧ ∃𝑥
∈ 𝑝 ∃𝑦 ∈ 𝑝 (𝑎(cgrG‘𝑔)〈“𝑥(𝑏‘1)𝑦”〉 ∧ 𝑥(𝑘‘(𝑏‘1))(𝑏‘0) ∧ 𝑦(𝑘‘(𝑏‘1))(𝑏‘2)))}) |
cinag 26314 | class inA |
cleag 26315 | class
≤∠ |
df-inag 26316 | ⊢ inA = (𝑔 ∈ V ↦ {〈𝑝, 𝑡〉 ∣ ((𝑝 ∈ (Base‘𝑔) ∧ 𝑡 ∈ ((Base‘𝑔) ↑𝑚 (0..^3))) ∧
(((𝑡‘0) ≠ (𝑡‘1) ∧ (𝑡‘2) ≠ (𝑡‘1) ∧ 𝑝 ≠ (𝑡‘1)) ∧ ∃𝑥 ∈ (Base‘𝑔)(𝑥 ∈ ((𝑡‘0)(Itv‘𝑔)(𝑡‘2)) ∧ (𝑥 = (𝑡‘1) ∨ 𝑥((hlG‘𝑔)‘(𝑡‘1))𝑝))))}) |
df-leag 26325 | ⊢ ≤∠ = (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 26344 | class eqltrG |
df-eqlg 26345 | ⊢ eqltrG = (𝑔 ∈ V ↦ {𝑥 ∈ ((Base‘𝑔) ↑𝑚 (0..^3)) ∣
𝑥(cgrG‘𝑔)〈“(𝑥‘1)(𝑥‘2)(𝑥‘0)”〉}) |
cttg 26352 | class toTG |
df-ttg 26353 | ⊢ toTG = (𝑤 ∈ V ↦ ⦋(𝑥 ∈ (Base‘𝑤), 𝑦 ∈ (Base‘𝑤) ↦ {𝑧 ∈ (Base‘𝑤) ∣ ∃𝑘 ∈ (0[,]1)(𝑧(-g‘𝑤)𝑥) = (𝑘( ·𝑠
‘𝑤)(𝑦(-g‘𝑤)𝑥))}) / 𝑖⦌((𝑤 sSet 〈(Itv‘ndx), 𝑖〉) sSet
〈(LineG‘ndx), (𝑥
∈ (Base‘𝑤),
𝑦 ∈ (Base‘𝑤) ↦ {𝑧 ∈ (Base‘𝑤) ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})〉)) |
cee 26367 | class 𝔼 |
cbtwn 26368 | class Btwn |
ccgr 26369 | class Cgr |
df-ee 26370 | ⊢ 𝔼 = (𝑛 ∈ ℕ ↦ (ℝ
↑𝑚 (1...𝑛))) |
df-btwn 26371 | ⊢ 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 26372 | ⊢ Cgr = {〈𝑥, 𝑦〉 ∣ ∃𝑛 ∈ ℕ ((𝑥 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑦 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛))) ∧ Σ𝑖 ∈ (1...𝑛)((((1st ‘𝑥)‘𝑖) − ((2nd ‘𝑥)‘𝑖))↑2) = Σ𝑖 ∈ (1...𝑛)((((1st ‘𝑦)‘𝑖) − ((2nd ‘𝑦)‘𝑖))↑2))} |
ceeng 26456 | class EEG |
df-eeng 26457 | ⊢ EEG = (𝑛 ∈ ℕ ↦
({〈(Base‘ndx), (𝔼‘𝑛)〉, 〈(dist‘ndx), (𝑥 ∈ (𝔼‘𝑛), 𝑦 ∈ (𝔼‘𝑛) ↦ Σ𝑖 ∈ (1...𝑛)(((𝑥‘𝑖) − (𝑦‘𝑖))↑2))〉} ∪
{〈(Itv‘ndx), (𝑥
∈ (𝔼‘𝑛),
𝑦 ∈
(𝔼‘𝑛) ↦
{𝑧 ∈
(𝔼‘𝑛) ∣
𝑧 Btwn 〈𝑥, 𝑦〉})〉, 〈(LineG‘ndx),
(𝑥 ∈
(𝔼‘𝑛), 𝑦 ∈ ((𝔼‘𝑛) ∖ {𝑥}) ↦ {𝑧 ∈ (𝔼‘𝑛) ∣ (𝑧 Btwn 〈𝑥, 𝑦〉 ∨ 𝑥 Btwn 〈𝑧, 𝑦〉 ∨ 𝑦 Btwn 〈𝑥, 𝑧〉)})〉})) |
cedgf 26467 | class .ef |
df-edgf 26468 | ⊢ .ef = Slot ;18 |
cvtx 26474 | class Vtx |
ciedg 26475 | class iEdg |
df-vtx 26476 | ⊢ Vtx = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st
‘𝑔),
(Base‘𝑔))) |
df-iedg 26477 | ⊢ iEdg = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (2nd
‘𝑔), (.ef‘𝑔))) |
cedg 26525 | class Edg |
df-edg 26526 | ⊢ Edg = (𝑔 ∈ V ↦ ran (iEdg‘𝑔)) |
cuhgr 26534 | class UHGraph |
cushgr 26535 | class USHGraph |
df-uhgr 26536 | ⊢ UHGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶(𝒫 𝑣 ∖ {∅})} |
df-ushgr 26537 | ⊢ USHGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→(𝒫 𝑣 ∖ {∅})} |
cupgr 26558 | class UPGraph |
cumgr 26559 | class UMGraph |
df-upgr 26560 | ⊢ UPGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣
(♯‘𝑥) ≤
2}} |
df-umgr 26561 | ⊢ UMGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣
(♯‘𝑥) =
2}} |
cuspgr 26626 | class USPGraph |
cusgr 26627 | class USGraph |
df-uspgr 26628 | ⊢ USPGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣
(♯‘𝑥) ≤
2}} |
df-usgr 26629 | ⊢ USGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣
(♯‘𝑥) =
2}} |
csubgr 26742 | class SubGraph |
df-subgr 26743 | ⊢ SubGraph = {〈𝑠, 𝑔〉 ∣ ((Vtx‘𝑠) ⊆ (Vtx‘𝑔) ∧ (iEdg‘𝑠) = ((iEdg‘𝑔) ↾ dom (iEdg‘𝑠)) ∧ (Edg‘𝑠) ⊆ 𝒫 (Vtx‘𝑠))} |
cfusgr 26791 | class FinUSGraph |
df-fusgr 26792 | ⊢ FinUSGraph = {𝑔 ∈ USGraph ∣ (Vtx‘𝑔) ∈ Fin} |
cnbgr 26807 | class NeighbVtx |
df-nbgr 26808 | ⊢ NeighbVtx = (𝑔 ∈ V, 𝑣 ∈ (Vtx‘𝑔) ↦ {𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣}) ∣ ∃𝑒 ∈ (Edg‘𝑔){𝑣, 𝑛} ⊆ 𝑒}) |
cuvtx 26860 | class UnivVtx |
df-uvtx 26861 | ⊢ UnivVtx = (𝑔 ∈ V ↦ {𝑣 ∈ (Vtx‘𝑔) ∣ ∀𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣})𝑛 ∈ (𝑔 NeighbVtx 𝑣)}) |
ccplgr 26884 | class ComplGraph |
ccusgr 26885 | class ComplUSGraph |
df-cplgr 26886 | ⊢ ComplGraph = {𝑔 ∣ (UnivVtx‘𝑔) = (Vtx‘𝑔)} |
df-cusgr 26887 | ⊢ ComplUSGraph = (USGraph ∩
ComplGraph) |
cvtxdg 26940 | class VtxDeg |
df-vtxdg 26941 | ⊢ VtxDeg = (𝑔 ∈ V ↦
⦋(Vtx‘𝑔) / 𝑣⦌⦋(iEdg‘𝑔) / 𝑒⦌(𝑢 ∈ 𝑣 ↦ ((♯‘{𝑥 ∈ dom 𝑒 ∣ 𝑢 ∈ (𝑒‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom 𝑒 ∣ (𝑒‘𝑥) = {𝑢}})))) |
crgr 27030 | class RegGraph |
crusgr 27031 | class RegUSGraph |
df-rgr 27032 | ⊢ RegGraph = {〈𝑔, 𝑘〉 ∣ (𝑘 ∈ ℕ0* ∧
∀𝑣 ∈
(Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 𝑘)} |
df-rusgr 27033 | ⊢ RegUSGraph = {〈𝑔, 𝑘〉 ∣ (𝑔 ∈ USGraph ∧ 𝑔RegGraph𝑘)} |
cewlks 27070 | class EdgWalks |
cwlks 27071 | class Walks |
cwlkson 27072 | class WalksOn |
df-ewlks 27073 | ⊢ 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 27074 | ⊢ Walks = (𝑔 ∈ V ↦ {〈𝑓, 𝑝〉 ∣ (𝑓 ∈ Word dom (iEdg‘𝑔) ∧ 𝑝:(0...(♯‘𝑓))⟶(Vtx‘𝑔) ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))if-((𝑝‘𝑘) = (𝑝‘(𝑘 + 1)), ((iEdg‘𝑔)‘(𝑓‘𝑘)) = {(𝑝‘𝑘)}, {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))} ⊆ ((iEdg‘𝑔)‘(𝑓‘𝑘))))}) |
df-wlkson 27075 | ⊢ WalksOn = (𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {〈𝑓, 𝑝〉 ∣ (𝑓(Walks‘𝑔)𝑝 ∧ (𝑝‘0) = 𝑎 ∧ (𝑝‘(♯‘𝑓)) = 𝑏)})) |
ctrls 27168 | class Trails |
ctrlson 27169 | class TrailsOn |
df-trls 27170 | ⊢ Trails = (𝑔 ∈ V ↦ {〈𝑓, 𝑝〉 ∣ (𝑓(Walks‘𝑔)𝑝 ∧ Fun ◡𝑓)}) |
df-trlson 27171 | ⊢ TrailsOn = (𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {〈𝑓, 𝑝〉 ∣ (𝑓(𝑎(WalksOn‘𝑔)𝑏)𝑝 ∧ 𝑓(Trails‘𝑔)𝑝)})) |
cpths 27191 | class Paths |
cspths 27192 | class SPaths |
cpthson 27193 | class PathsOn |
cspthson 27194 | class SPathsOn |
df-pths 27195 | ⊢ Paths = (𝑔 ∈ V ↦ {〈𝑓, 𝑝〉 ∣ (𝑓(Trails‘𝑔)𝑝 ∧ Fun ◡(𝑝 ↾ (1..^(♯‘𝑓))) ∧ ((𝑝 “ {0, (♯‘𝑓)}) ∩ (𝑝 “ (1..^(♯‘𝑓)))) =
∅)}) |
df-spths 27196 | ⊢ SPaths = (𝑔 ∈ V ↦ {〈𝑓, 𝑝〉 ∣ (𝑓(Trails‘𝑔)𝑝 ∧ Fun ◡𝑝)}) |
df-pthson 27197 | ⊢ PathsOn = (𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {〈𝑓, 𝑝〉 ∣ (𝑓(𝑎(TrailsOn‘𝑔)𝑏)𝑝 ∧ 𝑓(Paths‘𝑔)𝑝)})) |
df-spthson 27198 | ⊢ SPathsOn = (𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {〈𝑓, 𝑝〉 ∣ (𝑓(𝑎(TrailsOn‘𝑔)𝑏)𝑝 ∧ 𝑓(SPaths‘𝑔)𝑝)})) |
cclwlks 27249 | class ClWalks |
df-clwlks 27250 | ⊢ ClWalks = (𝑔 ∈ V ↦ {〈𝑓, 𝑝〉 ∣ (𝑓(Walks‘𝑔)𝑝 ∧ (𝑝‘0) = (𝑝‘(♯‘𝑓)))}) |
ccrcts 27263 | class Circuits |
ccycls 27264 | class Cycles |
df-crcts 27265 | ⊢ Circuits = (𝑔 ∈ V ↦ {〈𝑓, 𝑝〉 ∣ (𝑓(Trails‘𝑔)𝑝 ∧ (𝑝‘0) = (𝑝‘(♯‘𝑓)))}) |
df-cycls 27266 | ⊢ Cycles = (𝑔 ∈ V ↦ {〈𝑓, 𝑝〉 ∣ (𝑓(Paths‘𝑔)𝑝 ∧ (𝑝‘0) = (𝑝‘(♯‘𝑓)))}) |
cwwlks 27301 | class WWalks |
cwwlksn 27302 | class WWalksN |
cwwlksnon 27303 | class WWalksNOn |
cwwspthsn 27304 | class WSPathsN |
cwwspthsnon 27305 | class WSPathsNOn |
df-wwlks 27306 | ⊢ WWalks = (𝑔 ∈ V ↦ {𝑤 ∈ Word (Vtx‘𝑔) ∣ (𝑤 ≠ ∅ ∧ ∀𝑖 ∈
(0..^((♯‘𝑤)
− 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝑔))}) |
df-wwlksn 27307 | ⊢ WWalksN = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ {𝑤 ∈ (WWalks‘𝑔) ∣ (♯‘𝑤) = (𝑛 + 1)}) |
df-wwlksnon 27308 | ⊢ WWalksNOn = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {𝑤 ∈ (𝑛 WWalksN 𝑔) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤‘𝑛) = 𝑏)})) |
df-wspthsn 27309 | ⊢ WSPathsN = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ {𝑤 ∈ (𝑛 WWalksN 𝑔) ∣ ∃𝑓 𝑓(SPaths‘𝑔)𝑤}) |
df-wspthsnon 27310 | ⊢ 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 27477 | class ClWWalks |
df-clwwlk 27478 | ⊢ ClWWalks = (𝑔 ∈ V ↦ {𝑤 ∈ Word (Vtx‘𝑔) ∣ (𝑤 ≠ ∅ ∧ ∀𝑖 ∈
(0..^((♯‘𝑤)
− 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝑔) ∧ {(lastS‘𝑤), (𝑤‘0)} ∈ (Edg‘𝑔))}) |
cclwwlkn 27529 | class ClWWalksN |
df-clwwlkn 27530 | ⊢ ClWWalksN = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ {𝑤 ∈ (ClWWalks‘𝑔) ∣ (♯‘𝑤) = 𝑛}) |
cclwwlknon 27605 | class ClWWalksNOn |
df-clwwlknon 27606 | ⊢ ClWWalksNOn = (𝑔 ∈ V ↦ (𝑣 ∈ (Vtx‘𝑔), 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝑔) ∣ (𝑤‘0) = 𝑣})) |
cconngr 27705 | class ConnGraph |
df-conngr 27706 | ⊢ ConnGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣]∀𝑘 ∈ 𝑣 ∀𝑛 ∈ 𝑣 ∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝} |
ceupth 27716 | class EulerPaths |
df-eupth 27717 | ⊢ EulerPaths = (𝑔 ∈ V ↦ {〈𝑓, 𝑝〉 ∣ (𝑓(Trails‘𝑔)𝑝 ∧ 𝑓:(0..^(♯‘𝑓))–onto→dom (iEdg‘𝑔))}) |
cfrgr 27780 | class FriendGraph |
df-frgr 27781 | ⊢ FriendGraph = {𝑔 ∣ (𝑔 ∈ USGraph ∧
[(Vtx‘𝑔) /
𝑣][(Edg‘𝑔) / 𝑒]∀𝑘 ∈ 𝑣 ∀𝑙 ∈ (𝑣 ∖ {𝑘})∃!𝑥 ∈ 𝑣 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝑒)} |
cplig 28018 | class Plig |
df-plig 28019 | ⊢ Plig = {𝑥 ∣ (∀𝑎 ∈ ∪ 𝑥∀𝑏 ∈ ∪ 𝑥(𝑎 ≠ 𝑏 → ∃!𝑙 ∈ 𝑥 (𝑎 ∈ 𝑙 ∧ 𝑏 ∈ 𝑙)) ∧ ∀𝑙 ∈ 𝑥 ∃𝑎 ∈ ∪ 𝑥∃𝑏 ∈ ∪ 𝑥(𝑎 ≠ 𝑏 ∧ 𝑎 ∈ 𝑙 ∧ 𝑏 ∈ 𝑙) ∧ ∃𝑎 ∈ ∪ 𝑥∃𝑏 ∈ ∪ 𝑥∃𝑐 ∈ ∪ 𝑥∀𝑙 ∈ 𝑥 ¬ (𝑎 ∈ 𝑙 ∧ 𝑏 ∈ 𝑙 ∧ 𝑐 ∈ 𝑙))} |
cgr 28033 | class GrpOp |
cgi 28034 | class GId |
cgn 28035 | class inv |
cgs 28036 | class
/𝑔 |
df-grpo 28037 | ⊢ GrpOp = {𝑔 ∣ ∃𝑡(𝑔:(𝑡 × 𝑡)⟶𝑡 ∧ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 ∀𝑧 ∈ 𝑡 ((𝑥𝑔𝑦)𝑔𝑧) = (𝑥𝑔(𝑦𝑔𝑧)) ∧ ∃𝑢 ∈ 𝑡 ∀𝑥 ∈ 𝑡 ((𝑢𝑔𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑡 (𝑦𝑔𝑥) = 𝑢))} |
df-gid 28038 | ⊢ GId = (𝑔 ∈ V ↦ (℩𝑢 ∈ ran 𝑔∀𝑥 ∈ ran 𝑔((𝑢𝑔𝑥) = 𝑥 ∧ (𝑥𝑔𝑢) = 𝑥))) |
df-ginv 28039 | ⊢ inv = (𝑔 ∈ GrpOp ↦ (𝑥 ∈ ran 𝑔 ↦ (℩𝑧 ∈ ran 𝑔(𝑧𝑔𝑥) = (GId‘𝑔)))) |
df-gdiv 28040 | ⊢ /𝑔 = (𝑔 ∈ GrpOp ↦ (𝑥 ∈ ran 𝑔, 𝑦 ∈ ran 𝑔 ↦ (𝑥𝑔((inv‘𝑔)‘𝑦)))) |
cablo 28088 | class AbelOp |
df-ablo 28089 | ⊢ AbelOp = {𝑔 ∈ GrpOp ∣ ∀𝑥 ∈ ran 𝑔∀𝑦 ∈ ran 𝑔(𝑥𝑔𝑦) = (𝑦𝑔𝑥)} |
cvc 28102 | class
CVecOLD |
df-vc 28103 | ⊢ CVecOLD =
{〈𝑔, 𝑠〉 ∣ (𝑔 ∈ AbelOp ∧ 𝑠:(ℂ × ran 𝑔)⟶ran 𝑔 ∧ ∀𝑥 ∈ ran 𝑔((1𝑠𝑥) = 𝑥 ∧ ∀𝑦 ∈ ℂ (∀𝑧 ∈ ran 𝑔(𝑦𝑠(𝑥𝑔𝑧)) = ((𝑦𝑠𝑥)𝑔(𝑦𝑠𝑧)) ∧ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑠𝑥) = ((𝑦𝑠𝑥)𝑔(𝑧𝑠𝑥)) ∧ ((𝑦 · 𝑧)𝑠𝑥) = (𝑦𝑠(𝑧𝑠𝑥))))))} |
cnv 28128 | class NrmCVec |
cpv 28129 | class
+𝑣 |
cba 28130 | class BaseSet |
cns 28131 | class
·𝑠OLD |
cn0v 28132 | class 0vec |
cnsb 28133 | class
−𝑣 |
cnmcv 28134 | class
normCV |
cims 28135 | class IndMet |
df-nv 28136 | ⊢ NrmCVec = {〈〈𝑔, 𝑠〉, 𝑛〉 ∣ (〈𝑔, 𝑠〉 ∈ CVecOLD ∧ 𝑛:ran 𝑔⟶ℝ ∧ ∀𝑥 ∈ ran 𝑔(((𝑛‘𝑥) = 0 → 𝑥 = (GId‘𝑔)) ∧ ∀𝑦 ∈ ℂ (𝑛‘(𝑦𝑠𝑥)) = ((abs‘𝑦) · (𝑛‘𝑥)) ∧ ∀𝑦 ∈ ran 𝑔(𝑛‘(𝑥𝑔𝑦)) ≤ ((𝑛‘𝑥) + (𝑛‘𝑦))))} |
df-va 28139 | ⊢ +𝑣 =
(1st ∘ 1st ) |
df-ba 28140 | ⊢ BaseSet = (𝑥 ∈ V ↦ ran ( +𝑣
‘𝑥)) |
df-sm 28141 | ⊢
·𝑠OLD = (2nd ∘ 1st
) |
df-0v 28142 | ⊢ 0vec = (GId ∘
+𝑣 ) |
df-vs 28143 | ⊢ −𝑣 = (
/𝑔 ∘ +𝑣 ) |
df-nmcv 28144 | ⊢ normCV =
2nd |
df-ims 28145 | ⊢ IndMet = (𝑢 ∈ NrmCVec ↦
((normCV‘𝑢) ∘ ( −𝑣
‘𝑢))) |
cdip 28244 | class
·𝑖OLD |
df-dip 28245 | ⊢ ·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 28265 | class SubSp |
df-ssp 28266 | ⊢ SubSp = (𝑢 ∈ NrmCVec ↦ {𝑤 ∈ NrmCVec ∣ ((
+𝑣 ‘𝑤) ⊆ ( +𝑣
‘𝑢) ∧ (
·𝑠OLD ‘𝑤) ⊆ (
·𝑠OLD ‘𝑢) ∧ (normCV‘𝑤) ⊆
(normCV‘𝑢))}) |
clno 28284 | class LnOp |
cnmoo 28285 | class
normOpOLD |
cblo 28286 | class BLnOp |
c0o 28287 | class 0op |
df-lno 28288 | ⊢ 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 28289 | ⊢ 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 28290 | ⊢ BLnOp = (𝑢 ∈ NrmCVec, 𝑤 ∈ NrmCVec ↦ {𝑡 ∈ (𝑢 LnOp 𝑤) ∣ ((𝑢 normOpOLD 𝑤)‘𝑡) < +∞}) |
df-0o 28291 | ⊢ 0op = (𝑢 ∈ NrmCVec, 𝑤 ∈ NrmCVec ↦
((BaseSet‘𝑢) ×
{(0vec‘𝑤)})) |
caj 28292 | class adj |
chmo 28293 | class HmOp |
df-aj 28294 | ⊢ adj = (𝑢 ∈ NrmCVec, 𝑤 ∈ NrmCVec ↦ {〈𝑡, 𝑠〉 ∣ (𝑡:(BaseSet‘𝑢)⟶(BaseSet‘𝑤) ∧ 𝑠:(BaseSet‘𝑤)⟶(BaseSet‘𝑢) ∧ ∀𝑥 ∈ (BaseSet‘𝑢)∀𝑦 ∈ (BaseSet‘𝑤)((𝑡‘𝑥)(·𝑖OLD‘𝑤)𝑦) = (𝑥(·𝑖OLD‘𝑢)(𝑠‘𝑦)))}) |
df-hmo 28295 | ⊢ HmOp = (𝑢 ∈ NrmCVec ↦ {𝑡 ∈ dom (𝑢adj𝑢) ∣ ((𝑢adj𝑢)‘𝑡) = 𝑡}) |
ccphlo 28356 | class
CPreHilOLD |
df-ph 28357 | ⊢ CPreHilOLD = (NrmCVec
∩ {〈〈𝑔, 𝑠〉, 𝑛〉 ∣ ∀𝑥 ∈ ran 𝑔∀𝑦 ∈ ran 𝑔(((𝑛‘(𝑥𝑔𝑦))↑2) + ((𝑛‘(𝑥𝑔(-1𝑠𝑦)))↑2)) = (2 · (((𝑛‘𝑥)↑2) + ((𝑛‘𝑦)↑2)))}) |
ccbn 28407 | class CBan |
df-cbn 28408 | ⊢ CBan = {𝑢 ∈ NrmCVec ∣ (IndMet‘𝑢) ∈
(CMet‘(BaseSet‘𝑢))} |
chlo 28430 | class
CHilOLD |
df-hlo 28431 | ⊢ CHilOLD = (CBan ∩
CPreHilOLD) |
The
list of syntax, axioms (ax-) and definitions (df-) for the Hilbert Space Explorer starts here |
chba 28465 | class ℋ |
cva 28466 | class
+ℎ |
csm 28467 | class
·ℎ |
csp 28468 | class
·ih |
cno 28469 | class
normℎ |
c0v 28470 | class
0ℎ |
cmv 28471 | class
−ℎ |
ccauold 28472 | class Cauchy |
chli 28473 | class
⇝𝑣 |
csh 28474 | class
Sℋ |
cch 28475 | class
Cℋ |
cort 28476 | class ⊥ |
cph 28477 | class
+ℋ |
cspn 28478 | class span |
chj 28479 | class
∨ℋ |
chsup 28480 | class ∨ℋ |
c0h 28481 | class
0ℋ |
ccm 28482 | class
𝐶ℋ |
cpjh 28483 | class
projℎ |
chos 28484 | class +op |
chot 28485 | class
·op |
chod 28486 | class
−op |
chfs 28487 | class +fn |
chft 28488 | class
·fn |
ch0o 28489 | class
0hop |
chio 28490 | class Iop |
cnop 28491 | class
normop |
ccop 28492 | class ContOp |
clo 28493 | class LinOp |
cbo 28494 | class BndLinOp |
cuo 28495 | class UniOp |
cho 28496 | class HrmOp |
cnmf 28497 | class
normfn |
cnl 28498 | class null |
ccnfn 28499 | class ContFn |
clf 28500 | class LinFn |
cado 28501 | class
adjℎ |
cbr 28502 | class bra |
ck 28503 | class ketbra |
cleo 28504 | class
≤op |
cei 28505 | class eigvec |
cel 28506 | class eigval |
cspc 28507 | class Lambda |
cst 28508 | class States |
chst 28509 | class CHStates |
ccv 28510 | class
⋖ℋ |
cat 28511 | class HAtoms |
cmd 28512 | class
𝑀ℋ |
cdmd 28513 | class
𝑀ℋ* |
df-hnorm 28514 | ⊢ normℎ = (𝑥 ∈ dom dom
·ih ↦ (√‘(𝑥 ·ih 𝑥))) |
df-hba 28515 | ⊢ ℋ = (BaseSet‘〈〈
+ℎ , ·ℎ 〉,
normℎ〉) |
df-h0v 28516 | ⊢ 0ℎ =
(0vec‘〈〈 +ℎ ,
·ℎ 〉,
normℎ〉) |
df-hvsub 28517 | ⊢ −ℎ = (𝑥 ∈ ℋ, 𝑦 ∈ ℋ ↦ (𝑥 +ℎ (-1
·ℎ 𝑦))) |
df-hlim 28518 | ⊢ ⇝𝑣 = {〈𝑓, 𝑤〉 ∣ ((𝑓:ℕ⟶ ℋ ∧ 𝑤 ∈ ℋ) ∧
∀𝑥 ∈
ℝ+ ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ≥‘𝑦)(normℎ‘((𝑓‘𝑧) −ℎ 𝑤)) < 𝑥)} |
df-hcau 28519 | ⊢ Cauchy = {𝑓 ∈ ( ℋ ↑𝑚
ℕ) ∣ ∀𝑥
∈ ℝ+ ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ≥‘𝑦)(normℎ‘((𝑓‘𝑦) −ℎ (𝑓‘𝑧))) < 𝑥} |
ax-hilex 28545 | ⊢ ℋ ∈ V |
ax-hfvadd 28546 | ⊢ +ℎ :( ℋ ×
ℋ)⟶ ℋ |
ax-hvcom 28547 | ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 +ℎ 𝐵) = (𝐵 +ℎ 𝐴)) |
ax-hvass 28548 | ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 +ℎ 𝐵) +ℎ 𝐶) = (𝐴 +ℎ (𝐵 +ℎ 𝐶))) |
ax-hv0cl 28549 | ⊢ 0ℎ ∈
ℋ |
ax-hvaddid 28550 | ⊢ (𝐴 ∈ ℋ → (𝐴 +ℎ 0ℎ)
= 𝐴) |
ax-hfvmul 28551 | ⊢ ·ℎ :(ℂ
× ℋ)⟶ ℋ |
ax-hvmulid 28552 | ⊢ (𝐴 ∈ ℋ → (1
·ℎ 𝐴) = 𝐴) |
ax-hvmulass 28553 | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝐴 · 𝐵) ·ℎ 𝐶) = (𝐴 ·ℎ (𝐵
·ℎ 𝐶))) |
ax-hvdistr1 28554 | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 ·ℎ (𝐵 +ℎ 𝐶)) = ((𝐴 ·ℎ 𝐵) +ℎ (𝐴
·ℎ 𝐶))) |
ax-hvdistr2 28555 | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝐴 + 𝐵) ·ℎ 𝐶) = ((𝐴 ·ℎ 𝐶) +ℎ (𝐵
·ℎ 𝐶))) |
ax-hvmul0 28556 | ⊢ (𝐴 ∈ ℋ → (0
·ℎ 𝐴) = 0ℎ) |
ax-hfi 28625 | ⊢ ·ih :(
ℋ × ℋ)⟶ℂ |
ax-his1 28628 | ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ih 𝐵) = (∗‘(𝐵
·ih 𝐴))) |
ax-his2 28629 | ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 +ℎ 𝐵) ·ih 𝐶) = ((𝐴 ·ih 𝐶) + (𝐵 ·ih 𝐶))) |
ax-his3 28630 | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 ·ℎ 𝐵)
·ih 𝐶) = (𝐴 · (𝐵 ·ih 𝐶))) |
ax-his4 28631 | ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → 0 <
(𝐴
·ih 𝐴)) |
ax-hcompl 28748 | ⊢ (𝐹 ∈ Cauchy → ∃𝑥 ∈ ℋ 𝐹 ⇝𝑣
𝑥) |
df-sh 28753 | ⊢
Sℋ = {ℎ ∈ 𝒫 ℋ ∣
(0ℎ ∈ ℎ ∧ ( +ℎ “ (ℎ × ℎ)) ⊆ ℎ ∧ ( ·ℎ
“ (ℂ × ℎ))
⊆ ℎ)} |
df-ch 28767 | ⊢
Cℋ = {ℎ ∈ Sℋ
∣ ( ⇝𝑣 “ (ℎ ↑𝑚 ℕ)) ⊆
ℎ} |
df-oc 28798 | ⊢ ⊥ = (𝑥 ∈ 𝒫 ℋ ↦ {𝑦 ∈ ℋ ∣
∀𝑧 ∈ 𝑥 (𝑦 ·ih 𝑧) = 0}) |
df-ch0 28799 | ⊢ 0ℋ =
{0ℎ} |
df-shs 28856 | ⊢ +ℋ = (𝑥 ∈ Sℋ ,
𝑦 ∈
Sℋ ↦ ( +ℎ “ (𝑥 × 𝑦))) |
df-span 28857 | ⊢ span = (𝑥 ∈ 𝒫 ℋ ↦ ∩ {𝑦
∈ Sℋ ∣ 𝑥 ⊆ 𝑦}) |
df-chj 28858 | ⊢ ∨ℋ = (𝑥 ∈ 𝒫 ℋ, 𝑦 ∈ 𝒫 ℋ ↦
(⊥‘(⊥‘(𝑥 ∪ 𝑦)))) |
df-chsup 28859 | ⊢ ∨ℋ = (𝑥 ∈ 𝒫 𝒫 ℋ ↦
(⊥‘(⊥‘∪ 𝑥))) |
df-pjh 28943 | ⊢ projℎ = (ℎ ∈ Cℋ
↦ (𝑥 ∈ ℋ
↦ (℩𝑧
∈ ℎ ∃𝑦 ∈ (⊥‘ℎ)𝑥 = (𝑧 +ℎ 𝑦)))) |
df-cm 29131 | ⊢ 𝐶ℋ =
{〈𝑥, 𝑦〉 ∣ ((𝑥 ∈
Cℋ ∧ 𝑦 ∈ Cℋ )
∧ 𝑥 = ((𝑥 ∩ 𝑦) ∨ℋ (𝑥 ∩ (⊥‘𝑦))))} |
df-hosum 29278 | ⊢ +op = (𝑓 ∈ ( ℋ ↑𝑚
ℋ), 𝑔 ∈ (
ℋ ↑𝑚 ℋ) ↦ (𝑥 ∈ ℋ ↦ ((𝑓‘𝑥) +ℎ (𝑔‘𝑥)))) |
df-homul 29279 | ⊢ ·op = (𝑓 ∈ ℂ, 𝑔 ∈ ( ℋ
↑𝑚 ℋ) ↦ (𝑥 ∈ ℋ ↦ (𝑓 ·ℎ (𝑔‘𝑥)))) |
df-hodif 29280 | ⊢ −op = (𝑓 ∈ ( ℋ ↑𝑚
ℋ), 𝑔 ∈ (
ℋ ↑𝑚 ℋ) ↦ (𝑥 ∈ ℋ ↦ ((𝑓‘𝑥) −ℎ (𝑔‘𝑥)))) |
df-hfsum 29281 | ⊢ +fn = (𝑓 ∈ (ℂ ↑𝑚
ℋ), 𝑔 ∈ (ℂ
↑𝑚 ℋ) ↦ (𝑥 ∈ ℋ ↦ ((𝑓‘𝑥) + (𝑔‘𝑥)))) |
df-hfmul 29282 | ⊢ ·fn = (𝑓 ∈ ℂ, 𝑔 ∈ (ℂ
↑𝑚 ℋ) ↦ (𝑥 ∈ ℋ ↦ (𝑓 · (𝑔‘𝑥)))) |
df-h0op 29296 | ⊢ 0hop =
(projℎ‘0ℋ) |
df-iop 29297 | ⊢ Iop =
(projℎ‘ ℋ) |
df-nmop 29387 | ⊢ normop = (𝑡 ∈ ( ℋ ↑𝑚
ℋ) ↦ sup({𝑥
∣ ∃𝑧 ∈
ℋ ((normℎ‘𝑧) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑡‘𝑧)))}, ℝ*, <
)) |
df-cnop 29388 | ⊢ ContOp = {𝑡 ∈ ( ℋ ↑𝑚
ℋ) ∣ ∀𝑥
∈ ℋ ∀𝑦
∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ ℋ
((normℎ‘(𝑤 −ℎ 𝑥)) < 𝑧 →
(normℎ‘((𝑡‘𝑤) −ℎ (𝑡‘𝑥))) < 𝑦)} |
df-lnop 29389 | ⊢ LinOp = {𝑡 ∈ ( ℋ ↑𝑚
ℋ) ∣ ∀𝑥
∈ ℂ ∀𝑦
∈ ℋ ∀𝑧
∈ ℋ (𝑡‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑥 ·ℎ (𝑡‘𝑦)) +ℎ (𝑡‘𝑧))} |
df-bdop 29390 | ⊢ BndLinOp = {𝑡 ∈ LinOp ∣
(normop‘𝑡)
< +∞} |
df-unop 29391 | ⊢ UniOp = {𝑡 ∣ (𝑡: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑡‘𝑥) ·ih (𝑡‘𝑦)) = (𝑥 ·ih 𝑦))} |
df-hmop 29392 | ⊢ HrmOp = {𝑡 ∈ ( ℋ ↑𝑚
ℋ) ∣ ∀𝑥
∈ ℋ ∀𝑦
∈ ℋ (𝑥
·ih (𝑡‘𝑦)) = ((𝑡‘𝑥) ·ih 𝑦)} |
df-nmfn 29393 | ⊢ normfn = (𝑡 ∈ (ℂ ↑𝑚
ℋ) ↦ sup({𝑥
∣ ∃𝑧 ∈
ℋ ((normℎ‘𝑧) ≤ 1 ∧ 𝑥 = (abs‘(𝑡‘𝑧)))}, ℝ*, <
)) |
df-nlfn 29394 | ⊢ null = (𝑡 ∈ (ℂ ↑𝑚
ℋ) ↦ (◡𝑡 “ {0})) |
df-cnfn 29395 | ⊢ ContFn = {𝑡 ∈ (ℂ ↑𝑚
ℋ) ∣ ∀𝑥
∈ ℋ ∀𝑦
∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ ℋ
((normℎ‘(𝑤 −ℎ 𝑥)) < 𝑧 → (abs‘((𝑡‘𝑤) − (𝑡‘𝑥))) < 𝑦)} |
df-lnfn 29396 | ⊢ LinFn = {𝑡 ∈ (ℂ ↑𝑚
ℋ) ∣ ∀𝑥
∈ ℂ ∀𝑦
∈ ℋ ∀𝑧
∈ ℋ (𝑡‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑥 · (𝑡‘𝑦)) + (𝑡‘𝑧))} |
df-adjh 29397 | ⊢ adjℎ = {〈𝑡, 𝑢〉 ∣ (𝑡: ℋ⟶ ℋ ∧ 𝑢: ℋ⟶ ℋ ∧
∀𝑥 ∈ ℋ
∀𝑦 ∈ ℋ
((𝑡‘𝑥)
·ih 𝑦) = (𝑥 ·ih (𝑢‘𝑦)))} |
df-bra 29398 | ⊢ bra = (𝑥 ∈ ℋ ↦ (𝑦 ∈ ℋ ↦ (𝑦 ·ih 𝑥))) |
df-kb 29399 | ⊢ ketbra = (𝑥 ∈ ℋ, 𝑦 ∈ ℋ ↦ (𝑧 ∈ ℋ ↦ ((𝑧 ·ih 𝑦)
·ℎ 𝑥))) |
df-leop 29400 | ⊢ ≤op = {〈𝑡, 𝑢〉 ∣ ((𝑢 −op 𝑡) ∈ HrmOp ∧ ∀𝑥 ∈ ℋ 0 ≤ (((𝑢 −op 𝑡)‘𝑥) ·ih 𝑥))} |
df-eigvec 29401 | ⊢ eigvec = (𝑡 ∈ ( ℋ ↑𝑚
ℋ) ↦ {𝑥 ∈
( ℋ ∖ 0ℋ) ∣ ∃𝑧 ∈ ℂ (𝑡‘𝑥) = (𝑧 ·ℎ 𝑥)}) |
df-eigval 29402 | ⊢ eigval = (𝑡 ∈ ( ℋ ↑𝑚
ℋ) ↦ (𝑥 ∈
(eigvec‘𝑡) ↦
(((𝑡‘𝑥)
·ih 𝑥) / ((normℎ‘𝑥)↑2)))) |
df-spec 29403 | ⊢ Lambda = (𝑡 ∈ ( ℋ ↑𝑚
ℋ) ↦ {𝑥 ∈
ℂ ∣ ¬ (𝑡
−op (𝑥
·op ( I ↾ ℋ))): ℋ–1-1→ ℋ}) |
df-st 29759 | ⊢ States = {𝑓 ∈ ((0[,]1) ↑𝑚
Cℋ ) ∣ ((𝑓‘ ℋ) = 1 ∧ ∀𝑥 ∈
Cℋ ∀𝑦 ∈ Cℋ
(𝑥 ⊆
(⊥‘𝑦) →
(𝑓‘(𝑥 ∨ℋ 𝑦)) = ((𝑓‘𝑥) + (𝑓‘𝑦))))} |
df-hst 29760 | ⊢ CHStates = {𝑓 ∈ ( ℋ ↑𝑚
Cℋ ) ∣
((normℎ‘(𝑓‘ ℋ)) = 1 ∧ ∀𝑥 ∈
Cℋ ∀𝑦 ∈ Cℋ
(𝑥 ⊆
(⊥‘𝑦) →
(((𝑓‘𝑥)
·ih (𝑓‘𝑦)) = 0 ∧ (𝑓‘(𝑥 ∨ℋ 𝑦)) = ((𝑓‘𝑥) +ℎ (𝑓‘𝑦)))))} |
df-cv 29827 | ⊢ ⋖ℋ =
{〈𝑥, 𝑦〉 ∣ ((𝑥 ∈
Cℋ ∧ 𝑦 ∈ Cℋ )
∧ (𝑥 ⊊ 𝑦 ∧ ¬ ∃𝑧 ∈
Cℋ (𝑥 ⊊ 𝑧 ∧ 𝑧 ⊊ 𝑦)))} |
df-md 29828 | ⊢ 𝑀ℋ =
{〈𝑥, 𝑦〉 ∣ ((𝑥 ∈
Cℋ ∧ 𝑦 ∈ Cℋ )
∧ ∀𝑧 ∈
Cℋ (𝑧 ⊆ 𝑦 → ((𝑧 ∨ℋ 𝑥) ∩ 𝑦) = (𝑧 ∨ℋ (𝑥 ∩ 𝑦))))} |
df-dmd 29829 | ⊢ 𝑀ℋ* =
{〈𝑥, 𝑦〉 ∣ ((𝑥 ∈
Cℋ ∧ 𝑦 ∈ Cℋ )
∧ ∀𝑧 ∈
Cℋ (𝑦 ⊆ 𝑧 → ((𝑧 ∩ 𝑥) ∨ℋ 𝑦) = (𝑧 ∩ (𝑥 ∨ℋ 𝑦))))} |
df-at 29886 | ⊢ HAtoms = {𝑥 ∈ Cℋ
∣ 0ℋ ⋖ℋ 𝑥} |
The
list of syntax, axioms (ax-) and definitions (df-) for the User Mathboxes starts here |
w2reu 30013 | wff ∃!𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵𝜑 |
df-2reu 30014 | ⊢ (∃!𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵𝜑 ↔ (∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ ∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑)) |
cdp2 30282 | class _𝐴𝐵 |
df-dp2 30283 | ⊢ _𝐴𝐵 = (𝐴 + (𝐵 / ;10)) |
cdp 30299 | class . |
df-dp 30300 | ⊢ . = (𝑥 ∈ ℕ0, 𝑦 ∈ ℝ ↦ _𝑥𝑦) |
cxdiv 30328 | class
/𝑒 |
df-xdiv 30329 | ⊢ /𝑒 = (𝑥 ∈ ℝ*, 𝑦 ∈ (ℝ ∖ {0})
↦ (℩𝑧
∈ ℝ* (𝑦 ·e 𝑧) = 𝑥)) |
ax-xrssca 30370 | ⊢ ℝfld =
(Scalar‘ℝ*𝑠) |
ax-xrsvsca 30371 | ⊢ ·e = (
·𝑠
‘ℝ*𝑠) |
comnd 30394 | class oMnd |
cogrp 30395 | class oGrp |
df-omnd 30396 | ⊢ oMnd = {𝑔 ∈ Mnd ∣ [(Base‘𝑔) / 𝑣][(+g‘𝑔) / 𝑝][(le‘𝑔) / 𝑙](𝑔 ∈ Toset ∧ ∀𝑎 ∈ 𝑣 ∀𝑏 ∈ 𝑣 ∀𝑐 ∈ 𝑣 (𝑎𝑙𝑏 → (𝑎𝑝𝑐)𝑙(𝑏𝑝𝑐)))} |
df-ogrp 30397 | ⊢ oGrp = (Grp ∩ oMnd) |
csgns 30422 | class sgns |
df-sgns 30423 | ⊢ sgns = (𝑟 ∈ V ↦ (𝑥 ∈ (Base‘𝑟) ↦ if(𝑥 = (0g‘𝑟), 0, if((0g‘𝑟)(lt‘𝑟)𝑥, 1, -1)))) |
cinftm 30427 | class ⋘ |
carchi 30428 | class Archi |
df-inftm 30429 | ⊢ ⋘ = (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 30430 | ⊢ Archi = {𝑤 ∣ (⋘‘𝑤) = ∅} |
cslmd 30450 | class SLMod |
df-slmd 30451 | ⊢ SLMod = {𝑔 ∈ CMnd ∣
[(Base‘𝑔) /
𝑣][(+g‘𝑔) / 𝑎][(
·𝑠 ‘𝑔) / 𝑠][(Scalar‘𝑔) / 𝑓][(Base‘𝑓) / 𝑘][(+g‘𝑓) / 𝑝][(.r‘𝑓) / 𝑡](𝑓 ∈ SRing ∧ ∀𝑞 ∈ 𝑘 ∀𝑟 ∈ 𝑘 ∀𝑥 ∈ 𝑣 ∀𝑤 ∈ 𝑣 (((𝑟𝑠𝑤) ∈ 𝑣 ∧ (𝑟𝑠(𝑤𝑎𝑥)) = ((𝑟𝑠𝑤)𝑎(𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤)𝑎(𝑟𝑠𝑤))) ∧ (((𝑞𝑡𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠 |