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Mirrors > Home > MPE Home > Th. List > wlkp1lem7 | Structured version Visualization version GIF version |
Description: Lemma for wlkp1 27471. (Contributed by AV, 6-Mar-2021.) |
Ref | Expression |
---|---|
wlkp1.v | ⊢ 𝑉 = (Vtx‘𝐺) |
wlkp1.i | ⊢ 𝐼 = (iEdg‘𝐺) |
wlkp1.f | ⊢ (𝜑 → Fun 𝐼) |
wlkp1.a | ⊢ (𝜑 → 𝐼 ∈ Fin) |
wlkp1.b | ⊢ (𝜑 → 𝐵 ∈ 𝑊) |
wlkp1.c | ⊢ (𝜑 → 𝐶 ∈ 𝑉) |
wlkp1.d | ⊢ (𝜑 → ¬ 𝐵 ∈ dom 𝐼) |
wlkp1.w | ⊢ (𝜑 → 𝐹(Walks‘𝐺)𝑃) |
wlkp1.n | ⊢ 𝑁 = (♯‘𝐹) |
wlkp1.e | ⊢ (𝜑 → 𝐸 ∈ (Edg‘𝐺)) |
wlkp1.x | ⊢ (𝜑 → {(𝑃‘𝑁), 𝐶} ⊆ 𝐸) |
wlkp1.u | ⊢ (𝜑 → (iEdg‘𝑆) = (𝐼 ∪ {〈𝐵, 𝐸〉})) |
wlkp1.h | ⊢ 𝐻 = (𝐹 ∪ {〈𝑁, 𝐵〉}) |
wlkp1.q | ⊢ 𝑄 = (𝑃 ∪ {〈(𝑁 + 1), 𝐶〉}) |
wlkp1.s | ⊢ (𝜑 → (Vtx‘𝑆) = 𝑉) |
Ref | Expression |
---|---|
wlkp1lem7 | ⊢ (𝜑 → {(𝑄‘𝑁), (𝑄‘(𝑁 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻‘𝑁))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wlkp1.x | . . 3 ⊢ (𝜑 → {(𝑃‘𝑁), 𝐶} ⊆ 𝐸) | |
2 | fveq2 6645 | . . . . . 6 ⊢ (𝑘 = 𝑁 → (𝑄‘𝑘) = (𝑄‘𝑁)) | |
3 | fveq2 6645 | . . . . . 6 ⊢ (𝑘 = 𝑁 → (𝑃‘𝑘) = (𝑃‘𝑁)) | |
4 | 2, 3 | eqeq12d 2814 | . . . . 5 ⊢ (𝑘 = 𝑁 → ((𝑄‘𝑘) = (𝑃‘𝑘) ↔ (𝑄‘𝑁) = (𝑃‘𝑁))) |
5 | wlkp1.v | . . . . . 6 ⊢ 𝑉 = (Vtx‘𝐺) | |
6 | wlkp1.i | . . . . . 6 ⊢ 𝐼 = (iEdg‘𝐺) | |
7 | wlkp1.f | . . . . . 6 ⊢ (𝜑 → Fun 𝐼) | |
8 | wlkp1.a | . . . . . 6 ⊢ (𝜑 → 𝐼 ∈ Fin) | |
9 | wlkp1.b | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
10 | wlkp1.c | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ 𝑉) | |
11 | wlkp1.d | . . . . . 6 ⊢ (𝜑 → ¬ 𝐵 ∈ dom 𝐼) | |
12 | wlkp1.w | . . . . . 6 ⊢ (𝜑 → 𝐹(Walks‘𝐺)𝑃) | |
13 | wlkp1.n | . . . . . 6 ⊢ 𝑁 = (♯‘𝐹) | |
14 | wlkp1.e | . . . . . 6 ⊢ (𝜑 → 𝐸 ∈ (Edg‘𝐺)) | |
15 | wlkp1.u | . . . . . 6 ⊢ (𝜑 → (iEdg‘𝑆) = (𝐼 ∪ {〈𝐵, 𝐸〉})) | |
16 | wlkp1.h | . . . . . 6 ⊢ 𝐻 = (𝐹 ∪ {〈𝑁, 𝐵〉}) | |
17 | wlkp1.q | . . . . . 6 ⊢ 𝑄 = (𝑃 ∪ {〈(𝑁 + 1), 𝐶〉}) | |
18 | wlkp1.s | . . . . . 6 ⊢ (𝜑 → (Vtx‘𝑆) = 𝑉) | |
19 | 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 1, 15, 16, 17, 18 | wlkp1lem5 27467 | . . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ (0...𝑁)(𝑄‘𝑘) = (𝑃‘𝑘)) |
20 | wlkcl 27405 | . . . . . 6 ⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝐹) ∈ ℕ0) | |
21 | 13 | eqcomi 2807 | . . . . . . . 8 ⊢ (♯‘𝐹) = 𝑁 |
22 | 21 | eleq1i 2880 | . . . . . . 7 ⊢ ((♯‘𝐹) ∈ ℕ0 ↔ 𝑁 ∈ ℕ0) |
23 | nn0fz0 13000 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ0 ↔ 𝑁 ∈ (0...𝑁)) | |
24 | 22, 23 | sylbb 222 | . . . . . 6 ⊢ ((♯‘𝐹) ∈ ℕ0 → 𝑁 ∈ (0...𝑁)) |
25 | 12, 20, 24 | 3syl 18 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ (0...𝑁)) |
26 | 4, 19, 25 | rspcdva 3573 | . . . 4 ⊢ (𝜑 → (𝑄‘𝑁) = (𝑃‘𝑁)) |
27 | 17 | fveq1i 6646 | . . . . 5 ⊢ (𝑄‘(𝑁 + 1)) = ((𝑃 ∪ {〈(𝑁 + 1), 𝐶〉})‘(𝑁 + 1)) |
28 | ovex 7168 | . . . . . 6 ⊢ (𝑁 + 1) ∈ V | |
29 | 5, 6, 7, 8, 9, 10, 11, 12, 13 | wlkp1lem1 27463 | . . . . . 6 ⊢ (𝜑 → ¬ (𝑁 + 1) ∈ dom 𝑃) |
30 | fsnunfv 6926 | . . . . . 6 ⊢ (((𝑁 + 1) ∈ V ∧ 𝐶 ∈ 𝑉 ∧ ¬ (𝑁 + 1) ∈ dom 𝑃) → ((𝑃 ∪ {〈(𝑁 + 1), 𝐶〉})‘(𝑁 + 1)) = 𝐶) | |
31 | 28, 10, 29, 30 | mp3an2i 1463 | . . . . 5 ⊢ (𝜑 → ((𝑃 ∪ {〈(𝑁 + 1), 𝐶〉})‘(𝑁 + 1)) = 𝐶) |
32 | 27, 31 | syl5eq 2845 | . . . 4 ⊢ (𝜑 → (𝑄‘(𝑁 + 1)) = 𝐶) |
33 | 26, 32 | preq12d 4637 | . . 3 ⊢ (𝜑 → {(𝑄‘𝑁), (𝑄‘(𝑁 + 1))} = {(𝑃‘𝑁), 𝐶}) |
34 | fsnunfv 6926 | . . . 4 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐸 ∈ (Edg‘𝐺) ∧ ¬ 𝐵 ∈ dom 𝐼) → ((𝐼 ∪ {〈𝐵, 𝐸〉})‘𝐵) = 𝐸) | |
35 | 9, 14, 11, 34 | syl3anc 1368 | . . 3 ⊢ (𝜑 → ((𝐼 ∪ {〈𝐵, 𝐸〉})‘𝐵) = 𝐸) |
36 | 1, 33, 35 | 3sstr4d 3962 | . 2 ⊢ (𝜑 → {(𝑄‘𝑁), (𝑄‘(𝑁 + 1))} ⊆ ((𝐼 ∪ {〈𝐵, 𝐸〉})‘𝐵)) |
37 | 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 1, 15, 16 | wlkp1lem3 27465 | . 2 ⊢ (𝜑 → ((iEdg‘𝑆)‘(𝐻‘𝑁)) = ((𝐼 ∪ {〈𝐵, 𝐸〉})‘𝐵)) |
38 | 36, 37 | sseqtrrd 3956 | 1 ⊢ (𝜑 → {(𝑄‘𝑁), (𝑄‘(𝑁 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻‘𝑁))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ∪ cun 3879 ⊆ wss 3881 {csn 4525 {cpr 4527 〈cop 4531 class class class wbr 5030 dom cdm 5519 Fun wfun 6318 ‘cfv 6324 (class class class)co 7135 Fincfn 8492 0cc0 10526 1c1 10527 + caddc 10529 ℕ0cn0 11885 ...cfz 12885 ♯chash 13686 Vtxcvtx 26789 iEdgciedg 26790 Edgcedg 26840 Walkscwlks 27386 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-ifp 1059 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-fzo 13029 df-hash 13687 df-word 13858 df-wlks 27389 |
This theorem is referenced by: wlkp1lem8 27470 |
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