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Mirrors > Home > MPE Home > Th. List > wlkp1lem3 | Structured version Visualization version GIF version |
Description: Lemma for wlkp1 27390. (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 ∈ V) |
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〉}) |
Ref | Expression |
---|---|
wlkp1lem3 | ⊢ (𝜑 → ((iEdg‘𝑆)‘(𝐻‘𝑁)) = ((𝐼 ∪ {〈𝐵, 𝐸〉})‘𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wlkp1.u | . 2 ⊢ (𝜑 → (iEdg‘𝑆) = (𝐼 ∪ {〈𝐵, 𝐸〉})) | |
2 | wlkp1.h | . . . . 5 ⊢ 𝐻 = (𝐹 ∪ {〈𝑁, 𝐵〉}) | |
3 | 2 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝐻 = (𝐹 ∪ {〈𝑁, 𝐵〉})) |
4 | 3 | fveq1d 6665 | . . 3 ⊢ (𝜑 → (𝐻‘𝑁) = ((𝐹 ∪ {〈𝑁, 𝐵〉})‘𝑁)) |
5 | wlkp1.n | . . . . 5 ⊢ 𝑁 = (♯‘𝐹) | |
6 | 5 | fvexi 6677 | . . . 4 ⊢ 𝑁 ∈ V |
7 | wlkp1.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ V) | |
8 | wlkp1.w | . . . . 5 ⊢ (𝜑 → 𝐹(Walks‘𝐺)𝑃) | |
9 | wlkp1.i | . . . . . 6 ⊢ 𝐼 = (iEdg‘𝐺) | |
10 | 9 | wlkf 27323 | . . . . 5 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝐹 ∈ Word dom 𝐼) |
11 | lencl 13871 | . . . . . 6 ⊢ (𝐹 ∈ Word dom 𝐼 → (♯‘𝐹) ∈ ℕ0) | |
12 | wrddm 13856 | . . . . . 6 ⊢ (𝐹 ∈ Word dom 𝐼 → dom 𝐹 = (0..^(♯‘𝐹))) | |
13 | fzonel 13039 | . . . . . . 7 ⊢ ¬ (♯‘𝐹) ∈ (0..^(♯‘𝐹)) | |
14 | 5 | a1i 11 | . . . . . . . 8 ⊢ (((♯‘𝐹) ∈ ℕ0 ∧ dom 𝐹 = (0..^(♯‘𝐹))) → 𝑁 = (♯‘𝐹)) |
15 | simpr 485 | . . . . . . . 8 ⊢ (((♯‘𝐹) ∈ ℕ0 ∧ dom 𝐹 = (0..^(♯‘𝐹))) → dom 𝐹 = (0..^(♯‘𝐹))) | |
16 | 14, 15 | eleq12d 2904 | . . . . . . 7 ⊢ (((♯‘𝐹) ∈ ℕ0 ∧ dom 𝐹 = (0..^(♯‘𝐹))) → (𝑁 ∈ dom 𝐹 ↔ (♯‘𝐹) ∈ (0..^(♯‘𝐹)))) |
17 | 13, 16 | mtbiri 328 | . . . . . 6 ⊢ (((♯‘𝐹) ∈ ℕ0 ∧ dom 𝐹 = (0..^(♯‘𝐹))) → ¬ 𝑁 ∈ dom 𝐹) |
18 | 11, 12, 17 | syl2anc 584 | . . . . 5 ⊢ (𝐹 ∈ Word dom 𝐼 → ¬ 𝑁 ∈ dom 𝐹) |
19 | 8, 10, 18 | 3syl 18 | . . . 4 ⊢ (𝜑 → ¬ 𝑁 ∈ dom 𝐹) |
20 | fsnunfv 6941 | . . . 4 ⊢ ((𝑁 ∈ V ∧ 𝐵 ∈ V ∧ ¬ 𝑁 ∈ dom 𝐹) → ((𝐹 ∪ {〈𝑁, 𝐵〉})‘𝑁) = 𝐵) | |
21 | 6, 7, 19, 20 | mp3an2i 1457 | . . 3 ⊢ (𝜑 → ((𝐹 ∪ {〈𝑁, 𝐵〉})‘𝑁) = 𝐵) |
22 | 4, 21 | eqtrd 2853 | . 2 ⊢ (𝜑 → (𝐻‘𝑁) = 𝐵) |
23 | 1, 22 | fveq12d 6670 | 1 ⊢ (𝜑 → ((iEdg‘𝑆)‘(𝐻‘𝑁)) = ((𝐼 ∪ {〈𝐵, 𝐸〉})‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 Vcvv 3492 ∪ cun 3931 ⊆ wss 3933 {csn 4557 {cpr 4559 〈cop 4563 class class class wbr 5057 dom cdm 5548 Fun wfun 6342 ‘cfv 6348 (class class class)co 7145 Fincfn 8497 0cc0 10525 ℕ0cn0 11885 ..^cfzo 13021 ♯chash 13678 Word cword 13849 Vtxcvtx 26708 iEdgciedg 26709 Edgcedg 26759 Walkscwlks 27305 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-ifp 1055 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12881 df-fzo 13022 df-hash 13679 df-word 13850 df-wlks 27308 |
This theorem is referenced by: wlkp1lem7 27388 wlkp1lem8 27389 |
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