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Mirrors > Home > MPE Home > Th. List > wlkp1lem3 | Structured version Visualization version GIF version |
Description: Lemma for wlkp1 26982. (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 6435 | . . 3 ⊢ (𝜑 → (𝐻‘𝑁) = ((𝐹 ∪ {〈𝑁, 𝐵〉})‘𝑁)) |
5 | wlkp1.n | . . . . 5 ⊢ 𝑁 = (♯‘𝐹) | |
6 | 5 | fvexi 6447 | . . . 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 26912 | . . . . 5 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝐹 ∈ Word dom 𝐼) |
11 | lencl 13593 | . . . . . 6 ⊢ (𝐹 ∈ Word dom 𝐼 → (♯‘𝐹) ∈ ℕ0) | |
12 | wrddm 13581 | . . . . . 6 ⊢ (𝐹 ∈ Word dom 𝐼 → dom 𝐹 = (0..^(♯‘𝐹))) | |
13 | fzonel 12778 | . . . . . . 7 ⊢ ¬ (♯‘𝐹) ∈ (0..^(♯‘𝐹)) | |
14 | 5 | a1i 11 | . . . . . . . 8 ⊢ (((♯‘𝐹) ∈ ℕ0 ∧ dom 𝐹 = (0..^(♯‘𝐹))) → 𝑁 = (♯‘𝐹)) |
15 | simpr 479 | . . . . . . . 8 ⊢ (((♯‘𝐹) ∈ ℕ0 ∧ dom 𝐹 = (0..^(♯‘𝐹))) → dom 𝐹 = (0..^(♯‘𝐹))) | |
16 | 14, 15 | eleq12d 2900 | . . . . . . 7 ⊢ (((♯‘𝐹) ∈ ℕ0 ∧ dom 𝐹 = (0..^(♯‘𝐹))) → (𝑁 ∈ dom 𝐹 ↔ (♯‘𝐹) ∈ (0..^(♯‘𝐹)))) |
17 | 13, 16 | mtbiri 319 | . . . . . 6 ⊢ (((♯‘𝐹) ∈ ℕ0 ∧ dom 𝐹 = (0..^(♯‘𝐹))) → ¬ 𝑁 ∈ dom 𝐹) |
18 | 11, 12, 17 | syl2anc 581 | . . . . 5 ⊢ (𝐹 ∈ Word dom 𝐼 → ¬ 𝑁 ∈ dom 𝐹) |
19 | 8, 10, 18 | 3syl 18 | . . . 4 ⊢ (𝜑 → ¬ 𝑁 ∈ dom 𝐹) |
20 | fsnunfv 6709 | . . . 4 ⊢ ((𝑁 ∈ V ∧ 𝐵 ∈ V ∧ ¬ 𝑁 ∈ dom 𝐹) → ((𝐹 ∪ {〈𝑁, 𝐵〉})‘𝑁) = 𝐵) | |
21 | 6, 7, 19, 20 | mp3an2i 1596 | . . 3 ⊢ (𝜑 → ((𝐹 ∪ {〈𝑁, 𝐵〉})‘𝑁) = 𝐵) |
22 | 4, 21 | eqtrd 2861 | . 2 ⊢ (𝜑 → (𝐻‘𝑁) = 𝐵) |
23 | 1, 22 | fveq12d 6440 | 1 ⊢ (𝜑 → ((iEdg‘𝑆)‘(𝐻‘𝑁)) = ((𝐼 ∪ {〈𝐵, 𝐸〉})‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 386 = wceq 1658 ∈ wcel 2166 Vcvv 3414 ∪ cun 3796 ⊆ wss 3798 {csn 4397 {cpr 4399 〈cop 4403 class class class wbr 4873 dom cdm 5342 Fun wfun 6117 ‘cfv 6123 (class class class)co 6905 Fincfn 8222 0cc0 10252 ℕ0cn0 11618 ..^cfzo 12760 ♯chash 13410 Word cword 13574 Vtxcvtx 26294 iEdgciedg 26295 Edgcedg 26345 Walkscwlks 26894 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-ifp 1092 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-int 4698 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-1st 7428 df-2nd 7429 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-1o 7826 df-oadd 7830 df-er 8009 df-map 8124 df-pm 8125 df-en 8223 df-dom 8224 df-sdom 8225 df-fin 8226 df-card 9078 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-nn 11351 df-n0 11619 df-z 11705 df-uz 11969 df-fz 12620 df-fzo 12761 df-hash 13411 df-word 13575 df-wlks 26897 |
This theorem is referenced by: wlkp1lem7 26980 wlkp1lem8 26981 |
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