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Mirrors > Home > MPE Home > Th. List > wlkp1lem2 | Structured version Visualization version GIF version |
Description: Lemma for wlkp1 27555. (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〉}) |
Ref | Expression |
---|---|
wlkp1lem2 | ⊢ (𝜑 → (♯‘𝐻) = (𝑁 + 1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wlkp1.h | . . . 4 ⊢ 𝐻 = (𝐹 ∪ {〈𝑁, 𝐵〉}) | |
2 | 1 | fveq2i 6654 | . . 3 ⊢ (♯‘𝐻) = (♯‘(𝐹 ∪ {〈𝑁, 𝐵〉})) |
3 | 2 | a1i 11 | . 2 ⊢ (𝜑 → (♯‘𝐻) = (♯‘(𝐹 ∪ {〈𝑁, 𝐵〉}))) |
4 | opex 5317 | . . 3 ⊢ 〈𝑁, 𝐵〉 ∈ V | |
5 | wlkp1.w | . . . . 5 ⊢ (𝜑 → 𝐹(Walks‘𝐺)𝑃) | |
6 | wlkp1.i | . . . . . 6 ⊢ 𝐼 = (iEdg‘𝐺) | |
7 | 6 | wlkf 27488 | . . . . 5 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝐹 ∈ Word dom 𝐼) |
8 | wrdfin 13916 | . . . . 5 ⊢ (𝐹 ∈ Word dom 𝐼 → 𝐹 ∈ Fin) | |
9 | 5, 7, 8 | 3syl 18 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ Fin) |
10 | wlkp1.n | . . . . . 6 ⊢ 𝑁 = (♯‘𝐹) | |
11 | fzonel 13085 | . . . . . . . 8 ⊢ ¬ (♯‘𝐹) ∈ (0..^(♯‘𝐹)) | |
12 | 11 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → ¬ (♯‘𝐹) ∈ (0..^(♯‘𝐹))) |
13 | eleq1 2838 | . . . . . . . 8 ⊢ (𝑁 = (♯‘𝐹) → (𝑁 ∈ (0..^(♯‘𝐹)) ↔ (♯‘𝐹) ∈ (0..^(♯‘𝐹)))) | |
14 | 13 | notbid 322 | . . . . . . 7 ⊢ (𝑁 = (♯‘𝐹) → (¬ 𝑁 ∈ (0..^(♯‘𝐹)) ↔ ¬ (♯‘𝐹) ∈ (0..^(♯‘𝐹)))) |
15 | 12, 14 | syl5ibr 249 | . . . . . 6 ⊢ (𝑁 = (♯‘𝐹) → (𝜑 → ¬ 𝑁 ∈ (0..^(♯‘𝐹)))) |
16 | 10, 15 | ax-mp 5 | . . . . 5 ⊢ (𝜑 → ¬ 𝑁 ∈ (0..^(♯‘𝐹))) |
17 | wrdfn 13912 | . . . . . . 7 ⊢ (𝐹 ∈ Word dom 𝐼 → 𝐹 Fn (0..^(♯‘𝐹))) | |
18 | 5, 7, 17 | 3syl 18 | . . . . . 6 ⊢ (𝜑 → 𝐹 Fn (0..^(♯‘𝐹))) |
19 | fnop 6435 | . . . . . . 7 ⊢ ((𝐹 Fn (0..^(♯‘𝐹)) ∧ 〈𝑁, 𝐵〉 ∈ 𝐹) → 𝑁 ∈ (0..^(♯‘𝐹))) | |
20 | 19 | ex 417 | . . . . . 6 ⊢ (𝐹 Fn (0..^(♯‘𝐹)) → (〈𝑁, 𝐵〉 ∈ 𝐹 → 𝑁 ∈ (0..^(♯‘𝐹)))) |
21 | 18, 20 | syl 17 | . . . . 5 ⊢ (𝜑 → (〈𝑁, 𝐵〉 ∈ 𝐹 → 𝑁 ∈ (0..^(♯‘𝐹)))) |
22 | 16, 21 | mtod 201 | . . . 4 ⊢ (𝜑 → ¬ 〈𝑁, 𝐵〉 ∈ 𝐹) |
23 | 9, 22 | jca 516 | . . 3 ⊢ (𝜑 → (𝐹 ∈ Fin ∧ ¬ 〈𝑁, 𝐵〉 ∈ 𝐹)) |
24 | hashunsng 13788 | . . 3 ⊢ (〈𝑁, 𝐵〉 ∈ V → ((𝐹 ∈ Fin ∧ ¬ 〈𝑁, 𝐵〉 ∈ 𝐹) → (♯‘(𝐹 ∪ {〈𝑁, 𝐵〉})) = ((♯‘𝐹) + 1))) | |
25 | 4, 23, 24 | mpsyl 68 | . 2 ⊢ (𝜑 → (♯‘(𝐹 ∪ {〈𝑁, 𝐵〉})) = ((♯‘𝐹) + 1)) |
26 | 10 | eqcomi 2768 | . . . 4 ⊢ (♯‘𝐹) = 𝑁 |
27 | 26 | a1i 11 | . . 3 ⊢ (𝜑 → (♯‘𝐹) = 𝑁) |
28 | 27 | oveq1d 7158 | . 2 ⊢ (𝜑 → ((♯‘𝐹) + 1) = (𝑁 + 1)) |
29 | 3, 25, 28 | 3eqtrd 2798 | 1 ⊢ (𝜑 → (♯‘𝐻) = (𝑁 + 1)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 400 = wceq 1539 ∈ wcel 2112 Vcvv 3407 ∪ cun 3852 ⊆ wss 3854 {csn 4515 {cpr 4517 〈cop 4521 class class class wbr 5025 dom cdm 5517 Fun wfun 6322 Fn wfn 6323 ‘cfv 6328 (class class class)co 7143 Fincfn 8520 0cc0 10560 1c1 10561 + caddc 10563 ..^cfzo 13067 ♯chash 13725 Word cword 13898 Vtxcvtx 26873 iEdgciedg 26874 Edgcedg 26924 Walkscwlks 27470 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-rep 5149 ax-sep 5162 ax-nul 5169 ax-pow 5227 ax-pr 5291 ax-un 7452 ax-cnex 10616 ax-resscn 10617 ax-1cn 10618 ax-icn 10619 ax-addcl 10620 ax-addrcl 10621 ax-mulcl 10622 ax-mulrcl 10623 ax-mulcom 10624 ax-addass 10625 ax-mulass 10626 ax-distr 10627 ax-i2m1 10628 ax-1ne0 10629 ax-1rid 10630 ax-rnegex 10631 ax-rrecex 10632 ax-cnre 10633 ax-pre-lttri 10634 ax-pre-lttrn 10635 ax-pre-ltadd 10636 ax-pre-mulgt0 10637 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-ifp 1060 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2071 df-mo 2558 df-eu 2589 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2899 df-ne 2950 df-nel 3054 df-ral 3073 df-rex 3074 df-reu 3075 df-rab 3077 df-v 3409 df-sbc 3694 df-csb 3802 df-dif 3857 df-un 3859 df-in 3861 df-ss 3871 df-pss 3873 df-nul 4222 df-if 4414 df-pw 4489 df-sn 4516 df-pr 4518 df-tp 4520 df-op 4522 df-uni 4792 df-int 4832 df-iun 4878 df-br 5026 df-opab 5088 df-mpt 5106 df-tr 5132 df-id 5423 df-eprel 5428 df-po 5436 df-so 5437 df-fr 5476 df-we 5478 df-xp 5523 df-rel 5524 df-cnv 5525 df-co 5526 df-dm 5527 df-rn 5528 df-res 5529 df-ima 5530 df-pred 6119 df-ord 6165 df-on 6166 df-lim 6167 df-suc 6168 df-iota 6287 df-fun 6330 df-fn 6331 df-f 6332 df-f1 6333 df-fo 6334 df-f1o 6335 df-fv 6336 df-riota 7101 df-ov 7146 df-oprab 7147 df-mpo 7148 df-om 7573 df-1st 7686 df-2nd 7687 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-oadd 8109 df-er 8292 df-map 8411 df-en 8521 df-dom 8522 df-sdom 8523 df-fin 8524 df-dju 9348 df-card 9386 df-pnf 10700 df-mnf 10701 df-xr 10702 df-ltxr 10703 df-le 10704 df-sub 10895 df-neg 10896 df-nn 11660 df-n0 11920 df-z 12006 df-uz 12268 df-fz 12925 df-fzo 13068 df-hash 13726 df-word 13899 df-wlks 27473 |
This theorem is referenced by: wlkp1lem8 27554 wlkp1 27555 eupthp1 28085 |
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