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Mirrors > Home > MPE Home > Th. List > wlkp1lem5 | Structured version Visualization version GIF version |
Description: Lemma for wlkp1 27463. (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〉}) |
wlkp1.q | ⊢ 𝑄 = (𝑃 ∪ {〈(𝑁 + 1), 𝐶〉}) |
wlkp1.s | ⊢ (𝜑 → (Vtx‘𝑆) = 𝑉) |
Ref | Expression |
---|---|
wlkp1lem5 | ⊢ (𝜑 → ∀𝑘 ∈ (0...𝑁)(𝑄‘𝑘) = (𝑃‘𝑘)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wlkp1.q | . . . 4 ⊢ 𝑄 = (𝑃 ∪ {〈(𝑁 + 1), 𝐶〉}) | |
2 | 1 | fveq1i 6671 | . . 3 ⊢ (𝑄‘𝑘) = ((𝑃 ∪ {〈(𝑁 + 1), 𝐶〉})‘𝑘) |
3 | fzp1nel 12992 | . . . . . . . . 9 ⊢ ¬ (𝑁 + 1) ∈ (0...𝑁) | |
4 | eleq1 2900 | . . . . . . . . . . 11 ⊢ (𝑘 = (𝑁 + 1) → (𝑘 ∈ (0...𝑁) ↔ (𝑁 + 1) ∈ (0...𝑁))) | |
5 | 4 | notbid 320 | . . . . . . . . . 10 ⊢ (𝑘 = (𝑁 + 1) → (¬ 𝑘 ∈ (0...𝑁) ↔ ¬ (𝑁 + 1) ∈ (0...𝑁))) |
6 | 5 | eqcoms 2829 | . . . . . . . . 9 ⊢ ((𝑁 + 1) = 𝑘 → (¬ 𝑘 ∈ (0...𝑁) ↔ ¬ (𝑁 + 1) ∈ (0...𝑁))) |
7 | 3, 6 | mpbiri 260 | . . . . . . . 8 ⊢ ((𝑁 + 1) = 𝑘 → ¬ 𝑘 ∈ (0...𝑁)) |
8 | 7 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → ((𝑁 + 1) = 𝑘 → ¬ 𝑘 ∈ (0...𝑁))) |
9 | 8 | con2d 136 | . . . . . 6 ⊢ (𝜑 → (𝑘 ∈ (0...𝑁) → ¬ (𝑁 + 1) = 𝑘)) |
10 | 9 | imp 409 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ¬ (𝑁 + 1) = 𝑘) |
11 | 10 | neqned 3023 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝑁 + 1) ≠ 𝑘) |
12 | fvunsn 6941 | . . . 4 ⊢ ((𝑁 + 1) ≠ 𝑘 → ((𝑃 ∪ {〈(𝑁 + 1), 𝐶〉})‘𝑘) = (𝑃‘𝑘)) | |
13 | 11, 12 | syl 17 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ((𝑃 ∪ {〈(𝑁 + 1), 𝐶〉})‘𝑘) = (𝑃‘𝑘)) |
14 | 2, 13 | syl5eq 2868 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝑄‘𝑘) = (𝑃‘𝑘)) |
15 | 14 | ralrimiva 3182 | 1 ⊢ (𝜑 → ∀𝑘 ∈ (0...𝑁)(𝑄‘𝑘) = (𝑃‘𝑘)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ∀wral 3138 Vcvv 3494 ∪ cun 3934 ⊆ wss 3936 {csn 4567 {cpr 4569 〈cop 4573 class class class wbr 5066 dom cdm 5555 Fun wfun 6349 ‘cfv 6355 (class class class)co 7156 Fincfn 8509 0cc0 10537 1c1 10538 + caddc 10540 ...cfz 12893 ♯chash 13691 Vtxcvtx 26781 iEdgciedg 26782 Edgcedg 26832 Walkscwlks 27378 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-po 5474 df-so 5475 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-1st 7689 df-2nd 7690 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-z 11983 df-fz 12894 |
This theorem is referenced by: wlkp1lem6 27460 wlkp1lem7 27461 wlkp1lem8 27462 eupth2eucrct 27996 |
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