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Mirrors > Home > MPE Home > Th. List > wlkp1lem2 | Structured version Visualization version GIF version |
Description: Lemma for wlkp1 27465. (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 |
---|---|
wlkp1lem2 | ⊢ (𝜑 → (♯‘𝐻) = (𝑁 + 1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wlkp1.h | . . . 4 ⊢ 𝐻 = (𝐹 ∪ {〈𝑁, 𝐵〉}) | |
2 | 1 | fveq2i 6675 | . . 3 ⊢ (♯‘𝐻) = (♯‘(𝐹 ∪ {〈𝑁, 𝐵〉})) |
3 | 2 | a1i 11 | . 2 ⊢ (𝜑 → (♯‘𝐻) = (♯‘(𝐹 ∪ {〈𝑁, 𝐵〉}))) |
4 | opex 5358 | . . 3 ⊢ 〈𝑁, 𝐵〉 ∈ V | |
5 | wlkp1.w | . . . . 5 ⊢ (𝜑 → 𝐹(Walks‘𝐺)𝑃) | |
6 | wlkp1.i | . . . . . 6 ⊢ 𝐼 = (iEdg‘𝐺) | |
7 | 6 | wlkf 27398 | . . . . 5 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝐹 ∈ Word dom 𝐼) |
8 | wrdfin 13884 | . . . . 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 13054 | . . . . . . . 8 ⊢ ¬ (♯‘𝐹) ∈ (0..^(♯‘𝐹)) | |
12 | 11 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → ¬ (♯‘𝐹) ∈ (0..^(♯‘𝐹))) |
13 | eleq1 2902 | . . . . . . . 8 ⊢ (𝑁 = (♯‘𝐹) → (𝑁 ∈ (0..^(♯‘𝐹)) ↔ (♯‘𝐹) ∈ (0..^(♯‘𝐹)))) | |
14 | 13 | notbid 320 | . . . . . . 7 ⊢ (𝑁 = (♯‘𝐹) → (¬ 𝑁 ∈ (0..^(♯‘𝐹)) ↔ ¬ (♯‘𝐹) ∈ (0..^(♯‘𝐹)))) |
15 | 12, 14 | syl5ibr 248 | . . . . . 6 ⊢ (𝑁 = (♯‘𝐹) → (𝜑 → ¬ 𝑁 ∈ (0..^(♯‘𝐹)))) |
16 | 10, 15 | ax-mp 5 | . . . . 5 ⊢ (𝜑 → ¬ 𝑁 ∈ (0..^(♯‘𝐹))) |
17 | wrdfn 13879 | . . . . . . 7 ⊢ (𝐹 ∈ Word dom 𝐼 → 𝐹 Fn (0..^(♯‘𝐹))) | |
18 | 5, 7, 17 | 3syl 18 | . . . . . 6 ⊢ (𝜑 → 𝐹 Fn (0..^(♯‘𝐹))) |
19 | fnop 6462 | . . . . . . 7 ⊢ ((𝐹 Fn (0..^(♯‘𝐹)) ∧ 〈𝑁, 𝐵〉 ∈ 𝐹) → 𝑁 ∈ (0..^(♯‘𝐹))) | |
20 | 19 | ex 415 | . . . . . 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 200 | . . . 4 ⊢ (𝜑 → ¬ 〈𝑁, 𝐵〉 ∈ 𝐹) |
23 | 9, 22 | jca 514 | . . 3 ⊢ (𝜑 → (𝐹 ∈ Fin ∧ ¬ 〈𝑁, 𝐵〉 ∈ 𝐹)) |
24 | hashunsng 13756 | . . 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 2832 | . . . 4 ⊢ (♯‘𝐹) = 𝑁 |
27 | 26 | a1i 11 | . . 3 ⊢ (𝜑 → (♯‘𝐹) = 𝑁) |
28 | 27 | oveq1d 7173 | . 2 ⊢ (𝜑 → ((♯‘𝐹) + 1) = (𝑁 + 1)) |
29 | 3, 25, 28 | 3eqtrd 2862 | 1 ⊢ (𝜑 → (♯‘𝐻) = (𝑁 + 1)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3496 ∪ cun 3936 ⊆ wss 3938 {csn 4569 {cpr 4571 〈cop 4575 class class class wbr 5068 dom cdm 5557 Fun wfun 6351 Fn wfn 6352 ‘cfv 6357 (class class class)co 7158 Fincfn 8511 0cc0 10539 1c1 10540 + caddc 10542 ..^cfzo 13036 ♯chash 13693 Word cword 13864 Vtxcvtx 26783 iEdgciedg 26784 Edgcedg 26834 Walkscwlks 27380 |
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 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ifp 1058 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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-dju 9332 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-fzo 13037 df-hash 13694 df-word 13865 df-wlks 27383 |
This theorem is referenced by: wlkp1lem8 27464 wlkp1 27465 eupthp1 27997 |
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