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| Description: Lemma for wlkp1 29699. (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 6909 | . . 3 ⊢ (♯‘𝐻) = (♯‘(𝐹 ∪ {〈𝑁, 𝐵〉})) | 
| 3 | 2 | a1i 11 | . 2 ⊢ (𝜑 → (♯‘𝐻) = (♯‘(𝐹 ∪ {〈𝑁, 𝐵〉}))) | 
| 4 | opex 5469 | . . 3 ⊢ 〈𝑁, 𝐵〉 ∈ V | |
| 5 | wlkp1.w | . . . . 5 ⊢ (𝜑 → 𝐹(Walks‘𝐺)𝑃) | |
| 6 | wlkp1.i | . . . . . 6 ⊢ 𝐼 = (iEdg‘𝐺) | |
| 7 | 6 | wlkf 29632 | . . . . 5 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝐹 ∈ Word dom 𝐼) | 
| 8 | wrdfin 14570 | . . . . 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 13713 | . . . . . . . 8 ⊢ ¬ (♯‘𝐹) ∈ (0..^(♯‘𝐹)) | |
| 12 | 11 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → ¬ (♯‘𝐹) ∈ (0..^(♯‘𝐹))) | 
| 13 | eleq1 2829 | . . . . . . . 8 ⊢ (𝑁 = (♯‘𝐹) → (𝑁 ∈ (0..^(♯‘𝐹)) ↔ (♯‘𝐹) ∈ (0..^(♯‘𝐹)))) | |
| 14 | 13 | notbid 318 | . . . . . . 7 ⊢ (𝑁 = (♯‘𝐹) → (¬ 𝑁 ∈ (0..^(♯‘𝐹)) ↔ ¬ (♯‘𝐹) ∈ (0..^(♯‘𝐹)))) | 
| 15 | 12, 14 | imbitrrid 246 | . . . . . 6 ⊢ (𝑁 = (♯‘𝐹) → (𝜑 → ¬ 𝑁 ∈ (0..^(♯‘𝐹)))) | 
| 16 | 10, 15 | ax-mp 5 | . . . . 5 ⊢ (𝜑 → ¬ 𝑁 ∈ (0..^(♯‘𝐹))) | 
| 17 | wrdfn 14566 | . . . . . 6 ⊢ (𝐹 ∈ Word dom 𝐼 → 𝐹 Fn (0..^(♯‘𝐹))) | |
| 18 | fnop 6677 | . . . . . . 7 ⊢ ((𝐹 Fn (0..^(♯‘𝐹)) ∧ 〈𝑁, 𝐵〉 ∈ 𝐹) → 𝑁 ∈ (0..^(♯‘𝐹))) | |
| 19 | 18 | ex 412 | . . . . . 6 ⊢ (𝐹 Fn (0..^(♯‘𝐹)) → (〈𝑁, 𝐵〉 ∈ 𝐹 → 𝑁 ∈ (0..^(♯‘𝐹)))) | 
| 20 | 5, 7, 17, 19 | 4syl 19 | . . . . 5 ⊢ (𝜑 → (〈𝑁, 𝐵〉 ∈ 𝐹 → 𝑁 ∈ (0..^(♯‘𝐹)))) | 
| 21 | 16, 20 | mtod 198 | . . . 4 ⊢ (𝜑 → ¬ 〈𝑁, 𝐵〉 ∈ 𝐹) | 
| 22 | 9, 21 | jca 511 | . . 3 ⊢ (𝜑 → (𝐹 ∈ Fin ∧ ¬ 〈𝑁, 𝐵〉 ∈ 𝐹)) | 
| 23 | hashunsng 14431 | . . 3 ⊢ (〈𝑁, 𝐵〉 ∈ V → ((𝐹 ∈ Fin ∧ ¬ 〈𝑁, 𝐵〉 ∈ 𝐹) → (♯‘(𝐹 ∪ {〈𝑁, 𝐵〉})) = ((♯‘𝐹) + 1))) | |
| 24 | 4, 22, 23 | mpsyl 68 | . 2 ⊢ (𝜑 → (♯‘(𝐹 ∪ {〈𝑁, 𝐵〉})) = ((♯‘𝐹) + 1)) | 
| 25 | 10 | eqcomi 2746 | . . . 4 ⊢ (♯‘𝐹) = 𝑁 | 
| 26 | 25 | a1i 11 | . . 3 ⊢ (𝜑 → (♯‘𝐹) = 𝑁) | 
| 27 | 26 | oveq1d 7446 | . 2 ⊢ (𝜑 → ((♯‘𝐹) + 1) = (𝑁 + 1)) | 
| 28 | 3, 24, 27 | 3eqtrd 2781 | 1 ⊢ (𝜑 → (♯‘𝐻) = (𝑁 + 1)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∪ cun 3949 ⊆ wss 3951 {csn 4626 {cpr 4628 〈cop 4632 class class class wbr 5143 dom cdm 5685 Fun wfun 6555 Fn wfn 6556 ‘cfv 6561 (class class class)co 7431 Fincfn 8985 0cc0 11155 1c1 11156 + caddc 11158 ..^cfzo 13694 ♯chash 14369 Word cword 14552 Vtxcvtx 29013 iEdgciedg 29014 Edgcedg 29064 Walkscwlks 29614 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-ifp 1064 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-oadd 8510 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-dju 9941 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-fzo 13695 df-hash 14370 df-word 14553 df-wlks 29617 | 
| This theorem is referenced by: wlkp1lem8 29698 wlkp1 29699 eupthp1 30235 | 
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