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| Mirrors > Home > MPE Home > Th. List > wlkp1lem7 | Structured version Visualization version GIF version | ||
| 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〉}) |
| wlkp1.q | ⊢ 𝑄 = (𝑃 ∪ {〈(𝑁 + 1), 𝐶〉}) |
| wlkp1.s | ⊢ (𝜑 → (Vtx‘𝑆) = 𝑉) |
| Ref | Expression |
|---|---|
| wlkp1lem7 | ⊢ (𝜑 → {(𝑄‘𝑁), (𝑄‘(𝑁 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻‘𝑁))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | wlkp1.x | . . 3 ⊢ (𝜑 → {(𝑃‘𝑁), 𝐶} ⊆ 𝐸) | |
| 2 | fveq2 6906 | . . . . . 6 ⊢ (𝑘 = 𝑁 → (𝑄‘𝑘) = (𝑄‘𝑁)) | |
| 3 | fveq2 6906 | . . . . . 6 ⊢ (𝑘 = 𝑁 → (𝑃‘𝑘) = (𝑃‘𝑁)) | |
| 4 | 2, 3 | eqeq12d 2753 | . . . . 5 ⊢ (𝑘 = 𝑁 → ((𝑄‘𝑘) = (𝑃‘𝑘) ↔ (𝑄‘𝑁) = (𝑃‘𝑁))) |
| 5 | wlkp1.v | . . . . . 6 ⊢ 𝑉 = (Vtx‘𝐺) | |
| 6 | wlkp1.i | . . . . . 6 ⊢ 𝐼 = (iEdg‘𝐺) | |
| 7 | wlkp1.f | . . . . . 6 ⊢ (𝜑 → Fun 𝐼) | |
| 8 | wlkp1.a | . . . . . 6 ⊢ (𝜑 → 𝐼 ∈ Fin) | |
| 9 | wlkp1.b | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
| 10 | wlkp1.c | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ 𝑉) | |
| 11 | wlkp1.d | . . . . . 6 ⊢ (𝜑 → ¬ 𝐵 ∈ dom 𝐼) | |
| 12 | wlkp1.w | . . . . . 6 ⊢ (𝜑 → 𝐹(Walks‘𝐺)𝑃) | |
| 13 | wlkp1.n | . . . . . 6 ⊢ 𝑁 = (♯‘𝐹) | |
| 14 | wlkp1.e | . . . . . 6 ⊢ (𝜑 → 𝐸 ∈ (Edg‘𝐺)) | |
| 15 | wlkp1.u | . . . . . 6 ⊢ (𝜑 → (iEdg‘𝑆) = (𝐼 ∪ {〈𝐵, 𝐸〉})) | |
| 16 | wlkp1.h | . . . . . 6 ⊢ 𝐻 = (𝐹 ∪ {〈𝑁, 𝐵〉}) | |
| 17 | wlkp1.q | . . . . . 6 ⊢ 𝑄 = (𝑃 ∪ {〈(𝑁 + 1), 𝐶〉}) | |
| 18 | wlkp1.s | . . . . . 6 ⊢ (𝜑 → (Vtx‘𝑆) = 𝑉) | |
| 19 | 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 1, 15, 16, 17, 18 | wlkp1lem5 29695 | . . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ (0...𝑁)(𝑄‘𝑘) = (𝑃‘𝑘)) |
| 20 | wlkcl 29633 | . . . . . 6 ⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝐹) ∈ ℕ0) | |
| 21 | 13 | eqcomi 2746 | . . . . . . . 8 ⊢ (♯‘𝐹) = 𝑁 |
| 22 | 21 | eleq1i 2832 | . . . . . . 7 ⊢ ((♯‘𝐹) ∈ ℕ0 ↔ 𝑁 ∈ ℕ0) |
| 23 | nn0fz0 13665 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ0 ↔ 𝑁 ∈ (0...𝑁)) | |
| 24 | 22, 23 | sylbb 219 | . . . . . 6 ⊢ ((♯‘𝐹) ∈ ℕ0 → 𝑁 ∈ (0...𝑁)) |
| 25 | 12, 20, 24 | 3syl 18 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ (0...𝑁)) |
| 26 | 4, 19, 25 | rspcdva 3623 | . . . 4 ⊢ (𝜑 → (𝑄‘𝑁) = (𝑃‘𝑁)) |
| 27 | 17 | fveq1i 6907 | . . . . 5 ⊢ (𝑄‘(𝑁 + 1)) = ((𝑃 ∪ {〈(𝑁 + 1), 𝐶〉})‘(𝑁 + 1)) |
| 28 | ovex 7464 | . . . . . 6 ⊢ (𝑁 + 1) ∈ V | |
| 29 | 5, 6, 7, 8, 9, 10, 11, 12, 13 | wlkp1lem1 29691 | . . . . . 6 ⊢ (𝜑 → ¬ (𝑁 + 1) ∈ dom 𝑃) |
| 30 | fsnunfv 7207 | . . . . . 6 ⊢ (((𝑁 + 1) ∈ V ∧ 𝐶 ∈ 𝑉 ∧ ¬ (𝑁 + 1) ∈ dom 𝑃) → ((𝑃 ∪ {〈(𝑁 + 1), 𝐶〉})‘(𝑁 + 1)) = 𝐶) | |
| 31 | 28, 10, 29, 30 | mp3an2i 1468 | . . . . 5 ⊢ (𝜑 → ((𝑃 ∪ {〈(𝑁 + 1), 𝐶〉})‘(𝑁 + 1)) = 𝐶) |
| 32 | 27, 31 | eqtrid 2789 | . . . 4 ⊢ (𝜑 → (𝑄‘(𝑁 + 1)) = 𝐶) |
| 33 | 26, 32 | preq12d 4741 | . . 3 ⊢ (𝜑 → {(𝑄‘𝑁), (𝑄‘(𝑁 + 1))} = {(𝑃‘𝑁), 𝐶}) |
| 34 | fsnunfv 7207 | . . . 4 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐸 ∈ (Edg‘𝐺) ∧ ¬ 𝐵 ∈ dom 𝐼) → ((𝐼 ∪ {〈𝐵, 𝐸〉})‘𝐵) = 𝐸) | |
| 35 | 9, 14, 11, 34 | syl3anc 1373 | . . 3 ⊢ (𝜑 → ((𝐼 ∪ {〈𝐵, 𝐸〉})‘𝐵) = 𝐸) |
| 36 | 1, 33, 35 | 3sstr4d 4039 | . 2 ⊢ (𝜑 → {(𝑄‘𝑁), (𝑄‘(𝑁 + 1))} ⊆ ((𝐼 ∪ {〈𝐵, 𝐸〉})‘𝐵)) |
| 37 | 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 1, 15, 16 | wlkp1lem3 29693 | . 2 ⊢ (𝜑 → ((iEdg‘𝑆)‘(𝐻‘𝑁)) = ((𝐼 ∪ {〈𝐵, 𝐸〉})‘𝐵)) |
| 38 | 36, 37 | sseqtrrd 4021 | 1 ⊢ (𝜑 → {(𝑄‘𝑁), (𝑄‘(𝑁 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻‘𝑁))) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 = 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 ‘cfv 6561 (class class class)co 7431 Fincfn 8985 0cc0 11155 1c1 11156 + caddc 11158 ℕ0cn0 12526 ...cfz 13547 ♯chash 14369 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-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 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 |
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