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Mirrors > Home > MPE Home > Th. List > wlkp1lem7 | Structured version Visualization version GIF version |
Description: Lemma for wlkp1 27390. (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 |
---|---|
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 6663 | . . . . . 6 ⊢ (𝑘 = 𝑁 → (𝑄‘𝑘) = (𝑄‘𝑁)) | |
3 | fveq2 6663 | . . . . . 6 ⊢ (𝑘 = 𝑁 → (𝑃‘𝑘) = (𝑃‘𝑁)) | |
4 | 2, 3 | eqeq12d 2834 | . . . . 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 ∈ V) | |
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 27386 | . . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ (0...𝑁)(𝑄‘𝑘) = (𝑃‘𝑘)) |
20 | wlkcl 27324 | . . . . . 6 ⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝐹) ∈ ℕ0) | |
21 | 13 | eqcomi 2827 | . . . . . . . 8 ⊢ (♯‘𝐹) = 𝑁 |
22 | 21 | eleq1i 2900 | . . . . . . 7 ⊢ ((♯‘𝐹) ∈ ℕ0 ↔ 𝑁 ∈ ℕ0) |
23 | nn0fz0 12993 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ0 ↔ 𝑁 ∈ (0...𝑁)) | |
24 | 22, 23 | sylbb 220 | . . . . . 6 ⊢ ((♯‘𝐹) ∈ ℕ0 → 𝑁 ∈ (0...𝑁)) |
25 | 12, 20, 24 | 3syl 18 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ (0...𝑁)) |
26 | 4, 19, 25 | rspcdva 3622 | . . . 4 ⊢ (𝜑 → (𝑄‘𝑁) = (𝑃‘𝑁)) |
27 | 17 | fveq1i 6664 | . . . . 5 ⊢ (𝑄‘(𝑁 + 1)) = ((𝑃 ∪ {〈(𝑁 + 1), 𝐶〉})‘(𝑁 + 1)) |
28 | ovex 7178 | . . . . . 6 ⊢ (𝑁 + 1) ∈ V | |
29 | 5, 6, 7, 8, 9, 10, 11, 12, 13 | wlkp1lem1 27382 | . . . . . 6 ⊢ (𝜑 → ¬ (𝑁 + 1) ∈ dom 𝑃) |
30 | fsnunfv 6941 | . . . . . 6 ⊢ (((𝑁 + 1) ∈ V ∧ 𝐶 ∈ 𝑉 ∧ ¬ (𝑁 + 1) ∈ dom 𝑃) → ((𝑃 ∪ {〈(𝑁 + 1), 𝐶〉})‘(𝑁 + 1)) = 𝐶) | |
31 | 28, 10, 29, 30 | mp3an2i 1457 | . . . . 5 ⊢ (𝜑 → ((𝑃 ∪ {〈(𝑁 + 1), 𝐶〉})‘(𝑁 + 1)) = 𝐶) |
32 | 27, 31 | syl5eq 2865 | . . . 4 ⊢ (𝜑 → (𝑄‘(𝑁 + 1)) = 𝐶) |
33 | 26, 32 | preq12d 4669 | . . 3 ⊢ (𝜑 → {(𝑄‘𝑁), (𝑄‘(𝑁 + 1))} = {(𝑃‘𝑁), 𝐶}) |
34 | fsnunfv 6941 | . . . 4 ⊢ ((𝐵 ∈ V ∧ 𝐸 ∈ (Edg‘𝐺) ∧ ¬ 𝐵 ∈ dom 𝐼) → ((𝐼 ∪ {〈𝐵, 𝐸〉})‘𝐵) = 𝐸) | |
35 | 9, 14, 11, 34 | syl3anc 1363 | . . 3 ⊢ (𝜑 → ((𝐼 ∪ {〈𝐵, 𝐸〉})‘𝐵) = 𝐸) |
36 | 1, 33, 35 | 3sstr4d 4011 | . 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 27384 | . 2 ⊢ (𝜑 → ((iEdg‘𝑆)‘(𝐻‘𝑁)) = ((𝐼 ∪ {〈𝐵, 𝐸〉})‘𝐵)) |
38 | 36, 37 | sseqtrrd 4005 | 1 ⊢ (𝜑 → {(𝑄‘𝑁), (𝑄‘(𝑁 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻‘𝑁))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1528 ∈ wcel 2105 Vcvv 3492 ∪ cun 3931 ⊆ wss 3933 {csn 4557 {cpr 4559 〈cop 4563 class class class wbr 5057 dom cdm 5548 Fun wfun 6342 ‘cfv 6348 (class class class)co 7145 Fincfn 8497 0cc0 10525 1c1 10526 + caddc 10528 ℕ0cn0 11885 ...cfz 12880 ♯chash 13678 Vtxcvtx 26708 iEdgciedg 26709 Edgcedg 26759 Walkscwlks 27305 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-ifp 1055 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12881 df-fzo 13022 df-hash 13679 df-word 13850 df-wlks 27308 |
This theorem is referenced by: wlkp1lem8 27389 |
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