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Mirrors > Home > MPE Home > Th. List > wlkp1lem7 | Structured version Visualization version GIF version |
Description: Lemma for wlkp1 27801. (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 6739 | . . . . . 6 ⊢ (𝑘 = 𝑁 → (𝑄‘𝑘) = (𝑄‘𝑁)) | |
3 | fveq2 6739 | . . . . . 6 ⊢ (𝑘 = 𝑁 → (𝑃‘𝑘) = (𝑃‘𝑁)) | |
4 | 2, 3 | eqeq12d 2755 | . . . . 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 27797 | . . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ (0...𝑁)(𝑄‘𝑘) = (𝑃‘𝑘)) |
20 | wlkcl 27735 | . . . . . 6 ⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝐹) ∈ ℕ0) | |
21 | 13 | eqcomi 2748 | . . . . . . . 8 ⊢ (♯‘𝐹) = 𝑁 |
22 | 21 | eleq1i 2830 | . . . . . . 7 ⊢ ((♯‘𝐹) ∈ ℕ0 ↔ 𝑁 ∈ ℕ0) |
23 | nn0fz0 13240 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ0 ↔ 𝑁 ∈ (0...𝑁)) | |
24 | 22, 23 | sylbb 222 | . . . . . 6 ⊢ ((♯‘𝐹) ∈ ℕ0 → 𝑁 ∈ (0...𝑁)) |
25 | 12, 20, 24 | 3syl 18 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ (0...𝑁)) |
26 | 4, 19, 25 | rspcdva 3554 | . . . 4 ⊢ (𝜑 → (𝑄‘𝑁) = (𝑃‘𝑁)) |
27 | 17 | fveq1i 6740 | . . . . 5 ⊢ (𝑄‘(𝑁 + 1)) = ((𝑃 ∪ {〈(𝑁 + 1), 𝐶〉})‘(𝑁 + 1)) |
28 | ovex 7268 | . . . . . 6 ⊢ (𝑁 + 1) ∈ V | |
29 | 5, 6, 7, 8, 9, 10, 11, 12, 13 | wlkp1lem1 27793 | . . . . . 6 ⊢ (𝜑 → ¬ (𝑁 + 1) ∈ dom 𝑃) |
30 | fsnunfv 7024 | . . . . . 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 | syl5eq 2792 | . . . 4 ⊢ (𝜑 → (𝑄‘(𝑁 + 1)) = 𝐶) |
33 | 26, 32 | preq12d 4674 | . . 3 ⊢ (𝜑 → {(𝑄‘𝑁), (𝑄‘(𝑁 + 1))} = {(𝑃‘𝑁), 𝐶}) |
34 | fsnunfv 7024 | . . . 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 3965 | . 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 27795 | . 2 ⊢ (𝜑 → ((iEdg‘𝑆)‘(𝐻‘𝑁)) = ((𝐼 ∪ {〈𝐵, 𝐸〉})‘𝐵)) |
38 | 36, 37 | sseqtrrd 3959 | 1 ⊢ (𝜑 → {(𝑄‘𝑁), (𝑄‘(𝑁 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻‘𝑁))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1543 ∈ wcel 2112 Vcvv 3423 ∪ cun 3881 ⊆ wss 3883 {csn 4558 {cpr 4560 〈cop 4564 class class class wbr 5070 dom cdm 5569 Fun wfun 6395 ‘cfv 6401 (class class class)co 7235 Fincfn 8650 0cc0 10759 1c1 10760 + caddc 10762 ℕ0cn0 12120 ...cfz 13125 ♯chash 13929 Vtxcvtx 27119 iEdgciedg 27120 Edgcedg 27170 Walkscwlks 27716 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2710 ax-rep 5196 ax-sep 5209 ax-nul 5216 ax-pow 5275 ax-pr 5339 ax-un 7545 ax-cnex 10815 ax-resscn 10816 ax-1cn 10817 ax-icn 10818 ax-addcl 10819 ax-addrcl 10820 ax-mulcl 10821 ax-mulrcl 10822 ax-mulcom 10823 ax-addass 10824 ax-mulass 10825 ax-distr 10826 ax-i2m1 10827 ax-1ne0 10828 ax-1rid 10829 ax-rnegex 10830 ax-rrecex 10831 ax-cnre 10832 ax-pre-lttri 10833 ax-pre-lttrn 10834 ax-pre-ltadd 10835 ax-pre-mulgt0 10836 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-ifp 1064 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2818 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3071 df-rab 3073 df-v 3425 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4255 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5153 df-tr 5179 df-id 5472 df-eprel 5478 df-po 5486 df-so 5487 df-fr 5527 df-we 5529 df-xp 5575 df-rel 5576 df-cnv 5577 df-co 5578 df-dm 5579 df-rn 5580 df-res 5581 df-ima 5582 df-pred 6179 df-ord 6237 df-on 6238 df-lim 6239 df-suc 6240 df-iota 6359 df-fun 6403 df-fn 6404 df-f 6405 df-f1 6406 df-fo 6407 df-f1o 6408 df-fv 6409 df-riota 7192 df-ov 7238 df-oprab 7239 df-mpo 7240 df-om 7667 df-1st 7783 df-2nd 7784 df-wrecs 8071 df-recs 8132 df-rdg 8170 df-1o 8226 df-er 8415 df-map 8534 df-en 8651 df-dom 8652 df-sdom 8653 df-fin 8654 df-card 9585 df-pnf 10899 df-mnf 10900 df-xr 10901 df-ltxr 10902 df-le 10903 df-sub 11094 df-neg 11095 df-nn 11861 df-n0 12121 df-z 12207 df-uz 12469 df-fz 13126 df-fzo 13269 df-hash 13930 df-word 14103 df-wlks 27719 |
This theorem is referenced by: wlkp1lem8 27800 |
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