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Mirrors > Home > MPE Home > Th. List > Mathboxes > 1wlkreslem | Structured version Visualization version GIF version |
Description: Lemma for 1wlkres 40879. (Contributed by AV, 5-Mar-2021.) |
Ref | Expression |
---|---|
1wlkres.v | ⊢ 𝑉 = (Vtx‘𝐺) |
1wlkres.i | ⊢ 𝐼 = (iEdg‘𝐺) |
1wlkres.d | ⊢ (𝜑 → 𝐹(1Walks‘𝐺)𝑃) |
1wlkres.n | ⊢ (𝜑 → 𝑁 ∈ (0..^(#‘𝐹))) |
1wlkres.s | ⊢ (𝜑 → (Vtx‘𝑆) = 𝑉) |
1wlkres.e | ⊢ (𝜑 → (iEdg‘𝑆) = (𝐼 ↾ (𝐹 “ (0..^𝑁)))) |
1wlkres.h | ⊢ 𝐻 = (𝐹 ↾ (0..^𝑁)) |
1wlkres.q | ⊢ 𝑄 = (𝑃 ↾ (0...𝑁)) |
Ref | Expression |
---|---|
1wlkreslem | ⊢ (𝜑 → (𝑆 ∈ V ∧ 𝐻 ∈ V ∧ 𝑄 ∈ V)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-1 6 | . . 3 ⊢ (𝑆 ∈ V → (𝜑 → 𝑆 ∈ V)) | |
2 | df-nel 2783 | . . . 4 ⊢ (𝑆 ∉ V ↔ ¬ 𝑆 ∈ V) | |
3 | 1wlkres.d | . . . . . 6 ⊢ (𝜑 → 𝐹(1Walks‘𝐺)𝑃) | |
4 | df-br 4584 | . . . . . . 7 ⊢ (𝐹(1Walks‘𝐺)𝑃 ↔ 〈𝐹, 𝑃〉 ∈ (1Walks‘𝐺)) | |
5 | ne0i 3880 | . . . . . . . 8 ⊢ (〈𝐹, 𝑃〉 ∈ (1Walks‘𝐺) → (1Walks‘𝐺) ≠ ∅) | |
6 | 1wlkres.s | . . . . . . . . . . . . . 14 ⊢ (𝜑 → (Vtx‘𝑆) = 𝑉) | |
7 | 1wlkres.v | . . . . . . . . . . . . . 14 ⊢ 𝑉 = (Vtx‘𝐺) | |
8 | 6, 7 | syl6eq 2660 | . . . . . . . . . . . . 13 ⊢ (𝜑 → (Vtx‘𝑆) = (Vtx‘𝐺)) |
9 | 8 | anim1i 590 | . . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑆 ∉ V) → ((Vtx‘𝑆) = (Vtx‘𝐺) ∧ 𝑆 ∉ V)) |
10 | 9 | ancomd 466 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑆 ∉ V) → (𝑆 ∉ V ∧ (Vtx‘𝑆) = (Vtx‘𝐺))) |
11 | 1wlk0prc 40862 | . . . . . . . . . . 11 ⊢ ((𝑆 ∉ V ∧ (Vtx‘𝑆) = (Vtx‘𝐺)) → (1Walks‘𝐺) = ∅) | |
12 | eqneqall 2793 | . . . . . . . . . . 11 ⊢ ((1Walks‘𝐺) = ∅ → ((1Walks‘𝐺) ≠ ∅ → 𝑆 ∈ V)) | |
13 | 10, 11, 12 | 3syl 18 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑆 ∉ V) → ((1Walks‘𝐺) ≠ ∅ → 𝑆 ∈ V)) |
14 | 13 | expcom 450 | . . . . . . . . 9 ⊢ (𝑆 ∉ V → (𝜑 → ((1Walks‘𝐺) ≠ ∅ → 𝑆 ∈ V))) |
15 | 14 | com13 86 | . . . . . . . 8 ⊢ ((1Walks‘𝐺) ≠ ∅ → (𝜑 → (𝑆 ∉ V → 𝑆 ∈ V))) |
16 | 5, 15 | syl 17 | . . . . . . 7 ⊢ (〈𝐹, 𝑃〉 ∈ (1Walks‘𝐺) → (𝜑 → (𝑆 ∉ V → 𝑆 ∈ V))) |
17 | 4, 16 | sylbi 206 | . . . . . 6 ⊢ (𝐹(1Walks‘𝐺)𝑃 → (𝜑 → (𝑆 ∉ V → 𝑆 ∈ V))) |
18 | 3, 17 | mpcom 37 | . . . . 5 ⊢ (𝜑 → (𝑆 ∉ V → 𝑆 ∈ V)) |
19 | 18 | com12 32 | . . . 4 ⊢ (𝑆 ∉ V → (𝜑 → 𝑆 ∈ V)) |
20 | 2, 19 | sylbir 224 | . . 3 ⊢ (¬ 𝑆 ∈ V → (𝜑 → 𝑆 ∈ V)) |
21 | 1, 20 | pm2.61i 175 | . 2 ⊢ (𝜑 → 𝑆 ∈ V) |
22 | 1wlkres.h | . . 3 ⊢ 𝐻 = (𝐹 ↾ (0..^𝑁)) | |
23 | 1wlkres.i | . . . . . 6 ⊢ 𝐼 = (iEdg‘𝐺) | |
24 | 23 | 1wlkf 40819 | . . . . 5 ⊢ (𝐹(1Walks‘𝐺)𝑃 → 𝐹 ∈ Word dom 𝐼) |
25 | wrdf 13165 | . . . . . 6 ⊢ (𝐹 ∈ Word dom 𝐼 → 𝐹:(0..^(#‘𝐹))⟶dom 𝐼) | |
26 | 25 | ffund 5962 | . . . . 5 ⊢ (𝐹 ∈ Word dom 𝐼 → Fun 𝐹) |
27 | 3, 24, 26 | 3syl 18 | . . . 4 ⊢ (𝜑 → Fun 𝐹) |
28 | ovex 6577 | . . . 4 ⊢ (0..^𝑁) ∈ V | |
29 | resfunexg 6384 | . . . 4 ⊢ ((Fun 𝐹 ∧ (0..^𝑁) ∈ V) → (𝐹 ↾ (0..^𝑁)) ∈ V) | |
30 | 27, 28, 29 | sylancl 693 | . . 3 ⊢ (𝜑 → (𝐹 ↾ (0..^𝑁)) ∈ V) |
31 | 22, 30 | syl5eqel 2692 | . 2 ⊢ (𝜑 → 𝐻 ∈ V) |
32 | 1wlkres.q | . . 3 ⊢ 𝑄 = (𝑃 ↾ (0...𝑁)) | |
33 | 7 | 1wlkp 40821 | . . . . 5 ⊢ (𝐹(1Walks‘𝐺)𝑃 → 𝑃:(0...(#‘𝐹))⟶𝑉) |
34 | ffun 5961 | . . . . 5 ⊢ (𝑃:(0...(#‘𝐹))⟶𝑉 → Fun 𝑃) | |
35 | 3, 33, 34 | 3syl 18 | . . . 4 ⊢ (𝜑 → Fun 𝑃) |
36 | ovex 6577 | . . . 4 ⊢ (0...𝑁) ∈ V | |
37 | resfunexg 6384 | . . . 4 ⊢ ((Fun 𝑃 ∧ (0...𝑁) ∈ V) → (𝑃 ↾ (0...𝑁)) ∈ V) | |
38 | 35, 36, 37 | sylancl 693 | . . 3 ⊢ (𝜑 → (𝑃 ↾ (0...𝑁)) ∈ V) |
39 | 32, 38 | syl5eqel 2692 | . 2 ⊢ (𝜑 → 𝑄 ∈ V) |
40 | 21, 31, 39 | 3jca 1235 | 1 ⊢ (𝜑 → (𝑆 ∈ V ∧ 𝐻 ∈ V ∧ 𝑄 ∈ V)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∉ wnel 2781 Vcvv 3173 ∅c0 3874 〈cop 4131 class class class wbr 4583 dom cdm 5038 ↾ cres 5040 “ cima 5041 Fun wfun 5798 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 0cc0 9815 ...cfz 12197 ..^cfzo 12334 #chash 12979 Word cword 13146 Vtxcvtx 25673 iEdgciedg 25674 1Walksc1wlks 40796 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-ifp 1007 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-hash 12980 df-word 13154 df-1wlks 40800 |
This theorem is referenced by: 1wlkres 40879 trlres 40908 eupthres 41383 |
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