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Mirrors > Home > MPE Home > Th. List > 2wlkd | Structured version Visualization version GIF version |
Description: Construction of a walk from two given edges in a graph. (Contributed by Alexander van der Vekens, 5-Feb-2018.) (Revised by AV, 23-Jan-2021.) (Proof shortened by AV, 14-Feb-2021.) (Revised by AV, 24-Mar-2021.) |
Ref | Expression |
---|---|
2wlkd.p | ⊢ 𝑃 = 〈“𝐴𝐵𝐶”〉 |
2wlkd.f | ⊢ 𝐹 = 〈“𝐽𝐾”〉 |
2wlkd.s | ⊢ (𝜑 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) |
2wlkd.n | ⊢ (𝜑 → (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶)) |
2wlkd.e | ⊢ (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾))) |
2wlkd.v | ⊢ 𝑉 = (Vtx‘𝐺) |
2wlkd.i | ⊢ 𝐼 = (iEdg‘𝐺) |
Ref | Expression |
---|---|
2wlkd | ⊢ (𝜑 → 𝐹(Walks‘𝐺)𝑃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2wlkd.p | . . . 4 ⊢ 𝑃 = 〈“𝐴𝐵𝐶”〉 | |
2 | s3cli 14344 | . . . 4 ⊢ 〈“𝐴𝐵𝐶”〉 ∈ Word V | |
3 | 1, 2 | eqeltri 2830 | . . 3 ⊢ 𝑃 ∈ Word V |
4 | 3 | a1i 11 | . 2 ⊢ (𝜑 → 𝑃 ∈ Word V) |
5 | 2wlkd.f | . . . 4 ⊢ 𝐹 = 〈“𝐽𝐾”〉 | |
6 | s2cli 14343 | . . . 4 ⊢ 〈“𝐽𝐾”〉 ∈ Word V | |
7 | 5, 6 | eqeltri 2830 | . . 3 ⊢ 𝐹 ∈ Word V |
8 | 7 | a1i 11 | . 2 ⊢ (𝜑 → 𝐹 ∈ Word V) |
9 | 1, 5 | 2wlkdlem1 27875 | . . 3 ⊢ (♯‘𝑃) = ((♯‘𝐹) + 1) |
10 | 9 | a1i 11 | . 2 ⊢ (𝜑 → (♯‘𝑃) = ((♯‘𝐹) + 1)) |
11 | 2wlkd.s | . . 3 ⊢ (𝜑 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) | |
12 | 2wlkd.n | . . 3 ⊢ (𝜑 → (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶)) | |
13 | 2wlkd.e | . . 3 ⊢ (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾))) | |
14 | 1, 5, 11, 12, 13 | 2wlkdlem10 27885 | . 2 ⊢ (𝜑 → ∀𝑘 ∈ (0..^(♯‘𝐹)){(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘))) |
15 | 1, 5, 11, 12 | 2wlkdlem5 27879 | . 2 ⊢ (𝜑 → ∀𝑘 ∈ (0..^(♯‘𝐹))(𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1))) |
16 | 2wlkd.v | . . . . 5 ⊢ 𝑉 = (Vtx‘𝐺) | |
17 | 16 | 1vgrex 26959 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐺 ∈ V) |
18 | 17 | 3ad2ant1 1134 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → 𝐺 ∈ V) |
19 | 11, 18 | syl 17 | . 2 ⊢ (𝜑 → 𝐺 ∈ V) |
20 | 2wlkd.i | . 2 ⊢ 𝐼 = (iEdg‘𝐺) | |
21 | 1, 5, 11 | 2wlkdlem4 27878 | . 2 ⊢ (𝜑 → ∀𝑘 ∈ (0...(♯‘𝐹))(𝑃‘𝑘) ∈ 𝑉) |
22 | 4, 8, 10, 14, 15, 19, 16, 20, 21 | wlkd 27640 | 1 ⊢ (𝜑 → 𝐹(Walks‘𝐺)𝑃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1088 = wceq 1542 ∈ wcel 2114 ≠ wne 2935 Vcvv 3400 ⊆ wss 3853 {cpr 4528 class class class wbr 5040 ‘cfv 6349 (class class class)co 7182 1c1 10628 + caddc 10630 ♯chash 13794 Word cword 13967 〈“cs2 14304 〈“cs3 14305 Vtxcvtx 26953 iEdgciedg 26954 Walkscwlks 27550 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7491 ax-cnex 10683 ax-resscn 10684 ax-1cn 10685 ax-icn 10686 ax-addcl 10687 ax-addrcl 10688 ax-mulcl 10689 ax-mulrcl 10690 ax-mulcom 10691 ax-addass 10692 ax-mulass 10693 ax-distr 10694 ax-i2m1 10695 ax-1ne0 10696 ax-1rid 10697 ax-rnegex 10698 ax-rrecex 10699 ax-cnre 10700 ax-pre-lttri 10701 ax-pre-lttrn 10702 ax-pre-ltadd 10703 ax-pre-mulgt0 10704 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-ifp 1063 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-tp 4531 df-op 4533 df-uni 4807 df-int 4847 df-iun 4893 df-br 5041 df-opab 5103 df-mpt 5121 df-tr 5147 df-id 5439 df-eprel 5444 df-po 5452 df-so 5453 df-fr 5493 df-we 5495 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-pred 6139 df-ord 6185 df-on 6186 df-lim 6187 df-suc 6188 df-iota 6307 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7139 df-ov 7185 df-oprab 7186 df-mpo 7187 df-om 7612 df-1st 7726 df-2nd 7727 df-wrecs 7988 df-recs 8049 df-rdg 8087 df-1o 8143 df-er 8332 df-map 8451 df-en 8568 df-dom 8569 df-sdom 8570 df-fin 8571 df-card 9453 df-pnf 10767 df-mnf 10768 df-xr 10769 df-ltxr 10770 df-le 10771 df-sub 10962 df-neg 10963 df-nn 11729 df-2 11791 df-3 11792 df-n0 11989 df-z 12075 df-uz 12337 df-fz 12994 df-fzo 13137 df-hash 13795 df-word 13968 df-concat 14024 df-s1 14051 df-s2 14311 df-s3 14312 df-wlks 27553 |
This theorem is referenced by: 2wlkond 27887 2trld 27888 umgr2adedgwlk 27895 |
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