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Mirrors > Home > MPE Home > Th. List > 2wlkond | Structured version Visualization version GIF version |
Description: A walk of length 2 from one vertex to another, different vertex via a third vertex. (Contributed by Alexander van der Vekens, 6-Dec-2017.) (Revised by AV, 30-Jan-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 |
---|---|
2wlkond | ⊢ (𝜑 → 𝐹(𝐴(WalksOn‘𝐺)𝐶)𝑃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2wlkd.p | . . 3 ⊢ 𝑃 = 〈“𝐴𝐵𝐶”〉 | |
2 | 2wlkd.f | . . 3 ⊢ 𝐹 = 〈“𝐽𝐾”〉 | |
3 | 2wlkd.s | . . 3 ⊢ (𝜑 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) | |
4 | 2wlkd.n | . . 3 ⊢ (𝜑 → (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶)) | |
5 | 2wlkd.e | . . 3 ⊢ (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾))) | |
6 | 2wlkd.v | . . 3 ⊢ 𝑉 = (Vtx‘𝐺) | |
7 | 2wlkd.i | . . 3 ⊢ 𝐼 = (iEdg‘𝐺) | |
8 | 1, 2, 3, 4, 5, 6, 7 | 2wlkd 27714 | . 2 ⊢ (𝜑 → 𝐹(Walks‘𝐺)𝑃) |
9 | 3 | simp1d 1138 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
10 | 1 | fveq1i 6670 | . . . 4 ⊢ (𝑃‘0) = (〈“𝐴𝐵𝐶”〉‘0) |
11 | s3fv0 14252 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → (〈“𝐴𝐵𝐶”〉‘0) = 𝐴) | |
12 | 10, 11 | syl5eq 2868 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝑃‘0) = 𝐴) |
13 | 9, 12 | syl 17 | . 2 ⊢ (𝜑 → (𝑃‘0) = 𝐴) |
14 | 2 | fveq2i 6672 | . . . . 5 ⊢ (♯‘𝐹) = (♯‘〈“𝐽𝐾”〉) |
15 | s2len 14250 | . . . . 5 ⊢ (♯‘〈“𝐽𝐾”〉) = 2 | |
16 | 14, 15 | eqtri 2844 | . . . 4 ⊢ (♯‘𝐹) = 2 |
17 | 1, 16 | fveq12i 6675 | . . 3 ⊢ (𝑃‘(♯‘𝐹)) = (〈“𝐴𝐵𝐶”〉‘2) |
18 | 3 | simp3d 1140 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑉) |
19 | s3fv2 14254 | . . . 4 ⊢ (𝐶 ∈ 𝑉 → (〈“𝐴𝐵𝐶”〉‘2) = 𝐶) | |
20 | 18, 19 | syl 17 | . . 3 ⊢ (𝜑 → (〈“𝐴𝐵𝐶”〉‘2) = 𝐶) |
21 | 17, 20 | syl5eq 2868 | . 2 ⊢ (𝜑 → (𝑃‘(♯‘𝐹)) = 𝐶) |
22 | 3simpb 1145 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) | |
23 | 3, 22 | syl 17 | . . 3 ⊢ (𝜑 → (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) |
24 | s2cli 14241 | . . . . 5 ⊢ 〈“𝐽𝐾”〉 ∈ Word V | |
25 | 2, 24 | eqeltri 2909 | . . . 4 ⊢ 𝐹 ∈ Word V |
26 | s3cli 14242 | . . . . 5 ⊢ 〈“𝐴𝐵𝐶”〉 ∈ Word V | |
27 | 1, 26 | eqeltri 2909 | . . . 4 ⊢ 𝑃 ∈ Word V |
28 | 25, 27 | pm3.2i 473 | . . 3 ⊢ (𝐹 ∈ Word V ∧ 𝑃 ∈ Word V) |
29 | 6 | iswlkon 27438 | . . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ (𝐹 ∈ Word V ∧ 𝑃 ∈ Word V)) → (𝐹(𝐴(WalksOn‘𝐺)𝐶)𝑃 ↔ (𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(♯‘𝐹)) = 𝐶))) |
30 | 23, 28, 29 | sylancl 588 | . 2 ⊢ (𝜑 → (𝐹(𝐴(WalksOn‘𝐺)𝐶)𝑃 ↔ (𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(♯‘𝐹)) = 𝐶))) |
31 | 8, 13, 21, 30 | mpbir3and 1338 | 1 ⊢ (𝜑 → 𝐹(𝐴(WalksOn‘𝐺)𝐶)𝑃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 Vcvv 3494 ⊆ wss 3935 {cpr 4568 class class class wbr 5065 ‘cfv 6354 (class class class)co 7155 0cc0 10536 2c2 11691 ♯chash 13689 Word cword 13860 〈“cs2 14202 〈“cs3 14203 Vtxcvtx 26780 iEdgciedg 26781 Walkscwlks 27377 WalksOncwlkson 27378 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ifp 1058 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-1st 7688 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-oadd 8105 df-er 8288 df-map 8407 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-card 9367 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-nn 11638 df-2 11699 df-3 11700 df-n0 11897 df-z 11981 df-uz 12243 df-fz 12892 df-fzo 13033 df-hash 13690 df-word 13861 df-concat 13922 df-s1 13949 df-s2 14209 df-s3 14210 df-wlks 27380 df-wlkson 27381 |
This theorem is referenced by: 2trlond 27717 umgr2adedgwlkon 27724 |
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