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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 27265 | . 2 ⊢ (𝜑 → 𝐹(Walks‘𝐺)𝑃) |
9 | 3 | simp1d 1176 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
10 | 1 | fveq1i 6434 | . . . 4 ⊢ (𝑃‘0) = (〈“𝐴𝐵𝐶”〉‘0) |
11 | s3fv0 14012 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → (〈“𝐴𝐵𝐶”〉‘0) = 𝐴) | |
12 | 10, 11 | syl5eq 2873 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝑃‘0) = 𝐴) |
13 | 9, 12 | syl 17 | . 2 ⊢ (𝜑 → (𝑃‘0) = 𝐴) |
14 | 2 | fveq2i 6436 | . . . . 5 ⊢ (♯‘𝐹) = (♯‘〈“𝐽𝐾”〉) |
15 | s2len 14010 | . . . . 5 ⊢ (♯‘〈“𝐽𝐾”〉) = 2 | |
16 | 14, 15 | eqtri 2849 | . . . 4 ⊢ (♯‘𝐹) = 2 |
17 | 1, 16 | fveq12i 6439 | . . 3 ⊢ (𝑃‘(♯‘𝐹)) = (〈“𝐴𝐵𝐶”〉‘2) |
18 | 3 | simp3d 1178 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑉) |
19 | s3fv2 14014 | . . . 4 ⊢ (𝐶 ∈ 𝑉 → (〈“𝐴𝐵𝐶”〉‘2) = 𝐶) | |
20 | 18, 19 | syl 17 | . . 3 ⊢ (𝜑 → (〈“𝐴𝐵𝐶”〉‘2) = 𝐶) |
21 | 17, 20 | syl5eq 2873 | . 2 ⊢ (𝜑 → (𝑃‘(♯‘𝐹)) = 𝐶) |
22 | 3simpb 1184 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) | |
23 | 3, 22 | syl 17 | . . 3 ⊢ (𝜑 → (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) |
24 | s2cli 14001 | . . . . 5 ⊢ 〈“𝐽𝐾”〉 ∈ Word V | |
25 | 2, 24 | eqeltri 2902 | . . . 4 ⊢ 𝐹 ∈ Word V |
26 | s3cli 14002 | . . . . 5 ⊢ 〈“𝐴𝐵𝐶”〉 ∈ Word V | |
27 | 1, 26 | eqeltri 2902 | . . . 4 ⊢ 𝑃 ∈ Word V |
28 | 25, 27 | pm3.2i 464 | . . 3 ⊢ (𝐹 ∈ Word V ∧ 𝑃 ∈ Word V) |
29 | 6 | iswlkon 26954 | . . 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 580 | . 2 ⊢ (𝜑 → (𝐹(𝐴(WalksOn‘𝐺)𝐶)𝑃 ↔ (𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(♯‘𝐹)) = 𝐶))) |
31 | 8, 13, 21, 30 | mpbir3and 1446 | 1 ⊢ (𝜑 → 𝐹(𝐴(WalksOn‘𝐺)𝐶)𝑃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 ∧ w3a 1111 = wceq 1656 ∈ wcel 2164 ≠ wne 2999 Vcvv 3414 ⊆ wss 3798 {cpr 4399 class class class wbr 4873 ‘cfv 6123 (class class class)co 6905 0cc0 10252 2c2 11406 ♯chash 13410 Word cword 13574 〈“cs2 13962 〈“cs3 13963 Vtxcvtx 26294 iEdgciedg 26295 Walkscwlks 26894 WalksOncwlkson 26895 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-ifp 1090 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-int 4698 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-1st 7428 df-2nd 7429 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-1o 7826 df-oadd 7830 df-er 8009 df-map 8124 df-pm 8125 df-en 8223 df-dom 8224 df-sdom 8225 df-fin 8226 df-card 9078 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-nn 11351 df-2 11414 df-3 11415 df-n0 11619 df-z 11705 df-uz 11969 df-fz 12620 df-fzo 12761 df-hash 13411 df-word 13575 df-concat 13631 df-s1 13656 df-s2 13969 df-s3 13970 df-wlks 26897 df-wlkson 26898 |
This theorem is referenced by: 2trlond 27268 umgr2adedgwlkon 27275 |
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