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Mirrors > Home > MPE Home > Th. List > 2trlond | Structured version Visualization version GIF version |
Description: A trail 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) |
2trld.n | ⊢ (𝜑 → 𝐽 ≠ 𝐾) |
Ref | Expression |
---|---|
2trlond | ⊢ (𝜑 → 𝐹(𝐴(TrailsOn‘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 | 2wlkond 27710 | . 2 ⊢ (𝜑 → 𝐹(𝐴(WalksOn‘𝐺)𝐶)𝑃) |
9 | 2trld.n | . . 3 ⊢ (𝜑 → 𝐽 ≠ 𝐾) | |
10 | 1, 2, 3, 4, 5, 6, 7, 9 | 2trld 27711 | . 2 ⊢ (𝜑 → 𝐹(Trails‘𝐺)𝑃) |
11 | 3 | simp1d 1138 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
12 | 3 | simp3d 1140 | . . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑉) |
13 | s2cli 14236 | . . . . 5 ⊢ 〈“𝐽𝐾”〉 ∈ Word V | |
14 | 2, 13 | eqeltri 2909 | . . . 4 ⊢ 𝐹 ∈ Word V |
15 | 14 | a1i 11 | . . 3 ⊢ (𝜑 → 𝐹 ∈ Word V) |
16 | s3cli 14237 | . . . . 5 ⊢ 〈“𝐴𝐵𝐶”〉 ∈ Word V | |
17 | 1, 16 | eqeltri 2909 | . . . 4 ⊢ 𝑃 ∈ Word V |
18 | 17 | a1i 11 | . . 3 ⊢ (𝜑 → 𝑃 ∈ Word V) |
19 | 6 | istrlson 27482 | . . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ (𝐹 ∈ Word V ∧ 𝑃 ∈ Word V)) → (𝐹(𝐴(TrailsOn‘𝐺)𝐶)𝑃 ↔ (𝐹(𝐴(WalksOn‘𝐺)𝐶)𝑃 ∧ 𝐹(Trails‘𝐺)𝑃))) |
20 | 11, 12, 15, 18, 19 | syl22anc 836 | . 2 ⊢ (𝜑 → (𝐹(𝐴(TrailsOn‘𝐺)𝐶)𝑃 ↔ (𝐹(𝐴(WalksOn‘𝐺)𝐶)𝑃 ∧ 𝐹(Trails‘𝐺)𝑃))) |
21 | 8, 10, 20 | mpbir2and 711 | 1 ⊢ (𝜑 → 𝐹(𝐴(TrailsOn‘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 4562 class class class wbr 5058 ‘cfv 6349 (class class class)co 7150 Word cword 13855 〈“cs2 14197 〈“cs3 14198 Vtxcvtx 26775 iEdgciedg 26776 WalksOncwlkson 27373 Trailsctrls 27466 TrailsOnctrlson 27467 |
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 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
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 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12887 df-fzo 13028 df-hash 13685 df-word 13856 df-concat 13917 df-s1 13944 df-s2 14204 df-s3 14205 df-wlks 27375 df-wlkson 27376 df-trls 27468 df-trlson 27469 |
This theorem is referenced by: 2pthond 27715 |
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