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Mirrors > Home > MPE Home > Th. List > 2pthond | Structured version Visualization version GIF version |
Description: A simple path 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, 24-Jan-2021.) (Proof shortened 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) |
2spthd.n | ⊢ (𝜑 → 𝐴 ≠ 𝐶) |
Ref | Expression |
---|---|
2pthond | ⊢ (𝜑 → 𝐹(𝐴(SPathsOn‘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 | 2trld.n | . . 3 ⊢ (𝜑 → 𝐽 ≠ 𝐾) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | 2trlond 27720 | . 2 ⊢ (𝜑 → 𝐹(𝐴(TrailsOn‘𝐺)𝐶)𝑃) |
10 | 2spthd.n | . . 3 ⊢ (𝜑 → 𝐴 ≠ 𝐶) | |
11 | 1, 2, 3, 4, 5, 6, 7, 8, 10 | 2spthd 27722 | . 2 ⊢ (𝜑 → 𝐹(SPaths‘𝐺)𝑃) |
12 | 3simpb 1145 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) | |
13 | 3, 12 | syl 17 | . . 3 ⊢ (𝜑 → (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) |
14 | s2cli 14244 | . . . . 5 ⊢ 〈“𝐽𝐾”〉 ∈ Word V | |
15 | 2, 14 | eqeltri 2911 | . . . 4 ⊢ 𝐹 ∈ Word V |
16 | s3cli 14245 | . . . . 5 ⊢ 〈“𝐴𝐵𝐶”〉 ∈ Word V | |
17 | 1, 16 | eqeltri 2911 | . . . 4 ⊢ 𝑃 ∈ Word V |
18 | 15, 17 | pm3.2i 473 | . . 3 ⊢ (𝐹 ∈ Word V ∧ 𝑃 ∈ Word V) |
19 | 6 | isspthson 27526 | . . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ (𝐹 ∈ Word V ∧ 𝑃 ∈ Word V)) → (𝐹(𝐴(SPathsOn‘𝐺)𝐶)𝑃 ↔ (𝐹(𝐴(TrailsOn‘𝐺)𝐶)𝑃 ∧ 𝐹(SPaths‘𝐺)𝑃))) |
20 | 13, 18, 19 | sylancl 588 | . 2 ⊢ (𝜑 → (𝐹(𝐴(SPathsOn‘𝐺)𝐶)𝑃 ↔ (𝐹(𝐴(TrailsOn‘𝐺)𝐶)𝑃 ∧ 𝐹(SPaths‘𝐺)𝑃))) |
21 | 9, 11, 20 | mpbir2and 711 | 1 ⊢ (𝜑 → 𝐹(𝐴(SPathsOn‘𝐺)𝐶)𝑃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 Vcvv 3496 ⊆ wss 3938 {cpr 4571 class class class wbr 5068 ‘cfv 6357 (class class class)co 7158 Word cword 13864 〈“cs2 14205 〈“cs3 14206 Vtxcvtx 26783 iEdgciedg 26784 TrailsOnctrlson 27475 SPathscspths 27496 SPathsOncspthson 27498 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
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 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-3 11704 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-fzo 13037 df-hash 13694 df-word 13865 df-concat 13925 df-s1 13952 df-s2 14212 df-s3 14213 df-wlks 27383 df-wlkson 27384 df-trls 27476 df-trlson 27477 df-pths 27499 df-spths 27500 df-spthson 27502 |
This theorem is referenced by: 2pthon3v 27724 |
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