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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 27717 | . 2 ⊢ (𝜑 → 𝐹(Walks‘𝐺)𝑃) |
9 | 3 | simp1d 1138 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
10 | 1 | fveq1i 6673 | . . . 4 ⊢ (𝑃‘0) = (〈“𝐴𝐵𝐶”〉‘0) |
11 | s3fv0 14255 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → (〈“𝐴𝐵𝐶”〉‘0) = 𝐴) | |
12 | 10, 11 | syl5eq 2870 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝑃‘0) = 𝐴) |
13 | 9, 12 | syl 17 | . 2 ⊢ (𝜑 → (𝑃‘0) = 𝐴) |
14 | 2 | fveq2i 6675 | . . . . 5 ⊢ (♯‘𝐹) = (♯‘〈“𝐽𝐾”〉) |
15 | s2len 14253 | . . . . 5 ⊢ (♯‘〈“𝐽𝐾”〉) = 2 | |
16 | 14, 15 | eqtri 2846 | . . . 4 ⊢ (♯‘𝐹) = 2 |
17 | 1, 16 | fveq12i 6678 | . . 3 ⊢ (𝑃‘(♯‘𝐹)) = (〈“𝐴𝐵𝐶”〉‘2) |
18 | 3 | simp3d 1140 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑉) |
19 | s3fv2 14257 | . . . 4 ⊢ (𝐶 ∈ 𝑉 → (〈“𝐴𝐵𝐶”〉‘2) = 𝐶) | |
20 | 18, 19 | syl 17 | . . 3 ⊢ (𝜑 → (〈“𝐴𝐵𝐶”〉‘2) = 𝐶) |
21 | 17, 20 | syl5eq 2870 | . 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 14244 | . . . . 5 ⊢ 〈“𝐽𝐾”〉 ∈ Word V | |
25 | 2, 24 | eqeltri 2911 | . . . 4 ⊢ 𝐹 ∈ Word V |
26 | s3cli 14245 | . . . . 5 ⊢ 〈“𝐴𝐵𝐶”〉 ∈ Word V | |
27 | 1, 26 | eqeltri 2911 | . . . 4 ⊢ 𝑃 ∈ Word V |
28 | 25, 27 | pm3.2i 473 | . . 3 ⊢ (𝐹 ∈ Word V ∧ 𝑃 ∈ Word V) |
29 | 6 | iswlkon 27441 | . . 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 1537 ∈ wcel 2114 ≠ wne 3018 Vcvv 3496 ⊆ wss 3938 {cpr 4571 class class class wbr 5068 ‘cfv 6357 (class class class)co 7158 0cc0 10539 2c2 11695 ♯chash 13693 Word cword 13864 〈“cs2 14205 〈“cs3 14206 Vtxcvtx 26783 iEdgciedg 26784 Walkscwlks 27380 WalksOncwlkson 27381 |
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 |
This theorem is referenced by: 2trlond 27720 umgr2adedgwlkon 27727 |
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