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Mirrors > Home > MPE Home > Th. List > umgr2adedgwlk | Structured version Visualization version GIF version |
Description: In a multigraph, two adjacent edges form a walk of length 2. (Contributed by Alexander van der Vekens, 18-Feb-2018.) (Revised by AV, 29-Jan-2021.) |
Ref | Expression |
---|---|
umgr2adedgwlk.e | ⊢ 𝐸 = (Edg‘𝐺) |
umgr2adedgwlk.i | ⊢ 𝐼 = (iEdg‘𝐺) |
umgr2adedgwlk.f | ⊢ 𝐹 = 〈“𝐽𝐾”〉 |
umgr2adedgwlk.p | ⊢ 𝑃 = 〈“𝐴𝐵𝐶”〉 |
umgr2adedgwlk.g | ⊢ (𝜑 → 𝐺 ∈ UMGraph) |
umgr2adedgwlk.a | ⊢ (𝜑 → ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸)) |
umgr2adedgwlk.j | ⊢ (𝜑 → (𝐼‘𝐽) = {𝐴, 𝐵}) |
umgr2adedgwlk.k | ⊢ (𝜑 → (𝐼‘𝐾) = {𝐵, 𝐶}) |
Ref | Expression |
---|---|
umgr2adedgwlk | ⊢ (𝜑 → (𝐹(Walks‘𝐺)𝑃 ∧ (♯‘𝐹) = 2 ∧ (𝐴 = (𝑃‘0) ∧ 𝐵 = (𝑃‘1) ∧ 𝐶 = (𝑃‘2)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | umgr2adedgwlk.p | . . 3 ⊢ 𝑃 = 〈“𝐴𝐵𝐶”〉 | |
2 | umgr2adedgwlk.f | . . 3 ⊢ 𝐹 = 〈“𝐽𝐾”〉 | |
3 | umgr2adedgwlk.g | . . . . . 6 ⊢ (𝜑 → 𝐺 ∈ UMGraph) | |
4 | umgr2adedgwlk.a | . . . . . 6 ⊢ (𝜑 → ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸)) | |
5 | 3anass 1092 | . . . . . 6 ⊢ ((𝐺 ∈ UMGraph ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) ↔ (𝐺 ∈ UMGraph ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸))) | |
6 | 3, 4, 5 | sylanbrc 586 | . . . . 5 ⊢ (𝜑 → (𝐺 ∈ UMGraph ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸)) |
7 | umgr2adedgwlk.e | . . . . . 6 ⊢ 𝐸 = (Edg‘𝐺) | |
8 | 7 | umgr2adedgwlklem 27730 | . . . . 5 ⊢ ((𝐺 ∈ UMGraph ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) → ((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶) ∧ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺) ∧ 𝐶 ∈ (Vtx‘𝐺)))) |
9 | 6, 8 | syl 17 | . . . 4 ⊢ (𝜑 → ((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶) ∧ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺) ∧ 𝐶 ∈ (Vtx‘𝐺)))) |
10 | 9 | simprd 499 | . . 3 ⊢ (𝜑 → (𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺) ∧ 𝐶 ∈ (Vtx‘𝐺))) |
11 | 9 | simpld 498 | . . 3 ⊢ (𝜑 → (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶)) |
12 | ssid 3937 | . . . . 5 ⊢ {𝐴, 𝐵} ⊆ {𝐴, 𝐵} | |
13 | umgr2adedgwlk.j | . . . . 5 ⊢ (𝜑 → (𝐼‘𝐽) = {𝐴, 𝐵}) | |
14 | 12, 13 | sseqtrrid 3968 | . . . 4 ⊢ (𝜑 → {𝐴, 𝐵} ⊆ (𝐼‘𝐽)) |
15 | ssid 3937 | . . . . 5 ⊢ {𝐵, 𝐶} ⊆ {𝐵, 𝐶} | |
16 | umgr2adedgwlk.k | . . . . 5 ⊢ (𝜑 → (𝐼‘𝐾) = {𝐵, 𝐶}) | |
17 | 15, 16 | sseqtrrid 3968 | . . . 4 ⊢ (𝜑 → {𝐵, 𝐶} ⊆ (𝐼‘𝐾)) |
18 | 14, 17 | jca 515 | . . 3 ⊢ (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾))) |
19 | eqid 2798 | . . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
20 | umgr2adedgwlk.i | . . 3 ⊢ 𝐼 = (iEdg‘𝐺) | |
21 | 1, 2, 10, 11, 18, 19, 20 | 2wlkd 27722 | . 2 ⊢ (𝜑 → 𝐹(Walks‘𝐺)𝑃) |
22 | 2 | fveq2i 6648 | . . . 4 ⊢ (♯‘𝐹) = (♯‘〈“𝐽𝐾”〉) |
23 | s2len 14242 | . . . 4 ⊢ (♯‘〈“𝐽𝐾”〉) = 2 | |
24 | 22, 23 | eqtri 2821 | . . 3 ⊢ (♯‘𝐹) = 2 |
25 | 24 | a1i 11 | . 2 ⊢ (𝜑 → (♯‘𝐹) = 2) |
26 | s3fv0 14244 | . . . . 5 ⊢ (𝐴 ∈ (Vtx‘𝐺) → (〈“𝐴𝐵𝐶”〉‘0) = 𝐴) | |
27 | s3fv1 14245 | . . . . 5 ⊢ (𝐵 ∈ (Vtx‘𝐺) → (〈“𝐴𝐵𝐶”〉‘1) = 𝐵) | |
28 | s3fv2 14246 | . . . . 5 ⊢ (𝐶 ∈ (Vtx‘𝐺) → (〈“𝐴𝐵𝐶”〉‘2) = 𝐶) | |
29 | 26, 27, 28 | 3anim123i 1148 | . . . 4 ⊢ ((𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺) ∧ 𝐶 ∈ (Vtx‘𝐺)) → ((〈“𝐴𝐵𝐶”〉‘0) = 𝐴 ∧ (〈“𝐴𝐵𝐶”〉‘1) = 𝐵 ∧ (〈“𝐴𝐵𝐶”〉‘2) = 𝐶)) |
30 | 10, 29 | syl 17 | . . 3 ⊢ (𝜑 → ((〈“𝐴𝐵𝐶”〉‘0) = 𝐴 ∧ (〈“𝐴𝐵𝐶”〉‘1) = 𝐵 ∧ (〈“𝐴𝐵𝐶”〉‘2) = 𝐶)) |
31 | 1 | fveq1i 6646 | . . . . . 6 ⊢ (𝑃‘0) = (〈“𝐴𝐵𝐶”〉‘0) |
32 | 31 | eqeq2i 2811 | . . . . 5 ⊢ (𝐴 = (𝑃‘0) ↔ 𝐴 = (〈“𝐴𝐵𝐶”〉‘0)) |
33 | eqcom 2805 | . . . . 5 ⊢ (𝐴 = (〈“𝐴𝐵𝐶”〉‘0) ↔ (〈“𝐴𝐵𝐶”〉‘0) = 𝐴) | |
34 | 32, 33 | bitri 278 | . . . 4 ⊢ (𝐴 = (𝑃‘0) ↔ (〈“𝐴𝐵𝐶”〉‘0) = 𝐴) |
35 | 1 | fveq1i 6646 | . . . . . 6 ⊢ (𝑃‘1) = (〈“𝐴𝐵𝐶”〉‘1) |
36 | 35 | eqeq2i 2811 | . . . . 5 ⊢ (𝐵 = (𝑃‘1) ↔ 𝐵 = (〈“𝐴𝐵𝐶”〉‘1)) |
37 | eqcom 2805 | . . . . 5 ⊢ (𝐵 = (〈“𝐴𝐵𝐶”〉‘1) ↔ (〈“𝐴𝐵𝐶”〉‘1) = 𝐵) | |
38 | 36, 37 | bitri 278 | . . . 4 ⊢ (𝐵 = (𝑃‘1) ↔ (〈“𝐴𝐵𝐶”〉‘1) = 𝐵) |
39 | 1 | fveq1i 6646 | . . . . . 6 ⊢ (𝑃‘2) = (〈“𝐴𝐵𝐶”〉‘2) |
40 | 39 | eqeq2i 2811 | . . . . 5 ⊢ (𝐶 = (𝑃‘2) ↔ 𝐶 = (〈“𝐴𝐵𝐶”〉‘2)) |
41 | eqcom 2805 | . . . . 5 ⊢ (𝐶 = (〈“𝐴𝐵𝐶”〉‘2) ↔ (〈“𝐴𝐵𝐶”〉‘2) = 𝐶) | |
42 | 40, 41 | bitri 278 | . . . 4 ⊢ (𝐶 = (𝑃‘2) ↔ (〈“𝐴𝐵𝐶”〉‘2) = 𝐶) |
43 | 34, 38, 42 | 3anbi123i 1152 | . . 3 ⊢ ((𝐴 = (𝑃‘0) ∧ 𝐵 = (𝑃‘1) ∧ 𝐶 = (𝑃‘2)) ↔ ((〈“𝐴𝐵𝐶”〉‘0) = 𝐴 ∧ (〈“𝐴𝐵𝐶”〉‘1) = 𝐵 ∧ (〈“𝐴𝐵𝐶”〉‘2) = 𝐶)) |
44 | 30, 43 | sylibr 237 | . 2 ⊢ (𝜑 → (𝐴 = (𝑃‘0) ∧ 𝐵 = (𝑃‘1) ∧ 𝐶 = (𝑃‘2))) |
45 | 21, 25, 44 | 3jca 1125 | 1 ⊢ (𝜑 → (𝐹(Walks‘𝐺)𝑃 ∧ (♯‘𝐹) = 2 ∧ (𝐴 = (𝑃‘0) ∧ 𝐵 = (𝑃‘1) ∧ 𝐶 = (𝑃‘2)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ⊆ wss 3881 {cpr 4527 class class class wbr 5030 ‘cfv 6324 0cc0 10526 1c1 10527 2c2 11680 ♯chash 13686 〈“cs2 14194 〈“cs3 14195 Vtxcvtx 26789 iEdgciedg 26790 Edgcedg 26840 UMGraphcumgr 26874 Walkscwlks 27386 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-ifp 1059 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-dju 9314 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-fzo 13029 df-hash 13687 df-word 13858 df-concat 13914 df-s1 13941 df-s2 14201 df-s3 14202 df-edg 26841 df-umgr 26876 df-wlks 27389 |
This theorem is referenced by: umgr2adedgwlkonALT 27733 umgr2wlk 27735 |
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