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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 1093 | . . . . . 6 ⊢ ((𝐺 ∈ UMGraph ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) ↔ (𝐺 ∈ UMGraph ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸))) | |
6 | 3, 4, 5 | sylanbrc 582 | . . . . 5 ⊢ (𝜑 → (𝐺 ∈ UMGraph ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸)) |
7 | umgr2adedgwlk.e | . . . . . 6 ⊢ 𝐸 = (Edg‘𝐺) | |
8 | 7 | umgr2adedgwlklem 29742 | . . . . 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 495 | . . 3 ⊢ (𝜑 → (𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺) ∧ 𝐶 ∈ (Vtx‘𝐺))) |
11 | 9 | simpld 494 | . . 3 ⊢ (𝜑 → (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶)) |
12 | ssid 4000 | . . . . 5 ⊢ {𝐴, 𝐵} ⊆ {𝐴, 𝐵} | |
13 | umgr2adedgwlk.j | . . . . 5 ⊢ (𝜑 → (𝐼‘𝐽) = {𝐴, 𝐵}) | |
14 | 12, 13 | sseqtrrid 4031 | . . . 4 ⊢ (𝜑 → {𝐴, 𝐵} ⊆ (𝐼‘𝐽)) |
15 | ssid 4000 | . . . . 5 ⊢ {𝐵, 𝐶} ⊆ {𝐵, 𝐶} | |
16 | umgr2adedgwlk.k | . . . . 5 ⊢ (𝜑 → (𝐼‘𝐾) = {𝐵, 𝐶}) | |
17 | 15, 16 | sseqtrrid 4031 | . . . 4 ⊢ (𝜑 → {𝐵, 𝐶} ⊆ (𝐼‘𝐾)) |
18 | 14, 17 | jca 511 | . . 3 ⊢ (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾))) |
19 | eqid 2727 | . . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
20 | umgr2adedgwlk.i | . . 3 ⊢ 𝐼 = (iEdg‘𝐺) | |
21 | 1, 2, 10, 11, 18, 19, 20 | 2wlkd 29734 | . 2 ⊢ (𝜑 → 𝐹(Walks‘𝐺)𝑃) |
22 | 2 | fveq2i 6894 | . . . 4 ⊢ (♯‘𝐹) = (♯‘〈“𝐽𝐾”〉) |
23 | s2len 14864 | . . . 4 ⊢ (♯‘〈“𝐽𝐾”〉) = 2 | |
24 | 22, 23 | eqtri 2755 | . . 3 ⊢ (♯‘𝐹) = 2 |
25 | 24 | a1i 11 | . 2 ⊢ (𝜑 → (♯‘𝐹) = 2) |
26 | s3fv0 14866 | . . . . 5 ⊢ (𝐴 ∈ (Vtx‘𝐺) → (〈“𝐴𝐵𝐶”〉‘0) = 𝐴) | |
27 | s3fv1 14867 | . . . . 5 ⊢ (𝐵 ∈ (Vtx‘𝐺) → (〈“𝐴𝐵𝐶”〉‘1) = 𝐵) | |
28 | s3fv2 14868 | . . . . 5 ⊢ (𝐶 ∈ (Vtx‘𝐺) → (〈“𝐴𝐵𝐶”〉‘2) = 𝐶) | |
29 | 26, 27, 28 | 3anim123i 1149 | . . . 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 6892 | . . . . . 6 ⊢ (𝑃‘0) = (〈“𝐴𝐵𝐶”〉‘0) |
32 | 31 | eqeq2i 2740 | . . . . 5 ⊢ (𝐴 = (𝑃‘0) ↔ 𝐴 = (〈“𝐴𝐵𝐶”〉‘0)) |
33 | eqcom 2734 | . . . . 5 ⊢ (𝐴 = (〈“𝐴𝐵𝐶”〉‘0) ↔ (〈“𝐴𝐵𝐶”〉‘0) = 𝐴) | |
34 | 32, 33 | bitri 275 | . . . 4 ⊢ (𝐴 = (𝑃‘0) ↔ (〈“𝐴𝐵𝐶”〉‘0) = 𝐴) |
35 | 1 | fveq1i 6892 | . . . . . 6 ⊢ (𝑃‘1) = (〈“𝐴𝐵𝐶”〉‘1) |
36 | 35 | eqeq2i 2740 | . . . . 5 ⊢ (𝐵 = (𝑃‘1) ↔ 𝐵 = (〈“𝐴𝐵𝐶”〉‘1)) |
37 | eqcom 2734 | . . . . 5 ⊢ (𝐵 = (〈“𝐴𝐵𝐶”〉‘1) ↔ (〈“𝐴𝐵𝐶”〉‘1) = 𝐵) | |
38 | 36, 37 | bitri 275 | . . . 4 ⊢ (𝐵 = (𝑃‘1) ↔ (〈“𝐴𝐵𝐶”〉‘1) = 𝐵) |
39 | 1 | fveq1i 6892 | . . . . . 6 ⊢ (𝑃‘2) = (〈“𝐴𝐵𝐶”〉‘2) |
40 | 39 | eqeq2i 2740 | . . . . 5 ⊢ (𝐶 = (𝑃‘2) ↔ 𝐶 = (〈“𝐴𝐵𝐶”〉‘2)) |
41 | eqcom 2734 | . . . . 5 ⊢ (𝐶 = (〈“𝐴𝐵𝐶”〉‘2) ↔ (〈“𝐴𝐵𝐶”〉‘2) = 𝐶) | |
42 | 40, 41 | bitri 275 | . . . 4 ⊢ (𝐶 = (𝑃‘2) ↔ (〈“𝐴𝐵𝐶”〉‘2) = 𝐶) |
43 | 34, 38, 42 | 3anbi123i 1153 | . . 3 ⊢ ((𝐴 = (𝑃‘0) ∧ 𝐵 = (𝑃‘1) ∧ 𝐶 = (𝑃‘2)) ↔ ((〈“𝐴𝐵𝐶”〉‘0) = 𝐴 ∧ (〈“𝐴𝐵𝐶”〉‘1) = 𝐵 ∧ (〈“𝐴𝐵𝐶”〉‘2) = 𝐶)) |
44 | 30, 43 | sylibr 233 | . 2 ⊢ (𝜑 → (𝐴 = (𝑃‘0) ∧ 𝐵 = (𝑃‘1) ∧ 𝐶 = (𝑃‘2))) |
45 | 21, 25, 44 | 3jca 1126 | 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 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ≠ wne 2935 ⊆ wss 3944 {cpr 4626 class class class wbr 5142 ‘cfv 6542 0cc0 11130 1c1 11131 2c2 12289 ♯chash 14313 〈“cs2 14816 〈“cs3 14817 Vtxcvtx 28796 iEdgciedg 28797 Edgcedg 28847 UMGraphcumgr 28881 Walkscwlks 29397 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-ifp 1062 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-oadd 8484 df-er 8718 df-map 8838 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-dju 9916 df-card 9954 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-nn 12235 df-2 12297 df-3 12298 df-n0 12495 df-z 12581 df-uz 12845 df-fz 13509 df-fzo 13652 df-hash 14314 df-word 14489 df-concat 14545 df-s1 14570 df-s2 14823 df-s3 14824 df-edg 28848 df-umgr 28883 df-wlks 29400 |
This theorem is referenced by: umgr2adedgwlkonALT 29745 umgr2wlk 29747 |
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