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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 1120 | . . . . . 6 ⊢ ((𝐺 ∈ UMGraph ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) ↔ (𝐺 ∈ UMGraph ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸))) | |
6 | 3, 4, 5 | sylanbrc 578 | . . . . 5 ⊢ (𝜑 → (𝐺 ∈ UMGraph ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸)) |
7 | umgr2adedgwlk.e | . . . . . 6 ⊢ 𝐸 = (Edg‘𝐺) | |
8 | 7 | umgr2adedgwlklem 27280 | . . . . 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 491 | . . 3 ⊢ (𝜑 → (𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺) ∧ 𝐶 ∈ (Vtx‘𝐺))) |
11 | 9 | simpld 490 | . . 3 ⊢ (𝜑 → (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶)) |
12 | ssid 3848 | . . . . 5 ⊢ {𝐴, 𝐵} ⊆ {𝐴, 𝐵} | |
13 | umgr2adedgwlk.j | . . . . 5 ⊢ (𝜑 → (𝐼‘𝐽) = {𝐴, 𝐵}) | |
14 | 12, 13 | syl5sseqr 3879 | . . . 4 ⊢ (𝜑 → {𝐴, 𝐵} ⊆ (𝐼‘𝐽)) |
15 | ssid 3848 | . . . . 5 ⊢ {𝐵, 𝐶} ⊆ {𝐵, 𝐶} | |
16 | umgr2adedgwlk.k | . . . . 5 ⊢ (𝜑 → (𝐼‘𝐾) = {𝐵, 𝐶}) | |
17 | 15, 16 | syl5sseqr 3879 | . . . 4 ⊢ (𝜑 → {𝐵, 𝐶} ⊆ (𝐼‘𝐾)) |
18 | 14, 17 | jca 507 | . . 3 ⊢ (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾))) |
19 | eqid 2825 | . . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
20 | umgr2adedgwlk.i | . . 3 ⊢ 𝐼 = (iEdg‘𝐺) | |
21 | 1, 2, 10, 11, 18, 19, 20 | 2wlkd 27272 | . 2 ⊢ (𝜑 → 𝐹(Walks‘𝐺)𝑃) |
22 | 2 | fveq2i 6440 | . . . 4 ⊢ (♯‘𝐹) = (♯‘〈“𝐽𝐾”〉) |
23 | s2len 14017 | . . . 4 ⊢ (♯‘〈“𝐽𝐾”〉) = 2 | |
24 | 22, 23 | eqtri 2849 | . . 3 ⊢ (♯‘𝐹) = 2 |
25 | 24 | a1i 11 | . 2 ⊢ (𝜑 → (♯‘𝐹) = 2) |
26 | s3fv0 14019 | . . . . 5 ⊢ (𝐴 ∈ (Vtx‘𝐺) → (〈“𝐴𝐵𝐶”〉‘0) = 𝐴) | |
27 | s3fv1 14020 | . . . . 5 ⊢ (𝐵 ∈ (Vtx‘𝐺) → (〈“𝐴𝐵𝐶”〉‘1) = 𝐵) | |
28 | s3fv2 14021 | . . . . 5 ⊢ (𝐶 ∈ (Vtx‘𝐺) → (〈“𝐴𝐵𝐶”〉‘2) = 𝐶) | |
29 | 26, 27, 28 | 3anim123i 1194 | . . . 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 6438 | . . . . . 6 ⊢ (𝑃‘0) = (〈“𝐴𝐵𝐶”〉‘0) |
32 | 31 | eqeq2i 2837 | . . . . 5 ⊢ (𝐴 = (𝑃‘0) ↔ 𝐴 = (〈“𝐴𝐵𝐶”〉‘0)) |
33 | eqcom 2832 | . . . . 5 ⊢ (𝐴 = (〈“𝐴𝐵𝐶”〉‘0) ↔ (〈“𝐴𝐵𝐶”〉‘0) = 𝐴) | |
34 | 32, 33 | bitri 267 | . . . 4 ⊢ (𝐴 = (𝑃‘0) ↔ (〈“𝐴𝐵𝐶”〉‘0) = 𝐴) |
35 | 1 | fveq1i 6438 | . . . . . 6 ⊢ (𝑃‘1) = (〈“𝐴𝐵𝐶”〉‘1) |
36 | 35 | eqeq2i 2837 | . . . . 5 ⊢ (𝐵 = (𝑃‘1) ↔ 𝐵 = (〈“𝐴𝐵𝐶”〉‘1)) |
37 | eqcom 2832 | . . . . 5 ⊢ (𝐵 = (〈“𝐴𝐵𝐶”〉‘1) ↔ (〈“𝐴𝐵𝐶”〉‘1) = 𝐵) | |
38 | 36, 37 | bitri 267 | . . . 4 ⊢ (𝐵 = (𝑃‘1) ↔ (〈“𝐴𝐵𝐶”〉‘1) = 𝐵) |
39 | 1 | fveq1i 6438 | . . . . . 6 ⊢ (𝑃‘2) = (〈“𝐴𝐵𝐶”〉‘2) |
40 | 39 | eqeq2i 2837 | . . . . 5 ⊢ (𝐶 = (𝑃‘2) ↔ 𝐶 = (〈“𝐴𝐵𝐶”〉‘2)) |
41 | eqcom 2832 | . . . . 5 ⊢ (𝐶 = (〈“𝐴𝐵𝐶”〉‘2) ↔ (〈“𝐴𝐵𝐶”〉‘2) = 𝐶) | |
42 | 40, 41 | bitri 267 | . . . 4 ⊢ (𝐶 = (𝑃‘2) ↔ (〈“𝐴𝐵𝐶”〉‘2) = 𝐶) |
43 | 34, 38, 42 | 3anbi123i 1198 | . . 3 ⊢ ((𝐴 = (𝑃‘0) ∧ 𝐵 = (𝑃‘1) ∧ 𝐶 = (𝑃‘2)) ↔ ((〈“𝐴𝐵𝐶”〉‘0) = 𝐴 ∧ (〈“𝐴𝐵𝐶”〉‘1) = 𝐵 ∧ (〈“𝐴𝐵𝐶”〉‘2) = 𝐶)) |
44 | 30, 43 | sylibr 226 | . 2 ⊢ (𝜑 → (𝐴 = (𝑃‘0) ∧ 𝐵 = (𝑃‘1) ∧ 𝐶 = (𝑃‘2))) |
45 | 21, 25, 44 | 3jca 1162 | 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 386 ∧ w3a 1111 = wceq 1656 ∈ wcel 2164 ≠ wne 2999 ⊆ wss 3798 {cpr 4401 class class class wbr 4875 ‘cfv 6127 0cc0 10259 1c1 10260 2c2 11413 ♯chash 13417 〈“cs2 13969 〈“cs3 13970 Vtxcvtx 26301 iEdgciedg 26302 Edgcedg 26352 UMGraphcumgr 26386 Walkscwlks 26901 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-ifp 1090 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-int 4700 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-om 7332 df-1st 7433 df-2nd 7434 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-1o 7831 df-oadd 7835 df-er 8014 df-map 8129 df-pm 8130 df-en 8229 df-dom 8230 df-sdom 8231 df-fin 8232 df-card 9085 df-cda 9312 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-nn 11358 df-2 11421 df-3 11422 df-n0 11626 df-z 11712 df-uz 11976 df-fz 12627 df-fzo 12768 df-hash 13418 df-word 13582 df-concat 13638 df-s1 13663 df-s2 13976 df-s3 13977 df-edg 26353 df-umgr 26388 df-wlks 26904 |
This theorem is referenced by: umgr2adedgwlkonALT 27283 umgr2wlk 27285 |
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