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Mirrors > Home > MPE Home > Th. List > umgr2adedgwlkon | Structured version Visualization version GIF version |
Description: In a multigraph, two adjacent edges form a walk between two vertices. (Contributed by Alexander van der Vekens, 18-Feb-2018.) (Revised by AV, 30-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 |
---|---|
umgr2adedgwlkon | ⊢ (𝜑 → 𝐹(𝐴(WalksOn‘𝐺)𝐶)𝑃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | umgr2adedgwlk.p | . 2 ⊢ 𝑃 = 〈“𝐴𝐵𝐶”〉 | |
2 | umgr2adedgwlk.f | . 2 ⊢ 𝐹 = 〈“𝐽𝐾”〉 | |
3 | umgr2adedgwlk.g | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ UMGraph) | |
4 | umgr2adedgwlk.a | . . . . 5 ⊢ (𝜑 → ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸)) | |
5 | 3anass 1097 | . . . . 5 ⊢ ((𝐺 ∈ UMGraph ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) ↔ (𝐺 ∈ UMGraph ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸))) | |
6 | 3, 4, 5 | sylanbrc 586 | . . . 4 ⊢ (𝜑 → (𝐺 ∈ UMGraph ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸)) |
7 | umgr2adedgwlk.e | . . . . 5 ⊢ 𝐸 = (Edg‘𝐺) | |
8 | 7 | umgr2adedgwlklem 28000 | . . . 4 ⊢ ((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 | . . 3 ⊢ (𝜑 → ((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶) ∧ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺) ∧ 𝐶 ∈ (Vtx‘𝐺)))) |
10 | 9 | simprd 499 | . 2 ⊢ (𝜑 → (𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺) ∧ 𝐶 ∈ (Vtx‘𝐺))) |
11 | 9 | simpld 498 | . 2 ⊢ (𝜑 → (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶)) |
12 | ssid 3913 | . . . 4 ⊢ {𝐴, 𝐵} ⊆ {𝐴, 𝐵} | |
13 | umgr2adedgwlk.j | . . . 4 ⊢ (𝜑 → (𝐼‘𝐽) = {𝐴, 𝐵}) | |
14 | 12, 13 | sseqtrrid 3944 | . . 3 ⊢ (𝜑 → {𝐴, 𝐵} ⊆ (𝐼‘𝐽)) |
15 | ssid 3913 | . . . 4 ⊢ {𝐵, 𝐶} ⊆ {𝐵, 𝐶} | |
16 | umgr2adedgwlk.k | . . . 4 ⊢ (𝜑 → (𝐼‘𝐾) = {𝐵, 𝐶}) | |
17 | 15, 16 | sseqtrrid 3944 | . . 3 ⊢ (𝜑 → {𝐵, 𝐶} ⊆ (𝐼‘𝐾)) |
18 | 14, 17 | jca 515 | . 2 ⊢ (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾))) |
19 | eqid 2734 | . 2 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
20 | umgr2adedgwlk.i | . 2 ⊢ 𝐼 = (iEdg‘𝐺) | |
21 | 1, 2, 10, 11, 18, 19, 20 | 2wlkond 27993 | 1 ⊢ (𝜑 → 𝐹(𝐴(WalksOn‘𝐺)𝐶)𝑃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2110 ≠ wne 2935 ⊆ wss 3857 {cpr 4533 class class class wbr 5043 ‘cfv 6369 (class class class)co 7202 〈“cs2 14389 〈“cs3 14390 Vtxcvtx 27059 iEdgciedg 27060 Edgcedg 27110 UMGraphcumgr 27144 WalksOncwlkson 27657 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-ifp 1064 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-int 4850 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-om 7634 df-1st 7750 df-2nd 7751 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-1o 8191 df-oadd 8195 df-er 8380 df-map 8499 df-en 8616 df-dom 8617 df-sdom 8618 df-fin 8619 df-dju 9500 df-card 9538 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-nn 11814 df-2 11876 df-3 11877 df-n0 12074 df-z 12160 df-uz 12422 df-fz 13079 df-fzo 13222 df-hash 13880 df-word 14053 df-concat 14109 df-s1 14136 df-s2 14396 df-s3 14397 df-edg 27111 df-umgr 27146 df-wlks 27659 df-wlkson 27660 |
This theorem is referenced by: (None) |
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