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Mirrors > Home > MPE Home > Th. List > 3cyclpd | Structured version Visualization version GIF version |
Description: Construction of a 3-cycle from three given edges in a graph, containing an endpoint of one of these edges. (Contributed by Alexander van der Vekens, 17-Nov-2017.) (Revised by AV, 10-Feb-2021.) (Revised by AV, 24-Mar-2021.) |
Ref | Expression |
---|---|
3wlkd.p | ⊢ 𝑃 = 〈“𝐴𝐵𝐶𝐷”〉 |
3wlkd.f | ⊢ 𝐹 = 〈“𝐽𝐾𝐿”〉 |
3wlkd.s | ⊢ (𝜑 → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉))) |
3wlkd.n | ⊢ (𝜑 → ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ∧ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷) ∧ 𝐶 ≠ 𝐷)) |
3wlkd.e | ⊢ (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾) ∧ {𝐶, 𝐷} ⊆ (𝐼‘𝐿))) |
3wlkd.v | ⊢ 𝑉 = (Vtx‘𝐺) |
3wlkd.i | ⊢ 𝐼 = (iEdg‘𝐺) |
3trld.n | ⊢ (𝜑 → (𝐽 ≠ 𝐾 ∧ 𝐽 ≠ 𝐿 ∧ 𝐾 ≠ 𝐿)) |
3cycld.e | ⊢ (𝜑 → 𝐴 = 𝐷) |
Ref | Expression |
---|---|
3cyclpd | ⊢ (𝜑 → (𝐹(Cycles‘𝐺)𝑃 ∧ (♯‘𝐹) = 3 ∧ (𝑃‘0) = 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3wlkd.p | . . 3 ⊢ 𝑃 = 〈“𝐴𝐵𝐶𝐷”〉 | |
2 | 3wlkd.f | . . 3 ⊢ 𝐹 = 〈“𝐽𝐾𝐿”〉 | |
3 | 3wlkd.s | . . 3 ⊢ (𝜑 → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉))) | |
4 | 3wlkd.n | . . 3 ⊢ (𝜑 → ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ∧ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷) ∧ 𝐶 ≠ 𝐷)) | |
5 | 3wlkd.e | . . 3 ⊢ (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾) ∧ {𝐶, 𝐷} ⊆ (𝐼‘𝐿))) | |
6 | 3wlkd.v | . . 3 ⊢ 𝑉 = (Vtx‘𝐺) | |
7 | 3wlkd.i | . . 3 ⊢ 𝐼 = (iEdg‘𝐺) | |
8 | 3trld.n | . . 3 ⊢ (𝜑 → (𝐽 ≠ 𝐾 ∧ 𝐽 ≠ 𝐿 ∧ 𝐾 ≠ 𝐿)) | |
9 | 3cycld.e | . . 3 ⊢ (𝜑 → 𝐴 = 𝐷) | |
10 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | 3cycld 28215 | . 2 ⊢ (𝜑 → 𝐹(Cycles‘𝐺)𝑃) |
11 | 2 | fveq2i 6698 | . . . 4 ⊢ (♯‘𝐹) = (♯‘〈“𝐽𝐾𝐿”〉) |
12 | s3len 14424 | . . . 4 ⊢ (♯‘〈“𝐽𝐾𝐿”〉) = 3 | |
13 | 11, 12 | eqtri 2759 | . . 3 ⊢ (♯‘𝐹) = 3 |
14 | 13 | a1i 11 | . 2 ⊢ (𝜑 → (♯‘𝐹) = 3) |
15 | 1 | fveq1i 6696 | . . . . 5 ⊢ (𝑃‘0) = (〈“𝐴𝐵𝐶𝐷”〉‘0) |
16 | s4fv0 14425 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → (〈“𝐴𝐵𝐶𝐷”〉‘0) = 𝐴) | |
17 | 15, 16 | syl5eq 2783 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝑃‘0) = 𝐴) |
18 | 17 | ad2antrr 726 | . . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (𝑃‘0) = 𝐴) |
19 | 3, 18 | syl 17 | . 2 ⊢ (𝜑 → (𝑃‘0) = 𝐴) |
20 | 10, 14, 19 | 3jca 1130 | 1 ⊢ (𝜑 → (𝐹(Cycles‘𝐺)𝑃 ∧ (♯‘𝐹) = 3 ∧ (𝑃‘0) = 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2112 ≠ wne 2932 ⊆ wss 3853 {cpr 4529 class class class wbr 5039 ‘cfv 6358 0cc0 10694 3c3 11851 ♯chash 13861 〈“cs3 14372 〈“cs4 14373 Vtxcvtx 27041 iEdgciedg 27042 Cyclesccycls 27826 |
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 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 |
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 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-int 4846 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-er 8369 df-map 8488 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-card 9520 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-nn 11796 df-2 11858 df-3 11859 df-4 11860 df-n0 12056 df-z 12142 df-uz 12404 df-fz 13061 df-fzo 13204 df-hash 13862 df-word 14035 df-concat 14091 df-s1 14118 df-s2 14378 df-s3 14379 df-s4 14380 df-wlks 27641 df-trls 27734 df-pths 27757 df-cycls 27828 |
This theorem is referenced by: uhgr3cyclex 28219 |
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