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Mirrors > Home > MPE Home > Th. List > 3cycld | Structured version Visualization version GIF version |
Description: Construction of a 3-cycle from three given edges in a graph. (Contributed by Alexander van der Vekens, 13-Nov-2017.) (Revised by AV, 10-Feb-2021.) (Revised by AV, 24-Mar-2021.) (Proof shortened by AV, 30-Oct-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 |
---|---|
3cycld | ⊢ (𝜑 → 𝐹(Cycles‘𝐺)𝑃) |
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 | 1, 2, 3, 4, 5, 6, 7, 8 | 3pthd 27959 | . 2 ⊢ (𝜑 → 𝐹(Paths‘𝐺)𝑃) |
10 | 3cycld.e | . . 3 ⊢ (𝜑 → 𝐴 = 𝐷) | |
11 | 1 | fveq1i 6646 | . . . . . 6 ⊢ (𝑃‘0) = (〈“𝐴𝐵𝐶𝐷”〉‘0) |
12 | s4fv0 14248 | . . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (〈“𝐴𝐵𝐶𝐷”〉‘0) = 𝐴) | |
13 | 11, 12 | syl5eq 2845 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝑃‘0) = 𝐴) |
14 | 13 | ad3antrrr 729 | . . . 4 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) ∧ 𝐴 = 𝐷) → (𝑃‘0) = 𝐴) |
15 | simpr 488 | . . . 4 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) ∧ 𝐴 = 𝐷) → 𝐴 = 𝐷) | |
16 | 2 | fveq2i 6648 | . . . . . . . . 9 ⊢ (♯‘𝐹) = (♯‘〈“𝐽𝐾𝐿”〉) |
17 | s3len 14247 | . . . . . . . . 9 ⊢ (♯‘〈“𝐽𝐾𝐿”〉) = 3 | |
18 | 16, 17 | eqtri 2821 | . . . . . . . 8 ⊢ (♯‘𝐹) = 3 |
19 | 1, 18 | fveq12i 6651 | . . . . . . 7 ⊢ (𝑃‘(♯‘𝐹)) = (〈“𝐴𝐵𝐶𝐷”〉‘3) |
20 | s4fv3 14251 | . . . . . . 7 ⊢ (𝐷 ∈ 𝑉 → (〈“𝐴𝐵𝐶𝐷”〉‘3) = 𝐷) | |
21 | 19, 20 | syl5req 2846 | . . . . . 6 ⊢ (𝐷 ∈ 𝑉 → 𝐷 = (𝑃‘(♯‘𝐹))) |
22 | 21 | adantl 485 | . . . . 5 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉) → 𝐷 = (𝑃‘(♯‘𝐹))) |
23 | 22 | ad2antlr 726 | . . . 4 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) ∧ 𝐴 = 𝐷) → 𝐷 = (𝑃‘(♯‘𝐹))) |
24 | 14, 15, 23 | 3eqtrd 2837 | . . 3 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) ∧ 𝐴 = 𝐷) → (𝑃‘0) = (𝑃‘(♯‘𝐹))) |
25 | 3, 10, 24 | syl2anc 587 | . 2 ⊢ (𝜑 → (𝑃‘0) = (𝑃‘(♯‘𝐹))) |
26 | iscycl 27580 | . 2 ⊢ (𝐹(Cycles‘𝐺)𝑃 ↔ (𝐹(Paths‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹)))) | |
27 | 9, 25, 26 | sylanbrc 586 | 1 ⊢ (𝜑 → 𝐹(Cycles‘𝐺)𝑃) |
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 3c3 11681 ♯chash 13686 〈“cs3 14195 〈“cs4 14196 Vtxcvtx 26789 iEdgciedg 26790 Pathscpths 27501 Cyclesccycls 27574 |
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-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-4 11690 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-s4 14203 df-wlks 27389 df-trls 27482 df-pths 27505 df-cycls 27576 |
This theorem is referenced by: 3cyclpd 27964 |
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