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Mirrors > Home > MPE Home > Th. List > Mathboxes > 2cycld | Structured version Visualization version GIF version |
Description: Construction of a 2-cycle from two given edges in a graph. (Contributed by BTernaryTau, 16-Oct-2023.) |
Ref | Expression |
---|---|
2cycld.1 | ⊢ 𝑃 = 〈“𝐴𝐵𝐶”〉 |
2cycld.2 | ⊢ 𝐹 = 〈“𝐽𝐾”〉 |
2cycld.3 | ⊢ (𝜑 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) |
2cycld.4 | ⊢ (𝜑 → (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶)) |
2cycld.5 | ⊢ (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾))) |
2cycld.6 | ⊢ 𝑉 = (Vtx‘𝐺) |
2cycld.7 | ⊢ 𝐼 = (iEdg‘𝐺) |
2cycld.8 | ⊢ (𝜑 → 𝐽 ≠ 𝐾) |
2cycld.9 | ⊢ (𝜑 → 𝐴 = 𝐶) |
Ref | Expression |
---|---|
2cycld | ⊢ (𝜑 → 𝐹(Cycles‘𝐺)𝑃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2cycld.1 | . . 3 ⊢ 𝑃 = 〈“𝐴𝐵𝐶”〉 | |
2 | 2cycld.2 | . . 3 ⊢ 𝐹 = 〈“𝐽𝐾”〉 | |
3 | 2cycld.3 | . . 3 ⊢ (𝜑 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) | |
4 | 2cycld.4 | . . 3 ⊢ (𝜑 → (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶)) | |
5 | 2cycld.5 | . . 3 ⊢ (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾))) | |
6 | 2cycld.6 | . . 3 ⊢ 𝑉 = (Vtx‘𝐺) | |
7 | 2cycld.7 | . . 3 ⊢ 𝐼 = (iEdg‘𝐺) | |
8 | 2cycld.8 | . . 3 ⊢ (𝜑 → 𝐽 ≠ 𝐾) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | 2pthd 27870 | . 2 ⊢ (𝜑 → 𝐹(Paths‘𝐺)𝑃) |
10 | 2cycld.9 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐶) | |
11 | 1 | fveq1i 6669 | . . . . . . 7 ⊢ (𝑃‘0) = (〈“𝐴𝐵𝐶”〉‘0) |
12 | s3fv0 14335 | . . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (〈“𝐴𝐵𝐶”〉‘0) = 𝐴) | |
13 | 11, 12 | syl5eq 2785 | . . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝑃‘0) = 𝐴) |
14 | 13 | 3ad2ant1 1134 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝑃‘0) = 𝐴) |
15 | 14 | adantr 484 | . . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝐴 = 𝐶) → (𝑃‘0) = 𝐴) |
16 | simpr 488 | . . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝐴 = 𝐶) → 𝐴 = 𝐶) | |
17 | 2 | fveq2i 6671 | . . . . . . . . 9 ⊢ (♯‘𝐹) = (♯‘〈“𝐽𝐾”〉) |
18 | s2len 14333 | . . . . . . . . 9 ⊢ (♯‘〈“𝐽𝐾”〉) = 2 | |
19 | 17, 18 | eqtri 2761 | . . . . . . . 8 ⊢ (♯‘𝐹) = 2 |
20 | 1, 19 | fveq12i 6674 | . . . . . . 7 ⊢ (𝑃‘(♯‘𝐹)) = (〈“𝐴𝐵𝐶”〉‘2) |
21 | s3fv2 14337 | . . . . . . 7 ⊢ (𝐶 ∈ 𝑉 → (〈“𝐴𝐵𝐶”〉‘2) = 𝐶) | |
22 | 20, 21 | eqtr2id 2786 | . . . . . 6 ⊢ (𝐶 ∈ 𝑉 → 𝐶 = (𝑃‘(♯‘𝐹))) |
23 | 22 | 3ad2ant3 1136 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → 𝐶 = (𝑃‘(♯‘𝐹))) |
24 | 23 | adantr 484 | . . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝐴 = 𝐶) → 𝐶 = (𝑃‘(♯‘𝐹))) |
25 | 15, 16, 24 | 3eqtrd 2777 | . . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝐴 = 𝐶) → (𝑃‘0) = (𝑃‘(♯‘𝐹))) |
26 | 3, 10, 25 | syl2anc 587 | . 2 ⊢ (𝜑 → (𝑃‘0) = (𝑃‘(♯‘𝐹))) |
27 | iscycl 27724 | . 2 ⊢ (𝐹(Cycles‘𝐺)𝑃 ↔ (𝐹(Paths‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹)))) | |
28 | 9, 26, 27 | sylanbrc 586 | 1 ⊢ (𝜑 → 𝐹(Cycles‘𝐺)𝑃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1088 = wceq 1542 ∈ wcel 2113 ≠ wne 2934 ⊆ wss 3841 {cpr 4515 class class class wbr 5027 ‘cfv 6333 0cc0 10608 2c2 11764 ♯chash 13775 〈“cs2 14285 〈“cs3 14286 Vtxcvtx 26933 iEdgciedg 26934 Pathscpths 27645 Cyclesccycls 27718 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-rep 5151 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 ax-cnex 10664 ax-resscn 10665 ax-1cn 10666 ax-icn 10667 ax-addcl 10668 ax-addrcl 10669 ax-mulcl 10670 ax-mulrcl 10671 ax-mulcom 10672 ax-addass 10673 ax-mulass 10674 ax-distr 10675 ax-i2m1 10676 ax-1ne0 10677 ax-1rid 10678 ax-rnegex 10679 ax-rrecex 10680 ax-cnre 10681 ax-pre-lttri 10682 ax-pre-lttrn 10683 ax-pre-ltadd 10684 ax-pre-mulgt0 10685 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-ifp 1063 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-pss 3860 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-tp 4518 df-op 4520 df-uni 4794 df-int 4834 df-iun 4880 df-br 5028 df-opab 5090 df-mpt 5108 df-tr 5134 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 6123 df-ord 6169 df-on 6170 df-lim 6171 df-suc 6172 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-riota 7121 df-ov 7167 df-oprab 7168 df-mpo 7169 df-om 7594 df-1st 7707 df-2nd 7708 df-wrecs 7969 df-recs 8030 df-rdg 8068 df-1o 8124 df-er 8313 df-map 8432 df-en 8549 df-dom 8550 df-sdom 8551 df-fin 8552 df-card 9434 df-pnf 10748 df-mnf 10749 df-xr 10750 df-ltxr 10751 df-le 10752 df-sub 10943 df-neg 10944 df-nn 11710 df-2 11772 df-3 11773 df-n0 11970 df-z 12056 df-uz 12318 df-fz 12975 df-fzo 13118 df-hash 13776 df-word 13949 df-concat 14005 df-s1 14032 df-s2 14292 df-s3 14293 df-wlks 27533 df-trls 27626 df-pths 27649 df-cycls 27720 |
This theorem is referenced by: 2cycl2d 32664 |
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