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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 27717 | . 2 ⊢ (𝜑 → 𝐹(Paths‘𝐺)𝑃) |
10 | 2cycld.9 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐶) | |
11 | 1 | fveq1i 6664 | . . . . . . 7 ⊢ (𝑃‘0) = (〈“𝐴𝐵𝐶”〉‘0) |
12 | s3fv0 14248 | . . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (〈“𝐴𝐵𝐶”〉‘0) = 𝐴) | |
13 | 11, 12 | syl5eq 2867 | . . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝑃‘0) = 𝐴) |
14 | 13 | 3ad2ant1 1128 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝑃‘0) = 𝐴) |
15 | 14 | adantr 483 | . . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝐴 = 𝐶) → (𝑃‘0) = 𝐴) |
16 | simpr 487 | . . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝐴 = 𝐶) → 𝐴 = 𝐶) | |
17 | 2 | fveq2i 6666 | . . . . . . . . 9 ⊢ (♯‘𝐹) = (♯‘〈“𝐽𝐾”〉) |
18 | s2len 14246 | . . . . . . . . 9 ⊢ (♯‘〈“𝐽𝐾”〉) = 2 | |
19 | 17, 18 | eqtri 2843 | . . . . . . . 8 ⊢ (♯‘𝐹) = 2 |
20 | 1, 19 | fveq12i 6669 | . . . . . . 7 ⊢ (𝑃‘(♯‘𝐹)) = (〈“𝐴𝐵𝐶”〉‘2) |
21 | s3fv2 14250 | . . . . . . 7 ⊢ (𝐶 ∈ 𝑉 → (〈“𝐴𝐵𝐶”〉‘2) = 𝐶) | |
22 | 20, 21 | syl5req 2868 | . . . . . 6 ⊢ (𝐶 ∈ 𝑉 → 𝐶 = (𝑃‘(♯‘𝐹))) |
23 | 22 | 3ad2ant3 1130 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → 𝐶 = (𝑃‘(♯‘𝐹))) |
24 | 23 | adantr 483 | . . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝐴 = 𝐶) → 𝐶 = (𝑃‘(♯‘𝐹))) |
25 | 15, 16, 24 | 3eqtrd 2859 | . . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝐴 = 𝐶) → (𝑃‘0) = (𝑃‘(♯‘𝐹))) |
26 | 3, 10, 25 | syl2anc 586 | . 2 ⊢ (𝜑 → (𝑃‘0) = (𝑃‘(♯‘𝐹))) |
27 | iscycl 27570 | . 2 ⊢ (𝐹(Cycles‘𝐺)𝑃 ↔ (𝐹(Paths‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹)))) | |
28 | 9, 26, 27 | sylanbrc 585 | 1 ⊢ (𝜑 → 𝐹(Cycles‘𝐺)𝑃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1082 = wceq 1536 ∈ wcel 2113 ≠ wne 3015 ⊆ wss 3929 {cpr 4562 class class class wbr 5059 ‘cfv 6348 0cc0 10530 2c2 11686 ♯chash 13687 〈“cs2 14198 〈“cs3 14199 Vtxcvtx 26779 iEdgciedg 26780 Pathscpths 27491 Cyclesccycls 27564 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-cnex 10586 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 ax-pre-mulgt0 10607 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ifp 1058 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-om 7574 df-1st 7682 df-2nd 7683 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-1o 8095 df-oadd 8099 df-er 8282 df-map 8401 df-en 8503 df-dom 8504 df-sdom 8505 df-fin 8506 df-card 9361 df-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 df-le 10674 df-sub 10865 df-neg 10866 df-nn 11632 df-2 11694 df-3 11695 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12890 df-fzo 13031 df-hash 13688 df-word 13859 df-concat 13918 df-s1 13945 df-s2 14205 df-s3 14206 df-wlks 27379 df-trls 27472 df-pths 27495 df-cycls 27566 |
This theorem is referenced by: 2cycl2d 32407 |
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