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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 29970 | . 2 ⊢ (𝜑 → 𝐹(Paths‘𝐺)𝑃) |
10 | 2cycld.9 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐶) | |
11 | 1 | fveq1i 6908 | . . . . . . 7 ⊢ (𝑃‘0) = (〈“𝐴𝐵𝐶”〉‘0) |
12 | s3fv0 14927 | . . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (〈“𝐴𝐵𝐶”〉‘0) = 𝐴) | |
13 | 11, 12 | eqtrid 2787 | . . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝑃‘0) = 𝐴) |
14 | 13 | 3ad2ant1 1132 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝑃‘0) = 𝐴) |
15 | 14 | adantr 480 | . . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝐴 = 𝐶) → (𝑃‘0) = 𝐴) |
16 | simpr 484 | . . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝐴 = 𝐶) → 𝐴 = 𝐶) | |
17 | 2 | fveq2i 6910 | . . . . . . . . 9 ⊢ (♯‘𝐹) = (♯‘〈“𝐽𝐾”〉) |
18 | s2len 14925 | . . . . . . . . 9 ⊢ (♯‘〈“𝐽𝐾”〉) = 2 | |
19 | 17, 18 | eqtri 2763 | . . . . . . . 8 ⊢ (♯‘𝐹) = 2 |
20 | 1, 19 | fveq12i 6913 | . . . . . . 7 ⊢ (𝑃‘(♯‘𝐹)) = (〈“𝐴𝐵𝐶”〉‘2) |
21 | s3fv2 14929 | . . . . . . 7 ⊢ (𝐶 ∈ 𝑉 → (〈“𝐴𝐵𝐶”〉‘2) = 𝐶) | |
22 | 20, 21 | eqtr2id 2788 | . . . . . 6 ⊢ (𝐶 ∈ 𝑉 → 𝐶 = (𝑃‘(♯‘𝐹))) |
23 | 22 | 3ad2ant3 1134 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → 𝐶 = (𝑃‘(♯‘𝐹))) |
24 | 23 | adantr 480 | . . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝐴 = 𝐶) → 𝐶 = (𝑃‘(♯‘𝐹))) |
25 | 15, 16, 24 | 3eqtrd 2779 | . . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝐴 = 𝐶) → (𝑃‘0) = (𝑃‘(♯‘𝐹))) |
26 | 3, 10, 25 | syl2anc 584 | . 2 ⊢ (𝜑 → (𝑃‘0) = (𝑃‘(♯‘𝐹))) |
27 | iscycl 29824 | . 2 ⊢ (𝐹(Cycles‘𝐺)𝑃 ↔ (𝐹(Paths‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹)))) | |
28 | 9, 26, 27 | sylanbrc 583 | 1 ⊢ (𝜑 → 𝐹(Cycles‘𝐺)𝑃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ⊆ wss 3963 {cpr 4633 class class class wbr 5148 ‘cfv 6563 0cc0 11153 2c2 12319 ♯chash 14366 〈“cs2 14877 〈“cs3 14878 Vtxcvtx 29028 iEdgciedg 29029 Pathscpths 29745 Cyclesccycls 29818 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-ifp 1063 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12612 df-uz 12877 df-fz 13545 df-fzo 13692 df-hash 14367 df-word 14550 df-concat 14606 df-s1 14631 df-s2 14884 df-s3 14885 df-wlks 29632 df-trls 29725 df-pths 29749 df-cycls 29820 |
This theorem is referenced by: 2cycl2d 35124 |
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