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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 | β’ π = β¨βπ΄π΅πΆπ·ββ© |
3wlkd.f | β’ πΉ = β¨βπ½πΎπΏββ© |
3wlkd.s | β’ (π β ((π΄ β π β§ π΅ β π) β§ (πΆ β π β§ π· β π))) |
3wlkd.n | β’ (π β ((π΄ β π΅ β§ π΄ β πΆ) β§ (π΅ β πΆ β§ π΅ β π·) β§ πΆ β π·)) |
3wlkd.e | β’ (π β ({π΄, π΅} β (πΌβπ½) β§ {π΅, πΆ} β (πΌβπΎ) β§ {πΆ, π·} β (πΌβπΏ))) |
3wlkd.v | β’ π = (VtxβπΊ) |
3wlkd.i | β’ πΌ = (iEdgβπΊ) |
3trld.n | β’ (π β (π½ β πΎ β§ π½ β πΏ β§ πΎ β πΏ)) |
3cycld.e | β’ (π β π΄ = π·) |
Ref | Expression |
---|---|
3cycld | β’ (π β πΉ(CyclesβπΊ)π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3wlkd.p | . . 3 β’ π = β¨βπ΄π΅πΆπ·ββ© | |
2 | 3wlkd.f | . . 3 β’ πΉ = β¨βπ½πΎπΏββ© | |
3 | 3wlkd.s | . . 3 β’ (π β ((π΄ β π β§ π΅ β π) β§ (πΆ β π β§ π· β π))) | |
4 | 3wlkd.n | . . 3 β’ (π β ((π΄ β π΅ β§ π΄ β πΆ) β§ (π΅ β πΆ β§ π΅ β π·) β§ πΆ β π·)) | |
5 | 3wlkd.e | . . 3 β’ (π β ({π΄, π΅} β (πΌβπ½) β§ {π΅, πΆ} β (πΌβπΎ) β§ {πΆ, π·} β (πΌβπΏ))) | |
6 | 3wlkd.v | . . 3 β’ π = (VtxβπΊ) | |
7 | 3wlkd.i | . . 3 β’ πΌ = (iEdgβπΊ) | |
8 | 3trld.n | . . 3 β’ (π β (π½ β πΎ β§ π½ β πΏ β§ πΎ β πΏ)) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | 3pthd 29922 | . 2 β’ (π β πΉ(PathsβπΊ)π) |
10 | 3cycld.e | . . 3 β’ (π β π΄ = π·) | |
11 | 1 | fveq1i 6883 | . . . . . 6 β’ (πβ0) = (β¨βπ΄π΅πΆπ·ββ©β0) |
12 | s4fv0 14848 | . . . . . 6 β’ (π΄ β π β (β¨βπ΄π΅πΆπ·ββ©β0) = π΄) | |
13 | 11, 12 | eqtrid 2776 | . . . . 5 β’ (π΄ β π β (πβ0) = π΄) |
14 | 13 | ad3antrrr 727 | . . . 4 β’ ((((π΄ β π β§ π΅ β π) β§ (πΆ β π β§ π· β π)) β§ π΄ = π·) β (πβ0) = π΄) |
15 | simpr 484 | . . . 4 β’ ((((π΄ β π β§ π΅ β π) β§ (πΆ β π β§ π· β π)) β§ π΄ = π·) β π΄ = π·) | |
16 | 2 | fveq2i 6885 | . . . . . . . . 9 β’ (β―βπΉ) = (β―ββ¨βπ½πΎπΏββ©) |
17 | s3len 14847 | . . . . . . . . 9 β’ (β―ββ¨βπ½πΎπΏββ©) = 3 | |
18 | 16, 17 | eqtri 2752 | . . . . . . . 8 β’ (β―βπΉ) = 3 |
19 | 1, 18 | fveq12i 6888 | . . . . . . 7 β’ (πβ(β―βπΉ)) = (β¨βπ΄π΅πΆπ·ββ©β3) |
20 | s4fv3 14851 | . . . . . . 7 β’ (π· β π β (β¨βπ΄π΅πΆπ·ββ©β3) = π·) | |
21 | 19, 20 | eqtr2id 2777 | . . . . . 6 β’ (π· β π β π· = (πβ(β―βπΉ))) |
22 | 21 | adantl 481 | . . . . 5 β’ ((πΆ β π β§ π· β π) β π· = (πβ(β―βπΉ))) |
23 | 22 | ad2antlr 724 | . . . 4 β’ ((((π΄ β π β§ π΅ β π) β§ (πΆ β π β§ π· β π)) β§ π΄ = π·) β π· = (πβ(β―βπΉ))) |
24 | 14, 15, 23 | 3eqtrd 2768 | . . 3 β’ ((((π΄ β π β§ π΅ β π) β§ (πΆ β π β§ π· β π)) β§ π΄ = π·) β (πβ0) = (πβ(β―βπΉ))) |
25 | 3, 10, 24 | syl2anc 583 | . 2 β’ (π β (πβ0) = (πβ(β―βπΉ))) |
26 | iscycl 29543 | . 2 β’ (πΉ(CyclesβπΊ)π β (πΉ(PathsβπΊ)π β§ (πβ0) = (πβ(β―βπΉ)))) | |
27 | 9, 25, 26 | sylanbrc 582 | 1 β’ (π β πΉ(CyclesβπΊ)π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 β§ w3a 1084 = wceq 1533 β wcel 2098 β wne 2932 β wss 3941 {cpr 4623 class class class wbr 5139 βcfv 6534 0cc0 11107 3c3 12267 β―chash 14291 β¨βcs3 14795 β¨βcs4 14796 Vtxcvtx 28750 iEdgciedg 28751 Pathscpths 29464 Cyclesccycls 29537 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-ifp 1060 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-tp 4626 df-op 4628 df-uni 4901 df-int 4942 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8700 df-map 8819 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-card 9931 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-n0 12472 df-z 12558 df-uz 12822 df-fz 13486 df-fzo 13629 df-hash 14292 df-word 14467 df-concat 14523 df-s1 14548 df-s2 14801 df-s3 14802 df-s4 14803 df-wlks 29351 df-trls 29444 df-pths 29468 df-cycls 29539 |
This theorem is referenced by: 3cyclpd 29927 |
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