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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 29997 | . 2 β’ (π β πΉ(PathsβπΊ)π) |
10 | 3cycld.e | . . 3 β’ (π β π΄ = π·) | |
11 | 1 | fveq1i 6898 | . . . . . 6 β’ (πβ0) = (β¨βπ΄π΅πΆπ·ββ©β0) |
12 | s4fv0 14879 | . . . . . 6 β’ (π΄ β π β (β¨βπ΄π΅πΆπ·ββ©β0) = π΄) | |
13 | 11, 12 | eqtrid 2780 | . . . . 5 β’ (π΄ β π β (πβ0) = π΄) |
14 | 13 | ad3antrrr 729 | . . . 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 6900 | . . . . . . . . 9 β’ (β―βπΉ) = (β―ββ¨βπ½πΎπΏββ©) |
17 | s3len 14878 | . . . . . . . . 9 β’ (β―ββ¨βπ½πΎπΏββ©) = 3 | |
18 | 16, 17 | eqtri 2756 | . . . . . . . 8 β’ (β―βπΉ) = 3 |
19 | 1, 18 | fveq12i 6903 | . . . . . . 7 β’ (πβ(β―βπΉ)) = (β¨βπ΄π΅πΆπ·ββ©β3) |
20 | s4fv3 14882 | . . . . . . 7 β’ (π· β π β (β¨βπ΄π΅πΆπ·ββ©β3) = π·) | |
21 | 19, 20 | eqtr2id 2781 | . . . . . 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 726 | . . . 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 2772 | . . 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 29618 | . 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 1085 = wceq 1534 β wcel 2099 β wne 2937 β wss 3947 {cpr 4631 class class class wbr 5148 βcfv 6548 0cc0 11139 3c3 12299 β―chash 14322 β¨βcs3 14826 β¨βcs4 14827 Vtxcvtx 28822 iEdgciedg 28823 Pathscpths 29539 Cyclesccycls 29612 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-ifp 1062 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-card 9963 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-nn 12244 df-2 12306 df-3 12307 df-4 12308 df-n0 12504 df-z 12590 df-uz 12854 df-fz 13518 df-fzo 13661 df-hash 14323 df-word 14498 df-concat 14554 df-s1 14579 df-s2 14832 df-s3 14833 df-s4 14834 df-wlks 29426 df-trls 29519 df-pths 29543 df-cycls 29614 |
This theorem is referenced by: 3cyclpd 30002 |
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