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Mirrors > Home > MPE Home > Th. List > 3pthd | Structured version Visualization version GIF version |
Description: A path of length 3 from one vertex to another vertex via a third vertex. (Contributed by Alexander van der Vekens, 6-Dec-2017.) (Revised by AV, 10-Feb-2021.) (Revised by AV, 24-Mar-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)) |
Ref | Expression |
---|---|
3pthd | β’ (π β πΉ(PathsβπΊ)π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3wlkd.p | . . . 4 β’ π = β¨βπ΄π΅πΆπ·ββ© | |
2 | s4cli 14832 | . . . 4 β’ β¨βπ΄π΅πΆπ·ββ© β Word V | |
3 | 1, 2 | eqeltri 2829 | . . 3 β’ π β Word V |
4 | 3 | a1i 11 | . 2 β’ (π β π β Word V) |
5 | 3wlkd.f | . . . . 5 β’ πΉ = β¨βπ½πΎπΏββ© | |
6 | 5 | fveq2i 6894 | . . . 4 β’ (β―βπΉ) = (β―ββ¨βπ½πΎπΏββ©) |
7 | s3len 14844 | . . . 4 β’ (β―ββ¨βπ½πΎπΏββ©) = 3 | |
8 | 6, 7 | eqtri 2760 | . . 3 β’ (β―βπΉ) = 3 |
9 | 4m1e3 12340 | . . 3 β’ (4 β 1) = 3 | |
10 | 1 | fveq2i 6894 | . . . . 5 β’ (β―βπ) = (β―ββ¨βπ΄π΅πΆπ·ββ©) |
11 | s4len 14849 | . . . . 5 β’ (β―ββ¨βπ΄π΅πΆπ·ββ©) = 4 | |
12 | 10, 11 | eqtr2i 2761 | . . . 4 β’ 4 = (β―βπ) |
13 | 12 | oveq1i 7418 | . . 3 β’ (4 β 1) = ((β―βπ) β 1) |
14 | 8, 9, 13 | 3eqtr2i 2766 | . 2 β’ (β―βπΉ) = ((β―βπ) β 1) |
15 | 3wlkd.s | . . 3 β’ (π β ((π΄ β π β§ π΅ β π) β§ (πΆ β π β§ π· β π))) | |
16 | 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·)) | |
17 | 1, 5, 15, 16 | 3pthdlem1 29414 | . 2 β’ (π β βπ β (0..^(β―βπ))βπ β (1..^(β―βπΉ))(π β π β (πβπ) β (πβπ))) |
18 | eqid 2732 | . 2 β’ (β―βπΉ) = (β―βπΉ) | |
19 | 3wlkd.e | . . 3 β’ (π β ({π΄, π΅} β (πΌβπ½) β§ {π΅, πΆ} β (πΌβπΎ) β§ {πΆ, π·} β (πΌβπΏ))) | |
20 | 3wlkd.v | . . 3 β’ π = (VtxβπΊ) | |
21 | 3wlkd.i | . . 3 β’ πΌ = (iEdgβπΊ) | |
22 | 3trld.n | . . 3 β’ (π β (π½ β πΎ β§ π½ β πΏ β§ πΎ β πΏ)) | |
23 | 1, 5, 15, 16, 19, 20, 21, 22 | 3trld 29422 | . 2 β’ (π β πΉ(TrailsβπΊ)π) |
24 | 4, 14, 17, 18, 23 | pthd 29023 | 1 β’ (π β πΉ(PathsβπΊ)π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 396 β§ w3a 1087 = wceq 1541 β wcel 2106 β wne 2940 Vcvv 3474 β wss 3948 {cpr 4630 class class class wbr 5148 βcfv 6543 (class class class)co 7408 1c1 11110 β cmin 11443 3c3 12267 4c4 12268 β―chash 14289 Word cword 14463 β¨βcs3 14792 β¨βcs4 14793 Vtxcvtx 28253 iEdgciedg 28254 Pathscpths 28966 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-ifp 1062 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7855 df-1st 7974 df-2nd 7975 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-1o 8465 df-er 8702 df-map 8821 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-card 9933 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 13484 df-fzo 13627 df-hash 14290 df-word 14464 df-concat 14520 df-s1 14545 df-s2 14798 df-s3 14799 df-s4 14800 df-wlks 28853 df-trls 28946 df-pths 28970 |
This theorem is referenced by: 3pthond 29425 3cycld 29428 |
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