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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 14865 | . . . 4 β’ β¨βπ΄π΅πΆπ·ββ© β Word V | |
3 | 1, 2 | eqeltri 2825 | . . 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 6900 | . . . 4 β’ (β―βπΉ) = (β―ββ¨βπ½πΎπΏββ©) |
7 | s3len 14877 | . . . 4 β’ (β―ββ¨βπ½πΎπΏββ©) = 3 | |
8 | 6, 7 | eqtri 2756 | . . 3 β’ (β―βπΉ) = 3 |
9 | 4m1e3 12371 | . . 3 β’ (4 β 1) = 3 | |
10 | 1 | fveq2i 6900 | . . . . 5 β’ (β―βπ) = (β―ββ¨βπ΄π΅πΆπ·ββ©) |
11 | s4len 14882 | . . . . 5 β’ (β―ββ¨βπ΄π΅πΆπ·ββ©) = 4 | |
12 | 10, 11 | eqtr2i 2757 | . . . 4 β’ 4 = (β―βπ) |
13 | 12 | oveq1i 7430 | . . 3 β’ (4 β 1) = ((β―βπ) β 1) |
14 | 8, 9, 13 | 3eqtr2i 2762 | . 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 29973 | . 2 β’ (π β βπ β (0..^(β―βπ))βπ β (1..^(β―βπΉ))(π β π β (πβπ) β (πβπ))) |
18 | eqid 2728 | . 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 29981 | . 2 β’ (π β πΉ(TrailsβπΊ)π) |
24 | 4, 14, 17, 18, 23 | pthd 29582 | 1 β’ (π β πΉ(PathsβπΊ)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 Vcvv 3471 β wss 3947 {cpr 4631 class class class wbr 5148 βcfv 6548 (class class class)co 7420 1c1 11139 β cmin 11474 3c3 12298 4c4 12299 β―chash 14321 Word cword 14496 β¨βcs3 14825 β¨βcs4 14826 Vtxcvtx 28808 iEdgciedg 28809 Pathscpths 29525 |
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 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 |
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 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-1o 8486 df-er 8724 df-map 8846 df-en 8964 df-dom 8965 df-sdom 8966 df-fin 8967 df-card 9962 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-n0 12503 df-z 12589 df-uz 12853 df-fz 13517 df-fzo 13660 df-hash 14322 df-word 14497 df-concat 14553 df-s1 14578 df-s2 14831 df-s3 14832 df-s4 14833 df-wlks 29412 df-trls 29505 df-pths 29529 |
This theorem is referenced by: 3pthond 29984 3cycld 29987 |
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