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Mirrors > Home > MPE Home > Th. List > 3trlond | Structured version Visualization version GIF version |
Description: A trail of length 3 from one vertex to another, different vertex via a third vertex. (Contributed by AV, 8-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 |
---|---|
3trlond | β’ (π β πΉ(π΄(TrailsOnβπΊ)π·)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 | 1, 2, 3, 4, 5, 6, 7 | 3wlkond 29424 | . 2 β’ (π β πΉ(π΄(WalksOnβπΊ)π·)π) |
9 | 3trld.n | . . 3 β’ (π β (π½ β πΎ β§ π½ β πΏ β§ πΎ β πΏ)) | |
10 | 1, 2, 3, 4, 5, 6, 7, 9 | 3trld 29425 | . 2 β’ (π β πΉ(TrailsβπΊ)π) |
11 | 3 | simplld 767 | . . 3 β’ (π β π΄ β π) |
12 | 3 | simprrd 773 | . . 3 β’ (π β π· β π) |
13 | s3cli 14832 | . . . . . 6 β’ β¨βπ½πΎπΏββ© β Word V | |
14 | 2, 13 | eqeltri 2830 | . . . . 5 β’ πΉ β Word V |
15 | s4cli 14833 | . . . . . 6 β’ β¨βπ΄π΅πΆπ·ββ© β Word V | |
16 | 1, 15 | eqeltri 2830 | . . . . 5 β’ π β Word V |
17 | 14, 16 | pm3.2i 472 | . . . 4 β’ (πΉ β Word V β§ π β Word V) |
18 | 17 | a1i 11 | . . 3 β’ (π β (πΉ β Word V β§ π β Word V)) |
19 | 6 | istrlson 28964 | . . 3 β’ (((π΄ β π β§ π· β π) β§ (πΉ β Word V β§ π β Word V)) β (πΉ(π΄(TrailsOnβπΊ)π·)π β (πΉ(π΄(WalksOnβπΊ)π·)π β§ πΉ(TrailsβπΊ)π))) |
20 | 11, 12, 18, 19 | syl21anc 837 | . 2 β’ (π β (πΉ(π΄(TrailsOnβπΊ)π·)π β (πΉ(π΄(WalksOnβπΊ)π·)π β§ πΉ(TrailsβπΊ)π))) |
21 | 8, 10, 20 | mpbir2and 712 | 1 β’ (π β πΉ(π΄(TrailsOnβπΊ)π·)π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 397 β§ w3a 1088 = wceq 1542 β wcel 2107 β wne 2941 Vcvv 3475 β wss 3949 {cpr 4631 class class class wbr 5149 βcfv 6544 (class class class)co 7409 Word cword 14464 β¨βcs3 14793 β¨βcs4 14794 Vtxcvtx 28256 iEdgciedg 28257 WalksOncwlkson 28854 Trailsctrls 28947 TrailsOnctrlson 28948 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-ifp 1063 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-er 8703 df-map 8822 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-card 9934 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-nn 12213 df-2 12275 df-3 12276 df-4 12277 df-n0 12473 df-z 12559 df-uz 12823 df-fz 13485 df-fzo 13628 df-hash 14291 df-word 14465 df-lsw 14513 df-concat 14521 df-s1 14546 df-s2 14799 df-s3 14800 df-s4 14801 df-wlks 28856 df-wlkson 28857 df-trls 28949 df-trlson 28950 |
This theorem is referenced by: 3pthond 29428 3spthond 29430 |
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