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Mirrors > Home > MPE Home > Th. List > 3spthd | Structured version Visualization version GIF version |
Description: A simple path of length 3 from one vertex to another, different vertex via a third vertex. (Contributed 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)) |
3spthd.n | β’ (π β π΄ β π·) |
Ref | Expression |
---|---|
3spthd | β’ (π β πΉ(SPathsβπΊ)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 | 3trld 29425 | . 2 β’ (π β πΉ(TrailsβπΊ)π) |
10 | simpr 486 | . . 3 β’ ((π β§ πΉ(TrailsβπΊ)π) β πΉ(TrailsβπΊ)π) | |
11 | 3spthd.n | . . . . . . . . 9 β’ (π β π΄ β π·) | |
12 | df-3an 1090 | . . . . . . . . . . 11 β’ ((π΄ β π΅ β§ π΄ β πΆ β§ π΄ β π·) β ((π΄ β π΅ β§ π΄ β πΆ) β§ π΄ β π·)) | |
13 | 12 | simplbi2 502 | . . . . . . . . . 10 β’ ((π΄ β π΅ β§ π΄ β πΆ) β (π΄ β π· β (π΄ β π΅ β§ π΄ β πΆ β§ π΄ β π·))) |
14 | 13 | 3ad2ant1 1134 | . . . . . . . . 9 β’ (((π΄ β π΅ β§ π΄ β πΆ) β§ (π΅ β πΆ β§ π΅ β π·) β§ πΆ β π·) β (π΄ β π· β (π΄ β π΅ β§ π΄ β πΆ β§ π΄ β π·))) |
15 | 11, 14 | mpan9 508 | . . . . . . . 8 β’ ((π β§ ((π΄ β π΅ β§ π΄ β πΆ) β§ (π΅ β πΆ β§ π΅ β π·) β§ πΆ β π·)) β (π΄ β π΅ β§ π΄ β πΆ β§ π΄ β π·)) |
16 | simpr2 1196 | . . . . . . . 8 β’ ((π β§ ((π΄ β π΅ β§ π΄ β πΆ) β§ (π΅ β πΆ β§ π΅ β π·) β§ πΆ β π·)) β (π΅ β πΆ β§ π΅ β π·)) | |
17 | simpr3 1197 | . . . . . . . 8 β’ ((π β§ ((π΄ β π΅ β§ π΄ β πΆ) β§ (π΅ β πΆ β§ π΅ β π·) β§ πΆ β π·)) β πΆ β π·) | |
18 | 15, 16, 17 | 3jca 1129 | . . . . . . 7 β’ ((π β§ ((π΄ β π΅ β§ π΄ β πΆ) β§ (π΅ β πΆ β§ π΅ β π·) β§ πΆ β π·)) β ((π΄ β π΅ β§ π΄ β πΆ β§ π΄ β π·) β§ (π΅ β πΆ β§ π΅ β π·) β§ πΆ β π·)) |
19 | 4, 18 | mpdan 686 | . . . . . 6 β’ (π β ((π΄ β π΅ β§ π΄ β πΆ β§ π΄ β π·) β§ (π΅ β πΆ β§ π΅ β π·) β§ πΆ β π·)) |
20 | funcnvs4 14866 | . . . . . 6 β’ ((((π΄ β π β§ π΅ β π) β§ (πΆ β π β§ π· β π)) β§ ((π΄ β π΅ β§ π΄ β πΆ β§ π΄ β π·) β§ (π΅ β πΆ β§ π΅ β π·) β§ πΆ β π·)) β Fun β‘β¨βπ΄π΅πΆπ·ββ©) | |
21 | 3, 19, 20 | syl2anc 585 | . . . . 5 β’ (π β Fun β‘β¨βπ΄π΅πΆπ·ββ©) |
22 | 21 | adantr 482 | . . . 4 β’ ((π β§ πΉ(TrailsβπΊ)π) β Fun β‘β¨βπ΄π΅πΆπ·ββ©) |
23 | 1 | a1i 11 | . . . . . 6 β’ ((π β§ πΉ(TrailsβπΊ)π) β π = β¨βπ΄π΅πΆπ·ββ©) |
24 | 23 | cnveqd 5876 | . . . . 5 β’ ((π β§ πΉ(TrailsβπΊ)π) β β‘π = β‘β¨βπ΄π΅πΆπ·ββ©) |
25 | 24 | funeqd 6571 | . . . 4 β’ ((π β§ πΉ(TrailsβπΊ)π) β (Fun β‘π β Fun β‘β¨βπ΄π΅πΆπ·ββ©)) |
26 | 22, 25 | mpbird 257 | . . 3 β’ ((π β§ πΉ(TrailsβπΊ)π) β Fun β‘π) |
27 | isspth 28981 | . . 3 β’ (πΉ(SPathsβπΊ)π β (πΉ(TrailsβπΊ)π β§ Fun β‘π)) | |
28 | 10, 26, 27 | sylanbrc 584 | . 2 β’ ((π β§ πΉ(TrailsβπΊ)π) β πΉ(SPathsβπΊ)π) |
29 | 9, 28 | mpdan 686 | 1 β’ (π β πΉ(SPathsβπΊ)π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 β§ w3a 1088 = wceq 1542 β wcel 2107 β wne 2941 β wss 3949 {cpr 4631 class class class wbr 5149 β‘ccnv 5676 Fun wfun 6538 βcfv 6544 β¨βcs3 14793 β¨βcs4 14794 Vtxcvtx 28256 iEdgciedg 28257 Trailsctrls 28947 SPathscspths 28970 |
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-concat 14521 df-s1 14546 df-s2 14799 df-s3 14800 df-s4 14801 df-wlks 28856 df-trls 28949 df-spths 28974 |
This theorem is referenced by: 3spthond 29430 |
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