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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 29930 | . 2 β’ (π β πΉ(TrailsβπΊ)π) |
10 | simpr 484 | . . 3 β’ ((π β§ πΉ(TrailsβπΊ)π) β πΉ(TrailsβπΊ)π) | |
11 | 3spthd.n | . . . . . . . . 9 β’ (π β π΄ β π·) | |
12 | df-3an 1086 | . . . . . . . . . . 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 500 | . . . . . . . . . 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 1130 | . . . . . . . . 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 506 | . . . . . . . 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 1192 | . . . . . . . 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 1193 | . . . . . . . 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 1125 | . . . . . . 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 684 | . . . . . 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 14870 | . . . . . 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 583 | . . . . 5 β’ (π β Fun β‘β¨βπ΄π΅πΆπ·ββ©) |
22 | 21 | adantr 480 | . . . 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 5868 | . . . . 5 β’ ((π β§ πΉ(TrailsβπΊ)π) β β‘π = β‘β¨βπ΄π΅πΆπ·ββ©) |
25 | 24 | funeqd 6563 | . . . 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 29486 | . . 3 β’ (πΉ(SPathsβπΊ)π β (πΉ(TrailsβπΊ)π β§ Fun β‘π)) | |
28 | 10, 26, 27 | sylanbrc 582 | . 2 β’ ((π β§ πΉ(TrailsβπΊ)π) β πΉ(SPathsβπΊ)π) |
29 | 9, 28 | mpdan 684 | 1 β’ (π β πΉ(SPathsβπΊ)π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 β§ w3a 1084 = wceq 1533 β wcel 2098 β wne 2934 β wss 3943 {cpr 4625 class class class wbr 5141 β‘ccnv 5668 Fun wfun 6530 βcfv 6536 β¨βcs3 14797 β¨βcs4 14798 Vtxcvtx 28760 iEdgciedg 28761 Trailsctrls 29452 SPathscspths 29475 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 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 396 df-or 845 df-ifp 1060 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-er 8702 df-map 8821 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-card 9933 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-nn 12214 df-2 12276 df-3 12277 df-4 12278 df-n0 12474 df-z 12560 df-uz 12824 df-fz 13488 df-fzo 13631 df-hash 14294 df-word 14469 df-concat 14525 df-s1 14550 df-s2 14803 df-s3 14804 df-s4 14805 df-wlks 29361 df-trls 29454 df-spths 29479 |
This theorem is referenced by: 3spthond 29935 |
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