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Mirrors > Home > MPE Home > Th. List > 3spthond | 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.) |
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 |
---|---|
3spthond | β’ (π β πΉ(π΄(SPathsOnβπΊ)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 | 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 | 3trlond 29935 | . 2 β’ (π β πΉ(π΄(TrailsOnβπΊ)π·)π) |
10 | 3spthd.n | . . 3 β’ (π β π΄ β π·) | |
11 | 1, 2, 3, 4, 5, 6, 7, 8, 10 | 3spthd 29938 | . 2 β’ (π β πΉ(SPathsβπΊ)π) |
12 | 3 | simplld 765 | . . 3 β’ (π β π΄ β π) |
13 | 3 | simprrd 771 | . . 3 β’ (π β π· β π) |
14 | s3cli 14838 | . . . . . 6 β’ β¨βπ½πΎπΏββ© β Word V | |
15 | 2, 14 | eqeltri 2823 | . . . . 5 β’ πΉ β Word V |
16 | s4cli 14839 | . . . . . 6 β’ β¨βπ΄π΅πΆπ·ββ© β Word V | |
17 | 1, 16 | eqeltri 2823 | . . . . 5 β’ π β Word V |
18 | 15, 17 | pm3.2i 470 | . . . 4 β’ (πΉ β Word V β§ π β Word V) |
19 | 18 | a1i 11 | . . 3 β’ (π β (πΉ β Word V β§ π β Word V)) |
20 | 6 | isspthson 29509 | . . 3 β’ (((π΄ β π β§ π· β π) β§ (πΉ β Word V β§ π β Word V)) β (πΉ(π΄(SPathsOnβπΊ)π·)π β (πΉ(π΄(TrailsOnβπΊ)π·)π β§ πΉ(SPathsβπΊ)π))) |
21 | 12, 13, 19, 20 | syl21anc 835 | . 2 β’ (π β (πΉ(π΄(SPathsOnβπΊ)π·)π β (πΉ(π΄(TrailsOnβπΊ)π·)π β§ πΉ(SPathsβπΊ)π))) |
22 | 9, 11, 21 | mpbir2and 710 | 1 β’ (π β πΉ(π΄(SPathsOnβπΊ)π·)π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 395 β§ w3a 1084 = wceq 1533 β wcel 2098 β wne 2934 Vcvv 3468 β wss 3943 {cpr 4625 class class class wbr 5141 βcfv 6537 (class class class)co 7405 Word cword 14470 β¨βcs3 14799 β¨βcs4 14800 Vtxcvtx 28764 iEdgciedg 28765 TrailsOnctrlson 29457 SPathscspths 29479 SPathsOncspthson 29481 |
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 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
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 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-n0 12477 df-z 12563 df-uz 12827 df-fz 13491 df-fzo 13634 df-hash 14296 df-word 14471 df-lsw 14519 df-concat 14527 df-s1 14552 df-s2 14805 df-s3 14806 df-s4 14807 df-wlks 29365 df-wlkson 29366 df-trls 29458 df-trlson 29459 df-spths 29483 df-spthson 29485 |
This theorem is referenced by: (None) |
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