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Mirrors > Home > MPE Home > Th. List > 3pthond | Structured version Visualization version GIF version |
Description: A 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)) |
Ref | Expression |
---|---|
3pthond | β’ (π β πΉ(π΄(PathsOnβπΊ)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 29120 | . 2 β’ (π β πΉ(π΄(TrailsOnβπΊ)π·)π) |
10 | 1, 2, 3, 4, 5, 6, 7, 8 | 3pthd 29121 | . 2 β’ (π β πΉ(PathsβπΊ)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 14771 | . . . . . 6 β’ β¨βπ½πΎπΏββ© β Word V | |
14 | 2, 13 | eqeltri 2834 | . . . . 5 β’ πΉ β Word V |
15 | s4cli 14772 | . . . . . 6 β’ β¨βπ΄π΅πΆπ·ββ© β Word V | |
16 | 1, 15 | eqeltri 2834 | . . . . 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 | ispthson 28693 | . . 3 β’ (((π΄ β π β§ π· β π) β§ (πΉ β Word V β§ π β Word V)) β (πΉ(π΄(PathsOnβπΊ)π·)π β (πΉ(π΄(TrailsOnβπΊ)π·)π β§ πΉ(PathsβπΊ)π))) |
20 | 11, 12, 18, 19 | syl21anc 837 | . 2 β’ (π β (πΉ(π΄(PathsOnβπΊ)π·)π β (πΉ(π΄(TrailsOnβπΊ)π·)π β§ πΉ(PathsβπΊ)π))) |
21 | 9, 10, 20 | mpbir2and 712 | 1 β’ (π β πΉ(π΄(PathsOnβπΊ)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 2944 Vcvv 3446 β wss 3911 {cpr 4589 class class class wbr 5106 βcfv 6497 (class class class)co 7358 Word cword 14403 β¨βcs3 14732 β¨βcs4 14733 Vtxcvtx 27950 iEdgciedg 27951 TrailsOnctrlson 28642 Pathscpths 28663 PathsOncpthson 28665 |
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 2708 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11108 ax-resscn 11109 ax-1cn 11110 ax-icn 11111 ax-addcl 11112 ax-addrcl 11113 ax-mulcl 11114 ax-mulrcl 11115 ax-mulcom 11116 ax-addass 11117 ax-mulass 11118 ax-distr 11119 ax-i2m1 11120 ax-1ne0 11121 ax-1rid 11122 ax-rnegex 11123 ax-rrecex 11124 ax-cnre 11125 ax-pre-lttri 11126 ax-pre-lttrn 11127 ax-pre-ltadd 11128 ax-pre-mulgt0 11129 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8649 df-map 8768 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-card 9876 df-pnf 11192 df-mnf 11193 df-xr 11194 df-ltxr 11195 df-le 11196 df-sub 11388 df-neg 11389 df-nn 12155 df-2 12217 df-3 12218 df-4 12219 df-n0 12415 df-z 12501 df-uz 12765 df-fz 13426 df-fzo 13569 df-hash 14232 df-word 14404 df-lsw 14452 df-concat 14460 df-s1 14485 df-s2 14738 df-s3 14739 df-s4 14740 df-wlks 28550 df-wlkson 28551 df-trls 28643 df-trlson 28644 df-pths 28667 df-pthson 28669 |
This theorem is referenced by: (None) |
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