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| Mirrors > Home > MPE Home > Th. List > 0pthon | Structured version Visualization version GIF version | ||
| Description: A path of length 0 from a vertex to itself. (Contributed by Alexander van der Vekens, 3-Dec-2017.) (Revised by AV, 20-Jan-2021.) (Revised by AV, 30-Oct-2021.) |
| Ref | Expression |
|---|---|
| 0pthon.v | β’ π = (VtxβπΊ) |
| Ref | Expression |
|---|---|
| 0pthon | β’ ((π:(0...0)βΆπ β§ (πβ0) = π) β β (π(PathsOnβπΊ)π)π) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 0pthon.v | . . 3 β’ π = (VtxβπΊ) | |
| 2 | 1 | 0trlon 29644 | . 2 β’ ((π:(0...0)βΆπ β§ (πβ0) = π) β β (π(TrailsOnβπΊ)π)π) |
| 3 | simpl 481 | . . 3 β’ ((π:(0...0)βΆπ β§ (πβ0) = π) β π:(0...0)βΆπ) | |
| 4 | id 22 | . . . . . . 7 β’ (π:(0...0)βΆπ β π:(0...0)βΆπ) | |
| 5 | 0z 12573 | . . . . . . . 8 β’ 0 β β€ | |
| 6 | elfz3 13515 | . . . . . . . 8 β’ (0 β β€ β 0 β (0...0)) | |
| 7 | 5, 6 | mp1i 13 | . . . . . . 7 β’ (π:(0...0)βΆπ β 0 β (0...0)) |
| 8 | 4, 7 | ffvelcdmd 7086 | . . . . . 6 β’ (π:(0...0)βΆπ β (πβ0) β π) |
| 9 | 8 | adantr 479 | . . . . 5 β’ ((π:(0...0)βΆπ β§ (πβ0) = π) β (πβ0) β π) |
| 10 | eleq1 2819 | . . . . . 6 β’ ((πβ0) = π β ((πβ0) β π β π β π)) | |
| 11 | 10 | adantl 480 | . . . . 5 β’ ((π:(0...0)βΆπ β§ (πβ0) = π) β ((πβ0) β π β π β π)) |
| 12 | 9, 11 | mpbid 231 | . . . 4 β’ ((π:(0...0)βΆπ β§ (πβ0) = π) β π β π) |
| 13 | 1 | 1vgrex 28529 | . . . 4 β’ (π β π β πΊ β V) |
| 14 | 1 | 0pth 29645 | . . . 4 β’ (πΊ β V β (β (PathsβπΊ)π β π:(0...0)βΆπ)) |
| 15 | 12, 13, 14 | 3syl 18 | . . 3 β’ ((π:(0...0)βΆπ β§ (πβ0) = π) β (β (PathsβπΊ)π β π:(0...0)βΆπ)) |
| 16 | 3, 15 | mpbird 256 | . 2 β’ ((π:(0...0)βΆπ β§ (πβ0) = π) β β (PathsβπΊ)π) |
| 17 | 1 | 0wlkonlem1 29638 | . . 3 β’ ((π:(0...0)βΆπ β§ (πβ0) = π) β (π β π β§ π β π)) |
| 18 | 1 | 0wlkonlem2 29639 | . . . 4 β’ ((π:(0...0)βΆπ β§ (πβ0) = π) β π β (π βpm (0...0))) |
| 19 | 0ex 5306 | . . . 4 β’ β β V | |
| 20 | 18, 19 | jctil 518 | . . 3 β’ ((π:(0...0)βΆπ β§ (πβ0) = π) β (β β V β§ π β (π βpm (0...0)))) |
| 21 | 1 | ispthson 29266 | . . 3 β’ (((π β π β§ π β π) β§ (β β V β§ π β (π βpm (0...0)))) β (β (π(PathsOnβπΊ)π)π β (β (π(TrailsOnβπΊ)π)π β§ β (PathsβπΊ)π))) |
| 22 | 17, 20, 21 | syl2anc 582 | . 2 β’ ((π:(0...0)βΆπ β§ (πβ0) = π) β (β (π(PathsOnβπΊ)π)π β (β (π(TrailsOnβπΊ)π)π β§ β (PathsβπΊ)π))) |
| 23 | 2, 16, 22 | mpbir2and 709 | 1 β’ ((π:(0...0)βΆπ β§ (πβ0) = π) β β (π(PathsOnβπΊ)π)π) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β wb 205 β§ wa 394 = wceq 1539 β wcel 2104 Vcvv 3472 β c0 4321 class class class wbr 5147 βΆwf 6538 βcfv 6542 (class class class)co 7411 βpm cpm 8823 0cc0 11112 β€cz 12562 ...cfz 13488 Vtxcvtx 28523 TrailsOnctrlson 29215 Pathscpths 29236 PathsOncpthson 29238 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 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 395 df-or 844 df-ifp 1060 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-er 8705 df-map 8824 df-pm 8825 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-n0 12477 df-z 12563 df-uz 12827 df-fz 13489 df-fzo 13632 df-hash 14295 df-word 14469 df-wlks 29123 df-wlkson 29124 df-trls 29216 df-trlson 29217 df-pths 29240 df-pthson 29242 |
| This theorem is referenced by: 0pthon1 29648 |
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