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| Mirrors > Home > MPE Home > Th. List > 0trlon | Structured version Visualization version GIF version | ||
| Description: A trail of length 0 from a vertex to itself. (Contributed by Alexander van der Vekens, 2-Dec-2017.) (Revised by AV, 8-Jan-2021.) (Revised by AV, 23-Mar-2021.) (Proof shortened by AV, 30-Oct-2021.) |
| Ref | Expression |
|---|---|
| 0wlk.v | β’ π = (VtxβπΊ) |
| Ref | Expression |
|---|---|
| 0trlon | β’ ((π:(0...0)βΆπ β§ (πβ0) = π) β β (π(TrailsOnβπΊ)π)π) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 0wlk.v | . . 3 β’ π = (VtxβπΊ) | |
| 2 | 1 | 0wlkon 29638 | . 2 β’ ((π:(0...0)βΆπ β§ (πβ0) = π) β β (π(WalksOnβπΊ)π)π) |
| 3 | simpl 481 | . . 3 β’ ((π:(0...0)βΆπ β§ (πβ0) = π) β π:(0...0)βΆπ) | |
| 4 | 1 | 0wlkonlem1 29636 | . . . 4 β’ ((π:(0...0)βΆπ β§ (πβ0) = π) β (π β π β§ π β π)) |
| 5 | 1 | 1vgrex 28527 | . . . . 5 β’ (π β π β πΊ β V) |
| 6 | 5 | adantr 479 | . . . 4 β’ ((π β π β§ π β π) β πΊ β V) |
| 7 | 1 | 0trl 29640 | . . . 4 β’ (πΊ β V β (β (TrailsβπΊ)π β π:(0...0)βΆπ)) |
| 8 | 4, 6, 7 | 3syl 18 | . . 3 β’ ((π:(0...0)βΆπ β§ (πβ0) = π) β (β (TrailsβπΊ)π β π:(0...0)βΆπ)) |
| 9 | 3, 8 | mpbird 256 | . 2 β’ ((π:(0...0)βΆπ β§ (πβ0) = π) β β (TrailsβπΊ)π) |
| 10 | 0ex 5308 | . . . 4 β’ β β V | |
| 11 | 10 | a1i 11 | . . 3 β’ ((π:(0...0)βΆπ β§ (πβ0) = π) β β β V) |
| 12 | 1 | 0wlkonlem2 29637 | . . 3 β’ ((π:(0...0)βΆπ β§ (πβ0) = π) β π β (π βpm (0...0))) |
| 13 | 1 | istrlson 29229 | . . 3 β’ (((π β π β§ π β π) β§ (β β V β§ π β (π βpm (0...0)))) β (β (π(TrailsOnβπΊ)π)π β (β (π(WalksOnβπΊ)π)π β§ β (TrailsβπΊ)π))) |
| 14 | 4, 11, 12, 13 | syl12anc 833 | . 2 β’ ((π:(0...0)βΆπ β§ (πβ0) = π) β (β (π(TrailsOnβπΊ)π)π β (β (π(WalksOnβπΊ)π)π β§ β (TrailsβπΊ)π))) |
| 15 | 2, 9, 14 | mpbir2and 709 | 1 β’ ((π:(0...0)βΆπ β§ (πβ0) = π) β β (π(TrailsOnβπΊ)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 4323 class class class wbr 5149 βΆwf 6540 βcfv 6544 (class class class)co 7413 βpm cpm 8825 0cc0 11114 ...cfz 13490 Vtxcvtx 28521 WalksOncwlkson 29119 Trailsctrls 29212 TrailsOnctrlson 29213 |
| 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 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7729 ax-cnex 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191 |
| 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 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-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 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-1o 8470 df-er 8707 df-map 8826 df-pm 8827 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-card 9938 df-pnf 11256 df-mnf 11257 df-xr 11258 df-ltxr 11259 df-le 11260 df-sub 11452 df-neg 11453 df-nn 12219 df-n0 12479 df-z 12565 df-uz 12829 df-fz 13491 df-fzo 13634 df-hash 14297 df-word 14471 df-wlks 29121 df-wlkson 29122 df-trls 29214 df-trlson 29215 |
| This theorem is referenced by: 0pthon 29645 |
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