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| Mirrors > Home > MPE Home > Th. List > pthonpth | Structured version Visualization version GIF version | ||
| Description: A path is a path between its endpoints. (Contributed by AV, 31-Jan-2021.) |
| Ref | Expression |
|---|---|
| pthonpth | β’ (πΉ(PathsβπΊ)π β πΉ((πβ0)(PathsOnβπΊ)(πβ(β―βπΉ)))π) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | pthistrl 29417 | . . 3 β’ (πΉ(PathsβπΊ)π β πΉ(TrailsβπΊ)π) | |
| 2 | trlontrl 29403 | . . 3 β’ (πΉ(TrailsβπΊ)π β πΉ((πβ0)(TrailsOnβπΊ)(πβ(β―βπΉ)))π) | |
| 3 | 1, 2 | syl 17 | . 2 β’ (πΉ(PathsβπΊ)π β πΉ((πβ0)(TrailsOnβπΊ)(πβ(β―βπΉ)))π) |
| 4 | id 22 | . 2 β’ (πΉ(PathsβπΊ)π β πΉ(PathsβπΊ)π) | |
| 5 | pthiswlk 29419 | . . 3 β’ (πΉ(PathsβπΊ)π β πΉ(WalksβπΊ)π) | |
| 6 | eqid 2724 | . . . . 5 β’ (VtxβπΊ) = (VtxβπΊ) | |
| 7 | 6 | wlkepvtx 29352 | . . . 4 β’ (πΉ(WalksβπΊ)π β ((πβ0) β (VtxβπΊ) β§ (πβ(β―βπΉ)) β (VtxβπΊ))) |
| 8 | wlkv 29304 | . . . . 5 β’ (πΉ(WalksβπΊ)π β (πΊ β V β§ πΉ β V β§ π β V)) | |
| 9 | 3simpc 1147 | . . . . 5 β’ ((πΊ β V β§ πΉ β V β§ π β V) β (πΉ β V β§ π β V)) | |
| 10 | 8, 9 | syl 17 | . . . 4 β’ (πΉ(WalksβπΊ)π β (πΉ β V β§ π β V)) |
| 11 | 7, 10 | jca 511 | . . 3 β’ (πΉ(WalksβπΊ)π β (((πβ0) β (VtxβπΊ) β§ (πβ(β―βπΉ)) β (VtxβπΊ)) β§ (πΉ β V β§ π β V))) |
| 12 | 6 | ispthson 29434 | . . 3 β’ ((((πβ0) β (VtxβπΊ) β§ (πβ(β―βπΉ)) β (VtxβπΊ)) β§ (πΉ β V β§ π β V)) β (πΉ((πβ0)(PathsOnβπΊ)(πβ(β―βπΉ)))π β (πΉ((πβ0)(TrailsOnβπΊ)(πβ(β―βπΉ)))π β§ πΉ(PathsβπΊ)π))) |
| 13 | 5, 11, 12 | 3syl 18 | . 2 β’ (πΉ(PathsβπΊ)π β (πΉ((πβ0)(PathsOnβπΊ)(πβ(β―βπΉ)))π β (πΉ((πβ0)(TrailsOnβπΊ)(πβ(β―βπΉ)))π β§ πΉ(PathsβπΊ)π))) |
| 14 | 3, 4, 13 | mpbir2and 710 | 1 β’ (πΉ(PathsβπΊ)π β πΉ((πβ0)(PathsOnβπΊ)(πβ(β―βπΉ)))π) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β wb 205 β§ wa 395 β§ w3a 1084 β wcel 2098 Vcvv 3466 class class class wbr 5138 βcfv 6533 (class class class)co 7401 0cc0 11105 β―chash 14286 Vtxcvtx 28691 Walkscwlks 29288 Trailsctrls 29382 TrailsOnctrlson 29383 Pathscpths 29404 PathsOncpthson 29406 |
| 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 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 |
| 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 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-1st 7968 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-er 8698 df-map 8817 df-en 8935 df-dom 8936 df-sdom 8937 df-fin 8938 df-card 9929 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-n0 12469 df-z 12555 df-uz 12819 df-fz 13481 df-fzo 13624 df-hash 14287 df-word 14461 df-wlks 29291 df-wlkson 29292 df-trls 29384 df-trlson 29385 df-pths 29408 df-pthson 29410 |
| This theorem is referenced by: cyclispthon 29493 |
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