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Mirrors > Home > MPE Home > Th. List > spthispth | Structured version Visualization version GIF version |
Description: A simple path is a path (in an undirected graph). (Contributed by Alexander van der Vekens, 21-Oct-2017.) (Revised by AV, 9-Jan-2021.) (Proof shortened by AV, 30-Oct-2021.) |
Ref | Expression |
---|---|
spthispth | β’ (πΉ(SPathsβπΊ)π β πΉ(PathsβπΊ)π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 481 | . . 3 β’ ((πΉ(TrailsβπΊ)π β§ Fun β‘π) β πΉ(TrailsβπΊ)π) | |
2 | funres11 6625 | . . . 4 β’ (Fun β‘π β Fun β‘(π βΎ (1..^(β―βπΉ)))) | |
3 | 2 | adantl 480 | . . 3 β’ ((πΉ(TrailsβπΊ)π β§ Fun β‘π) β Fun β‘(π βΎ (1..^(β―βπΉ)))) |
4 | imain 6633 | . . . . 5 β’ (Fun β‘π β (π β ({0, (β―βπΉ)} β© (1..^(β―βπΉ)))) = ((π β {0, (β―βπΉ)}) β© (π β (1..^(β―βπΉ))))) | |
5 | 1e0p1 12749 | . . . . . . . . . 10 β’ 1 = (0 + 1) | |
6 | 5 | oveq1i 7426 | . . . . . . . . 9 β’ (1..^(β―βπΉ)) = ((0 + 1)..^(β―βπΉ)) |
7 | 6 | ineq2i 4203 | . . . . . . . 8 β’ ({0, (β―βπΉ)} β© (1..^(β―βπΉ))) = ({0, (β―βπΉ)} β© ((0 + 1)..^(β―βπΉ))) |
8 | 0z 12599 | . . . . . . . . 9 β’ 0 β β€ | |
9 | prinfzo0 13703 | . . . . . . . . 9 β’ (0 β β€ β ({0, (β―βπΉ)} β© ((0 + 1)..^(β―βπΉ))) = β ) | |
10 | 8, 9 | ax-mp 5 | . . . . . . . 8 β’ ({0, (β―βπΉ)} β© ((0 + 1)..^(β―βπΉ))) = β |
11 | 7, 10 | eqtri 2753 | . . . . . . 7 β’ ({0, (β―βπΉ)} β© (1..^(β―βπΉ))) = β |
12 | 11 | imaeq2i 6056 | . . . . . 6 β’ (π β ({0, (β―βπΉ)} β© (1..^(β―βπΉ)))) = (π β β ) |
13 | ima0 6075 | . . . . . 6 β’ (π β β ) = β | |
14 | 12, 13 | eqtri 2753 | . . . . 5 β’ (π β ({0, (β―βπΉ)} β© (1..^(β―βπΉ)))) = β |
15 | 4, 14 | eqtr3di 2780 | . . . 4 β’ (Fun β‘π β ((π β {0, (β―βπΉ)}) β© (π β (1..^(β―βπΉ)))) = β ) |
16 | 15 | adantl 480 | . . 3 β’ ((πΉ(TrailsβπΊ)π β§ Fun β‘π) β ((π β {0, (β―βπΉ)}) β© (π β (1..^(β―βπΉ)))) = β ) |
17 | 1, 3, 16 | 3jca 1125 | . 2 β’ ((πΉ(TrailsβπΊ)π β§ Fun β‘π) β (πΉ(TrailsβπΊ)π β§ Fun β‘(π βΎ (1..^(β―βπΉ))) β§ ((π β {0, (β―βπΉ)}) β© (π β (1..^(β―βπΉ)))) = β )) |
18 | isspth 29582 | . 2 β’ (πΉ(SPathsβπΊ)π β (πΉ(TrailsβπΊ)π β§ Fun β‘π)) | |
19 | ispth 29581 | . 2 β’ (πΉ(PathsβπΊ)π β (πΉ(TrailsβπΊ)π β§ Fun β‘(π βΎ (1..^(β―βπΉ))) β§ ((π β {0, (β―βπΉ)}) β© (π β (1..^(β―βπΉ)))) = β )) | |
20 | 17, 18, 19 | 3imtr4i 291 | 1 β’ (πΉ(SPathsβπΊ)π β πΉ(PathsβπΊ)π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 β§ w3a 1084 = wceq 1533 β wcel 2098 β© cin 3938 β c0 4318 {cpr 4626 class class class wbr 5143 β‘ccnv 5671 βΎ cres 5674 β cima 5675 Fun wfun 6537 βcfv 6543 (class class class)co 7416 0cc0 11138 1c1 11139 + caddc 11141 β€cz 12588 ..^cfzo 13659 β―chash 14321 Trailsctrls 29548 Pathscpths 29570 SPathscspths 29571 |
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 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 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 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7991 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-er 8723 df-en 8963 df-dom 8964 df-sdom 8965 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-n0 12503 df-z 12589 df-uz 12853 df-fz 13517 df-fzo 13660 df-trls 29550 df-pths 29574 df-spths 29575 |
This theorem is referenced by: spthiswlk 29586 isspthonpth 29607 spthonpthon 29609 usgr2trlspth 29619 usgr2pthspth 29620 wspthsnonn0vne 29772 spthcycl 34796 |
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