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Mirrors > Home > MPE Home > Th. List > spthdep | Structured version Visualization version GIF version |
Description: A simple path (at least of length 1) has different start and end points (in an undirected graph). (Contributed by AV, 31-Jan-2021.) (Proof shortened by AV, 30-Oct-2021.) |
Ref | Expression |
---|---|
spthdep | β’ ((πΉ(SPathsβπΊ)π β§ (β―βπΉ) β 0) β (πβ0) β (πβ(β―βπΉ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isspth 29558 | . . 3 β’ (πΉ(SPathsβπΊ)π β (πΉ(TrailsβπΊ)π β§ Fun β‘π)) | |
2 | trliswlk 29531 | . . . . . . . . 9 β’ (πΉ(TrailsβπΊ)π β πΉ(WalksβπΊ)π) | |
3 | eqid 2728 | . . . . . . . . . 10 β’ (VtxβπΊ) = (VtxβπΊ) | |
4 | 3 | wlkp 29450 | . . . . . . . . 9 β’ (πΉ(WalksβπΊ)π β π:(0...(β―βπΉ))βΆ(VtxβπΊ)) |
5 | 2, 4 | syl 17 | . . . . . . . 8 β’ (πΉ(TrailsβπΊ)π β π:(0...(β―βπΉ))βΆ(VtxβπΊ)) |
6 | 5 | anim1i 613 | . . . . . . 7 β’ ((πΉ(TrailsβπΊ)π β§ Fun β‘π) β (π:(0...(β―βπΉ))βΆ(VtxβπΊ) β§ Fun β‘π)) |
7 | df-f1 6558 | . . . . . . 7 β’ (π:(0...(β―βπΉ))β1-1β(VtxβπΊ) β (π:(0...(β―βπΉ))βΆ(VtxβπΊ) β§ Fun β‘π)) | |
8 | 6, 7 | sylibr 233 | . . . . . 6 β’ ((πΉ(TrailsβπΊ)π β§ Fun β‘π) β π:(0...(β―βπΉ))β1-1β(VtxβπΊ)) |
9 | wlkcl 29449 | . . . . . . . 8 β’ (πΉ(WalksβπΊ)π β (β―βπΉ) β β0) | |
10 | nn0fz0 13639 | . . . . . . . . . 10 β’ ((β―βπΉ) β β0 β (β―βπΉ) β (0...(β―βπΉ))) | |
11 | 10 | biimpi 215 | . . . . . . . . 9 β’ ((β―βπΉ) β β0 β (β―βπΉ) β (0...(β―βπΉ))) |
12 | 0elfz 13638 | . . . . . . . . 9 β’ ((β―βπΉ) β β0 β 0 β (0...(β―βπΉ))) | |
13 | 11, 12 | jca 510 | . . . . . . . 8 β’ ((β―βπΉ) β β0 β ((β―βπΉ) β (0...(β―βπΉ)) β§ 0 β (0...(β―βπΉ)))) |
14 | 2, 9, 13 | 3syl 18 | . . . . . . 7 β’ (πΉ(TrailsβπΊ)π β ((β―βπΉ) β (0...(β―βπΉ)) β§ 0 β (0...(β―βπΉ)))) |
15 | 14 | adantr 479 | . . . . . 6 β’ ((πΉ(TrailsβπΊ)π β§ Fun β‘π) β ((β―βπΉ) β (0...(β―βπΉ)) β§ 0 β (0...(β―βπΉ)))) |
16 | 8, 15 | jca 510 | . . . . 5 β’ ((πΉ(TrailsβπΊ)π β§ Fun β‘π) β (π:(0...(β―βπΉ))β1-1β(VtxβπΊ) β§ ((β―βπΉ) β (0...(β―βπΉ)) β§ 0 β (0...(β―βπΉ))))) |
17 | eqcom 2735 | . . . . . 6 β’ ((πβ0) = (πβ(β―βπΉ)) β (πβ(β―βπΉ)) = (πβ0)) | |
18 | f1veqaeq 7273 | . . . . . 6 β’ ((π:(0...(β―βπΉ))β1-1β(VtxβπΊ) β§ ((β―βπΉ) β (0...(β―βπΉ)) β§ 0 β (0...(β―βπΉ)))) β ((πβ(β―βπΉ)) = (πβ0) β (β―βπΉ) = 0)) | |
19 | 17, 18 | biimtrid 241 | . . . . 5 β’ ((π:(0...(β―βπΉ))β1-1β(VtxβπΊ) β§ ((β―βπΉ) β (0...(β―βπΉ)) β§ 0 β (0...(β―βπΉ)))) β ((πβ0) = (πβ(β―βπΉ)) β (β―βπΉ) = 0)) |
20 | 16, 19 | syl 17 | . . . 4 β’ ((πΉ(TrailsβπΊ)π β§ Fun β‘π) β ((πβ0) = (πβ(β―βπΉ)) β (β―βπΉ) = 0)) |
21 | 20 | necon3d 2958 | . . 3 β’ ((πΉ(TrailsβπΊ)π β§ Fun β‘π) β ((β―βπΉ) β 0 β (πβ0) β (πβ(β―βπΉ)))) |
22 | 1, 21 | sylbi 216 | . 2 β’ (πΉ(SPathsβπΊ)π β ((β―βπΉ) β 0 β (πβ0) β (πβ(β―βπΉ)))) |
23 | 22 | imp 405 | 1 β’ ((πΉ(SPathsβπΊ)π β§ (β―βπΉ) β 0) β (πβ0) β (πβ(β―βπΉ))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 β wne 2937 class class class wbr 5152 β‘ccnv 5681 Fun wfun 6547 βΆwf 6549 β1-1βwf1 6550 βcfv 6553 (class class class)co 7426 0cc0 11146 β0cn0 12510 ...cfz 13524 β―chash 14329 Vtxcvtx 28829 Walkscwlks 29430 Trailsctrls 29524 SPathscspths 29547 |
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 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-ifp 1061 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-er 8731 df-map 8853 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-card 9970 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-n0 12511 df-z 12597 df-uz 12861 df-fz 13525 df-fzo 13668 df-hash 14330 df-word 14505 df-wlks 29433 df-trls 29526 df-spths 29551 |
This theorem is referenced by: cyclnspth 29634 |
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