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Theorem spthdep 29568
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.)
Assertion
Ref Expression
spthdep ((𝐹(SPathsβ€˜πΊ)𝑃 ∧ (β™―β€˜πΉ) β‰  0) β†’ (π‘ƒβ€˜0) β‰  (π‘ƒβ€˜(β™―β€˜πΉ)))
Proof of Theorem spthdep
StepHypRef Expression
1 isspth 29558 . . 3 (𝐹(SPathsβ€˜πΊ)𝑃 ↔ (𝐹(Trailsβ€˜πΊ)𝑃 ∧ Fun ◑𝑃))
2 trliswlk 29531 . . . . . . . . 9 (𝐹(Trailsβ€˜πΊ)𝑃 β†’ 𝐹(Walksβ€˜πΊ)𝑃)
3 eqid 2728 . . . . . . . . . 10 (Vtxβ€˜πΊ) = (Vtxβ€˜πΊ)
43wlkp 29450 . . . . . . . . 9 (𝐹(Walksβ€˜πΊ)𝑃 β†’ 𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ))
52, 4syl 17 . . . . . . . 8 (𝐹(Trailsβ€˜πΊ)𝑃 β†’ 𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ))
65anim1i 613 . . . . . . 7 ((𝐹(Trailsβ€˜πΊ)𝑃 ∧ Fun ◑𝑃) β†’ (𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) ∧ Fun ◑𝑃))
7 df-f1 6558 . . . . . . 7 (𝑃:(0...(β™―β€˜πΉ))–1-1β†’(Vtxβ€˜πΊ) ↔ (𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) ∧ Fun ◑𝑃))
86, 7sylibr 233 . . . . . 6 ((𝐹(Trailsβ€˜πΊ)𝑃 ∧ Fun ◑𝑃) β†’ 𝑃:(0...(β™―β€˜πΉ))–1-1β†’(Vtxβ€˜πΊ))
9 wlkcl 29449 . . . . . . . 8 (𝐹(Walksβ€˜πΊ)𝑃 β†’ (β™―β€˜πΉ) ∈ β„•0)
10 nn0fz0 13639 . . . . . . . . . 10 ((β™―β€˜πΉ) ∈ β„•0 ↔ (β™―β€˜πΉ) ∈ (0...(β™―β€˜πΉ)))
1110biimpi 215 . . . . . . . . 9 ((β™―β€˜πΉ) ∈ β„•0 β†’ (β™―β€˜πΉ) ∈ (0...(β™―β€˜πΉ)))
12 0elfz 13638 . . . . . . . . 9 ((β™―β€˜πΉ) ∈ β„•0 β†’ 0 ∈ (0...(β™―β€˜πΉ)))
1311, 12jca 510 . . . . . . . 8 ((β™―β€˜πΉ) ∈ β„•0 β†’ ((β™―β€˜πΉ) ∈ (0...(β™―β€˜πΉ)) ∧ 0 ∈ (0...(β™―β€˜πΉ))))
142, 9, 133syl 18 . . . . . . 7 (𝐹(Trailsβ€˜πΊ)𝑃 β†’ ((β™―β€˜πΉ) ∈ (0...(β™―β€˜πΉ)) ∧ 0 ∈ (0...(β™―β€˜πΉ))))
1514adantr 479 . . . . . 6 ((𝐹(Trailsβ€˜πΊ)𝑃 ∧ Fun ◑𝑃) β†’ ((β™―β€˜πΉ) ∈ (0...(β™―β€˜πΉ)) ∧ 0 ∈ (0...(β™―β€˜πΉ))))
168, 15jca 510 . . . . 5 ((𝐹(Trailsβ€˜πΊ)𝑃 ∧ Fun ◑𝑃) β†’ (𝑃:(0...(β™―β€˜πΉ))–1-1β†’(Vtxβ€˜πΊ) ∧ ((β™―β€˜πΉ) ∈ (0...(β™―β€˜πΉ)) ∧ 0 ∈ (0...(β™―β€˜πΉ)))))
17 eqcom 2735 . . . . . 6 ((π‘ƒβ€˜0) = (π‘ƒβ€˜(β™―β€˜πΉ)) ↔ (π‘ƒβ€˜(β™―β€˜πΉ)) = (π‘ƒβ€˜0))
18 f1veqaeq 7273 . . . . . 6 ((𝑃:(0...(β™―β€˜πΉ))–1-1β†’(Vtxβ€˜πΊ) ∧ ((β™―β€˜πΉ) ∈ (0...(β™―β€˜πΉ)) ∧ 0 ∈ (0...(β™―β€˜πΉ)))) β†’ ((π‘ƒβ€˜(β™―β€˜πΉ)) = (π‘ƒβ€˜0) β†’ (β™―β€˜πΉ) = 0))
1917, 18biimtrid 241 . . . . 5 ((𝑃:(0...(β™―β€˜πΉ))–1-1β†’(Vtxβ€˜πΊ) ∧ ((β™―β€˜πΉ) ∈ (0...(β™―β€˜πΉ)) ∧ 0 ∈ (0...(β™―β€˜πΉ)))) β†’ ((π‘ƒβ€˜0) = (π‘ƒβ€˜(β™―β€˜πΉ)) β†’ (β™―β€˜πΉ) = 0))
2016, 19syl 17 . . . 4 ((𝐹(Trailsβ€˜πΊ)𝑃 ∧ Fun ◑𝑃) β†’ ((π‘ƒβ€˜0) = (π‘ƒβ€˜(β™―β€˜πΉ)) β†’ (β™―β€˜πΉ) = 0))
2120necon3d 2958 . . 3 ((𝐹(Trailsβ€˜πΊ)𝑃 ∧ Fun ◑𝑃) β†’ ((β™―β€˜πΉ) β‰  0 β†’ (π‘ƒβ€˜0) β‰  (π‘ƒβ€˜(β™―β€˜πΉ))))
221, 21sylbi 216 . 2 (𝐹(SPathsβ€˜πΊ)𝑃 β†’ ((β™―β€˜πΉ) β‰  0 β†’ (π‘ƒβ€˜0) β‰  (π‘ƒβ€˜(β™―β€˜πΉ))))
2322imp 405 1 ((𝐹(SPathsβ€˜πΊ)𝑃 ∧ (β™―β€˜πΉ) β‰  0) β†’ (π‘ƒβ€˜0) β‰  (π‘ƒβ€˜(β™―β€˜πΉ)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098   β‰  wne 2937   class class class wbr 5152  β—‘ccnv 5681  Fun wfun 6547  βŸΆwf 6549  β€“1-1β†’wf1 6550  β€˜cfv 6553  (class class class)co 7426  0cc0 11146  β„•0cn0 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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