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Theorem spthdep 28980
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 28970 . . 3 (𝐹(SPathsβ€˜πΊ)𝑃 ↔ (𝐹(Trailsβ€˜πΊ)𝑃 ∧ Fun ◑𝑃))
2 trliswlk 28943 . . . . . . . . 9 (𝐹(Trailsβ€˜πΊ)𝑃 β†’ 𝐹(Walksβ€˜πΊ)𝑃)
3 eqid 2732 . . . . . . . . . 10 (Vtxβ€˜πΊ) = (Vtxβ€˜πΊ)
43wlkp 28862 . . . . . . . . 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 615 . . . . . . 7 ((𝐹(Trailsβ€˜πΊ)𝑃 ∧ Fun ◑𝑃) β†’ (𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) ∧ Fun ◑𝑃))
7 df-f1 6545 . . . . . . 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 28861 . . . . . . . 8 (𝐹(Walksβ€˜πΊ)𝑃 β†’ (β™―β€˜πΉ) ∈ β„•0)
10 nn0fz0 13595 . . . . . . . . . 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 13594 . . . . . . . . 9 ((β™―β€˜πΉ) ∈ β„•0 β†’ 0 ∈ (0...(β™―β€˜πΉ)))
1311, 12jca 512 . . . . . . . 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 481 . . . . . 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 512 . . . . 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 2739 . . . . . 6 ((π‘ƒβ€˜0) = (π‘ƒβ€˜(β™―β€˜πΉ)) ↔ (π‘ƒβ€˜(β™―β€˜πΉ)) = (π‘ƒβ€˜0))
18 f1veqaeq 7252 . . . . . 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 2961 . . 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 407 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 396   = wceq 1541   ∈ wcel 2106   β‰  wne 2940   class class class wbr 5147  β—‘ccnv 5674  Fun wfun 6534  βŸΆwf 6536  β€“1-1β†’wf1 6537  β€˜cfv 6540  (class class class)co 7405  0cc0 11106  β„•0cn0 12468  ...cfz 13480  β™―chash 14286  Vtxcvtx 28245  Walkscwlks 28842  Trailsctrls 28936  SPathscspths 28959
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-ifp 1062  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-er 8699  df-map 8818  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-card 9930  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 28845  df-trls 28938  df-spths 28963
This theorem is referenced by:  cyclnspth  29046
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