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Theorem spthdep 29495
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 29485 . . 3 (𝐹(SPathsβ€˜πΊ)𝑃 ↔ (𝐹(Trailsβ€˜πΊ)𝑃 ∧ Fun ◑𝑃))
2 trliswlk 29458 . . . . . . . . 9 (𝐹(Trailsβ€˜πΊ)𝑃 β†’ 𝐹(Walksβ€˜πΊ)𝑃)
3 eqid 2726 . . . . . . . . . 10 (Vtxβ€˜πΊ) = (Vtxβ€˜πΊ)
43wlkp 29377 . . . . . . . . 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 614 . . . . . . 7 ((𝐹(Trailsβ€˜πΊ)𝑃 ∧ Fun ◑𝑃) β†’ (𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) ∧ Fun ◑𝑃))
7 df-f1 6541 . . . . . . 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 29376 . . . . . . . 8 (𝐹(Walksβ€˜πΊ)𝑃 β†’ (β™―β€˜πΉ) ∈ β„•0)
10 nn0fz0 13602 . . . . . . . . . 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 13601 . . . . . . . . 9 ((β™―β€˜πΉ) ∈ β„•0 β†’ 0 ∈ (0...(β™―β€˜πΉ)))
1311, 12jca 511 . . . . . . . 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 480 . . . . . 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 511 . . . . 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 2733 . . . . . 6 ((π‘ƒβ€˜0) = (π‘ƒβ€˜(β™―β€˜πΉ)) ↔ (π‘ƒβ€˜(β™―β€˜πΉ)) = (π‘ƒβ€˜0))
18 f1veqaeq 7251 . . . . . 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 2955 . . 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 406 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 395   = wceq 1533   ∈ wcel 2098   β‰  wne 2934   class class class wbr 5141  β—‘ccnv 5668  Fun wfun 6530  βŸΆwf 6532  β€“1-1β†’wf1 6533  β€˜cfv 6536  (class class class)co 7404  0cc0 11109  β„•0cn0 12473  ...cfz 13487  β™―chash 14292  Vtxcvtx 28759  Walkscwlks 29357  Trailsctrls 29451  SPathscspths 29474
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 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721  ax-cnex 11165  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185  ax-pre-mulgt0 11186
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 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6293  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7360  df-ov 7407  df-oprab 7408  df-mpo 7409  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8264  df-wrecs 8295  df-recs 8369  df-rdg 8408  df-1o 8464  df-er 8702  df-map 8821  df-en 8939  df-dom 8940  df-sdom 8941  df-fin 8942  df-card 9933  df-pnf 11251  df-mnf 11252  df-xr 11253  df-ltxr 11254  df-le 11255  df-sub 11447  df-neg 11448  df-nn 12214  df-n0 12474  df-z 12560  df-uz 12824  df-fz 13488  df-fzo 13631  df-hash 14293  df-word 14468  df-wlks 29360  df-trls 29453  df-spths 29478
This theorem is referenced by:  cyclnspth  29561
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