MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  wlkon2n0 Structured version   Visualization version   GIF version
Theorem wlkon2n0 29191
Description: The length of a walk between two different vertices is not 0 (i.e. is at least 1). (Contributed by AV, 3-Apr-2021.)
Assertion
Ref Expression
wlkon2n0 ((𝐹(𝐴(WalksOnβ€˜πΊ)𝐡)𝑃 ∧ 𝐴 β‰  𝐡) β†’ (β™―β€˜πΉ) β‰  0)
Proof of Theorem wlkon2n0
StepHypRef Expression
1 eqid 2731 . . . . 5 (Vtxβ€˜πΊ) = (Vtxβ€˜πΊ)
21wlkonprop 29183 . . . 4 (𝐹(𝐴(WalksOnβ€˜πΊ)𝐡)𝑃 β†’ ((𝐺 ∈ V ∧ 𝐴 ∈ (Vtxβ€˜πΊ) ∧ 𝐡 ∈ (Vtxβ€˜πΊ)) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐹(Walksβ€˜πΊ)𝑃 ∧ (π‘ƒβ€˜0) = 𝐴 ∧ (π‘ƒβ€˜(β™―β€˜πΉ)) = 𝐡)))
3 fveqeq2 6900 . . . . . . . . 9 ((β™―β€˜πΉ) = 0 β†’ ((π‘ƒβ€˜(β™―β€˜πΉ)) = 𝐡 ↔ (π‘ƒβ€˜0) = 𝐡))
43anbi2d 628 . . . . . . . 8 ((β™―β€˜πΉ) = 0 β†’ (((π‘ƒβ€˜0) = 𝐴 ∧ (π‘ƒβ€˜(β™―β€˜πΉ)) = 𝐡) ↔ ((π‘ƒβ€˜0) = 𝐴 ∧ (π‘ƒβ€˜0) = 𝐡)))
5 eqtr2 2755 . . . . . . . . 9 (((π‘ƒβ€˜0) = 𝐴 ∧ (π‘ƒβ€˜0) = 𝐡) β†’ 𝐴 = 𝐡)
6 nne 2943 . . . . . . . . 9 (Β¬ 𝐴 β‰  𝐡 ↔ 𝐴 = 𝐡)
75, 6sylibr 233 . . . . . . . 8 (((π‘ƒβ€˜0) = 𝐴 ∧ (π‘ƒβ€˜0) = 𝐡) β†’ Β¬ 𝐴 β‰  𝐡)
84, 7syl6bi 253 . . . . . . 7 ((β™―β€˜πΉ) = 0 β†’ (((π‘ƒβ€˜0) = 𝐴 ∧ (π‘ƒβ€˜(β™―β€˜πΉ)) = 𝐡) β†’ Β¬ 𝐴 β‰  𝐡))
98com12 32 . . . . . 6 (((π‘ƒβ€˜0) = 𝐴 ∧ (π‘ƒβ€˜(β™―β€˜πΉ)) = 𝐡) β†’ ((β™―β€˜πΉ) = 0 β†’ Β¬ 𝐴 β‰  𝐡))
1093adant1 1129 . . . . 5 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ (π‘ƒβ€˜0) = 𝐴 ∧ (π‘ƒβ€˜(β™―β€˜πΉ)) = 𝐡) β†’ ((β™―β€˜πΉ) = 0 β†’ Β¬ 𝐴 β‰  𝐡))
11103ad2ant3 1134 . . . 4 (((𝐺 ∈ V ∧ 𝐴 ∈ (Vtxβ€˜πΊ) ∧ 𝐡 ∈ (Vtxβ€˜πΊ)) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐹(Walksβ€˜πΊ)𝑃 ∧ (π‘ƒβ€˜0) = 𝐴 ∧ (π‘ƒβ€˜(β™―β€˜πΉ)) = 𝐡)) β†’ ((β™―β€˜πΉ) = 0 β†’ Β¬ 𝐴 β‰  𝐡))
122, 11syl 17 . . 3 (𝐹(𝐴(WalksOnβ€˜πΊ)𝐡)𝑃 β†’ ((β™―β€˜πΉ) = 0 β†’ Β¬ 𝐴 β‰  𝐡))
1312necon2ad 2954 . 2 (𝐹(𝐴(WalksOnβ€˜πΊ)𝐡)𝑃 β†’ (𝐴 β‰  𝐡 β†’ (β™―β€˜πΉ) β‰  0))
1413imp 406 1 ((𝐹(𝐴(WalksOnβ€˜πΊ)𝐡)𝑃 ∧ 𝐴 β‰  𝐡) β†’ (β™―β€˜πΉ) β‰  0)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105   β‰  wne 2939  Vcvv 3473   class class class wbr 5148  β€˜cfv 6543  (class class class)co 7412  0cc0 11114  β™―chash 14295  Vtxcvtx 28524  Walkscwlks 29121  WalksOncwlkson 29122
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  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-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7979  df-2nd 7980  df-wlkson 29125
This theorem is referenced by:  conngrv2edg  29716
  Copyright terms: Public domain W3C validator