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Mirrors  >  Home  >  MPE Home  >  Th. List  >  uhgrwkspth Structured version   Visualization version   GIF version
Theorem uhgrwkspth 29012
Description: Any walk of length 1 between two different vertices is a simple path. (Contributed by AV, 25-Jan-2021.) (Proof shortened by AV, 31-Oct-2021.) (Revised by AV, 7-Jul-2022.)
Assertion
Ref Expression
uhgrwkspth ((𝐺 ∈ π‘Š ∧ (β™―β€˜πΉ) = 1 ∧ 𝐴 β‰  𝐡) β†’ (𝐹(𝐴(WalksOnβ€˜πΊ)𝐡)𝑃 ↔ 𝐹(𝐴(SPathsOnβ€˜πΊ)𝐡)𝑃))
Proof of Theorem uhgrwkspth
StepHypRef Expression
1 simpl31 1255 . . . . . . . 8 ((((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)) ∧ (𝐺 ∈ π‘Š ∧ (β™―β€˜πΉ) = 1 ∧ 𝐴 β‰  𝐡)) β†’ 𝐹(Walksβ€˜πΊ)𝑃)
2 uhgrwkspthlem1 29010 . . . . . . . . . . . . . 14 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ (β™―β€˜πΉ) = 1) β†’ Fun ◑𝐹)
32expcom 415 . . . . . . . . . . . . 13 ((β™―β€˜πΉ) = 1 β†’ (𝐹(Walksβ€˜πΊ)𝑃 β†’ Fun ◑𝐹))
433ad2ant2 1135 . . . . . . . . . . . 12 ((𝐺 ∈ π‘Š ∧ (β™―β€˜πΉ) = 1 ∧ 𝐴 β‰  𝐡) β†’ (𝐹(Walksβ€˜πΊ)𝑃 β†’ Fun ◑𝐹))
54com12 32 . . . . . . . . . . 11 (𝐹(Walksβ€˜πΊ)𝑃 β†’ ((𝐺 ∈ π‘Š ∧ (β™―β€˜πΉ) = 1 ∧ 𝐴 β‰  𝐡) β†’ Fun ◑𝐹))
653ad2ant1 1134 . . . . . . . . . 10 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ (π‘ƒβ€˜0) = 𝐴 ∧ (π‘ƒβ€˜(β™―β€˜πΉ)) = 𝐡) β†’ ((𝐺 ∈ π‘Š ∧ (β™―β€˜πΉ) = 1 ∧ 𝐴 β‰  𝐡) β†’ Fun ◑𝐹))
763ad2ant3 1136 . . . . . . . . 9 (((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†’ ((𝐺 ∈ π‘Š ∧ (β™―β€˜πΉ) = 1 ∧ 𝐴 β‰  𝐡) β†’ Fun ◑𝐹))
87imp 408 . . . . . . . 8 ((((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)) ∧ (𝐺 ∈ π‘Š ∧ (β™―β€˜πΉ) = 1 ∧ 𝐴 β‰  𝐡)) β†’ Fun ◑𝐹)
9 istrl 28953 . . . . . . . 8 (𝐹(Trailsβ€˜πΊ)𝑃 ↔ (𝐹(Walksβ€˜πΊ)𝑃 ∧ Fun ◑𝐹))
101, 8, 9sylanbrc 584 . . . . . . 7 ((((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)) ∧ (𝐺 ∈ π‘Š ∧ (β™―β€˜πΉ) = 1 ∧ 𝐴 β‰  𝐡)) β†’ 𝐹(Trailsβ€˜πΊ)𝑃)
11 3simpc 1151 . . . . . . . . 9 ((𝐺 ∈ π‘Š ∧ (β™―β€˜πΉ) = 1 ∧ 𝐴 β‰  𝐡) β†’ ((β™―β€˜πΉ) = 1 ∧ 𝐴 β‰  𝐡))
1211adantl 483 . . . . . . . 8 ((((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)) ∧ (𝐺 ∈ π‘Š ∧ (β™―β€˜πΉ) = 1 ∧ 𝐴 β‰  𝐡)) β†’ ((β™―β€˜πΉ) = 1 ∧ 𝐴 β‰  𝐡))
13 3simpc 1151 . . . . . . . . . 10 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ (π‘ƒβ€˜0) = 𝐴 ∧ (π‘ƒβ€˜(β™―β€˜πΉ)) = 𝐡) β†’ ((π‘ƒβ€˜0) = 𝐴 ∧ (π‘ƒβ€˜(β™―β€˜πΉ)) = 𝐡))
14133ad2ant3 1136 . . . . . . . . 9 (((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†’ ((π‘ƒβ€˜0) = 𝐴 ∧ (π‘ƒβ€˜(β™―β€˜πΉ)) = 𝐡))
1514adantr 482 . . . . . . . 8 ((((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)) ∧ (𝐺 ∈ π‘Š ∧ (β™―β€˜πΉ) = 1 ∧ 𝐴 β‰  𝐡)) β†’ ((π‘ƒβ€˜0) = 𝐴 ∧ (π‘ƒβ€˜(β™―β€˜πΉ)) = 𝐡))
16 uhgrwkspthlem2 29011 . . . . . . . 8 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ ((β™―β€˜πΉ) = 1 ∧ 𝐴 β‰  𝐡) ∧ ((π‘ƒβ€˜0) = 𝐴 ∧ (π‘ƒβ€˜(β™―β€˜πΉ)) = 𝐡)) β†’ Fun ◑𝑃)
171, 12, 15, 16syl3anc 1372 . . . . . . 7 ((((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)) ∧ (𝐺 ∈ π‘Š ∧ (β™―β€˜πΉ) = 1 ∧ 𝐴 β‰  𝐡)) β†’ Fun ◑𝑃)
18 isspth 28981 . . . . . . 7 (𝐹(SPathsβ€˜πΊ)𝑃 ↔ (𝐹(Trailsβ€˜πΊ)𝑃 ∧ Fun ◑𝑃))
1910, 17, 18sylanbrc 584 . . . . . 6 ((((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)) ∧ (𝐺 ∈ π‘Š ∧ (β™―β€˜πΉ) = 1 ∧ 𝐴 β‰  𝐡)) β†’ 𝐹(SPathsβ€˜πΊ)𝑃)
20 3anass 1096 . . . . . 6 ((𝐹(SPathsβ€˜πΊ)𝑃 ∧ (π‘ƒβ€˜0) = 𝐴 ∧ (π‘ƒβ€˜(β™―β€˜πΉ)) = 𝐡) ↔ (𝐹(SPathsβ€˜πΊ)𝑃 ∧ ((π‘ƒβ€˜0) = 𝐴 ∧ (π‘ƒβ€˜(β™―β€˜πΉ)) = 𝐡)))
2119, 15, 20sylanbrc 584 . . . . 5 ((((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)) ∧ (𝐺 ∈ π‘Š ∧ (β™―β€˜πΉ) = 1 ∧ 𝐴 β‰  𝐡)) β†’ (𝐹(SPathsβ€˜πΊ)𝑃 ∧ (π‘ƒβ€˜0) = 𝐴 ∧ (π‘ƒβ€˜(β™―β€˜πΉ)) = 𝐡))
22 3simpa 1149 . . . . . . 7 (((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†’ ((𝐴 ∈ (Vtxβ€˜πΊ) ∧ 𝐡 ∈ (Vtxβ€˜πΊ)) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)))
2322adantr 482 . . . . . 6 ((((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)) ∧ (𝐺 ∈ π‘Š ∧ (β™―β€˜πΉ) = 1 ∧ 𝐴 β‰  𝐡)) β†’ ((𝐴 ∈ (Vtxβ€˜πΊ) ∧ 𝐡 ∈ (Vtxβ€˜πΊ)) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)))
24 eqid 2733 . . . . . . 7 (Vtxβ€˜πΊ) = (Vtxβ€˜πΊ)
2524isspthonpth 29006 . . . . . 6 (((𝐴 ∈ (Vtxβ€˜πΊ) ∧ 𝐡 ∈ (Vtxβ€˜πΊ)) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) β†’ (𝐹(𝐴(SPathsOnβ€˜πΊ)𝐡)𝑃 ↔ (𝐹(SPathsβ€˜πΊ)𝑃 ∧ (π‘ƒβ€˜0) = 𝐴 ∧ (π‘ƒβ€˜(β™―β€˜πΉ)) = 𝐡)))
2623, 25syl 17 . . . . 5 ((((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)) ∧ (𝐺 ∈ π‘Š ∧ (β™―β€˜πΉ) = 1 ∧ 𝐴 β‰  𝐡)) β†’ (𝐹(𝐴(SPathsOnβ€˜πΊ)𝐡)𝑃 ↔ (𝐹(SPathsβ€˜πΊ)𝑃 ∧ (π‘ƒβ€˜0) = 𝐴 ∧ (π‘ƒβ€˜(β™―β€˜πΉ)) = 𝐡)))
2721, 26mpbird 257 . . . 4 ((((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)) ∧ (𝐺 ∈ π‘Š ∧ (β™―β€˜πΉ) = 1 ∧ 𝐴 β‰  𝐡)) β†’ 𝐹(𝐴(SPathsOnβ€˜πΊ)𝐡)𝑃)
2827ex 414 . . 3 (((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†’ ((𝐺 ∈ π‘Š ∧ (β™―β€˜πΉ) = 1 ∧ 𝐴 β‰  𝐡) β†’ 𝐹(𝐴(SPathsOnβ€˜πΊ)𝐡)𝑃))
2924wlkonprop 28915 . . . 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)))
30 3simpc 1151 . . . . 5 ((𝐺 ∈ V ∧ 𝐴 ∈ (Vtxβ€˜πΊ) ∧ 𝐡 ∈ (Vtxβ€˜πΊ)) β†’ (𝐴 ∈ (Vtxβ€˜πΊ) ∧ 𝐡 ∈ (Vtxβ€˜πΊ)))
31303anim1i 1153 . . . 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†’ ((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)))
3229, 31syl 17 . . 3 (𝐹(𝐴(WalksOnβ€˜πΊ)𝐡)𝑃 β†’ ((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)))
3328, 32syl11 33 . 2 ((𝐺 ∈ π‘Š ∧ (β™―β€˜πΉ) = 1 ∧ 𝐴 β‰  𝐡) β†’ (𝐹(𝐴(WalksOnβ€˜πΊ)𝐡)𝑃 β†’ 𝐹(𝐴(SPathsOnβ€˜πΊ)𝐡)𝑃))
34 spthonpthon 29008 . . 3 (𝐹(𝐴(SPathsOnβ€˜πΊ)𝐡)𝑃 β†’ 𝐹(𝐴(PathsOnβ€˜πΊ)𝐡)𝑃)
35 pthontrlon 29004 . . 3 (𝐹(𝐴(PathsOnβ€˜πΊ)𝐡)𝑃 β†’ 𝐹(𝐴(TrailsOnβ€˜πΊ)𝐡)𝑃)
36 trlsonwlkon 28967 . . 3 (𝐹(𝐴(TrailsOnβ€˜πΊ)𝐡)𝑃 β†’ 𝐹(𝐴(WalksOnβ€˜πΊ)𝐡)𝑃)
3734, 35, 363syl 18 . 2 (𝐹(𝐴(SPathsOnβ€˜πΊ)𝐡)𝑃 β†’ 𝐹(𝐴(WalksOnβ€˜πΊ)𝐡)𝑃)
3833, 37impbid1 224 1 ((𝐺 ∈ π‘Š ∧ (β™―β€˜πΉ) = 1 ∧ 𝐴 β‰  𝐡) β†’ (𝐹(𝐴(WalksOnβ€˜πΊ)𝐡)𝑃 ↔ 𝐹(𝐴(SPathsOnβ€˜πΊ)𝐡)𝑃))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2941  Vcvv 3475   class class class wbr 5149  β—‘ccnv 5676  Fun wfun 6538  β€˜cfv 6544  (class class class)co 7409  0cc0 11110  1c1 11111  β™―chash 14290  Vtxcvtx 28256  Walkscwlks 28853  WalksOncwlkson 28854  Trailsctrls 28947  TrailsOnctrlson 28948  SPathscspths 28970  PathsOncpthson 28971  SPathsOncspthson 28972
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-ifp 1063  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-er 8703  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-card 9934  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-n0 12473  df-z 12559  df-uz 12823  df-fz 13485  df-fzo 13628  df-hash 14291  df-word 14465  df-concat 14521  df-s1 14546  df-s2 14799  df-wlks 28856  df-wlkson 28857  df-trls 28949  df-trlson 28950  df-pths 28973  df-spths 28974  df-pthson 28975  df-spthson 28976
This theorem is referenced by: (None)
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