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Mirrors  >  Home  >  MPE Home  >  Th. List  >  uhgrwkspth Structured version   Visualization version   GIF version
Theorem uhgrwkspth 29279
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 1252 . . . . . . . 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 29277 . . . . . . . . . . . . . 14 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ (β™―β€˜πΉ) = 1) β†’ Fun ◑𝐹)
32expcom 412 . . . . . . . . . . . . 13 ((β™―β€˜πΉ) = 1 β†’ (𝐹(Walksβ€˜πΊ)𝑃 β†’ Fun ◑𝐹))
433ad2ant2 1132 . . . . . . . . . . . 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 1131 . . . . . . . . . 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 1133 . . . . . . . . 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 405 . . . . . . . 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 29220 . . . . . . . 8 (𝐹(Trailsβ€˜πΊ)𝑃 ↔ (𝐹(Walksβ€˜πΊ)𝑃 ∧ Fun ◑𝐹))
101, 8, 9sylanbrc 581 . . . . . . 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 1148 . . . . . . . . 9 ((𝐺 ∈ π‘Š ∧ (β™―β€˜πΉ) = 1 ∧ 𝐴 β‰  𝐡) β†’ ((β™―β€˜πΉ) = 1 ∧ 𝐴 β‰  𝐡))
1211adantl 480 . . . . . . . 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 1148 . . . . . . . . . 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 1133 . . . . . . . . 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 479 . . . . . . . 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 29278 . . . . . . . 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 1369 . . . . . . 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 29248 . . . . . . 7 (𝐹(SPathsβ€˜πΊ)𝑃 ↔ (𝐹(Trailsβ€˜πΊ)𝑃 ∧ Fun ◑𝑃))
1910, 17, 18sylanbrc 581 . . . . . 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 1093 . . . . . 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 581 . . . . 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 1146 . . . . . . 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 479 . . . . . 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 2730 . . . . . . 7 (Vtxβ€˜πΊ) = (Vtxβ€˜πΊ)
2524isspthonpth 29273 . . . . . 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 256 . . . 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 411 . . 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 29182 . . . 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 1148 . . . . 5 ((𝐺 ∈ V ∧ 𝐴 ∈ (Vtxβ€˜πΊ) ∧ 𝐡 ∈ (Vtxβ€˜πΊ)) β†’ (𝐴 ∈ (Vtxβ€˜πΊ) ∧ 𝐡 ∈ (Vtxβ€˜πΊ)))
31303anim1i 1150 . . . 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 29275 . . 3 (𝐹(𝐴(SPathsOnβ€˜πΊ)𝐡)𝑃 β†’ 𝐹(𝐴(PathsOnβ€˜πΊ)𝐡)𝑃)
35 pthontrlon 29271 . . 3 (𝐹(𝐴(PathsOnβ€˜πΊ)𝐡)𝑃 β†’ 𝐹(𝐴(TrailsOnβ€˜πΊ)𝐡)𝑃)
36 trlsonwlkon 29234 . . 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 394   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104   β‰  wne 2938  Vcvv 3472   class class class wbr 5147  β—‘ccnv 5674  Fun wfun 6536  β€˜cfv 6542  (class class class)co 7411  0cc0 11112  1c1 11113  β™―chash 14294  Vtxcvtx 28523  Walkscwlks 29120  WalksOncwlkson 29121  Trailsctrls 29214  TrailsOnctrlson 29215  SPathscspths 29237  PathsOncpthson 29238  SPathsOncspthson 29239
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-ifp 1060  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  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 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-er 8705  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-card 9936  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-n0 12477  df-z 12563  df-uz 12827  df-fz 13489  df-fzo 13632  df-hash 14295  df-word 14469  df-concat 14525  df-s1 14550  df-s2 14803  df-wlks 29123  df-wlkson 29124  df-trls 29216  df-trlson 29217  df-pths 29240  df-spths 29241  df-pthson 29242  df-spthson 29243
This theorem is referenced by: (None)
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