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Theorem usgr2wlkspth 29547
Description: In a simple graph, any walk of length 2 between two different vertices is a simple path. (Contributed by Alexander van der Vekens, 2-Mar-2018.) (Revised by AV, 27-Jan-2021.) (Proof shortened by AV, 31-Oct-2021.)
Assertion
Ref Expression
usgr2wlkspth ((𝐺 ∈ USGraph ∧ (β™―β€˜πΉ) = 2 ∧ 𝐴 β‰  𝐡) β†’ (𝐹(𝐴(WalksOnβ€˜πΊ)𝐡)𝑃 ↔ 𝐹(𝐴(SPathsOnβ€˜πΊ)𝐡)𝑃))
Proof of Theorem usgr2wlkspth
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 ∈ USGraph ∧ (β™―β€˜πΉ) = 2 ∧ 𝐴 β‰  𝐡)) β†’ 𝐹(Walksβ€˜πΊ)𝑃)
2 simp2 1135 . . . . . . . . . . . . . 14 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ (π‘ƒβ€˜0) = 𝐴 ∧ (π‘ƒβ€˜(β™―β€˜πΉ)) = 𝐡) β†’ (π‘ƒβ€˜0) = 𝐴)
3 simp3 1136 . . . . . . . . . . . . . 14 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ (π‘ƒβ€˜0) = 𝐴 ∧ (π‘ƒβ€˜(β™―β€˜πΉ)) = 𝐡) β†’ (π‘ƒβ€˜(β™―β€˜πΉ)) = 𝐡)
42, 3neeq12d 2997 . . . . . . . . . . . . 13 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ (π‘ƒβ€˜0) = 𝐴 ∧ (π‘ƒβ€˜(β™―β€˜πΉ)) = 𝐡) β†’ ((π‘ƒβ€˜0) β‰  (π‘ƒβ€˜(β™―β€˜πΉ)) ↔ 𝐴 β‰  𝐡))
54bicomd 222 . . . . . . . . . . . 12 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ (π‘ƒβ€˜0) = 𝐴 ∧ (π‘ƒβ€˜(β™―β€˜πΉ)) = 𝐡) β†’ (𝐴 β‰  𝐡 ↔ (π‘ƒβ€˜0) β‰  (π‘ƒβ€˜(β™―β€˜πΉ))))
653anbi3d 1439 . . . . . . . . . . 11 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ (π‘ƒβ€˜0) = 𝐴 ∧ (π‘ƒβ€˜(β™―β€˜πΉ)) = 𝐡) β†’ ((𝐺 ∈ USGraph ∧ (β™―β€˜πΉ) = 2 ∧ 𝐴 β‰  𝐡) ↔ (𝐺 ∈ USGraph ∧ (β™―β€˜πΉ) = 2 ∧ (π‘ƒβ€˜0) β‰  (π‘ƒβ€˜(β™―β€˜πΉ)))))
7 usgr2wlkspthlem1 29545 . . . . . . . . . . . . 13 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ (𝐺 ∈ USGraph ∧ (β™―β€˜πΉ) = 2 ∧ (π‘ƒβ€˜0) β‰  (π‘ƒβ€˜(β™―β€˜πΉ)))) β†’ Fun ◑𝐹)
87ex 412 . . . . . . . . . . . 12 (𝐹(Walksβ€˜πΊ)𝑃 β†’ ((𝐺 ∈ USGraph ∧ (β™―β€˜πΉ) = 2 ∧ (π‘ƒβ€˜0) β‰  (π‘ƒβ€˜(β™―β€˜πΉ))) β†’ Fun ◑𝐹))
983ad2ant1 1131 . . . . . . . . . . 11 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ (π‘ƒβ€˜0) = 𝐴 ∧ (π‘ƒβ€˜(β™―β€˜πΉ)) = 𝐡) β†’ ((𝐺 ∈ USGraph ∧ (β™―β€˜πΉ) = 2 ∧ (π‘ƒβ€˜0) β‰  (π‘ƒβ€˜(β™―β€˜πΉ))) β†’ Fun ◑𝐹))
106, 9sylbid 239 . . . . . . . . . 10 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ (π‘ƒβ€˜0) = 𝐴 ∧ (π‘ƒβ€˜(β™―β€˜πΉ)) = 𝐡) β†’ ((𝐺 ∈ USGraph ∧ (β™―β€˜πΉ) = 2 ∧ 𝐴 β‰  𝐡) β†’ Fun ◑𝐹))
11103ad2ant3 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 ∈ USGraph ∧ (β™―β€˜πΉ) = 2 ∧ 𝐴 β‰  𝐡) β†’ Fun ◑𝐹))
1211imp 406 . . . . . . . 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 ∈ USGraph ∧ (β™―β€˜πΉ) = 2 ∧ 𝐴 β‰  𝐡)) β†’ Fun ◑𝐹)
13 istrl 29484 . . . . . . . 8 (𝐹(Trailsβ€˜πΊ)𝑃 ↔ (𝐹(Walksβ€˜πΊ)𝑃 ∧ Fun ◑𝐹))
141, 12, 13sylanbrc 582 . . . . . . 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 ∈ USGraph ∧ (β™―β€˜πΉ) = 2 ∧ 𝐴 β‰  𝐡)) β†’ 𝐹(Trailsβ€˜πΊ)𝑃)
15 usgr2wlkspthlem2 29546 . . . . . . . . . . . 12 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ (𝐺 ∈ USGraph ∧ (β™―β€˜πΉ) = 2 ∧ (π‘ƒβ€˜0) β‰  (π‘ƒβ€˜(β™―β€˜πΉ)))) β†’ Fun ◑𝑃)
1615ex 412 . . . . . . . . . . 11 (𝐹(Walksβ€˜πΊ)𝑃 β†’ ((𝐺 ∈ USGraph ∧ (β™―β€˜πΉ) = 2 ∧ (π‘ƒβ€˜0) β‰  (π‘ƒβ€˜(β™―β€˜πΉ))) β†’ Fun ◑𝑃))
17163ad2ant1 1131 . . . . . . . . . 10 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ (π‘ƒβ€˜0) = 𝐴 ∧ (π‘ƒβ€˜(β™―β€˜πΉ)) = 𝐡) β†’ ((𝐺 ∈ USGraph ∧ (β™―β€˜πΉ) = 2 ∧ (π‘ƒβ€˜0) β‰  (π‘ƒβ€˜(β™―β€˜πΉ))) β†’ Fun ◑𝑃))
186, 17sylbid 239 . . . . . . . . 9 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ (π‘ƒβ€˜0) = 𝐴 ∧ (π‘ƒβ€˜(β™―β€˜πΉ)) = 𝐡) β†’ ((𝐺 ∈ USGraph ∧ (β™―β€˜πΉ) = 2 ∧ 𝐴 β‰  𝐡) β†’ Fun ◑𝑃))
19183ad2ant3 1133 . . . . . . . 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)) β†’ ((𝐺 ∈ USGraph ∧ (β™―β€˜πΉ) = 2 ∧ 𝐴 β‰  𝐡) β†’ Fun ◑𝑃))
2019imp 406 . . . . . . 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 ∈ USGraph ∧ (β™―β€˜πΉ) = 2 ∧ 𝐴 β‰  𝐡)) β†’ Fun ◑𝑃)
21 isspth 29512 . . . . . . 7 (𝐹(SPathsβ€˜πΊ)𝑃 ↔ (𝐹(Trailsβ€˜πΊ)𝑃 ∧ Fun ◑𝑃))
2214, 20, 21sylanbrc 582 . . . . . 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 ∈ USGraph ∧ (β™―β€˜πΉ) = 2 ∧ 𝐴 β‰  𝐡)) β†’ 𝐹(SPathsβ€˜πΊ)𝑃)
23 3simpc 1148 . . . . . . . 8 ((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))
24233ad2ant3 1133 . . . . . . 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†’ ((π‘ƒβ€˜0) = 𝐴 ∧ (π‘ƒβ€˜(β™―β€˜πΉ)) = 𝐡))
2524adantr 480 . . . . . 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 ∈ USGraph ∧ (β™―β€˜πΉ) = 2 ∧ 𝐴 β‰  𝐡)) β†’ ((π‘ƒβ€˜0) = 𝐴 ∧ (π‘ƒβ€˜(β™―β€˜πΉ)) = 𝐡))
26 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)))
2722, 25, 26sylanbrc 582 . . . . 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 ∈ USGraph ∧ (β™―β€˜πΉ) = 2 ∧ 𝐴 β‰  𝐡)) β†’ (𝐹(SPathsβ€˜πΊ)𝑃 ∧ (π‘ƒβ€˜0) = 𝐴 ∧ (π‘ƒβ€˜(β™―β€˜πΉ)) = 𝐡))
28 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)))
2928adantr 480 . . . . . 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 ∈ USGraph ∧ (β™―β€˜πΉ) = 2 ∧ 𝐴 β‰  𝐡)) β†’ ((𝐴 ∈ (Vtxβ€˜πΊ) ∧ 𝐡 ∈ (Vtxβ€˜πΊ)) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)))
30 eqid 2727 . . . . . . 7 (Vtxβ€˜πΊ) = (Vtxβ€˜πΊ)
3130isspthonpth 29537 . . . . . 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)))
3229, 31syl 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 ∈ USGraph ∧ (β™―β€˜πΉ) = 2 ∧ 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)))
3327, 32mpbird 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 ∈ USGraph ∧ (β™―β€˜πΉ) = 2 ∧ 𝐴 β‰  𝐡)) β†’ 𝐹(𝐴(SPathsOnβ€˜πΊ)𝐡)𝑃)
3433ex 412 . . 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 ∈ USGraph ∧ (β™―β€˜πΉ) = 2 ∧ 𝐴 β‰  𝐡) β†’ 𝐹(𝐴(SPathsOnβ€˜πΊ)𝐡)𝑃))
3530wlkonprop 29446 . . . 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)))
36 3simpc 1148 . . . . 5 ((𝐺 ∈ V ∧ 𝐴 ∈ (Vtxβ€˜πΊ) ∧ 𝐡 ∈ (Vtxβ€˜πΊ)) β†’ (𝐴 ∈ (Vtxβ€˜πΊ) ∧ 𝐡 ∈ (Vtxβ€˜πΊ)))
37363anim1i 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)))
3835, 37syl 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)))
3934, 38syl11 33 . 2 ((𝐺 ∈ USGraph ∧ (β™―β€˜πΉ) = 2 ∧ 𝐴 β‰  𝐡) β†’ (𝐹(𝐴(WalksOnβ€˜πΊ)𝐡)𝑃 β†’ 𝐹(𝐴(SPathsOnβ€˜πΊ)𝐡)𝑃))
40 spthonpthon 29539 . . 3 (𝐹(𝐴(SPathsOnβ€˜πΊ)𝐡)𝑃 β†’ 𝐹(𝐴(PathsOnβ€˜πΊ)𝐡)𝑃)
41 pthontrlon 29535 . . 3 (𝐹(𝐴(PathsOnβ€˜πΊ)𝐡)𝑃 β†’ 𝐹(𝐴(TrailsOnβ€˜πΊ)𝐡)𝑃)
42 trlsonwlkon 29498 . . 3 (𝐹(𝐴(TrailsOnβ€˜πΊ)𝐡)𝑃 β†’ 𝐹(𝐴(WalksOnβ€˜πΊ)𝐡)𝑃)
4340, 41, 423syl 18 . 2 (𝐹(𝐴(SPathsOnβ€˜πΊ)𝐡)𝑃 β†’ 𝐹(𝐴(WalksOnβ€˜πΊ)𝐡)𝑃)
4439, 43impbid1 224 1 ((𝐺 ∈ USGraph ∧ (β™―β€˜πΉ) = 2 ∧ 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 395   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099   β‰  wne 2935  Vcvv 3469   class class class wbr 5142  β—‘ccnv 5671  Fun wfun 6536  β€˜cfv 6542  (class class class)co 7414  0cc0 11124  2c2 12283  β™―chash 14307  Vtxcvtx 28783  USGraphcusgr 28936  Walkscwlks 29384  WalksOncwlkson 29385  Trailsctrls 29478  TrailsOnctrlson 29479  SPathscspths 29501  PathsOncpthson 29502  SPathsOncspthson 29503
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7732  ax-cnex 11180  ax-resscn 11181  ax-1cn 11182  ax-icn 11183  ax-addcl 11184  ax-addrcl 11185  ax-mulcl 11186  ax-mulrcl 11187  ax-mulcom 11188  ax-addass 11189  ax-mulass 11190  ax-distr 11191  ax-i2m1 11192  ax-1ne0 11193  ax-1rid 11194  ax-rnegex 11195  ax-rrecex 11196  ax-cnre 11197  ax-pre-lttri 11198  ax-pre-lttrn 11199  ax-pre-ltadd 11200  ax-pre-mulgt0 11201
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-ifp 1062  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-tp 4629  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  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 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7863  df-1st 7985  df-2nd 7986  df-frecs 8278  df-wrecs 8309  df-recs 8383  df-rdg 8422  df-1o 8478  df-2o 8479  df-oadd 8482  df-er 8716  df-map 8836  df-pm 8837  df-en 8954  df-dom 8955  df-sdom 8956  df-fin 8957  df-dju 9910  df-card 9948  df-pnf 11266  df-mnf 11267  df-xr 11268  df-ltxr 11269  df-le 11270  df-sub 11462  df-neg 11463  df-nn 12229  df-2 12291  df-3 12292  df-n0 12489  df-xnn0 12561  df-z 12575  df-uz 12839  df-fz 13503  df-fzo 13646  df-hash 14308  df-word 14483  df-concat 14539  df-s1 14564  df-s2 14817  df-s3 14818  df-edg 28835  df-uhgr 28845  df-upgr 28869  df-umgr 28870  df-uspgr 28937  df-usgr 28938  df-wlks 29387  df-wlkson 29388  df-trls 29480  df-trlson 29481  df-pths 29504  df-spths 29505  df-pthson 29506  df-spthson 29507
This theorem is referenced by:  usgr2trlspth  29549  wpthswwlks2on  29746
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