Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  uspgr2wlkeq Structured version   Visualization version   GIF version

Theorem uspgr2wlkeq 40854
 Description: Conditions for two walks within the same simple pseudograph being the same. It is sufficient that the vertices (in the same order) are identical. (Contributed by AV, 3-Jul-2018.) (Revised by AV, 14-Apr-2021.)
Assertion
Ref Expression
uspgr2wlkeq ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘𝐺)) ∧ 𝑁 = (#‘(1st𝐴))) → (𝐴 = 𝐵 ↔ (𝑁 = (#‘(1st𝐵)) ∧ ∀𝑦 ∈ (0...𝑁)((2nd𝐴)‘𝑦) = ((2nd𝐵)‘𝑦))))
Distinct variable groups:   𝑦,𝐴   𝑦,𝐵   𝑦,𝐺   𝑦,𝑁

Proof of Theorem uspgr2wlkeq
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 3anan32 1043 . . 3 ((𝑁 = (#‘(1st𝐵)) ∧ ∀𝑦 ∈ (0..^𝑁)((1st𝐴)‘𝑦) = ((1st𝐵)‘𝑦) ∧ ∀𝑦 ∈ (0...𝑁)((2nd𝐴)‘𝑦) = ((2nd𝐵)‘𝑦)) ↔ ((𝑁 = (#‘(1st𝐵)) ∧ ∀𝑦 ∈ (0...𝑁)((2nd𝐴)‘𝑦) = ((2nd𝐵)‘𝑦)) ∧ ∀𝑦 ∈ (0..^𝑁)((1st𝐴)‘𝑦) = ((1st𝐵)‘𝑦)))
21a1i 11 . 2 ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘𝐺)) ∧ 𝑁 = (#‘(1st𝐴))) → ((𝑁 = (#‘(1st𝐵)) ∧ ∀𝑦 ∈ (0..^𝑁)((1st𝐴)‘𝑦) = ((1st𝐵)‘𝑦) ∧ ∀𝑦 ∈ (0...𝑁)((2nd𝐴)‘𝑦) = ((2nd𝐵)‘𝑦)) ↔ ((𝑁 = (#‘(1st𝐵)) ∧ ∀𝑦 ∈ (0...𝑁)((2nd𝐴)‘𝑦) = ((2nd𝐵)‘𝑦)) ∧ ∀𝑦 ∈ (0..^𝑁)((1st𝐴)‘𝑦) = ((1st𝐵)‘𝑦))))
3 1wlkeq 40838 . . . 4 ((𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘𝐺) ∧ 𝑁 = (#‘(1st𝐴))) → (𝐴 = 𝐵 ↔ (𝑁 = (#‘(1st𝐵)) ∧ ∀𝑦 ∈ (0..^𝑁)((1st𝐴)‘𝑦) = ((1st𝐵)‘𝑦) ∧ ∀𝑦 ∈ (0...𝑁)((2nd𝐴)‘𝑦) = ((2nd𝐵)‘𝑦))))
433expa 1257 . . 3 (((𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘𝐺)) ∧ 𝑁 = (#‘(1st𝐴))) → (𝐴 = 𝐵 ↔ (𝑁 = (#‘(1st𝐵)) ∧ ∀𝑦 ∈ (0..^𝑁)((1st𝐴)‘𝑦) = ((1st𝐵)‘𝑦) ∧ ∀𝑦 ∈ (0...𝑁)((2nd𝐴)‘𝑦) = ((2nd𝐵)‘𝑦))))
543adant1 1072 . 2 ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘𝐺)) ∧ 𝑁 = (#‘(1st𝐴))) → (𝐴 = 𝐵 ↔ (𝑁 = (#‘(1st𝐵)) ∧ ∀𝑦 ∈ (0..^𝑁)((1st𝐴)‘𝑦) = ((1st𝐵)‘𝑦) ∧ ∀𝑦 ∈ (0...𝑁)((2nd𝐴)‘𝑦) = ((2nd𝐵)‘𝑦))))
6 fzofzp1 12431 . . . . . . . . . . . 12 (𝑥 ∈ (0..^𝑁) → (𝑥 + 1) ∈ (0...𝑁))
76adantl 481 . . . . . . . . . . 11 ((((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘𝐺)) ∧ 𝑁 = (#‘(1st𝐴))) ∧ 𝑁 = (#‘(1st𝐵))) ∧ 𝑥 ∈ (0..^𝑁)) → (𝑥 + 1) ∈ (0...𝑁))
8 fveq2 6103 . . . . . . . . . . . . 13 (𝑦 = (𝑥 + 1) → ((2nd𝐴)‘𝑦) = ((2nd𝐴)‘(𝑥 + 1)))
9 fveq2 6103 . . . . . . . . . . . . 13 (𝑦 = (𝑥 + 1) → ((2nd𝐵)‘𝑦) = ((2nd𝐵)‘(𝑥 + 1)))
108, 9eqeq12d 2625 . . . . . . . . . . . 12 (𝑦 = (𝑥 + 1) → (((2nd𝐴)‘𝑦) = ((2nd𝐵)‘𝑦) ↔ ((2nd𝐴)‘(𝑥 + 1)) = ((2nd𝐵)‘(𝑥 + 1))))
1110adantl 481 . . . . . . . . . . 11 (((((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘𝐺)) ∧ 𝑁 = (#‘(1st𝐴))) ∧ 𝑁 = (#‘(1st𝐵))) ∧ 𝑥 ∈ (0..^𝑁)) ∧ 𝑦 = (𝑥 + 1)) → (((2nd𝐴)‘𝑦) = ((2nd𝐵)‘𝑦) ↔ ((2nd𝐴)‘(𝑥 + 1)) = ((2nd𝐵)‘(𝑥 + 1))))
127, 11rspcdv 3285 . . . . . . . . . 10 ((((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘𝐺)) ∧ 𝑁 = (#‘(1st𝐴))) ∧ 𝑁 = (#‘(1st𝐵))) ∧ 𝑥 ∈ (0..^𝑁)) → (∀𝑦 ∈ (0...𝑁)((2nd𝐴)‘𝑦) = ((2nd𝐵)‘𝑦) → ((2nd𝐴)‘(𝑥 + 1)) = ((2nd𝐵)‘(𝑥 + 1))))
1312impancom 455 . . . . . . . . 9 ((((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘𝐺)) ∧ 𝑁 = (#‘(1st𝐴))) ∧ 𝑁 = (#‘(1st𝐵))) ∧ ∀𝑦 ∈ (0...𝑁)((2nd𝐴)‘𝑦) = ((2nd𝐵)‘𝑦)) → (𝑥 ∈ (0..^𝑁) → ((2nd𝐴)‘(𝑥 + 1)) = ((2nd𝐵)‘(𝑥 + 1))))
1413ralrimiv 2948 . . . . . . . 8 ((((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘𝐺)) ∧ 𝑁 = (#‘(1st𝐴))) ∧ 𝑁 = (#‘(1st𝐵))) ∧ ∀𝑦 ∈ (0...𝑁)((2nd𝐴)‘𝑦) = ((2nd𝐵)‘𝑦)) → ∀𝑥 ∈ (0..^𝑁)((2nd𝐴)‘(𝑥 + 1)) = ((2nd𝐵)‘(𝑥 + 1)))
15 oveq1 6556 . . . . . . . . . . 11 (𝑦 = 𝑥 → (𝑦 + 1) = (𝑥 + 1))
1615fveq2d 6107 . . . . . . . . . 10 (𝑦 = 𝑥 → ((2nd𝐴)‘(𝑦 + 1)) = ((2nd𝐴)‘(𝑥 + 1)))
1715fveq2d 6107 . . . . . . . . . 10 (𝑦 = 𝑥 → ((2nd𝐵)‘(𝑦 + 1)) = ((2nd𝐵)‘(𝑥 + 1)))
1816, 17eqeq12d 2625 . . . . . . . . 9 (𝑦 = 𝑥 → (((2nd𝐴)‘(𝑦 + 1)) = ((2nd𝐵)‘(𝑦 + 1)) ↔ ((2nd𝐴)‘(𝑥 + 1)) = ((2nd𝐵)‘(𝑥 + 1))))
1918cbvralv 3147 . . . . . . . 8 (∀𝑦 ∈ (0..^𝑁)((2nd𝐴)‘(𝑦 + 1)) = ((2nd𝐵)‘(𝑦 + 1)) ↔ ∀𝑥 ∈ (0..^𝑁)((2nd𝐴)‘(𝑥 + 1)) = ((2nd𝐵)‘(𝑥 + 1)))
2014, 19sylibr 223 . . . . . . 7 ((((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘𝐺)) ∧ 𝑁 = (#‘(1st𝐴))) ∧ 𝑁 = (#‘(1st𝐵))) ∧ ∀𝑦 ∈ (0...𝑁)((2nd𝐴)‘𝑦) = ((2nd𝐵)‘𝑦)) → ∀𝑦 ∈ (0..^𝑁)((2nd𝐴)‘(𝑦 + 1)) = ((2nd𝐵)‘(𝑦 + 1)))
21 fzossfz 12357 . . . . . . . . . 10 (0..^𝑁) ⊆ (0...𝑁)
22 ssralv 3629 . . . . . . . . . 10 ((0..^𝑁) ⊆ (0...𝑁) → (∀𝑦 ∈ (0...𝑁)((2nd𝐴)‘𝑦) = ((2nd𝐵)‘𝑦) → ∀𝑦 ∈ (0..^𝑁)((2nd𝐴)‘𝑦) = ((2nd𝐵)‘𝑦)))
2321, 22mp1i 13 . . . . . . . . 9 (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘𝐺)) ∧ 𝑁 = (#‘(1st𝐴))) ∧ 𝑁 = (#‘(1st𝐵))) → (∀𝑦 ∈ (0...𝑁)((2nd𝐴)‘𝑦) = ((2nd𝐵)‘𝑦) → ∀𝑦 ∈ (0..^𝑁)((2nd𝐴)‘𝑦) = ((2nd𝐵)‘𝑦)))
24 r19.26 3046 . . . . . . . . . . 11 (∀𝑦 ∈ (0..^𝑁)(((2nd𝐴)‘𝑦) = ((2nd𝐵)‘𝑦) ∧ ((2nd𝐴)‘(𝑦 + 1)) = ((2nd𝐵)‘(𝑦 + 1))) ↔ (∀𝑦 ∈ (0..^𝑁)((2nd𝐴)‘𝑦) = ((2nd𝐵)‘𝑦) ∧ ∀𝑦 ∈ (0..^𝑁)((2nd𝐴)‘(𝑦 + 1)) = ((2nd𝐵)‘(𝑦 + 1))))
25 preq12 4214 . . . . . . . . . . . . 13 ((((2nd𝐴)‘𝑦) = ((2nd𝐵)‘𝑦) ∧ ((2nd𝐴)‘(𝑦 + 1)) = ((2nd𝐵)‘(𝑦 + 1))) → {((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))} = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))})
2625a1i 11 . . . . . . . . . . . 12 (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘𝐺)) ∧ 𝑁 = (#‘(1st𝐴))) ∧ 𝑁 = (#‘(1st𝐵))) → ((((2nd𝐴)‘𝑦) = ((2nd𝐵)‘𝑦) ∧ ((2nd𝐴)‘(𝑦 + 1)) = ((2nd𝐵)‘(𝑦 + 1))) → {((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))} = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))}))
2726ralimdv 2946 . . . . . . . . . . 11 (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘𝐺)) ∧ 𝑁 = (#‘(1st𝐴))) ∧ 𝑁 = (#‘(1st𝐵))) → (∀𝑦 ∈ (0..^𝑁)(((2nd𝐴)‘𝑦) = ((2nd𝐵)‘𝑦) ∧ ((2nd𝐴)‘(𝑦 + 1)) = ((2nd𝐵)‘(𝑦 + 1))) → ∀𝑦 ∈ (0..^𝑁){((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))} = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))}))
2824, 27syl5bir 232 . . . . . . . . . 10 (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘𝐺)) ∧ 𝑁 = (#‘(1st𝐴))) ∧ 𝑁 = (#‘(1st𝐵))) → ((∀𝑦 ∈ (0..^𝑁)((2nd𝐴)‘𝑦) = ((2nd𝐵)‘𝑦) ∧ ∀𝑦 ∈ (0..^𝑁)((2nd𝐴)‘(𝑦 + 1)) = ((2nd𝐵)‘(𝑦 + 1))) → ∀𝑦 ∈ (0..^𝑁){((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))} = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))}))
2928expd 451 . . . . . . . . 9 (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘𝐺)) ∧ 𝑁 = (#‘(1st𝐴))) ∧ 𝑁 = (#‘(1st𝐵))) → (∀𝑦 ∈ (0..^𝑁)((2nd𝐴)‘𝑦) = ((2nd𝐵)‘𝑦) → (∀𝑦 ∈ (0..^𝑁)((2nd𝐴)‘(𝑦 + 1)) = ((2nd𝐵)‘(𝑦 + 1)) → ∀𝑦 ∈ (0..^𝑁){((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))} = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))})))
3023, 29syld 46 . . . . . . . 8 (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘𝐺)) ∧ 𝑁 = (#‘(1st𝐴))) ∧ 𝑁 = (#‘(1st𝐵))) → (∀𝑦 ∈ (0...𝑁)((2nd𝐴)‘𝑦) = ((2nd𝐵)‘𝑦) → (∀𝑦 ∈ (0..^𝑁)((2nd𝐴)‘(𝑦 + 1)) = ((2nd𝐵)‘(𝑦 + 1)) → ∀𝑦 ∈ (0..^𝑁){((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))} = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))})))
3130imp 444 . . . . . . 7 ((((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘𝐺)) ∧ 𝑁 = (#‘(1st𝐴))) ∧ 𝑁 = (#‘(1st𝐵))) ∧ ∀𝑦 ∈ (0...𝑁)((2nd𝐴)‘𝑦) = ((2nd𝐵)‘𝑦)) → (∀𝑦 ∈ (0..^𝑁)((2nd𝐴)‘(𝑦 + 1)) = ((2nd𝐵)‘(𝑦 + 1)) → ∀𝑦 ∈ (0..^𝑁){((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))} = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))}))
3220, 31mpd 15 . . . . . 6 ((((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘𝐺)) ∧ 𝑁 = (#‘(1st𝐴))) ∧ 𝑁 = (#‘(1st𝐵))) ∧ ∀𝑦 ∈ (0...𝑁)((2nd𝐴)‘𝑦) = ((2nd𝐵)‘𝑦)) → ∀𝑦 ∈ (0..^𝑁){((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))} = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))})
3332ex 449 . . . . 5 (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘𝐺)) ∧ 𝑁 = (#‘(1st𝐴))) ∧ 𝑁 = (#‘(1st𝐵))) → (∀𝑦 ∈ (0...𝑁)((2nd𝐴)‘𝑦) = ((2nd𝐵)‘𝑦) → ∀𝑦 ∈ (0..^𝑁){((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))} = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))}))
34 uspgrupgr 40406 . . . . . . . 8 (𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph )
35 eqid 2610 . . . . . . . . . 10 (Vtx‘𝐺) = (Vtx‘𝐺)
36 eqid 2610 . . . . . . . . . 10 (iEdg‘𝐺) = (iEdg‘𝐺)
37 eqid 2610 . . . . . . . . . 10 (1st𝐴) = (1st𝐴)
38 eqid 2610 . . . . . . . . . 10 (2nd𝐴) = (2nd𝐴)
3935, 36, 37, 38upgr1wlkcompim 40851 . . . . . . . . 9 ((𝐺 ∈ UPGraph ∧ 𝐴 ∈ (1Walks‘𝐺)) → ((1st𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝐴):(0...(#‘(1st𝐴)))⟶(Vtx‘𝐺) ∧ ∀𝑦 ∈ (0..^(#‘(1st𝐴)))((iEdg‘𝐺)‘((1st𝐴)‘𝑦)) = {((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))}))
4039ex 449 . . . . . . . 8 (𝐺 ∈ UPGraph → (𝐴 ∈ (1Walks‘𝐺) → ((1st𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝐴):(0...(#‘(1st𝐴)))⟶(Vtx‘𝐺) ∧ ∀𝑦 ∈ (0..^(#‘(1st𝐴)))((iEdg‘𝐺)‘((1st𝐴)‘𝑦)) = {((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))})))
4134, 40syl 17 . . . . . . 7 (𝐺 ∈ USPGraph → (𝐴 ∈ (1Walks‘𝐺) → ((1st𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝐴):(0...(#‘(1st𝐴)))⟶(Vtx‘𝐺) ∧ ∀𝑦 ∈ (0..^(#‘(1st𝐴)))((iEdg‘𝐺)‘((1st𝐴)‘𝑦)) = {((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))})))
42 eqid 2610 . . . . . . . . . 10 (1st𝐵) = (1st𝐵)
43 eqid 2610 . . . . . . . . . 10 (2nd𝐵) = (2nd𝐵)
4435, 36, 42, 43upgr1wlkcompim 40851 . . . . . . . . 9 ((𝐺 ∈ UPGraph ∧ 𝐵 ∈ (1Walks‘𝐺)) → ((1st𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝐵):(0...(#‘(1st𝐵)))⟶(Vtx‘𝐺) ∧ ∀𝑦 ∈ (0..^(#‘(1st𝐵)))((iEdg‘𝐺)‘((1st𝐵)‘𝑦)) = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))}))
4544ex 449 . . . . . . . 8 (𝐺 ∈ UPGraph → (𝐵 ∈ (1Walks‘𝐺) → ((1st𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝐵):(0...(#‘(1st𝐵)))⟶(Vtx‘𝐺) ∧ ∀𝑦 ∈ (0..^(#‘(1st𝐵)))((iEdg‘𝐺)‘((1st𝐵)‘𝑦)) = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))})))
4634, 45syl 17 . . . . . . 7 (𝐺 ∈ USPGraph → (𝐵 ∈ (1Walks‘𝐺) → ((1st𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝐵):(0...(#‘(1st𝐵)))⟶(Vtx‘𝐺) ∧ ∀𝑦 ∈ (0..^(#‘(1st𝐵)))((iEdg‘𝐺)‘((1st𝐵)‘𝑦)) = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))})))
47 oveq2 6557 . . . . . . . . . . . . . . . . . . 19 ((#‘(1st𝐵)) = 𝑁 → (0..^(#‘(1st𝐵))) = (0..^𝑁))
4847eqcoms 2618 . . . . . . . . . . . . . . . . . 18 (𝑁 = (#‘(1st𝐵)) → (0..^(#‘(1st𝐵))) = (0..^𝑁))
4948raleqdv 3121 . . . . . . . . . . . . . . . . 17 (𝑁 = (#‘(1st𝐵)) → (∀𝑦 ∈ (0..^(#‘(1st𝐵)))((iEdg‘𝐺)‘((1st𝐵)‘𝑦)) = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))} ↔ ∀𝑦 ∈ (0..^𝑁)((iEdg‘𝐺)‘((1st𝐵)‘𝑦)) = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))}))
50 oveq2 6557 . . . . . . . . . . . . . . . . . . 19 ((#‘(1st𝐴)) = 𝑁 → (0..^(#‘(1st𝐴))) = (0..^𝑁))
5150eqcoms 2618 . . . . . . . . . . . . . . . . . 18 (𝑁 = (#‘(1st𝐴)) → (0..^(#‘(1st𝐴))) = (0..^𝑁))
5251raleqdv 3121 . . . . . . . . . . . . . . . . 17 (𝑁 = (#‘(1st𝐴)) → (∀𝑦 ∈ (0..^(#‘(1st𝐴)))((iEdg‘𝐺)‘((1st𝐴)‘𝑦)) = {((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))} ↔ ∀𝑦 ∈ (0..^𝑁)((iEdg‘𝐺)‘((1st𝐴)‘𝑦)) = {((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))}))
5349, 52bi2anan9r 914 . . . . . . . . . . . . . . . 16 ((𝑁 = (#‘(1st𝐴)) ∧ 𝑁 = (#‘(1st𝐵))) → ((∀𝑦 ∈ (0..^(#‘(1st𝐵)))((iEdg‘𝐺)‘((1st𝐵)‘𝑦)) = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))} ∧ ∀𝑦 ∈ (0..^(#‘(1st𝐴)))((iEdg‘𝐺)‘((1st𝐴)‘𝑦)) = {((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))}) ↔ (∀𝑦 ∈ (0..^𝑁)((iEdg‘𝐺)‘((1st𝐵)‘𝑦)) = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))} ∧ ∀𝑦 ∈ (0..^𝑁)((iEdg‘𝐺)‘((1st𝐴)‘𝑦)) = {((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))})))
54 r19.26 3046 . . . . . . . . . . . . . . . . 17 (∀𝑦 ∈ (0..^𝑁)(((iEdg‘𝐺)‘((1st𝐵)‘𝑦)) = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))} ∧ ((iEdg‘𝐺)‘((1st𝐴)‘𝑦)) = {((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))}) ↔ (∀𝑦 ∈ (0..^𝑁)((iEdg‘𝐺)‘((1st𝐵)‘𝑦)) = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))} ∧ ∀𝑦 ∈ (0..^𝑁)((iEdg‘𝐺)‘((1st𝐴)‘𝑦)) = {((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))}))
55 eqeq2 2621 . . . . . . . . . . . . . . . . . . . . 21 ({((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))} = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))} → (((iEdg‘𝐺)‘((1st𝐴)‘𝑦)) = {((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))} ↔ ((iEdg‘𝐺)‘((1st𝐴)‘𝑦)) = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))}))
56 eqeq2 2621 . . . . . . . . . . . . . . . . . . . . . . 23 ({((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))} = ((iEdg‘𝐺)‘((1st𝐴)‘𝑦)) → (((iEdg‘𝐺)‘((1st𝐵)‘𝑦)) = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))} ↔ ((iEdg‘𝐺)‘((1st𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1st𝐴)‘𝑦))))
5756eqcoms 2618 . . . . . . . . . . . . . . . . . . . . . 22 (((iEdg‘𝐺)‘((1st𝐴)‘𝑦)) = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))} → (((iEdg‘𝐺)‘((1st𝐵)‘𝑦)) = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))} ↔ ((iEdg‘𝐺)‘((1st𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1st𝐴)‘𝑦))))
5857biimpd 218 . . . . . . . . . . . . . . . . . . . . 21 (((iEdg‘𝐺)‘((1st𝐴)‘𝑦)) = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))} → (((iEdg‘𝐺)‘((1st𝐵)‘𝑦)) = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))} → ((iEdg‘𝐺)‘((1st𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1st𝐴)‘𝑦))))
5955, 58syl6bi 242 . . . . . . . . . . . . . . . . . . . 20 ({((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))} = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))} → (((iEdg‘𝐺)‘((1st𝐴)‘𝑦)) = {((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))} → (((iEdg‘𝐺)‘((1st𝐵)‘𝑦)) = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))} → ((iEdg‘𝐺)‘((1st𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1st𝐴)‘𝑦)))))
6059com13 86 . . . . . . . . . . . . . . . . . . 19 (((iEdg‘𝐺)‘((1st𝐵)‘𝑦)) = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))} → (((iEdg‘𝐺)‘((1st𝐴)‘𝑦)) = {((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))} → ({((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))} = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))} → ((iEdg‘𝐺)‘((1st𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1st𝐴)‘𝑦)))))
6160imp 444 . . . . . . . . . . . . . . . . . 18 ((((iEdg‘𝐺)‘((1st𝐵)‘𝑦)) = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))} ∧ ((iEdg‘𝐺)‘((1st𝐴)‘𝑦)) = {((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))}) → ({((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))} = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))} → ((iEdg‘𝐺)‘((1st𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1st𝐴)‘𝑦))))
6261ral2imi 2931 . . . . . . . . . . . . . . . . 17 (∀𝑦 ∈ (0..^𝑁)(((iEdg‘𝐺)‘((1st𝐵)‘𝑦)) = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))} ∧ ((iEdg‘𝐺)‘((1st𝐴)‘𝑦)) = {((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))}) → (∀𝑦 ∈ (0..^𝑁){((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))} = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))} → ∀𝑦 ∈ (0..^𝑁)((iEdg‘𝐺)‘((1st𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1st𝐴)‘𝑦))))
6354, 62sylbir 224 . . . . . . . . . . . . . . . 16 ((∀𝑦 ∈ (0..^𝑁)((iEdg‘𝐺)‘((1st𝐵)‘𝑦)) = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))} ∧ ∀𝑦 ∈ (0..^𝑁)((iEdg‘𝐺)‘((1st𝐴)‘𝑦)) = {((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))}) → (∀𝑦 ∈ (0..^𝑁){((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))} = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))} → ∀𝑦 ∈ (0..^𝑁)((iEdg‘𝐺)‘((1st𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1st𝐴)‘𝑦))))
6453, 63syl6bi 242 . . . . . . . . . . . . . . 15 ((𝑁 = (#‘(1st𝐴)) ∧ 𝑁 = (#‘(1st𝐵))) → ((∀𝑦 ∈ (0..^(#‘(1st𝐵)))((iEdg‘𝐺)‘((1st𝐵)‘𝑦)) = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))} ∧ ∀𝑦 ∈ (0..^(#‘(1st𝐴)))((iEdg‘𝐺)‘((1st𝐴)‘𝑦)) = {((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))}) → (∀𝑦 ∈ (0..^𝑁){((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))} = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))} → ∀𝑦 ∈ (0..^𝑁)((iEdg‘𝐺)‘((1st𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1st𝐴)‘𝑦)))))
6564com12 32 . . . . . . . . . . . . . 14 ((∀𝑦 ∈ (0..^(#‘(1st𝐵)))((iEdg‘𝐺)‘((1st𝐵)‘𝑦)) = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))} ∧ ∀𝑦 ∈ (0..^(#‘(1st𝐴)))((iEdg‘𝐺)‘((1st𝐴)‘𝑦)) = {((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))}) → ((𝑁 = (#‘(1st𝐴)) ∧ 𝑁 = (#‘(1st𝐵))) → (∀𝑦 ∈ (0..^𝑁){((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))} = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))} → ∀𝑦 ∈ (0..^𝑁)((iEdg‘𝐺)‘((1st𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1st𝐴)‘𝑦)))))
6665ex 449 . . . . . . . . . . . . 13 (∀𝑦 ∈ (0..^(#‘(1st𝐵)))((iEdg‘𝐺)‘((1st𝐵)‘𝑦)) = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))} → (∀𝑦 ∈ (0..^(#‘(1st𝐴)))((iEdg‘𝐺)‘((1st𝐴)‘𝑦)) = {((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))} → ((𝑁 = (#‘(1st𝐴)) ∧ 𝑁 = (#‘(1st𝐵))) → (∀𝑦 ∈ (0..^𝑁){((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))} = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))} → ∀𝑦 ∈ (0..^𝑁)((iEdg‘𝐺)‘((1st𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1st𝐴)‘𝑦))))))
67663ad2ant3 1077 . . . . . . . . . . . 12 (((1st𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝐵):(0...(#‘(1st𝐵)))⟶(Vtx‘𝐺) ∧ ∀𝑦 ∈ (0..^(#‘(1st𝐵)))((iEdg‘𝐺)‘((1st𝐵)‘𝑦)) = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))}) → (∀𝑦 ∈ (0..^(#‘(1st𝐴)))((iEdg‘𝐺)‘((1st𝐴)‘𝑦)) = {((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))} → ((𝑁 = (#‘(1st𝐴)) ∧ 𝑁 = (#‘(1st𝐵))) → (∀𝑦 ∈ (0..^𝑁){((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))} = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))} → ∀𝑦 ∈ (0..^𝑁)((iEdg‘𝐺)‘((1st𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1st𝐴)‘𝑦))))))
6867com12 32 . . . . . . . . . . 11 (∀𝑦 ∈ (0..^(#‘(1st𝐴)))((iEdg‘𝐺)‘((1st𝐴)‘𝑦)) = {((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))} → (((1st𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝐵):(0...(#‘(1st𝐵)))⟶(Vtx‘𝐺) ∧ ∀𝑦 ∈ (0..^(#‘(1st𝐵)))((iEdg‘𝐺)‘((1st𝐵)‘𝑦)) = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))}) → ((𝑁 = (#‘(1st𝐴)) ∧ 𝑁 = (#‘(1st𝐵))) → (∀𝑦 ∈ (0..^𝑁){((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))} = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))} → ∀𝑦 ∈ (0..^𝑁)((iEdg‘𝐺)‘((1st𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1st𝐴)‘𝑦))))))
69683ad2ant3 1077 . . . . . . . . . 10 (((1st𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝐴):(0...(#‘(1st𝐴)))⟶(Vtx‘𝐺) ∧ ∀𝑦 ∈ (0..^(#‘(1st𝐴)))((iEdg‘𝐺)‘((1st𝐴)‘𝑦)) = {((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))}) → (((1st𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝐵):(0...(#‘(1st𝐵)))⟶(Vtx‘𝐺) ∧ ∀𝑦 ∈ (0..^(#‘(1st𝐵)))((iEdg‘𝐺)‘((1st𝐵)‘𝑦)) = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))}) → ((𝑁 = (#‘(1st𝐴)) ∧ 𝑁 = (#‘(1st𝐵))) → (∀𝑦 ∈ (0..^𝑁){((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))} = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))} → ∀𝑦 ∈ (0..^𝑁)((iEdg‘𝐺)‘((1st𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1st𝐴)‘𝑦))))))
7069imp 444 . . . . . . . . 9 ((((1st𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝐴):(0...(#‘(1st𝐴)))⟶(Vtx‘𝐺) ∧ ∀𝑦 ∈ (0..^(#‘(1st𝐴)))((iEdg‘𝐺)‘((1st𝐴)‘𝑦)) = {((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))}) ∧ ((1st𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝐵):(0...(#‘(1st𝐵)))⟶(Vtx‘𝐺) ∧ ∀𝑦 ∈ (0..^(#‘(1st𝐵)))((iEdg‘𝐺)‘((1st𝐵)‘𝑦)) = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))})) → ((𝑁 = (#‘(1st𝐴)) ∧ 𝑁 = (#‘(1st𝐵))) → (∀𝑦 ∈ (0..^𝑁){((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))} = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))} → ∀𝑦 ∈ (0..^𝑁)((iEdg‘𝐺)‘((1st𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1st𝐴)‘𝑦)))))
7170expd 451 . . . . . . . 8 ((((1st𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝐴):(0...(#‘(1st𝐴)))⟶(Vtx‘𝐺) ∧ ∀𝑦 ∈ (0..^(#‘(1st𝐴)))((iEdg‘𝐺)‘((1st𝐴)‘𝑦)) = {((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))}) ∧ ((1st𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝐵):(0...(#‘(1st𝐵)))⟶(Vtx‘𝐺) ∧ ∀𝑦 ∈ (0..^(#‘(1st𝐵)))((iEdg‘𝐺)‘((1st𝐵)‘𝑦)) = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))})) → (𝑁 = (#‘(1st𝐴)) → (𝑁 = (#‘(1st𝐵)) → (∀𝑦 ∈ (0..^𝑁){((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))} = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))} → ∀𝑦 ∈ (0..^𝑁)((iEdg‘𝐺)‘((1st𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1st𝐴)‘𝑦))))))
7271a1i 11 . . . . . . 7 (𝐺 ∈ USPGraph → ((((1st𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝐴):(0...(#‘(1st𝐴)))⟶(Vtx‘𝐺) ∧ ∀𝑦 ∈ (0..^(#‘(1st𝐴)))((iEdg‘𝐺)‘((1st𝐴)‘𝑦)) = {((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))}) ∧ ((1st𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝐵):(0...(#‘(1st𝐵)))⟶(Vtx‘𝐺) ∧ ∀𝑦 ∈ (0..^(#‘(1st𝐵)))((iEdg‘𝐺)‘((1st𝐵)‘𝑦)) = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))})) → (𝑁 = (#‘(1st𝐴)) → (𝑁 = (#‘(1st𝐵)) → (∀𝑦 ∈ (0..^𝑁){((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))} = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))} → ∀𝑦 ∈ (0..^𝑁)((iEdg‘𝐺)‘((1st𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1st𝐴)‘𝑦)))))))
7341, 46, 72syl2and 499 . . . . . 6 (𝐺 ∈ USPGraph → ((𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘𝐺)) → (𝑁 = (#‘(1st𝐴)) → (𝑁 = (#‘(1st𝐵)) → (∀𝑦 ∈ (0..^𝑁){((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))} = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))} → ∀𝑦 ∈ (0..^𝑁)((iEdg‘𝐺)‘((1st𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1st𝐴)‘𝑦)))))))
74733imp1 1272 . . . . 5 (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘𝐺)) ∧ 𝑁 = (#‘(1st𝐴))) ∧ 𝑁 = (#‘(1st𝐵))) → (∀𝑦 ∈ (0..^𝑁){((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))} = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))} → ∀𝑦 ∈ (0..^𝑁)((iEdg‘𝐺)‘((1st𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1st𝐴)‘𝑦))))
75 eqcom 2617 . . . . . . 7 (((iEdg‘𝐺)‘((1st𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1st𝐴)‘𝑦)) ↔ ((iEdg‘𝐺)‘((1st𝐴)‘𝑦)) = ((iEdg‘𝐺)‘((1st𝐵)‘𝑦)))
7636uspgrf1oedg 40403 . . . . . . . . . . . 12 (𝐺 ∈ USPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1-onto→(Edg‘𝐺))
77 f1of1 6049 . . . . . . . . . . . 12 ((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1-onto→(Edg‘𝐺) → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(Edg‘𝐺))
7876, 77syl 17 . . . . . . . . . . 11 (𝐺 ∈ USPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(Edg‘𝐺))
79 eqidd 2611 . . . . . . . . . . . 12 (𝐺 ∈ USPGraph → (iEdg‘𝐺) = (iEdg‘𝐺))
80 eqidd 2611 . . . . . . . . . . . 12 (𝐺 ∈ USPGraph → dom (iEdg‘𝐺) = dom (iEdg‘𝐺))
81 edgaval 25794 . . . . . . . . . . . . 13 (𝐺 ∈ USPGraph → (Edg‘𝐺) = ran (iEdg‘𝐺))
8281eqcomd 2616 . . . . . . . . . . . 12 (𝐺 ∈ USPGraph → ran (iEdg‘𝐺) = (Edg‘𝐺))
8379, 80, 82f1eq123d 6044 . . . . . . . . . . 11 (𝐺 ∈ USPGraph → ((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→ran (iEdg‘𝐺) ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(Edg‘𝐺)))
8478, 83mpbird 246 . . . . . . . . . 10 (𝐺 ∈ USPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→ran (iEdg‘𝐺))
85843ad2ant1 1075 . . . . . . . . 9 ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘𝐺)) ∧ 𝑁 = (#‘(1st𝐴))) → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→ran (iEdg‘𝐺))
8685adantr 480 . . . . . . . 8 (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘𝐺)) ∧ 𝑁 = (#‘(1st𝐴))) ∧ 𝑁 = (#‘(1st𝐵))) → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→ran (iEdg‘𝐺))
8735, 36, 37, 381wlkelwrd 40837 . . . . . . . . . . . . . . 15 (𝐴 ∈ (1Walks‘𝐺) → ((1st𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝐴):(0...(#‘(1st𝐴)))⟶(Vtx‘𝐺)))
8835, 36, 42, 431wlkelwrd 40837 . . . . . . . . . . . . . . 15 (𝐵 ∈ (1Walks‘𝐺) → ((1st𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝐵):(0...(#‘(1st𝐵)))⟶(Vtx‘𝐺)))
89 oveq2 6557 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑁 = (#‘(1st𝐴)) → (0..^𝑁) = (0..^(#‘(1st𝐴))))
9089eleq2d 2673 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑁 = (#‘(1st𝐴)) → (𝑦 ∈ (0..^𝑁) ↔ 𝑦 ∈ (0..^(#‘(1st𝐴)))))
91 wrdsymbcl 13173 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((1st𝐴) ∈ Word dom (iEdg‘𝐺) ∧ 𝑦 ∈ (0..^(#‘(1st𝐴)))) → ((1st𝐴)‘𝑦) ∈ dom (iEdg‘𝐺))
9291expcom 450 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 ∈ (0..^(#‘(1st𝐴))) → ((1st𝐴) ∈ Word dom (iEdg‘𝐺) → ((1st𝐴)‘𝑦) ∈ dom (iEdg‘𝐺)))
9390, 92syl6bi 242 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑁 = (#‘(1st𝐴)) → (𝑦 ∈ (0..^𝑁) → ((1st𝐴) ∈ Word dom (iEdg‘𝐺) → ((1st𝐴)‘𝑦) ∈ dom (iEdg‘𝐺))))
9493adantr 480 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑁 = (#‘(1st𝐴)) ∧ 𝑁 = (#‘(1st𝐵))) → (𝑦 ∈ (0..^𝑁) → ((1st𝐴) ∈ Word dom (iEdg‘𝐺) → ((1st𝐴)‘𝑦) ∈ dom (iEdg‘𝐺))))
9594imp 444 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑁 = (#‘(1st𝐴)) ∧ 𝑁 = (#‘(1st𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → ((1st𝐴) ∈ Word dom (iEdg‘𝐺) → ((1st𝐴)‘𝑦) ∈ dom (iEdg‘𝐺)))
9695com12 32 . . . . . . . . . . . . . . . . . . . . . 22 ((1st𝐴) ∈ Word dom (iEdg‘𝐺) → (((𝑁 = (#‘(1st𝐴)) ∧ 𝑁 = (#‘(1st𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → ((1st𝐴)‘𝑦) ∈ dom (iEdg‘𝐺)))
9796adantl 481 . . . . . . . . . . . . . . . . . . . . 21 (((1st𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (1st𝐴) ∈ Word dom (iEdg‘𝐺)) → (((𝑁 = (#‘(1st𝐴)) ∧ 𝑁 = (#‘(1st𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → ((1st𝐴)‘𝑦) ∈ dom (iEdg‘𝐺)))
98 oveq2 6557 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑁 = (#‘(1st𝐵)) → (0..^𝑁) = (0..^(#‘(1st𝐵))))
9998eleq2d 2673 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑁 = (#‘(1st𝐵)) → (𝑦 ∈ (0..^𝑁) ↔ 𝑦 ∈ (0..^(#‘(1st𝐵)))))
100 wrdsymbcl 13173 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((1st𝐵) ∈ Word dom (iEdg‘𝐺) ∧ 𝑦 ∈ (0..^(#‘(1st𝐵)))) → ((1st𝐵)‘𝑦) ∈ dom (iEdg‘𝐺))
101100expcom 450 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 ∈ (0..^(#‘(1st𝐵))) → ((1st𝐵) ∈ Word dom (iEdg‘𝐺) → ((1st𝐵)‘𝑦) ∈ dom (iEdg‘𝐺)))
10299, 101syl6bi 242 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑁 = (#‘(1st𝐵)) → (𝑦 ∈ (0..^𝑁) → ((1st𝐵) ∈ Word dom (iEdg‘𝐺) → ((1st𝐵)‘𝑦) ∈ dom (iEdg‘𝐺))))
103102adantl 481 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑁 = (#‘(1st𝐴)) ∧ 𝑁 = (#‘(1st𝐵))) → (𝑦 ∈ (0..^𝑁) → ((1st𝐵) ∈ Word dom (iEdg‘𝐺) → ((1st𝐵)‘𝑦) ∈ dom (iEdg‘𝐺))))
104103imp 444 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑁 = (#‘(1st𝐴)) ∧ 𝑁 = (#‘(1st𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → ((1st𝐵) ∈ Word dom (iEdg‘𝐺) → ((1st𝐵)‘𝑦) ∈ dom (iEdg‘𝐺)))
105104com12 32 . . . . . . . . . . . . . . . . . . . . . 22 ((1st𝐵) ∈ Word dom (iEdg‘𝐺) → (((𝑁 = (#‘(1st𝐴)) ∧ 𝑁 = (#‘(1st𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → ((1st𝐵)‘𝑦) ∈ dom (iEdg‘𝐺)))
106105adantr 480 . . . . . . . . . . . . . . . . . . . . 21 (((1st𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (1st𝐴) ∈ Word dom (iEdg‘𝐺)) → (((𝑁 = (#‘(1st𝐴)) ∧ 𝑁 = (#‘(1st𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → ((1st𝐵)‘𝑦) ∈ dom (iEdg‘𝐺)))
10797, 106jcad 554 . . . . . . . . . . . . . . . . . . . 20 (((1st𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (1st𝐴) ∈ Word dom (iEdg‘𝐺)) → (((𝑁 = (#‘(1st𝐴)) ∧ 𝑁 = (#‘(1st𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → (((1st𝐴)‘𝑦) ∈ dom (iEdg‘𝐺) ∧ ((1st𝐵)‘𝑦) ∈ dom (iEdg‘𝐺))))
108107ex 449 . . . . . . . . . . . . . . . . . . 19 ((1st𝐵) ∈ Word dom (iEdg‘𝐺) → ((1st𝐴) ∈ Word dom (iEdg‘𝐺) → (((𝑁 = (#‘(1st𝐴)) ∧ 𝑁 = (#‘(1st𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → (((1st𝐴)‘𝑦) ∈ dom (iEdg‘𝐺) ∧ ((1st𝐵)‘𝑦) ∈ dom (iEdg‘𝐺)))))
109108adantr 480 . . . . . . . . . . . . . . . . . 18 (((1st𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝐵):(0...(#‘(1st𝐵)))⟶(Vtx‘𝐺)) → ((1st𝐴) ∈ Word dom (iEdg‘𝐺) → (((𝑁 = (#‘(1st𝐴)) ∧ 𝑁 = (#‘(1st𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → (((1st𝐴)‘𝑦) ∈ dom (iEdg‘𝐺) ∧ ((1st𝐵)‘𝑦) ∈ dom (iEdg‘𝐺)))))
110109com12 32 . . . . . . . . . . . . . . . . 17 ((1st𝐴) ∈ Word dom (iEdg‘𝐺) → (((1st𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝐵):(0...(#‘(1st𝐵)))⟶(Vtx‘𝐺)) → (((𝑁 = (#‘(1st𝐴)) ∧ 𝑁 = (#‘(1st𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → (((1st𝐴)‘𝑦) ∈ dom (iEdg‘𝐺) ∧ ((1st𝐵)‘𝑦) ∈ dom (iEdg‘𝐺)))))
111110adantr 480 . . . . . . . . . . . . . . . 16 (((1st𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝐴):(0...(#‘(1st𝐴)))⟶(Vtx‘𝐺)) → (((1st𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝐵):(0...(#‘(1st𝐵)))⟶(Vtx‘𝐺)) → (((𝑁 = (#‘(1st𝐴)) ∧ 𝑁 = (#‘(1st𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → (((1st𝐴)‘𝑦) ∈ dom (iEdg‘𝐺) ∧ ((1st𝐵)‘𝑦) ∈ dom (iEdg‘𝐺)))))
112111imp 444 . . . . . . . . . . . . . . 15 ((((1st𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝐴):(0...(#‘(1st𝐴)))⟶(Vtx‘𝐺)) ∧ ((1st𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝐵):(0...(#‘(1st𝐵)))⟶(Vtx‘𝐺))) → (((𝑁 = (#‘(1st𝐴)) ∧ 𝑁 = (#‘(1st𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → (((1st𝐴)‘𝑦) ∈ dom (iEdg‘𝐺) ∧ ((1st𝐵)‘𝑦) ∈ dom (iEdg‘𝐺))))
11387, 88, 112syl2an 493 . . . . . . . . . . . . . 14 ((𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘𝐺)) → (((𝑁 = (#‘(1st𝐴)) ∧ 𝑁 = (#‘(1st𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → (((1st𝐴)‘𝑦) ∈ dom (iEdg‘𝐺) ∧ ((1st𝐵)‘𝑦) ∈ dom (iEdg‘𝐺))))
114113expd 451 . . . . . . . . . . . . 13 ((𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘𝐺)) → ((𝑁 = (#‘(1st𝐴)) ∧ 𝑁 = (#‘(1st𝐵))) → (𝑦 ∈ (0..^𝑁) → (((1st𝐴)‘𝑦) ∈ dom (iEdg‘𝐺) ∧ ((1st𝐵)‘𝑦) ∈ dom (iEdg‘𝐺)))))
115114expd 451 . . . . . . . . . . . 12 ((𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘𝐺)) → (𝑁 = (#‘(1st𝐴)) → (𝑁 = (#‘(1st𝐵)) → (𝑦 ∈ (0..^𝑁) → (((1st𝐴)‘𝑦) ∈ dom (iEdg‘𝐺) ∧ ((1st𝐵)‘𝑦) ∈ dom (iEdg‘𝐺))))))
116115imp 444 . . . . . . . . . . 11 (((𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘𝐺)) ∧ 𝑁 = (#‘(1st𝐴))) → (𝑁 = (#‘(1st𝐵)) → (𝑦 ∈ (0..^𝑁) → (((1st𝐴)‘𝑦) ∈ dom (iEdg‘𝐺) ∧ ((1st𝐵)‘𝑦) ∈ dom (iEdg‘𝐺)))))
1171163adant1 1072 . . . . . . . . . 10 ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘𝐺)) ∧ 𝑁 = (#‘(1st𝐴))) → (𝑁 = (#‘(1st𝐵)) → (𝑦 ∈ (0..^𝑁) → (((1st𝐴)‘𝑦) ∈ dom (iEdg‘𝐺) ∧ ((1st𝐵)‘𝑦) ∈ dom (iEdg‘𝐺)))))
118117imp 444 . . . . . . . . 9 (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘𝐺)) ∧ 𝑁 = (#‘(1st𝐴))) ∧ 𝑁 = (#‘(1st𝐵))) → (𝑦 ∈ (0..^𝑁) → (((1st𝐴)‘𝑦) ∈ dom (iEdg‘𝐺) ∧ ((1st𝐵)‘𝑦) ∈ dom (iEdg‘𝐺))))
119118imp 444 . . . . . . . 8 ((((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘𝐺)) ∧ 𝑁 = (#‘(1st𝐴))) ∧ 𝑁 = (#‘(1st𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → (((1st𝐴)‘𝑦) ∈ dom (iEdg‘𝐺) ∧ ((1st𝐵)‘𝑦) ∈ dom (iEdg‘𝐺)))
120 f1veqaeq 6418 . . . . . . . 8 (((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→ran (iEdg‘𝐺) ∧ (((1st𝐴)‘𝑦) ∈ dom (iEdg‘𝐺) ∧ ((1st𝐵)‘𝑦) ∈ dom (iEdg‘𝐺))) → (((iEdg‘𝐺)‘((1st𝐴)‘𝑦)) = ((iEdg‘𝐺)‘((1st𝐵)‘𝑦)) → ((1st𝐴)‘𝑦) = ((1st𝐵)‘𝑦)))
12186, 119, 120syl2an2r 872 . . . . . . 7 ((((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘𝐺)) ∧ 𝑁 = (#‘(1st𝐴))) ∧ 𝑁 = (#‘(1st𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → (((iEdg‘𝐺)‘((1st𝐴)‘𝑦)) = ((iEdg‘𝐺)‘((1st𝐵)‘𝑦)) → ((1st𝐴)‘𝑦) = ((1st𝐵)‘𝑦)))
12275, 121syl5bi 231 . . . . . 6 ((((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘𝐺)) ∧ 𝑁 = (#‘(1st𝐴))) ∧ 𝑁 = (#‘(1st𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → (((iEdg‘𝐺)‘((1st𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1st𝐴)‘𝑦)) → ((1st𝐴)‘𝑦) = ((1st𝐵)‘𝑦)))
123122ralimdva 2945 . . . . 5 (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘𝐺)) ∧ 𝑁 = (#‘(1st𝐴))) ∧ 𝑁 = (#‘(1st𝐵))) → (∀𝑦 ∈ (0..^𝑁)((iEdg‘𝐺)‘((1st𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1st𝐴)‘𝑦)) → ∀𝑦 ∈ (0..^𝑁)((1st𝐴)‘𝑦) = ((1st𝐵)‘𝑦)))
12433, 74, 1233syld 58 . . . 4 (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘𝐺)) ∧ 𝑁 = (#‘(1st𝐴))) ∧ 𝑁 = (#‘(1st𝐵))) → (∀𝑦 ∈ (0...𝑁)((2nd𝐴)‘𝑦) = ((2nd𝐵)‘𝑦) → ∀𝑦 ∈ (0..^𝑁)((1st𝐴)‘𝑦) = ((1st𝐵)‘𝑦)))
125124expimpd 627 . . 3 ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘𝐺)) ∧ 𝑁 = (#‘(1st𝐴))) → ((𝑁 = (#‘(1st𝐵)) ∧ ∀𝑦 ∈ (0...𝑁)((2nd𝐴)‘𝑦) = ((2nd𝐵)‘𝑦)) → ∀𝑦 ∈ (0..^𝑁)((1st𝐴)‘𝑦) = ((1st𝐵)‘𝑦)))
126125pm4.71d 664 . 2 ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘𝐺)) ∧ 𝑁 = (#‘(1st𝐴))) → ((𝑁 = (#‘(1st𝐵)) ∧ ∀𝑦 ∈ (0...𝑁)((2nd𝐴)‘𝑦) = ((2nd𝐵)‘𝑦)) ↔ ((𝑁 = (#‘(1st𝐵)) ∧ ∀𝑦 ∈ (0...𝑁)((2nd𝐴)‘𝑦) = ((2nd𝐵)‘𝑦)) ∧ ∀𝑦 ∈ (0..^𝑁)((1st𝐴)‘𝑦) = ((1st𝐵)‘𝑦))))
1272, 5, 1263bitr4d 299 1 ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘𝐺)) ∧ 𝑁 = (#‘(1st𝐴))) → (𝐴 = 𝐵 ↔ (𝑁 = (#‘(1st𝐵)) ∧ ∀𝑦 ∈ (0...𝑁)((2nd𝐴)‘𝑦) = ((2nd𝐵)‘𝑦))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896   ⊆ wss 3540  {cpr 4127  dom cdm 5038  ran crn 5039  ⟶wf 5800  –1-1→wf1 5801  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549  1st c1st 7057  2nd c2nd 7058  0cc0 9815  1c1 9816   + caddc 9818  ...cfz 12197  ..^cfzo 12334  #chash 12979  Word cword 13146  Vtxcvtx 25673  iEdgciedg 25674   UPGraph cupgr 25747  Edgcedga 25792   USPGraph cuspgr 40378  1Walksc1wlks 40796 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-ifp 1007  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-uhgr 25724  df-upgr 25749  df-edga 25793  df-uspgr 40380  df-1wlks 40800  df-wlks 40801 This theorem is referenced by:  uspgr2wlkeq2  40855  clwlksf1clwwlk  41276
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