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Theorem uspgr2wlkeq 27343
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 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘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 1091 . . 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 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘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 wlkeq 27331 . . . 4 ((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺) ∧ 𝑁 = (♯‘(1st𝐴))) → (𝐴 = 𝐵 ↔ (𝑁 = (♯‘(1st𝐵)) ∧ ∀𝑦 ∈ (0..^𝑁)((1st𝐴)‘𝑦) = ((1st𝐵)‘𝑦) ∧ ∀𝑦 ∈ (0...𝑁)((2nd𝐴)‘𝑦) = ((2nd𝐵)‘𝑦))))
433expa 1112 . . 3 (((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1st𝐴))) → (𝐴 = 𝐵 ↔ (𝑁 = (♯‘(1st𝐵)) ∧ ∀𝑦 ∈ (0..^𝑁)((1st𝐴)‘𝑦) = ((1st𝐵)‘𝑦) ∧ ∀𝑦 ∈ (0...𝑁)((2nd𝐴)‘𝑦) = ((2nd𝐵)‘𝑦))))
543adant1 1124 . 2 ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1st𝐴))) → (𝐴 = 𝐵 ↔ (𝑁 = (♯‘(1st𝐵)) ∧ ∀𝑦 ∈ (0..^𝑁)((1st𝐴)‘𝑦) = ((1st𝐵)‘𝑦) ∧ ∀𝑦 ∈ (0...𝑁)((2nd𝐴)‘𝑦) = ((2nd𝐵)‘𝑦))))
6 fzofzp1 13127 . . . . . . . . . . . 12 (𝑥 ∈ (0..^𝑁) → (𝑥 + 1) ∈ (0...𝑁))
76adantl 482 . . . . . . . . . . 11 ((((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1st𝐴))) ∧ 𝑁 = (♯‘(1st𝐵))) ∧ 𝑥 ∈ (0..^𝑁)) → (𝑥 + 1) ∈ (0...𝑁))
8 fveq2 6666 . . . . . . . . . . . . 13 (𝑦 = (𝑥 + 1) → ((2nd𝐴)‘𝑦) = ((2nd𝐴)‘(𝑥 + 1)))
9 fveq2 6666 . . . . . . . . . . . . 13 (𝑦 = (𝑥 + 1) → ((2nd𝐵)‘𝑦) = ((2nd𝐵)‘(𝑥 + 1)))
108, 9eqeq12d 2841 . . . . . . . . . . . 12 (𝑦 = (𝑥 + 1) → (((2nd𝐴)‘𝑦) = ((2nd𝐵)‘𝑦) ↔ ((2nd𝐴)‘(𝑥 + 1)) = ((2nd𝐵)‘(𝑥 + 1))))
1110adantl 482 . . . . . . . . . . 11 (((((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1st𝐴))) ∧ 𝑁 = (♯‘(1st𝐵))) ∧ 𝑥 ∈ (0..^𝑁)) ∧ 𝑦 = (𝑥 + 1)) → (((2nd𝐴)‘𝑦) = ((2nd𝐵)‘𝑦) ↔ ((2nd𝐴)‘(𝑥 + 1)) = ((2nd𝐵)‘(𝑥 + 1))))
127, 11rspcdv 3618 . . . . . . . . . 10 ((((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1st𝐴))) ∧ 𝑁 = (♯‘(1st𝐵))) ∧ 𝑥 ∈ (0..^𝑁)) → (∀𝑦 ∈ (0...𝑁)((2nd𝐴)‘𝑦) = ((2nd𝐵)‘𝑦) → ((2nd𝐴)‘(𝑥 + 1)) = ((2nd𝐵)‘(𝑥 + 1))))
1312impancom 452 . . . . . . . . 9 ((((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1st𝐴))) ∧ 𝑁 = (♯‘(1st𝐵))) ∧ ∀𝑦 ∈ (0...𝑁)((2nd𝐴)‘𝑦) = ((2nd𝐵)‘𝑦)) → (𝑥 ∈ (0..^𝑁) → ((2nd𝐴)‘(𝑥 + 1)) = ((2nd𝐵)‘(𝑥 + 1))))
1413ralrimiv 3185 . . . . . . . 8 ((((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1st𝐴))) ∧ 𝑁 = (♯‘(1st𝐵))) ∧ ∀𝑦 ∈ (0...𝑁)((2nd𝐴)‘𝑦) = ((2nd𝐵)‘𝑦)) → ∀𝑥 ∈ (0..^𝑁)((2nd𝐴)‘(𝑥 + 1)) = ((2nd𝐵)‘(𝑥 + 1)))
15 fvoveq1 7174 . . . . . . . . . 10 (𝑦 = 𝑥 → ((2nd𝐴)‘(𝑦 + 1)) = ((2nd𝐴)‘(𝑥 + 1)))
16 fvoveq1 7174 . . . . . . . . . 10 (𝑦 = 𝑥 → ((2nd𝐵)‘(𝑦 + 1)) = ((2nd𝐵)‘(𝑥 + 1)))
1715, 16eqeq12d 2841 . . . . . . . . 9 (𝑦 = 𝑥 → (((2nd𝐴)‘(𝑦 + 1)) = ((2nd𝐵)‘(𝑦 + 1)) ↔ ((2nd𝐴)‘(𝑥 + 1)) = ((2nd𝐵)‘(𝑥 + 1))))
1817cbvralv 3457 . . . . . . . 8 (∀𝑦 ∈ (0..^𝑁)((2nd𝐴)‘(𝑦 + 1)) = ((2nd𝐵)‘(𝑦 + 1)) ↔ ∀𝑥 ∈ (0..^𝑁)((2nd𝐴)‘(𝑥 + 1)) = ((2nd𝐵)‘(𝑥 + 1)))
1914, 18sylibr 235 . . . . . . 7 ((((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1st𝐴))) ∧ 𝑁 = (♯‘(1st𝐵))) ∧ ∀𝑦 ∈ (0...𝑁)((2nd𝐴)‘𝑦) = ((2nd𝐵)‘𝑦)) → ∀𝑦 ∈ (0..^𝑁)((2nd𝐴)‘(𝑦 + 1)) = ((2nd𝐵)‘(𝑦 + 1)))
20 fzossfz 13049 . . . . . . . . . 10 (0..^𝑁) ⊆ (0...𝑁)
21 ssralv 4036 . . . . . . . . . 10 ((0..^𝑁) ⊆ (0...𝑁) → (∀𝑦 ∈ (0...𝑁)((2nd𝐴)‘𝑦) = ((2nd𝐵)‘𝑦) → ∀𝑦 ∈ (0..^𝑁)((2nd𝐴)‘𝑦) = ((2nd𝐵)‘𝑦)))
2220, 21mp1i 13 . . . . . . . . 9 (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1st𝐴))) ∧ 𝑁 = (♯‘(1st𝐵))) → (∀𝑦 ∈ (0...𝑁)((2nd𝐴)‘𝑦) = ((2nd𝐵)‘𝑦) → ∀𝑦 ∈ (0..^𝑁)((2nd𝐴)‘𝑦) = ((2nd𝐵)‘𝑦)))
23 r19.26 3174 . . . . . . . . . . 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))))
24 preq12 4669 . . . . . . . . . . . . 13 ((((2nd𝐴)‘𝑦) = ((2nd𝐵)‘𝑦) ∧ ((2nd𝐴)‘(𝑦 + 1)) = ((2nd𝐵)‘(𝑦 + 1))) → {((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))} = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))})
2524a1i 11 . . . . . . . . . . . 12 (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘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))}))
2625ralimdv 3182 . . . . . . . . . . 11 (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘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))}))
2723, 26syl5bir 244 . . . . . . . . . 10 (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘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))}))
2827expd 416 . . . . . . . . 9 (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘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))})))
2922, 28syld 47 . . . . . . . 8 (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘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))})))
3029imp 407 . . . . . . 7 ((((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘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))}))
3119, 30mpd 15 . . . . . 6 ((((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘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))})
3231ex 413 . . . . 5 (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘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))}))
33 uspgrupgr 26877 . . . . . . . 8 (𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph)
34 eqid 2825 . . . . . . . . . 10 (Vtx‘𝐺) = (Vtx‘𝐺)
35 eqid 2825 . . . . . . . . . 10 (iEdg‘𝐺) = (iEdg‘𝐺)
36 eqid 2825 . . . . . . . . . 10 (1st𝐴) = (1st𝐴)
37 eqid 2825 . . . . . . . . . 10 (2nd𝐴) = (2nd𝐴)
3834, 35, 36, 37upgrwlkcompim 27340 . . . . . . . . 9 ((𝐺 ∈ UPGraph ∧ 𝐴 ∈ (Walks‘𝐺)) → ((1st𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝐴):(0...(♯‘(1st𝐴)))⟶(Vtx‘𝐺) ∧ ∀𝑦 ∈ (0..^(♯‘(1st𝐴)))((iEdg‘𝐺)‘((1st𝐴)‘𝑦)) = {((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))}))
3938ex 413 . . . . . . . 8 (𝐺 ∈ UPGraph → (𝐴 ∈ (Walks‘𝐺) → ((1st𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝐴):(0...(♯‘(1st𝐴)))⟶(Vtx‘𝐺) ∧ ∀𝑦 ∈ (0..^(♯‘(1st𝐴)))((iEdg‘𝐺)‘((1st𝐴)‘𝑦)) = {((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))})))
4033, 39syl 17 . . . . . . 7 (𝐺 ∈ USPGraph → (𝐴 ∈ (Walks‘𝐺) → ((1st𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝐴):(0...(♯‘(1st𝐴)))⟶(Vtx‘𝐺) ∧ ∀𝑦 ∈ (0..^(♯‘(1st𝐴)))((iEdg‘𝐺)‘((1st𝐴)‘𝑦)) = {((2nd𝐴)‘𝑦), ((2nd𝐴)‘(𝑦 + 1))})))
41 eqid 2825 . . . . . . . . . 10 (1st𝐵) = (1st𝐵)
42 eqid 2825 . . . . . . . . . 10 (2nd𝐵) = (2nd𝐵)
4334, 35, 41, 42upgrwlkcompim 27340 . . . . . . . . 9 ((𝐺 ∈ UPGraph ∧ 𝐵 ∈ (Walks‘𝐺)) → ((1st𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝐵):(0...(♯‘(1st𝐵)))⟶(Vtx‘𝐺) ∧ ∀𝑦 ∈ (0..^(♯‘(1st𝐵)))((iEdg‘𝐺)‘((1st𝐵)‘𝑦)) = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))}))
4443ex 413 . . . . . . . 8 (𝐺 ∈ UPGraph → (𝐵 ∈ (Walks‘𝐺) → ((1st𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝐵):(0...(♯‘(1st𝐵)))⟶(Vtx‘𝐺) ∧ ∀𝑦 ∈ (0..^(♯‘(1st𝐵)))((iEdg‘𝐺)‘((1st𝐵)‘𝑦)) = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))})))
4533, 44syl 17 . . . . . . 7 (𝐺 ∈ USPGraph → (𝐵 ∈ (Walks‘𝐺) → ((1st𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝐵):(0...(♯‘(1st𝐵)))⟶(Vtx‘𝐺) ∧ ∀𝑦 ∈ (0..^(♯‘(1st𝐵)))((iEdg‘𝐺)‘((1st𝐵)‘𝑦)) = {((2nd𝐵)‘𝑦), ((2nd𝐵)‘(𝑦 + 1))})))
46 oveq2 7159 . . . . . . . . . . . . . . . . . . 19 ((♯‘(1st𝐵)) = 𝑁 → (0..^(♯‘(1st𝐵))) = (0..^𝑁))
4746eqcoms 2833 . . . . . . . . . . . . . . . . . 18 (𝑁 = (♯‘(1st𝐵)) → (0..^(♯‘(1st𝐵))) = (0..^𝑁))
4847raleqdv 3420 . . . . . . . . . . . . . . . . 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))}))
49 oveq2 7159 . . . . . . . . . . . . . . . . . . 19 ((♯‘(1st𝐴)) = 𝑁 → (0..^(♯‘(1st𝐴))) = (0..^𝑁))
5049eqcoms 2833 . . . . . . . . . . . . . . . . . 18 (𝑁 = (♯‘(1st𝐴)) → (0..^(♯‘(1st𝐴))) = (0..^𝑁))
5150raleqdv 3420 . . . . . . . . . . . . . . . . 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))}))
5248, 51bi2anan9r 636 . . . . . . . . . . . . . . . 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))})))
53 r19.26 3174 . . . . . . . . . . . . . . . . 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))}))
54 eqeq2 2837 . . . . . . . . . . . . . . . . . . . . 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))}))
55 eqeq2 2837 . . . . . . . . . . . . . . . . . . . . . . 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))))
5655eqcoms 2833 . . . . . . . . . . . . . . . . . . . . . 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))))
5756biimpd 230 . . . . . . . . . . . . . . . . . . . . 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))))
5854, 57syl6bi 254 . . . . . . . . . . . . . . . . . . . 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)))))
5958com13 88 . . . . . . . . . . . . . . . . . . 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)))))
6059imp 407 . . . . . . . . . . . . . . . . . 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))))
6160ral2imi 3160 . . . . . . . . . . . . . . . . 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))))
6253, 61sylbir 236 . . . . . . . . . . . . . . . 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))))
6352, 62syl6bi 254 . . . . . . . . . . . . . . 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)))))
6463com12 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)))))
6564ex 413 . . . . . . . . . . . . 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))))))
66653ad2ant3 1129 . . . . . . . . . . . 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))))))
6766com12 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))))))
68673ad2ant3 1129 . . . . . . . . . 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))))))
6968imp 407 . . . . . . . . 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)))))
7069expd 416 . . . . . . . 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))))))
7170a1i 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)))))))
7240, 45, 71syl2and 607 . . . . . 6 (𝐺 ∈ USPGraph → ((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘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)))))))
73723imp1 1341 . . . . 5 (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘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))))
74 eqcom 2832 . . . . . . 7 (((iEdg‘𝐺)‘((1st𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1st𝐴)‘𝑦)) ↔ ((iEdg‘𝐺)‘((1st𝐴)‘𝑦)) = ((iEdg‘𝐺)‘((1st𝐵)‘𝑦)))
7535uspgrf1oedg 26874 . . . . . . . . . . . 12 (𝐺 ∈ USPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1-onto→(Edg‘𝐺))
76 f1of1 6610 . . . . . . . . . . . 12 ((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1-onto→(Edg‘𝐺) → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(Edg‘𝐺))
7775, 76syl 17 . . . . . . . . . . 11 (𝐺 ∈ USPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(Edg‘𝐺))
78 eqidd 2826 . . . . . . . . . . . 12 (𝐺 ∈ USPGraph → (iEdg‘𝐺) = (iEdg‘𝐺))
79 eqidd 2826 . . . . . . . . . . . 12 (𝐺 ∈ USPGraph → dom (iEdg‘𝐺) = dom (iEdg‘𝐺))
80 edgval 26750 . . . . . . . . . . . . . 14 (Edg‘𝐺) = ran (iEdg‘𝐺)
8180eqcomi 2834 . . . . . . . . . . . . 13 ran (iEdg‘𝐺) = (Edg‘𝐺)
8281a1i 11 . . . . . . . . . . . 12 (𝐺 ∈ USPGraph → ran (iEdg‘𝐺) = (Edg‘𝐺))
8378, 79, 82f1eq123d 6604 . . . . . . . . . . 11 (𝐺 ∈ USPGraph → ((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→ran (iEdg‘𝐺) ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(Edg‘𝐺)))
8477, 83mpbird 258 . . . . . . . . . 10 (𝐺 ∈ USPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→ran (iEdg‘𝐺))
85843ad2ant1 1127 . . . . . . . . 9 ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1st𝐴))) → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→ran (iEdg‘𝐺))
8685adantr 481 . . . . . . . 8 (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1st𝐴))) ∧ 𝑁 = (♯‘(1st𝐵))) → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→ran (iEdg‘𝐺))
8734, 35, 36, 37wlkelwrd 27330 . . . . . . . . . . . . . . 15 (𝐴 ∈ (Walks‘𝐺) → ((1st𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝐴):(0...(♯‘(1st𝐴)))⟶(Vtx‘𝐺)))
8834, 35, 41, 42wlkelwrd 27330 . . . . . . . . . . . . . . 15 (𝐵 ∈ (Walks‘𝐺) → ((1st𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝐵):(0...(♯‘(1st𝐵)))⟶(Vtx‘𝐺)))
89 oveq2 7159 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑁 = (♯‘(1st𝐴)) → (0..^𝑁) = (0..^(♯‘(1st𝐴))))
9089eleq2d 2902 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑁 = (♯‘(1st𝐴)) → (𝑦 ∈ (0..^𝑁) ↔ 𝑦 ∈ (0..^(♯‘(1st𝐴)))))
91 wrdsymbcl 13868 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((1st𝐴) ∈ Word dom (iEdg‘𝐺) ∧ 𝑦 ∈ (0..^(♯‘(1st𝐴)))) → ((1st𝐴)‘𝑦) ∈ dom (iEdg‘𝐺))
9291expcom 414 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 ∈ (0..^(♯‘(1st𝐴))) → ((1st𝐴) ∈ Word dom (iEdg‘𝐺) → ((1st𝐴)‘𝑦) ∈ dom (iEdg‘𝐺)))
9390, 92syl6bi 254 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑁 = (♯‘(1st𝐴)) → (𝑦 ∈ (0..^𝑁) → ((1st𝐴) ∈ Word dom (iEdg‘𝐺) → ((1st𝐴)‘𝑦) ∈ dom (iEdg‘𝐺))))
9493adantr 481 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑁 = (♯‘(1st𝐴)) ∧ 𝑁 = (♯‘(1st𝐵))) → (𝑦 ∈ (0..^𝑁) → ((1st𝐴) ∈ Word dom (iEdg‘𝐺) → ((1st𝐴)‘𝑦) ∈ dom (iEdg‘𝐺))))
9594imp 407 . . . . . . . . . . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . . . . . . . . . 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 7159 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑁 = (♯‘(1st𝐵)) → (0..^𝑁) = (0..^(♯‘(1st𝐵))))
9998eleq2d 2902 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑁 = (♯‘(1st𝐵)) → (𝑦 ∈ (0..^𝑁) ↔ 𝑦 ∈ (0..^(♯‘(1st𝐵)))))
100 wrdsymbcl 13868 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((1st𝐵) ∈ Word dom (iEdg‘𝐺) ∧ 𝑦 ∈ (0..^(♯‘(1st𝐵)))) → ((1st𝐵)‘𝑦) ∈ dom (iEdg‘𝐺))
101100expcom 414 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 ∈ (0..^(♯‘(1st𝐵))) → ((1st𝐵) ∈ Word dom (iEdg‘𝐺) → ((1st𝐵)‘𝑦) ∈ dom (iEdg‘𝐺)))
10299, 101syl6bi 254 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑁 = (♯‘(1st𝐵)) → (𝑦 ∈ (0..^𝑁) → ((1st𝐵) ∈ Word dom (iEdg‘𝐺) → ((1st𝐵)‘𝑦) ∈ dom (iEdg‘𝐺))))
103102adantl 482 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑁 = (♯‘(1st𝐴)) ∧ 𝑁 = (♯‘(1st𝐵))) → (𝑦 ∈ (0..^𝑁) → ((1st𝐵) ∈ Word dom (iEdg‘𝐺) → ((1st𝐵)‘𝑦) ∈ dom (iEdg‘𝐺))))
104103imp 407 . . . . . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . . . . 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 513 . . . . . . . . . . . . . . . . . . . 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 413 . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . 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 407 . . . . . . . . . . . . . . 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 595 . . . . . . . . . . . . . 14 ((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) → (((𝑁 = (♯‘(1st𝐴)) ∧ 𝑁 = (♯‘(1st𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → (((1st𝐴)‘𝑦) ∈ dom (iEdg‘𝐺) ∧ ((1st𝐵)‘𝑦) ∈ dom (iEdg‘𝐺))))
114113expd 416 . . . . . . . . . . . . 13 ((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) → ((𝑁 = (♯‘(1st𝐴)) ∧ 𝑁 = (♯‘(1st𝐵))) → (𝑦 ∈ (0..^𝑁) → (((1st𝐴)‘𝑦) ∈ dom (iEdg‘𝐺) ∧ ((1st𝐵)‘𝑦) ∈ dom (iEdg‘𝐺)))))
115114expd 416 . . . . . . . . . . . 12 ((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) → (𝑁 = (♯‘(1st𝐴)) → (𝑁 = (♯‘(1st𝐵)) → (𝑦 ∈ (0..^𝑁) → (((1st𝐴)‘𝑦) ∈ dom (iEdg‘𝐺) ∧ ((1st𝐵)‘𝑦) ∈ dom (iEdg‘𝐺))))))
116115imp 407 . . . . . . . . . . 11 (((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1st𝐴))) → (𝑁 = (♯‘(1st𝐵)) → (𝑦 ∈ (0..^𝑁) → (((1st𝐴)‘𝑦) ∈ dom (iEdg‘𝐺) ∧ ((1st𝐵)‘𝑦) ∈ dom (iEdg‘𝐺)))))
1171163adant1 1124 . . . . . . . . . 10 ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1st𝐴))) → (𝑁 = (♯‘(1st𝐵)) → (𝑦 ∈ (0..^𝑁) → (((1st𝐴)‘𝑦) ∈ dom (iEdg‘𝐺) ∧ ((1st𝐵)‘𝑦) ∈ dom (iEdg‘𝐺)))))
118117imp 407 . . . . . . . . 9 (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1st𝐴))) ∧ 𝑁 = (♯‘(1st𝐵))) → (𝑦 ∈ (0..^𝑁) → (((1st𝐴)‘𝑦) ∈ dom (iEdg‘𝐺) ∧ ((1st𝐵)‘𝑦) ∈ dom (iEdg‘𝐺))))
119118imp 407 . . . . . . . 8 ((((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1st𝐴))) ∧ 𝑁 = (♯‘(1st𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → (((1st𝐴)‘𝑦) ∈ dom (iEdg‘𝐺) ∧ ((1st𝐵)‘𝑦) ∈ dom (iEdg‘𝐺)))
120 f1veqaeq 7012 . . . . . . . 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 681 . . . . . . 7 ((((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1st𝐴))) ∧ 𝑁 = (♯‘(1st𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → (((iEdg‘𝐺)‘((1st𝐴)‘𝑦)) = ((iEdg‘𝐺)‘((1st𝐵)‘𝑦)) → ((1st𝐴)‘𝑦) = ((1st𝐵)‘𝑦)))
12274, 121syl5bi 243 . . . . . 6 ((((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1st𝐴))) ∧ 𝑁 = (♯‘(1st𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → (((iEdg‘𝐺)‘((1st𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1st𝐴)‘𝑦)) → ((1st𝐴)‘𝑦) = ((1st𝐵)‘𝑦)))
123122ralimdva 3181 . . . . 5 (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1st𝐴))) ∧ 𝑁 = (♯‘(1st𝐵))) → (∀𝑦 ∈ (0..^𝑁)((iEdg‘𝐺)‘((1st𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1st𝐴)‘𝑦)) → ∀𝑦 ∈ (0..^𝑁)((1st𝐴)‘𝑦) = ((1st𝐵)‘𝑦)))
12432, 73, 1233syld 60 . . . 4 (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1st𝐴))) ∧ 𝑁 = (♯‘(1st𝐵))) → (∀𝑦 ∈ (0...𝑁)((2nd𝐴)‘𝑦) = ((2nd𝐵)‘𝑦) → ∀𝑦 ∈ (0..^𝑁)((1st𝐴)‘𝑦) = ((1st𝐵)‘𝑦)))
125124expimpd 454 . . 3 ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1st𝐴))) → ((𝑁 = (♯‘(1st𝐵)) ∧ ∀𝑦 ∈ (0...𝑁)((2nd𝐴)‘𝑦) = ((2nd𝐵)‘𝑦)) → ∀𝑦 ∈ (0..^𝑁)((1st𝐴)‘𝑦) = ((1st𝐵)‘𝑦)))
126125pm4.71d 562 . 2 ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘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 312 1 ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘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 207  wa 396  w3a 1081   = wceq 1530  wcel 2107  wral 3142  wss 3939  {cpr 4565  dom cdm 5553  ran crn 5554  wf 6347  1-1wf1 6348  1-1-ontowf1o 6350  cfv 6351  (class class class)co 7151  1st c1st 7681  2nd c2nd 7682  0cc0 10529  1c1 10530   + caddc 10532  ...cfz 12885  ..^cfzo 13026  chash 13683  Word cword 13854  Vtxcvtx 26697  iEdgciedg 26698  Edgcedg 26748  UPGraphcupgr 26781  USPGraphcuspgr 26849  Walkscwlks 27294
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-13 2385  ax-ext 2797  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-ifp 1057  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-nel 3128  df-ral 3147  df-rex 3148  df-reu 3149  df-rmo 3150  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4837  df-int 4874  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-tr 5169  df-id 5458  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-we 5514  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-pred 6145  df-ord 6191  df-on 6192  df-lim 6193  df-suc 6194  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-om 7572  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-2o 8097  df-oadd 8100  df-er 8282  df-map 8401  df-pm 8402  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-dju 9322  df-card 9360  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11631  df-2 11692  df-n0 11890  df-xnn0 11960  df-z 11974  df-uz 12236  df-fz 12886  df-fzo 13027  df-hash 13684  df-word 13855  df-edg 26749  df-uhgr 26759  df-upgr 26783  df-uspgr 26851  df-wlks 27297
This theorem is referenced by:  uspgr2wlkeq2  27344  clwlkclwwlkf1  27704
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