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Theorem uspgr2wlkeq 26777
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 1082 . . 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 26764 . . . 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 1111 . . 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 12773 . . . . . . . . . . . 12 (𝑥 ∈ (0..^𝑁) → (𝑥 + 1) ∈ (0...𝑁))
76adantl 467 . . . . . . . . . . 11 ((((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1st𝐴))) ∧ 𝑁 = (♯‘(1st𝐵))) ∧ 𝑥 ∈ (0..^𝑁)) → (𝑥 + 1) ∈ (0...𝑁))
8 fveq2 6332 . . . . . . . . . . . . 13 (𝑦 = (𝑥 + 1) → ((2nd𝐴)‘𝑦) = ((2nd𝐴)‘(𝑥 + 1)))
9 fveq2 6332 . . . . . . . . . . . . 13 (𝑦 = (𝑥 + 1) → ((2nd𝐵)‘𝑦) = ((2nd𝐵)‘(𝑥 + 1)))
108, 9eqeq12d 2786 . . . . . . . . . . . 12 (𝑦 = (𝑥 + 1) → (((2nd𝐴)‘𝑦) = ((2nd𝐵)‘𝑦) ↔ ((2nd𝐴)‘(𝑥 + 1)) = ((2nd𝐵)‘(𝑥 + 1))))
1110adantl 467 . . . . . . . . . . 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 3463 . . . . . . . . . 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 439 . . . . . . . . 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 3114 . . . . . . . 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 6816 . . . . . . . . . 10 (𝑦 = 𝑥 → ((2nd𝐴)‘(𝑦 + 1)) = ((2nd𝐴)‘(𝑥 + 1)))
16 fvoveq1 6816 . . . . . . . . . 10 (𝑦 = 𝑥 → ((2nd𝐵)‘(𝑦 + 1)) = ((2nd𝐵)‘(𝑥 + 1)))
1715, 16eqeq12d 2786 . . . . . . . . 9 (𝑦 = 𝑥 → (((2nd𝐴)‘(𝑦 + 1)) = ((2nd𝐵)‘(𝑦 + 1)) ↔ ((2nd𝐴)‘(𝑥 + 1)) = ((2nd𝐵)‘(𝑥 + 1))))
1817cbvralv 3320 . . . . . . . 8 (∀𝑦 ∈ (0..^𝑁)((2nd𝐴)‘(𝑦 + 1)) = ((2nd𝐵)‘(𝑦 + 1)) ↔ ∀𝑥 ∈ (0..^𝑁)((2nd𝐴)‘(𝑥 + 1)) = ((2nd𝐵)‘(𝑥 + 1)))
1914, 18sylibr 224 . . . . . . 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 12696 . . . . . . . . . 10 (0..^𝑁) ⊆ (0...𝑁)
21 ssralv 3815 . . . . . . . . . 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 3212 . . . . . . . . . . 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 4406 . . . . . . . . . . . . 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 3112 . . . . . . . . . . 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 233 . . . . . . . . . 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 400 . . . . . . . . 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 393 . . . . . . 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 397 . . . . 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 26293 . . . . . . . 8 (𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph)
34 eqid 2771 . . . . . . . . . 10 (Vtx‘𝐺) = (Vtx‘𝐺)
35 eqid 2771 . . . . . . . . . 10 (iEdg‘𝐺) = (iEdg‘𝐺)
36 eqid 2771 . . . . . . . . . 10 (1st𝐴) = (1st𝐴)
37 eqid 2771 . . . . . . . . . 10 (2nd𝐴) = (2nd𝐴)
3834, 35, 36, 37upgrwlkcompim 26774 . . . . . . . . 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 397 . . . . . . . 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 2771 . . . . . . . . . 10 (1st𝐵) = (1st𝐵)
42 eqid 2771 . . . . . . . . . 10 (2nd𝐵) = (2nd𝐵)
4334, 35, 41, 42upgrwlkcompim 26774 . . . . . . . . 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 397 . . . . . . . 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 6801 . . . . . . . . . . . . . . . . . . 19 ((♯‘(1st𝐵)) = 𝑁 → (0..^(♯‘(1st𝐵))) = (0..^𝑁))
4746eqcoms 2779 . . . . . . . . . . . . . . . . . 18 (𝑁 = (♯‘(1st𝐵)) → (0..^(♯‘(1st𝐵))) = (0..^𝑁))
4847raleqdv 3293 . . . . . . . . . . . . . . . . 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 6801 . . . . . . . . . . . . . . . . . . 19 ((♯‘(1st𝐴)) = 𝑁 → (0..^(♯‘(1st𝐴))) = (0..^𝑁))
5049eqcoms 2779 . . . . . . . . . . . . . . . . . 18 (𝑁 = (♯‘(1st𝐴)) → (0..^(♯‘(1st𝐴))) = (0..^𝑁))
5150raleqdv 3293 . . . . . . . . . . . . . . . . 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 621 . . . . . . . . . . . . . . . 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 3212 . . . . . . . . . . . . . . . . 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 2782 . . . . . . . . . . . . . . . . . . . . 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 2782 . . . . . . . . . . . . . . . . . . . . . . 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 2779 . . . . . . . . . . . . . . . . . . . . . 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 219 . . . . . . . . . . . . . . . . . . . . 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 243 . . . . . . . . . . . . . . . . . . . 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 393 . . . . . . . . . . . . . . . . . 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 3096 . . . . . . . . . . . . . . . . 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 225 . . . . . . . . . . . . . . . 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 243 . . . . . . . . . . . . . . 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 397 . . . . . . . . . . . . 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 393 . . . . . . . . 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 400 . . . . . . . 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 595 . . . . . 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 1440 . . . . 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 2778 . . . . . . 7 (((iEdg‘𝐺)‘((1st𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1st𝐴)‘𝑦)) ↔ ((iEdg‘𝐺)‘((1st𝐴)‘𝑦)) = ((iEdg‘𝐺)‘((1st𝐵)‘𝑦)))
7535uspgrf1oedg 26290 . . . . . . . . . . . 12 (𝐺 ∈ USPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1-onto→(Edg‘𝐺))
76 f1of1 6277 . . . . . . . . . . . 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 2772 . . . . . . . . . . . 12 (𝐺 ∈ USPGraph → (iEdg‘𝐺) = (iEdg‘𝐺))
79 eqidd 2772 . . . . . . . . . . . 12 (𝐺 ∈ USPGraph → dom (iEdg‘𝐺) = dom (iEdg‘𝐺))
80 edgval 26162 . . . . . . . . . . . . . 14 (Edg‘𝐺) = ran (iEdg‘𝐺)
8180eqcomi 2780 . . . . . . . . . . . . 13 ran (iEdg‘𝐺) = (Edg‘𝐺)
8281a1i 11 . . . . . . . . . . . 12 (𝐺 ∈ USPGraph → ran (iEdg‘𝐺) = (Edg‘𝐺))
8378, 79, 82f1eq123d 6272 . . . . . . . . . . 11 (𝐺 ∈ USPGraph → ((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→ran (iEdg‘𝐺) ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(Edg‘𝐺)))
8477, 83mpbird 247 . . . . . . . . . 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 466 . . . . . . . 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 26763 . . . . . . . . . . . . . . 15 (𝐴 ∈ (Walks‘𝐺) → ((1st𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝐴):(0...(♯‘(1st𝐴)))⟶(Vtx‘𝐺)))
8834, 35, 41, 42wlkelwrd 26763 . . . . . . . . . . . . . . 15 (𝐵 ∈ (Walks‘𝐺) → ((1st𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝐵):(0...(♯‘(1st𝐵)))⟶(Vtx‘𝐺)))
89 oveq2 6801 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑁 = (♯‘(1st𝐴)) → (0..^𝑁) = (0..^(♯‘(1st𝐴))))
9089eleq2d 2836 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑁 = (♯‘(1st𝐴)) → (𝑦 ∈ (0..^𝑁) ↔ 𝑦 ∈ (0..^(♯‘(1st𝐴)))))
91 wrdsymbcl 13514 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((1st𝐴) ∈ Word dom (iEdg‘𝐺) ∧ 𝑦 ∈ (0..^(♯‘(1st𝐴)))) → ((1st𝐴)‘𝑦) ∈ dom (iEdg‘𝐺))
9291expcom 398 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 ∈ (0..^(♯‘(1st𝐴))) → ((1st𝐴) ∈ Word dom (iEdg‘𝐺) → ((1st𝐴)‘𝑦) ∈ dom (iEdg‘𝐺)))
9390, 92syl6bi 243 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑁 = (♯‘(1st𝐴)) → (𝑦 ∈ (0..^𝑁) → ((1st𝐴) ∈ Word dom (iEdg‘𝐺) → ((1st𝐴)‘𝑦) ∈ dom (iEdg‘𝐺))))
9493adantr 466 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑁 = (♯‘(1st𝐴)) ∧ 𝑁 = (♯‘(1st𝐵))) → (𝑦 ∈ (0..^𝑁) → ((1st𝐴) ∈ Word dom (iEdg‘𝐺) → ((1st𝐴)‘𝑦) ∈ dom (iEdg‘𝐺))))
9594imp 393 . . . . . . . . . . . . . . . . . . . . . . 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 467 . . . . . . . . . . . . . . . . . . . . 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 6801 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑁 = (♯‘(1st𝐵)) → (0..^𝑁) = (0..^(♯‘(1st𝐵))))
9998eleq2d 2836 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑁 = (♯‘(1st𝐵)) → (𝑦 ∈ (0..^𝑁) ↔ 𝑦 ∈ (0..^(♯‘(1st𝐵)))))
100 wrdsymbcl 13514 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((1st𝐵) ∈ Word dom (iEdg‘𝐺) ∧ 𝑦 ∈ (0..^(♯‘(1st𝐵)))) → ((1st𝐵)‘𝑦) ∈ dom (iEdg‘𝐺))
101100expcom 398 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 ∈ (0..^(♯‘(1st𝐵))) → ((1st𝐵) ∈ Word dom (iEdg‘𝐺) → ((1st𝐵)‘𝑦) ∈ dom (iEdg‘𝐺)))
10299, 101syl6bi 243 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑁 = (♯‘(1st𝐵)) → (𝑦 ∈ (0..^𝑁) → ((1st𝐵) ∈ Word dom (iEdg‘𝐺) → ((1st𝐵)‘𝑦) ∈ dom (iEdg‘𝐺))))
103102adantl 467 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑁 = (♯‘(1st𝐴)) ∧ 𝑁 = (♯‘(1st𝐵))) → (𝑦 ∈ (0..^𝑁) → ((1st𝐵) ∈ Word dom (iEdg‘𝐺) → ((1st𝐵)‘𝑦) ∈ dom (iEdg‘𝐺))))
104103imp 393 . . . . . . . . . . . . . . . . . . . . . . 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 466 . . . . . . . . . . . . . . . . . . . . 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 502 . . . . . . . . . . . . . . . . . . . 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 397 . . . . . . . . . . . . . . . . . . 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 466 . . . . . . . . . . . . . . . . . 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 466 . . . . . . . . . . . . . . . 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 393 . . . . . . . . . . . . . . 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 583 . . . . . . . . . . . . . 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 400 . . . . . . . . . . . . 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 400 . . . . . . . . . . . 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 393 . . . . . . . . . . 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 393 . . . . . . . . 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 393 . . . . . . . 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 6657 . . . . . . . 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 664 . . . . . . 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 232 . . . . . 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 3111 . . . . 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 441 . . 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 551 . 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 300 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 196  wa 382  w3a 1071   = wceq 1631  wcel 2145  wral 3061  wss 3723  {cpr 4318  dom cdm 5249  ran crn 5250  wf 6027  1-1wf1 6028  1-1-ontowf1o 6030  cfv 6031  (class class class)co 6793  1st c1st 7313  2nd c2nd 7314  0cc0 10138  1c1 10139   + caddc 10141  ...cfz 12533  ..^cfzo 12673  chash 13321  Word cword 13487  Vtxcvtx 26095  iEdgciedg 26096  Edgcedg 26160  UPGraphcupgr 26196  USPGraphcuspgr 26265  Walkscwlks 26727
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-cnex 10194  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-mulcom 10202  ax-addass 10203  ax-mulass 10204  ax-distr 10205  ax-i2m1 10206  ax-1ne0 10207  ax-1rid 10208  ax-rnegex 10209  ax-rrecex 10210  ax-cnre 10211  ax-pre-lttri 10212  ax-pre-lttrn 10213  ax-pre-ltadd 10214  ax-pre-mulgt0 10215
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-ifp 1050  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-2o 7714  df-oadd 7717  df-er 7896  df-map 8011  df-pm 8012  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113  df-card 8965  df-cda 9192  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-sub 10470  df-neg 10471  df-nn 11223  df-2 11281  df-n0 11495  df-xnn0 11566  df-z 11580  df-uz 11889  df-fz 12534  df-fzo 12674  df-hash 13322  df-word 13495  df-edg 26161  df-uhgr 26174  df-upgr 26198  df-uspgr 26267  df-wlks 26730
This theorem is referenced by:  uspgr2wlkeq2  26778  clwlkclwwlkf1  27160  clwlksf1clwwlkOLD  27250
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