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Theorem uspgr2wlkeq 29678
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 1096 . . 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 29666 . . . 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 1117 . . 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 1129 . 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 13799 . . . . . . . . . . . 12 (𝑥 ∈ (0..^𝑁) → (𝑥 + 1) ∈ (0...𝑁))
76adantl 481 . . . . . . . . . . 11 ((((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1st𝐴))) ∧ 𝑁 = (♯‘(1st𝐵))) ∧ 𝑥 ∈ (0..^𝑁)) → (𝑥 + 1) ∈ (0...𝑁))
8 fveq2 6906 . . . . . . . . . . . . 13 (𝑦 = (𝑥 + 1) → ((2nd𝐴)‘𝑦) = ((2nd𝐴)‘(𝑥 + 1)))
9 fveq2 6906 . . . . . . . . . . . . 13 (𝑦 = (𝑥 + 1) → ((2nd𝐵)‘𝑦) = ((2nd𝐵)‘(𝑥 + 1)))
108, 9eqeq12d 2750 . . . . . . . . . . . 12 (𝑦 = (𝑥 + 1) → (((2nd𝐴)‘𝑦) = ((2nd𝐵)‘𝑦) ↔ ((2nd𝐴)‘(𝑥 + 1)) = ((2nd𝐵)‘(𝑥 + 1))))
1110adantl 481 . . . . . . . . . . 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 3613 . . . . . . . . . 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 451 . . . . . . . . 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 3142 . . . . . . . 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 7453 . . . . . . . . . 10 (𝑦 = 𝑥 → ((2nd𝐴)‘(𝑦 + 1)) = ((2nd𝐴)‘(𝑥 + 1)))
16 fvoveq1 7453 . . . . . . . . . 10 (𝑦 = 𝑥 → ((2nd𝐵)‘(𝑦 + 1)) = ((2nd𝐵)‘(𝑥 + 1)))
1715, 16eqeq12d 2750 . . . . . . . . 9 (𝑦 = 𝑥 → (((2nd𝐴)‘(𝑦 + 1)) = ((2nd𝐵)‘(𝑦 + 1)) ↔ ((2nd𝐴)‘(𝑥 + 1)) = ((2nd𝐵)‘(𝑥 + 1))))
1817cbvralvw 3234 . . . . . . . 8 (∀𝑦 ∈ (0..^𝑁)((2nd𝐴)‘(𝑦 + 1)) = ((2nd𝐵)‘(𝑦 + 1)) ↔ ∀𝑥 ∈ (0..^𝑁)((2nd𝐴)‘(𝑥 + 1)) = ((2nd𝐵)‘(𝑥 + 1)))
1914, 18sylibr 234 . . . . . . 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 13714 . . . . . . . . . 10 (0..^𝑁) ⊆ (0...𝑁)
21 ssralv 4063 . . . . . . . . . 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 3108 . . . . . . . . . . 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 4739 . . . . . . . . . . . . 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 3166 . . . . . . . . . . 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, 26biimtrrid 243 . . . . . . . . . 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 415 . . . . . . . . 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 406 . . . . . . 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 412 . . . . 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 29209 . . . . . . . 8 (𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph)
34 eqid 2734 . . . . . . . . . 10 (Vtx‘𝐺) = (Vtx‘𝐺)
35 eqid 2734 . . . . . . . . . 10 (iEdg‘𝐺) = (iEdg‘𝐺)
36 eqid 2734 . . . . . . . . . 10 (1st𝐴) = (1st𝐴)
37 eqid 2734 . . . . . . . . . 10 (2nd𝐴) = (2nd𝐴)
3834, 35, 36, 37upgrwlkcompim 29675 . . . . . . . . 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 412 . . . . . . . 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 2734 . . . . . . . . . 10 (1st𝐵) = (1st𝐵)
42 eqid 2734 . . . . . . . . . 10 (2nd𝐵) = (2nd𝐵)
4334, 35, 41, 42upgrwlkcompim 29675 . . . . . . . . 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 412 . . . . . . . 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 7438 . . . . . . . . . . . . . . . . . . 19 ((♯‘(1st𝐵)) = 𝑁 → (0..^(♯‘(1st𝐵))) = (0..^𝑁))
4746eqcoms 2742 . . . . . . . . . . . . . . . . . 18 (𝑁 = (♯‘(1st𝐵)) → (0..^(♯‘(1st𝐵))) = (0..^𝑁))
4847raleqdv 3323 . . . . . . . . . . . . . . . . 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 7438 . . . . . . . . . . . . . . . . . . 19 ((♯‘(1st𝐴)) = 𝑁 → (0..^(♯‘(1st𝐴))) = (0..^𝑁))
5049eqcoms 2742 . . . . . . . . . . . . . . . . . 18 (𝑁 = (♯‘(1st𝐴)) → (0..^(♯‘(1st𝐴))) = (0..^𝑁))
5150raleqdv 3323 . . . . . . . . . . . . . . . . 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 639 . . . . . . . . . . . . . . . 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 3108 . . . . . . . . . . . . . . . . 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 2746 . . . . . . . . . . . . . . . . . . . . 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 2746 . . . . . . . . . . . . . . . . . . . . . . 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 2742 . . . . . . . . . . . . . . . . . . . . . 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 229 . . . . . . . . . . . . . . . . . . . . 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, 57biimtrdi 253 . . . . . . . . . . . . . . . . . . . 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 406 . . . . . . . . . . . . . . . . . 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 3082 . . . . . . . . . . . . . . . . 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 235 . . . . . . . . . . . . . . . 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, 62biimtrdi 253 . . . . . . . . . . . . . . 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 412 . . . . . . . . . . . . 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 1134 . . . . . . . . . . . 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 1134 . . . . . . . . . 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 406 . . . . . . . . 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 415 . . . . . . . 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 608 . . . . . 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 1346 . . . . 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 2741 . . . . . . 7 (((iEdg‘𝐺)‘((1st𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1st𝐴)‘𝑦)) ↔ ((iEdg‘𝐺)‘((1st𝐴)‘𝑦)) = ((iEdg‘𝐺)‘((1st𝐵)‘𝑦)))
7535uspgrf1oedg 29204 . . . . . . . . . . . 12 (𝐺 ∈ USPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1-onto→(Edg‘𝐺))
76 f1of1 6847 . . . . . . . . . . . 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 2735 . . . . . . . . . . . 12 (𝐺 ∈ USPGraph → (iEdg‘𝐺) = (iEdg‘𝐺))
79 eqidd 2735 . . . . . . . . . . . 12 (𝐺 ∈ USPGraph → dom (iEdg‘𝐺) = dom (iEdg‘𝐺))
80 edgval 29080 . . . . . . . . . . . . . 14 (Edg‘𝐺) = ran (iEdg‘𝐺)
8180eqcomi 2743 . . . . . . . . . . . . 13 ran (iEdg‘𝐺) = (Edg‘𝐺)
8281a1i 11 . . . . . . . . . . . 12 (𝐺 ∈ USPGraph → ran (iEdg‘𝐺) = (Edg‘𝐺))
8378, 79, 82f1eq123d 6840 . . . . . . . . . . 11 (𝐺 ∈ USPGraph → ((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→ran (iEdg‘𝐺) ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(Edg‘𝐺)))
8477, 83mpbird 257 . . . . . . . . . 10 (𝐺 ∈ USPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→ran (iEdg‘𝐺))
85843ad2ant1 1132 . . . . . . . . 9 ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1st𝐴))) → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→ran (iEdg‘𝐺))
8685adantr 480 . . . . . . . 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 29665 . . . . . . . . . . . . . . 15 (𝐴 ∈ (Walks‘𝐺) → ((1st𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝐴):(0...(♯‘(1st𝐴)))⟶(Vtx‘𝐺)))
8834, 35, 41, 42wlkelwrd 29665 . . . . . . . . . . . . . . 15 (𝐵 ∈ (Walks‘𝐺) → ((1st𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝐵):(0...(♯‘(1st𝐵)))⟶(Vtx‘𝐺)))
89 oveq2 7438 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑁 = (♯‘(1st𝐴)) → (0..^𝑁) = (0..^(♯‘(1st𝐴))))
9089eleq2d 2824 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑁 = (♯‘(1st𝐴)) → (𝑦 ∈ (0..^𝑁) ↔ 𝑦 ∈ (0..^(♯‘(1st𝐴)))))
91 wrdsymbcl 14561 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((1st𝐴) ∈ Word dom (iEdg‘𝐺) ∧ 𝑦 ∈ (0..^(♯‘(1st𝐴)))) → ((1st𝐴)‘𝑦) ∈ dom (iEdg‘𝐺))
9291expcom 413 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 ∈ (0..^(♯‘(1st𝐴))) → ((1st𝐴) ∈ Word dom (iEdg‘𝐺) → ((1st𝐴)‘𝑦) ∈ dom (iEdg‘𝐺)))
9390, 92biimtrdi 253 . . . . . . . . . . . . . . . . . . . . . . . . 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 406 . . . . . . . . . . . . . . . . . . . . . . 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 7438 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑁 = (♯‘(1st𝐵)) → (0..^𝑁) = (0..^(♯‘(1st𝐵))))
9998eleq2d 2824 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑁 = (♯‘(1st𝐵)) → (𝑦 ∈ (0..^𝑁) ↔ 𝑦 ∈ (0..^(♯‘(1st𝐵)))))
100 wrdsymbcl 14561 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((1st𝐵) ∈ Word dom (iEdg‘𝐺) ∧ 𝑦 ∈ (0..^(♯‘(1st𝐵)))) → ((1st𝐵)‘𝑦) ∈ dom (iEdg‘𝐺))
101100expcom 413 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 ∈ (0..^(♯‘(1st𝐵))) → ((1st𝐵) ∈ Word dom (iEdg‘𝐺) → ((1st𝐵)‘𝑦) ∈ dom (iEdg‘𝐺)))
10299, 101biimtrdi 253 . . . . . . . . . . . . . . . . . . . . . . . . 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 406 . . . . . . . . . . . . . . . . . . . . . . 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 512 . . . . . . . . . . . . . . . . . . . 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 412 . . . . . . . . . . . . . . . . . . 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 406 . . . . . . . . . . . . . . 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 596 . . . . . . . . . . . . . 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 415 . . . . . . . . . . . . 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 415 . . . . . . . . . . . 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 406 . . . . . . . . . . 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 1129 . . . . . . . . . 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 406 . . . . . . . . 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 406 . . . . . . . 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 7276 . . . . . . . 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 685 . . . . . . 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, 121biimtrid 242 . . . . . 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 3164 . . . . 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 453 . . 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 561 . 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 311 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 206  wa 395  w3a 1086   = wceq 1536  wcel 2105  wral 3058  wss 3962  {cpr 4632  dom cdm 5688  ran crn 5689  wf 6558  1-1wf1 6559  1-1-ontowf1o 6561  cfv 6562  (class class class)co 7430  1st c1st 8010  2nd c2nd 8011  0cc0 11152  1c1 11153   + caddc 11155  ...cfz 13543  ..^cfzo 13690  chash 14365  Word cword 14548  Vtxcvtx 29027  iEdgciedg 29028  Edgcedg 29078  UPGraphcupgr 29111  USPGraphcuspgr 29179  Walkscwlks 29628
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ifp 1063  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-2o 8505  df-oadd 8508  df-er 8743  df-map 8866  df-pm 8867  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-dju 9938  df-card 9976  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-nn 12264  df-2 12326  df-n0 12524  df-xnn0 12597  df-z 12611  df-uz 12876  df-fz 13544  df-fzo 13691  df-hash 14366  df-word 14549  df-edg 29079  df-uhgr 29089  df-upgr 29113  df-uspgr 29181  df-wlks 29631
This theorem is referenced by:  uspgr2wlkeq2  29679  clwlkclwwlkf1  30038
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