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Theorem wlkeq 28679
Description: Conditions for two walks (within the same graph) being the same. (Contributed by AV, 1-Jul-2018.) (Revised by AV, 16-May-2019.) (Revised by AV, 14-Apr-2021.)
Assertion
Ref Expression
wlkeq ((𝐴 ∈ (Walksβ€˜πΊ) ∧ 𝐡 ∈ (Walksβ€˜πΊ) ∧ 𝑁 = (β™―β€˜(1st β€˜π΄))) β†’ (𝐴 = 𝐡 ↔ (𝑁 = (β™―β€˜(1st β€˜π΅)) ∧ βˆ€π‘₯ ∈ (0..^𝑁)((1st β€˜π΄)β€˜π‘₯) = ((1st β€˜π΅)β€˜π‘₯) ∧ βˆ€π‘₯ ∈ (0...𝑁)((2nd β€˜π΄)β€˜π‘₯) = ((2nd β€˜π΅)β€˜π‘₯))))
Distinct variable groups:   π‘₯,𝐴   π‘₯,𝐡   π‘₯,𝑁
Allowed substitution hint:   𝐺(π‘₯)

Proof of Theorem wlkeq
StepHypRef Expression
1 eqid 2731 . . . . . . 7 (Vtxβ€˜πΊ) = (Vtxβ€˜πΊ)
2 eqid 2731 . . . . . . 7 (iEdgβ€˜πΊ) = (iEdgβ€˜πΊ)
3 eqid 2731 . . . . . . 7 (1st β€˜π΄) = (1st β€˜π΄)
4 eqid 2731 . . . . . . 7 (2nd β€˜π΄) = (2nd β€˜π΄)
51, 2, 3, 4wlkelwrd 28678 . . . . . 6 (𝐴 ∈ (Walksβ€˜πΊ) β†’ ((1st β€˜π΄) ∈ Word dom (iEdgβ€˜πΊ) ∧ (2nd β€˜π΄):(0...(β™―β€˜(1st β€˜π΄)))⟢(Vtxβ€˜πΊ)))
6 eqid 2731 . . . . . . 7 (1st β€˜π΅) = (1st β€˜π΅)
7 eqid 2731 . . . . . . 7 (2nd β€˜π΅) = (2nd β€˜π΅)
81, 2, 6, 7wlkelwrd 28678 . . . . . 6 (𝐡 ∈ (Walksβ€˜πΊ) β†’ ((1st β€˜π΅) ∈ Word dom (iEdgβ€˜πΊ) ∧ (2nd β€˜π΅):(0...(β™―β€˜(1st β€˜π΅)))⟢(Vtxβ€˜πΊ)))
95, 8anim12i 613 . . . . 5 ((𝐴 ∈ (Walksβ€˜πΊ) ∧ 𝐡 ∈ (Walksβ€˜πΊ)) β†’ (((1st β€˜π΄) ∈ Word dom (iEdgβ€˜πΊ) ∧ (2nd β€˜π΄):(0...(β™―β€˜(1st β€˜π΄)))⟢(Vtxβ€˜πΊ)) ∧ ((1st β€˜π΅) ∈ Word dom (iEdgβ€˜πΊ) ∧ (2nd β€˜π΅):(0...(β™―β€˜(1st β€˜π΅)))⟢(Vtxβ€˜πΊ))))
10 wlkop 28673 . . . . . . 7 (𝐴 ∈ (Walksβ€˜πΊ) β†’ 𝐴 = ⟨(1st β€˜π΄), (2nd β€˜π΄)⟩)
11 eleq1 2820 . . . . . . . 8 (𝐴 = ⟨(1st β€˜π΄), (2nd β€˜π΄)⟩ β†’ (𝐴 ∈ (Walksβ€˜πΊ) ↔ ⟨(1st β€˜π΄), (2nd β€˜π΄)⟩ ∈ (Walksβ€˜πΊ)))
12 df-br 5126 . . . . . . . . 9 ((1st β€˜π΄)(Walksβ€˜πΊ)(2nd β€˜π΄) ↔ ⟨(1st β€˜π΄), (2nd β€˜π΄)⟩ ∈ (Walksβ€˜πΊ))
13 wlklenvm1 28667 . . . . . . . . 9 ((1st β€˜π΄)(Walksβ€˜πΊ)(2nd β€˜π΄) β†’ (β™―β€˜(1st β€˜π΄)) = ((β™―β€˜(2nd β€˜π΄)) βˆ’ 1))
1412, 13sylbir 234 . . . . . . . 8 (⟨(1st β€˜π΄), (2nd β€˜π΄)⟩ ∈ (Walksβ€˜πΊ) β†’ (β™―β€˜(1st β€˜π΄)) = ((β™―β€˜(2nd β€˜π΄)) βˆ’ 1))
1511, 14syl6bi 252 . . . . . . 7 (𝐴 = ⟨(1st β€˜π΄), (2nd β€˜π΄)⟩ β†’ (𝐴 ∈ (Walksβ€˜πΊ) β†’ (β™―β€˜(1st β€˜π΄)) = ((β™―β€˜(2nd β€˜π΄)) βˆ’ 1)))
1610, 15mpcom 38 . . . . . 6 (𝐴 ∈ (Walksβ€˜πΊ) β†’ (β™―β€˜(1st β€˜π΄)) = ((β™―β€˜(2nd β€˜π΄)) βˆ’ 1))
17 wlkop 28673 . . . . . . 7 (𝐡 ∈ (Walksβ€˜πΊ) β†’ 𝐡 = ⟨(1st β€˜π΅), (2nd β€˜π΅)⟩)
18 eleq1 2820 . . . . . . . 8 (𝐡 = ⟨(1st β€˜π΅), (2nd β€˜π΅)⟩ β†’ (𝐡 ∈ (Walksβ€˜πΊ) ↔ ⟨(1st β€˜π΅), (2nd β€˜π΅)⟩ ∈ (Walksβ€˜πΊ)))
19 df-br 5126 . . . . . . . . 9 ((1st β€˜π΅)(Walksβ€˜πΊ)(2nd β€˜π΅) ↔ ⟨(1st β€˜π΅), (2nd β€˜π΅)⟩ ∈ (Walksβ€˜πΊ))
20 wlklenvm1 28667 . . . . . . . . 9 ((1st β€˜π΅)(Walksβ€˜πΊ)(2nd β€˜π΅) β†’ (β™―β€˜(1st β€˜π΅)) = ((β™―β€˜(2nd β€˜π΅)) βˆ’ 1))
2119, 20sylbir 234 . . . . . . . 8 (⟨(1st β€˜π΅), (2nd β€˜π΅)⟩ ∈ (Walksβ€˜πΊ) β†’ (β™―β€˜(1st β€˜π΅)) = ((β™―β€˜(2nd β€˜π΅)) βˆ’ 1))
2218, 21syl6bi 252 . . . . . . 7 (𝐡 = ⟨(1st β€˜π΅), (2nd β€˜π΅)⟩ β†’ (𝐡 ∈ (Walksβ€˜πΊ) β†’ (β™―β€˜(1st β€˜π΅)) = ((β™―β€˜(2nd β€˜π΅)) βˆ’ 1)))
2317, 22mpcom 38 . . . . . 6 (𝐡 ∈ (Walksβ€˜πΊ) β†’ (β™―β€˜(1st β€˜π΅)) = ((β™―β€˜(2nd β€˜π΅)) βˆ’ 1))
2416, 23anim12i 613 . . . . 5 ((𝐴 ∈ (Walksβ€˜πΊ) ∧ 𝐡 ∈ (Walksβ€˜πΊ)) β†’ ((β™―β€˜(1st β€˜π΄)) = ((β™―β€˜(2nd β€˜π΄)) βˆ’ 1) ∧ (β™―β€˜(1st β€˜π΅)) = ((β™―β€˜(2nd β€˜π΅)) βˆ’ 1)))
25 eqwrd 14472 . . . . . . . 8 (((1st β€˜π΄) ∈ Word dom (iEdgβ€˜πΊ) ∧ (1st β€˜π΅) ∈ Word dom (iEdgβ€˜πΊ)) β†’ ((1st β€˜π΄) = (1st β€˜π΅) ↔ ((β™―β€˜(1st β€˜π΄)) = (β™―β€˜(1st β€˜π΅)) ∧ βˆ€π‘₯ ∈ (0..^(β™―β€˜(1st β€˜π΄)))((1st β€˜π΄)β€˜π‘₯) = ((1st β€˜π΅)β€˜π‘₯))))
2625ad2ant2r 745 . . . . . . 7 ((((1st β€˜π΄) ∈ Word dom (iEdgβ€˜πΊ) ∧ (2nd β€˜π΄):(0...(β™―β€˜(1st β€˜π΄)))⟢(Vtxβ€˜πΊ)) ∧ ((1st β€˜π΅) ∈ Word dom (iEdgβ€˜πΊ) ∧ (2nd β€˜π΅):(0...(β™―β€˜(1st β€˜π΅)))⟢(Vtxβ€˜πΊ))) β†’ ((1st β€˜π΄) = (1st β€˜π΅) ↔ ((β™―β€˜(1st β€˜π΄)) = (β™―β€˜(1st β€˜π΅)) ∧ βˆ€π‘₯ ∈ (0..^(β™―β€˜(1st β€˜π΄)))((1st β€˜π΄)β€˜π‘₯) = ((1st β€˜π΅)β€˜π‘₯))))
2726adantr 481 . . . . . 6 (((((1st β€˜π΄) ∈ Word dom (iEdgβ€˜πΊ) ∧ (2nd β€˜π΄):(0...(β™―β€˜(1st β€˜π΄)))⟢(Vtxβ€˜πΊ)) ∧ ((1st β€˜π΅) ∈ Word dom (iEdgβ€˜πΊ) ∧ (2nd β€˜π΅):(0...(β™―β€˜(1st β€˜π΅)))⟢(Vtxβ€˜πΊ))) ∧ ((β™―β€˜(1st β€˜π΄)) = ((β™―β€˜(2nd β€˜π΄)) βˆ’ 1) ∧ (β™―β€˜(1st β€˜π΅)) = ((β™―β€˜(2nd β€˜π΅)) βˆ’ 1))) β†’ ((1st β€˜π΄) = (1st β€˜π΅) ↔ ((β™―β€˜(1st β€˜π΄)) = (β™―β€˜(1st β€˜π΅)) ∧ βˆ€π‘₯ ∈ (0..^(β™―β€˜(1st β€˜π΄)))((1st β€˜π΄)β€˜π‘₯) = ((1st β€˜π΅)β€˜π‘₯))))
28 lencl 14448 . . . . . . . . 9 ((1st β€˜π΄) ∈ Word dom (iEdgβ€˜πΊ) β†’ (β™―β€˜(1st β€˜π΄)) ∈ β„•0)
2928adantr 481 . . . . . . . 8 (((1st β€˜π΄) ∈ Word dom (iEdgβ€˜πΊ) ∧ (2nd β€˜π΄):(0...(β™―β€˜(1st β€˜π΄)))⟢(Vtxβ€˜πΊ)) β†’ (β™―β€˜(1st β€˜π΄)) ∈ β„•0)
30 simpr 485 . . . . . . . 8 (((1st β€˜π΄) ∈ Word dom (iEdgβ€˜πΊ) ∧ (2nd β€˜π΄):(0...(β™―β€˜(1st β€˜π΄)))⟢(Vtxβ€˜πΊ)) β†’ (2nd β€˜π΄):(0...(β™―β€˜(1st β€˜π΄)))⟢(Vtxβ€˜πΊ))
31 simpr 485 . . . . . . . 8 (((1st β€˜π΅) ∈ Word dom (iEdgβ€˜πΊ) ∧ (2nd β€˜π΅):(0...(β™―β€˜(1st β€˜π΅)))⟢(Vtxβ€˜πΊ)) β†’ (2nd β€˜π΅):(0...(β™―β€˜(1st β€˜π΅)))⟢(Vtxβ€˜πΊ))
32 2ffzeq 13587 . . . . . . . 8 (((β™―β€˜(1st β€˜π΄)) ∈ β„•0 ∧ (2nd β€˜π΄):(0...(β™―β€˜(1st β€˜π΄)))⟢(Vtxβ€˜πΊ) ∧ (2nd β€˜π΅):(0...(β™―β€˜(1st β€˜π΅)))⟢(Vtxβ€˜πΊ)) β†’ ((2nd β€˜π΄) = (2nd β€˜π΅) ↔ ((β™―β€˜(1st β€˜π΄)) = (β™―β€˜(1st β€˜π΅)) ∧ βˆ€π‘₯ ∈ (0...(β™―β€˜(1st β€˜π΄)))((2nd β€˜π΄)β€˜π‘₯) = ((2nd β€˜π΅)β€˜π‘₯))))
3329, 30, 31, 32syl2an3an 1422 . . . . . . 7 ((((1st β€˜π΄) ∈ Word dom (iEdgβ€˜πΊ) ∧ (2nd β€˜π΄):(0...(β™―β€˜(1st β€˜π΄)))⟢(Vtxβ€˜πΊ)) ∧ ((1st β€˜π΅) ∈ Word dom (iEdgβ€˜πΊ) ∧ (2nd β€˜π΅):(0...(β™―β€˜(1st β€˜π΅)))⟢(Vtxβ€˜πΊ))) β†’ ((2nd β€˜π΄) = (2nd β€˜π΅) ↔ ((β™―β€˜(1st β€˜π΄)) = (β™―β€˜(1st β€˜π΅)) ∧ βˆ€π‘₯ ∈ (0...(β™―β€˜(1st β€˜π΄)))((2nd β€˜π΄)β€˜π‘₯) = ((2nd β€˜π΅)β€˜π‘₯))))
3433adantr 481 . . . . . 6 (((((1st β€˜π΄) ∈ Word dom (iEdgβ€˜πΊ) ∧ (2nd β€˜π΄):(0...(β™―β€˜(1st β€˜π΄)))⟢(Vtxβ€˜πΊ)) ∧ ((1st β€˜π΅) ∈ Word dom (iEdgβ€˜πΊ) ∧ (2nd β€˜π΅):(0...(β™―β€˜(1st β€˜π΅)))⟢(Vtxβ€˜πΊ))) ∧ ((β™―β€˜(1st β€˜π΄)) = ((β™―β€˜(2nd β€˜π΄)) βˆ’ 1) ∧ (β™―β€˜(1st β€˜π΅)) = ((β™―β€˜(2nd β€˜π΅)) βˆ’ 1))) β†’ ((2nd β€˜π΄) = (2nd β€˜π΅) ↔ ((β™―β€˜(1st β€˜π΄)) = (β™―β€˜(1st β€˜π΅)) ∧ βˆ€π‘₯ ∈ (0...(β™―β€˜(1st β€˜π΄)))((2nd β€˜π΄)β€˜π‘₯) = ((2nd β€˜π΅)β€˜π‘₯))))
3527, 34anbi12d 631 . . . . 5 (((((1st β€˜π΄) ∈ Word dom (iEdgβ€˜πΊ) ∧ (2nd β€˜π΄):(0...(β™―β€˜(1st β€˜π΄)))⟢(Vtxβ€˜πΊ)) ∧ ((1st β€˜π΅) ∈ Word dom (iEdgβ€˜πΊ) ∧ (2nd β€˜π΅):(0...(β™―β€˜(1st β€˜π΅)))⟢(Vtxβ€˜πΊ))) ∧ ((β™―β€˜(1st β€˜π΄)) = ((β™―β€˜(2nd β€˜π΄)) βˆ’ 1) ∧ (β™―β€˜(1st β€˜π΅)) = ((β™―β€˜(2nd β€˜π΅)) βˆ’ 1))) β†’ (((1st β€˜π΄) = (1st β€˜π΅) ∧ (2nd β€˜π΄) = (2nd β€˜π΅)) ↔ (((β™―β€˜(1st β€˜π΄)) = (β™―β€˜(1st β€˜π΅)) ∧ βˆ€π‘₯ ∈ (0..^(β™―β€˜(1st β€˜π΄)))((1st β€˜π΄)β€˜π‘₯) = ((1st β€˜π΅)β€˜π‘₯)) ∧ ((β™―β€˜(1st β€˜π΄)) = (β™―β€˜(1st β€˜π΅)) ∧ βˆ€π‘₯ ∈ (0...(β™―β€˜(1st β€˜π΄)))((2nd β€˜π΄)β€˜π‘₯) = ((2nd β€˜π΅)β€˜π‘₯)))))
369, 24, 35syl2anc 584 . . . 4 ((𝐴 ∈ (Walksβ€˜πΊ) ∧ 𝐡 ∈ (Walksβ€˜πΊ)) β†’ (((1st β€˜π΄) = (1st β€˜π΅) ∧ (2nd β€˜π΄) = (2nd β€˜π΅)) ↔ (((β™―β€˜(1st β€˜π΄)) = (β™―β€˜(1st β€˜π΅)) ∧ βˆ€π‘₯ ∈ (0..^(β™―β€˜(1st β€˜π΄)))((1st β€˜π΄)β€˜π‘₯) = ((1st β€˜π΅)β€˜π‘₯)) ∧ ((β™―β€˜(1st β€˜π΄)) = (β™―β€˜(1st β€˜π΅)) ∧ βˆ€π‘₯ ∈ (0...(β™―β€˜(1st β€˜π΄)))((2nd β€˜π΄)β€˜π‘₯) = ((2nd β€˜π΅)β€˜π‘₯)))))
37363adant3 1132 . . 3 ((𝐴 ∈ (Walksβ€˜πΊ) ∧ 𝐡 ∈ (Walksβ€˜πΊ) ∧ 𝑁 = (β™―β€˜(1st β€˜π΄))) β†’ (((1st β€˜π΄) = (1st β€˜π΅) ∧ (2nd β€˜π΄) = (2nd β€˜π΅)) ↔ (((β™―β€˜(1st β€˜π΄)) = (β™―β€˜(1st β€˜π΅)) ∧ βˆ€π‘₯ ∈ (0..^(β™―β€˜(1st β€˜π΄)))((1st β€˜π΄)β€˜π‘₯) = ((1st β€˜π΅)β€˜π‘₯)) ∧ ((β™―β€˜(1st β€˜π΄)) = (β™―β€˜(1st β€˜π΅)) ∧ βˆ€π‘₯ ∈ (0...(β™―β€˜(1st β€˜π΄)))((2nd β€˜π΄)β€˜π‘₯) = ((2nd β€˜π΅)β€˜π‘₯)))))
38 eqeq1 2735 . . . . . . 7 (𝑁 = (β™―β€˜(1st β€˜π΄)) β†’ (𝑁 = (β™―β€˜(1st β€˜π΅)) ↔ (β™―β€˜(1st β€˜π΄)) = (β™―β€˜(1st β€˜π΅))))
39 oveq2 7385 . . . . . . . 8 (𝑁 = (β™―β€˜(1st β€˜π΄)) β†’ (0..^𝑁) = (0..^(β™―β€˜(1st β€˜π΄))))
4039raleqdv 3324 . . . . . . 7 (𝑁 = (β™―β€˜(1st β€˜π΄)) β†’ (βˆ€π‘₯ ∈ (0..^𝑁)((1st β€˜π΄)β€˜π‘₯) = ((1st β€˜π΅)β€˜π‘₯) ↔ βˆ€π‘₯ ∈ (0..^(β™―β€˜(1st β€˜π΄)))((1st β€˜π΄)β€˜π‘₯) = ((1st β€˜π΅)β€˜π‘₯)))
4138, 40anbi12d 631 . . . . . 6 (𝑁 = (β™―β€˜(1st β€˜π΄)) β†’ ((𝑁 = (β™―β€˜(1st β€˜π΅)) ∧ βˆ€π‘₯ ∈ (0..^𝑁)((1st β€˜π΄)β€˜π‘₯) = ((1st β€˜π΅)β€˜π‘₯)) ↔ ((β™―β€˜(1st β€˜π΄)) = (β™―β€˜(1st β€˜π΅)) ∧ βˆ€π‘₯ ∈ (0..^(β™―β€˜(1st β€˜π΄)))((1st β€˜π΄)β€˜π‘₯) = ((1st β€˜π΅)β€˜π‘₯))))
42 oveq2 7385 . . . . . . . 8 (𝑁 = (β™―β€˜(1st β€˜π΄)) β†’ (0...𝑁) = (0...(β™―β€˜(1st β€˜π΄))))
4342raleqdv 3324 . . . . . . 7 (𝑁 = (β™―β€˜(1st β€˜π΄)) β†’ (βˆ€π‘₯ ∈ (0...𝑁)((2nd β€˜π΄)β€˜π‘₯) = ((2nd β€˜π΅)β€˜π‘₯) ↔ βˆ€π‘₯ ∈ (0...(β™―β€˜(1st β€˜π΄)))((2nd β€˜π΄)β€˜π‘₯) = ((2nd β€˜π΅)β€˜π‘₯)))
4438, 43anbi12d 631 . . . . . 6 (𝑁 = (β™―β€˜(1st β€˜π΄)) β†’ ((𝑁 = (β™―β€˜(1st β€˜π΅)) ∧ βˆ€π‘₯ ∈ (0...𝑁)((2nd β€˜π΄)β€˜π‘₯) = ((2nd β€˜π΅)β€˜π‘₯)) ↔ ((β™―β€˜(1st β€˜π΄)) = (β™―β€˜(1st β€˜π΅)) ∧ βˆ€π‘₯ ∈ (0...(β™―β€˜(1st β€˜π΄)))((2nd β€˜π΄)β€˜π‘₯) = ((2nd β€˜π΅)β€˜π‘₯))))
4541, 44anbi12d 631 . . . . 5 (𝑁 = (β™―β€˜(1st β€˜π΄)) β†’ (((𝑁 = (β™―β€˜(1st β€˜π΅)) ∧ βˆ€π‘₯ ∈ (0..^𝑁)((1st β€˜π΄)β€˜π‘₯) = ((1st β€˜π΅)β€˜π‘₯)) ∧ (𝑁 = (β™―β€˜(1st β€˜π΅)) ∧ βˆ€π‘₯ ∈ (0...𝑁)((2nd β€˜π΄)β€˜π‘₯) = ((2nd β€˜π΅)β€˜π‘₯))) ↔ (((β™―β€˜(1st β€˜π΄)) = (β™―β€˜(1st β€˜π΅)) ∧ βˆ€π‘₯ ∈ (0..^(β™―β€˜(1st β€˜π΄)))((1st β€˜π΄)β€˜π‘₯) = ((1st β€˜π΅)β€˜π‘₯)) ∧ ((β™―β€˜(1st β€˜π΄)) = (β™―β€˜(1st β€˜π΅)) ∧ βˆ€π‘₯ ∈ (0...(β™―β€˜(1st β€˜π΄)))((2nd β€˜π΄)β€˜π‘₯) = ((2nd β€˜π΅)β€˜π‘₯)))))
4645bibi2d 342 . . . 4 (𝑁 = (β™―β€˜(1st β€˜π΄)) β†’ ((((1st β€˜π΄) = (1st β€˜π΅) ∧ (2nd β€˜π΄) = (2nd β€˜π΅)) ↔ ((𝑁 = (β™―β€˜(1st β€˜π΅)) ∧ βˆ€π‘₯ ∈ (0..^𝑁)((1st β€˜π΄)β€˜π‘₯) = ((1st β€˜π΅)β€˜π‘₯)) ∧ (𝑁 = (β™―β€˜(1st β€˜π΅)) ∧ βˆ€π‘₯ ∈ (0...𝑁)((2nd β€˜π΄)β€˜π‘₯) = ((2nd β€˜π΅)β€˜π‘₯)))) ↔ (((1st β€˜π΄) = (1st β€˜π΅) ∧ (2nd β€˜π΄) = (2nd β€˜π΅)) ↔ (((β™―β€˜(1st β€˜π΄)) = (β™―β€˜(1st β€˜π΅)) ∧ βˆ€π‘₯ ∈ (0..^(β™―β€˜(1st β€˜π΄)))((1st β€˜π΄)β€˜π‘₯) = ((1st β€˜π΅)β€˜π‘₯)) ∧ ((β™―β€˜(1st β€˜π΄)) = (β™―β€˜(1st β€˜π΅)) ∧ βˆ€π‘₯ ∈ (0...(β™―β€˜(1st β€˜π΄)))((2nd β€˜π΄)β€˜π‘₯) = ((2nd β€˜π΅)β€˜π‘₯))))))
47463ad2ant3 1135 . . 3 ((𝐴 ∈ (Walksβ€˜πΊ) ∧ 𝐡 ∈ (Walksβ€˜πΊ) ∧ 𝑁 = (β™―β€˜(1st β€˜π΄))) β†’ ((((1st β€˜π΄) = (1st β€˜π΅) ∧ (2nd β€˜π΄) = (2nd β€˜π΅)) ↔ ((𝑁 = (β™―β€˜(1st β€˜π΅)) ∧ βˆ€π‘₯ ∈ (0..^𝑁)((1st β€˜π΄)β€˜π‘₯) = ((1st β€˜π΅)β€˜π‘₯)) ∧ (𝑁 = (β™―β€˜(1st β€˜π΅)) ∧ βˆ€π‘₯ ∈ (0...𝑁)((2nd β€˜π΄)β€˜π‘₯) = ((2nd β€˜π΅)β€˜π‘₯)))) ↔ (((1st β€˜π΄) = (1st β€˜π΅) ∧ (2nd β€˜π΄) = (2nd β€˜π΅)) ↔ (((β™―β€˜(1st β€˜π΄)) = (β™―β€˜(1st β€˜π΅)) ∧ βˆ€π‘₯ ∈ (0..^(β™―β€˜(1st β€˜π΄)))((1st β€˜π΄)β€˜π‘₯) = ((1st β€˜π΅)β€˜π‘₯)) ∧ ((β™―β€˜(1st β€˜π΄)) = (β™―β€˜(1st β€˜π΅)) ∧ βˆ€π‘₯ ∈ (0...(β™―β€˜(1st β€˜π΄)))((2nd β€˜π΄)β€˜π‘₯) = ((2nd β€˜π΅)β€˜π‘₯))))))
4837, 47mpbird 256 . 2 ((𝐴 ∈ (Walksβ€˜πΊ) ∧ 𝐡 ∈ (Walksβ€˜πΊ) ∧ 𝑁 = (β™―β€˜(1st β€˜π΄))) β†’ (((1st β€˜π΄) = (1st β€˜π΅) ∧ (2nd β€˜π΄) = (2nd β€˜π΅)) ↔ ((𝑁 = (β™―β€˜(1st β€˜π΅)) ∧ βˆ€π‘₯ ∈ (0..^𝑁)((1st β€˜π΄)β€˜π‘₯) = ((1st β€˜π΅)β€˜π‘₯)) ∧ (𝑁 = (β™―β€˜(1st β€˜π΅)) ∧ βˆ€π‘₯ ∈ (0...𝑁)((2nd β€˜π΄)β€˜π‘₯) = ((2nd β€˜π΅)β€˜π‘₯)))))
49 1st2ndb 7981 . . . . 5 (𝐴 ∈ (V Γ— V) ↔ 𝐴 = ⟨(1st β€˜π΄), (2nd β€˜π΄)⟩)
5010, 49sylibr 233 . . . 4 (𝐴 ∈ (Walksβ€˜πΊ) β†’ 𝐴 ∈ (V Γ— V))
51 1st2ndb 7981 . . . . 5 (𝐡 ∈ (V Γ— V) ↔ 𝐡 = ⟨(1st β€˜π΅), (2nd β€˜π΅)⟩)
5217, 51sylibr 233 . . . 4 (𝐡 ∈ (Walksβ€˜πΊ) β†’ 𝐡 ∈ (V Γ— V))
53 xpopth 7982 . . . 4 ((𝐴 ∈ (V Γ— V) ∧ 𝐡 ∈ (V Γ— V)) β†’ (((1st β€˜π΄) = (1st β€˜π΅) ∧ (2nd β€˜π΄) = (2nd β€˜π΅)) ↔ 𝐴 = 𝐡))
5450, 52, 53syl2an 596 . . 3 ((𝐴 ∈ (Walksβ€˜πΊ) ∧ 𝐡 ∈ (Walksβ€˜πΊ)) β†’ (((1st β€˜π΄) = (1st β€˜π΅) ∧ (2nd β€˜π΄) = (2nd β€˜π΅)) ↔ 𝐴 = 𝐡))
55543adant3 1132 . 2 ((𝐴 ∈ (Walksβ€˜πΊ) ∧ 𝐡 ∈ (Walksβ€˜πΊ) ∧ 𝑁 = (β™―β€˜(1st β€˜π΄))) β†’ (((1st β€˜π΄) = (1st β€˜π΅) ∧ (2nd β€˜π΄) = (2nd β€˜π΅)) ↔ 𝐴 = 𝐡))
56 3anass 1095 . . . 4 ((𝑁 = (β™―β€˜(1st β€˜π΅)) ∧ βˆ€π‘₯ ∈ (0..^𝑁)((1st β€˜π΄)β€˜π‘₯) = ((1st β€˜π΅)β€˜π‘₯) ∧ βˆ€π‘₯ ∈ (0...𝑁)((2nd β€˜π΄)β€˜π‘₯) = ((2nd β€˜π΅)β€˜π‘₯)) ↔ (𝑁 = (β™―β€˜(1st β€˜π΅)) ∧ (βˆ€π‘₯ ∈ (0..^𝑁)((1st β€˜π΄)β€˜π‘₯) = ((1st β€˜π΅)β€˜π‘₯) ∧ βˆ€π‘₯ ∈ (0...𝑁)((2nd β€˜π΄)β€˜π‘₯) = ((2nd β€˜π΅)β€˜π‘₯))))
57 anandi 674 . . . 4 ((𝑁 = (β™―β€˜(1st β€˜π΅)) ∧ (βˆ€π‘₯ ∈ (0..^𝑁)((1st β€˜π΄)β€˜π‘₯) = ((1st β€˜π΅)β€˜π‘₯) ∧ βˆ€π‘₯ ∈ (0...𝑁)((2nd β€˜π΄)β€˜π‘₯) = ((2nd β€˜π΅)β€˜π‘₯))) ↔ ((𝑁 = (β™―β€˜(1st β€˜π΅)) ∧ βˆ€π‘₯ ∈ (0..^𝑁)((1st β€˜π΄)β€˜π‘₯) = ((1st β€˜π΅)β€˜π‘₯)) ∧ (𝑁 = (β™―β€˜(1st β€˜π΅)) ∧ βˆ€π‘₯ ∈ (0...𝑁)((2nd β€˜π΄)β€˜π‘₯) = ((2nd β€˜π΅)β€˜π‘₯))))
5856, 57bitr2i 275 . . 3 (((𝑁 = (β™―β€˜(1st β€˜π΅)) ∧ βˆ€π‘₯ ∈ (0..^𝑁)((1st β€˜π΄)β€˜π‘₯) = ((1st β€˜π΅)β€˜π‘₯)) ∧ (𝑁 = (β™―β€˜(1st β€˜π΅)) ∧ βˆ€π‘₯ ∈ (0...𝑁)((2nd β€˜π΄)β€˜π‘₯) = ((2nd β€˜π΅)β€˜π‘₯))) ↔ (𝑁 = (β™―β€˜(1st β€˜π΅)) ∧ βˆ€π‘₯ ∈ (0..^𝑁)((1st β€˜π΄)β€˜π‘₯) = ((1st β€˜π΅)β€˜π‘₯) ∧ βˆ€π‘₯ ∈ (0...𝑁)((2nd β€˜π΄)β€˜π‘₯) = ((2nd β€˜π΅)β€˜π‘₯)))
5958a1i 11 . 2 ((𝐴 ∈ (Walksβ€˜πΊ) ∧ 𝐡 ∈ (Walksβ€˜πΊ) ∧ 𝑁 = (β™―β€˜(1st β€˜π΄))) β†’ (((𝑁 = (β™―β€˜(1st β€˜π΅)) ∧ βˆ€π‘₯ ∈ (0..^𝑁)((1st β€˜π΄)β€˜π‘₯) = ((1st β€˜π΅)β€˜π‘₯)) ∧ (𝑁 = (β™―β€˜(1st β€˜π΅)) ∧ βˆ€π‘₯ ∈ (0...𝑁)((2nd β€˜π΄)β€˜π‘₯) = ((2nd β€˜π΅)β€˜π‘₯))) ↔ (𝑁 = (β™―β€˜(1st β€˜π΅)) ∧ βˆ€π‘₯ ∈ (0..^𝑁)((1st β€˜π΄)β€˜π‘₯) = ((1st β€˜π΅)β€˜π‘₯) ∧ βˆ€π‘₯ ∈ (0...𝑁)((2nd β€˜π΄)β€˜π‘₯) = ((2nd β€˜π΅)β€˜π‘₯))))
6048, 55, 593bitr3d 308 1 ((𝐴 ∈ (Walksβ€˜πΊ) ∧ 𝐡 ∈ (Walksβ€˜πΊ) ∧ 𝑁 = (β™―β€˜(1st β€˜π΄))) β†’ (𝐴 = 𝐡 ↔ (𝑁 = (β™―β€˜(1st β€˜π΅)) ∧ βˆ€π‘₯ ∈ (0..^𝑁)((1st β€˜π΄)β€˜π‘₯) = ((1st β€˜π΅)β€˜π‘₯) ∧ βˆ€π‘₯ ∈ (0...𝑁)((2nd β€˜π΄)β€˜π‘₯) = ((2nd β€˜π΅)β€˜π‘₯))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆ€wral 3060  Vcvv 3459  βŸ¨cop 4612   class class class wbr 5125   Γ— cxp 5651  dom cdm 5653  βŸΆwf 6512  β€˜cfv 6516  (class class class)co 7377  1st c1st 7939  2nd c2nd 7940  0cc0 11075  1c1 11076   βˆ’ cmin 11409  β„•0cn0 12437  ...cfz 13449  ..^cfzo 13592  β™―chash 14255  Word cword 14429  Vtxcvtx 28044  iEdgciedg 28045  Walkscwlks 28641
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5262  ax-sep 5276  ax-nul 5283  ax-pow 5340  ax-pr 5404  ax-un 7692  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-ifp 1062  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3365  df-rab 3419  df-v 3461  df-sbc 3758  df-csb 3874  df-dif 3931  df-un 3933  df-in 3935  df-ss 3945  df-pss 3947  df-nul 4303  df-if 4507  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4886  df-int 4928  df-iun 4976  df-br 5126  df-opab 5188  df-mpt 5209  df-tr 5243  df-id 5551  df-eprel 5557  df-po 5565  df-so 5566  df-fr 5608  df-we 5610  df-xp 5659  df-rel 5660  df-cnv 5661  df-co 5662  df-dm 5663  df-rn 5664  df-res 5665  df-ima 5666  df-pred 6273  df-ord 6340  df-on 6341  df-lim 6342  df-suc 6343  df-iota 6468  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-riota 7333  df-ov 7380  df-oprab 7381  df-mpo 7382  df-om 7823  df-1st 7941  df-2nd 7942  df-frecs 8232  df-wrecs 8263  df-recs 8337  df-rdg 8376  df-1o 8432  df-er 8670  df-map 8789  df-pm 8790  df-en 8906  df-dom 8907  df-sdom 8908  df-fin 8909  df-card 9899  df-pnf 11215  df-mnf 11216  df-xr 11217  df-ltxr 11218  df-le 11219  df-sub 11411  df-neg 11412  df-nn 12178  df-n0 12438  df-z 12524  df-uz 12788  df-fz 13450  df-fzo 13593  df-hash 14256  df-word 14430  df-wlks 28644
This theorem is referenced by:  uspgr2wlkeq  28691
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