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Theorem clwlkclwwlkf1lem3 28999
Description: Lemma 3 for clwlkclwwlkf1 29003. (Contributed by Alexander van der Vekens, 5-Jul-2018.) (Revised by AV, 30-Oct-2022.)
Hypotheses
Ref Expression
clwlkclwwlkf.c 𝐢 = {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ 1 ≀ (β™―β€˜(1st β€˜π‘€))}
clwlkclwwlkf.a 𝐴 = (1st β€˜π‘ˆ)
clwlkclwwlkf.b 𝐡 = (2nd β€˜π‘ˆ)
clwlkclwwlkf.d 𝐷 = (1st β€˜π‘Š)
clwlkclwwlkf.e 𝐸 = (2nd β€˜π‘Š)
Assertion
Ref Expression
clwlkclwwlkf1lem3 ((π‘ˆ ∈ 𝐢 ∧ π‘Š ∈ 𝐢 ∧ (𝐡 prefix (β™―β€˜π΄)) = (𝐸 prefix (β™―β€˜π·))) β†’ βˆ€π‘– ∈ (0...(β™―β€˜π΄))(π΅β€˜π‘–) = (πΈβ€˜π‘–))
Distinct variable groups:   𝑖,𝐺   𝑀,𝐺   𝑀,𝐴   𝑀,π‘ˆ   𝐴,𝑖   𝐡,𝑖   𝐷,𝑖   𝑀,𝐷   𝑖,𝐸   𝑀,π‘Š
Allowed substitution hints:   𝐡(𝑀)   𝐢(𝑀,𝑖)   π‘ˆ(𝑖)   𝐸(𝑀)   π‘Š(𝑖)

Proof of Theorem clwlkclwwlkf1lem3
StepHypRef Expression
1 clwlkclwwlkf.c . . . . 5 𝐢 = {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ 1 ≀ (β™―β€˜(1st β€˜π‘€))}
2 clwlkclwwlkf.a . . . . 5 𝐴 = (1st β€˜π‘ˆ)
3 clwlkclwwlkf.b . . . . 5 𝐡 = (2nd β€˜π‘ˆ)
4 clwlkclwwlkf.d . . . . 5 𝐷 = (1st β€˜π‘Š)
5 clwlkclwwlkf.e . . . . 5 𝐸 = (2nd β€˜π‘Š)
61, 2, 3, 4, 5clwlkclwwlkf1lem2 28998 . . . 4 ((π‘ˆ ∈ 𝐢 ∧ π‘Š ∈ 𝐢 ∧ (𝐡 prefix (β™―β€˜π΄)) = (𝐸 prefix (β™―β€˜π·))) β†’ ((β™―β€˜π΄) = (β™―β€˜π·) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜π΄))(π΅β€˜π‘–) = (πΈβ€˜π‘–)))
7 simprr 772 . . . . 5 (((π‘ˆ ∈ 𝐢 ∧ π‘Š ∈ 𝐢 ∧ (𝐡 prefix (β™―β€˜π΄)) = (𝐸 prefix (β™―β€˜π·))) ∧ ((β™―β€˜π΄) = (β™―β€˜π·) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜π΄))(π΅β€˜π‘–) = (πΈβ€˜π‘–))) β†’ βˆ€π‘– ∈ (0..^(β™―β€˜π΄))(π΅β€˜π‘–) = (πΈβ€˜π‘–))
81, 2, 3clwlkclwwlkflem 28997 . . . . . . . . 9 (π‘ˆ ∈ 𝐢 β†’ (𝐴(Walksβ€˜πΊ)𝐡 ∧ (π΅β€˜0) = (π΅β€˜(β™―β€˜π΄)) ∧ (β™―β€˜π΄) ∈ β„•))
91, 4, 5clwlkclwwlkflem 28997 . . . . . . . . 9 (π‘Š ∈ 𝐢 β†’ (𝐷(Walksβ€˜πΊ)𝐸 ∧ (πΈβ€˜0) = (πΈβ€˜(β™―β€˜π·)) ∧ (β™―β€˜π·) ∈ β„•))
10 lbfzo0 13621 . . . . . . . . . . . . . . . 16 (0 ∈ (0..^(β™―β€˜π΄)) ↔ (β™―β€˜π΄) ∈ β„•)
1110biimpri 227 . . . . . . . . . . . . . . 15 ((β™―β€˜π΄) ∈ β„• β†’ 0 ∈ (0..^(β™―β€˜π΄)))
12113ad2ant3 1136 . . . . . . . . . . . . . 14 ((𝐴(Walksβ€˜πΊ)𝐡 ∧ (π΅β€˜0) = (π΅β€˜(β™―β€˜π΄)) ∧ (β™―β€˜π΄) ∈ β„•) β†’ 0 ∈ (0..^(β™―β€˜π΄)))
1312adantr 482 . . . . . . . . . . . . 13 (((𝐴(Walksβ€˜πΊ)𝐡 ∧ (π΅β€˜0) = (π΅β€˜(β™―β€˜π΄)) ∧ (β™―β€˜π΄) ∈ β„•) ∧ (𝐷(Walksβ€˜πΊ)𝐸 ∧ (πΈβ€˜0) = (πΈβ€˜(β™―β€˜π·)) ∧ (β™―β€˜π·) ∈ β„•)) β†’ 0 ∈ (0..^(β™―β€˜π΄)))
1413adantr 482 . . . . . . . . . . . 12 ((((𝐴(Walksβ€˜πΊ)𝐡 ∧ (π΅β€˜0) = (π΅β€˜(β™―β€˜π΄)) ∧ (β™―β€˜π΄) ∈ β„•) ∧ (𝐷(Walksβ€˜πΊ)𝐸 ∧ (πΈβ€˜0) = (πΈβ€˜(β™―β€˜π·)) ∧ (β™―β€˜π·) ∈ β„•)) ∧ (β™―β€˜π΄) = (β™―β€˜π·)) β†’ 0 ∈ (0..^(β™―β€˜π΄)))
15 fveq2 6846 . . . . . . . . . . . . . 14 (𝑖 = 0 β†’ (π΅β€˜π‘–) = (π΅β€˜0))
16 fveq2 6846 . . . . . . . . . . . . . 14 (𝑖 = 0 β†’ (πΈβ€˜π‘–) = (πΈβ€˜0))
1715, 16eqeq12d 2749 . . . . . . . . . . . . 13 (𝑖 = 0 β†’ ((π΅β€˜π‘–) = (πΈβ€˜π‘–) ↔ (π΅β€˜0) = (πΈβ€˜0)))
1817rspcv 3579 . . . . . . . . . . . 12 (0 ∈ (0..^(β™―β€˜π΄)) β†’ (βˆ€π‘– ∈ (0..^(β™―β€˜π΄))(π΅β€˜π‘–) = (πΈβ€˜π‘–) β†’ (π΅β€˜0) = (πΈβ€˜0)))
1914, 18syl 17 . . . . . . . . . . 11 ((((𝐴(Walksβ€˜πΊ)𝐡 ∧ (π΅β€˜0) = (π΅β€˜(β™―β€˜π΄)) ∧ (β™―β€˜π΄) ∈ β„•) ∧ (𝐷(Walksβ€˜πΊ)𝐸 ∧ (πΈβ€˜0) = (πΈβ€˜(β™―β€˜π·)) ∧ (β™―β€˜π·) ∈ β„•)) ∧ (β™―β€˜π΄) = (β™―β€˜π·)) β†’ (βˆ€π‘– ∈ (0..^(β™―β€˜π΄))(π΅β€˜π‘–) = (πΈβ€˜π‘–) β†’ (π΅β€˜0) = (πΈβ€˜0)))
20 simpl 484 . . . . . . . . . . . . . . . . . . . . . . 23 (((π΅β€˜(β™―β€˜π΄)) = (π΅β€˜0) ∧ ((π΅β€˜0) = (πΈβ€˜0) ∧ (πΈβ€˜0) = (πΈβ€˜(β™―β€˜π·)))) β†’ (π΅β€˜(β™―β€˜π΄)) = (π΅β€˜0))
21 eqtr 2756 . . . . . . . . . . . . . . . . . . . . . . . 24 (((π΅β€˜0) = (πΈβ€˜0) ∧ (πΈβ€˜0) = (πΈβ€˜(β™―β€˜π·))) β†’ (π΅β€˜0) = (πΈβ€˜(β™―β€˜π·)))
2221adantl 483 . . . . . . . . . . . . . . . . . . . . . . 23 (((π΅β€˜(β™―β€˜π΄)) = (π΅β€˜0) ∧ ((π΅β€˜0) = (πΈβ€˜0) ∧ (πΈβ€˜0) = (πΈβ€˜(β™―β€˜π·)))) β†’ (π΅β€˜0) = (πΈβ€˜(β™―β€˜π·)))
2320, 22eqtrd 2773 . . . . . . . . . . . . . . . . . . . . . 22 (((π΅β€˜(β™―β€˜π΄)) = (π΅β€˜0) ∧ ((π΅β€˜0) = (πΈβ€˜0) ∧ (πΈβ€˜0) = (πΈβ€˜(β™―β€˜π·)))) β†’ (π΅β€˜(β™―β€˜π΄)) = (πΈβ€˜(β™―β€˜π·)))
2423exp32 422 . . . . . . . . . . . . . . . . . . . . 21 ((π΅β€˜(β™―β€˜π΄)) = (π΅β€˜0) β†’ ((π΅β€˜0) = (πΈβ€˜0) β†’ ((πΈβ€˜0) = (πΈβ€˜(β™―β€˜π·)) β†’ (π΅β€˜(β™―β€˜π΄)) = (πΈβ€˜(β™―β€˜π·)))))
2524com23 86 . . . . . . . . . . . . . . . . . . . 20 ((π΅β€˜(β™―β€˜π΄)) = (π΅β€˜0) β†’ ((πΈβ€˜0) = (πΈβ€˜(β™―β€˜π·)) β†’ ((π΅β€˜0) = (πΈβ€˜0) β†’ (π΅β€˜(β™―β€˜π΄)) = (πΈβ€˜(β™―β€˜π·)))))
2625eqcoms 2741 . . . . . . . . . . . . . . . . . . 19 ((π΅β€˜0) = (π΅β€˜(β™―β€˜π΄)) β†’ ((πΈβ€˜0) = (πΈβ€˜(β™―β€˜π·)) β†’ ((π΅β€˜0) = (πΈβ€˜0) β†’ (π΅β€˜(β™―β€˜π΄)) = (πΈβ€˜(β™―β€˜π·)))))
27263ad2ant2 1135 . . . . . . . . . . . . . . . . . 18 ((𝐴(Walksβ€˜πΊ)𝐡 ∧ (π΅β€˜0) = (π΅β€˜(β™―β€˜π΄)) ∧ (β™―β€˜π΄) ∈ β„•) β†’ ((πΈβ€˜0) = (πΈβ€˜(β™―β€˜π·)) β†’ ((π΅β€˜0) = (πΈβ€˜0) β†’ (π΅β€˜(β™―β€˜π΄)) = (πΈβ€˜(β™―β€˜π·)))))
2827com12 32 . . . . . . . . . . . . . . . . 17 ((πΈβ€˜0) = (πΈβ€˜(β™―β€˜π·)) β†’ ((𝐴(Walksβ€˜πΊ)𝐡 ∧ (π΅β€˜0) = (π΅β€˜(β™―β€˜π΄)) ∧ (β™―β€˜π΄) ∈ β„•) β†’ ((π΅β€˜0) = (πΈβ€˜0) β†’ (π΅β€˜(β™―β€˜π΄)) = (πΈβ€˜(β™―β€˜π·)))))
29283ad2ant2 1135 . . . . . . . . . . . . . . . 16 ((𝐷(Walksβ€˜πΊ)𝐸 ∧ (πΈβ€˜0) = (πΈβ€˜(β™―β€˜π·)) ∧ (β™―β€˜π·) ∈ β„•) β†’ ((𝐴(Walksβ€˜πΊ)𝐡 ∧ (π΅β€˜0) = (π΅β€˜(β™―β€˜π΄)) ∧ (β™―β€˜π΄) ∈ β„•) β†’ ((π΅β€˜0) = (πΈβ€˜0) β†’ (π΅β€˜(β™―β€˜π΄)) = (πΈβ€˜(β™―β€˜π·)))))
3029impcom 409 . . . . . . . . . . . . . . 15 (((𝐴(Walksβ€˜πΊ)𝐡 ∧ (π΅β€˜0) = (π΅β€˜(β™―β€˜π΄)) ∧ (β™―β€˜π΄) ∈ β„•) ∧ (𝐷(Walksβ€˜πΊ)𝐸 ∧ (πΈβ€˜0) = (πΈβ€˜(β™―β€˜π·)) ∧ (β™―β€˜π·) ∈ β„•)) β†’ ((π΅β€˜0) = (πΈβ€˜0) β†’ (π΅β€˜(β™―β€˜π΄)) = (πΈβ€˜(β™―β€˜π·))))
3130adantr 482 . . . . . . . . . . . . . 14 ((((𝐴(Walksβ€˜πΊ)𝐡 ∧ (π΅β€˜0) = (π΅β€˜(β™―β€˜π΄)) ∧ (β™―β€˜π΄) ∈ β„•) ∧ (𝐷(Walksβ€˜πΊ)𝐸 ∧ (πΈβ€˜0) = (πΈβ€˜(β™―β€˜π·)) ∧ (β™―β€˜π·) ∈ β„•)) ∧ (β™―β€˜π΄) = (β™―β€˜π·)) β†’ ((π΅β€˜0) = (πΈβ€˜0) β†’ (π΅β€˜(β™―β€˜π΄)) = (πΈβ€˜(β™―β€˜π·))))
3231imp 408 . . . . . . . . . . . . 13 (((((𝐴(Walksβ€˜πΊ)𝐡 ∧ (π΅β€˜0) = (π΅β€˜(β™―β€˜π΄)) ∧ (β™―β€˜π΄) ∈ β„•) ∧ (𝐷(Walksβ€˜πΊ)𝐸 ∧ (πΈβ€˜0) = (πΈβ€˜(β™―β€˜π·)) ∧ (β™―β€˜π·) ∈ β„•)) ∧ (β™―β€˜π΄) = (β™―β€˜π·)) ∧ (π΅β€˜0) = (πΈβ€˜0)) β†’ (π΅β€˜(β™―β€˜π΄)) = (πΈβ€˜(β™―β€˜π·)))
33 fveq2 6846 . . . . . . . . . . . . . . . 16 ((β™―β€˜π·) = (β™―β€˜π΄) β†’ (πΈβ€˜(β™―β€˜π·)) = (πΈβ€˜(β™―β€˜π΄)))
3433eqcoms 2741 . . . . . . . . . . . . . . 15 ((β™―β€˜π΄) = (β™―β€˜π·) β†’ (πΈβ€˜(β™―β€˜π·)) = (πΈβ€˜(β™―β€˜π΄)))
3534adantl 483 . . . . . . . . . . . . . 14 ((((𝐴(Walksβ€˜πΊ)𝐡 ∧ (π΅β€˜0) = (π΅β€˜(β™―β€˜π΄)) ∧ (β™―β€˜π΄) ∈ β„•) ∧ (𝐷(Walksβ€˜πΊ)𝐸 ∧ (πΈβ€˜0) = (πΈβ€˜(β™―β€˜π·)) ∧ (β™―β€˜π·) ∈ β„•)) ∧ (β™―β€˜π΄) = (β™―β€˜π·)) β†’ (πΈβ€˜(β™―β€˜π·)) = (πΈβ€˜(β™―β€˜π΄)))
3635adantr 482 . . . . . . . . . . . . 13 (((((𝐴(Walksβ€˜πΊ)𝐡 ∧ (π΅β€˜0) = (π΅β€˜(β™―β€˜π΄)) ∧ (β™―β€˜π΄) ∈ β„•) ∧ (𝐷(Walksβ€˜πΊ)𝐸 ∧ (πΈβ€˜0) = (πΈβ€˜(β™―β€˜π·)) ∧ (β™―β€˜π·) ∈ β„•)) ∧ (β™―β€˜π΄) = (β™―β€˜π·)) ∧ (π΅β€˜0) = (πΈβ€˜0)) β†’ (πΈβ€˜(β™―β€˜π·)) = (πΈβ€˜(β™―β€˜π΄)))
3732, 36eqtrd 2773 . . . . . . . . . . . 12 (((((𝐴(Walksβ€˜πΊ)𝐡 ∧ (π΅β€˜0) = (π΅β€˜(β™―β€˜π΄)) ∧ (β™―β€˜π΄) ∈ β„•) ∧ (𝐷(Walksβ€˜πΊ)𝐸 ∧ (πΈβ€˜0) = (πΈβ€˜(β™―β€˜π·)) ∧ (β™―β€˜π·) ∈ β„•)) ∧ (β™―β€˜π΄) = (β™―β€˜π·)) ∧ (π΅β€˜0) = (πΈβ€˜0)) β†’ (π΅β€˜(β™―β€˜π΄)) = (πΈβ€˜(β™―β€˜π΄)))
3837ex 414 . . . . . . . . . . 11 ((((𝐴(Walksβ€˜πΊ)𝐡 ∧ (π΅β€˜0) = (π΅β€˜(β™―β€˜π΄)) ∧ (β™―β€˜π΄) ∈ β„•) ∧ (𝐷(Walksβ€˜πΊ)𝐸 ∧ (πΈβ€˜0) = (πΈβ€˜(β™―β€˜π·)) ∧ (β™―β€˜π·) ∈ β„•)) ∧ (β™―β€˜π΄) = (β™―β€˜π·)) β†’ ((π΅β€˜0) = (πΈβ€˜0) β†’ (π΅β€˜(β™―β€˜π΄)) = (πΈβ€˜(β™―β€˜π΄))))
3919, 38syld 47 . . . . . . . . . 10 ((((𝐴(Walksβ€˜πΊ)𝐡 ∧ (π΅β€˜0) = (π΅β€˜(β™―β€˜π΄)) ∧ (β™―β€˜π΄) ∈ β„•) ∧ (𝐷(Walksβ€˜πΊ)𝐸 ∧ (πΈβ€˜0) = (πΈβ€˜(β™―β€˜π·)) ∧ (β™―β€˜π·) ∈ β„•)) ∧ (β™―β€˜π΄) = (β™―β€˜π·)) β†’ (βˆ€π‘– ∈ (0..^(β™―β€˜π΄))(π΅β€˜π‘–) = (πΈβ€˜π‘–) β†’ (π΅β€˜(β™―β€˜π΄)) = (πΈβ€˜(β™―β€˜π΄))))
4039ex 414 . . . . . . . . 9 (((𝐴(Walksβ€˜πΊ)𝐡 ∧ (π΅β€˜0) = (π΅β€˜(β™―β€˜π΄)) ∧ (β™―β€˜π΄) ∈ β„•) ∧ (𝐷(Walksβ€˜πΊ)𝐸 ∧ (πΈβ€˜0) = (πΈβ€˜(β™―β€˜π·)) ∧ (β™―β€˜π·) ∈ β„•)) β†’ ((β™―β€˜π΄) = (β™―β€˜π·) β†’ (βˆ€π‘– ∈ (0..^(β™―β€˜π΄))(π΅β€˜π‘–) = (πΈβ€˜π‘–) β†’ (π΅β€˜(β™―β€˜π΄)) = (πΈβ€˜(β™―β€˜π΄)))))
418, 9, 40syl2an 597 . . . . . . . 8 ((π‘ˆ ∈ 𝐢 ∧ π‘Š ∈ 𝐢) β†’ ((β™―β€˜π΄) = (β™―β€˜π·) β†’ (βˆ€π‘– ∈ (0..^(β™―β€˜π΄))(π΅β€˜π‘–) = (πΈβ€˜π‘–) β†’ (π΅β€˜(β™―β€˜π΄)) = (πΈβ€˜(β™―β€˜π΄)))))
4241impd 412 . . . . . . 7 ((π‘ˆ ∈ 𝐢 ∧ π‘Š ∈ 𝐢) β†’ (((β™―β€˜π΄) = (β™―β€˜π·) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜π΄))(π΅β€˜π‘–) = (πΈβ€˜π‘–)) β†’ (π΅β€˜(β™―β€˜π΄)) = (πΈβ€˜(β™―β€˜π΄))))
43423adant3 1133 . . . . . 6 ((π‘ˆ ∈ 𝐢 ∧ π‘Š ∈ 𝐢 ∧ (𝐡 prefix (β™―β€˜π΄)) = (𝐸 prefix (β™―β€˜π·))) β†’ (((β™―β€˜π΄) = (β™―β€˜π·) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜π΄))(π΅β€˜π‘–) = (πΈβ€˜π‘–)) β†’ (π΅β€˜(β™―β€˜π΄)) = (πΈβ€˜(β™―β€˜π΄))))
4443imp 408 . . . . 5 (((π‘ˆ ∈ 𝐢 ∧ π‘Š ∈ 𝐢 ∧ (𝐡 prefix (β™―β€˜π΄)) = (𝐸 prefix (β™―β€˜π·))) ∧ ((β™―β€˜π΄) = (β™―β€˜π·) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜π΄))(π΅β€˜π‘–) = (πΈβ€˜π‘–))) β†’ (π΅β€˜(β™―β€˜π΄)) = (πΈβ€˜(β™―β€˜π΄)))
457, 44jca 513 . . . 4 (((π‘ˆ ∈ 𝐢 ∧ π‘Š ∈ 𝐢 ∧ (𝐡 prefix (β™―β€˜π΄)) = (𝐸 prefix (β™―β€˜π·))) ∧ ((β™―β€˜π΄) = (β™―β€˜π·) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜π΄))(π΅β€˜π‘–) = (πΈβ€˜π‘–))) β†’ (βˆ€π‘– ∈ (0..^(β™―β€˜π΄))(π΅β€˜π‘–) = (πΈβ€˜π‘–) ∧ (π΅β€˜(β™―β€˜π΄)) = (πΈβ€˜(β™―β€˜π΄))))
466, 45mpdan 686 . . 3 ((π‘ˆ ∈ 𝐢 ∧ π‘Š ∈ 𝐢 ∧ (𝐡 prefix (β™―β€˜π΄)) = (𝐸 prefix (β™―β€˜π·))) β†’ (βˆ€π‘– ∈ (0..^(β™―β€˜π΄))(π΅β€˜π‘–) = (πΈβ€˜π‘–) ∧ (π΅β€˜(β™―β€˜π΄)) = (πΈβ€˜(β™―β€˜π΄))))
47 fvex 6859 . . . 4 (β™―β€˜π΄) ∈ V
48 fveq2 6846 . . . . . 6 (𝑖 = (β™―β€˜π΄) β†’ (π΅β€˜π‘–) = (π΅β€˜(β™―β€˜π΄)))
49 fveq2 6846 . . . . . 6 (𝑖 = (β™―β€˜π΄) β†’ (πΈβ€˜π‘–) = (πΈβ€˜(β™―β€˜π΄)))
5048, 49eqeq12d 2749 . . . . 5 (𝑖 = (β™―β€˜π΄) β†’ ((π΅β€˜π‘–) = (πΈβ€˜π‘–) ↔ (π΅β€˜(β™―β€˜π΄)) = (πΈβ€˜(β™―β€˜π΄))))
5150ralunsn 4855 . . . 4 ((β™―β€˜π΄) ∈ V β†’ (βˆ€π‘– ∈ ((0..^(β™―β€˜π΄)) βˆͺ {(β™―β€˜π΄)})(π΅β€˜π‘–) = (πΈβ€˜π‘–) ↔ (βˆ€π‘– ∈ (0..^(β™―β€˜π΄))(π΅β€˜π‘–) = (πΈβ€˜π‘–) ∧ (π΅β€˜(β™―β€˜π΄)) = (πΈβ€˜(β™―β€˜π΄)))))
5247, 51ax-mp 5 . . 3 (βˆ€π‘– ∈ ((0..^(β™―β€˜π΄)) βˆͺ {(β™―β€˜π΄)})(π΅β€˜π‘–) = (πΈβ€˜π‘–) ↔ (βˆ€π‘– ∈ (0..^(β™―β€˜π΄))(π΅β€˜π‘–) = (πΈβ€˜π‘–) ∧ (π΅β€˜(β™―β€˜π΄)) = (πΈβ€˜(β™―β€˜π΄))))
5346, 52sylibr 233 . 2 ((π‘ˆ ∈ 𝐢 ∧ π‘Š ∈ 𝐢 ∧ (𝐡 prefix (β™―β€˜π΄)) = (𝐸 prefix (β™―β€˜π·))) β†’ βˆ€π‘– ∈ ((0..^(β™―β€˜π΄)) βˆͺ {(β™―β€˜π΄)})(π΅β€˜π‘–) = (πΈβ€˜π‘–))
54 nnnn0 12428 . . . . . . . 8 ((β™―β€˜π΄) ∈ β„• β†’ (β™―β€˜π΄) ∈ β„•0)
55 elnn0uz 12816 . . . . . . . 8 ((β™―β€˜π΄) ∈ β„•0 ↔ (β™―β€˜π΄) ∈ (β„€β‰₯β€˜0))
5654, 55sylib 217 . . . . . . 7 ((β™―β€˜π΄) ∈ β„• β†’ (β™―β€˜π΄) ∈ (β„€β‰₯β€˜0))
57563ad2ant3 1136 . . . . . 6 ((𝐴(Walksβ€˜πΊ)𝐡 ∧ (π΅β€˜0) = (π΅β€˜(β™―β€˜π΄)) ∧ (β™―β€˜π΄) ∈ β„•) β†’ (β™―β€˜π΄) ∈ (β„€β‰₯β€˜0))
588, 57syl 17 . . . . 5 (π‘ˆ ∈ 𝐢 β†’ (β™―β€˜π΄) ∈ (β„€β‰₯β€˜0))
59583ad2ant1 1134 . . . 4 ((π‘ˆ ∈ 𝐢 ∧ π‘Š ∈ 𝐢 ∧ (𝐡 prefix (β™―β€˜π΄)) = (𝐸 prefix (β™―β€˜π·))) β†’ (β™―β€˜π΄) ∈ (β„€β‰₯β€˜0))
60 fzisfzounsn 13693 . . . 4 ((β™―β€˜π΄) ∈ (β„€β‰₯β€˜0) β†’ (0...(β™―β€˜π΄)) = ((0..^(β™―β€˜π΄)) βˆͺ {(β™―β€˜π΄)}))
6159, 60syl 17 . . 3 ((π‘ˆ ∈ 𝐢 ∧ π‘Š ∈ 𝐢 ∧ (𝐡 prefix (β™―β€˜π΄)) = (𝐸 prefix (β™―β€˜π·))) β†’ (0...(β™―β€˜π΄)) = ((0..^(β™―β€˜π΄)) βˆͺ {(β™―β€˜π΄)}))
6261raleqdv 3312 . 2 ((π‘ˆ ∈ 𝐢 ∧ π‘Š ∈ 𝐢 ∧ (𝐡 prefix (β™―β€˜π΄)) = (𝐸 prefix (β™―β€˜π·))) β†’ (βˆ€π‘– ∈ (0...(β™―β€˜π΄))(π΅β€˜π‘–) = (πΈβ€˜π‘–) ↔ βˆ€π‘– ∈ ((0..^(β™―β€˜π΄)) βˆͺ {(β™―β€˜π΄)})(π΅β€˜π‘–) = (πΈβ€˜π‘–)))
6353, 62mpbird 257 1 ((π‘ˆ ∈ 𝐢 ∧ π‘Š ∈ 𝐢 ∧ (𝐡 prefix (β™―β€˜π΄)) = (𝐸 prefix (β™―β€˜π·))) β†’ βˆ€π‘– ∈ (0...(β™―β€˜π΄))(π΅β€˜π‘–) = (πΈβ€˜π‘–))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061  {crab 3406  Vcvv 3447   βˆͺ cun 3912  {csn 4590   class class class wbr 5109  β€˜cfv 6500  (class class class)co 7361  1st c1st 7923  2nd c2nd 7924  0cc0 11059  1c1 11060   ≀ cle 11198  β„•cn 12161  β„•0cn0 12421  β„€β‰₯cuz 12771  ...cfz 13433  ..^cfzo 13576  β™―chash 14239   prefix cpfx 14567  Walkscwlks 28593  ClWalkscclwlks 28767
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-cnex 11115  ax-resscn 11116  ax-1cn 11117  ax-icn 11118  ax-addcl 11119  ax-addrcl 11120  ax-mulcl 11121  ax-mulrcl 11122  ax-mulcom 11123  ax-addass 11124  ax-mulass 11125  ax-distr 11126  ax-i2m1 11127  ax-1ne0 11128  ax-1rid 11129  ax-rnegex 11130  ax-rrecex 11131  ax-cnre 11132  ax-pre-lttri 11133  ax-pre-lttrn 11134  ax-pre-ltadd 11135  ax-pre-mulgt0 11136
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-ifp 1063  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-int 4912  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7317  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-1st 7925  df-2nd 7926  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-1o 8416  df-er 8654  df-map 8773  df-en 8890  df-dom 8891  df-sdom 8892  df-fin 8893  df-card 9883  df-pnf 11199  df-mnf 11200  df-xr 11201  df-ltxr 11202  df-le 11203  df-sub 11395  df-neg 11396  df-nn 12162  df-n0 12422  df-z 12508  df-uz 12772  df-fz 13434  df-fzo 13577  df-hash 14240  df-word 14412  df-substr 14538  df-pfx 14568  df-wlks 28596  df-clwlks 28768
This theorem is referenced by:  clwlkclwwlkf1  29003
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