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Theorem clwlkclwwlkf1lem3 29259
Description: Lemma 3 for clwlkclwwlkf1 29263. (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 29258 . . . 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 29257 . . . . . . . . 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 29257 . . . . . . . . 9 (π‘Š ∈ 𝐢 β†’ (𝐷(Walksβ€˜πΊ)𝐸 ∧ (πΈβ€˜0) = (πΈβ€˜(β™―β€˜π·)) ∧ (β™―β€˜π·) ∈ β„•))
10 lbfzo0 13672 . . . . . . . . . . . . . . . 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 6892 . . . . . . . . . . . . . 14 (𝑖 = 0 β†’ (π΅β€˜π‘–) = (π΅β€˜0))
16 fveq2 6892 . . . . . . . . . . . . . 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 3609 . . . . . . . . . . . 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 6892 . . . . . . . . . . . . . . . 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 6905 . . . 4 (β™―β€˜π΄) ∈ V
48 fveq2 6892 . . . . . 6 (𝑖 = (β™―β€˜π΄) β†’ (π΅β€˜π‘–) = (π΅β€˜(β™―β€˜π΄)))
49 fveq2 6892 . . . . . 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 4895 . . . 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 12479 . . . . . . . 8 ((β™―β€˜π΄) ∈ β„• β†’ (β™―β€˜π΄) ∈ β„•0)
55 elnn0uz 12867 . . . . . . . 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 13744 . . . 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 3326 . 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 3062  {crab 3433  Vcvv 3475   βˆͺ cun 3947  {csn 4629   class class class wbr 5149  β€˜cfv 6544  (class class class)co 7409  1st c1st 7973  2nd c2nd 7974  0cc0 11110  1c1 11111   ≀ cle 11249  β„•cn 12212  β„•0cn0 12472  β„€β‰₯cuz 12822  ...cfz 13484  ..^cfzo 13627  β™―chash 14290   prefix cpfx 14620  Walkscwlks 28853  ClWalkscclwlks 29027
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 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187
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 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-er 8703  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-card 9934  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-n0 12473  df-z 12559  df-uz 12823  df-fz 13485  df-fzo 13628  df-hash 14291  df-word 14465  df-substr 14591  df-pfx 14621  df-wlks 28856  df-clwlks 29028
This theorem is referenced by:  clwlkclwwlkf1  29263
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