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Theorem clwlkclwwlkf1lem3 29256
Description: Lemma 3 for clwlkclwwlkf1 29260. (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 29255 . . . 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 771 . . . . 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 29254 . . . . . . . . 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 29254 . . . . . . . . 9 (π‘Š ∈ 𝐢 β†’ (𝐷(Walksβ€˜πΊ)𝐸 ∧ (πΈβ€˜0) = (πΈβ€˜(β™―β€˜π·)) ∧ (β™―β€˜π·) ∈ β„•))
10 lbfzo0 13671 . . . . . . . . . . . . . . . 16 (0 ∈ (0..^(β™―β€˜π΄)) ↔ (β™―β€˜π΄) ∈ β„•)
1110biimpri 227 . . . . . . . . . . . . . . 15 ((β™―β€˜π΄) ∈ β„• β†’ 0 ∈ (0..^(β™―β€˜π΄)))
12113ad2ant3 1135 . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . 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 481 . . . . . . . . . . . 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 6891 . . . . . . . . . . . . . 14 (𝑖 = 0 β†’ (π΅β€˜π‘–) = (π΅β€˜0))
16 fveq2 6891 . . . . . . . . . . . . . 14 (𝑖 = 0 β†’ (πΈβ€˜π‘–) = (πΈβ€˜0))
1715, 16eqeq12d 2748 . . . . . . . . . . . . 13 (𝑖 = 0 β†’ ((π΅β€˜π‘–) = (πΈβ€˜π‘–) ↔ (π΅β€˜0) = (πΈβ€˜0)))
1817rspcv 3608 . . . . . . . . . . . 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 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€˜(β™―β€˜π΄)) = (π΅β€˜0))
21 eqtr 2755 . . . . . . . . . . . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . . . . . . . . . . . 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 2772 . . . . . . . . . . . . . . . . . . . . . 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 421 . . . . . . . . . . . . . . . . . . . . 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 2740 . . . . . . . . . . . . . . . . . . 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 1134 . . . . . . . . . . . . . . . . . 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 1134 . . . . . . . . . . . . . . . 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 408 . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . 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 407 . . . . . . . . . . . . 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 6891 . . . . . . . . . . . . . . . 16 ((β™―β€˜π·) = (β™―β€˜π΄) β†’ (πΈβ€˜(β™―β€˜π·)) = (πΈβ€˜(β™―β€˜π΄)))
3433eqcoms 2740 . . . . . . . . . . . . . . 15 ((β™―β€˜π΄) = (β™―β€˜π·) β†’ (πΈβ€˜(β™―β€˜π·)) = (πΈβ€˜(β™―β€˜π΄)))
3534adantl 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Έβ€˜(β™―β€˜π·)) = (πΈβ€˜(β™―β€˜π΄)))
3635adantr 481 . . . . . . . . . . . . 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 2772 . . . . . . . . . . . 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 413 . . . . . . . . . . 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 413 . . . . . . . . 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 596 . . . . . . . 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 411 . . . . . . 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 1132 . . . . . 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 407 . . . . 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 512 . . . 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 685 . . 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 6904 . . . 4 (β™―β€˜π΄) ∈ V
48 fveq2 6891 . . . . . 6 (𝑖 = (β™―β€˜π΄) β†’ (π΅β€˜π‘–) = (π΅β€˜(β™―β€˜π΄)))
49 fveq2 6891 . . . . . 6 (𝑖 = (β™―β€˜π΄) β†’ (πΈβ€˜π‘–) = (πΈβ€˜(β™―β€˜π΄)))
5048, 49eqeq12d 2748 . . . . 5 (𝑖 = (β™―β€˜π΄) β†’ ((π΅β€˜π‘–) = (πΈβ€˜π‘–) ↔ (π΅β€˜(β™―β€˜π΄)) = (πΈβ€˜(β™―β€˜π΄))))
5150ralunsn 4894 . . . 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 12478 . . . . . . . 8 ((β™―β€˜π΄) ∈ β„• β†’ (β™―β€˜π΄) ∈ β„•0)
55 elnn0uz 12866 . . . . . . . 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 1135 . . . . . 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 1133 . . . 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 13743 . . . 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 3325 . 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 256 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 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  {crab 3432  Vcvv 3474   βˆͺ cun 3946  {csn 4628   class class class wbr 5148  β€˜cfv 6543  (class class class)co 7408  1st c1st 7972  2nd c2nd 7973  0cc0 11109  1c1 11110   ≀ cle 11248  β„•cn 12211  β„•0cn0 12471  β„€β‰₯cuz 12821  ...cfz 13483  ..^cfzo 13626  β™―chash 14289   prefix cpfx 14619  Walkscwlks 28850  ClWalkscclwlks 29024
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 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724  ax-cnex 11165  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185  ax-pre-mulgt0 11186
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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7855  df-1st 7974  df-2nd 7975  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-1o 8465  df-er 8702  df-map 8821  df-en 8939  df-dom 8940  df-sdom 8941  df-fin 8942  df-card 9933  df-pnf 11249  df-mnf 11250  df-xr 11251  df-ltxr 11252  df-le 11253  df-sub 11445  df-neg 11446  df-nn 12212  df-n0 12472  df-z 12558  df-uz 12822  df-fz 13484  df-fzo 13627  df-hash 14290  df-word 14464  df-substr 14590  df-pfx 14620  df-wlks 28853  df-clwlks 29025
This theorem is referenced by:  clwlkclwwlkf1  29260
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