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Theorem clwlkclwwlkf1lem3 29524
Description: Lemma 3 for clwlkclwwlkf1 29528. (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 29523 . . . 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 769 . . . . 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 29522 . . . . . . . . 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 29522 . . . . . . . . 9 (π‘Š ∈ 𝐢 β†’ (𝐷(Walksβ€˜πΊ)𝐸 ∧ (πΈβ€˜0) = (πΈβ€˜(β™―β€˜π·)) ∧ (β™―β€˜π·) ∈ β„•))
10 lbfzo0 13678 . . . . . . . . . . . . . . . 16 (0 ∈ (0..^(β™―β€˜π΄)) ↔ (β™―β€˜π΄) ∈ β„•)
1110biimpri 227 . . . . . . . . . . . . . . 15 ((β™―β€˜π΄) ∈ β„• β†’ 0 ∈ (0..^(β™―β€˜π΄)))
12113ad2ant3 1133 . . . . . . . . . . . . . 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 479 . . . . . . . . . . . . 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 479 . . . . . . . . . . . 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 2746 . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . . . . . . 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 2753 . . . . . . . . . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . . . . . . . . . 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 2770 . . . . . . . . . . . . . . . . . . . . . 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 419 . . . . . . . . . . . . . . . . . . . . 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 2738 . . . . . . . . . . . . . . . . . . 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 1132 . . . . . . . . . . . . . . . . . 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 1132 . . . . . . . . . . . . . . . 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 406 . . . . . . . . . . . . . . 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 479 . . . . . . . . . . . . . 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 405 . . . . . . . . . . . . 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 2738 . . . . . . . . . . . . . . 15 ((β™―β€˜π΄) = (β™―β€˜π·) β†’ (πΈβ€˜(β™―β€˜π·)) = (πΈβ€˜(β™―β€˜π΄)))
3534adantl 480 . . . . . . . . . . . . . 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 479 . . . . . . . . . . . . 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 2770 . . . . . . . . . . . 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 411 . . . . . . . . . . 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 411 . . . . . . . . 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 594 . . . . . . . 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 409 . . . . . . 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 1130 . . . . . 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 405 . . . . 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 510 . . . 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 683 . . 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 2746 . . . . 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 12485 . . . . . . . 8 ((β™―β€˜π΄) ∈ β„• β†’ (β™―β€˜π΄) ∈ β„•0)
55 elnn0uz 12873 . . . . . . . 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 1133 . . . . . 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 1131 . . . 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 13750 . . . 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 3323 . 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 394   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104  βˆ€wral 3059  {crab 3430  Vcvv 3472   βˆͺ cun 3947  {csn 4629   class class class wbr 5149  β€˜cfv 6544  (class class class)co 7413  1st c1st 7977  2nd c2nd 7978  0cc0 11114  1c1 11115   ≀ cle 11255  β„•cn 12218  β„•0cn0 12478  β„€β‰₯cuz 12828  ...cfz 13490  ..^cfzo 13633  β™―chash 14296   prefix cpfx 14626  Walkscwlks 29118  ClWalkscclwlks 29292
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7729  ax-cnex 11170  ax-resscn 11171  ax-1cn 11172  ax-icn 11173  ax-addcl 11174  ax-addrcl 11175  ax-mulcl 11176  ax-mulrcl 11177  ax-mulcom 11178  ax-addass 11179  ax-mulass 11180  ax-distr 11181  ax-i2m1 11182  ax-1ne0 11183  ax-1rid 11184  ax-rnegex 11185  ax-rrecex 11186  ax-cnre 11187  ax-pre-lttri 11188  ax-pre-lttrn 11189  ax-pre-ltadd 11190  ax-pre-mulgt0 11191
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-ifp 1060  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  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 7369  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7860  df-1st 7979  df-2nd 7980  df-frecs 8270  df-wrecs 8301  df-recs 8375  df-rdg 8414  df-1o 8470  df-er 8707  df-map 8826  df-en 8944  df-dom 8945  df-sdom 8946  df-fin 8947  df-card 9938  df-pnf 11256  df-mnf 11257  df-xr 11258  df-ltxr 11259  df-le 11260  df-sub 11452  df-neg 11453  df-nn 12219  df-n0 12479  df-z 12565  df-uz 12829  df-fz 13491  df-fzo 13634  df-hash 14297  df-word 14471  df-substr 14597  df-pfx 14627  df-wlks 29121  df-clwlks 29293
This theorem is referenced by:  clwlkclwwlkf1  29528
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