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Theorem 2cshw 13410
Description: Cyclically shifting a word two times. (Contributed by AV, 7-Apr-2018.) (Revised by AV, 4-Jun-2018.) (Revised by AV, 31-Oct-2018.)
Assertion
Ref Expression
2cshw ((𝑊 ∈ Word 𝑉𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑊 cyclShift 𝑀) cyclShift 𝑁) = (𝑊 cyclShift (𝑀 + 𝑁)))

Proof of Theorem 2cshw
Dummy variable 𝑖 is distinct from all other variables.
StepHypRef Expression
1 cshwlen 13396 . . . . 5 ((𝑊 ∈ Word 𝑉𝑀 ∈ ℤ) → (#‘(𝑊 cyclShift 𝑀)) = (#‘𝑊))
213adant3 1074 . . . 4 ((𝑊 ∈ Word 𝑉𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (#‘(𝑊 cyclShift 𝑀)) = (#‘𝑊))
3 cshwcl 13395 . . . . . . 7 (𝑊 ∈ Word 𝑉 → (𝑊 cyclShift 𝑀) ∈ Word 𝑉)
43anim1i 590 . . . . . 6 ((𝑊 ∈ Word 𝑉𝑁 ∈ ℤ) → ((𝑊 cyclShift 𝑀) ∈ Word 𝑉𝑁 ∈ ℤ))
543adant2 1073 . . . . 5 ((𝑊 ∈ Word 𝑉𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑊 cyclShift 𝑀) ∈ Word 𝑉𝑁 ∈ ℤ))
6 cshwlen 13396 . . . . 5 (((𝑊 cyclShift 𝑀) ∈ Word 𝑉𝑁 ∈ ℤ) → (#‘((𝑊 cyclShift 𝑀) cyclShift 𝑁)) = (#‘(𝑊 cyclShift 𝑀)))
75, 6syl 17 . . . 4 ((𝑊 ∈ Word 𝑉𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (#‘((𝑊 cyclShift 𝑀) cyclShift 𝑁)) = (#‘(𝑊 cyclShift 𝑀)))
8 simp1 1054 . . . . . 6 ((𝑊 ∈ Word 𝑉𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑊 ∈ Word 𝑉)
9 zaddcl 11294 . . . . . . 7 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 + 𝑁) ∈ ℤ)
1093adant1 1072 . . . . . 6 ((𝑊 ∈ Word 𝑉𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 + 𝑁) ∈ ℤ)
118, 10jca 553 . . . . 5 ((𝑊 ∈ Word 𝑉𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑊 ∈ Word 𝑉 ∧ (𝑀 + 𝑁) ∈ ℤ))
12 cshwlen 13396 . . . . 5 ((𝑊 ∈ Word 𝑉 ∧ (𝑀 + 𝑁) ∈ ℤ) → (#‘(𝑊 cyclShift (𝑀 + 𝑁))) = (#‘𝑊))
1311, 12syl 17 . . . 4 ((𝑊 ∈ Word 𝑉𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (#‘(𝑊 cyclShift (𝑀 + 𝑁))) = (#‘𝑊))
142, 7, 133eqtr4d 2654 . . 3 ((𝑊 ∈ Word 𝑉𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (#‘((𝑊 cyclShift 𝑀) cyclShift 𝑁)) = (#‘(𝑊 cyclShift (𝑀 + 𝑁))))
157, 2eqtrd 2644 . . . . . . 7 ((𝑊 ∈ Word 𝑉𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (#‘((𝑊 cyclShift 𝑀) cyclShift 𝑁)) = (#‘𝑊))
1615oveq2d 6565 . . . . . 6 ((𝑊 ∈ Word 𝑉𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (0..^(#‘((𝑊 cyclShift 𝑀) cyclShift 𝑁))) = (0..^(#‘𝑊)))
1716eleq2d 2673 . . . . 5 ((𝑊 ∈ Word 𝑉𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑖 ∈ (0..^(#‘((𝑊 cyclShift 𝑀) cyclShift 𝑁))) ↔ 𝑖 ∈ (0..^(#‘𝑊))))
1833ad2ant1 1075 . . . . . . . . 9 ((𝑊 ∈ Word 𝑉𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑊 cyclShift 𝑀) ∈ Word 𝑉)
1918adantr 480 . . . . . . . 8 (((𝑊 ∈ Word 𝑉𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑖 ∈ (0..^(#‘𝑊))) → (𝑊 cyclShift 𝑀) ∈ Word 𝑉)
20 simp3 1056 . . . . . . . . 9 ((𝑊 ∈ Word 𝑉𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑁 ∈ ℤ)
2120adantr 480 . . . . . . . 8 (((𝑊 ∈ Word 𝑉𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑖 ∈ (0..^(#‘𝑊))) → 𝑁 ∈ ℤ)
222eqcomd 2616 . . . . . . . . . . 11 ((𝑊 ∈ Word 𝑉𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (#‘𝑊) = (#‘(𝑊 cyclShift 𝑀)))
2322oveq2d 6565 . . . . . . . . . 10 ((𝑊 ∈ Word 𝑉𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (0..^(#‘𝑊)) = (0..^(#‘(𝑊 cyclShift 𝑀))))
2423eleq2d 2673 . . . . . . . . 9 ((𝑊 ∈ Word 𝑉𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑖 ∈ (0..^(#‘𝑊)) ↔ 𝑖 ∈ (0..^(#‘(𝑊 cyclShift 𝑀)))))
2524biimpa 500 . . . . . . . 8 (((𝑊 ∈ Word 𝑉𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑖 ∈ (0..^(#‘𝑊))) → 𝑖 ∈ (0..^(#‘(𝑊 cyclShift 𝑀))))
26 cshwidxmod 13400 . . . . . . . 8 (((𝑊 cyclShift 𝑀) ∈ Word 𝑉𝑁 ∈ ℤ ∧ 𝑖 ∈ (0..^(#‘(𝑊 cyclShift 𝑀)))) → (((𝑊 cyclShift 𝑀) cyclShift 𝑁)‘𝑖) = ((𝑊 cyclShift 𝑀)‘((𝑖 + 𝑁) mod (#‘(𝑊 cyclShift 𝑀)))))
2719, 21, 25, 26syl3anc 1318 . . . . . . 7 (((𝑊 ∈ Word 𝑉𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑖 ∈ (0..^(#‘𝑊))) → (((𝑊 cyclShift 𝑀) cyclShift 𝑁)‘𝑖) = ((𝑊 cyclShift 𝑀)‘((𝑖 + 𝑁) mod (#‘(𝑊 cyclShift 𝑀)))))
288adantr 480 . . . . . . . 8 (((𝑊 ∈ Word 𝑉𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑖 ∈ (0..^(#‘𝑊))) → 𝑊 ∈ Word 𝑉)
29 simpl2 1058 . . . . . . . 8 (((𝑊 ∈ Word 𝑉𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑖 ∈ (0..^(#‘𝑊))) → 𝑀 ∈ ℤ)
30 elfzo0 12376 . . . . . . . . . . . 12 (𝑖 ∈ (0..^(#‘𝑊)) ↔ (𝑖 ∈ ℕ0 ∧ (#‘𝑊) ∈ ℕ ∧ 𝑖 < (#‘𝑊)))
31 nn0z 11277 . . . . . . . . . . . . . . . . 17 (𝑖 ∈ ℕ0𝑖 ∈ ℤ)
3231ad2antrr 758 . . . . . . . . . . . . . . . 16 (((𝑖 ∈ ℕ0 ∧ (#‘𝑊) ∈ ℕ) ∧ (𝑊 ∈ Word 𝑉𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → 𝑖 ∈ ℤ)
3320adantl 481 . . . . . . . . . . . . . . . 16 (((𝑖 ∈ ℕ0 ∧ (#‘𝑊) ∈ ℕ) ∧ (𝑊 ∈ Word 𝑉𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → 𝑁 ∈ ℤ)
3432, 33zaddcld 11362 . . . . . . . . . . . . . . 15 (((𝑖 ∈ ℕ0 ∧ (#‘𝑊) ∈ ℕ) ∧ (𝑊 ∈ Word 𝑉𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (𝑖 + 𝑁) ∈ ℤ)
35 simpr 476 . . . . . . . . . . . . . . . 16 ((𝑖 ∈ ℕ0 ∧ (#‘𝑊) ∈ ℕ) → (#‘𝑊) ∈ ℕ)
3635adantr 480 . . . . . . . . . . . . . . 15 (((𝑖 ∈ ℕ0 ∧ (#‘𝑊) ∈ ℕ) ∧ (𝑊 ∈ Word 𝑉𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (#‘𝑊) ∈ ℕ)
3734, 36jca 553 . . . . . . . . . . . . . 14 (((𝑖 ∈ ℕ0 ∧ (#‘𝑊) ∈ ℕ) ∧ (𝑊 ∈ Word 𝑉𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → ((𝑖 + 𝑁) ∈ ℤ ∧ (#‘𝑊) ∈ ℕ))
3837ex 449 . . . . . . . . . . . . 13 ((𝑖 ∈ ℕ0 ∧ (#‘𝑊) ∈ ℕ) → ((𝑊 ∈ Word 𝑉𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑖 + 𝑁) ∈ ℤ ∧ (#‘𝑊) ∈ ℕ)))
39383adant3 1074 . . . . . . . . . . . 12 ((𝑖 ∈ ℕ0 ∧ (#‘𝑊) ∈ ℕ ∧ 𝑖 < (#‘𝑊)) → ((𝑊 ∈ Word 𝑉𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑖 + 𝑁) ∈ ℤ ∧ (#‘𝑊) ∈ ℕ)))
4030, 39sylbi 206 . . . . . . . . . . 11 (𝑖 ∈ (0..^(#‘𝑊)) → ((𝑊 ∈ Word 𝑉𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑖 + 𝑁) ∈ ℤ ∧ (#‘𝑊) ∈ ℕ)))
4140impcom 445 . . . . . . . . . 10 (((𝑊 ∈ Word 𝑉𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑖 ∈ (0..^(#‘𝑊))) → ((𝑖 + 𝑁) ∈ ℤ ∧ (#‘𝑊) ∈ ℕ))
42 zmodfzo 12555 . . . . . . . . . 10 (((𝑖 + 𝑁) ∈ ℤ ∧ (#‘𝑊) ∈ ℕ) → ((𝑖 + 𝑁) mod (#‘𝑊)) ∈ (0..^(#‘𝑊)))
4341, 42syl 17 . . . . . . . . 9 (((𝑊 ∈ Word 𝑉𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑖 ∈ (0..^(#‘𝑊))) → ((𝑖 + 𝑁) mod (#‘𝑊)) ∈ (0..^(#‘𝑊)))
442oveq2d 6565 . . . . . . . . . . 11 ((𝑊 ∈ Word 𝑉𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑖 + 𝑁) mod (#‘(𝑊 cyclShift 𝑀))) = ((𝑖 + 𝑁) mod (#‘𝑊)))
4544eleq1d 2672 . . . . . . . . . 10 ((𝑊 ∈ Word 𝑉𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (((𝑖 + 𝑁) mod (#‘(𝑊 cyclShift 𝑀))) ∈ (0..^(#‘𝑊)) ↔ ((𝑖 + 𝑁) mod (#‘𝑊)) ∈ (0..^(#‘𝑊))))
4645adantr 480 . . . . . . . . 9 (((𝑊 ∈ Word 𝑉𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑖 ∈ (0..^(#‘𝑊))) → (((𝑖 + 𝑁) mod (#‘(𝑊 cyclShift 𝑀))) ∈ (0..^(#‘𝑊)) ↔ ((𝑖 + 𝑁) mod (#‘𝑊)) ∈ (0..^(#‘𝑊))))
4743, 46mpbird 246 . . . . . . . 8 (((𝑊 ∈ Word 𝑉𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑖 ∈ (0..^(#‘𝑊))) → ((𝑖 + 𝑁) mod (#‘(𝑊 cyclShift 𝑀))) ∈ (0..^(#‘𝑊)))
48 cshwidxmod 13400 . . . . . . . 8 ((𝑊 ∈ Word 𝑉𝑀 ∈ ℤ ∧ ((𝑖 + 𝑁) mod (#‘(𝑊 cyclShift 𝑀))) ∈ (0..^(#‘𝑊))) → ((𝑊 cyclShift 𝑀)‘((𝑖 + 𝑁) mod (#‘(𝑊 cyclShift 𝑀)))) = (𝑊‘((((𝑖 + 𝑁) mod (#‘(𝑊 cyclShift 𝑀))) + 𝑀) mod (#‘𝑊))))
4928, 29, 47, 48syl3anc 1318 . . . . . . 7 (((𝑊 ∈ Word 𝑉𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑖 ∈ (0..^(#‘𝑊))) → ((𝑊 cyclShift 𝑀)‘((𝑖 + 𝑁) mod (#‘(𝑊 cyclShift 𝑀)))) = (𝑊‘((((𝑖 + 𝑁) mod (#‘(𝑊 cyclShift 𝑀))) + 𝑀) mod (#‘𝑊))))
50 nn0re 11178 . . . . . . . . . . . . . . . . . . 19 (𝑖 ∈ ℕ0𝑖 ∈ ℝ)
5150ad2antrr 758 . . . . . . . . . . . . . . . . . 18 (((𝑖 ∈ ℕ0 ∧ (#‘𝑊) ∈ ℕ) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → 𝑖 ∈ ℝ)
52 zre 11258 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ ℤ → 𝑁 ∈ ℝ)
5352ad2antll 761 . . . . . . . . . . . . . . . . . 18 (((𝑖 ∈ ℕ0 ∧ (#‘𝑊) ∈ ℕ) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → 𝑁 ∈ ℝ)
5451, 53readdcld 9948 . . . . . . . . . . . . . . . . 17 (((𝑖 ∈ ℕ0 ∧ (#‘𝑊) ∈ ℕ) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (𝑖 + 𝑁) ∈ ℝ)
55 zre 11258 . . . . . . . . . . . . . . . . . 18 (𝑀 ∈ ℤ → 𝑀 ∈ ℝ)
5655ad2antrl 760 . . . . . . . . . . . . . . . . 17 (((𝑖 ∈ ℕ0 ∧ (#‘𝑊) ∈ ℕ) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → 𝑀 ∈ ℝ)
57 nnrp 11718 . . . . . . . . . . . . . . . . . . 19 ((#‘𝑊) ∈ ℕ → (#‘𝑊) ∈ ℝ+)
5857adantl 481 . . . . . . . . . . . . . . . . . 18 ((𝑖 ∈ ℕ0 ∧ (#‘𝑊) ∈ ℕ) → (#‘𝑊) ∈ ℝ+)
5958adantr 480 . . . . . . . . . . . . . . . . 17 (((𝑖 ∈ ℕ0 ∧ (#‘𝑊) ∈ ℕ) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (#‘𝑊) ∈ ℝ+)
60 modaddmod 12571 . . . . . . . . . . . . . . . . 17 (((𝑖 + 𝑁) ∈ ℝ ∧ 𝑀 ∈ ℝ ∧ (#‘𝑊) ∈ ℝ+) → ((((𝑖 + 𝑁) mod (#‘𝑊)) + 𝑀) mod (#‘𝑊)) = (((𝑖 + 𝑁) + 𝑀) mod (#‘𝑊)))
6154, 56, 59, 60syl3anc 1318 . . . . . . . . . . . . . . . 16 (((𝑖 ∈ ℕ0 ∧ (#‘𝑊) ∈ ℕ) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → ((((𝑖 + 𝑁) mod (#‘𝑊)) + 𝑀) mod (#‘𝑊)) = (((𝑖 + 𝑁) + 𝑀) mod (#‘𝑊)))
62 nn0cn 11179 . . . . . . . . . . . . . . . . . . . 20 (𝑖 ∈ ℕ0𝑖 ∈ ℂ)
6362ad2antrr 758 . . . . . . . . . . . . . . . . . . 19 (((𝑖 ∈ ℕ0 ∧ (#‘𝑊) ∈ ℕ) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → 𝑖 ∈ ℂ)
64 zcn 11259 . . . . . . . . . . . . . . . . . . . 20 (𝑀 ∈ ℤ → 𝑀 ∈ ℂ)
6564ad2antrl 760 . . . . . . . . . . . . . . . . . . 19 (((𝑖 ∈ ℕ0 ∧ (#‘𝑊) ∈ ℕ) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → 𝑀 ∈ ℂ)
66 zcn 11259 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℤ → 𝑁 ∈ ℂ)
6766ad2antll 761 . . . . . . . . . . . . . . . . . . 19 (((𝑖 ∈ ℕ0 ∧ (#‘𝑊) ∈ ℕ) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → 𝑁 ∈ ℂ)
68 add32r 10134 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ) → (𝑖 + (𝑀 + 𝑁)) = ((𝑖 + 𝑁) + 𝑀))
6963, 65, 67, 68syl3anc 1318 . . . . . . . . . . . . . . . . . 18 (((𝑖 ∈ ℕ0 ∧ (#‘𝑊) ∈ ℕ) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (𝑖 + (𝑀 + 𝑁)) = ((𝑖 + 𝑁) + 𝑀))
7069eqcomd 2616 . . . . . . . . . . . . . . . . 17 (((𝑖 ∈ ℕ0 ∧ (#‘𝑊) ∈ ℕ) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → ((𝑖 + 𝑁) + 𝑀) = (𝑖 + (𝑀 + 𝑁)))
7170oveq1d 6564 . . . . . . . . . . . . . . . 16 (((𝑖 ∈ ℕ0 ∧ (#‘𝑊) ∈ ℕ) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (((𝑖 + 𝑁) + 𝑀) mod (#‘𝑊)) = ((𝑖 + (𝑀 + 𝑁)) mod (#‘𝑊)))
7261, 71eqtrd 2644 . . . . . . . . . . . . . . 15 (((𝑖 ∈ ℕ0 ∧ (#‘𝑊) ∈ ℕ) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → ((((𝑖 + 𝑁) mod (#‘𝑊)) + 𝑀) mod (#‘𝑊)) = ((𝑖 + (𝑀 + 𝑁)) mod (#‘𝑊)))
7372ex 449 . . . . . . . . . . . . . 14 ((𝑖 ∈ ℕ0 ∧ (#‘𝑊) ∈ ℕ) → ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((((𝑖 + 𝑁) mod (#‘𝑊)) + 𝑀) mod (#‘𝑊)) = ((𝑖 + (𝑀 + 𝑁)) mod (#‘𝑊))))
74733adant3 1074 . . . . . . . . . . . . 13 ((𝑖 ∈ ℕ0 ∧ (#‘𝑊) ∈ ℕ ∧ 𝑖 < (#‘𝑊)) → ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((((𝑖 + 𝑁) mod (#‘𝑊)) + 𝑀) mod (#‘𝑊)) = ((𝑖 + (𝑀 + 𝑁)) mod (#‘𝑊))))
7530, 74sylbi 206 . . . . . . . . . . . 12 (𝑖 ∈ (0..^(#‘𝑊)) → ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((((𝑖 + 𝑁) mod (#‘𝑊)) + 𝑀) mod (#‘𝑊)) = ((𝑖 + (𝑀 + 𝑁)) mod (#‘𝑊))))
7675com12 32 . . . . . . . . . . 11 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑖 ∈ (0..^(#‘𝑊)) → ((((𝑖 + 𝑁) mod (#‘𝑊)) + 𝑀) mod (#‘𝑊)) = ((𝑖 + (𝑀 + 𝑁)) mod (#‘𝑊))))
77763adant1 1072 . . . . . . . . . 10 ((𝑊 ∈ Word 𝑉𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑖 ∈ (0..^(#‘𝑊)) → ((((𝑖 + 𝑁) mod (#‘𝑊)) + 𝑀) mod (#‘𝑊)) = ((𝑖 + (𝑀 + 𝑁)) mod (#‘𝑊))))
7877imp 444 . . . . . . . . 9 (((𝑊 ∈ Word 𝑉𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑖 ∈ (0..^(#‘𝑊))) → ((((𝑖 + 𝑁) mod (#‘𝑊)) + 𝑀) mod (#‘𝑊)) = ((𝑖 + (𝑀 + 𝑁)) mod (#‘𝑊)))
7978fveq2d 6107 . . . . . . . 8 (((𝑊 ∈ Word 𝑉𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑖 ∈ (0..^(#‘𝑊))) → (𝑊‘((((𝑖 + 𝑁) mod (#‘𝑊)) + 𝑀) mod (#‘𝑊))) = (𝑊‘((𝑖 + (𝑀 + 𝑁)) mod (#‘𝑊))))
802adantr 480 . . . . . . . . . . . 12 (((𝑊 ∈ Word 𝑉𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑖 ∈ (0..^(#‘𝑊))) → (#‘(𝑊 cyclShift 𝑀)) = (#‘𝑊))
8180oveq2d 6565 . . . . . . . . . . 11 (((𝑊 ∈ Word 𝑉𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑖 ∈ (0..^(#‘𝑊))) → ((𝑖 + 𝑁) mod (#‘(𝑊 cyclShift 𝑀))) = ((𝑖 + 𝑁) mod (#‘𝑊)))
8281oveq1d 6564 . . . . . . . . . 10 (((𝑊 ∈ Word 𝑉𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑖 ∈ (0..^(#‘𝑊))) → (((𝑖 + 𝑁) mod (#‘(𝑊 cyclShift 𝑀))) + 𝑀) = (((𝑖 + 𝑁) mod (#‘𝑊)) + 𝑀))
8382oveq1d 6564 . . . . . . . . 9 (((𝑊 ∈ Word 𝑉𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑖 ∈ (0..^(#‘𝑊))) → ((((𝑖 + 𝑁) mod (#‘(𝑊 cyclShift 𝑀))) + 𝑀) mod (#‘𝑊)) = ((((𝑖 + 𝑁) mod (#‘𝑊)) + 𝑀) mod (#‘𝑊)))
8483fveq2d 6107 . . . . . . . 8 (((𝑊 ∈ Word 𝑉𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑖 ∈ (0..^(#‘𝑊))) → (𝑊‘((((𝑖 + 𝑁) mod (#‘(𝑊 cyclShift 𝑀))) + 𝑀) mod (#‘𝑊))) = (𝑊‘((((𝑖 + 𝑁) mod (#‘𝑊)) + 𝑀) mod (#‘𝑊))))
8510adantr 480 . . . . . . . . 9 (((𝑊 ∈ Word 𝑉𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑖 ∈ (0..^(#‘𝑊))) → (𝑀 + 𝑁) ∈ ℤ)
86 simpr 476 . . . . . . . . 9 (((𝑊 ∈ Word 𝑉𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑖 ∈ (0..^(#‘𝑊))) → 𝑖 ∈ (0..^(#‘𝑊)))
87 cshwidxmod 13400 . . . . . . . . 9 ((𝑊 ∈ Word 𝑉 ∧ (𝑀 + 𝑁) ∈ ℤ ∧ 𝑖 ∈ (0..^(#‘𝑊))) → ((𝑊 cyclShift (𝑀 + 𝑁))‘𝑖) = (𝑊‘((𝑖 + (𝑀 + 𝑁)) mod (#‘𝑊))))
8828, 85, 86, 87syl3anc 1318 . . . . . . . 8 (((𝑊 ∈ Word 𝑉𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑖 ∈ (0..^(#‘𝑊))) → ((𝑊 cyclShift (𝑀 + 𝑁))‘𝑖) = (𝑊‘((𝑖 + (𝑀 + 𝑁)) mod (#‘𝑊))))
8979, 84, 883eqtr4d 2654 . . . . . . 7 (((𝑊 ∈ Word 𝑉𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑖 ∈ (0..^(#‘𝑊))) → (𝑊‘((((𝑖 + 𝑁) mod (#‘(𝑊 cyclShift 𝑀))) + 𝑀) mod (#‘𝑊))) = ((𝑊 cyclShift (𝑀 + 𝑁))‘𝑖))
9027, 49, 893eqtrd 2648 . . . . . 6 (((𝑊 ∈ Word 𝑉𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑖 ∈ (0..^(#‘𝑊))) → (((𝑊 cyclShift 𝑀) cyclShift 𝑁)‘𝑖) = ((𝑊 cyclShift (𝑀 + 𝑁))‘𝑖))
9190ex 449 . . . . 5 ((𝑊 ∈ Word 𝑉𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑖 ∈ (0..^(#‘𝑊)) → (((𝑊 cyclShift 𝑀) cyclShift 𝑁)‘𝑖) = ((𝑊 cyclShift (𝑀 + 𝑁))‘𝑖)))
9217, 91sylbid 229 . . . 4 ((𝑊 ∈ Word 𝑉𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑖 ∈ (0..^(#‘((𝑊 cyclShift 𝑀) cyclShift 𝑁))) → (((𝑊 cyclShift 𝑀) cyclShift 𝑁)‘𝑖) = ((𝑊 cyclShift (𝑀 + 𝑁))‘𝑖)))
9392ralrimiv 2948 . . 3 ((𝑊 ∈ Word 𝑉𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ∀𝑖 ∈ (0..^(#‘((𝑊 cyclShift 𝑀) cyclShift 𝑁)))(((𝑊 cyclShift 𝑀) cyclShift 𝑁)‘𝑖) = ((𝑊 cyclShift (𝑀 + 𝑁))‘𝑖))
9414, 93jca 553 . 2 ((𝑊 ∈ Word 𝑉𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((#‘((𝑊 cyclShift 𝑀) cyclShift 𝑁)) = (#‘(𝑊 cyclShift (𝑀 + 𝑁))) ∧ ∀𝑖 ∈ (0..^(#‘((𝑊 cyclShift 𝑀) cyclShift 𝑁)))(((𝑊 cyclShift 𝑀) cyclShift 𝑁)‘𝑖) = ((𝑊 cyclShift (𝑀 + 𝑁))‘𝑖)))
95 cshwcl 13395 . . . . . 6 ((𝑊 cyclShift 𝑀) ∈ Word 𝑉 → ((𝑊 cyclShift 𝑀) cyclShift 𝑁) ∈ Word 𝑉)
963, 95syl 17 . . . . 5 (𝑊 ∈ Word 𝑉 → ((𝑊 cyclShift 𝑀) cyclShift 𝑁) ∈ Word 𝑉)
97 cshwcl 13395 . . . . 5 (𝑊 ∈ Word 𝑉 → (𝑊 cyclShift (𝑀 + 𝑁)) ∈ Word 𝑉)
9896, 97jca 553 . . . 4 (𝑊 ∈ Word 𝑉 → (((𝑊 cyclShift 𝑀) cyclShift 𝑁) ∈ Word 𝑉 ∧ (𝑊 cyclShift (𝑀 + 𝑁)) ∈ Word 𝑉))
99983ad2ant1 1075 . . 3 ((𝑊 ∈ Word 𝑉𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (((𝑊 cyclShift 𝑀) cyclShift 𝑁) ∈ Word 𝑉 ∧ (𝑊 cyclShift (𝑀 + 𝑁)) ∈ Word 𝑉))
100 eqwrd 13201 . . 3 ((((𝑊 cyclShift 𝑀) cyclShift 𝑁) ∈ Word 𝑉 ∧ (𝑊 cyclShift (𝑀 + 𝑁)) ∈ Word 𝑉) → (((𝑊 cyclShift 𝑀) cyclShift 𝑁) = (𝑊 cyclShift (𝑀 + 𝑁)) ↔ ((#‘((𝑊 cyclShift 𝑀) cyclShift 𝑁)) = (#‘(𝑊 cyclShift (𝑀 + 𝑁))) ∧ ∀𝑖 ∈ (0..^(#‘((𝑊 cyclShift 𝑀) cyclShift 𝑁)))(((𝑊 cyclShift 𝑀) cyclShift 𝑁)‘𝑖) = ((𝑊 cyclShift (𝑀 + 𝑁))‘𝑖))))
10199, 100syl 17 . 2 ((𝑊 ∈ Word 𝑉𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (((𝑊 cyclShift 𝑀) cyclShift 𝑁) = (𝑊 cyclShift (𝑀 + 𝑁)) ↔ ((#‘((𝑊 cyclShift 𝑀) cyclShift 𝑁)) = (#‘(𝑊 cyclShift (𝑀 + 𝑁))) ∧ ∀𝑖 ∈ (0..^(#‘((𝑊 cyclShift 𝑀) cyclShift 𝑁)))(((𝑊 cyclShift 𝑀) cyclShift 𝑁)‘𝑖) = ((𝑊 cyclShift (𝑀 + 𝑁))‘𝑖))))
10294, 101mpbird 246 1 ((𝑊 ∈ Word 𝑉𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑊 cyclShift 𝑀) cyclShift 𝑁) = (𝑊 cyclShift (𝑀 + 𝑁)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383  w3a 1031   = wceq 1475  wcel 1977  wral 2896   class class class wbr 4583  cfv 5804  (class class class)co 6549  cc 9813  cr 9814  0cc0 9815   + caddc 9818   < clt 9953  cn 10897  0cn0 11169  cz 11254  +crp 11708  ..^cfzo 12334   mod cmo 12530  #chash 12979  Word cword 13146   cyclShift ccsh 13385
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-inf 8232  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-rp 11709  df-fz 12198  df-fzo 12335  df-fl 12455  df-mod 12531  df-hash 12980  df-word 13154  df-concat 13156  df-substr 13158  df-csh 13386
This theorem is referenced by:  2cshwid  13411  2cshwcom  13413  cshweqdif2  13416  2cshwcshw  13422  cshwcshid  13424  cshwcsh2id  13425  cshwshashlem2  15641
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