Theorem 2cshw 13559

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 𝑉 ∧ 𝑀 ∈ 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.

Step	Hyp	Ref	Expression
1		cshwlen 13545	. . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ℤ) → (#‘(𝑊 cyclShift 𝑀)) = (#‘𝑊))
2	1	3adant3 1081	. . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (#‘(𝑊 cyclShift 𝑀)) = (#‘𝑊))
3		cshwcl 13544	. . . . . . 7 ⊢ (𝑊 ∈ Word 𝑉 → (𝑊 cyclShift 𝑀) ∈ Word 𝑉)
4	3	anim1i 592	. . . . . 6 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℤ) → ((𝑊 cyclShift 𝑀) ∈ Word 𝑉 ∧ 𝑁 ∈ ℤ))
5	4	3adant2 1080	. . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑊 cyclShift 𝑀) ∈ Word 𝑉 ∧ 𝑁 ∈ ℤ))
6		cshwlen 13545	. . . . 5 ⊢ (((𝑊 cyclShift 𝑀) ∈ Word 𝑉 ∧ 𝑁 ∈ ℤ) → (#‘((𝑊 cyclShift 𝑀) cyclShift 𝑁)) = (#‘(𝑊 cyclShift 𝑀)))
7	5, 6	syl 17	. . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (#‘((𝑊 cyclShift 𝑀) cyclShift 𝑁)) = (#‘(𝑊 cyclShift 𝑀)))
8		simp1 1061	. . . . . 6 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑊 ∈ Word 𝑉)
9		zaddcl 11417	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 + 𝑁) ∈ ℤ)
10	9	3adant1 1079	. . . . . 6 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 + 𝑁) ∈ ℤ)
11	8, 10	jca 554	. . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑊 ∈ Word 𝑉 ∧ (𝑀 + 𝑁) ∈ ℤ))
12		cshwlen 13545	. . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (𝑀 + 𝑁) ∈ ℤ) → (#‘(𝑊 cyclShift (𝑀 + 𝑁))) = (#‘𝑊))
13	11, 12	syl 17	. . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (#‘(𝑊 cyclShift (𝑀 + 𝑁))) = (#‘𝑊))
14	2, 7, 13	3eqtr4d 2666	. . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (#‘((𝑊 cyclShift 𝑀) cyclShift 𝑁)) = (#‘(𝑊 cyclShift (𝑀 + 𝑁))))
15	7, 2	eqtrd 2656	. . . . . . 7 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (#‘((𝑊 cyclShift 𝑀) cyclShift 𝑁)) = (#‘𝑊))
16	15	oveq2d 6666	. . . . . 6 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (0..^(#‘((𝑊 cyclShift 𝑀) cyclShift 𝑁))) = (0..^(#‘𝑊)))
17	16	eleq2d 2687	. . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑖 ∈ (0..^(#‘((𝑊 cyclShift 𝑀) cyclShift 𝑁))) ↔ 𝑖 ∈ (0..^(#‘𝑊))))
18	3	3ad2ant1 1082	. . . . . . . . 9 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑊 cyclShift 𝑀) ∈ Word 𝑉)
19	18	adantr 481	. . . . . . . 8 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑖 ∈ (0..^(#‘𝑊))) → (𝑊 cyclShift 𝑀) ∈ Word 𝑉)
20		simp3 1063	. . . . . . . . 9 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑁 ∈ ℤ)
21	20	adantr 481	. . . . . . . 8 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑖 ∈ (0..^(#‘𝑊))) → 𝑁 ∈ ℤ)
22	2	eqcomd 2628	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (#‘𝑊) = (#‘(𝑊 cyclShift 𝑀)))
23	22	oveq2d 6666	. . . . . . . . . 10 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (0..^(#‘𝑊)) = (0..^(#‘(𝑊 cyclShift 𝑀))))
24	23	eleq2d 2687	. . . . . . . . 9 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑖 ∈ (0..^(#‘𝑊)) ↔ 𝑖 ∈ (0..^(#‘(𝑊 cyclShift 𝑀)))))
25	24	biimpa 501	. . . . . . . 8 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑖 ∈ (0..^(#‘𝑊))) → 𝑖 ∈ (0..^(#‘(𝑊 cyclShift 𝑀))))
26		cshwidxmod 13549	. . . . . . . 8 ⊢ (((𝑊 cyclShift 𝑀) ∈ Word 𝑉 ∧ 𝑁 ∈ ℤ ∧ 𝑖 ∈ (0..^(#‘(𝑊 cyclShift 𝑀)))) → (((𝑊 cyclShift 𝑀) cyclShift 𝑁)‘𝑖) = ((𝑊 cyclShift 𝑀)‘((𝑖 + 𝑁) mod (#‘(𝑊 cyclShift 𝑀)))))
27	19, 21, 25, 26	syl3anc 1326	. . . . . . 7 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑖 ∈ (0..^(#‘𝑊))) → (((𝑊 cyclShift 𝑀) cyclShift 𝑁)‘𝑖) = ((𝑊 cyclShift 𝑀)‘((𝑖 + 𝑁) mod (#‘(𝑊 cyclShift 𝑀)))))
28	8	adantr 481	. . . . . . . 8 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑖 ∈ (0..^(#‘𝑊))) → 𝑊 ∈ Word 𝑉)
29		simpl2 1065	. . . . . . . 8 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑖 ∈ (0..^(#‘𝑊))) → 𝑀 ∈ ℤ)
30		elfzo0 12508	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (0..^(#‘𝑊)) ↔ (𝑖 ∈ ℕ0 ∧ (#‘𝑊) ∈ ℕ ∧ 𝑖 < (#‘𝑊)))
31		nn0z 11400	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ ℕ0 → 𝑖 ∈ ℤ)
32	31	ad2antrr 762	. . . . . . . . . . . . . . . 16 ⊢ (((𝑖 ∈ ℕ0 ∧ (#‘𝑊) ∈ ℕ) ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → 𝑖 ∈ ℤ)
33	20	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ (((𝑖 ∈ ℕ0 ∧ (#‘𝑊) ∈ ℕ) ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → 𝑁 ∈ ℤ)
34	32, 33	zaddcld 11486	. . . . . . . . . . . . . . 15 ⊢ (((𝑖 ∈ ℕ0 ∧ (#‘𝑊) ∈ ℕ) ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (𝑖 + 𝑁) ∈ ℤ)
35		simpr 477	. . . . . . . . . . . . . . . 16 ⊢ ((𝑖 ∈ ℕ0 ∧ (#‘𝑊) ∈ ℕ) → (#‘𝑊) ∈ ℕ)
36	35	adantr 481	. . . . . . . . . . . . . . 15 ⊢ (((𝑖 ∈ ℕ0 ∧ (#‘𝑊) ∈ ℕ) ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (#‘𝑊) ∈ ℕ)
37	34, 36	jca 554	. . . . . . . . . . . . . 14 ⊢ (((𝑖 ∈ ℕ0 ∧ (#‘𝑊) ∈ ℕ) ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → ((𝑖 + 𝑁) ∈ ℤ ∧ (#‘𝑊) ∈ ℕ))
38	37	ex 450	. . . . . . . . . . . . 13 ⊢ ((𝑖 ∈ ℕ0 ∧ (#‘𝑊) ∈ ℕ) → ((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑖 + 𝑁) ∈ ℤ ∧ (#‘𝑊) ∈ ℕ)))
39	38	3adant3 1081	. . . . . . . . . . . 12 ⊢ ((𝑖 ∈ ℕ0 ∧ (#‘𝑊) ∈ ℕ ∧ 𝑖 < (#‘𝑊)) → ((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑖 + 𝑁) ∈ ℤ ∧ (#‘𝑊) ∈ ℕ)))
40	30, 39	sylbi 207	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (0..^(#‘𝑊)) → ((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑖 + 𝑁) ∈ ℤ ∧ (#‘𝑊) ∈ ℕ)))
41	40	impcom 446	. . . . . . . . . 10 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑖 ∈ (0..^(#‘𝑊))) → ((𝑖 + 𝑁) ∈ ℤ ∧ (#‘𝑊) ∈ ℕ))
42		zmodfzo 12693	. . . . . . . . . 10 ⊢ (((𝑖 + 𝑁) ∈ ℤ ∧ (#‘𝑊) ∈ ℕ) → ((𝑖 + 𝑁) mod (#‘𝑊)) ∈ (0..^(#‘𝑊)))
43	41, 42	syl 17	. . . . . . . . 9 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑖 ∈ (0..^(#‘𝑊))) → ((𝑖 + 𝑁) mod (#‘𝑊)) ∈ (0..^(#‘𝑊)))
44	2	oveq2d 6666	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑖 + 𝑁) mod (#‘(𝑊 cyclShift 𝑀))) = ((𝑖 + 𝑁) mod (#‘𝑊)))
45	44	eleq1d 2686	. . . . . . . . . 10 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (((𝑖 + 𝑁) mod (#‘(𝑊 cyclShift 𝑀))) ∈ (0..^(#‘𝑊)) ↔ ((𝑖 + 𝑁) mod (#‘𝑊)) ∈ (0..^(#‘𝑊))))
46	45	adantr 481	. . . . . . . . 9 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑖 ∈ (0..^(#‘𝑊))) → (((𝑖 + 𝑁) mod (#‘(𝑊 cyclShift 𝑀))) ∈ (0..^(#‘𝑊)) ↔ ((𝑖 + 𝑁) mod (#‘𝑊)) ∈ (0..^(#‘𝑊))))
47	43, 46	mpbird 247	. . . . . . . 8 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑖 ∈ (0..^(#‘𝑊))) → ((𝑖 + 𝑁) mod (#‘(𝑊 cyclShift 𝑀))) ∈ (0..^(#‘𝑊)))
48		cshwidxmod 13549	. . . . . . . 8 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ 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))))
49	28, 29, 47, 48	syl3anc 1326	. . . . . . 7 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ 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 11301	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 ∈ ℕ0 → 𝑖 ∈ ℝ)
51	50	ad2antrr 762	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑖 ∈ ℕ0 ∧ (#‘𝑊) ∈ ℕ) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → 𝑖 ∈ ℝ)
52		zre 11381	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ)
53	52	ad2antll 765	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑖 ∈ ℕ0 ∧ (#‘𝑊) ∈ ℕ) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → 𝑁 ∈ ℝ)
54	51, 53	readdcld 10069	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑖 ∈ ℕ0 ∧ (#‘𝑊) ∈ ℕ) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (𝑖 + 𝑁) ∈ ℝ)
55		zre 11381	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ)
56	55	ad2antrl 764	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑖 ∈ ℕ0 ∧ (#‘𝑊) ∈ ℕ) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → 𝑀 ∈ ℝ)
57		nnrp 11842	. . . . . . . . . . . . . . . . . . 19 ⊢ ((#‘𝑊) ∈ ℕ → (#‘𝑊) ∈ ℝ+)
58	57	adantl 482	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 ∈ ℕ0 ∧ (#‘𝑊) ∈ ℕ) → (#‘𝑊) ∈ ℝ+)
59	58	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑖 ∈ ℕ0 ∧ (#‘𝑊) ∈ ℕ) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (#‘𝑊) ∈ ℝ+)
60		modaddmod 12709	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑖 + 𝑁) ∈ ℝ ∧ 𝑀 ∈ ℝ ∧ (#‘𝑊) ∈ ℝ+) → ((((𝑖 + 𝑁) mod (#‘𝑊)) + 𝑀) mod (#‘𝑊)) = (((𝑖 + 𝑁) + 𝑀) mod (#‘𝑊)))
61	54, 56, 59, 60	syl3anc 1326	. . . . . . . . . . . . . . . 16 ⊢ (((𝑖 ∈ ℕ0 ∧ (#‘𝑊) ∈ ℕ) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → ((((𝑖 + 𝑁) mod (#‘𝑊)) + 𝑀) mod (#‘𝑊)) = (((𝑖 + 𝑁) + 𝑀) mod (#‘𝑊)))
62		nn0cn 11302	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 ∈ ℕ0 → 𝑖 ∈ ℂ)
63	62	ad2antrr 762	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑖 ∈ ℕ0 ∧ (#‘𝑊) ∈ ℕ) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → 𝑖 ∈ ℂ)
64		zcn 11382	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℂ)
65	64	ad2antrl 764	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑖 ∈ ℕ0 ∧ (#‘𝑊) ∈ ℕ) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → 𝑀 ∈ ℂ)
66		zcn 11382	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ)
67	66	ad2antll 765	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑖 ∈ ℕ0 ∧ (#‘𝑊) ∈ ℕ) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → 𝑁 ∈ ℂ)
68		add32r 10255	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑖 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ) → (𝑖 + (𝑀 + 𝑁)) = ((𝑖 + 𝑁) + 𝑀))
69	63, 65, 67, 68	syl3anc 1326	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑖 ∈ ℕ0 ∧ (#‘𝑊) ∈ ℕ) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (𝑖 + (𝑀 + 𝑁)) = ((𝑖 + 𝑁) + 𝑀))
70	69	eqcomd 2628	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑖 ∈ ℕ0 ∧ (#‘𝑊) ∈ ℕ) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → ((𝑖 + 𝑁) + 𝑀) = (𝑖 + (𝑀 + 𝑁)))
71	70	oveq1d 6665	. . . . . . . . . . . . . . . 16 ⊢ (((𝑖 ∈ ℕ0 ∧ (#‘𝑊) ∈ ℕ) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (((𝑖 + 𝑁) + 𝑀) mod (#‘𝑊)) = ((𝑖 + (𝑀 + 𝑁)) mod (#‘𝑊)))
72	61, 71	eqtrd 2656	. . . . . . . . . . . . . . 15 ⊢ (((𝑖 ∈ ℕ0 ∧ (#‘𝑊) ∈ ℕ) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → ((((𝑖 + 𝑁) mod (#‘𝑊)) + 𝑀) mod (#‘𝑊)) = ((𝑖 + (𝑀 + 𝑁)) mod (#‘𝑊)))
73	72	ex 450	. . . . . . . . . . . . . 14 ⊢ ((𝑖 ∈ ℕ0 ∧ (#‘𝑊) ∈ ℕ) → ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((((𝑖 + 𝑁) mod (#‘𝑊)) + 𝑀) mod (#‘𝑊)) = ((𝑖 + (𝑀 + 𝑁)) mod (#‘𝑊))))
74	73	3adant3 1081	. . . . . . . . . . . . 13 ⊢ ((𝑖 ∈ ℕ0 ∧ (#‘𝑊) ∈ ℕ ∧ 𝑖 < (#‘𝑊)) → ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((((𝑖 + 𝑁) mod (#‘𝑊)) + 𝑀) mod (#‘𝑊)) = ((𝑖 + (𝑀 + 𝑁)) mod (#‘𝑊))))
75	30, 74	sylbi 207	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (0..^(#‘𝑊)) → ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((((𝑖 + 𝑁) mod (#‘𝑊)) + 𝑀) mod (#‘𝑊)) = ((𝑖 + (𝑀 + 𝑁)) mod (#‘𝑊))))
76	75	com12 32	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑖 ∈ (0..^(#‘𝑊)) → ((((𝑖 + 𝑁) mod (#‘𝑊)) + 𝑀) mod (#‘𝑊)) = ((𝑖 + (𝑀 + 𝑁)) mod (#‘𝑊))))
77	76	3adant1 1079	. . . . . . . . . 10 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑖 ∈ (0..^(#‘𝑊)) → ((((𝑖 + 𝑁) mod (#‘𝑊)) + 𝑀) mod (#‘𝑊)) = ((𝑖 + (𝑀 + 𝑁)) mod (#‘𝑊))))
78	77	imp 445	. . . . . . . . 9 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑖 ∈ (0..^(#‘𝑊))) → ((((𝑖 + 𝑁) mod (#‘𝑊)) + 𝑀) mod (#‘𝑊)) = ((𝑖 + (𝑀 + 𝑁)) mod (#‘𝑊)))
79	78	fveq2d 6195	. . . . . . . 8 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑖 ∈ (0..^(#‘𝑊))) → (𝑊‘((((𝑖 + 𝑁) mod (#‘𝑊)) + 𝑀) mod (#‘𝑊))) = (𝑊‘((𝑖 + (𝑀 + 𝑁)) mod (#‘𝑊))))
80	2	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑖 ∈ (0..^(#‘𝑊))) → (#‘(𝑊 cyclShift 𝑀)) = (#‘𝑊))
81	80	oveq2d 6666	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑖 ∈ (0..^(#‘𝑊))) → ((𝑖 + 𝑁) mod (#‘(𝑊 cyclShift 𝑀))) = ((𝑖 + 𝑁) mod (#‘𝑊)))
82	81	oveq1d 6665	. . . . . . . . . 10 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑖 ∈ (0..^(#‘𝑊))) → (((𝑖 + 𝑁) mod (#‘(𝑊 cyclShift 𝑀))) + 𝑀) = (((𝑖 + 𝑁) mod (#‘𝑊)) + 𝑀))
83	82	oveq1d 6665	. . . . . . . . 9 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑖 ∈ (0..^(#‘𝑊))) → ((((𝑖 + 𝑁) mod (#‘(𝑊 cyclShift 𝑀))) + 𝑀) mod (#‘𝑊)) = ((((𝑖 + 𝑁) mod (#‘𝑊)) + 𝑀) mod (#‘𝑊)))
84	83	fveq2d 6195	. . . . . . . 8 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑖 ∈ (0..^(#‘𝑊))) → (𝑊‘((((𝑖 + 𝑁) mod (#‘(𝑊 cyclShift 𝑀))) + 𝑀) mod (#‘𝑊))) = (𝑊‘((((𝑖 + 𝑁) mod (#‘𝑊)) + 𝑀) mod (#‘𝑊))))
85	10	adantr 481	. . . . . . . . 9 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑖 ∈ (0..^(#‘𝑊))) → (𝑀 + 𝑁) ∈ ℤ)
86		simpr 477	. . . . . . . . 9 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑖 ∈ (0..^(#‘𝑊))) → 𝑖 ∈ (0..^(#‘𝑊)))
87		cshwidxmod 13549	. . . . . . . . 9 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (𝑀 + 𝑁) ∈ ℤ ∧ 𝑖 ∈ (0..^(#‘𝑊))) → ((𝑊 cyclShift (𝑀 + 𝑁))‘𝑖) = (𝑊‘((𝑖 + (𝑀 + 𝑁)) mod (#‘𝑊))))
88	28, 85, 86, 87	syl3anc 1326	. . . . . . . 8 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑖 ∈ (0..^(#‘𝑊))) → ((𝑊 cyclShift (𝑀 + 𝑁))‘𝑖) = (𝑊‘((𝑖 + (𝑀 + 𝑁)) mod (#‘𝑊))))
89	79, 84, 88	3eqtr4d 2666	. . . . . . 7 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑖 ∈ (0..^(#‘𝑊))) → (𝑊‘((((𝑖 + 𝑁) mod (#‘(𝑊 cyclShift 𝑀))) + 𝑀) mod (#‘𝑊))) = ((𝑊 cyclShift (𝑀 + 𝑁))‘𝑖))
90	27, 49, 89	3eqtrd 2660	. . . . . 6 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑖 ∈ (0..^(#‘𝑊))) → (((𝑊 cyclShift 𝑀) cyclShift 𝑁)‘𝑖) = ((𝑊 cyclShift (𝑀 + 𝑁))‘𝑖))
91	90	ex 450	. . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑖 ∈ (0..^(#‘𝑊)) → (((𝑊 cyclShift 𝑀) cyclShift 𝑁)‘𝑖) = ((𝑊 cyclShift (𝑀 + 𝑁))‘𝑖)))
92	17, 91	sylbid 230	. . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑖 ∈ (0..^(#‘((𝑊 cyclShift 𝑀) cyclShift 𝑁))) → (((𝑊 cyclShift 𝑀) cyclShift 𝑁)‘𝑖) = ((𝑊 cyclShift (𝑀 + 𝑁))‘𝑖)))
93	92	ralrimiv 2965	. . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ∀𝑖 ∈ (0..^(#‘((𝑊 cyclShift 𝑀) cyclShift 𝑁)))(((𝑊 cyclShift 𝑀) cyclShift 𝑁)‘𝑖) = ((𝑊 cyclShift (𝑀 + 𝑁))‘𝑖))
94	14, 93	jca 554	. 2 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ 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 13544	. . . . . 6 ⊢ ((𝑊 cyclShift 𝑀) ∈ Word 𝑉 → ((𝑊 cyclShift 𝑀) cyclShift 𝑁) ∈ Word 𝑉)
96	3, 95	syl 17	. . . . 5 ⊢ (𝑊 ∈ Word 𝑉 → ((𝑊 cyclShift 𝑀) cyclShift 𝑁) ∈ Word 𝑉)
97		cshwcl 13544	. . . . 5 ⊢ (𝑊 ∈ Word 𝑉 → (𝑊 cyclShift (𝑀 + 𝑁)) ∈ Word 𝑉)
98	96, 97	jca 554	. . . 4 ⊢ (𝑊 ∈ Word 𝑉 → (((𝑊 cyclShift 𝑀) cyclShift 𝑁) ∈ Word 𝑉 ∧ (𝑊 cyclShift (𝑀 + 𝑁)) ∈ Word 𝑉))
99	98	3ad2ant1 1082	. . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (((𝑊 cyclShift 𝑀) cyclShift 𝑁) ∈ Word 𝑉 ∧ (𝑊 cyclShift (𝑀 + 𝑁)) ∈ Word 𝑉))
100		eqwrd 13346	. . 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))))
101	99, 100	syl 17	. 2 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ 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))))
102	94, 101	mpbird 247	1 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑊 cyclShift 𝑀) cyclShift 𝑁) = (𝑊 cyclShift (𝑀 + 𝑁)))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∀wral 2912   class class class wbr 4653  ‘cfv 5888  (class class class)co 6650  ℂcc 9934  ℝcr 9935  0cc0 9936   + caddc 9939   < clt 10074  ℕcn 11020  ℕ₀cn0 11292  ℤcz 11377  ℝ⁺crp 11832  ..^cfzo 12465   mod cmo 12668  #chash 13117  Word cword 13291   cyclShift ccsh 13534

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-hash 13118  df-word 13299  df-concat 13301  df-substr 13303  df-csh 13535

This theorem is referenced by:  2cshwid  13560  2cshwcom  13562  cshweqdif2  13565  2cshwcshw  13571  cshwcshid  13573  cshwcsh2id  13574  cshwshashlem2  15803