Theorem cshimadifsn0 13427

Description: The image of a cyclically shifted word under its domain without its upper bound is the image of a cyclically shifted word under its domain without the number of shifted symbols. (Contributed by AV, 19-Mar-2021.)

Assertion

Ref	Expression
cshimadifsn0	⊢ ((𝐹 ∈ Word 𝑆 ∧ 𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) → (𝐹 “ ((0..^𝑁) ∖ {𝐽})) = ((𝐹 cyclShift (𝐽 + 1)) “ (0..^(𝑁 − 1))))

Proof of Theorem cshimadifsn0

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cshimadifsn 13426	. 2 ⊢ ((𝐹 ∈ Word 𝑆 ∧ 𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) → (𝐹 “ ((0..^𝑁) ∖ {𝐽})) = ((𝐹 cyclShift 𝐽) “ (1..^𝑁)))
2		elfzoel2 12338	. . . . . . . 8 ⊢ (𝐽 ∈ (0..^𝑁) → 𝑁 ∈ ℤ)
3		elfzom1elp1fzo1 12434	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℤ ∧ 𝑦 ∈ (0..^(𝑁 − 1))) → (𝑦 + 1) ∈ (1..^𝑁))
4	3	ex 449	. . . . . . . 8 ⊢ (𝑁 ∈ ℤ → (𝑦 ∈ (0..^(𝑁 − 1)) → (𝑦 + 1) ∈ (1..^𝑁)))
5	2, 4	syl 17	. . . . . . 7 ⊢ (𝐽 ∈ (0..^𝑁) → (𝑦 ∈ (0..^(𝑁 − 1)) → (𝑦 + 1) ∈ (1..^𝑁)))
6	5	3ad2ant3 1077	. . . . . 6 ⊢ ((𝐹 ∈ Word 𝑆 ∧ 𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) → (𝑦 ∈ (0..^(𝑁 − 1)) → (𝑦 + 1) ∈ (1..^𝑁)))
7	6	imp 444	. . . . 5 ⊢ (((𝐹 ∈ Word 𝑆 ∧ 𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ 𝑦 ∈ (0..^(𝑁 − 1))) → (𝑦 + 1) ∈ (1..^𝑁))
8		elfzo1elm1fzo0 12435	. . . . . . 7 ⊢ (𝑥 ∈ (1..^𝑁) → (𝑥 − 1) ∈ (0..^(𝑁 − 1)))
9	8	adantl 481	. . . . . 6 ⊢ (((𝐹 ∈ Word 𝑆 ∧ 𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ 𝑥 ∈ (1..^𝑁)) → (𝑥 − 1) ∈ (0..^(𝑁 − 1)))
10		oveq1 6556	. . . . . . . 8 ⊢ (𝑦 = (𝑥 − 1) → (𝑦 + 1) = ((𝑥 − 1) + 1))
11	10	eqeq2d 2620	. . . . . . 7 ⊢ (𝑦 = (𝑥 − 1) → (𝑥 = (𝑦 + 1) ↔ 𝑥 = ((𝑥 − 1) + 1)))
12	11	adantl 481	. . . . . 6 ⊢ ((((𝐹 ∈ Word 𝑆 ∧ 𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ 𝑥 ∈ (1..^𝑁)) ∧ 𝑦 = (𝑥 − 1)) → (𝑥 = (𝑦 + 1) ↔ 𝑥 = ((𝑥 − 1) + 1)))
13		elfzoelz 12339	. . . . . . . . . 10 ⊢ (𝑥 ∈ (1..^𝑁) → 𝑥 ∈ ℤ)
14	13	zcnd 11359	. . . . . . . . 9 ⊢ (𝑥 ∈ (1..^𝑁) → 𝑥 ∈ ℂ)
15		npcan1 10334	. . . . . . . . 9 ⊢ (𝑥 ∈ ℂ → ((𝑥 − 1) + 1) = 𝑥)
16	14, 15	syl 17	. . . . . . . 8 ⊢ (𝑥 ∈ (1..^𝑁) → ((𝑥 − 1) + 1) = 𝑥)
17	16	eqcomd 2616	. . . . . . 7 ⊢ (𝑥 ∈ (1..^𝑁) → 𝑥 = ((𝑥 − 1) + 1))
18	17	adantl 481	. . . . . 6 ⊢ (((𝐹 ∈ Word 𝑆 ∧ 𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ 𝑥 ∈ (1..^𝑁)) → 𝑥 = ((𝑥 − 1) + 1))
19	9, 12, 18	rspcedvd 3289	. . . . 5 ⊢ (((𝐹 ∈ Word 𝑆 ∧ 𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ 𝑥 ∈ (1..^𝑁)) → ∃𝑦 ∈ (0..^(𝑁 − 1))𝑥 = (𝑦 + 1))
20		fveq2 6103	. . . . . . . 8 ⊢ (𝑥 = (𝑦 + 1) → ((𝐹 cyclShift 𝐽)‘𝑥) = ((𝐹 cyclShift 𝐽)‘(𝑦 + 1)))
21	20	3ad2ant3 1077	. . . . . . 7 ⊢ (((𝐹 ∈ Word 𝑆 ∧ 𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ 𝑦 ∈ (0..^(𝑁 − 1)) ∧ 𝑥 = (𝑦 + 1)) → ((𝐹 cyclShift 𝐽)‘𝑥) = ((𝐹 cyclShift 𝐽)‘(𝑦 + 1)))
22		elfzoelz 12339	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (0..^(𝑁 − 1)) → 𝑦 ∈ ℤ)
23	22	zcnd 11359	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (0..^(𝑁 − 1)) → 𝑦 ∈ ℂ)
24	23	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ Word 𝑆 ∧ 𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ 𝑦 ∈ (0..^(𝑁 − 1))) → 𝑦 ∈ ℂ)
25		elfzoelz 12339	. . . . . . . . . . . . . . 15 ⊢ (𝐽 ∈ (0..^𝑁) → 𝐽 ∈ ℤ)
26	25	zcnd 11359	. . . . . . . . . . . . . 14 ⊢ (𝐽 ∈ (0..^𝑁) → 𝐽 ∈ ℂ)
27	26	3ad2ant3 1077	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ Word 𝑆 ∧ 𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) → 𝐽 ∈ ℂ)
28	27	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ Word 𝑆 ∧ 𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ 𝑦 ∈ (0..^(𝑁 − 1))) → 𝐽 ∈ ℂ)
29		1cnd 9935	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ Word 𝑆 ∧ 𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ 𝑦 ∈ (0..^(𝑁 − 1))) → 1 ∈ ℂ)
30		add32r 10134	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℂ ∧ 𝐽 ∈ ℂ ∧ 1 ∈ ℂ) → (𝑦 + (𝐽 + 1)) = ((𝑦 + 1) + 𝐽))
31	24, 28, 29, 30	syl3anc 1318	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ Word 𝑆 ∧ 𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ 𝑦 ∈ (0..^(𝑁 − 1))) → (𝑦 + (𝐽 + 1)) = ((𝑦 + 1) + 𝐽))
32	31	oveq1d 6564	. . . . . . . . . 10 ⊢ (((𝐹 ∈ Word 𝑆 ∧ 𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ 𝑦 ∈ (0..^(𝑁 − 1))) → ((𝑦 + (𝐽 + 1)) mod (#‘𝐹)) = (((𝑦 + 1) + 𝐽) mod (#‘𝐹)))
33	32	fveq2d 6107	. . . . . . . . 9 ⊢ (((𝐹 ∈ Word 𝑆 ∧ 𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ 𝑦 ∈ (0..^(𝑁 − 1))) → (𝐹‘((𝑦 + (𝐽 + 1)) mod (#‘𝐹))) = (𝐹‘(((𝑦 + 1) + 𝐽) mod (#‘𝐹))))
34		simpl1 1057	. . . . . . . . . 10 ⊢ (((𝐹 ∈ Word 𝑆 ∧ 𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ 𝑦 ∈ (0..^(𝑁 − 1))) → 𝐹 ∈ Word 𝑆)
35	25	peano2zd 11361	. . . . . . . . . . . 12 ⊢ (𝐽 ∈ (0..^𝑁) → (𝐽 + 1) ∈ ℤ)
36	35	3ad2ant3 1077	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ Word 𝑆 ∧ 𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) → (𝐽 + 1) ∈ ℤ)
37	36	adantr 480	. . . . . . . . . 10 ⊢ (((𝐹 ∈ Word 𝑆 ∧ 𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ 𝑦 ∈ (0..^(𝑁 − 1))) → (𝐽 + 1) ∈ ℤ)
38		fzossrbm1 12366	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ℤ → (0..^(𝑁 − 1)) ⊆ (0..^𝑁))
39	2, 38	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝐽 ∈ (0..^𝑁) → (0..^(𝑁 − 1)) ⊆ (0..^𝑁))
40	39	sseld 3567	. . . . . . . . . . . . 13 ⊢ (𝐽 ∈ (0..^𝑁) → (𝑦 ∈ (0..^(𝑁 − 1)) → 𝑦 ∈ (0..^𝑁)))
41	40	3ad2ant3 1077	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ Word 𝑆 ∧ 𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) → (𝑦 ∈ (0..^(𝑁 − 1)) → 𝑦 ∈ (0..^𝑁)))
42	41	imp 444	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ Word 𝑆 ∧ 𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ 𝑦 ∈ (0..^(𝑁 − 1))) → 𝑦 ∈ (0..^𝑁))
43		oveq2 6557	. . . . . . . . . . . . . 14 ⊢ (𝑁 = (#‘𝐹) → (0..^𝑁) = (0..^(#‘𝐹)))
44	43	eleq2d 2673	. . . . . . . . . . . . 13 ⊢ (𝑁 = (#‘𝐹) → (𝑦 ∈ (0..^𝑁) ↔ 𝑦 ∈ (0..^(#‘𝐹))))
45	44	3ad2ant2 1076	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ Word 𝑆 ∧ 𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) → (𝑦 ∈ (0..^𝑁) ↔ 𝑦 ∈ (0..^(#‘𝐹))))
46	45	adantr 480	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ Word 𝑆 ∧ 𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ 𝑦 ∈ (0..^(𝑁 − 1))) → (𝑦 ∈ (0..^𝑁) ↔ 𝑦 ∈ (0..^(#‘𝐹))))
47	42, 46	mpbid 221	. . . . . . . . . 10 ⊢ (((𝐹 ∈ Word 𝑆 ∧ 𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ 𝑦 ∈ (0..^(𝑁 − 1))) → 𝑦 ∈ (0..^(#‘𝐹)))
48		cshwidxmod 13400	. . . . . . . . . 10 ⊢ ((𝐹 ∈ Word 𝑆 ∧ (𝐽 + 1) ∈ ℤ ∧ 𝑦 ∈ (0..^(#‘𝐹))) → ((𝐹 cyclShift (𝐽 + 1))‘𝑦) = (𝐹‘((𝑦 + (𝐽 + 1)) mod (#‘𝐹))))
49	34, 37, 47, 48	syl3anc 1318	. . . . . . . . 9 ⊢ (((𝐹 ∈ Word 𝑆 ∧ 𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ 𝑦 ∈ (0..^(𝑁 − 1))) → ((𝐹 cyclShift (𝐽 + 1))‘𝑦) = (𝐹‘((𝑦 + (𝐽 + 1)) mod (#‘𝐹))))
50	25	3ad2ant3 1077	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ Word 𝑆 ∧ 𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) → 𝐽 ∈ ℤ)
51	50	adantr 480	. . . . . . . . . 10 ⊢ (((𝐹 ∈ Word 𝑆 ∧ 𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ 𝑦 ∈ (0..^(𝑁 − 1))) → 𝐽 ∈ ℤ)
52		fzo0ss1 12367	. . . . . . . . . . . 12 ⊢ (1..^𝑁) ⊆ (0..^𝑁)
53	2	3ad2ant3 1077	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ Word 𝑆 ∧ 𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) → 𝑁 ∈ ℤ)
54	53, 3	sylan 487	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ Word 𝑆 ∧ 𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ 𝑦 ∈ (0..^(𝑁 − 1))) → (𝑦 + 1) ∈ (1..^𝑁))
55	52, 54	sseldi 3566	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ Word 𝑆 ∧ 𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ 𝑦 ∈ (0..^(𝑁 − 1))) → (𝑦 + 1) ∈ (0..^𝑁))
56	43	eleq2d 2673	. . . . . . . . . . . . 13 ⊢ (𝑁 = (#‘𝐹) → ((𝑦 + 1) ∈ (0..^𝑁) ↔ (𝑦 + 1) ∈ (0..^(#‘𝐹))))
57	56	3ad2ant2 1076	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ Word 𝑆 ∧ 𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) → ((𝑦 + 1) ∈ (0..^𝑁) ↔ (𝑦 + 1) ∈ (0..^(#‘𝐹))))
58	57	adantr 480	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ Word 𝑆 ∧ 𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ 𝑦 ∈ (0..^(𝑁 − 1))) → ((𝑦 + 1) ∈ (0..^𝑁) ↔ (𝑦 + 1) ∈ (0..^(#‘𝐹))))
59	55, 58	mpbid 221	. . . . . . . . . 10 ⊢ (((𝐹 ∈ Word 𝑆 ∧ 𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ 𝑦 ∈ (0..^(𝑁 − 1))) → (𝑦 + 1) ∈ (0..^(#‘𝐹)))
60		cshwidxmod 13400	. . . . . . . . . 10 ⊢ ((𝐹 ∈ Word 𝑆 ∧ 𝐽 ∈ ℤ ∧ (𝑦 + 1) ∈ (0..^(#‘𝐹))) → ((𝐹 cyclShift 𝐽)‘(𝑦 + 1)) = (𝐹‘(((𝑦 + 1) + 𝐽) mod (#‘𝐹))))
61	34, 51, 59, 60	syl3anc 1318	. . . . . . . . 9 ⊢ (((𝐹 ∈ Word 𝑆 ∧ 𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ 𝑦 ∈ (0..^(𝑁 − 1))) → ((𝐹 cyclShift 𝐽)‘(𝑦 + 1)) = (𝐹‘(((𝑦 + 1) + 𝐽) mod (#‘𝐹))))
62	33, 49, 61	3eqtr4rd 2655	. . . . . . . 8 ⊢ (((𝐹 ∈ Word 𝑆 ∧ 𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ 𝑦 ∈ (0..^(𝑁 − 1))) → ((𝐹 cyclShift 𝐽)‘(𝑦 + 1)) = ((𝐹 cyclShift (𝐽 + 1))‘𝑦))
63	62	3adant3 1074	. . . . . . 7 ⊢ (((𝐹 ∈ Word 𝑆 ∧ 𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ 𝑦 ∈ (0..^(𝑁 − 1)) ∧ 𝑥 = (𝑦 + 1)) → ((𝐹 cyclShift 𝐽)‘(𝑦 + 1)) = ((𝐹 cyclShift (𝐽 + 1))‘𝑦))
64	21, 63	eqtrd 2644	. . . . . 6 ⊢ (((𝐹 ∈ Word 𝑆 ∧ 𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ 𝑦 ∈ (0..^(𝑁 − 1)) ∧ 𝑥 = (𝑦 + 1)) → ((𝐹 cyclShift 𝐽)‘𝑥) = ((𝐹 cyclShift (𝐽 + 1))‘𝑦))
65	64	eqeq1d 2612	. . . . 5 ⊢ (((𝐹 ∈ Word 𝑆 ∧ 𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ 𝑦 ∈ (0..^(𝑁 − 1)) ∧ 𝑥 = (𝑦 + 1)) → (((𝐹 cyclShift 𝐽)‘𝑥) = 𝑧 ↔ ((𝐹 cyclShift (𝐽 + 1))‘𝑦) = 𝑧))
66	7, 19, 65	rexxfrd2 4811	. . . 4 ⊢ ((𝐹 ∈ Word 𝑆 ∧ 𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) → (∃𝑥 ∈ (1..^𝑁)((𝐹 cyclShift 𝐽)‘𝑥) = 𝑧 ↔ ∃𝑦 ∈ (0..^(𝑁 − 1))((𝐹 cyclShift (𝐽 + 1))‘𝑦) = 𝑧))
67	66	abbidv 2728	. . 3 ⊢ ((𝐹 ∈ Word 𝑆 ∧ 𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) → {𝑧 ∣ ∃𝑥 ∈ (1..^𝑁)((𝐹 cyclShift 𝐽)‘𝑥) = 𝑧} = {𝑧 ∣ ∃𝑦 ∈ (0..^(𝑁 − 1))((𝐹 cyclShift (𝐽 + 1))‘𝑦) = 𝑧})
68	25	anim2i 591	. . . . . . 7 ⊢ ((𝐹 ∈ Word 𝑆 ∧ 𝐽 ∈ (0..^𝑁)) → (𝐹 ∈ Word 𝑆 ∧ 𝐽 ∈ ℤ))
69	68	3adant2 1073	. . . . . 6 ⊢ ((𝐹 ∈ Word 𝑆 ∧ 𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) → (𝐹 ∈ Word 𝑆 ∧ 𝐽 ∈ ℤ))
70		cshwfn 13398	. . . . . 6 ⊢ ((𝐹 ∈ Word 𝑆 ∧ 𝐽 ∈ ℤ) → (𝐹 cyclShift 𝐽) Fn (0..^(#‘𝐹)))
71	69, 70	syl 17	. . . . 5 ⊢ ((𝐹 ∈ Word 𝑆 ∧ 𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) → (𝐹 cyclShift 𝐽) Fn (0..^(#‘𝐹)))
72		fnfun 5902	. . . . . . 7 ⊢ ((𝐹 cyclShift 𝐽) Fn (0..^(#‘𝐹)) → Fun (𝐹 cyclShift 𝐽))
73	72	adantl 481	. . . . . 6 ⊢ (((𝐹 ∈ Word 𝑆 ∧ 𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ (𝐹 cyclShift 𝐽) Fn (0..^(#‘𝐹))) → Fun (𝐹 cyclShift 𝐽))
74	43	3ad2ant2 1076	. . . . . . . . 9 ⊢ ((𝐹 ∈ Word 𝑆 ∧ 𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) → (0..^𝑁) = (0..^(#‘𝐹)))
75	52, 74	syl5sseq 3616	. . . . . . . 8 ⊢ ((𝐹 ∈ Word 𝑆 ∧ 𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) → (1..^𝑁) ⊆ (0..^(#‘𝐹)))
76	75	adantr 480	. . . . . . 7 ⊢ (((𝐹 ∈ Word 𝑆 ∧ 𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ (𝐹 cyclShift 𝐽) Fn (0..^(#‘𝐹))) → (1..^𝑁) ⊆ (0..^(#‘𝐹)))
77		fndm 5904	. . . . . . . 8 ⊢ ((𝐹 cyclShift 𝐽) Fn (0..^(#‘𝐹)) → dom (𝐹 cyclShift 𝐽) = (0..^(#‘𝐹)))
78	77	adantl 481	. . . . . . 7 ⊢ (((𝐹 ∈ Word 𝑆 ∧ 𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ (𝐹 cyclShift 𝐽) Fn (0..^(#‘𝐹))) → dom (𝐹 cyclShift 𝐽) = (0..^(#‘𝐹)))
79	76, 78	sseqtr4d 3605	. . . . . 6 ⊢ (((𝐹 ∈ Word 𝑆 ∧ 𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ (𝐹 cyclShift 𝐽) Fn (0..^(#‘𝐹))) → (1..^𝑁) ⊆ dom (𝐹 cyclShift 𝐽))
80	73, 79	jca 553	. . . . 5 ⊢ (((𝐹 ∈ Word 𝑆 ∧ 𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ (𝐹 cyclShift 𝐽) Fn (0..^(#‘𝐹))) → (Fun (𝐹 cyclShift 𝐽) ∧ (1..^𝑁) ⊆ dom (𝐹 cyclShift 𝐽)))
81	71, 80	mpdan 699	. . . 4 ⊢ ((𝐹 ∈ Word 𝑆 ∧ 𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) → (Fun (𝐹 cyclShift 𝐽) ∧ (1..^𝑁) ⊆ dom (𝐹 cyclShift 𝐽)))
82		dfimafn 6155	. . . 4 ⊢ ((Fun (𝐹 cyclShift 𝐽) ∧ (1..^𝑁) ⊆ dom (𝐹 cyclShift 𝐽)) → ((𝐹 cyclShift 𝐽) “ (1..^𝑁)) = {𝑧 ∣ ∃𝑥 ∈ (1..^𝑁)((𝐹 cyclShift 𝐽)‘𝑥) = 𝑧})
83	81, 82	syl 17	. . 3 ⊢ ((𝐹 ∈ Word 𝑆 ∧ 𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) → ((𝐹 cyclShift 𝐽) “ (1..^𝑁)) = {𝑧 ∣ ∃𝑥 ∈ (1..^𝑁)((𝐹 cyclShift 𝐽)‘𝑥) = 𝑧})
84	35	anim2i 591	. . . . . . 7 ⊢ ((𝐹 ∈ Word 𝑆 ∧ 𝐽 ∈ (0..^𝑁)) → (𝐹 ∈ Word 𝑆 ∧ (𝐽 + 1) ∈ ℤ))
85	84	3adant2 1073	. . . . . 6 ⊢ ((𝐹 ∈ Word 𝑆 ∧ 𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) → (𝐹 ∈ Word 𝑆 ∧ (𝐽 + 1) ∈ ℤ))
86		cshwfn 13398	. . . . . 6 ⊢ ((𝐹 ∈ Word 𝑆 ∧ (𝐽 + 1) ∈ ℤ) → (𝐹 cyclShift (𝐽 + 1)) Fn (0..^(#‘𝐹)))
87	85, 86	syl 17	. . . . 5 ⊢ ((𝐹 ∈ Word 𝑆 ∧ 𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) → (𝐹 cyclShift (𝐽 + 1)) Fn (0..^(#‘𝐹)))
88		fnfun 5902	. . . . . . 7 ⊢ ((𝐹 cyclShift (𝐽 + 1)) Fn (0..^(#‘𝐹)) → Fun (𝐹 cyclShift (𝐽 + 1)))
89	88	adantl 481	. . . . . 6 ⊢ (((𝐹 ∈ Word 𝑆 ∧ 𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ (𝐹 cyclShift (𝐽 + 1)) Fn (0..^(#‘𝐹))) → Fun (𝐹 cyclShift (𝐽 + 1)))
90	39	3ad2ant3 1077	. . . . . . . . 9 ⊢ ((𝐹 ∈ Word 𝑆 ∧ 𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) → (0..^(𝑁 − 1)) ⊆ (0..^𝑁))
91		oveq2 6557	. . . . . . . . . . 11 ⊢ ((#‘𝐹) = 𝑁 → (0..^(#‘𝐹)) = (0..^𝑁))
92	91	eqcoms 2618	. . . . . . . . . 10 ⊢ (𝑁 = (#‘𝐹) → (0..^(#‘𝐹)) = (0..^𝑁))
93	92	3ad2ant2 1076	. . . . . . . . 9 ⊢ ((𝐹 ∈ Word 𝑆 ∧ 𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) → (0..^(#‘𝐹)) = (0..^𝑁))
94	90, 93	sseqtr4d 3605	. . . . . . . 8 ⊢ ((𝐹 ∈ Word 𝑆 ∧ 𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) → (0..^(𝑁 − 1)) ⊆ (0..^(#‘𝐹)))
95	94	adantr 480	. . . . . . 7 ⊢ (((𝐹 ∈ Word 𝑆 ∧ 𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ (𝐹 cyclShift (𝐽 + 1)) Fn (0..^(#‘𝐹))) → (0..^(𝑁 − 1)) ⊆ (0..^(#‘𝐹)))
96		fndm 5904	. . . . . . . 8 ⊢ ((𝐹 cyclShift (𝐽 + 1)) Fn (0..^(#‘𝐹)) → dom (𝐹 cyclShift (𝐽 + 1)) = (0..^(#‘𝐹)))
97	96	adantl 481	. . . . . . 7 ⊢ (((𝐹 ∈ Word 𝑆 ∧ 𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ (𝐹 cyclShift (𝐽 + 1)) Fn (0..^(#‘𝐹))) → dom (𝐹 cyclShift (𝐽 + 1)) = (0..^(#‘𝐹)))
98	95, 97	sseqtr4d 3605	. . . . . 6 ⊢ (((𝐹 ∈ Word 𝑆 ∧ 𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ (𝐹 cyclShift (𝐽 + 1)) Fn (0..^(#‘𝐹))) → (0..^(𝑁 − 1)) ⊆ dom (𝐹 cyclShift (𝐽 + 1)))
99	89, 98	jca 553	. . . . 5 ⊢ (((𝐹 ∈ Word 𝑆 ∧ 𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ (𝐹 cyclShift (𝐽 + 1)) Fn (0..^(#‘𝐹))) → (Fun (𝐹 cyclShift (𝐽 + 1)) ∧ (0..^(𝑁 − 1)) ⊆ dom (𝐹 cyclShift (𝐽 + 1))))
100	87, 99	mpdan 699	. . . 4 ⊢ ((𝐹 ∈ Word 𝑆 ∧ 𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) → (Fun (𝐹 cyclShift (𝐽 + 1)) ∧ (0..^(𝑁 − 1)) ⊆ dom (𝐹 cyclShift (𝐽 + 1))))
101		dfimafn 6155	. . . 4 ⊢ ((Fun (𝐹 cyclShift (𝐽 + 1)) ∧ (0..^(𝑁 − 1)) ⊆ dom (𝐹 cyclShift (𝐽 + 1))) → ((𝐹 cyclShift (𝐽 + 1)) “ (0..^(𝑁 − 1))) = {𝑧 ∣ ∃𝑦 ∈ (0..^(𝑁 − 1))((𝐹 cyclShift (𝐽 + 1))‘𝑦) = 𝑧})
102	100, 101	syl 17	. . 3 ⊢ ((𝐹 ∈ Word 𝑆 ∧ 𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) → ((𝐹 cyclShift (𝐽 + 1)) “ (0..^(𝑁 − 1))) = {𝑧 ∣ ∃𝑦 ∈ (0..^(𝑁 − 1))((𝐹 cyclShift (𝐽 + 1))‘𝑦) = 𝑧})
103	67, 83, 102	3eqtr4d 2654	. 2 ⊢ ((𝐹 ∈ Word 𝑆 ∧ 𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) → ((𝐹 cyclShift 𝐽) “ (1..^𝑁)) = ((𝐹 cyclShift (𝐽 + 1)) “ (0..^(𝑁 − 1))))
104	1, 103	eqtrd 2644	1 ⊢ ((𝐹 ∈ Word 𝑆 ∧ 𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) → (𝐹 “ ((0..^𝑁) ∖ {𝐽})) = ((𝐹 cyclShift (𝐽 + 1)) “ (0..^(𝑁 − 1))))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  {cab 2596  ∃wrex 2897   ∖ cdif 3537   ⊆ wss 3540  {csn 4125  dom cdm 5038   “ cima 5041  Fun wfun 5798   Fn wfn 5799  ‘cfv 5804  (class class class)co 6549  ℂcc 9813  0cc0 9815  1c1 9816   + caddc 9818   − cmin 10145  ℤcz 11254  ..^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-ico 12052  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:  eucrct2eupth  41413