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Theorem cshweqdif2 14172

Description: If cyclically shifting two words (of the same length) results in the same word, cyclically shifting one of the words by the difference of the numbers of shifts results in the other word. (Contributed by AV, 21-Apr-2018.) (Revised by AV, 6-Jun-2018.) (Revised by AV, 1-Nov-2018.)

**Assertion**
Ref	Expression
cshweqdif2	⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → ((𝑊 cyclShift 𝑁) = (𝑈 cyclShift 𝑀) → (𝑈 cyclShift (𝑀 − 𝑁)) = 𝑊))

**Proof of Theorem cshweqdif2**
Step	Hyp	Ref	Expression
1		simpr 488	. . . . . . . . 9 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) → 𝑈 ∈ Word 𝑉)
2	1	adantr 484	. . . . . . . 8 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → 𝑈 ∈ Word 𝑉)
3		zsubcl 12012	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 − 𝑁) ∈ ℤ)
4	3	ancoms 462	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑀 − 𝑁) ∈ ℤ)
5	4	adantl 485	. . . . . . . 8 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → (𝑀 − 𝑁) ∈ ℤ)
6		simpr 488	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → 𝑀 ∈ ℤ)
7	6	adantl 485	. . . . . . . 8 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → 𝑀 ∈ ℤ)
8	2, 5, 7	3jca 1125	. . . . . . 7 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → (𝑈 ∈ Word 𝑉 ∧ (𝑀 − 𝑁) ∈ ℤ ∧ 𝑀 ∈ ℤ))
9	8	adantr 484	. . . . . 6 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑊 cyclShift 𝑁) = (𝑈 cyclShift 𝑀)) → (𝑈 ∈ Word 𝑉 ∧ (𝑀 − 𝑁) ∈ ℤ ∧ 𝑀 ∈ ℤ))
10		3cshw 14171	. . . . . 6 ⊢ ((𝑈 ∈ Word 𝑉 ∧ (𝑀 − 𝑁) ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑈 cyclShift (𝑀 − 𝑁)) = (((𝑈 cyclShift 𝑀) cyclShift (𝑀 − 𝑁)) cyclShift ((♯‘𝑈) − 𝑀)))
11	9, 10	syl 17	. . . . 5 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑊 cyclShift 𝑁) = (𝑈 cyclShift 𝑀)) → (𝑈 cyclShift (𝑀 − 𝑁)) = (((𝑈 cyclShift 𝑀) cyclShift (𝑀 − 𝑁)) cyclShift ((♯‘𝑈) − 𝑀)))
12		simpl 486	. . . . . . . . . 10 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → (𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉))
13	12	ancomd 465	. . . . . . . . 9 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → (𝑈 ∈ Word 𝑉 ∧ 𝑊 ∈ Word 𝑉))
14	13	adantr 484	. . . . . . . 8 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑊 cyclShift 𝑁) = (𝑈 cyclShift 𝑀)) → (𝑈 ∈ Word 𝑉 ∧ 𝑊 ∈ Word 𝑉))
15		simpr 488	. . . . . . . . . 10 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ))
16	15	ancomd 465	. . . . . . . . 9 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ))
17	16	adantr 484	. . . . . . . 8 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑊 cyclShift 𝑁) = (𝑈 cyclShift 𝑀)) → (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ))
18		simpr 488	. . . . . . . . 9 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑊 cyclShift 𝑁) = (𝑈 cyclShift 𝑀)) → (𝑊 cyclShift 𝑁) = (𝑈 cyclShift 𝑀))
19	18	eqcomd 2804	. . . . . . . 8 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑊 cyclShift 𝑁) = (𝑈 cyclShift 𝑀)) → (𝑈 cyclShift 𝑀) = (𝑊 cyclShift 𝑁))
20		cshwleneq 14170	. . . . . . . 8 ⊢ (((𝑈 ∈ Word 𝑉 ∧ 𝑊 ∈ Word 𝑉) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑈 cyclShift 𝑀) = (𝑊 cyclShift 𝑁)) → (♯‘𝑈) = (♯‘𝑊))
21	14, 17, 19, 20	syl3anc 1368	. . . . . . 7 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑊 cyclShift 𝑁) = (𝑈 cyclShift 𝑀)) → (♯‘𝑈) = (♯‘𝑊))
22	21	oveq1d 7150	. . . . . 6 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑊 cyclShift 𝑁) = (𝑈 cyclShift 𝑀)) → ((♯‘𝑈) − 𝑀) = ((♯‘𝑊) − 𝑀))
23	22	oveq2d 7151	. . . . 5 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑊 cyclShift 𝑁) = (𝑈 cyclShift 𝑀)) → (((𝑈 cyclShift 𝑀) cyclShift (𝑀 − 𝑁)) cyclShift ((♯‘𝑈) − 𝑀)) = (((𝑈 cyclShift 𝑀) cyclShift (𝑀 − 𝑁)) cyclShift ((♯‘𝑊) − 𝑀)))
24	11, 23	eqtrd 2833	. . . 4 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑊 cyclShift 𝑁) = (𝑈 cyclShift 𝑀)) → (𝑈 cyclShift (𝑀 − 𝑁)) = (((𝑈 cyclShift 𝑀) cyclShift (𝑀 − 𝑁)) cyclShift ((♯‘𝑊) − 𝑀)))
25	19	oveq1d 7150	. . . . . 6 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑊 cyclShift 𝑁) = (𝑈 cyclShift 𝑀)) → ((𝑈 cyclShift 𝑀) cyclShift (𝑀 − 𝑁)) = ((𝑊 cyclShift 𝑁) cyclShift (𝑀 − 𝑁)))
26		simpl 486	. . . . . . . . . 10 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) → 𝑊 ∈ Word 𝑉)
27	26	adantr 484	. . . . . . . . 9 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → 𝑊 ∈ Word 𝑉)
28		simpl 486	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → 𝑁 ∈ ℤ)
29	28	adantl 485	. . . . . . . . 9 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → 𝑁 ∈ ℤ)
30	27, 29, 5	3jca 1125	. . . . . . . 8 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → (𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℤ ∧ (𝑀 − 𝑁) ∈ ℤ))
31	30	adantr 484	. . . . . . 7 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑊 cyclShift 𝑁) = (𝑈 cyclShift 𝑀)) → (𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℤ ∧ (𝑀 − 𝑁) ∈ ℤ))
32		2cshw 14166	. . . . . . 7 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℤ ∧ (𝑀 − 𝑁) ∈ ℤ) → ((𝑊 cyclShift 𝑁) cyclShift (𝑀 − 𝑁)) = (𝑊 cyclShift (𝑁 + (𝑀 − 𝑁))))
33	31, 32	syl 17	. . . . . 6 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑊 cyclShift 𝑁) = (𝑈 cyclShift 𝑀)) → ((𝑊 cyclShift 𝑁) cyclShift (𝑀 − 𝑁)) = (𝑊 cyclShift (𝑁 + (𝑀 − 𝑁))))
34		zcn 11974	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ)
35		zcn 11974	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℂ)
36	34, 35	anim12i 615	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑁 ∈ ℂ ∧ 𝑀 ∈ ℂ))
37	36	adantl 485	. . . . . . . . 9 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → (𝑁 ∈ ℂ ∧ 𝑀 ∈ ℂ))
38	37	adantr 484	. . . . . . . 8 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑊 cyclShift 𝑁) = (𝑈 cyclShift 𝑀)) → (𝑁 ∈ ℂ ∧ 𝑀 ∈ ℂ))
39		pncan3 10883	. . . . . . . 8 ⊢ ((𝑁 ∈ ℂ ∧ 𝑀 ∈ ℂ) → (𝑁 + (𝑀 − 𝑁)) = 𝑀)
40	38, 39	syl 17	. . . . . . 7 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑊 cyclShift 𝑁) = (𝑈 cyclShift 𝑀)) → (𝑁 + (𝑀 − 𝑁)) = 𝑀)
41	40	oveq2d 7151	. . . . . 6 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑊 cyclShift 𝑁) = (𝑈 cyclShift 𝑀)) → (𝑊 cyclShift (𝑁 + (𝑀 − 𝑁))) = (𝑊 cyclShift 𝑀))
42	25, 33, 41	3eqtrd 2837	. . . . 5 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑊 cyclShift 𝑁) = (𝑈 cyclShift 𝑀)) → ((𝑈 cyclShift 𝑀) cyclShift (𝑀 − 𝑁)) = (𝑊 cyclShift 𝑀))
43	42	oveq1d 7150	. . . 4 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑊 cyclShift 𝑁) = (𝑈 cyclShift 𝑀)) → (((𝑈 cyclShift 𝑀) cyclShift (𝑀 − 𝑁)) cyclShift ((♯‘𝑊) − 𝑀)) = ((𝑊 cyclShift 𝑀) cyclShift ((♯‘𝑊) − 𝑀)))
44		lencl 13876	. . . . . . . . . 10 ⊢ (𝑊 ∈ Word 𝑉 → (♯‘𝑊) ∈ ℕ₀)
45	44	nn0zd 12073	. . . . . . . . 9 ⊢ (𝑊 ∈ Word 𝑉 → (♯‘𝑊) ∈ ℤ)
46	45	adantr 484	. . . . . . . 8 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) → (♯‘𝑊) ∈ ℤ)
47		zsubcl 12012	. . . . . . . 8 ⊢ (((♯‘𝑊) ∈ ℤ ∧ 𝑀 ∈ ℤ) → ((♯‘𝑊) − 𝑀) ∈ ℤ)
48	46, 6, 47	syl2an 598	. . . . . . 7 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → ((♯‘𝑊) − 𝑀) ∈ ℤ)
49	27, 7, 48	3jca 1125	. . . . . 6 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → (𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ℤ ∧ ((♯‘𝑊) − 𝑀) ∈ ℤ))
50	49	adantr 484	. . . . 5 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑊 cyclShift 𝑁) = (𝑈 cyclShift 𝑀)) → (𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ℤ ∧ ((♯‘𝑊) − 𝑀) ∈ ℤ))
51		2cshw 14166	. . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ℤ ∧ ((♯‘𝑊) − 𝑀) ∈ ℤ) → ((𝑊 cyclShift 𝑀) cyclShift ((♯‘𝑊) − 𝑀)) = (𝑊 cyclShift (𝑀 + ((♯‘𝑊) − 𝑀))))
52	50, 51	syl 17	. . . 4 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑊 cyclShift 𝑁) = (𝑈 cyclShift 𝑀)) → ((𝑊 cyclShift 𝑀) cyclShift ((♯‘𝑊) − 𝑀)) = (𝑊 cyclShift (𝑀 + ((♯‘𝑊) − 𝑀))))
53	24, 43, 52	3eqtrd 2837	. . 3 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑊 cyclShift 𝑁) = (𝑈 cyclShift 𝑀)) → (𝑈 cyclShift (𝑀 − 𝑁)) = (𝑊 cyclShift (𝑀 + ((♯‘𝑊) − 𝑀))))
54	44	nn0cnd 11945	. . . . . . . . 9 ⊢ (𝑊 ∈ Word 𝑉 → (♯‘𝑊) ∈ ℂ)
55	54	adantr 484	. . . . . . . 8 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) → (♯‘𝑊) ∈ ℂ)
56	35	adantl 485	. . . . . . . 8 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → 𝑀 ∈ ℂ)
57	55, 56	anim12i 615	. . . . . . 7 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → ((♯‘𝑊) ∈ ℂ ∧ 𝑀 ∈ ℂ))
58	57	ancomd 465	. . . . . 6 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → (𝑀 ∈ ℂ ∧ (♯‘𝑊) ∈ ℂ))
59	58	adantr 484	. . . . 5 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑊 cyclShift 𝑁) = (𝑈 cyclShift 𝑀)) → (𝑀 ∈ ℂ ∧ (♯‘𝑊) ∈ ℂ))
60		pncan3 10883	. . . . 5 ⊢ ((𝑀 ∈ ℂ ∧ (♯‘𝑊) ∈ ℂ) → (𝑀 + ((♯‘𝑊) − 𝑀)) = (♯‘𝑊))
61	59, 60	syl 17	. . . 4 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑊 cyclShift 𝑁) = (𝑈 cyclShift 𝑀)) → (𝑀 + ((♯‘𝑊) − 𝑀)) = (♯‘𝑊))
62	61	oveq2d 7151	. . 3 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑊 cyclShift 𝑁) = (𝑈 cyclShift 𝑀)) → (𝑊 cyclShift (𝑀 + ((♯‘𝑊) − 𝑀))) = (𝑊 cyclShift (♯‘𝑊)))
63		cshwn 14150	. . . . 5 ⊢ (𝑊 ∈ Word 𝑉 → (𝑊 cyclShift (♯‘𝑊)) = 𝑊)
64	27, 63	syl 17	. . . 4 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → (𝑊 cyclShift (♯‘𝑊)) = 𝑊)
65	64	adantr 484	. . 3 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑊 cyclShift 𝑁) = (𝑈 cyclShift 𝑀)) → (𝑊 cyclShift (♯‘𝑊)) = 𝑊)
66	53, 62, 65	3eqtrd 2837	. 2 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑊 cyclShift 𝑁) = (𝑈 cyclShift 𝑀)) → (𝑈 cyclShift (𝑀 − 𝑁)) = 𝑊)
67	66	ex 416	1 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → ((𝑊 cyclShift 𝑁) = (𝑈 cyclShift 𝑀) → (𝑈 cyclShift (𝑀 − 𝑁)) = 𝑊))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ‘cfv 6324 (class class class)co 7135 ℂcc 10524 + caddc 10529 − cmin 10859 ℤcz 11969 ♯chash 13686 Word cword 13857 cyclShift ccsh 14141

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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-sup 8890 df-inf 8891 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-fz 12886 df-fzo 13029 df-fl 13157 df-mod 13233 df-hash 13687 df-word 13858 df-concat 13914 df-substr 13994 df-pfx 14024 df-csh 14142

This theorem is referenced by: cshweqdifid 14173

W3C validator