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Theorem cshwcsh2id 14181

Description: A cyclically shifted word can be reconstructed by cyclically shifting it again twice. Lemma for erclwwlktr 27807 and erclwwlkntr 27856. (Contributed by AV, 9-Apr-2018.) (Revised by AV, 11-Jun-2018.) (Proof shortened by AV, 3-Nov-2018.)

**Hypotheses**
Ref	Expression
cshwcsh2id.1	⊢ (𝜑 → 𝑧 ∈ Word 𝑉)
cshwcsh2id.2	⊢ (𝜑 → ((♯‘𝑦) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑦)))

**Assertion**
Ref	Expression
cshwcsh2id	⊢ (𝜑 → (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚)) ∧ (𝑘 ∈ (0...(♯‘𝑧)) ∧ 𝑦 = (𝑧 cyclShift 𝑘))) → ∃𝑛 ∈ (0...(♯‘𝑧))𝑥 = (𝑧 cyclShift 𝑛)))

Distinct variable group: 𝑘,𝑚,𝑛,𝑥,𝑧 Allowed substitution hints: 𝜑(𝑥,𝑦,𝑧,𝑘,𝑚,𝑛) 𝑉(𝑥,𝑦,𝑧,𝑘,𝑚,𝑛)

**Proof of Theorem cshwcsh2id**
Step	Hyp	Ref	Expression
1		oveq1 7142	. . . . . . . . 9 ⊢ (𝑦 = (𝑧 cyclShift 𝑘) → (𝑦 cyclShift 𝑚) = ((𝑧 cyclShift 𝑘) cyclShift 𝑚))
2	1	eqeq2d 2809	. . . . . . . 8 ⊢ (𝑦 = (𝑧 cyclShift 𝑘) → (𝑥 = (𝑦 cyclShift 𝑚) ↔ 𝑥 = ((𝑧 cyclShift 𝑘) cyclShift 𝑚)))
3	2	anbi2d 631	. . . . . . 7 ⊢ (𝑦 = (𝑧 cyclShift 𝑘) → ((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚)) ↔ (𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑥 = ((𝑧 cyclShift 𝑘) cyclShift 𝑚))))
4	3	adantr 484	. . . . . 6 ⊢ ((𝑦 = (𝑧 cyclShift 𝑘) ∧ 𝑘 ∈ (0...(♯‘𝑧))) → ((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚)) ↔ (𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑥 = ((𝑧 cyclShift 𝑘) cyclShift 𝑚))))
5		elfznn0 12995	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ (0...(♯‘𝑧)) → 𝑘 ∈ ℕ₀)
6		elfznn0 12995	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 ∈ (0...(♯‘𝑦)) → 𝑚 ∈ ℕ₀)
7		nn0addcl 11920	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑘 ∈ ℕ₀ ∧ 𝑚 ∈ ℕ₀) → (𝑘 + 𝑚) ∈ ℕ₀)
8	5, 6, 7	syl2anr 599	. . . . . . . . . . . . . . . 16 ⊢ ((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) → (𝑘 + 𝑚) ∈ ℕ₀)
9	8	adantr 484	. . . . . . . . . . . . . . 15 ⊢ (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ ((𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) → (𝑘 + 𝑚) ∈ ℕ₀)
10		elfz3nn0 12996	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ (0...(♯‘𝑧)) → (♯‘𝑧) ∈ ℕ₀)
11	10	ad2antlr 726	. . . . . . . . . . . . . . 15 ⊢ (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ ((𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) → (♯‘𝑧) ∈ ℕ₀)
12		simprl 770	. . . . . . . . . . . . . . 15 ⊢ (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ ((𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) → (𝑘 + 𝑚) ≤ (♯‘𝑧))
13		elfz2nn0 12993	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 + 𝑚) ∈ (0...(♯‘𝑧)) ↔ ((𝑘 + 𝑚) ∈ ℕ₀ ∧ (♯‘𝑧) ∈ ℕ₀ ∧ (𝑘 + 𝑚) ≤ (♯‘𝑧)))
14	9, 11, 12, 13	syl3anbrc 1340	. . . . . . . . . . . . . 14 ⊢ (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ ((𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) → (𝑘 + 𝑚) ∈ (0...(♯‘𝑧)))
15	14	adantr 484	. . . . . . . . . . . . 13 ⊢ ((((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ ((𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) ∧ 𝑥 = ((𝑧 cyclShift 𝑘) cyclShift 𝑚)) → (𝑘 + 𝑚) ∈ (0...(♯‘𝑧)))
16		cshwcsh2id.1	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑧 ∈ Word 𝑉)
17	16	adantl 485	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑) → 𝑧 ∈ Word 𝑉)
18	17	adantl 485	. . . . . . . . . . . . . . . 16 ⊢ (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ ((𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) → 𝑧 ∈ Word 𝑉)
19		elfzelz 12902	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ (0...(♯‘𝑧)) → 𝑘 ∈ ℤ)
20	19	ad2antlr 726	. . . . . . . . . . . . . . . 16 ⊢ (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ ((𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) → 𝑘 ∈ ℤ)
21		elfzelz 12902	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 ∈ (0...(♯‘𝑦)) → 𝑚 ∈ ℤ)
22	21	adantr 484	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) → 𝑚 ∈ ℤ)
23	22	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ ((𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) → 𝑚 ∈ ℤ)
24		2cshw 14166	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ Word 𝑉 ∧ 𝑘 ∈ ℤ ∧ 𝑚 ∈ ℤ) → ((𝑧 cyclShift 𝑘) cyclShift 𝑚) = (𝑧 cyclShift (𝑘 + 𝑚)))
25	18, 20, 23, 24	syl3anc 1368	. . . . . . . . . . . . . . 15 ⊢ (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ ((𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) → ((𝑧 cyclShift 𝑘) cyclShift 𝑚) = (𝑧 cyclShift (𝑘 + 𝑚)))
26	25	eqeq2d 2809	. . . . . . . . . . . . . 14 ⊢ (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ ((𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) → (𝑥 = ((𝑧 cyclShift 𝑘) cyclShift 𝑚) ↔ 𝑥 = (𝑧 cyclShift (𝑘 + 𝑚))))
27	26	biimpa 480	. . . . . . . . . . . . 13 ⊢ ((((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ ((𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) ∧ 𝑥 = ((𝑧 cyclShift 𝑘) cyclShift 𝑚)) → 𝑥 = (𝑧 cyclShift (𝑘 + 𝑚)))
28	15, 27	jca 515	. . . . . . . . . . . 12 ⊢ ((((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ ((𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) ∧ 𝑥 = ((𝑧 cyclShift 𝑘) cyclShift 𝑚)) → ((𝑘 + 𝑚) ∈ (0...(♯‘𝑧)) ∧ 𝑥 = (𝑧 cyclShift (𝑘 + 𝑚))))
29	28	exp41 438	. . . . . . . . . . 11 ⊢ (𝑚 ∈ (0...(♯‘𝑦)) → (𝑘 ∈ (0...(♯‘𝑧)) → (((𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑) → (𝑥 = ((𝑧 cyclShift 𝑘) cyclShift 𝑚) → ((𝑘 + 𝑚) ∈ (0...(♯‘𝑧)) ∧ 𝑥 = (𝑧 cyclShift (𝑘 + 𝑚)))))))
30	29	com23 86	. . . . . . . . . 10 ⊢ (𝑚 ∈ (0...(♯‘𝑦)) → (((𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑) → (𝑘 ∈ (0...(♯‘𝑧)) → (𝑥 = ((𝑧 cyclShift 𝑘) cyclShift 𝑚) → ((𝑘 + 𝑚) ∈ (0...(♯‘𝑧)) ∧ 𝑥 = (𝑧 cyclShift (𝑘 + 𝑚)))))))
31	30	com24 95	. . . . . . . . 9 ⊢ (𝑚 ∈ (0...(♯‘𝑦)) → (𝑥 = ((𝑧 cyclShift 𝑘) cyclShift 𝑚) → (𝑘 ∈ (0...(♯‘𝑧)) → (((𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑) → ((𝑘 + 𝑚) ∈ (0...(♯‘𝑧)) ∧ 𝑥 = (𝑧 cyclShift (𝑘 + 𝑚)))))))
32	31	imp 410	. . . . . . . 8 ⊢ ((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑥 = ((𝑧 cyclShift 𝑘) cyclShift 𝑚)) → (𝑘 ∈ (0...(♯‘𝑧)) → (((𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑) → ((𝑘 + 𝑚) ∈ (0...(♯‘𝑧)) ∧ 𝑥 = (𝑧 cyclShift (𝑘 + 𝑚))))))
33	32	com12 32	. . . . . . 7 ⊢ (𝑘 ∈ (0...(♯‘𝑧)) → ((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑥 = ((𝑧 cyclShift 𝑘) cyclShift 𝑚)) → (((𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑) → ((𝑘 + 𝑚) ∈ (0...(♯‘𝑧)) ∧ 𝑥 = (𝑧 cyclShift (𝑘 + 𝑚))))))
34	33	adantl 485	. . . . . 6 ⊢ ((𝑦 = (𝑧 cyclShift 𝑘) ∧ 𝑘 ∈ (0...(♯‘𝑧))) → ((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑥 = ((𝑧 cyclShift 𝑘) cyclShift 𝑚)) → (((𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑) → ((𝑘 + 𝑚) ∈ (0...(♯‘𝑧)) ∧ 𝑥 = (𝑧 cyclShift (𝑘 + 𝑚))))))
35	4, 34	sylbid 243	. . . . 5 ⊢ ((𝑦 = (𝑧 cyclShift 𝑘) ∧ 𝑘 ∈ (0...(♯‘𝑧))) → ((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚)) → (((𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑) → ((𝑘 + 𝑚) ∈ (0...(♯‘𝑧)) ∧ 𝑥 = (𝑧 cyclShift (𝑘 + 𝑚))))))
36	35	ancoms 462	. . . 4 ⊢ ((𝑘 ∈ (0...(♯‘𝑧)) ∧ 𝑦 = (𝑧 cyclShift 𝑘)) → ((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚)) → (((𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑) → ((𝑘 + 𝑚) ∈ (0...(♯‘𝑧)) ∧ 𝑥 = (𝑧 cyclShift (𝑘 + 𝑚))))))
37	36	impcom 411	. . 3 ⊢ (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚)) ∧ (𝑘 ∈ (0...(♯‘𝑧)) ∧ 𝑦 = (𝑧 cyclShift 𝑘))) → (((𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑) → ((𝑘 + 𝑚) ∈ (0...(♯‘𝑧)) ∧ 𝑥 = (𝑧 cyclShift (𝑘 + 𝑚)))))
38		oveq2 7143	. . . 4 ⊢ (𝑛 = (𝑘 + 𝑚) → (𝑧 cyclShift 𝑛) = (𝑧 cyclShift (𝑘 + 𝑚)))
39	38	rspceeqv 3586	. . 3 ⊢ (((𝑘 + 𝑚) ∈ (0...(♯‘𝑧)) ∧ 𝑥 = (𝑧 cyclShift (𝑘 + 𝑚))) → ∃𝑛 ∈ (0...(♯‘𝑧))𝑥 = (𝑧 cyclShift 𝑛))
40	37, 39	syl6com 37	. 2 ⊢ (((𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑) → (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚)) ∧ (𝑘 ∈ (0...(♯‘𝑧)) ∧ 𝑦 = (𝑧 cyclShift 𝑘))) → ∃𝑛 ∈ (0...(♯‘𝑧))𝑥 = (𝑧 cyclShift 𝑛)))
41		elfz2 12892	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ (0...(♯‘𝑧)) ↔ ((0 ∈ ℤ ∧ (♯‘𝑧) ∈ ℤ ∧ 𝑘 ∈ ℤ) ∧ (0 ≤ 𝑘 ∧ 𝑘 ≤ (♯‘𝑧))))
42		nn0z 11993	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑚 ∈ ℕ₀ → 𝑚 ∈ ℤ)
43		zaddcl 12010	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑘 ∈ ℤ ∧ 𝑚 ∈ ℤ) → (𝑘 + 𝑚) ∈ ℤ)
44	43	ex 416	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑘 ∈ ℤ → (𝑚 ∈ ℤ → (𝑘 + 𝑚) ∈ ℤ))
45	44	adantl 485	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((♯‘𝑧) ∈ ℤ ∧ 𝑘 ∈ ℤ) → (𝑚 ∈ ℤ → (𝑘 + 𝑚) ∈ ℤ))
46	45	impcom 411	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑚 ∈ ℤ ∧ ((♯‘𝑧) ∈ ℤ ∧ 𝑘 ∈ ℤ)) → (𝑘 + 𝑚) ∈ ℤ)
47		simprl 770	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑚 ∈ ℤ ∧ ((♯‘𝑧) ∈ ℤ ∧ 𝑘 ∈ ℤ)) → (♯‘𝑧) ∈ ℤ)
48	46, 47	zsubcld 12080	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑚 ∈ ℤ ∧ ((♯‘𝑧) ∈ ℤ ∧ 𝑘 ∈ ℤ)) → ((𝑘 + 𝑚) − (♯‘𝑧)) ∈ ℤ)
49	48	ex 416	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑚 ∈ ℤ → (((♯‘𝑧) ∈ ℤ ∧ 𝑘 ∈ ℤ) → ((𝑘 + 𝑚) − (♯‘𝑧)) ∈ ℤ))
50	42, 49	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑚 ∈ ℕ₀ → (((♯‘𝑧) ∈ ℤ ∧ 𝑘 ∈ ℤ) → ((𝑘 + 𝑚) − (♯‘𝑧)) ∈ ℤ))
51	50	com12 32	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((♯‘𝑧) ∈ ℤ ∧ 𝑘 ∈ ℤ) → (𝑚 ∈ ℕ₀ → ((𝑘 + 𝑚) − (♯‘𝑧)) ∈ ℤ))
52	51	3adant1 1127	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((0 ∈ ℤ ∧ (♯‘𝑧) ∈ ℤ ∧ 𝑘 ∈ ℤ) → (𝑚 ∈ ℕ₀ → ((𝑘 + 𝑚) − (♯‘𝑧)) ∈ ℤ))
53	52	adantr 484	. . . . . . . . . . . . . . . . . . 19 ⊢ (((0 ∈ ℤ ∧ (♯‘𝑧) ∈ ℤ ∧ 𝑘 ∈ ℤ) ∧ (0 ≤ 𝑘 ∧ 𝑘 ≤ (♯‘𝑧))) → (𝑚 ∈ ℕ₀ → ((𝑘 + 𝑚) − (♯‘𝑧)) ∈ ℤ))
54	41, 53	sylbi 220	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ (0...(♯‘𝑧)) → (𝑚 ∈ ℕ₀ → ((𝑘 + 𝑚) − (♯‘𝑧)) ∈ ℤ))
55	6, 54	mpan9 510	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) → ((𝑘 + 𝑚) − (♯‘𝑧)) ∈ ℤ)
56	55	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) → ((𝑘 + 𝑚) − (♯‘𝑧)) ∈ ℤ)
57		elfz2nn0 12993	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 ∈ (0...(♯‘𝑧)) ↔ (𝑘 ∈ ℕ₀ ∧ (♯‘𝑧) ∈ ℕ₀ ∧ 𝑘 ≤ (♯‘𝑧)))
58		nn0re 11894	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑘 ∈ ℕ₀ → 𝑘 ∈ ℝ)
59		nn0re 11894	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((♯‘𝑧) ∈ ℕ₀ → (♯‘𝑧) ∈ ℝ)
60	58, 59	anim12i 615	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑘 ∈ ℕ₀ ∧ (♯‘𝑧) ∈ ℕ₀) → (𝑘 ∈ ℝ ∧ (♯‘𝑧) ∈ ℝ))
61		nn0re 11894	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑚 ∈ ℕ₀ → 𝑚 ∈ ℝ)
62	60, 61	anim12i 615	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑘 ∈ ℕ₀ ∧ (♯‘𝑧) ∈ ℕ₀) ∧ 𝑚 ∈ ℕ₀) → ((𝑘 ∈ ℝ ∧ (♯‘𝑧) ∈ ℝ) ∧ 𝑚 ∈ ℝ))
63		simplr 768	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑘 ∈ ℝ ∧ (♯‘𝑧) ∈ ℝ) ∧ 𝑚 ∈ ℝ) → (♯‘𝑧) ∈ ℝ)
64		readdcl 10609	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑘 ∈ ℝ ∧ 𝑚 ∈ ℝ) → (𝑘 + 𝑚) ∈ ℝ)
65	64	adantlr 714	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑘 ∈ ℝ ∧ (♯‘𝑧) ∈ ℝ) ∧ 𝑚 ∈ ℝ) → (𝑘 + 𝑚) ∈ ℝ)
66	63, 65	ltnled 10776	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑘 ∈ ℝ ∧ (♯‘𝑧) ∈ ℝ) ∧ 𝑚 ∈ ℝ) → ((♯‘𝑧) < (𝑘 + 𝑚) ↔ ¬ (𝑘 + 𝑚) ≤ (♯‘𝑧)))
67	63, 65	posdifd 11216	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑘 ∈ ℝ ∧ (♯‘𝑧) ∈ ℝ) ∧ 𝑚 ∈ ℝ) → ((♯‘𝑧) < (𝑘 + 𝑚) ↔ 0 < ((𝑘 + 𝑚) − (♯‘𝑧))))
68	67	biimpd 232	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑘 ∈ ℝ ∧ (♯‘𝑧) ∈ ℝ) ∧ 𝑚 ∈ ℝ) → ((♯‘𝑧) < (𝑘 + 𝑚) → 0 < ((𝑘 + 𝑚) − (♯‘𝑧))))
69	66, 68	sylbird 263	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑘 ∈ ℝ ∧ (♯‘𝑧) ∈ ℝ) ∧ 𝑚 ∈ ℝ) → (¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) → 0 < ((𝑘 + 𝑚) − (♯‘𝑧))))
70	62, 69	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑘 ∈ ℕ₀ ∧ (♯‘𝑧) ∈ ℕ₀) ∧ 𝑚 ∈ ℕ₀) → (¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) → 0 < ((𝑘 + 𝑚) − (♯‘𝑧))))
71	70	ex 416	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑘 ∈ ℕ₀ ∧ (♯‘𝑧) ∈ ℕ₀) → (𝑚 ∈ ℕ₀ → (¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) → 0 < ((𝑘 + 𝑚) − (♯‘𝑧)))))
72	71	3adant3 1129	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑘 ∈ ℕ₀ ∧ (♯‘𝑧) ∈ ℕ₀ ∧ 𝑘 ≤ (♯‘𝑧)) → (𝑚 ∈ ℕ₀ → (¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) → 0 < ((𝑘 + 𝑚) − (♯‘𝑧)))))
73	57, 72	sylbi 220	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ (0...(♯‘𝑧)) → (𝑚 ∈ ℕ₀ → (¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) → 0 < ((𝑘 + 𝑚) − (♯‘𝑧)))))
74	6, 73	mpan9 510	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) → (¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) → 0 < ((𝑘 + 𝑚) − (♯‘𝑧))))
75	74	com12 32	. . . . . . . . . . . . . . . . . 18 ⊢ (¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) → ((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) → 0 < ((𝑘 + 𝑚) − (♯‘𝑧))))
76	75	adantr 484	. . . . . . . . . . . . . . . . 17 ⊢ ((¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑) → ((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) → 0 < ((𝑘 + 𝑚) − (♯‘𝑧))))
77	76	impcom 411	. . . . . . . . . . . . . . . 16 ⊢ (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) → 0 < ((𝑘 + 𝑚) − (♯‘𝑧)))
78		elnnz 11979	. . . . . . . . . . . . . . . 16 ⊢ (((𝑘 + 𝑚) − (♯‘𝑧)) ∈ ℕ ↔ (((𝑘 + 𝑚) − (♯‘𝑧)) ∈ ℤ ∧ 0 < ((𝑘 + 𝑚) − (♯‘𝑧))))
79	56, 77, 78	sylanbrc 586	. . . . . . . . . . . . . . 15 ⊢ (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) → ((𝑘 + 𝑚) − (♯‘𝑧)) ∈ ℕ)
80	79	nnnn0d 11943	. . . . . . . . . . . . . 14 ⊢ (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) → ((𝑘 + 𝑚) − (♯‘𝑧)) ∈ ℕ₀)
81	10	ad2antlr 726	. . . . . . . . . . . . . 14 ⊢ (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) → (♯‘𝑧) ∈ ℕ₀)
82		cshwcsh2id.2	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((♯‘𝑦) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑦)))
83		oveq2 7143	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((♯‘𝑦) = (♯‘𝑧) → (0...(♯‘𝑦)) = (0...(♯‘𝑧)))
84	83	eleq2d 2875	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((♯‘𝑦) = (♯‘𝑧) → (𝑚 ∈ (0...(♯‘𝑦)) ↔ 𝑚 ∈ (0...(♯‘𝑧))))
85	84	anbi1d 632	. . . . . . . . . . . . . . . . . . 19 ⊢ ((♯‘𝑦) = (♯‘𝑧) → ((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ↔ (𝑚 ∈ (0...(♯‘𝑧)) ∧ 𝑘 ∈ (0...(♯‘𝑧)))))
86		elfz2nn0 12993	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑚 ∈ (0...(♯‘𝑧)) ↔ (𝑚 ∈ ℕ₀ ∧ (♯‘𝑧) ∈ ℕ₀ ∧ 𝑚 ≤ (♯‘𝑧)))
87	58	adantr 484	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑘 ∈ ℕ₀ ∧ (♯‘𝑧) ∈ ℕ₀) → 𝑘 ∈ ℝ)
88	87, 61	anim12i 615	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝑘 ∈ ℕ₀ ∧ (♯‘𝑧) ∈ ℕ₀) ∧ 𝑚 ∈ ℕ₀) → (𝑘 ∈ ℝ ∧ 𝑚 ∈ ℝ))
89	59, 59	jca 515	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((♯‘𝑧) ∈ ℕ₀ → ((♯‘𝑧) ∈ ℝ ∧ (♯‘𝑧) ∈ ℝ))
90	89	ad2antlr 726	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝑘 ∈ ℕ₀ ∧ (♯‘𝑧) ∈ ℕ₀) ∧ 𝑚 ∈ ℕ₀) → ((♯‘𝑧) ∈ ℝ ∧ (♯‘𝑧) ∈ ℝ))
91		le2add 11111	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝑘 ∈ ℝ ∧ 𝑚 ∈ ℝ) ∧ ((♯‘𝑧) ∈ ℝ ∧ (♯‘𝑧) ∈ ℝ)) → ((𝑘 ≤ (♯‘𝑧) ∧ 𝑚 ≤ (♯‘𝑧)) → (𝑘 + 𝑚) ≤ ((♯‘𝑧) + (♯‘𝑧))))
92	88, 90, 91	syl2anc 587	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝑘 ∈ ℕ₀ ∧ (♯‘𝑧) ∈ ℕ₀) ∧ 𝑚 ∈ ℕ₀) → ((𝑘 ≤ (♯‘𝑧) ∧ 𝑚 ≤ (♯‘𝑧)) → (𝑘 + 𝑚) ≤ ((♯‘𝑧) + (♯‘𝑧))))
93		nn0readdcl 11949	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑘 ∈ ℕ₀ ∧ 𝑚 ∈ ℕ₀) → (𝑘 + 𝑚) ∈ ℝ)
94	93	adantlr 714	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝑘 ∈ ℕ₀ ∧ (♯‘𝑧) ∈ ℕ₀) ∧ 𝑚 ∈ ℕ₀) → (𝑘 + 𝑚) ∈ ℝ)
95	59	ad2antlr 726	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝑘 ∈ ℕ₀ ∧ (♯‘𝑧) ∈ ℕ₀) ∧ 𝑚 ∈ ℕ₀) → (♯‘𝑧) ∈ ℝ)
96	94, 95, 95	lesubadd2d 11228	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝑘 ∈ ℕ₀ ∧ (♯‘𝑧) ∈ ℕ₀) ∧ 𝑚 ∈ ℕ₀) → (((𝑘 + 𝑚) − (♯‘𝑧)) ≤ (♯‘𝑧) ↔ (𝑘 + 𝑚) ≤ ((♯‘𝑧) + (♯‘𝑧))))
97	92, 96	sylibrd 262	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝑘 ∈ ℕ₀ ∧ (♯‘𝑧) ∈ ℕ₀) ∧ 𝑚 ∈ ℕ₀) → ((𝑘 ≤ (♯‘𝑧) ∧ 𝑚 ≤ (♯‘𝑧)) → ((𝑘 + 𝑚) − (♯‘𝑧)) ≤ (♯‘𝑧)))
98	97	expcomd 420	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑘 ∈ ℕ₀ ∧ (♯‘𝑧) ∈ ℕ₀) ∧ 𝑚 ∈ ℕ₀) → (𝑚 ≤ (♯‘𝑧) → (𝑘 ≤ (♯‘𝑧) → ((𝑘 + 𝑚) − (♯‘𝑧)) ≤ (♯‘𝑧))))
99	98	ex 416	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑘 ∈ ℕ₀ ∧ (♯‘𝑧) ∈ ℕ₀) → (𝑚 ∈ ℕ₀ → (𝑚 ≤ (♯‘𝑧) → (𝑘 ≤ (♯‘𝑧) → ((𝑘 + 𝑚) − (♯‘𝑧)) ≤ (♯‘𝑧)))))
100	99	com24 95	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑘 ∈ ℕ₀ ∧ (♯‘𝑧) ∈ ℕ₀) → (𝑘 ≤ (♯‘𝑧) → (𝑚 ≤ (♯‘𝑧) → (𝑚 ∈ ℕ₀ → ((𝑘 + 𝑚) − (♯‘𝑧)) ≤ (♯‘𝑧)))))
101	100	3impia 1114	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑘 ∈ ℕ₀ ∧ (♯‘𝑧) ∈ ℕ₀ ∧ 𝑘 ≤ (♯‘𝑧)) → (𝑚 ≤ (♯‘𝑧) → (𝑚 ∈ ℕ₀ → ((𝑘 + 𝑚) − (♯‘𝑧)) ≤ (♯‘𝑧))))
102	101	com13 88	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑚 ∈ ℕ₀ → (𝑚 ≤ (♯‘𝑧) → ((𝑘 ∈ ℕ₀ ∧ (♯‘𝑧) ∈ ℕ₀ ∧ 𝑘 ≤ (♯‘𝑧)) → ((𝑘 + 𝑚) − (♯‘𝑧)) ≤ (♯‘𝑧))))
103	102	imp 410	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑚 ∈ ℕ₀ ∧ 𝑚 ≤ (♯‘𝑧)) → ((𝑘 ∈ ℕ₀ ∧ (♯‘𝑧) ∈ ℕ₀ ∧ 𝑘 ≤ (♯‘𝑧)) → ((𝑘 + 𝑚) − (♯‘𝑧)) ≤ (♯‘𝑧)))
104	57, 103	syl5bi 245	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑚 ∈ ℕ₀ ∧ 𝑚 ≤ (♯‘𝑧)) → (𝑘 ∈ (0...(♯‘𝑧)) → ((𝑘 + 𝑚) − (♯‘𝑧)) ≤ (♯‘𝑧)))
105	104	3adant2 1128	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑚 ∈ ℕ₀ ∧ (♯‘𝑧) ∈ ℕ₀ ∧ 𝑚 ≤ (♯‘𝑧)) → (𝑘 ∈ (0...(♯‘𝑧)) → ((𝑘 + 𝑚) − (♯‘𝑧)) ≤ (♯‘𝑧)))
106	86, 105	sylbi 220	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑚 ∈ (0...(♯‘𝑧)) → (𝑘 ∈ (0...(♯‘𝑧)) → ((𝑘 + 𝑚) − (♯‘𝑧)) ≤ (♯‘𝑧)))
107	106	imp 410	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑚 ∈ (0...(♯‘𝑧)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) → ((𝑘 + 𝑚) − (♯‘𝑧)) ≤ (♯‘𝑧))
108	85, 107	syl6bi 256	. . . . . . . . . . . . . . . . . 18 ⊢ ((♯‘𝑦) = (♯‘𝑧) → ((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) → ((𝑘 + 𝑚) − (♯‘𝑧)) ≤ (♯‘𝑧)))
109	108	adantr 484	. . . . . . . . . . . . . . . . 17 ⊢ (((♯‘𝑦) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑦)) → ((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) → ((𝑘 + 𝑚) − (♯‘𝑧)) ≤ (♯‘𝑧)))
110	82, 109	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) → ((𝑘 + 𝑚) − (♯‘𝑧)) ≤ (♯‘𝑧)))
111	110	adantl 485	. . . . . . . . . . . . . . 15 ⊢ ((¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑) → ((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) → ((𝑘 + 𝑚) − (♯‘𝑧)) ≤ (♯‘𝑧)))
112	111	impcom 411	. . . . . . . . . . . . . 14 ⊢ (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) → ((𝑘 + 𝑚) − (♯‘𝑧)) ≤ (♯‘𝑧))
113		elfz2nn0 12993	. . . . . . . . . . . . . 14 ⊢ (((𝑘 + 𝑚) − (♯‘𝑧)) ∈ (0...(♯‘𝑧)) ↔ (((𝑘 + 𝑚) − (♯‘𝑧)) ∈ ℕ₀ ∧ (♯‘𝑧) ∈ ℕ₀ ∧ ((𝑘 + 𝑚) − (♯‘𝑧)) ≤ (♯‘𝑧)))
114	80, 81, 112, 113	syl3anbrc 1340	. . . . . . . . . . . . 13 ⊢ (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) → ((𝑘 + 𝑚) − (♯‘𝑧)) ∈ (0...(♯‘𝑧)))
115	114	adantr 484	. . . . . . . . . . . 12 ⊢ ((((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) ∧ 𝑥 = ((𝑧 cyclShift 𝑘) cyclShift 𝑚)) → ((𝑘 + 𝑚) − (♯‘𝑧)) ∈ (0...(♯‘𝑧)))
116	16	adantl 485	. . . . . . . . . . . . . . . . 17 ⊢ ((¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑) → 𝑧 ∈ Word 𝑉)
117	116	adantl 485	. . . . . . . . . . . . . . . 16 ⊢ (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) → 𝑧 ∈ Word 𝑉)
118	19	ad2antlr 726	. . . . . . . . . . . . . . . 16 ⊢ (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) → 𝑘 ∈ ℤ)
119	22	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) → 𝑚 ∈ ℤ)
120	117, 118, 119, 24	syl3anc 1368	. . . . . . . . . . . . . . 15 ⊢ (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) → ((𝑧 cyclShift 𝑘) cyclShift 𝑚) = (𝑧 cyclShift (𝑘 + 𝑚)))
121	19, 21, 43	syl2anr 599	. . . . . . . . . . . . . . . 16 ⊢ ((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) → (𝑘 + 𝑚) ∈ ℤ)
122		cshwsublen 14149	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ Word 𝑉 ∧ (𝑘 + 𝑚) ∈ ℤ) → (𝑧 cyclShift (𝑘 + 𝑚)) = (𝑧 cyclShift ((𝑘 + 𝑚) − (♯‘𝑧))))
123	116, 121, 122	syl2anr 599	. . . . . . . . . . . . . . 15 ⊢ (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) → (𝑧 cyclShift (𝑘 + 𝑚)) = (𝑧 cyclShift ((𝑘 + 𝑚) − (♯‘𝑧))))
124	120, 123	eqtrd 2833	. . . . . . . . . . . . . 14 ⊢ (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) → ((𝑧 cyclShift 𝑘) cyclShift 𝑚) = (𝑧 cyclShift ((𝑘 + 𝑚) − (♯‘𝑧))))
125	124	eqeq2d 2809	. . . . . . . . . . . . 13 ⊢ (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) → (𝑥 = ((𝑧 cyclShift 𝑘) cyclShift 𝑚) ↔ 𝑥 = (𝑧 cyclShift ((𝑘 + 𝑚) − (♯‘𝑧)))))
126	125	biimpa 480	. . . . . . . . . . . 12 ⊢ ((((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) ∧ 𝑥 = ((𝑧 cyclShift 𝑘) cyclShift 𝑚)) → 𝑥 = (𝑧 cyclShift ((𝑘 + 𝑚) − (♯‘𝑧))))
127	115, 126	jca 515	. . . . . . . . . . 11 ⊢ ((((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) ∧ 𝑥 = ((𝑧 cyclShift 𝑘) cyclShift 𝑚)) → (((𝑘 + 𝑚) − (♯‘𝑧)) ∈ (0...(♯‘𝑧)) ∧ 𝑥 = (𝑧 cyclShift ((𝑘 + 𝑚) − (♯‘𝑧)))))
128	127	exp41 438	. . . . . . . . . 10 ⊢ (𝑚 ∈ (0...(♯‘𝑦)) → (𝑘 ∈ (0...(♯‘𝑧)) → ((¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑) → (𝑥 = ((𝑧 cyclShift 𝑘) cyclShift 𝑚) → (((𝑘 + 𝑚) − (♯‘𝑧)) ∈ (0...(♯‘𝑧)) ∧ 𝑥 = (𝑧 cyclShift ((𝑘 + 𝑚) − (♯‘𝑧))))))))
129	128	com23 86	. . . . . . . . 9 ⊢ (𝑚 ∈ (0...(♯‘𝑦)) → ((¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑) → (𝑘 ∈ (0...(♯‘𝑧)) → (𝑥 = ((𝑧 cyclShift 𝑘) cyclShift 𝑚) → (((𝑘 + 𝑚) − (♯‘𝑧)) ∈ (0...(♯‘𝑧)) ∧ 𝑥 = (𝑧 cyclShift ((𝑘 + 𝑚) − (♯‘𝑧))))))))
130	129	com24 95	. . . . . . . 8 ⊢ (𝑚 ∈ (0...(♯‘𝑦)) → (𝑥 = ((𝑧 cyclShift 𝑘) cyclShift 𝑚) → (𝑘 ∈ (0...(♯‘𝑧)) → ((¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑) → (((𝑘 + 𝑚) − (♯‘𝑧)) ∈ (0...(♯‘𝑧)) ∧ 𝑥 = (𝑧 cyclShift ((𝑘 + 𝑚) − (♯‘𝑧))))))))
131	130	imp 410	. . . . . . 7 ⊢ ((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑥 = ((𝑧 cyclShift 𝑘) cyclShift 𝑚)) → (𝑘 ∈ (0...(♯‘𝑧)) → ((¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑) → (((𝑘 + 𝑚) − (♯‘𝑧)) ∈ (0...(♯‘𝑧)) ∧ 𝑥 = (𝑧 cyclShift ((𝑘 + 𝑚) − (♯‘𝑧)))))))
132	3, 131	syl6bi 256	. . . . . 6 ⊢ (𝑦 = (𝑧 cyclShift 𝑘) → ((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚)) → (𝑘 ∈ (0...(♯‘𝑧)) → ((¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑) → (((𝑘 + 𝑚) − (♯‘𝑧)) ∈ (0...(♯‘𝑧)) ∧ 𝑥 = (𝑧 cyclShift ((𝑘 + 𝑚) − (♯‘𝑧))))))))
133	132	com23 86	. . . . 5 ⊢ (𝑦 = (𝑧 cyclShift 𝑘) → (𝑘 ∈ (0...(♯‘𝑧)) → ((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚)) → ((¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑) → (((𝑘 + 𝑚) − (♯‘𝑧)) ∈ (0...(♯‘𝑧)) ∧ 𝑥 = (𝑧 cyclShift ((𝑘 + 𝑚) − (♯‘𝑧))))))))
134	133	impcom 411	. . . 4 ⊢ ((𝑘 ∈ (0...(♯‘𝑧)) ∧ 𝑦 = (𝑧 cyclShift 𝑘)) → ((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚)) → ((¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑) → (((𝑘 + 𝑚) − (♯‘𝑧)) ∈ (0...(♯‘𝑧)) ∧ 𝑥 = (𝑧 cyclShift ((𝑘 + 𝑚) − (♯‘𝑧)))))))
135	134	impcom 411	. . 3 ⊢ (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚)) ∧ (𝑘 ∈ (0...(♯‘𝑧)) ∧ 𝑦 = (𝑧 cyclShift 𝑘))) → ((¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑) → (((𝑘 + 𝑚) − (♯‘𝑧)) ∈ (0...(♯‘𝑧)) ∧ 𝑥 = (𝑧 cyclShift ((𝑘 + 𝑚) − (♯‘𝑧))))))
136		oveq2 7143	. . . 4 ⊢ (𝑛 = ((𝑘 + 𝑚) − (♯‘𝑧)) → (𝑧 cyclShift 𝑛) = (𝑧 cyclShift ((𝑘 + 𝑚) − (♯‘𝑧))))
137	136	rspceeqv 3586	. . 3 ⊢ ((((𝑘 + 𝑚) − (♯‘𝑧)) ∈ (0...(♯‘𝑧)) ∧ 𝑥 = (𝑧 cyclShift ((𝑘 + 𝑚) − (♯‘𝑧)))) → ∃𝑛 ∈ (0...(♯‘𝑧))𝑥 = (𝑧 cyclShift 𝑛))
138	135, 137	syl6com 37	. 2 ⊢ ((¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑) → (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚)) ∧ (𝑘 ∈ (0...(♯‘𝑧)) ∧ 𝑦 = (𝑧 cyclShift 𝑘))) → ∃𝑛 ∈ (0...(♯‘𝑧))𝑥 = (𝑧 cyclShift 𝑛)))
139	40, 138	pm2.61ian 811	1 ⊢ (𝜑 → (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚)) ∧ (𝑘 ∈ (0...(♯‘𝑧)) ∧ 𝑦 = (𝑧 cyclShift 𝑘))) → ∃𝑛 ∈ (0...(♯‘𝑧))𝑥 = (𝑧 cyclShift 𝑛)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ∃wrex 3107 class class class wbr 5030 ‘cfv 6324 (class class class)co 7135 ℝcr 10525 0cc0 10526 + caddc 10529 < clt 10664 ≤ cle 10665 − cmin 10859 ℕcn 11625 ℕ₀cn0 11885 ℤcz 11969 ...cfz 12885 ♯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: erclwwlktr 27807 erclwwlkntr 27856

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