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Theorem cshwcsh2id 13425

Description: A cyclically shifted word can be reconstructed by cyclically shifting it again twice. Lemma for erclwwlktr 26343 and erclwwlkntr 26355. (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 6556	. . . . . . . . 9 ⊢ (𝑦 = (𝑧 cyclShift 𝑘) → (𝑦 cyclShift 𝑚) = ((𝑧 cyclShift 𝑘) cyclShift 𝑚))
2	1	eqeq2d 2620	. . . . . . . 8 ⊢ (𝑦 = (𝑧 cyclShift 𝑘) → (𝑥 = (𝑦 cyclShift 𝑚) ↔ 𝑥 = ((𝑧 cyclShift 𝑘) cyclShift 𝑚)))
3	2	anbi2d 736	. . . . . . 7 ⊢ (𝑦 = (𝑧 cyclShift 𝑘) → ((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚)) ↔ (𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑥 = ((𝑧 cyclShift 𝑘) cyclShift 𝑚))))
4	3	adantr 480	. . . . . 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 12302	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ (0...(#‘𝑧)) → 𝑘 ∈ ℕ₀)
6		elfznn0 12302	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 ∈ (0...(#‘𝑦)) → 𝑚 ∈ ℕ₀)
7		nn0addcl 11205	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑘 ∈ ℕ₀ ∧ 𝑚 ∈ ℕ₀) → (𝑘 + 𝑚) ∈ ℕ₀)
8	5, 6, 7	syl2anr 494	. . . . . . . . . . . . . . . 16 ⊢ ((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) → (𝑘 + 𝑚) ∈ ℕ₀)
9	8	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ ((𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → (𝑘 + 𝑚) ∈ ℕ₀)
10		elfz3nn0 12303	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ (0...(#‘𝑧)) → (#‘𝑧) ∈ ℕ₀)
11	10	ad2antlr 759	. . . . . . . . . . . . . . 15 ⊢ (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ ((𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → (#‘𝑧) ∈ ℕ₀)
12		simprl 790	. . . . . . . . . . . . . . 15 ⊢ (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ ((𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → (𝑘 + 𝑚) ≤ (#‘𝑧))
13		elfz2nn0 12300	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 + 𝑚) ∈ (0...(#‘𝑧)) ↔ ((𝑘 + 𝑚) ∈ ℕ₀ ∧ (#‘𝑧) ∈ ℕ₀ ∧ (𝑘 + 𝑚) ≤ (#‘𝑧)))
14	9, 11, 12, 13	syl3anbrc 1239	. . . . . . . . . . . . . 14 ⊢ (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ ((𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → (𝑘 + 𝑚) ∈ (0...(#‘𝑧)))
15	14	adantr 480	. . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑) → 𝑧 ∈ Word 𝑉)
18	17	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ ((𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → 𝑧 ∈ Word 𝑉)
19		elfzelz 12213	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ (0...(#‘𝑧)) → 𝑘 ∈ ℤ)
20	19	ad2antlr 759	. . . . . . . . . . . . . . . 16 ⊢ (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ ((𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → 𝑘 ∈ ℤ)
21		elfzelz 12213	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 ∈ (0...(#‘𝑦)) → 𝑚 ∈ ℤ)
22	21	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) → 𝑚 ∈ ℤ)
23	22	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ ((𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → 𝑚 ∈ ℤ)
24		2cshw 13410	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ Word 𝑉 ∧ 𝑘 ∈ ℤ ∧ 𝑚 ∈ ℤ) → ((𝑧 cyclShift 𝑘) cyclShift 𝑚) = (𝑧 cyclShift (𝑘 + 𝑚)))
25	18, 20, 23, 24	syl3anc 1318	. . . . . . . . . . . . . . 15 ⊢ (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ ((𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → ((𝑧 cyclShift 𝑘) cyclShift 𝑚) = (𝑧 cyclShift (𝑘 + 𝑚)))
26	25	eqeq2d 2620	. . . . . . . . . . . . . 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 500	. . . . . . . . . . . . 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 553	. . . . . . . . . . . 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 636	. . . . . . . . . . 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 84	. . . . . . . . . 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 93	. . . . . . . . 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 444	. . . . . . . 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 481	. . . . . 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 229	. . . . 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 468	. . . 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 445	. . 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 6557	. . . . 5 ⊢ (𝑛 = (𝑘 + 𝑚) → (𝑧 cyclShift 𝑛) = (𝑧 cyclShift (𝑘 + 𝑚)))
39	38	eqeq2d 2620	. . . 4 ⊢ (𝑛 = (𝑘 + 𝑚) → (𝑥 = (𝑧 cyclShift 𝑛) ↔ 𝑥 = (𝑧 cyclShift (𝑘 + 𝑚))))
40	39	rspcev 3282	. . 3 ⊢ (((𝑘 + 𝑚) ∈ (0...(#‘𝑧)) ∧ 𝑥 = (𝑧 cyclShift (𝑘 + 𝑚))) → ∃𝑛 ∈ (0...(#‘𝑧))𝑥 = (𝑧 cyclShift 𝑛))
41	37, 40	syl6com 36	. 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)))
42		elfz2 12204	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ (0...(#‘𝑧)) ↔ ((0 ∈ ℤ ∧ (#‘𝑧) ∈ ℤ ∧ 𝑘 ∈ ℤ) ∧ (0 ≤ 𝑘 ∧ 𝑘 ≤ (#‘𝑧))))
43		nn0z 11277	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑚 ∈ ℕ₀ → 𝑚 ∈ ℤ)
44		zaddcl 11294	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑘 ∈ ℤ ∧ 𝑚 ∈ ℤ) → (𝑘 + 𝑚) ∈ ℤ)
45	44	ex 449	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑘 ∈ ℤ → (𝑚 ∈ ℤ → (𝑘 + 𝑚) ∈ ℤ))
46	45	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((#‘𝑧) ∈ ℤ ∧ 𝑘 ∈ ℤ) → (𝑚 ∈ ℤ → (𝑘 + 𝑚) ∈ ℤ))
47	46	impcom 445	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑚 ∈ ℤ ∧ ((#‘𝑧) ∈ ℤ ∧ 𝑘 ∈ ℤ)) → (𝑘 + 𝑚) ∈ ℤ)
48		simprl 790	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑚 ∈ ℤ ∧ ((#‘𝑧) ∈ ℤ ∧ 𝑘 ∈ ℤ)) → (#‘𝑧) ∈ ℤ)
49	47, 48	zsubcld 11363	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑚 ∈ ℤ ∧ ((#‘𝑧) ∈ ℤ ∧ 𝑘 ∈ ℤ)) → ((𝑘 + 𝑚) − (#‘𝑧)) ∈ ℤ)
50	49	ex 449	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑚 ∈ ℤ → (((#‘𝑧) ∈ ℤ ∧ 𝑘 ∈ ℤ) → ((𝑘 + 𝑚) − (#‘𝑧)) ∈ ℤ))
51	43, 50	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑚 ∈ ℕ₀ → (((#‘𝑧) ∈ ℤ ∧ 𝑘 ∈ ℤ) → ((𝑘 + 𝑚) − (#‘𝑧)) ∈ ℤ))
52	51	com12 32	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((#‘𝑧) ∈ ℤ ∧ 𝑘 ∈ ℤ) → (𝑚 ∈ ℕ₀ → ((𝑘 + 𝑚) − (#‘𝑧)) ∈ ℤ))
53	52	3adant1 1072	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((0 ∈ ℤ ∧ (#‘𝑧) ∈ ℤ ∧ 𝑘 ∈ ℤ) → (𝑚 ∈ ℕ₀ → ((𝑘 + 𝑚) − (#‘𝑧)) ∈ ℤ))
54	53	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ (((0 ∈ ℤ ∧ (#‘𝑧) ∈ ℤ ∧ 𝑘 ∈ ℤ) ∧ (0 ≤ 𝑘 ∧ 𝑘 ≤ (#‘𝑧))) → (𝑚 ∈ ℕ₀ → ((𝑘 + 𝑚) − (#‘𝑧)) ∈ ℤ))
55	42, 54	sylbi 206	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ (0...(#‘𝑧)) → (𝑚 ∈ ℕ₀ → ((𝑘 + 𝑚) − (#‘𝑧)) ∈ ℤ))
56	6, 55	mpan9 485	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) → ((𝑘 + 𝑚) − (#‘𝑧)) ∈ ℤ)
57	56	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → ((𝑘 + 𝑚) − (#‘𝑧)) ∈ ℤ)
58		elfz2nn0 12300	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 ∈ (0...(#‘𝑧)) ↔ (𝑘 ∈ ℕ₀ ∧ (#‘𝑧) ∈ ℕ₀ ∧ 𝑘 ≤ (#‘𝑧)))
59		nn0re 11178	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑘 ∈ ℕ₀ → 𝑘 ∈ ℝ)
60		nn0re 11178	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((#‘𝑧) ∈ ℕ₀ → (#‘𝑧) ∈ ℝ)
61	59, 60	anim12i 588	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑘 ∈ ℕ₀ ∧ (#‘𝑧) ∈ ℕ₀) → (𝑘 ∈ ℝ ∧ (#‘𝑧) ∈ ℝ))
62		nn0re 11178	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑚 ∈ ℕ₀ → 𝑚 ∈ ℝ)
63	61, 62	anim12i 588	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑘 ∈ ℕ₀ ∧ (#‘𝑧) ∈ ℕ₀) ∧ 𝑚 ∈ ℕ₀) → ((𝑘 ∈ ℝ ∧ (#‘𝑧) ∈ ℝ) ∧ 𝑚 ∈ ℝ))
64		simplr 788	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑘 ∈ ℝ ∧ (#‘𝑧) ∈ ℝ) ∧ 𝑚 ∈ ℝ) → (#‘𝑧) ∈ ℝ)
65		readdcl 9898	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑘 ∈ ℝ ∧ 𝑚 ∈ ℝ) → (𝑘 + 𝑚) ∈ ℝ)
66	65	adantlr 747	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑘 ∈ ℝ ∧ (#‘𝑧) ∈ ℝ) ∧ 𝑚 ∈ ℝ) → (𝑘 + 𝑚) ∈ ℝ)
67	64, 66	ltnled 10063	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑘 ∈ ℝ ∧ (#‘𝑧) ∈ ℝ) ∧ 𝑚 ∈ ℝ) → ((#‘𝑧) < (𝑘 + 𝑚) ↔ ¬ (𝑘 + 𝑚) ≤ (#‘𝑧)))
68	64, 66	posdifd 10493	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑘 ∈ ℝ ∧ (#‘𝑧) ∈ ℝ) ∧ 𝑚 ∈ ℝ) → ((#‘𝑧) < (𝑘 + 𝑚) ↔ 0 < ((𝑘 + 𝑚) − (#‘𝑧))))
69	68	biimpd 218	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑘 ∈ ℝ ∧ (#‘𝑧) ∈ ℝ) ∧ 𝑚 ∈ ℝ) → ((#‘𝑧) < (𝑘 + 𝑚) → 0 < ((𝑘 + 𝑚) − (#‘𝑧))))
70	67, 69	sylbird 249	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑘 ∈ ℝ ∧ (#‘𝑧) ∈ ℝ) ∧ 𝑚 ∈ ℝ) → (¬ (𝑘 + 𝑚) ≤ (#‘𝑧) → 0 < ((𝑘 + 𝑚) − (#‘𝑧))))
71	63, 70	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑘 ∈ ℕ₀ ∧ (#‘𝑧) ∈ ℕ₀) ∧ 𝑚 ∈ ℕ₀) → (¬ (𝑘 + 𝑚) ≤ (#‘𝑧) → 0 < ((𝑘 + 𝑚) − (#‘𝑧))))
72	71	ex 449	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑘 ∈ ℕ₀ ∧ (#‘𝑧) ∈ ℕ₀) → (𝑚 ∈ ℕ₀ → (¬ (𝑘 + 𝑚) ≤ (#‘𝑧) → 0 < ((𝑘 + 𝑚) − (#‘𝑧)))))
73	72	3adant3 1074	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑘 ∈ ℕ₀ ∧ (#‘𝑧) ∈ ℕ₀ ∧ 𝑘 ≤ (#‘𝑧)) → (𝑚 ∈ ℕ₀ → (¬ (𝑘 + 𝑚) ≤ (#‘𝑧) → 0 < ((𝑘 + 𝑚) − (#‘𝑧)))))
74	58, 73	sylbi 206	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ (0...(#‘𝑧)) → (𝑚 ∈ ℕ₀ → (¬ (𝑘 + 𝑚) ≤ (#‘𝑧) → 0 < ((𝑘 + 𝑚) − (#‘𝑧)))))
75	6, 74	mpan9 485	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) → (¬ (𝑘 + 𝑚) ≤ (#‘𝑧) → 0 < ((𝑘 + 𝑚) − (#‘𝑧))))
76	75	com12 32	. . . . . . . . . . . . . . . . . 18 ⊢ (¬ (𝑘 + 𝑚) ≤ (#‘𝑧) → ((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) → 0 < ((𝑘 + 𝑚) − (#‘𝑧))))
77	76	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑) → ((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) → 0 < ((𝑘 + 𝑚) − (#‘𝑧))))
78	77	impcom 445	. . . . . . . . . . . . . . . 16 ⊢ (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → 0 < ((𝑘 + 𝑚) − (#‘𝑧)))
79		elnnz 11264	. . . . . . . . . . . . . . . 16 ⊢ (((𝑘 + 𝑚) − (#‘𝑧)) ∈ ℕ ↔ (((𝑘 + 𝑚) − (#‘𝑧)) ∈ ℤ ∧ 0 < ((𝑘 + 𝑚) − (#‘𝑧))))
80	57, 78, 79	sylanbrc 695	. . . . . . . . . . . . . . 15 ⊢ (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → ((𝑘 + 𝑚) − (#‘𝑧)) ∈ ℕ)
81	80	nnnn0d 11228	. . . . . . . . . . . . . 14 ⊢ (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → ((𝑘 + 𝑚) − (#‘𝑧)) ∈ ℕ₀)
82	10	ad2antlr 759	. . . . . . . . . . . . . 14 ⊢ (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → (#‘𝑧) ∈ ℕ₀)
83		cshwcsh2id.2	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦)))
84		oveq2 6557	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((#‘𝑦) = (#‘𝑧) → (0...(#‘𝑦)) = (0...(#‘𝑧)))
85	84	eleq2d 2673	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((#‘𝑦) = (#‘𝑧) → (𝑚 ∈ (0...(#‘𝑦)) ↔ 𝑚 ∈ (0...(#‘𝑧))))
86	85	anbi1d 737	. . . . . . . . . . . . . . . . . . 19 ⊢ ((#‘𝑦) = (#‘𝑧) → ((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ↔ (𝑚 ∈ (0...(#‘𝑧)) ∧ 𝑘 ∈ (0...(#‘𝑧)))))
87		elfz2nn0 12300	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑚 ∈ (0...(#‘𝑧)) ↔ (𝑚 ∈ ℕ₀ ∧ (#‘𝑧) ∈ ℕ₀ ∧ 𝑚 ≤ (#‘𝑧)))
88	59	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑘 ∈ ℕ₀ ∧ (#‘𝑧) ∈ ℕ₀) → 𝑘 ∈ ℝ)
89	88, 62	anim12i 588	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝑘 ∈ ℕ₀ ∧ (#‘𝑧) ∈ ℕ₀) ∧ 𝑚 ∈ ℕ₀) → (𝑘 ∈ ℝ ∧ 𝑚 ∈ ℝ))
90	60, 60	jca 553	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((#‘𝑧) ∈ ℕ₀ → ((#‘𝑧) ∈ ℝ ∧ (#‘𝑧) ∈ ℝ))
91	90	ad2antlr 759	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝑘 ∈ ℕ₀ ∧ (#‘𝑧) ∈ ℕ₀) ∧ 𝑚 ∈ ℕ₀) → ((#‘𝑧) ∈ ℝ ∧ (#‘𝑧) ∈ ℝ))
92		le2add 10389	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝑘 ∈ ℝ ∧ 𝑚 ∈ ℝ) ∧ ((#‘𝑧) ∈ ℝ ∧ (#‘𝑧) ∈ ℝ)) → ((𝑘 ≤ (#‘𝑧) ∧ 𝑚 ≤ (#‘𝑧)) → (𝑘 + 𝑚) ≤ ((#‘𝑧) + (#‘𝑧))))
93	89, 91, 92	syl2anc 691	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝑘 ∈ ℕ₀ ∧ (#‘𝑧) ∈ ℕ₀) ∧ 𝑚 ∈ ℕ₀) → ((𝑘 ≤ (#‘𝑧) ∧ 𝑚 ≤ (#‘𝑧)) → (𝑘 + 𝑚) ≤ ((#‘𝑧) + (#‘𝑧))))
94		nn0readdcl 11234	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑘 ∈ ℕ₀ ∧ 𝑚 ∈ ℕ₀) → (𝑘 + 𝑚) ∈ ℝ)
95	94	adantlr 747	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝑘 ∈ ℕ₀ ∧ (#‘𝑧) ∈ ℕ₀) ∧ 𝑚 ∈ ℕ₀) → (𝑘 + 𝑚) ∈ ℝ)
96	60	ad2antlr 759	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝑘 ∈ ℕ₀ ∧ (#‘𝑧) ∈ ℕ₀) ∧ 𝑚 ∈ ℕ₀) → (#‘𝑧) ∈ ℝ)
97	95, 96, 96	lesubadd2d 10505	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝑘 ∈ ℕ₀ ∧ (#‘𝑧) ∈ ℕ₀) ∧ 𝑚 ∈ ℕ₀) → (((𝑘 + 𝑚) − (#‘𝑧)) ≤ (#‘𝑧) ↔ (𝑘 + 𝑚) ≤ ((#‘𝑧) + (#‘𝑧))))
98	93, 97	sylibrd 248	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝑘 ∈ ℕ₀ ∧ (#‘𝑧) ∈ ℕ₀) ∧ 𝑚 ∈ ℕ₀) → ((𝑘 ≤ (#‘𝑧) ∧ 𝑚 ≤ (#‘𝑧)) → ((𝑘 + 𝑚) − (#‘𝑧)) ≤ (#‘𝑧)))
99	98	expcomd 453	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑘 ∈ ℕ₀ ∧ (#‘𝑧) ∈ ℕ₀) ∧ 𝑚 ∈ ℕ₀) → (𝑚 ≤ (#‘𝑧) → (𝑘 ≤ (#‘𝑧) → ((𝑘 + 𝑚) − (#‘𝑧)) ≤ (#‘𝑧))))
100	99	ex 449	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑘 ∈ ℕ₀ ∧ (#‘𝑧) ∈ ℕ₀) → (𝑚 ∈ ℕ₀ → (𝑚 ≤ (#‘𝑧) → (𝑘 ≤ (#‘𝑧) → ((𝑘 + 𝑚) − (#‘𝑧)) ≤ (#‘𝑧)))))
101	100	com24 93	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑘 ∈ ℕ₀ ∧ (#‘𝑧) ∈ ℕ₀) → (𝑘 ≤ (#‘𝑧) → (𝑚 ≤ (#‘𝑧) → (𝑚 ∈ ℕ₀ → ((𝑘 + 𝑚) − (#‘𝑧)) ≤ (#‘𝑧)))))
102	101	3impia 1253	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑘 ∈ ℕ₀ ∧ (#‘𝑧) ∈ ℕ₀ ∧ 𝑘 ≤ (#‘𝑧)) → (𝑚 ≤ (#‘𝑧) → (𝑚 ∈ ℕ₀ → ((𝑘 + 𝑚) − (#‘𝑧)) ≤ (#‘𝑧))))
103	102	com13 86	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑚 ∈ ℕ₀ → (𝑚 ≤ (#‘𝑧) → ((𝑘 ∈ ℕ₀ ∧ (#‘𝑧) ∈ ℕ₀ ∧ 𝑘 ≤ (#‘𝑧)) → ((𝑘 + 𝑚) − (#‘𝑧)) ≤ (#‘𝑧))))
104	103	imp 444	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑚 ∈ ℕ₀ ∧ 𝑚 ≤ (#‘𝑧)) → ((𝑘 ∈ ℕ₀ ∧ (#‘𝑧) ∈ ℕ₀ ∧ 𝑘 ≤ (#‘𝑧)) → ((𝑘 + 𝑚) − (#‘𝑧)) ≤ (#‘𝑧)))
105	58, 104	syl5bi 231	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑚 ∈ ℕ₀ ∧ 𝑚 ≤ (#‘𝑧)) → (𝑘 ∈ (0...(#‘𝑧)) → ((𝑘 + 𝑚) − (#‘𝑧)) ≤ (#‘𝑧)))
106	105	3adant2 1073	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑚 ∈ ℕ₀ ∧ (#‘𝑧) ∈ ℕ₀ ∧ 𝑚 ≤ (#‘𝑧)) → (𝑘 ∈ (0...(#‘𝑧)) → ((𝑘 + 𝑚) − (#‘𝑧)) ≤ (#‘𝑧)))
107	87, 106	sylbi 206	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑚 ∈ (0...(#‘𝑧)) → (𝑘 ∈ (0...(#‘𝑧)) → ((𝑘 + 𝑚) − (#‘𝑧)) ≤ (#‘𝑧)))
108	107	imp 444	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑚 ∈ (0...(#‘𝑧)) ∧ 𝑘 ∈ (0...(#‘𝑧))) → ((𝑘 + 𝑚) − (#‘𝑧)) ≤ (#‘𝑧))
109	86, 108	syl6bi 242	. . . . . . . . . . . . . . . . . 18 ⊢ ((#‘𝑦) = (#‘𝑧) → ((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) → ((𝑘 + 𝑚) − (#‘𝑧)) ≤ (#‘𝑧)))
110	109	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ (((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦)) → ((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) → ((𝑘 + 𝑚) − (#‘𝑧)) ≤ (#‘𝑧)))
111	83, 110	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) → ((𝑘 + 𝑚) − (#‘𝑧)) ≤ (#‘𝑧)))
112	111	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑) → ((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) → ((𝑘 + 𝑚) − (#‘𝑧)) ≤ (#‘𝑧)))
113	112	impcom 445	. . . . . . . . . . . . . 14 ⊢ (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → ((𝑘 + 𝑚) − (#‘𝑧)) ≤ (#‘𝑧))
114		elfz2nn0 12300	. . . . . . . . . . . . . 14 ⊢ (((𝑘 + 𝑚) − (#‘𝑧)) ∈ (0...(#‘𝑧)) ↔ (((𝑘 + 𝑚) − (#‘𝑧)) ∈ ℕ₀ ∧ (#‘𝑧) ∈ ℕ₀ ∧ ((𝑘 + 𝑚) − (#‘𝑧)) ≤ (#‘𝑧)))
115	81, 82, 113, 114	syl3anbrc 1239	. . . . . . . . . . . . 13 ⊢ (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → ((𝑘 + 𝑚) − (#‘𝑧)) ∈ (0...(#‘𝑧)))
116	115	adantr 480	. . . . . . . . . . . 12 ⊢ ((((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) ∧ 𝑥 = ((𝑧 cyclShift 𝑘) cyclShift 𝑚)) → ((𝑘 + 𝑚) − (#‘𝑧)) ∈ (0...(#‘𝑧)))
117	16	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑) → 𝑧 ∈ Word 𝑉)
118	117	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → 𝑧 ∈ Word 𝑉)
119	19	ad2antlr 759	. . . . . . . . . . . . . . . 16 ⊢ (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → 𝑘 ∈ ℤ)
120	22	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → 𝑚 ∈ ℤ)
121	118, 119, 120, 24	syl3anc 1318	. . . . . . . . . . . . . . 15 ⊢ (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → ((𝑧 cyclShift 𝑘) cyclShift 𝑚) = (𝑧 cyclShift (𝑘 + 𝑚)))
122	19, 21, 44	syl2anr 494	. . . . . . . . . . . . . . . 16 ⊢ ((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) → (𝑘 + 𝑚) ∈ ℤ)
123		cshwsublen 13393	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ Word 𝑉 ∧ (𝑘 + 𝑚) ∈ ℤ) → (𝑧 cyclShift (𝑘 + 𝑚)) = (𝑧 cyclShift ((𝑘 + 𝑚) − (#‘𝑧))))
124	117, 122, 123	syl2anr 494	. . . . . . . . . . . . . . 15 ⊢ (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → (𝑧 cyclShift (𝑘 + 𝑚)) = (𝑧 cyclShift ((𝑘 + 𝑚) − (#‘𝑧))))
125	121, 124	eqtrd 2644	. . . . . . . . . . . . . 14 ⊢ (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → ((𝑧 cyclShift 𝑘) cyclShift 𝑚) = (𝑧 cyclShift ((𝑘 + 𝑚) − (#‘𝑧))))
126	125	eqeq2d 2620	. . . . . . . . . . . . 13 ⊢ (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → (𝑥 = ((𝑧 cyclShift 𝑘) cyclShift 𝑚) ↔ 𝑥 = (𝑧 cyclShift ((𝑘 + 𝑚) − (#‘𝑧)))))
127	126	biimpa 500	. . . . . . . . . . . 12 ⊢ ((((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) ∧ 𝑥 = ((𝑧 cyclShift 𝑘) cyclShift 𝑚)) → 𝑥 = (𝑧 cyclShift ((𝑘 + 𝑚) − (#‘𝑧))))
128	116, 127	jca 553	. . . . . . . . . . 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)))))
129	128	exp41 636	. . . . . . . . . 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))))))))
130	129	com23 84	. . . . . . . . 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))))))))
131	130	com24 93	. . . . . . . 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))))))))
132	131	imp 444	. . . . . . 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)))))))
133	3, 132	syl6bi 242	. . . . . 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))))))))
134	133	com23 84	. . . . 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))))))))
135	134	impcom 445	. . . 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)))))))
136	135	impcom 445	. . 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))))))
137		oveq2 6557	. . . . 5 ⊢ (𝑛 = ((𝑘 + 𝑚) − (#‘𝑧)) → (𝑧 cyclShift 𝑛) = (𝑧 cyclShift ((𝑘 + 𝑚) − (#‘𝑧))))
138	137	eqeq2d 2620	. . . 4 ⊢ (𝑛 = ((𝑘 + 𝑚) − (#‘𝑧)) → (𝑥 = (𝑧 cyclShift 𝑛) ↔ 𝑥 = (𝑧 cyclShift ((𝑘 + 𝑚) − (#‘𝑧)))))
139	138	rspcev 3282	. . 3 ⊢ ((((𝑘 + 𝑚) − (#‘𝑧)) ∈ (0...(#‘𝑧)) ∧ 𝑥 = (𝑧 cyclShift ((𝑘 + 𝑚) − (#‘𝑧)))) → ∃𝑛 ∈ (0...(#‘𝑧))𝑥 = (𝑧 cyclShift 𝑛))
140	136, 139	syl6com 36	. 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)))
141	41, 140	pm2.61ian 827	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 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 ℝcr 9814 0cc0 9815 + caddc 9818 < clt 9953 ≤ cle 9954 − cmin 10145 ℕcn 10897 ℕ₀cn0 11169 ℤcz 11254 ...cfz 12197 #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: erclwwlktr 26343 erclwwlkntr 26355 erclwwlkstr 41243 erclwwlksntr 41255

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