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Theorem cshwcsh2id 14800

Description: A cyclically shifted word can be reconstructed by cyclically shifting it again twice. Lemma for erclwwlktr 29957 and erclwwlkntr 30006. (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 7396	. . . . . . . . 9 ⊢ (𝑦 = (𝑧 cyclShift 𝑘) → (𝑦 cyclShift 𝑚) = ((𝑧 cyclShift 𝑘) cyclShift 𝑚))
2	1	eqeq2d 2741	. . . . . . . 8 ⊢ (𝑦 = (𝑧 cyclShift 𝑘) → (𝑥 = (𝑦 cyclShift 𝑚) ↔ 𝑥 = ((𝑧 cyclShift 𝑘) cyclShift 𝑚)))
3	2	anbi2d 630	. . . . . . 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 13587	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ (0...(♯‘𝑧)) → 𝑘 ∈ ℕ₀)
6		elfznn0 13587	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 ∈ (0...(♯‘𝑦)) → 𝑚 ∈ ℕ₀)
7		nn0addcl 12483	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑘 ∈ ℕ₀ ∧ 𝑚 ∈ ℕ₀) → (𝑘 + 𝑚) ∈ ℕ₀)
8	5, 6, 7	syl2anr 597	. . . . . . . . . . . . . . . 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 13588	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ (0...(♯‘𝑧)) → (♯‘𝑧) ∈ ℕ₀)
11	10	ad2antlr 727	. . . . . . . . . . . . . . 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 13585	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 + 𝑚) ∈ (0...(♯‘𝑧)) ↔ ((𝑘 + 𝑚) ∈ ℕ₀ ∧ (♯‘𝑧) ∈ ℕ₀ ∧ (𝑘 + 𝑚) ≤ (♯‘𝑧)))
14	9, 11, 12, 13	syl3anbrc 1344	. . . . . . . . . . . . . 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 13491	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ (0...(♯‘𝑧)) → 𝑘 ∈ ℤ)
20	19	ad2antlr 727	. . . . . . . . . . . . . . . 16 ⊢ (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ ((𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) → 𝑘 ∈ ℤ)
21		elfzelz 13491	. . . . . . . . . . . . . . . . . 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 14784	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ Word 𝑉 ∧ 𝑘 ∈ ℤ ∧ 𝑚 ∈ ℤ) → ((𝑧 cyclShift 𝑘) cyclShift 𝑚) = (𝑧 cyclShift (𝑘 + 𝑚)))
25	18, 20, 23, 24	syl3anc 1373	. . . . . . . . . . . . . . 15 ⊢ (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ ((𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) → ((𝑧 cyclShift 𝑘) cyclShift 𝑚) = (𝑧 cyclShift (𝑘 + 𝑚)))
26	25	eqeq2d 2741	. . . . . . . . . . . . . 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 476	. . . . . . . . . . . . 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 511	. . . . . . . . . . . 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 434	. . . . . . . . . . 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 406	. . . . . . . 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 240	. . . . 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 458	. . . 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 407	. . 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 7397	. . . 4 ⊢ (𝑛 = (𝑘 + 𝑚) → (𝑧 cyclShift 𝑛) = (𝑧 cyclShift (𝑘 + 𝑚)))
39	38	rspceeqv 3614	. . 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 13481	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ (0...(♯‘𝑧)) ↔ ((0 ∈ ℤ ∧ (♯‘𝑧) ∈ ℤ ∧ 𝑘 ∈ ℤ) ∧ (0 ≤ 𝑘 ∧ 𝑘 ≤ (♯‘𝑧))))
42		nn0z 12560	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑚 ∈ ℕ₀ → 𝑚 ∈ ℤ)
43		zaddcl 12579	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑘 ∈ ℤ ∧ 𝑚 ∈ ℤ) → (𝑘 + 𝑚) ∈ ℤ)
44	43	ex 412	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑘 ∈ ℤ → (𝑚 ∈ ℤ → (𝑘 + 𝑚) ∈ ℤ))
45	44	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((♯‘𝑧) ∈ ℤ ∧ 𝑘 ∈ ℤ) → (𝑚 ∈ ℤ → (𝑘 + 𝑚) ∈ ℤ))
46	45	impcom 407	. . . . . . . . . . . . . . . . . . . . . . . . 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 12649	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑚 ∈ ℤ ∧ ((♯‘𝑧) ∈ ℤ ∧ 𝑘 ∈ ℤ)) → ((𝑘 + 𝑚) − (♯‘𝑧)) ∈ ℤ)
49	48	ex 412	. . . . . . . . . . . . . . . . . . . . . . 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 1130	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((0 ∈ ℤ ∧ (♯‘𝑧) ∈ ℤ ∧ 𝑘 ∈ ℤ) → (𝑚 ∈ ℕ₀ → ((𝑘 + 𝑚) − (♯‘𝑧)) ∈ ℤ))
53	52	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ (((0 ∈ ℤ ∧ (♯‘𝑧) ∈ ℤ ∧ 𝑘 ∈ ℤ) ∧ (0 ≤ 𝑘 ∧ 𝑘 ≤ (♯‘𝑧))) → (𝑚 ∈ ℕ₀ → ((𝑘 + 𝑚) − (♯‘𝑧)) ∈ ℤ))
54	41, 53	sylbi 217	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ (0...(♯‘𝑧)) → (𝑚 ∈ ℕ₀ → ((𝑘 + 𝑚) − (♯‘𝑧)) ∈ ℤ))
55	6, 54	mpan9 506	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) → ((𝑘 + 𝑚) − (♯‘𝑧)) ∈ ℤ)
56	55	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) → ((𝑘 + 𝑚) − (♯‘𝑧)) ∈ ℤ)
57		elfz2nn0 13585	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 ∈ (0...(♯‘𝑧)) ↔ (𝑘 ∈ ℕ₀ ∧ (♯‘𝑧) ∈ ℕ₀ ∧ 𝑘 ≤ (♯‘𝑧)))
58		nn0re 12457	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑘 ∈ ℕ₀ → 𝑘 ∈ ℝ)
59		nn0re 12457	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((♯‘𝑧) ∈ ℕ₀ → (♯‘𝑧) ∈ ℝ)
60	58, 59	anim12i 613	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑘 ∈ ℕ₀ ∧ (♯‘𝑧) ∈ ℕ₀) → (𝑘 ∈ ℝ ∧ (♯‘𝑧) ∈ ℝ))
61		nn0re 12457	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑚 ∈ ℕ₀ → 𝑚 ∈ ℝ)
62	60, 61	anim12i 613	. . . . . . . . . . . . . . . . . . . . . . . 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 11157	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑘 ∈ ℝ ∧ 𝑚 ∈ ℝ) → (𝑘 + 𝑚) ∈ ℝ)
65	64	adantlr 715	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑘 ∈ ℝ ∧ (♯‘𝑧) ∈ ℝ) ∧ 𝑚 ∈ ℝ) → (𝑘 + 𝑚) ∈ ℝ)
66	63, 65	ltnled 11327	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑘 ∈ ℝ ∧ (♯‘𝑧) ∈ ℝ) ∧ 𝑚 ∈ ℝ) → ((♯‘𝑧) < (𝑘 + 𝑚) ↔ ¬ (𝑘 + 𝑚) ≤ (♯‘𝑧)))
67	63, 65	posdifd 11771	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑘 ∈ ℝ ∧ (♯‘𝑧) ∈ ℝ) ∧ 𝑚 ∈ ℝ) → ((♯‘𝑧) < (𝑘 + 𝑚) ↔ 0 < ((𝑘 + 𝑚) − (♯‘𝑧))))
68	67	biimpd 229	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑘 ∈ ℝ ∧ (♯‘𝑧) ∈ ℝ) ∧ 𝑚 ∈ ℝ) → ((♯‘𝑧) < (𝑘 + 𝑚) → 0 < ((𝑘 + 𝑚) − (♯‘𝑧))))
69	66, 68	sylbird 260	. . . . . . . . . . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑘 ∈ ℕ₀ ∧ (♯‘𝑧) ∈ ℕ₀) → (𝑚 ∈ ℕ₀ → (¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) → 0 < ((𝑘 + 𝑚) − (♯‘𝑧)))))
72	71	3adant3 1132	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑘 ∈ ℕ₀ ∧ (♯‘𝑧) ∈ ℕ₀ ∧ 𝑘 ≤ (♯‘𝑧)) → (𝑚 ∈ ℕ₀ → (¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) → 0 < ((𝑘 + 𝑚) − (♯‘𝑧)))))
73	57, 72	sylbi 217	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ (0...(♯‘𝑧)) → (𝑚 ∈ ℕ₀ → (¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) → 0 < ((𝑘 + 𝑚) − (♯‘𝑧)))))
74	6, 73	mpan9 506	. . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . 17 ⊢ ((¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑) → ((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) → 0 < ((𝑘 + 𝑚) − (♯‘𝑧))))
77	76	impcom 407	. . . . . . . . . . . . . . . 16 ⊢ (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) → 0 < ((𝑘 + 𝑚) − (♯‘𝑧)))
78		elnnz 12545	. . . . . . . . . . . . . . . 16 ⊢ (((𝑘 + 𝑚) − (♯‘𝑧)) ∈ ℕ ↔ (((𝑘 + 𝑚) − (♯‘𝑧)) ∈ ℤ ∧ 0 < ((𝑘 + 𝑚) − (♯‘𝑧))))
79	56, 77, 78	sylanbrc 583	. . . . . . . . . . . . . . 15 ⊢ (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) → ((𝑘 + 𝑚) − (♯‘𝑧)) ∈ ℕ)
80	79	nnnn0d 12509	. . . . . . . . . . . . . 14 ⊢ (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) → ((𝑘 + 𝑚) − (♯‘𝑧)) ∈ ℕ₀)
81	10	ad2antlr 727	. . . . . . . . . . . . . 14 ⊢ (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) → (♯‘𝑧) ∈ ℕ₀)
82		cshwcsh2id.2	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((♯‘𝑦) = (♯‘𝑧) ∧ (♯‘𝑥) = (♯‘𝑦)))
83		oveq2 7397	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((♯‘𝑦) = (♯‘𝑧) → (0...(♯‘𝑦)) = (0...(♯‘𝑧)))
84	83	eleq2d 2815	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((♯‘𝑦) = (♯‘𝑧) → (𝑚 ∈ (0...(♯‘𝑦)) ↔ 𝑚 ∈ (0...(♯‘𝑧))))
85	84	anbi1d 631	. . . . . . . . . . . . . . . . . . 19 ⊢ ((♯‘𝑦) = (♯‘𝑧) → ((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ↔ (𝑚 ∈ (0...(♯‘𝑧)) ∧ 𝑘 ∈ (0...(♯‘𝑧)))))
86		elfz2nn0 13585	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑚 ∈ (0...(♯‘𝑧)) ↔ (𝑚 ∈ ℕ₀ ∧ (♯‘𝑧) ∈ ℕ₀ ∧ 𝑚 ≤ (♯‘𝑧)))
87	58	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑘 ∈ ℕ₀ ∧ (♯‘𝑧) ∈ ℕ₀) → 𝑘 ∈ ℝ)
88	87, 61	anim12i 613	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝑘 ∈ ℕ₀ ∧ (♯‘𝑧) ∈ ℕ₀) ∧ 𝑚 ∈ ℕ₀) → (𝑘 ∈ ℝ ∧ 𝑚 ∈ ℝ))
89	59, 59	jca 511	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((♯‘𝑧) ∈ ℕ₀ → ((♯‘𝑧) ∈ ℝ ∧ (♯‘𝑧) ∈ ℝ))
90	89	ad2antlr 727	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝑘 ∈ ℕ₀ ∧ (♯‘𝑧) ∈ ℕ₀) ∧ 𝑚 ∈ ℕ₀) → ((♯‘𝑧) ∈ ℝ ∧ (♯‘𝑧) ∈ ℝ))
91		le2add 11666	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝑘 ∈ ℝ ∧ 𝑚 ∈ ℝ) ∧ ((♯‘𝑧) ∈ ℝ ∧ (♯‘𝑧) ∈ ℝ)) → ((𝑘 ≤ (♯‘𝑧) ∧ 𝑚 ≤ (♯‘𝑧)) → (𝑘 + 𝑚) ≤ ((♯‘𝑧) + (♯‘𝑧))))
92	88, 90, 91	syl2anc 584	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝑘 ∈ ℕ₀ ∧ (♯‘𝑧) ∈ ℕ₀) ∧ 𝑚 ∈ ℕ₀) → ((𝑘 ≤ (♯‘𝑧) ∧ 𝑚 ≤ (♯‘𝑧)) → (𝑘 + 𝑚) ≤ ((♯‘𝑧) + (♯‘𝑧))))
93		nn0readdcl 12515	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑘 ∈ ℕ₀ ∧ 𝑚 ∈ ℕ₀) → (𝑘 + 𝑚) ∈ ℝ)
94	93	adantlr 715	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝑘 ∈ ℕ₀ ∧ (♯‘𝑧) ∈ ℕ₀) ∧ 𝑚 ∈ ℕ₀) → (𝑘 + 𝑚) ∈ ℝ)
95	59	ad2antlr 727	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝑘 ∈ ℕ₀ ∧ (♯‘𝑧) ∈ ℕ₀) ∧ 𝑚 ∈ ℕ₀) → (♯‘𝑧) ∈ ℝ)
96	94, 95, 95	lesubadd2d 11783	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝑘 ∈ ℕ₀ ∧ (♯‘𝑧) ∈ ℕ₀) ∧ 𝑚 ∈ ℕ₀) → (((𝑘 + 𝑚) − (♯‘𝑧)) ≤ (♯‘𝑧) ↔ (𝑘 + 𝑚) ≤ ((♯‘𝑧) + (♯‘𝑧))))
97	92, 96	sylibrd 259	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝑘 ∈ ℕ₀ ∧ (♯‘𝑧) ∈ ℕ₀) ∧ 𝑚 ∈ ℕ₀) → ((𝑘 ≤ (♯‘𝑧) ∧ 𝑚 ≤ (♯‘𝑧)) → ((𝑘 + 𝑚) − (♯‘𝑧)) ≤ (♯‘𝑧)))
98	97	expcomd 416	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑘 ∈ ℕ₀ ∧ (♯‘𝑧) ∈ ℕ₀) ∧ 𝑚 ∈ ℕ₀) → (𝑚 ≤ (♯‘𝑧) → (𝑘 ≤ (♯‘𝑧) → ((𝑘 + 𝑚) − (♯‘𝑧)) ≤ (♯‘𝑧))))
99	98	ex 412	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 1117	. . . . . . . . . . . . . . . . . . . . . . . . 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 406	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑚 ∈ ℕ₀ ∧ 𝑚 ≤ (♯‘𝑧)) → ((𝑘 ∈ ℕ₀ ∧ (♯‘𝑧) ∈ ℕ₀ ∧ 𝑘 ≤ (♯‘𝑧)) → ((𝑘 + 𝑚) − (♯‘𝑧)) ≤ (♯‘𝑧)))
104	57, 103	biimtrid 242	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑚 ∈ ℕ₀ ∧ 𝑚 ≤ (♯‘𝑧)) → (𝑘 ∈ (0...(♯‘𝑧)) → ((𝑘 + 𝑚) − (♯‘𝑧)) ≤ (♯‘𝑧)))
105	104	3adant2 1131	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑚 ∈ ℕ₀ ∧ (♯‘𝑧) ∈ ℕ₀ ∧ 𝑚 ≤ (♯‘𝑧)) → (𝑘 ∈ (0...(♯‘𝑧)) → ((𝑘 + 𝑚) − (♯‘𝑧)) ≤ (♯‘𝑧)))
106	86, 105	sylbi 217	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑚 ∈ (0...(♯‘𝑧)) → (𝑘 ∈ (0...(♯‘𝑧)) → ((𝑘 + 𝑚) − (♯‘𝑧)) ≤ (♯‘𝑧)))
107	106	imp 406	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑚 ∈ (0...(♯‘𝑧)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) → ((𝑘 + 𝑚) − (♯‘𝑧)) ≤ (♯‘𝑧))
108	85, 107	biimtrdi 253	. . . . . . . . . . . . . . . . . 18 ⊢ ((♯‘𝑦) = (♯‘𝑧) → ((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) → ((𝑘 + 𝑚) − (♯‘𝑧)) ≤ (♯‘𝑧)))
109	108	adantr 480	. . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . 15 ⊢ ((¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑) → ((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) → ((𝑘 + 𝑚) − (♯‘𝑧)) ≤ (♯‘𝑧)))
112	111	impcom 407	. . . . . . . . . . . . . 14 ⊢ (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) → ((𝑘 + 𝑚) − (♯‘𝑧)) ≤ (♯‘𝑧))
113		elfz2nn0 13585	. . . . . . . . . . . . . 14 ⊢ (((𝑘 + 𝑚) − (♯‘𝑧)) ∈ (0...(♯‘𝑧)) ↔ (((𝑘 + 𝑚) − (♯‘𝑧)) ∈ ℕ₀ ∧ (♯‘𝑧) ∈ ℕ₀ ∧ ((𝑘 + 𝑚) − (♯‘𝑧)) ≤ (♯‘𝑧)))
114	80, 81, 112, 113	syl3anbrc 1344	. . . . . . . . . . . . 13 ⊢ (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) → ((𝑘 + 𝑚) − (♯‘𝑧)) ∈ (0...(♯‘𝑧)))
115	114	adantr 480	. . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . 17 ⊢ ((¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑) → 𝑧 ∈ Word 𝑉)
117	116	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) → 𝑧 ∈ Word 𝑉)
118	19	ad2antlr 727	. . . . . . . . . . . . . . . 16 ⊢ (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) → 𝑘 ∈ ℤ)
119	22	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) → 𝑚 ∈ ℤ)
120	117, 118, 119, 24	syl3anc 1373	. . . . . . . . . . . . . . 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 597	. . . . . . . . . . . . . . . 16 ⊢ ((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) → (𝑘 + 𝑚) ∈ ℤ)
122		cshwsublen 14767	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ Word 𝑉 ∧ (𝑘 + 𝑚) ∈ ℤ) → (𝑧 cyclShift (𝑘 + 𝑚)) = (𝑧 cyclShift ((𝑘 + 𝑚) − (♯‘𝑧))))
123	116, 121, 122	syl2anr 597	. . . . . . . . . . . . . . 15 ⊢ (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) → (𝑧 cyclShift (𝑘 + 𝑚)) = (𝑧 cyclShift ((𝑘 + 𝑚) − (♯‘𝑧))))
124	120, 123	eqtrd 2765	. . . . . . . . . . . . . 14 ⊢ (((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑘 ∈ (0...(♯‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (♯‘𝑧) ∧ 𝜑)) → ((𝑧 cyclShift 𝑘) cyclShift 𝑚) = (𝑧 cyclShift ((𝑘 + 𝑚) − (♯‘𝑧))))
125	124	eqeq2d 2741	. . . . . . . . . . . . 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 476	. . . . . . . . . . . 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 511	. . . . . . . . . . 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 434	. . . . . . . . . 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 406	. . . . . . 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	biimtrdi 253	. . . . . 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 407	. . . 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 407	. . 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 7397	. . . 4 ⊢ (𝑛 = ((𝑘 + 𝑚) − (♯‘𝑧)) → (𝑧 cyclShift 𝑛) = (𝑧 cyclShift ((𝑘 + 𝑚) − (♯‘𝑧))))
137	136	rspceeqv 3614	. . 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 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ∃wrex 3054 class class class wbr 5109 ‘cfv 6513 (class class class)co 7389 ℝcr 11073 0cc0 11074 + caddc 11077 < clt 11214 ≤ cle 11215 − cmin 11411 ℕcn 12187 ℕ₀cn0 12448 ℤcz 12535 ...cfz 13474 ♯chash 14301 Word cword 14484 cyclShift ccsh 14759

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5236 ax-sep 5253 ax-nul 5263 ax-pow 5322 ax-pr 5389 ax-un 7713 ax-cnex 11130 ax-resscn 11131 ax-1cn 11132 ax-icn 11133 ax-addcl 11134 ax-addrcl 11135 ax-mulcl 11136 ax-mulrcl 11137 ax-mulcom 11138 ax-addass 11139 ax-mulass 11140 ax-distr 11141 ax-i2m1 11142 ax-1ne0 11143 ax-1rid 11144 ax-rnegex 11145 ax-rrecex 11146 ax-cnre 11147 ax-pre-lttri 11148 ax-pre-lttrn 11149 ax-pre-ltadd 11150 ax-pre-mulgt0 11151 ax-pre-sup 11152

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3756 df-csb 3865 df-dif 3919 df-un 3921 df-in 3923 df-ss 3933 df-pss 3936 df-nul 4299 df-if 4491 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4874 df-int 4913 df-iun 4959 df-br 5110 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5535 df-eprel 5540 df-po 5548 df-so 5549 df-fr 5593 df-we 5595 df-xp 5646 df-rel 5647 df-cnv 5648 df-co 5649 df-dm 5650 df-rn 5651 df-res 5652 df-ima 5653 df-pred 6276 df-ord 6337 df-on 6338 df-lim 6339 df-suc 6340 df-iota 6466 df-fun 6515 df-fn 6516 df-f 6517 df-f1 6518 df-fo 6519 df-f1o 6520 df-fv 6521 df-riota 7346 df-ov 7392 df-oprab 7393 df-mpo 7394 df-om 7845 df-1st 7970 df-2nd 7971 df-frecs 8262 df-wrecs 8293 df-recs 8342 df-rdg 8380 df-1o 8436 df-er 8673 df-en 8921 df-dom 8922 df-sdom 8923 df-fin 8924 df-sup 9399 df-inf 9400 df-card 9898 df-pnf 11216 df-mnf 11217 df-xr 11218 df-ltxr 11219 df-le 11220 df-sub 11413 df-neg 11414 df-div 11842 df-nn 12188 df-2 12250 df-n0 12449 df-z 12536 df-uz 12800 df-rp 12958 df-fz 13475 df-fzo 13622 df-fl 13760 df-mod 13838 df-hash 14302 df-word 14485 df-concat 14542 df-substr 14612 df-pfx 14642 df-csh 14760

This theorem is referenced by: erclwwlktr 29957 erclwwlkntr 30006

W3C validator