cshwcsh2id - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > cshwcsh2id		Structured version Visualization version GIF version

Theorem cshwcsh2id 13574

Description: A cyclically shifted word can be reconstructed by cyclically shifting it again twice. Lemma for erclwwlkstr 26936 and erclwwlksntr 26948. (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 6657	. . . . . . . . 9 ⊢ (𝑦 = (𝑧 cyclShift 𝑘) → (𝑦 cyclShift 𝑚) = ((𝑧 cyclShift 𝑘) cyclShift 𝑚))
2	1	eqeq2d 2632	. . . . . . . 8 ⊢ (𝑦 = (𝑧 cyclShift 𝑘) → (𝑥 = (𝑦 cyclShift 𝑚) ↔ 𝑥 = ((𝑧 cyclShift 𝑘) cyclShift 𝑚)))
3	2	anbi2d 740	. . . . . . 7 ⊢ (𝑦 = (𝑧 cyclShift 𝑘) → ((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚)) ↔ (𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑥 = ((𝑧 cyclShift 𝑘) cyclShift 𝑚))))
4	3	adantr 481	. . . . . 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 12433	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ (0...(#‘𝑧)) → 𝑘 ∈ ℕ₀)
6		elfznn0 12433	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 ∈ (0...(#‘𝑦)) → 𝑚 ∈ ℕ₀)
7		nn0addcl 11328	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑘 ∈ ℕ₀ ∧ 𝑚 ∈ ℕ₀) → (𝑘 + 𝑚) ∈ ℕ₀)
8	5, 6, 7	syl2anr 495	. . . . . . . . . . . . . . . 16 ⊢ ((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) → (𝑘 + 𝑚) ∈ ℕ₀)
9	8	adantr 481	. . . . . . . . . . . . . . 15 ⊢ (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ ((𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → (𝑘 + 𝑚) ∈ ℕ₀)
10		elfz3nn0 12434	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ (0...(#‘𝑧)) → (#‘𝑧) ∈ ℕ₀)
11	10	ad2antlr 763	. . . . . . . . . . . . . . 15 ⊢ (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ ((𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → (#‘𝑧) ∈ ℕ₀)
12		simprl 794	. . . . . . . . . . . . . . 15 ⊢ (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ ((𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → (𝑘 + 𝑚) ≤ (#‘𝑧))
13		elfz2nn0 12431	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 + 𝑚) ∈ (0...(#‘𝑧)) ↔ ((𝑘 + 𝑚) ∈ ℕ₀ ∧ (#‘𝑧) ∈ ℕ₀ ∧ (𝑘 + 𝑚) ≤ (#‘𝑧)))
14	9, 11, 12, 13	syl3anbrc 1246	. . . . . . . . . . . . . 14 ⊢ (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ ((𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → (𝑘 + 𝑚) ∈ (0...(#‘𝑧)))
15	14	adantr 481	. . . . . . . . . . . . 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 482	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑) → 𝑧 ∈ Word 𝑉)
18	17	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ ((𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → 𝑧 ∈ Word 𝑉)
19		elfzelz 12342	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ (0...(#‘𝑧)) → 𝑘 ∈ ℤ)
20	19	ad2antlr 763	. . . . . . . . . . . . . . . 16 ⊢ (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ ((𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → 𝑘 ∈ ℤ)
21		elfzelz 12342	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 ∈ (0...(#‘𝑦)) → 𝑚 ∈ ℤ)
22	21	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) → 𝑚 ∈ ℤ)
23	22	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ ((𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → 𝑚 ∈ ℤ)
24		2cshw 13559	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ Word 𝑉 ∧ 𝑘 ∈ ℤ ∧ 𝑚 ∈ ℤ) → ((𝑧 cyclShift 𝑘) cyclShift 𝑚) = (𝑧 cyclShift (𝑘 + 𝑚)))
25	18, 20, 23, 24	syl3anc 1326	. . . . . . . . . . . . . . 15 ⊢ (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ ((𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → ((𝑧 cyclShift 𝑘) cyclShift 𝑚) = (𝑧 cyclShift (𝑘 + 𝑚)))
26	25	eqeq2d 2632	. . . . . . . . . . . . . 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 501	. . . . . . . . . . . . 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 554	. . . . . . . . . . . 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 638	. . . . . . . . . . 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 445	. . . . . . . 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 482	. . . . . 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 230	. . . . 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 469	. . . 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 446	. . 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 6658	. . . . 5 ⊢ (𝑛 = (𝑘 + 𝑚) → (𝑧 cyclShift 𝑛) = (𝑧 cyclShift (𝑘 + 𝑚)))
39	38	eqeq2d 2632	. . . 4 ⊢ (𝑛 = (𝑘 + 𝑚) → (𝑥 = (𝑧 cyclShift 𝑛) ↔ 𝑥 = (𝑧 cyclShift (𝑘 + 𝑚))))
40	39	rspcev 3309	. . 3 ⊢ (((𝑘 + 𝑚) ∈ (0...(#‘𝑧)) ∧ 𝑥 = (𝑧 cyclShift (𝑘 + 𝑚))) → ∃𝑛 ∈ (0...(#‘𝑧))𝑥 = (𝑧 cyclShift 𝑛))
41	37, 40	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)))
42		elfz2 12333	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ (0...(#‘𝑧)) ↔ ((0 ∈ ℤ ∧ (#‘𝑧) ∈ ℤ ∧ 𝑘 ∈ ℤ) ∧ (0 ≤ 𝑘 ∧ 𝑘 ≤ (#‘𝑧))))
43		nn0z 11400	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑚 ∈ ℕ₀ → 𝑚 ∈ ℤ)
44		zaddcl 11417	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑘 ∈ ℤ ∧ 𝑚 ∈ ℤ) → (𝑘 + 𝑚) ∈ ℤ)
45	44	ex 450	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑘 ∈ ℤ → (𝑚 ∈ ℤ → (𝑘 + 𝑚) ∈ ℤ))
46	45	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((#‘𝑧) ∈ ℤ ∧ 𝑘 ∈ ℤ) → (𝑚 ∈ ℤ → (𝑘 + 𝑚) ∈ ℤ))
47	46	impcom 446	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑚 ∈ ℤ ∧ ((#‘𝑧) ∈ ℤ ∧ 𝑘 ∈ ℤ)) → (𝑘 + 𝑚) ∈ ℤ)
48		simprl 794	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑚 ∈ ℤ ∧ ((#‘𝑧) ∈ ℤ ∧ 𝑘 ∈ ℤ)) → (#‘𝑧) ∈ ℤ)
49	47, 48	zsubcld 11487	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑚 ∈ ℤ ∧ ((#‘𝑧) ∈ ℤ ∧ 𝑘 ∈ ℤ)) → ((𝑘 + 𝑚) − (#‘𝑧)) ∈ ℤ)
50	49	ex 450	. . . . . . . . . . . . . . . . . . . . . . 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 1079	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((0 ∈ ℤ ∧ (#‘𝑧) ∈ ℤ ∧ 𝑘 ∈ ℤ) → (𝑚 ∈ ℕ₀ → ((𝑘 + 𝑚) − (#‘𝑧)) ∈ ℤ))
54	53	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ (((0 ∈ ℤ ∧ (#‘𝑧) ∈ ℤ ∧ 𝑘 ∈ ℤ) ∧ (0 ≤ 𝑘 ∧ 𝑘 ≤ (#‘𝑧))) → (𝑚 ∈ ℕ₀ → ((𝑘 + 𝑚) − (#‘𝑧)) ∈ ℤ))
55	42, 54	sylbi 207	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ (0...(#‘𝑧)) → (𝑚 ∈ ℕ₀ → ((𝑘 + 𝑚) − (#‘𝑧)) ∈ ℤ))
56	6, 55	mpan9 486	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) → ((𝑘 + 𝑚) − (#‘𝑧)) ∈ ℤ)
57	56	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → ((𝑘 + 𝑚) − (#‘𝑧)) ∈ ℤ)
58		elfz2nn0 12431	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 ∈ (0...(#‘𝑧)) ↔ (𝑘 ∈ ℕ₀ ∧ (#‘𝑧) ∈ ℕ₀ ∧ 𝑘 ≤ (#‘𝑧)))
59		nn0re 11301	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑘 ∈ ℕ₀ → 𝑘 ∈ ℝ)
60		nn0re 11301	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((#‘𝑧) ∈ ℕ₀ → (#‘𝑧) ∈ ℝ)
61	59, 60	anim12i 590	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑘 ∈ ℕ₀ ∧ (#‘𝑧) ∈ ℕ₀) → (𝑘 ∈ ℝ ∧ (#‘𝑧) ∈ ℝ))
62		nn0re 11301	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑚 ∈ ℕ₀ → 𝑚 ∈ ℝ)
63	61, 62	anim12i 590	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑘 ∈ ℕ₀ ∧ (#‘𝑧) ∈ ℕ₀) ∧ 𝑚 ∈ ℕ₀) → ((𝑘 ∈ ℝ ∧ (#‘𝑧) ∈ ℝ) ∧ 𝑚 ∈ ℝ))
64		simplr 792	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑘 ∈ ℝ ∧ (#‘𝑧) ∈ ℝ) ∧ 𝑚 ∈ ℝ) → (#‘𝑧) ∈ ℝ)
65		readdcl 10019	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑘 ∈ ℝ ∧ 𝑚 ∈ ℝ) → (𝑘 + 𝑚) ∈ ℝ)
66	65	adantlr 751	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑘 ∈ ℝ ∧ (#‘𝑧) ∈ ℝ) ∧ 𝑚 ∈ ℝ) → (𝑘 + 𝑚) ∈ ℝ)
67	64, 66	ltnled 10184	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑘 ∈ ℝ ∧ (#‘𝑧) ∈ ℝ) ∧ 𝑚 ∈ ℝ) → ((#‘𝑧) < (𝑘 + 𝑚) ↔ ¬ (𝑘 + 𝑚) ≤ (#‘𝑧)))
68	64, 66	posdifd 10614	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑘 ∈ ℝ ∧ (#‘𝑧) ∈ ℝ) ∧ 𝑚 ∈ ℝ) → ((#‘𝑧) < (𝑘 + 𝑚) ↔ 0 < ((𝑘 + 𝑚) − (#‘𝑧))))
69	68	biimpd 219	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑘 ∈ ℝ ∧ (#‘𝑧) ∈ ℝ) ∧ 𝑚 ∈ ℝ) → ((#‘𝑧) < (𝑘 + 𝑚) → 0 < ((𝑘 + 𝑚) − (#‘𝑧))))
70	67, 69	sylbird 250	. . . . . . . . . . . . . . . . . . . . . . . 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 450	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑘 ∈ ℕ₀ ∧ (#‘𝑧) ∈ ℕ₀) → (𝑚 ∈ ℕ₀ → (¬ (𝑘 + 𝑚) ≤ (#‘𝑧) → 0 < ((𝑘 + 𝑚) − (#‘𝑧)))))
73	72	3adant3 1081	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑘 ∈ ℕ₀ ∧ (#‘𝑧) ∈ ℕ₀ ∧ 𝑘 ≤ (#‘𝑧)) → (𝑚 ∈ ℕ₀ → (¬ (𝑘 + 𝑚) ≤ (#‘𝑧) → 0 < ((𝑘 + 𝑚) − (#‘𝑧)))))
74	58, 73	sylbi 207	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ (0...(#‘𝑧)) → (𝑚 ∈ ℕ₀ → (¬ (𝑘 + 𝑚) ≤ (#‘𝑧) → 0 < ((𝑘 + 𝑚) − (#‘𝑧)))))
75	6, 74	mpan9 486	. . . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . 17 ⊢ ((¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑) → ((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) → 0 < ((𝑘 + 𝑚) − (#‘𝑧))))
78	77	impcom 446	. . . . . . . . . . . . . . . 16 ⊢ (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → 0 < ((𝑘 + 𝑚) − (#‘𝑧)))
79		elnnz 11387	. . . . . . . . . . . . . . . 16 ⊢ (((𝑘 + 𝑚) − (#‘𝑧)) ∈ ℕ ↔ (((𝑘 + 𝑚) − (#‘𝑧)) ∈ ℤ ∧ 0 < ((𝑘 + 𝑚) − (#‘𝑧))))
80	57, 78, 79	sylanbrc 698	. . . . . . . . . . . . . . 15 ⊢ (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → ((𝑘 + 𝑚) − (#‘𝑧)) ∈ ℕ)
81	80	nnnn0d 11351	. . . . . . . . . . . . . 14 ⊢ (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → ((𝑘 + 𝑚) − (#‘𝑧)) ∈ ℕ₀)
82	10	ad2antlr 763	. . . . . . . . . . . . . 14 ⊢ (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → (#‘𝑧) ∈ ℕ₀)
83		cshwcsh2id.2	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦)))
84		oveq2 6658	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((#‘𝑦) = (#‘𝑧) → (0...(#‘𝑦)) = (0...(#‘𝑧)))
85	84	eleq2d 2687	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((#‘𝑦) = (#‘𝑧) → (𝑚 ∈ (0...(#‘𝑦)) ↔ 𝑚 ∈ (0...(#‘𝑧))))
86	85	anbi1d 741	. . . . . . . . . . . . . . . . . . 19 ⊢ ((#‘𝑦) = (#‘𝑧) → ((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ↔ (𝑚 ∈ (0...(#‘𝑧)) ∧ 𝑘 ∈ (0...(#‘𝑧)))))
87		elfz2nn0 12431	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑚 ∈ (0...(#‘𝑧)) ↔ (𝑚 ∈ ℕ₀ ∧ (#‘𝑧) ∈ ℕ₀ ∧ 𝑚 ≤ (#‘𝑧)))
88	59	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑘 ∈ ℕ₀ ∧ (#‘𝑧) ∈ ℕ₀) → 𝑘 ∈ ℝ)
89	88, 62	anim12i 590	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝑘 ∈ ℕ₀ ∧ (#‘𝑧) ∈ ℕ₀) ∧ 𝑚 ∈ ℕ₀) → (𝑘 ∈ ℝ ∧ 𝑚 ∈ ℝ))
90	60, 60	jca 554	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((#‘𝑧) ∈ ℕ₀ → ((#‘𝑧) ∈ ℝ ∧ (#‘𝑧) ∈ ℝ))
91	90	ad2antlr 763	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝑘 ∈ ℕ₀ ∧ (#‘𝑧) ∈ ℕ₀) ∧ 𝑚 ∈ ℕ₀) → ((#‘𝑧) ∈ ℝ ∧ (#‘𝑧) ∈ ℝ))
92		le2add 10510	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝑘 ∈ ℝ ∧ 𝑚 ∈ ℝ) ∧ ((#‘𝑧) ∈ ℝ ∧ (#‘𝑧) ∈ ℝ)) → ((𝑘 ≤ (#‘𝑧) ∧ 𝑚 ≤ (#‘𝑧)) → (𝑘 + 𝑚) ≤ ((#‘𝑧) + (#‘𝑧))))
93	89, 91, 92	syl2anc 693	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝑘 ∈ ℕ₀ ∧ (#‘𝑧) ∈ ℕ₀) ∧ 𝑚 ∈ ℕ₀) → ((𝑘 ≤ (#‘𝑧) ∧ 𝑚 ≤ (#‘𝑧)) → (𝑘 + 𝑚) ≤ ((#‘𝑧) + (#‘𝑧))))
94		nn0readdcl 11357	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑘 ∈ ℕ₀ ∧ 𝑚 ∈ ℕ₀) → (𝑘 + 𝑚) ∈ ℝ)
95	94	adantlr 751	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝑘 ∈ ℕ₀ ∧ (#‘𝑧) ∈ ℕ₀) ∧ 𝑚 ∈ ℕ₀) → (𝑘 + 𝑚) ∈ ℝ)
96	60	ad2antlr 763	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝑘 ∈ ℕ₀ ∧ (#‘𝑧) ∈ ℕ₀) ∧ 𝑚 ∈ ℕ₀) → (#‘𝑧) ∈ ℝ)
97	95, 96, 96	lesubadd2d 10626	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝑘 ∈ ℕ₀ ∧ (#‘𝑧) ∈ ℕ₀) ∧ 𝑚 ∈ ℕ₀) → (((𝑘 + 𝑚) − (#‘𝑧)) ≤ (#‘𝑧) ↔ (𝑘 + 𝑚) ≤ ((#‘𝑧) + (#‘𝑧))))
98	93, 97	sylibrd 249	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝑘 ∈ ℕ₀ ∧ (#‘𝑧) ∈ ℕ₀) ∧ 𝑚 ∈ ℕ₀) → ((𝑘 ≤ (#‘𝑧) ∧ 𝑚 ≤ (#‘𝑧)) → ((𝑘 + 𝑚) − (#‘𝑧)) ≤ (#‘𝑧)))
99	98	expcomd 454	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑘 ∈ ℕ₀ ∧ (#‘𝑧) ∈ ℕ₀) ∧ 𝑚 ∈ ℕ₀) → (𝑚 ≤ (#‘𝑧) → (𝑘 ≤ (#‘𝑧) → ((𝑘 + 𝑚) − (#‘𝑧)) ≤ (#‘𝑧))))
100	99	ex 450	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑘 ∈ ℕ₀ ∧ (#‘𝑧) ∈ ℕ₀) → (𝑚 ∈ ℕ₀ → (𝑚 ≤ (#‘𝑧) → (𝑘 ≤ (#‘𝑧) → ((𝑘 + 𝑚) − (#‘𝑧)) ≤ (#‘𝑧)))))
101	100	com24 95	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑘 ∈ ℕ₀ ∧ (#‘𝑧) ∈ ℕ₀) → (𝑘 ≤ (#‘𝑧) → (𝑚 ≤ (#‘𝑧) → (𝑚 ∈ ℕ₀ → ((𝑘 + 𝑚) − (#‘𝑧)) ≤ (#‘𝑧)))))
102	101	3impia 1261	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑘 ∈ ℕ₀ ∧ (#‘𝑧) ∈ ℕ₀ ∧ 𝑘 ≤ (#‘𝑧)) → (𝑚 ≤ (#‘𝑧) → (𝑚 ∈ ℕ₀ → ((𝑘 + 𝑚) − (#‘𝑧)) ≤ (#‘𝑧))))
103	102	com13 88	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑚 ∈ ℕ₀ → (𝑚 ≤ (#‘𝑧) → ((𝑘 ∈ ℕ₀ ∧ (#‘𝑧) ∈ ℕ₀ ∧ 𝑘 ≤ (#‘𝑧)) → ((𝑘 + 𝑚) − (#‘𝑧)) ≤ (#‘𝑧))))
104	103	imp 445	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑚 ∈ ℕ₀ ∧ 𝑚 ≤ (#‘𝑧)) → ((𝑘 ∈ ℕ₀ ∧ (#‘𝑧) ∈ ℕ₀ ∧ 𝑘 ≤ (#‘𝑧)) → ((𝑘 + 𝑚) − (#‘𝑧)) ≤ (#‘𝑧)))
105	58, 104	syl5bi 232	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑚 ∈ ℕ₀ ∧ 𝑚 ≤ (#‘𝑧)) → (𝑘 ∈ (0...(#‘𝑧)) → ((𝑘 + 𝑚) − (#‘𝑧)) ≤ (#‘𝑧)))
106	105	3adant2 1080	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑚 ∈ ℕ₀ ∧ (#‘𝑧) ∈ ℕ₀ ∧ 𝑚 ≤ (#‘𝑧)) → (𝑘 ∈ (0...(#‘𝑧)) → ((𝑘 + 𝑚) − (#‘𝑧)) ≤ (#‘𝑧)))
107	87, 106	sylbi 207	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑚 ∈ (0...(#‘𝑧)) → (𝑘 ∈ (0...(#‘𝑧)) → ((𝑘 + 𝑚) − (#‘𝑧)) ≤ (#‘𝑧)))
108	107	imp 445	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑚 ∈ (0...(#‘𝑧)) ∧ 𝑘 ∈ (0...(#‘𝑧))) → ((𝑘 + 𝑚) − (#‘𝑧)) ≤ (#‘𝑧))
109	86, 108	syl6bi 243	. . . . . . . . . . . . . . . . . 18 ⊢ ((#‘𝑦) = (#‘𝑧) → ((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) → ((𝑘 + 𝑚) − (#‘𝑧)) ≤ (#‘𝑧)))
110	109	adantr 481	. . . . . . . . . . . . . . . . 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 482	. . . . . . . . . . . . . . 15 ⊢ ((¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑) → ((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) → ((𝑘 + 𝑚) − (#‘𝑧)) ≤ (#‘𝑧)))
113	112	impcom 446	. . . . . . . . . . . . . 14 ⊢ (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → ((𝑘 + 𝑚) − (#‘𝑧)) ≤ (#‘𝑧))
114		elfz2nn0 12431	. . . . . . . . . . . . . 14 ⊢ (((𝑘 + 𝑚) − (#‘𝑧)) ∈ (0...(#‘𝑧)) ↔ (((𝑘 + 𝑚) − (#‘𝑧)) ∈ ℕ₀ ∧ (#‘𝑧) ∈ ℕ₀ ∧ ((𝑘 + 𝑚) − (#‘𝑧)) ≤ (#‘𝑧)))
115	81, 82, 113, 114	syl3anbrc 1246	. . . . . . . . . . . . 13 ⊢ (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → ((𝑘 + 𝑚) − (#‘𝑧)) ∈ (0...(#‘𝑧)))
116	115	adantr 481	. . . . . . . . . . . 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 482	. . . . . . . . . . . . . . . . 17 ⊢ ((¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑) → 𝑧 ∈ Word 𝑉)
118	117	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → 𝑧 ∈ Word 𝑉)
119	19	ad2antlr 763	. . . . . . . . . . . . . . . 16 ⊢ (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → 𝑘 ∈ ℤ)
120	22	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → 𝑚 ∈ ℤ)
121	118, 119, 120, 24	syl3anc 1326	. . . . . . . . . . . . . . 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 495	. . . . . . . . . . . . . . . 16 ⊢ ((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) → (𝑘 + 𝑚) ∈ ℤ)
123		cshwsublen 13542	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ Word 𝑉 ∧ (𝑘 + 𝑚) ∈ ℤ) → (𝑧 cyclShift (𝑘 + 𝑚)) = (𝑧 cyclShift ((𝑘 + 𝑚) − (#‘𝑧))))
124	117, 122, 123	syl2anr 495	. . . . . . . . . . . . . . 15 ⊢ (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → (𝑧 cyclShift (𝑘 + 𝑚)) = (𝑧 cyclShift ((𝑘 + 𝑚) − (#‘𝑧))))
125	121, 124	eqtrd 2656	. . . . . . . . . . . . . 14 ⊢ (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑘 ∈ (0...(#‘𝑧))) ∧ (¬ (𝑘 + 𝑚) ≤ (#‘𝑧) ∧ 𝜑)) → ((𝑧 cyclShift 𝑘) cyclShift 𝑚) = (𝑧 cyclShift ((𝑘 + 𝑚) − (#‘𝑧))))
126	125	eqeq2d 2632	. . . . . . . . . . . . 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 501	. . . . . . . . . . . 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 554	. . . . . . . . . . 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 638	. . . . . . . . . 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 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))))))))
131	130	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))))))))
132	131	imp 445	. . . . . . 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 243	. . . . . 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 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))))))))
135	134	impcom 446	. . . 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 446	. . 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 6658	. . . . 5 ⊢ (𝑛 = ((𝑘 + 𝑚) − (#‘𝑧)) → (𝑧 cyclShift 𝑛) = (𝑧 cyclShift ((𝑘 + 𝑚) − (#‘𝑧))))
138	137	eqeq2d 2632	. . . 4 ⊢ (𝑛 = ((𝑘 + 𝑚) − (#‘𝑧)) → (𝑥 = (𝑧 cyclShift 𝑛) ↔ 𝑥 = (𝑧 cyclShift ((𝑘 + 𝑚) − (#‘𝑧)))))
139	138	rspcev 3309	. . 3 ⊢ ((((𝑘 + 𝑚) − (#‘𝑧)) ∈ (0...(#‘𝑧)) ∧ 𝑥 = (𝑧 cyclShift ((𝑘 + 𝑚) − (#‘𝑧)))) → ∃𝑛 ∈ (0...(#‘𝑧))𝑥 = (𝑧 cyclShift 𝑛))
140	136, 139	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)))
141	41, 140	pm2.61ian 831	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 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ∃wrex 2913 class class class wbr 4653 ‘cfv 5888 (class class class)co 6650 ℝcr 9935 0cc0 9936 + caddc 9939 < clt 10074 ≤ cle 10075 − cmin 10266 ℕcn 11020 ℕ₀cn0 11292 ℤcz 11377 ...cfz 12326 #chash 13117 Word cword 13291 cyclShift ccsh 13534

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-inf 8349 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-fz 12327 df-fzo 12466 df-fl 12593 df-mod 12669 df-hash 13118 df-word 13299 df-concat 13301 df-substr 13303 df-csh 13535

This theorem is referenced by: erclwwlkstr 26936 erclwwlksntr 26948

W3C validator