Theorem swrdccatin12 13342

Description: The subword of a concatenation of two words within both of the concatenated words. (Contributed by Alexander van der Vekens, 5-Apr-2018.) (Revised by Alexander van der Vekens, 27-May-2018.)

Hypothesis

Ref	Expression
swrdccatin12.l	⊢ 𝐿 = (#‘𝐴)

Assertion

Ref	Expression
swrdccatin12	⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → ((𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵)))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ (𝐵 substr ⟨0, (𝑁 − 𝐿)⟩))))

Proof of Theorem swrdccatin12

Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ccatcl 13212	. . . . 5 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (𝐴 ++ 𝐵) ∈ Word 𝑉)
2	1	adantr 480	. . . 4 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → (𝐴 ++ 𝐵) ∈ Word 𝑉)
3		elfz0fzfz0 12313	. . . . 5 ⊢ ((𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵)))) → 𝑀 ∈ (0...𝑁))
4	3	adantl 481	. . . 4 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → 𝑀 ∈ (0...𝑁))
5		elfzuz2 12217	. . . . . . . . 9 ⊢ (𝑀 ∈ (0...𝐿) → 𝐿 ∈ (ℤ≥‘0))
6	5	adantl 481	. . . . . . . 8 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...𝐿)) → 𝐿 ∈ (ℤ≥‘0))
7		fzss1 12251	. . . . . . . 8 ⊢ (𝐿 ∈ (ℤ≥‘0) → (𝐿...(𝐿 + (#‘𝐵))) ⊆ (0...(𝐿 + (#‘𝐵))))
8	6, 7	syl 17	. . . . . . 7 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...𝐿)) → (𝐿...(𝐿 + (#‘𝐵))) ⊆ (0...(𝐿 + (#‘𝐵))))
9		ccatlen 13213	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (#‘(𝐴 ++ 𝐵)) = ((#‘𝐴) + (#‘𝐵)))
10		swrdccatin12.l	. . . . . . . . . . . 12 ⊢ 𝐿 = (#‘𝐴)
11	10	eqcomi 2619	. . . . . . . . . . 11 ⊢ (#‘𝐴) = 𝐿
12	11	oveq1i 6559	. . . . . . . . . 10 ⊢ ((#‘𝐴) + (#‘𝐵)) = (𝐿 + (#‘𝐵))
13	9, 12	syl6eq 2660	. . . . . . . . 9 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (#‘(𝐴 ++ 𝐵)) = (𝐿 + (#‘𝐵)))
14	13	adantr 480	. . . . . . . 8 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...𝐿)) → (#‘(𝐴 ++ 𝐵)) = (𝐿 + (#‘𝐵)))
15	14	oveq2d 6565	. . . . . . 7 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...𝐿)) → (0...(#‘(𝐴 ++ 𝐵))) = (0...(𝐿 + (#‘𝐵))))
16	8, 15	sseqtr4d 3605	. . . . . 6 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...𝐿)) → (𝐿...(𝐿 + (#‘𝐵))) ⊆ (0...(#‘(𝐴 ++ 𝐵))))
17	16	sseld 3567	. . . . 5 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...𝐿)) → (𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))) → 𝑁 ∈ (0...(#‘(𝐴 ++ 𝐵)))))
18	17	impr 647	. . . 4 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → 𝑁 ∈ (0...(#‘(𝐴 ++ 𝐵))))
19		swrdvalfn 13278	. . . 4 ⊢ (((𝐴 ++ 𝐵) ∈ Word 𝑉 ∧ 𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘(𝐴 ++ 𝐵)))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) Fn (0..^(𝑁 − 𝑀)))
20	2, 4, 18, 19	syl3anc 1318	. . 3 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) Fn (0..^(𝑁 − 𝑀)))
21		swrdcl 13271	. . . . . . 7 ⊢ (𝐴 ∈ Word 𝑉 → (𝐴 substr ⟨𝑀, 𝐿⟩) ∈ Word 𝑉)
22		swrdcl 13271	. . . . . . 7 ⊢ (𝐵 ∈ Word 𝑉 → (𝐵 substr ⟨0, (𝑁 − 𝐿)⟩) ∈ Word 𝑉)
23	21, 22	anim12i 588	. . . . . 6 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → ((𝐴 substr ⟨𝑀, 𝐿⟩) ∈ Word 𝑉 ∧ (𝐵 substr ⟨0, (𝑁 − 𝐿)⟩) ∈ Word 𝑉))
24	23	adantr 480	. . . . 5 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → ((𝐴 substr ⟨𝑀, 𝐿⟩) ∈ Word 𝑉 ∧ (𝐵 substr ⟨0, (𝑁 − 𝐿)⟩) ∈ Word 𝑉))
25		ccatvalfn 13218	. . . . 5 ⊢ (((𝐴 substr ⟨𝑀, 𝐿⟩) ∈ Word 𝑉 ∧ (𝐵 substr ⟨0, (𝑁 − 𝐿)⟩) ∈ Word 𝑉) → ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ (𝐵 substr ⟨0, (𝑁 − 𝐿)⟩)) Fn (0..^((#‘(𝐴 substr ⟨𝑀, 𝐿⟩)) + (#‘(𝐵 substr ⟨0, (𝑁 − 𝐿)⟩)))))
26	24, 25	syl 17	. . . 4 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ (𝐵 substr ⟨0, (𝑁 − 𝐿)⟩)) Fn (0..^((#‘(𝐴 substr ⟨𝑀, 𝐿⟩)) + (#‘(𝐵 substr ⟨0, (𝑁 − 𝐿)⟩)))))
27		simpll 786	. . . . . . . . 9 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → 𝐴 ∈ Word 𝑉)
28		simprl 790	. . . . . . . . 9 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → 𝑀 ∈ (0...𝐿))
29		lencl 13179	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ Word 𝑉 → (#‘𝐴) ∈ ℕ0)
30		elnn0uz 11601	. . . . . . . . . . . . . 14 ⊢ ((#‘𝐴) ∈ ℕ0 ↔ (#‘𝐴) ∈ (ℤ≥‘0))
31		eluzfz2 12220	. . . . . . . . . . . . . 14 ⊢ ((#‘𝐴) ∈ (ℤ≥‘0) → (#‘𝐴) ∈ (0...(#‘𝐴)))
32	30, 31	sylbi 206	. . . . . . . . . . . . 13 ⊢ ((#‘𝐴) ∈ ℕ0 → (#‘𝐴) ∈ (0...(#‘𝐴)))
33	10, 32	syl5eqel 2692	. . . . . . . . . . . 12 ⊢ ((#‘𝐴) ∈ ℕ0 → 𝐿 ∈ (0...(#‘𝐴)))
34	29, 33	syl 17	. . . . . . . . . . 11 ⊢ (𝐴 ∈ Word 𝑉 → 𝐿 ∈ (0...(#‘𝐴)))
35	34	adantr 480	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → 𝐿 ∈ (0...(#‘𝐴)))
36	35	adantr 480	. . . . . . . . 9 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → 𝐿 ∈ (0...(#‘𝐴)))
37		swrdlen 13275	. . . . . . . . 9 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝑀 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(#‘𝐴))) → (#‘(𝐴 substr ⟨𝑀, 𝐿⟩)) = (𝐿 − 𝑀))
38	27, 28, 36, 37	syl3anc 1318	. . . . . . . 8 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → (#‘(𝐴 substr ⟨𝑀, 𝐿⟩)) = (𝐿 − 𝑀))
39		simpr 476	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → 𝐵 ∈ Word 𝑉)
40	39	adantr 480	. . . . . . . . 9 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → 𝐵 ∈ Word 𝑉)
41		lencl 13179	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ Word 𝑉 → (#‘𝐵) ∈ ℕ0)
42	41	nn0zd 11356	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ Word 𝑉 → (#‘𝐵) ∈ ℤ)
43	42	adantl 481	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (#‘𝐵) ∈ ℤ)
44		simpr 476	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵)))) → 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))
45	43, 44	anim12i 588	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → ((#‘𝐵) ∈ ℤ ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵)))))
46		elfzmlbp 12319	. . . . . . . . . 10 ⊢ (((#‘𝐵) ∈ ℤ ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵)))) → (𝑁 − 𝐿) ∈ (0...(#‘𝐵)))
47	45, 46	syl 17	. . . . . . . . 9 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → (𝑁 − 𝐿) ∈ (0...(#‘𝐵)))
48		swrd0len 13274	. . . . . . . . 9 ⊢ ((𝐵 ∈ Word 𝑉 ∧ (𝑁 − 𝐿) ∈ (0...(#‘𝐵))) → (#‘(𝐵 substr ⟨0, (𝑁 − 𝐿)⟩)) = (𝑁 − 𝐿))
49	40, 47, 48	syl2anc 691	. . . . . . . 8 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → (#‘(𝐵 substr ⟨0, (𝑁 − 𝐿)⟩)) = (𝑁 − 𝐿))
50	38, 49	oveq12d 6567	. . . . . . 7 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → ((#‘(𝐴 substr ⟨𝑀, 𝐿⟩)) + (#‘(𝐵 substr ⟨0, (𝑁 − 𝐿)⟩))) = ((𝐿 − 𝑀) + (𝑁 − 𝐿)))
51		elfz2nn0 12300	. . . . . . . . . . 11 ⊢ (𝑀 ∈ (0...𝐿) ↔ (𝑀 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0 ∧ 𝑀 ≤ 𝐿))
52		elfzelz 12213	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))) → 𝑁 ∈ ℤ)
53		nn0cn 11179	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐿 ∈ ℕ0 → 𝐿 ∈ ℂ)
54	53	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0) → 𝐿 ∈ ℂ)
55	54	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℤ ∧ (𝑀 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0)) → 𝐿 ∈ ℂ)
56		nn0cn 11179	. . . . . . . . . . . . . . . . 17 ⊢ (𝑀 ∈ ℕ0 → 𝑀 ∈ ℂ)
57	56	ad2antrl 760	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℤ ∧ (𝑀 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0)) → 𝑀 ∈ ℂ)
58		zcn 11259	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ)
59	58	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℤ ∧ (𝑀 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0)) → 𝑁 ∈ ℂ)
60	55, 57, 59	3jca 1235	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℤ ∧ (𝑀 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0)) → (𝐿 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ))
61	60	ex 449	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℤ → ((𝑀 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0) → (𝐿 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ)))
62	52, 61	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))) → ((𝑀 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0) → (𝐿 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ)))
63	62	com12 32	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0) → (𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))) → (𝐿 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ)))
64	63	3adant3 1074	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0 ∧ 𝑀 ≤ 𝐿) → (𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))) → (𝐿 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ)))
65	51, 64	sylbi 206	. . . . . . . . . 10 ⊢ (𝑀 ∈ (0...𝐿) → (𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))) → (𝐿 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ)))
66	65	imp 444	. . . . . . . . 9 ⊢ ((𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵)))) → (𝐿 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ))
67	66	adantl 481	. . . . . . . 8 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → (𝐿 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ))
68		npncan3 10198	. . . . . . . 8 ⊢ ((𝐿 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ) → ((𝐿 − 𝑀) + (𝑁 − 𝐿)) = (𝑁 − 𝑀))
69	67, 68	syl 17	. . . . . . 7 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → ((𝐿 − 𝑀) + (𝑁 − 𝐿)) = (𝑁 − 𝑀))
70	50, 69	eqtr2d 2645	. . . . . 6 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → (𝑁 − 𝑀) = ((#‘(𝐴 substr ⟨𝑀, 𝐿⟩)) + (#‘(𝐵 substr ⟨0, (𝑁 − 𝐿)⟩))))
71	70	oveq2d 6565	. . . . 5 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → (0..^(𝑁 − 𝑀)) = (0..^((#‘(𝐴 substr ⟨𝑀, 𝐿⟩)) + (#‘(𝐵 substr ⟨0, (𝑁 − 𝐿)⟩)))))
72	71	fneq2d 5896	. . . 4 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → (((𝐴 substr ⟨𝑀, 𝐿⟩) ++ (𝐵 substr ⟨0, (𝑁 − 𝐿)⟩)) Fn (0..^(𝑁 − 𝑀)) ↔ ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ (𝐵 substr ⟨0, (𝑁 − 𝐿)⟩)) Fn (0..^((#‘(𝐴 substr ⟨𝑀, 𝐿⟩)) + (#‘(𝐵 substr ⟨0, (𝑁 − 𝐿)⟩))))))
73	26, 72	mpbird 246	. . 3 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ (𝐵 substr ⟨0, (𝑁 − 𝐿)⟩)) Fn (0..^(𝑁 − 𝑀)))
74		simprl 790	. . . . . 6 ⊢ ((𝑘 ∈ (0..^(𝐿 − 𝑀)) ∧ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀)))) → ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))))
75		simpr 476	. . . . . . . 8 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀))) → 𝑘 ∈ (0..^(𝑁 − 𝑀)))
76	75	anim2i 591	. . . . . . 7 ⊢ ((𝑘 ∈ (0..^(𝐿 − 𝑀)) ∧ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀)))) → (𝑘 ∈ (0..^(𝐿 − 𝑀)) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀))))
77	76	ancomd 466	. . . . . 6 ⊢ ((𝑘 ∈ (0..^(𝐿 − 𝑀)) ∧ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀)))) → (𝑘 ∈ (0..^(𝑁 − 𝑀)) ∧ 𝑘 ∈ (0..^(𝐿 − 𝑀))))
78	10	swrdccatin12lem3 13341	. . . . . 6 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → ((𝑘 ∈ (0..^(𝑁 − 𝑀)) ∧ 𝑘 ∈ (0..^(𝐿 − 𝑀))) → (((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩)‘𝑘) = ((𝐴 substr ⟨𝑀, 𝐿⟩)‘𝑘)))
79	74, 77, 78	sylc 63	. . . . 5 ⊢ ((𝑘 ∈ (0..^(𝐿 − 𝑀)) ∧ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀)))) → (((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩)‘𝑘) = ((𝐴 substr ⟨𝑀, 𝐿⟩)‘𝑘))
80	24	ad2antrl 760	. . . . . . 7 ⊢ ((𝑘 ∈ (0..^(𝐿 − 𝑀)) ∧ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀)))) → ((𝐴 substr ⟨𝑀, 𝐿⟩) ∈ Word 𝑉 ∧ (𝐵 substr ⟨0, (𝑁 − 𝐿)⟩) ∈ Word 𝑉))
81		simpl 472	. . . . . . . 8 ⊢ ((𝑘 ∈ (0..^(𝐿 − 𝑀)) ∧ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀)))) → 𝑘 ∈ (0..^(𝐿 − 𝑀)))
82		nn0fz0 12306	. . . . . . . . . . . . . . . 16 ⊢ ((#‘𝐴) ∈ ℕ0 ↔ (#‘𝐴) ∈ (0...(#‘𝐴)))
83	29, 82	sylib 207	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ Word 𝑉 → (#‘𝐴) ∈ (0...(#‘𝐴)))
84	10, 83	syl5eqel 2692	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ Word 𝑉 → 𝐿 ∈ (0...(#‘𝐴)))
85	84	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → 𝐿 ∈ (0...(#‘𝐴)))
86	85	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → 𝐿 ∈ (0...(#‘𝐴)))
87	27, 28, 86	3jca 1235	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → (𝐴 ∈ Word 𝑉 ∧ 𝑀 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(#‘𝐴))))
88	87	ad2antrl 760	. . . . . . . . . 10 ⊢ ((𝑘 ∈ (0..^(𝐿 − 𝑀)) ∧ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀)))) → (𝐴 ∈ Word 𝑉 ∧ 𝑀 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(#‘𝐴))))
89	88, 37	syl 17	. . . . . . . . 9 ⊢ ((𝑘 ∈ (0..^(𝐿 − 𝑀)) ∧ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀)))) → (#‘(𝐴 substr ⟨𝑀, 𝐿⟩)) = (𝐿 − 𝑀))
90	89	oveq2d 6565	. . . . . . . 8 ⊢ ((𝑘 ∈ (0..^(𝐿 − 𝑀)) ∧ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀)))) → (0..^(#‘(𝐴 substr ⟨𝑀, 𝐿⟩))) = (0..^(𝐿 − 𝑀)))
91	81, 90	eleqtrrd 2691	. . . . . . 7 ⊢ ((𝑘 ∈ (0..^(𝐿 − 𝑀)) ∧ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀)))) → 𝑘 ∈ (0..^(#‘(𝐴 substr ⟨𝑀, 𝐿⟩))))
92		df-3an 1033	. . . . . . 7 ⊢ (((𝐴 substr ⟨𝑀, 𝐿⟩) ∈ Word 𝑉 ∧ (𝐵 substr ⟨0, (𝑁 − 𝐿)⟩) ∈ Word 𝑉 ∧ 𝑘 ∈ (0..^(#‘(𝐴 substr ⟨𝑀, 𝐿⟩)))) ↔ (((𝐴 substr ⟨𝑀, 𝐿⟩) ∈ Word 𝑉 ∧ (𝐵 substr ⟨0, (𝑁 − 𝐿)⟩) ∈ Word 𝑉) ∧ 𝑘 ∈ (0..^(#‘(𝐴 substr ⟨𝑀, 𝐿⟩)))))
93	80, 91, 92	sylanbrc 695	. . . . . 6 ⊢ ((𝑘 ∈ (0..^(𝐿 − 𝑀)) ∧ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀)))) → ((𝐴 substr ⟨𝑀, 𝐿⟩) ∈ Word 𝑉 ∧ (𝐵 substr ⟨0, (𝑁 − 𝐿)⟩) ∈ Word 𝑉 ∧ 𝑘 ∈ (0..^(#‘(𝐴 substr ⟨𝑀, 𝐿⟩)))))
94		ccatval1 13214	. . . . . 6 ⊢ (((𝐴 substr ⟨𝑀, 𝐿⟩) ∈ Word 𝑉 ∧ (𝐵 substr ⟨0, (𝑁 − 𝐿)⟩) ∈ Word 𝑉 ∧ 𝑘 ∈ (0..^(#‘(𝐴 substr ⟨𝑀, 𝐿⟩)))) → (((𝐴 substr ⟨𝑀, 𝐿⟩) ++ (𝐵 substr ⟨0, (𝑁 − 𝐿)⟩))‘𝑘) = ((𝐴 substr ⟨𝑀, 𝐿⟩)‘𝑘))
95	93, 94	syl 17	. . . . 5 ⊢ ((𝑘 ∈ (0..^(𝐿 − 𝑀)) ∧ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀)))) → (((𝐴 substr ⟨𝑀, 𝐿⟩) ++ (𝐵 substr ⟨0, (𝑁 − 𝐿)⟩))‘𝑘) = ((𝐴 substr ⟨𝑀, 𝐿⟩)‘𝑘))
96	79, 95	eqtr4d 2647	. . . 4 ⊢ ((𝑘 ∈ (0..^(𝐿 − 𝑀)) ∧ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀)))) → (((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩)‘𝑘) = (((𝐴 substr ⟨𝑀, 𝐿⟩) ++ (𝐵 substr ⟨0, (𝑁 − 𝐿)⟩))‘𝑘))
97		simprl 790	. . . . . 6 ⊢ ((¬ 𝑘 ∈ (0..^(𝐿 − 𝑀)) ∧ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀)))) → ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))))
98	75	anim2i 591	. . . . . . 7 ⊢ ((¬ 𝑘 ∈ (0..^(𝐿 − 𝑀)) ∧ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀)))) → (¬ 𝑘 ∈ (0..^(𝐿 − 𝑀)) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀))))
99	98	ancomd 466	. . . . . 6 ⊢ ((¬ 𝑘 ∈ (0..^(𝐿 − 𝑀)) ∧ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀)))) → (𝑘 ∈ (0..^(𝑁 − 𝑀)) ∧ ¬ 𝑘 ∈ (0..^(𝐿 − 𝑀))))
100	10	swrdccatin12lem2 13340	. . . . . 6 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → ((𝑘 ∈ (0..^(𝑁 − 𝑀)) ∧ ¬ 𝑘 ∈ (0..^(𝐿 − 𝑀))) → (((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩)‘𝑘) = ((𝐵 substr ⟨0, (𝑁 − 𝐿)⟩)‘(𝑘 − (#‘(𝐴 substr ⟨𝑀, 𝐿⟩))))))
101	97, 99, 100	sylc 63	. . . . 5 ⊢ ((¬ 𝑘 ∈ (0..^(𝐿 − 𝑀)) ∧ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀)))) → (((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩)‘𝑘) = ((𝐵 substr ⟨0, (𝑁 − 𝐿)⟩)‘(𝑘 − (#‘(𝐴 substr ⟨𝑀, 𝐿⟩)))))
102	24	ad2antrl 760	. . . . . . 7 ⊢ ((¬ 𝑘 ∈ (0..^(𝐿 − 𝑀)) ∧ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀)))) → ((𝐴 substr ⟨𝑀, 𝐿⟩) ∈ Word 𝑉 ∧ (𝐵 substr ⟨0, (𝑁 − 𝐿)⟩) ∈ Word 𝑉))
103		elfzuz 12209	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))) → 𝑁 ∈ (ℤ≥‘𝐿))
104		eluzelz 11573	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ (ℤ≥‘𝐿) → 𝑁 ∈ ℤ)
105		simpll 786	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐿 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0) ∧ 𝑁 ∈ ℤ) → 𝐿 ∈ ℕ0)
106		simpr 476	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐿 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0) → 𝑀 ∈ ℕ0)
107	106	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐿 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0) ∧ 𝑁 ∈ ℤ) → 𝑀 ∈ ℕ0)
108		simpr 476	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐿 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0) ∧ 𝑁 ∈ ℤ) → 𝑁 ∈ ℤ)
109	105, 107, 108	3jca 1235	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐿 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0) ∧ 𝑁 ∈ ℤ) → (𝐿 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℤ))
110	109	ex 449	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐿 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0) → (𝑁 ∈ ℤ → (𝐿 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℤ)))
111	110	ancoms 468	. . . . . . . . . . . . . . . 16 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0) → (𝑁 ∈ ℤ → (𝐿 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℤ)))
112	111	3adant3 1074	. . . . . . . . . . . . . . 15 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0 ∧ 𝑀 ≤ 𝐿) → (𝑁 ∈ ℤ → (𝐿 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℤ)))
113	51, 112	sylbi 206	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ (0...𝐿) → (𝑁 ∈ ℤ → (𝐿 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℤ)))
114	104, 113	syl5com 31	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ (ℤ≥‘𝐿) → (𝑀 ∈ (0...𝐿) → (𝐿 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℤ)))
115	103, 114	syl 17	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))) → (𝑀 ∈ (0...𝐿) → (𝐿 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℤ)))
116	115	impcom 445	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵)))) → (𝐿 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℤ))
117	116	adantl 481	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → (𝐿 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℤ))
118	117	ad2antrl 760	. . . . . . . . 9 ⊢ ((¬ 𝑘 ∈ (0..^(𝐿 − 𝑀)) ∧ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀)))) → (𝐿 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℤ))
119		swrdccatin12lem1 13335	. . . . . . . . 9 ⊢ ((𝐿 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℤ) → ((𝑘 ∈ (0..^(𝑁 − 𝑀)) ∧ ¬ 𝑘 ∈ (0..^(𝐿 − 𝑀))) → 𝑘 ∈ ((𝐿 − 𝑀)..^((𝐿 − 𝑀) + (𝑁 − 𝐿)))))
120	118, 99, 119	sylc 63	. . . . . . . 8 ⊢ ((¬ 𝑘 ∈ (0..^(𝐿 − 𝑀)) ∧ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀)))) → 𝑘 ∈ ((𝐿 − 𝑀)..^((𝐿 − 𝑀) + (𝑁 − 𝐿))))
121	27, 28, 86, 37	syl3anc 1318	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → (#‘(𝐴 substr ⟨𝑀, 𝐿⟩)) = (𝐿 − 𝑀))
122	39	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))) ∧ (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉)) → 𝐵 ∈ Word 𝑉)
123	43	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))) ∧ (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉)) → (#‘𝐵) ∈ ℤ)
124		simpl 472	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))) ∧ (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉)) → 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))
125	123, 124, 46	syl2anc 691	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))) ∧ (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉)) → (𝑁 − 𝐿) ∈ (0...(#‘𝐵)))
126	122, 125	jca 553	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))) ∧ (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉)) → (𝐵 ∈ Word 𝑉 ∧ (𝑁 − 𝐿) ∈ (0...(#‘𝐵))))
127	126	ex 449	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))) → ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (𝐵 ∈ Word 𝑉 ∧ (𝑁 − 𝐿) ∈ (0...(#‘𝐵)))))
128	127	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵)))) → ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (𝐵 ∈ Word 𝑉 ∧ (𝑁 − 𝐿) ∈ (0...(#‘𝐵)))))
129	128	impcom 445	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → (𝐵 ∈ Word 𝑉 ∧ (𝑁 − 𝐿) ∈ (0...(#‘𝐵))))
130	129, 48	syl 17	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → (#‘(𝐵 substr ⟨0, (𝑁 − 𝐿)⟩)) = (𝑁 − 𝐿))
131	121, 130	oveq12d 6567	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → ((#‘(𝐴 substr ⟨𝑀, 𝐿⟩)) + (#‘(𝐵 substr ⟨0, (𝑁 − 𝐿)⟩))) = ((𝐿 − 𝑀) + (𝑁 − 𝐿)))
132	121, 131	oveq12d 6567	. . . . . . . . 9 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → ((#‘(𝐴 substr ⟨𝑀, 𝐿⟩))..^((#‘(𝐴 substr ⟨𝑀, 𝐿⟩)) + (#‘(𝐵 substr ⟨0, (𝑁 − 𝐿)⟩)))) = ((𝐿 − 𝑀)..^((𝐿 − 𝑀) + (𝑁 − 𝐿))))
133	132	ad2antrl 760	. . . . . . . 8 ⊢ ((¬ 𝑘 ∈ (0..^(𝐿 − 𝑀)) ∧ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀)))) → ((#‘(𝐴 substr ⟨𝑀, 𝐿⟩))..^((#‘(𝐴 substr ⟨𝑀, 𝐿⟩)) + (#‘(𝐵 substr ⟨0, (𝑁 − 𝐿)⟩)))) = ((𝐿 − 𝑀)..^((𝐿 − 𝑀) + (𝑁 − 𝐿))))
134	120, 133	eleqtrrd 2691	. . . . . . 7 ⊢ ((¬ 𝑘 ∈ (0..^(𝐿 − 𝑀)) ∧ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀)))) → 𝑘 ∈ ((#‘(𝐴 substr ⟨𝑀, 𝐿⟩))..^((#‘(𝐴 substr ⟨𝑀, 𝐿⟩)) + (#‘(𝐵 substr ⟨0, (𝑁 − 𝐿)⟩)))))
135		df-3an 1033	. . . . . . 7 ⊢ (((𝐴 substr ⟨𝑀, 𝐿⟩) ∈ Word 𝑉 ∧ (𝐵 substr ⟨0, (𝑁 − 𝐿)⟩) ∈ Word 𝑉 ∧ 𝑘 ∈ ((#‘(𝐴 substr ⟨𝑀, 𝐿⟩))..^((#‘(𝐴 substr ⟨𝑀, 𝐿⟩)) + (#‘(𝐵 substr ⟨0, (𝑁 − 𝐿)⟩))))) ↔ (((𝐴 substr ⟨𝑀, 𝐿⟩) ∈ Word 𝑉 ∧ (𝐵 substr ⟨0, (𝑁 − 𝐿)⟩) ∈ Word 𝑉) ∧ 𝑘 ∈ ((#‘(𝐴 substr ⟨𝑀, 𝐿⟩))..^((#‘(𝐴 substr ⟨𝑀, 𝐿⟩)) + (#‘(𝐵 substr ⟨0, (𝑁 − 𝐿)⟩))))))
136	102, 134, 135	sylanbrc 695	. . . . . 6 ⊢ ((¬ 𝑘 ∈ (0..^(𝐿 − 𝑀)) ∧ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀)))) → ((𝐴 substr ⟨𝑀, 𝐿⟩) ∈ Word 𝑉 ∧ (𝐵 substr ⟨0, (𝑁 − 𝐿)⟩) ∈ Word 𝑉 ∧ 𝑘 ∈ ((#‘(𝐴 substr ⟨𝑀, 𝐿⟩))..^((#‘(𝐴 substr ⟨𝑀, 𝐿⟩)) + (#‘(𝐵 substr ⟨0, (𝑁 − 𝐿)⟩))))))
137		ccatval2 13215	. . . . . 6 ⊢ (((𝐴 substr ⟨𝑀, 𝐿⟩) ∈ Word 𝑉 ∧ (𝐵 substr ⟨0, (𝑁 − 𝐿)⟩) ∈ Word 𝑉 ∧ 𝑘 ∈ ((#‘(𝐴 substr ⟨𝑀, 𝐿⟩))..^((#‘(𝐴 substr ⟨𝑀, 𝐿⟩)) + (#‘(𝐵 substr ⟨0, (𝑁 − 𝐿)⟩))))) → (((𝐴 substr ⟨𝑀, 𝐿⟩) ++ (𝐵 substr ⟨0, (𝑁 − 𝐿)⟩))‘𝑘) = ((𝐵 substr ⟨0, (𝑁 − 𝐿)⟩)‘(𝑘 − (#‘(𝐴 substr ⟨𝑀, 𝐿⟩)))))
138	136, 137	syl 17	. . . . 5 ⊢ ((¬ 𝑘 ∈ (0..^(𝐿 − 𝑀)) ∧ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀)))) → (((𝐴 substr ⟨𝑀, 𝐿⟩) ++ (𝐵 substr ⟨0, (𝑁 − 𝐿)⟩))‘𝑘) = ((𝐵 substr ⟨0, (𝑁 − 𝐿)⟩)‘(𝑘 − (#‘(𝐴 substr ⟨𝑀, 𝐿⟩)))))
139	101, 138	eqtr4d 2647	. . . 4 ⊢ ((¬ 𝑘 ∈ (0..^(𝐿 − 𝑀)) ∧ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀)))) → (((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩)‘𝑘) = (((𝐴 substr ⟨𝑀, 𝐿⟩) ++ (𝐵 substr ⟨0, (𝑁 − 𝐿)⟩))‘𝑘))
140	96, 139	pm2.61ian 827	. . 3 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) ∧ 𝑘 ∈ (0..^(𝑁 − 𝑀))) → (((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩)‘𝑘) = (((𝐴 substr ⟨𝑀, 𝐿⟩) ++ (𝐵 substr ⟨0, (𝑁 − 𝐿)⟩))‘𝑘))
141	20, 73, 140	eqfnfvd 6222	. 2 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ (𝐵 substr ⟨0, (𝑁 − 𝐿)⟩)))
142	141	ex 449	1 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → ((𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵)))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ (𝐵 substr ⟨0, (𝑁 − 𝐿)⟩))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ⊆ wss 3540  ⟨cop 4131   class class class wbr 4583   Fn wfn 5799  ‘cfv 5804  (class class class)co 6549  ℂcc 9813  0cc0 9815   + caddc 9818   ≤ cle 9954   − cmin 10145  ℕ₀cn0 11169  ℤcz 11254  ℤ_≥cuz 11563  ...cfz 12197  ..^cfzo 12334  #chash 12979  Word cword 13146   ++ cconcat 13148   substr csubstr 13150

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

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-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-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-concat 13156  df-substr 13158

This theorem is referenced by:  swrdccat3  13343  swrdccatin12d  13352