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Theorem swrdccat3blem 13346

Description: Lemma for swrdccat3b 13347. (Contributed by AV, 30-May-2018.)

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

**Assertion**
Ref	Expression
swrdccat3blem	⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...(𝐿 + (#‘𝐵)))) ∧ (𝐿 + (#‘𝐵)) ≤ 𝐿) → if(𝐿 ≤ 𝑀, (𝐵 substr ⟨(𝑀 − 𝐿), (#‘𝐵)⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ 𝐵)) = (𝐴 substr ⟨𝑀, (𝐿 + (#‘𝐵))⟩))

**Proof of Theorem swrdccat3blem**
Step	Hyp	Ref	Expression
1		lencl 13179	. . . . . . . 8 ⊢ (𝐵 ∈ Word 𝑉 → (#‘𝐵) ∈ ℕ₀)
2		nn0le0eq0 11198	. . . . . . . . 9 ⊢ ((#‘𝐵) ∈ ℕ₀ → ((#‘𝐵) ≤ 0 ↔ (#‘𝐵) = 0))
3	2	biimpd 218	. . . . . . . 8 ⊢ ((#‘𝐵) ∈ ℕ₀ → ((#‘𝐵) ≤ 0 → (#‘𝐵) = 0))
4	1, 3	syl 17	. . . . . . 7 ⊢ (𝐵 ∈ Word 𝑉 → ((#‘𝐵) ≤ 0 → (#‘𝐵) = 0))
5	4	adantl 481	. . . . . 6 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → ((#‘𝐵) ≤ 0 → (#‘𝐵) = 0))
6		hasheq0 13015	. . . . . . . . . . 11 ⊢ (𝐵 ∈ Word 𝑉 → ((#‘𝐵) = 0 ↔ 𝐵 = ∅))
7	6	biimpd 218	. . . . . . . . . 10 ⊢ (𝐵 ∈ Word 𝑉 → ((#‘𝐵) = 0 → 𝐵 = ∅))
8	7	adantl 481	. . . . . . . . 9 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → ((#‘𝐵) = 0 → 𝐵 = ∅))
9	8	imp 444	. . . . . . . 8 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (#‘𝐵) = 0) → 𝐵 = ∅)
10		lencl 13179	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ Word 𝑉 → (#‘𝐴) ∈ ℕ₀)
11		swrdccatin12.l	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐿 = (#‘𝐴)
12	11	eqcomi 2619	. . . . . . . . . . . . . . . . . 18 ⊢ (#‘𝐴) = 𝐿
13	12	eleq1i 2679	. . . . . . . . . . . . . . . . 17 ⊢ ((#‘𝐴) ∈ ℕ₀ ↔ 𝐿 ∈ ℕ₀)
14		nn0re 11178	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐿 ∈ ℕ₀ → 𝐿 ∈ ℝ)
15		elfz2nn0 12300	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑀 ∈ (0...(𝐿 + 0)) ↔ (𝑀 ∈ ℕ₀ ∧ (𝐿 + 0) ∈ ℕ₀ ∧ 𝑀 ≤ (𝐿 + 0)))
16		recn 9905	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝐿 ∈ ℝ → 𝐿 ∈ ℂ)
17	16	addid1d 10115	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐿 ∈ ℝ → (𝐿 + 0) = 𝐿)
18	17	breq2d 4595	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐿 ∈ ℝ → (𝑀 ≤ (𝐿 + 0) ↔ 𝑀 ≤ 𝐿))
19		nn0re 11178	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑀 ∈ ℕ₀ → 𝑀 ∈ ℝ)
20	19	anim1i 590	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝐿 ∈ ℝ) → (𝑀 ∈ ℝ ∧ 𝐿 ∈ ℝ))
21	20	ancoms 468	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝐿 ∈ ℝ ∧ 𝑀 ∈ ℕ₀) → (𝑀 ∈ ℝ ∧ 𝐿 ∈ ℝ))
22		letri3 10002	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑀 ∈ ℝ ∧ 𝐿 ∈ ℝ) → (𝑀 = 𝐿 ↔ (𝑀 ≤ 𝐿 ∧ 𝐿 ≤ 𝑀)))
23	21, 22	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝐿 ∈ ℝ ∧ 𝑀 ∈ ℕ₀) → (𝑀 = 𝐿 ↔ (𝑀 ≤ 𝐿 ∧ 𝐿 ≤ 𝑀)))
24	23	biimprd 237	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝐿 ∈ ℝ ∧ 𝑀 ∈ ℕ₀) → ((𝑀 ≤ 𝐿 ∧ 𝐿 ≤ 𝑀) → 𝑀 = 𝐿))
25	24	exp4b 630	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐿 ∈ ℝ → (𝑀 ∈ ℕ₀ → (𝑀 ≤ 𝐿 → (𝐿 ≤ 𝑀 → 𝑀 = 𝐿))))
26	25	com23 84	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐿 ∈ ℝ → (𝑀 ≤ 𝐿 → (𝑀 ∈ ℕ₀ → (𝐿 ≤ 𝑀 → 𝑀 = 𝐿))))
27	18, 26	sylbid 229	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐿 ∈ ℝ → (𝑀 ≤ (𝐿 + 0) → (𝑀 ∈ ℕ₀ → (𝐿 ≤ 𝑀 → 𝑀 = 𝐿))))
28	27	com3l 87	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑀 ≤ (𝐿 + 0) → (𝑀 ∈ ℕ₀ → (𝐿 ∈ ℝ → (𝐿 ≤ 𝑀 → 𝑀 = 𝐿))))
29	28	impcom 445	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑀 ≤ (𝐿 + 0)) → (𝐿 ∈ ℝ → (𝐿 ≤ 𝑀 → 𝑀 = 𝐿)))
30	29	3adant2 1073	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑀 ∈ ℕ₀ ∧ (𝐿 + 0) ∈ ℕ₀ ∧ 𝑀 ≤ (𝐿 + 0)) → (𝐿 ∈ ℝ → (𝐿 ≤ 𝑀 → 𝑀 = 𝐿)))
31	30	com12 32	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐿 ∈ ℝ → ((𝑀 ∈ ℕ₀ ∧ (𝐿 + 0) ∈ ℕ₀ ∧ 𝑀 ≤ (𝐿 + 0)) → (𝐿 ≤ 𝑀 → 𝑀 = 𝐿)))
32	15, 31	syl5bi 231	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐿 ∈ ℝ → (𝑀 ∈ (0...(𝐿 + 0)) → (𝐿 ≤ 𝑀 → 𝑀 = 𝐿)))
33	14, 32	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝐿 ∈ ℕ₀ → (𝑀 ∈ (0...(𝐿 + 0)) → (𝐿 ≤ 𝑀 → 𝑀 = 𝐿)))
34	13, 33	sylbi 206	. . . . . . . . . . . . . . . 16 ⊢ ((#‘𝐴) ∈ ℕ₀ → (𝑀 ∈ (0...(𝐿 + 0)) → (𝐿 ≤ 𝑀 → 𝑀 = 𝐿)))
35	10, 34	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ Word 𝑉 → (𝑀 ∈ (0...(𝐿 + 0)) → (𝐿 ≤ 𝑀 → 𝑀 = 𝐿)))
36	35	imp 444	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝑀 ∈ (0...(𝐿 + 0))) → (𝐿 ≤ 𝑀 → 𝑀 = 𝐿))
37		elfznn0 12302	. . . . . . . . . . . . . . . 16 ⊢ (𝑀 ∈ (0...(𝐿 + 0)) → 𝑀 ∈ ℕ₀)
38		swrd00 13270	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (∅ substr ⟨0, 0⟩) = ∅
39		swrd00 13270	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐴 substr ⟨𝐿, 𝐿⟩) = ∅
40	38, 39	eqtr4i 2635	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∅ substr ⟨0, 0⟩) = (𝐴 substr ⟨𝐿, 𝐿⟩)
41		nn0cn 11179	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐿 ∈ ℕ₀ → 𝐿 ∈ ℂ)
42	41	subidd 10259	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐿 ∈ ℕ₀ → (𝐿 − 𝐿) = 0)
43	42	opeq1d 4346	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐿 ∈ ℕ₀ → ⟨(𝐿 − 𝐿), 0⟩ = ⟨0, 0⟩)
44	43	oveq2d 6565	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐿 ∈ ℕ₀ → (∅ substr ⟨(𝐿 − 𝐿), 0⟩) = (∅ substr ⟨0, 0⟩))
45	41	addid1d 10115	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐿 ∈ ℕ₀ → (𝐿 + 0) = 𝐿)
46	45	opeq2d 4347	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐿 ∈ ℕ₀ → ⟨𝐿, (𝐿 + 0)⟩ = ⟨𝐿, 𝐿⟩)
47	46	oveq2d 6565	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐿 ∈ ℕ₀ → (𝐴 substr ⟨𝐿, (𝐿 + 0)⟩) = (𝐴 substr ⟨𝐿, 𝐿⟩))
48	40, 44, 47	3eqtr4a 2670	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐿 ∈ ℕ₀ → (∅ substr ⟨(𝐿 − 𝐿), 0⟩) = (𝐴 substr ⟨𝐿, (𝐿 + 0)⟩))
49	48	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑀 = 𝐿 → (𝐿 ∈ ℕ₀ → (∅ substr ⟨(𝐿 − 𝐿), 0⟩) = (𝐴 substr ⟨𝐿, (𝐿 + 0)⟩)))
50		eleq1 2676	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑀 = 𝐿 → (𝑀 ∈ ℕ₀ ↔ 𝐿 ∈ ℕ₀))
51		oveq1 6556	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑀 = 𝐿 → (𝑀 − 𝐿) = (𝐿 − 𝐿))
52	51	opeq1d 4346	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑀 = 𝐿 → ⟨(𝑀 − 𝐿), 0⟩ = ⟨(𝐿 − 𝐿), 0⟩)
53	52	oveq2d 6565	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑀 = 𝐿 → (∅ substr ⟨(𝑀 − 𝐿), 0⟩) = (∅ substr ⟨(𝐿 − 𝐿), 0⟩))
54		opeq1 4340	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑀 = 𝐿 → ⟨𝑀, (𝐿 + 0)⟩ = ⟨𝐿, (𝐿 + 0)⟩)
55	54	oveq2d 6565	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑀 = 𝐿 → (𝐴 substr ⟨𝑀, (𝐿 + 0)⟩) = (𝐴 substr ⟨𝐿, (𝐿 + 0)⟩))
56	53, 55	eqeq12d 2625	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑀 = 𝐿 → ((∅ substr ⟨(𝑀 − 𝐿), 0⟩) = (𝐴 substr ⟨𝑀, (𝐿 + 0)⟩) ↔ (∅ substr ⟨(𝐿 − 𝐿), 0⟩) = (𝐴 substr ⟨𝐿, (𝐿 + 0)⟩)))
57	49, 50, 56	3imtr4d 282	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑀 = 𝐿 → (𝑀 ∈ ℕ₀ → (∅ substr ⟨(𝑀 − 𝐿), 0⟩) = (𝐴 substr ⟨𝑀, (𝐿 + 0)⟩)))
58	57	com12 32	. . . . . . . . . . . . . . . . 17 ⊢ (𝑀 ∈ ℕ₀ → (𝑀 = 𝐿 → (∅ substr ⟨(𝑀 − 𝐿), 0⟩) = (𝐴 substr ⟨𝑀, (𝐿 + 0)⟩)))
59	58	a1d 25	. . . . . . . . . . . . . . . 16 ⊢ (𝑀 ∈ ℕ₀ → (𝐴 ∈ Word 𝑉 → (𝑀 = 𝐿 → (∅ substr ⟨(𝑀 − 𝐿), 0⟩) = (𝐴 substr ⟨𝑀, (𝐿 + 0)⟩))))
60	37, 59	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑀 ∈ (0...(𝐿 + 0)) → (𝐴 ∈ Word 𝑉 → (𝑀 = 𝐿 → (∅ substr ⟨(𝑀 − 𝐿), 0⟩) = (𝐴 substr ⟨𝑀, (𝐿 + 0)⟩))))
61	60	impcom 445	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝑀 ∈ (0...(𝐿 + 0))) → (𝑀 = 𝐿 → (∅ substr ⟨(𝑀 − 𝐿), 0⟩) = (𝐴 substr ⟨𝑀, (𝐿 + 0)⟩)))
62	36, 61	syld 46	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝑀 ∈ (0...(𝐿 + 0))) → (𝐿 ≤ 𝑀 → (∅ substr ⟨(𝑀 − 𝐿), 0⟩) = (𝐴 substr ⟨𝑀, (𝐿 + 0)⟩)))
63	62	imp 444	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝑀 ∈ (0...(𝐿 + 0))) ∧ 𝐿 ≤ 𝑀) → (∅ substr ⟨(𝑀 − 𝐿), 0⟩) = (𝐴 substr ⟨𝑀, (𝐿 + 0)⟩))
64		swrdcl 13271	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ Word 𝑉 → (𝐴 substr ⟨𝑀, 𝐿⟩) ∈ Word 𝑉)
65		ccatrid 13223	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 substr ⟨𝑀, 𝐿⟩) ∈ Word 𝑉 → ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ ∅) = (𝐴 substr ⟨𝑀, 𝐿⟩))
66	64, 65	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ Word 𝑉 → ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ ∅) = (𝐴 substr ⟨𝑀, 𝐿⟩))
67	13, 41	sylbi 206	. . . . . . . . . . . . . . . . . . 19 ⊢ ((#‘𝐴) ∈ ℕ₀ → 𝐿 ∈ ℂ)
68	10, 67	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ Word 𝑉 → 𝐿 ∈ ℂ)
69		addid1 10095	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐿 ∈ ℂ → (𝐿 + 0) = 𝐿)
70	69	eqcomd 2616	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐿 ∈ ℂ → 𝐿 = (𝐿 + 0))
71	68, 70	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ Word 𝑉 → 𝐿 = (𝐿 + 0))
72	71	opeq2d 4347	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ Word 𝑉 → ⟨𝑀, 𝐿⟩ = ⟨𝑀, (𝐿 + 0)⟩)
73	72	oveq2d 6565	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ Word 𝑉 → (𝐴 substr ⟨𝑀, 𝐿⟩) = (𝐴 substr ⟨𝑀, (𝐿 + 0)⟩))
74	66, 73	eqtrd 2644	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ Word 𝑉 → ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ ∅) = (𝐴 substr ⟨𝑀, (𝐿 + 0)⟩))
75	74	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝑀 ∈ (0...(𝐿 + 0))) → ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ ∅) = (𝐴 substr ⟨𝑀, (𝐿 + 0)⟩))
76	75	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝑀 ∈ (0...(𝐿 + 0))) ∧ ¬ 𝐿 ≤ 𝑀) → ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ ∅) = (𝐴 substr ⟨𝑀, (𝐿 + 0)⟩))
77	63, 76	ifeqda 4071	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝑀 ∈ (0...(𝐿 + 0))) → if(𝐿 ≤ 𝑀, (∅ substr ⟨(𝑀 − 𝐿), 0⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ ∅)) = (𝐴 substr ⟨𝑀, (𝐿 + 0)⟩))
78	77	ex 449	. . . . . . . . . 10 ⊢ (𝐴 ∈ Word 𝑉 → (𝑀 ∈ (0...(𝐿 + 0)) → if(𝐿 ≤ 𝑀, (∅ substr ⟨(𝑀 − 𝐿), 0⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ ∅)) = (𝐴 substr ⟨𝑀, (𝐿 + 0)⟩)))
79	78	ad3antrrr 762	. . . . . . . . 9 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (#‘𝐵) = 0) ∧ 𝐵 = ∅) → (𝑀 ∈ (0...(𝐿 + 0)) → if(𝐿 ≤ 𝑀, (∅ substr ⟨(𝑀 − 𝐿), 0⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ ∅)) = (𝐴 substr ⟨𝑀, (𝐿 + 0)⟩)))
80		oveq2 6557	. . . . . . . . . . . . . 14 ⊢ ((#‘𝐵) = 0 → (𝐿 + (#‘𝐵)) = (𝐿 + 0))
81	80	oveq2d 6565	. . . . . . . . . . . . 13 ⊢ ((#‘𝐵) = 0 → (0...(𝐿 + (#‘𝐵))) = (0...(𝐿 + 0)))
82	81	eleq2d 2673	. . . . . . . . . . . 12 ⊢ ((#‘𝐵) = 0 → (𝑀 ∈ (0...(𝐿 + (#‘𝐵))) ↔ 𝑀 ∈ (0...(𝐿 + 0))))
83	82	adantr 480	. . . . . . . . . . 11 ⊢ (((#‘𝐵) = 0 ∧ 𝐵 = ∅) → (𝑀 ∈ (0...(𝐿 + (#‘𝐵))) ↔ 𝑀 ∈ (0...(𝐿 + 0))))
84		simpr 476	. . . . . . . . . . . . . 14 ⊢ (((#‘𝐵) = 0 ∧ 𝐵 = ∅) → 𝐵 = ∅)
85		opeq2 4341	. . . . . . . . . . . . . . 15 ⊢ ((#‘𝐵) = 0 → ⟨(𝑀 − 𝐿), (#‘𝐵)⟩ = ⟨(𝑀 − 𝐿), 0⟩)
86	85	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((#‘𝐵) = 0 ∧ 𝐵 = ∅) → ⟨(𝑀 − 𝐿), (#‘𝐵)⟩ = ⟨(𝑀 − 𝐿), 0⟩)
87	84, 86	oveq12d 6567	. . . . . . . . . . . . 13 ⊢ (((#‘𝐵) = 0 ∧ 𝐵 = ∅) → (𝐵 substr ⟨(𝑀 − 𝐿), (#‘𝐵)⟩) = (∅ substr ⟨(𝑀 − 𝐿), 0⟩))
88		oveq2 6557	. . . . . . . . . . . . . 14 ⊢ (𝐵 = ∅ → ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ 𝐵) = ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ ∅))
89	88	adantl 481	. . . . . . . . . . . . 13 ⊢ (((#‘𝐵) = 0 ∧ 𝐵 = ∅) → ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ 𝐵) = ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ ∅))
90	87, 89	ifeq12d 4056	. . . . . . . . . . . 12 ⊢ (((#‘𝐵) = 0 ∧ 𝐵 = ∅) → if(𝐿 ≤ 𝑀, (𝐵 substr ⟨(𝑀 − 𝐿), (#‘𝐵)⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ 𝐵)) = if(𝐿 ≤ 𝑀, (∅ substr ⟨(𝑀 − 𝐿), 0⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ ∅)))
91	80	opeq2d 4347	. . . . . . . . . . . . . 14 ⊢ ((#‘𝐵) = 0 → ⟨𝑀, (𝐿 + (#‘𝐵))⟩ = ⟨𝑀, (𝐿 + 0)⟩)
92	91	oveq2d 6565	. . . . . . . . . . . . 13 ⊢ ((#‘𝐵) = 0 → (𝐴 substr ⟨𝑀, (𝐿 + (#‘𝐵))⟩) = (𝐴 substr ⟨𝑀, (𝐿 + 0)⟩))
93	92	adantr 480	. . . . . . . . . . . 12 ⊢ (((#‘𝐵) = 0 ∧ 𝐵 = ∅) → (𝐴 substr ⟨𝑀, (𝐿 + (#‘𝐵))⟩) = (𝐴 substr ⟨𝑀, (𝐿 + 0)⟩))
94	90, 93	eqeq12d 2625	. . . . . . . . . . 11 ⊢ (((#‘𝐵) = 0 ∧ 𝐵 = ∅) → (if(𝐿 ≤ 𝑀, (𝐵 substr ⟨(𝑀 − 𝐿), (#‘𝐵)⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ 𝐵)) = (𝐴 substr ⟨𝑀, (𝐿 + (#‘𝐵))⟩) ↔ if(𝐿 ≤ 𝑀, (∅ substr ⟨(𝑀 − 𝐿), 0⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ ∅)) = (𝐴 substr ⟨𝑀, (𝐿 + 0)⟩)))
95	83, 94	imbi12d 333	. . . . . . . . . 10 ⊢ (((#‘𝐵) = 0 ∧ 𝐵 = ∅) → ((𝑀 ∈ (0...(𝐿 + (#‘𝐵))) → if(𝐿 ≤ 𝑀, (𝐵 substr ⟨(𝑀 − 𝐿), (#‘𝐵)⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ 𝐵)) = (𝐴 substr ⟨𝑀, (𝐿 + (#‘𝐵))⟩)) ↔ (𝑀 ∈ (0...(𝐿 + 0)) → if(𝐿 ≤ 𝑀, (∅ substr ⟨(𝑀 − 𝐿), 0⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ ∅)) = (𝐴 substr ⟨𝑀, (𝐿 + 0)⟩))))
96	95	adantll 746	. . . . . . . . 9 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (#‘𝐵) = 0) ∧ 𝐵 = ∅) → ((𝑀 ∈ (0...(𝐿 + (#‘𝐵))) → if(𝐿 ≤ 𝑀, (𝐵 substr ⟨(𝑀 − 𝐿), (#‘𝐵)⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ 𝐵)) = (𝐴 substr ⟨𝑀, (𝐿 + (#‘𝐵))⟩)) ↔ (𝑀 ∈ (0...(𝐿 + 0)) → if(𝐿 ≤ 𝑀, (∅ substr ⟨(𝑀 − 𝐿), 0⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ ∅)) = (𝐴 substr ⟨𝑀, (𝐿 + 0)⟩))))
97	79, 96	mpbird 246	. . . . . . . 8 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (#‘𝐵) = 0) ∧ 𝐵 = ∅) → (𝑀 ∈ (0...(𝐿 + (#‘𝐵))) → if(𝐿 ≤ 𝑀, (𝐵 substr ⟨(𝑀 − 𝐿), (#‘𝐵)⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ 𝐵)) = (𝐴 substr ⟨𝑀, (𝐿 + (#‘𝐵))⟩)))
98	9, 97	mpdan 699	. . . . . . 7 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (#‘𝐵) = 0) → (𝑀 ∈ (0...(𝐿 + (#‘𝐵))) → if(𝐿 ≤ 𝑀, (𝐵 substr ⟨(𝑀 − 𝐿), (#‘𝐵)⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ 𝐵)) = (𝐴 substr ⟨𝑀, (𝐿 + (#‘𝐵))⟩)))
99	98	ex 449	. . . . . 6 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → ((#‘𝐵) = 0 → (𝑀 ∈ (0...(𝐿 + (#‘𝐵))) → if(𝐿 ≤ 𝑀, (𝐵 substr ⟨(𝑀 − 𝐿), (#‘𝐵)⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ 𝐵)) = (𝐴 substr ⟨𝑀, (𝐿 + (#‘𝐵))⟩))))
100	5, 99	syld 46	. . . . 5 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → ((#‘𝐵) ≤ 0 → (𝑀 ∈ (0...(𝐿 + (#‘𝐵))) → if(𝐿 ≤ 𝑀, (𝐵 substr ⟨(𝑀 − 𝐿), (#‘𝐵)⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ 𝐵)) = (𝐴 substr ⟨𝑀, (𝐿 + (#‘𝐵))⟩))))
101	100	com23 84	. . . 4 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (𝑀 ∈ (0...(𝐿 + (#‘𝐵))) → ((#‘𝐵) ≤ 0 → if(𝐿 ≤ 𝑀, (𝐵 substr ⟨(𝑀 − 𝐿), (#‘𝐵)⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ 𝐵)) = (𝐴 substr ⟨𝑀, (𝐿 + (#‘𝐵))⟩))))
102	101	imp 444	. . 3 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...(𝐿 + (#‘𝐵)))) → ((#‘𝐵) ≤ 0 → if(𝐿 ≤ 𝑀, (𝐵 substr ⟨(𝑀 − 𝐿), (#‘𝐵)⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ 𝐵)) = (𝐴 substr ⟨𝑀, (𝐿 + (#‘𝐵))⟩)))
103	102	adantr 480	. 2 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...(𝐿 + (#‘𝐵)))) ∧ (𝐿 + (#‘𝐵)) ≤ 𝐿) → ((#‘𝐵) ≤ 0 → if(𝐿 ≤ 𝑀, (𝐵 substr ⟨(𝑀 − 𝐿), (#‘𝐵)⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ 𝐵)) = (𝐴 substr ⟨𝑀, (𝐿 + (#‘𝐵))⟩)))
104	11	eleq1i 2679	. . . . . . . 8 ⊢ (𝐿 ∈ ℕ₀ ↔ (#‘𝐴) ∈ ℕ₀)
105	104, 14	sylbir 224	. . . . . . 7 ⊢ ((#‘𝐴) ∈ ℕ₀ → 𝐿 ∈ ℝ)
106	10, 105	syl 17	. . . . . 6 ⊢ (𝐴 ∈ Word 𝑉 → 𝐿 ∈ ℝ)
107	1	nn0red 11229	. . . . . 6 ⊢ (𝐵 ∈ Word 𝑉 → (#‘𝐵) ∈ ℝ)
108		leaddle0 10422	. . . . . 6 ⊢ ((𝐿 ∈ ℝ ∧ (#‘𝐵) ∈ ℝ) → ((𝐿 + (#‘𝐵)) ≤ 𝐿 ↔ (#‘𝐵) ≤ 0))
109	106, 107, 108	syl2an 493	. . . . 5 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → ((𝐿 + (#‘𝐵)) ≤ 𝐿 ↔ (#‘𝐵) ≤ 0))
110		pm2.24 120	. . . . 5 ⊢ ((#‘𝐵) ≤ 0 → (¬ (#‘𝐵) ≤ 0 → if(𝐿 ≤ 𝑀, (𝐵 substr ⟨(𝑀 − 𝐿), (#‘𝐵)⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ 𝐵)) = (𝐴 substr ⟨𝑀, (𝐿 + (#‘𝐵))⟩)))
111	109, 110	syl6bi 242	. . . 4 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → ((𝐿 + (#‘𝐵)) ≤ 𝐿 → (¬ (#‘𝐵) ≤ 0 → if(𝐿 ≤ 𝑀, (𝐵 substr ⟨(𝑀 − 𝐿), (#‘𝐵)⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ 𝐵)) = (𝐴 substr ⟨𝑀, (𝐿 + (#‘𝐵))⟩))))
112	111	adantr 480	. . 3 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...(𝐿 + (#‘𝐵)))) → ((𝐿 + (#‘𝐵)) ≤ 𝐿 → (¬ (#‘𝐵) ≤ 0 → if(𝐿 ≤ 𝑀, (𝐵 substr ⟨(𝑀 − 𝐿), (#‘𝐵)⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ 𝐵)) = (𝐴 substr ⟨𝑀, (𝐿 + (#‘𝐵))⟩))))
113	112	imp 444	. 2 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...(𝐿 + (#‘𝐵)))) ∧ (𝐿 + (#‘𝐵)) ≤ 𝐿) → (¬ (#‘𝐵) ≤ 0 → if(𝐿 ≤ 𝑀, (𝐵 substr ⟨(𝑀 − 𝐿), (#‘𝐵)⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ 𝐵)) = (𝐴 substr ⟨𝑀, (𝐿 + (#‘𝐵))⟩)))
114	103, 113	pm2.61d 169	1 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...(𝐿 + (#‘𝐵)))) ∧ (𝐿 + (#‘𝐵)) ≤ 𝐿) → if(𝐿 ≤ 𝑀, (𝐵 substr ⟨(𝑀 − 𝐿), (#‘𝐵)⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ 𝐵)) = (𝐴 substr ⟨𝑀, (𝐿 + (#‘𝐵))⟩))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∅c0 3874 ifcif 4036 ⟨cop 4131 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 ℝcr 9814 0cc0 9815 + caddc 9818 ≤ cle 9954 − cmin 10145 ℕ₀cn0 11169 ...cfz 12197 #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: swrdccat3b 13347

W3C validator