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Theorem swrdccat3blem 14663

Description: Lemma for swrdccat3b 14664. (Contributed by AV, 30-May-2018.)

**Hypothesis**
Ref	Expression
swrdccatin2.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 14458	. . . . . . . 8 ⊢ (𝐵 ∈ Word 𝑉 → (♯‘𝐵) ∈ ℕ₀)
2		nn0le0eq0 12430	. . . . . . . . 9 ⊢ ((♯‘𝐵) ∈ ℕ₀ → ((♯‘𝐵) ≤ 0 ↔ (♯‘𝐵) = 0))
3	2	biimpd 229	. . . . . . . 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 14288	. . . . . . . . . . 11 ⊢ (𝐵 ∈ Word 𝑉 → ((♯‘𝐵) = 0 ↔ 𝐵 = ∅))
7	6	biimpd 229	. . . . . . . . . 10 ⊢ (𝐵 ∈ Word 𝑉 → ((♯‘𝐵) = 0 → 𝐵 = ∅))
8	7	adantl 481	. . . . . . . . 9 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → ((♯‘𝐵) = 0 → 𝐵 = ∅))
9	8	imp 406	. . . . . . . 8 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (♯‘𝐵) = 0) → 𝐵 = ∅)
10		lencl 14458	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ Word 𝑉 → (♯‘𝐴) ∈ ℕ₀)
11		swrdccatin2.l	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐿 = (♯‘𝐴)
12	11	eqcomi 2738	. . . . . . . . . . . . . . . . . 18 ⊢ (♯‘𝐴) = 𝐿
13	12	eleq1i 2819	. . . . . . . . . . . . . . . . 17 ⊢ ((♯‘𝐴) ∈ ℕ₀ ↔ 𝐿 ∈ ℕ₀)
14		nn0re 12411	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐿 ∈ ℕ₀ → 𝐿 ∈ ℝ)
15		elfz2nn0 13539	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑀 ∈ (0...(𝐿 + 0)) ↔ (𝑀 ∈ ℕ₀ ∧ (𝐿 + 0) ∈ ℕ₀ ∧ 𝑀 ≤ (𝐿 + 0)))
16		recn 11118	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝐿 ∈ ℝ → 𝐿 ∈ ℂ)
17	16	addridd 11334	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐿 ∈ ℝ → (𝐿 + 0) = 𝐿)
18	17	breq2d 5107	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐿 ∈ ℝ → (𝑀 ≤ (𝐿 + 0) ↔ 𝑀 ≤ 𝐿))
19		nn0re 12411	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑀 ∈ ℕ₀ → 𝑀 ∈ ℝ)
20	19	anim1i 615	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝐿 ∈ ℝ) → (𝑀 ∈ ℝ ∧ 𝐿 ∈ ℝ))
21	20	ancoms 458	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝐿 ∈ ℝ ∧ 𝑀 ∈ ℕ₀) → (𝑀 ∈ ℝ ∧ 𝐿 ∈ ℝ))
22		letri3 11219	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 248	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝐿 ∈ ℝ ∧ 𝑀 ∈ ℕ₀) → ((𝑀 ≤ 𝐿 ∧ 𝐿 ≤ 𝑀) → 𝑀 = 𝐿))
25	24	exp4b 430	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐿 ∈ ℝ → (𝑀 ∈ ℕ₀ → (𝑀 ≤ 𝐿 → (𝐿 ≤ 𝑀 → 𝑀 = 𝐿))))
26	25	com23 86	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐿 ∈ ℝ → (𝑀 ≤ 𝐿 → (𝑀 ∈ ℕ₀ → (𝐿 ≤ 𝑀 → 𝑀 = 𝐿))))
27	18, 26	sylbid 240	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐿 ∈ ℝ → (𝑀 ≤ (𝐿 + 0) → (𝑀 ∈ ℕ₀ → (𝐿 ≤ 𝑀 → 𝑀 = 𝐿))))
28	27	com3l 89	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑀 ≤ (𝐿 + 0) → (𝑀 ∈ ℕ₀ → (𝐿 ∈ ℝ → (𝐿 ≤ 𝑀 → 𝑀 = 𝐿))))
29	28	impcom 407	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑀 ≤ (𝐿 + 0)) → (𝐿 ∈ ℝ → (𝐿 ≤ 𝑀 → 𝑀 = 𝐿)))
30	29	3adant2 1131	. . . . . . . . . . . . . . . . . . . 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	biimtrid 242	. . . . . . . . . . . . . . . . . 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 217	. . . . . . . . . . . . . . . 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 406	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝑀 ∈ (0...(𝐿 + 0))) → (𝐿 ≤ 𝑀 → 𝑀 = 𝐿))
37		elfznn0 13541	. . . . . . . . . . . . . . . 16 ⊢ (𝑀 ∈ (0...(𝐿 + 0)) → 𝑀 ∈ ℕ₀)
38		swrd00 14569	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (∅ substr ⟨0, 0⟩) = ∅
39		swrd00 14569	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐴 substr ⟨𝐿, 𝐿⟩) = ∅
40	38, 39	eqtr4i 2755	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∅ substr ⟨0, 0⟩) = (𝐴 substr ⟨𝐿, 𝐿⟩)
41		nn0cn 12412	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐿 ∈ ℕ₀ → 𝐿 ∈ ℂ)
42	41	subidd 11481	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐿 ∈ ℕ₀ → (𝐿 − 𝐿) = 0)
43	42	opeq1d 4833	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐿 ∈ ℕ₀ → ⟨(𝐿 − 𝐿), 0⟩ = ⟨0, 0⟩)
44	43	oveq2d 7369	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐿 ∈ ℕ₀ → (∅ substr ⟨(𝐿 − 𝐿), 0⟩) = (∅ substr ⟨0, 0⟩))
45	41	addridd 11334	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐿 ∈ ℕ₀ → (𝐿 + 0) = 𝐿)
46	45	opeq2d 4834	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐿 ∈ ℕ₀ → ⟨𝐿, (𝐿 + 0)⟩ = ⟨𝐿, 𝐿⟩)
47	46	oveq2d 7369	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐿 ∈ ℕ₀ → (𝐴 substr ⟨𝐿, (𝐿 + 0)⟩) = (𝐴 substr ⟨𝐿, 𝐿⟩))
48	40, 44, 47	3eqtr4a 2790	. . . . . . . . . . . . . . . . . . . 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 2816	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑀 = 𝐿 → (𝑀 ∈ ℕ₀ ↔ 𝐿 ∈ ℕ₀))
51		oveq1 7360	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑀 = 𝐿 → (𝑀 − 𝐿) = (𝐿 − 𝐿))
52	51	opeq1d 4833	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑀 = 𝐿 → ⟨(𝑀 − 𝐿), 0⟩ = ⟨(𝐿 − 𝐿), 0⟩)
53	52	oveq2d 7369	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑀 = 𝐿 → (∅ substr ⟨(𝑀 − 𝐿), 0⟩) = (∅ substr ⟨(𝐿 − 𝐿), 0⟩))
54		opeq1 4827	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑀 = 𝐿 → ⟨𝑀, (𝐿 + 0)⟩ = ⟨𝐿, (𝐿 + 0)⟩)
55	54	oveq2d 7369	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑀 = 𝐿 → (𝐴 substr ⟨𝑀, (𝐿 + 0)⟩) = (𝐴 substr ⟨𝐿, (𝐿 + 0)⟩))
56	53, 55	eqeq12d 2745	. . . . . . . . . . . . . . . . . . 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 294	. . . . . . . . . . . . . . . . . 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 407	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝑀 ∈ (0...(𝐿 + 0))) → (𝑀 = 𝐿 → (∅ substr ⟨(𝑀 − 𝐿), 0⟩) = (𝐴 substr ⟨𝑀, (𝐿 + 0)⟩)))
62	36, 61	syld 47	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝑀 ∈ (0...(𝐿 + 0))) → (𝐿 ≤ 𝑀 → (∅ substr ⟨(𝑀 − 𝐿), 0⟩) = (𝐴 substr ⟨𝑀, (𝐿 + 0)⟩)))
63	62	imp 406	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝑀 ∈ (0...(𝐿 + 0))) ∧ 𝐿 ≤ 𝑀) → (∅ substr ⟨(𝑀 − 𝐿), 0⟩) = (𝐴 substr ⟨𝑀, (𝐿 + 0)⟩))
64		swrdcl 14570	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ Word 𝑉 → (𝐴 substr ⟨𝑀, 𝐿⟩) ∈ Word 𝑉)
65		ccatrid 14512	. . . . . . . . . . . . . . . 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 217	. . . . . . . . . . . . . . . . . . 19 ⊢ ((♯‘𝐴) ∈ ℕ₀ → 𝐿 ∈ ℂ)
68	10, 67	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ Word 𝑉 → 𝐿 ∈ ℂ)
69		addrid 11314	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐿 ∈ ℂ → (𝐿 + 0) = 𝐿)
70	69	eqcomd 2735	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐿 ∈ ℂ → 𝐿 = (𝐿 + 0))
71	68, 70	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ Word 𝑉 → 𝐿 = (𝐿 + 0))
72	71	opeq2d 4834	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ Word 𝑉 → ⟨𝑀, 𝐿⟩ = ⟨𝑀, (𝐿 + 0)⟩)
73	72	oveq2d 7369	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ Word 𝑉 → (𝐴 substr ⟨𝑀, 𝐿⟩) = (𝐴 substr ⟨𝑀, (𝐿 + 0)⟩))
74	66, 73	eqtrd 2764	. . . . . . . . . . . . . 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 4515	. . . . . . . . . . 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 412	. . . . . . . . . 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 730	. . . . . . . . 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 7361	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝐵) = 0 → (𝐿 + (♯‘𝐵)) = (𝐿 + 0))
81	80	oveq2d 7369	. . . . . . . . . . . . 13 ⊢ ((♯‘𝐵) = 0 → (0...(𝐿 + (♯‘𝐵))) = (0...(𝐿 + 0)))
82	81	eleq2d 2814	. . . . . . . . . . . 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 484	. . . . . . . . . . . . . 14 ⊢ (((♯‘𝐵) = 0 ∧ 𝐵 = ∅) → 𝐵 = ∅)
85		opeq2 4828	. . . . . . . . . . . . . . 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 7371	. . . . . . . . . . . . 13 ⊢ (((♯‘𝐵) = 0 ∧ 𝐵 = ∅) → (𝐵 substr ⟨(𝑀 − 𝐿), (♯‘𝐵)⟩) = (∅ substr ⟨(𝑀 − 𝐿), 0⟩))
88		oveq2 7361	. . . . . . . . . . . . . 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 4500	. . . . . . . . . . . 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 4834	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝐵) = 0 → ⟨𝑀, (𝐿 + (♯‘𝐵))⟩ = ⟨𝑀, (𝐿 + 0)⟩)
92	91	oveq2d 7369	. . . . . . . . . . . . 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 2745	. . . . . . . . . . 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 344	. . . . . . . . . 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 714	. . . . . . . . 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 257	. . . . . . . 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 687	. . . . . . 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 412	. . . . . 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 47	. . . . 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 86	. . . 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 406	. . 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 2819	. . . . . . . 8 ⊢ (𝐿 ∈ ℕ₀ ↔ (♯‘𝐴) ∈ ℕ₀)
105	104, 14	sylbir 235	. . . . . . 7 ⊢ ((♯‘𝐴) ∈ ℕ₀ → 𝐿 ∈ ℝ)
106	10, 105	syl 17	. . . . . 6 ⊢ (𝐴 ∈ Word 𝑉 → 𝐿 ∈ ℝ)
107	1	nn0red 12464	. . . . . 6 ⊢ (𝐵 ∈ Word 𝑉 → (♯‘𝐵) ∈ ℝ)
108		leaddle0 11653	. . . . . 6 ⊢ ((𝐿 ∈ ℝ ∧ (♯‘𝐵) ∈ ℝ) → ((𝐿 + (♯‘𝐵)) ≤ 𝐿 ↔ (♯‘𝐵) ≤ 0))
109	106, 107, 108	syl2an 596	. . . . 5 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → ((𝐿 + (♯‘𝐵)) ≤ 𝐿 ↔ (♯‘𝐵) ≤ 0))
110		pm2.24 124	. . . . 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	biimtrdi 253	. . . 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 406	. 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 179	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 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ∅c0 4286 ifcif 4478 ⟨cop 4585 class class class wbr 5095 ‘cfv 6486 (class class class)co 7353 ℂcc 11026 ℝcr 11027 0cc0 11028 + caddc 11031 ≤ cle 11169 − cmin 11365 ℕ₀cn0 12402 ...cfz 13428 ♯chash 14255 Word cword 14438 ++ cconcat 14495 substr csubstr 14565

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 2701 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105

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 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-reu 3346 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4862 df-int 4900 df-iun 4946 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-riota 7310 df-ov 7356 df-oprab 7357 df-mpo 7358 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-er 8632 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-card 9854 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11367 df-neg 11368 df-nn 12147 df-n0 12403 df-z 12490 df-uz 12754 df-fz 13429 df-fzo 13576 df-hash 14256 df-word 14439 df-concat 14496 df-substr 14566

This theorem is referenced by: swrdccat3b 14664

W3C validator