Theorem smumullem 16551

Description: Lemma for smumul 16552. (Contributed by Mario Carneiro, 22-Sep-2016.)

Hypotheses

Ref	Expression
smumullem.a	⊢ (𝜑 → 𝐴 ∈ ℤ)
smumullem.b	⊢ (𝜑 → 𝐵 ∈ ℤ)
smumullem.n	⊢ (𝜑 → 𝑁 ∈ ℕ0)

Assertion

Ref	Expression
smumullem	⊢ (𝜑 → (((bits‘𝐴) ∩ (0..^𝑁)) smul (bits‘𝐵)) = (bits‘((𝐴 mod (2↑𝑁)) · 𝐵)))

Proof of Theorem smumullem

Dummy variables 𝑘 𝑛 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		smumullem.n	. 2 ⊢ (𝜑 → 𝑁 ∈ ℕ0)
2		oveq2 7420	. . . . . . . . . 10 ⊢ (𝑥 = 0 → (0..^𝑥) = (0..^0))
3		fzo0 13714	. . . . . . . . . 10 ⊢ (0..^0) = ∅
4	2, 3	eqtrdi 2814	. . . . . . . . 9 ⊢ (𝑥 = 0 → (0..^𝑥) = ∅)
5	4	ineq2d 4174	. . . . . . . 8 ⊢ (𝑥 = 0 → ((bits‘𝐴) ∩ (0..^𝑥)) = ((bits‘𝐴) ∩ ∅))
6		in0 4353	. . . . . . . 8 ⊢ ((bits‘𝐴) ∩ ∅) = ∅
7	5, 6	eqtrdi 2814	. . . . . . 7 ⊢ (𝑥 = 0 → ((bits‘𝐴) ∩ (0..^𝑥)) = ∅)
8	7	oveq1d 7427	. . . . . 6 ⊢ (𝑥 = 0 → (((bits‘𝐴) ∩ (0..^𝑥)) smul (bits‘𝐵)) = (∅ smul (bits‘𝐵)))
9		bitsss 16485	. . . . . . 7 ⊢ (bits‘𝐵) ⊆ ℕ0
10		smu02 16546	. . . . . . 7 ⊢ ((bits‘𝐵) ⊆ ℕ0 → (∅ smul (bits‘𝐵)) = ∅)
11	9, 10	ax-mp 5	. . . . . 6 ⊢ (∅ smul (bits‘𝐵)) = ∅
12	8, 11	eqtrdi 2814	. . . . 5 ⊢ (𝑥 = 0 → (((bits‘𝐴) ∩ (0..^𝑥)) smul (bits‘𝐵)) = ∅)
13		oveq2 7420	. . . . . . . 8 ⊢ (𝑥 = 0 → (2↑𝑥) = (2↑0))
14		2cn 12317	. . . . . . . . 9 ⊢ 2 ∈ ℂ
15		exp0 14103	. . . . . . . . 9 ⊢ (2 ∈ ℂ → (2↑0) = 1)
16	14, 15	ax-mp 5	. . . . . . . 8 ⊢ (2↑0) = 1
17	13, 16	eqtrdi 2814	. . . . . . 7 ⊢ (𝑥 = 0 → (2↑𝑥) = 1)
18	17	oveq2d 7428	. . . . . 6 ⊢ (𝑥 = 0 → (𝐴 mod (2↑𝑥)) = (𝐴 mod 1))
19	18	fvoveq1d 7434	. . . . 5 ⊢ (𝑥 = 0 → (bits‘((𝐴 mod (2↑𝑥)) · 𝐵)) = (bits‘((𝐴 mod 1) · 𝐵)))
20	12, 19	eqeq12d 2779	. . . 4 ⊢ (𝑥 = 0 → ((((bits‘𝐴) ∩ (0..^𝑥)) smul (bits‘𝐵)) = (bits‘((𝐴 mod (2↑𝑥)) · 𝐵)) ↔ ∅ = (bits‘((𝐴 mod 1) · 𝐵))))
21	20	imbi2d 343	. . 3 ⊢ (𝑥 = 0 → ((𝜑 → (((bits‘𝐴) ∩ (0..^𝑥)) smul (bits‘𝐵)) = (bits‘((𝐴 mod (2↑𝑥)) · 𝐵))) ↔ (𝜑 → ∅ = (bits‘((𝐴 mod 1) · 𝐵)))))
22		oveq2 7420	. . . . . . 7 ⊢ (𝑥 = 𝑘 → (0..^𝑥) = (0..^𝑘))
23	22	ineq2d 4174	. . . . . 6 ⊢ (𝑥 = 𝑘 → ((bits‘𝐴) ∩ (0..^𝑥)) = ((bits‘𝐴) ∩ (0..^𝑘)))
24	23	oveq1d 7427	. . . . 5 ⊢ (𝑥 = 𝑘 → (((bits‘𝐴) ∩ (0..^𝑥)) smul (bits‘𝐵)) = (((bits‘𝐴) ∩ (0..^𝑘)) smul (bits‘𝐵)))
25		oveq2 7420	. . . . . . 7 ⊢ (𝑥 = 𝑘 → (2↑𝑥) = (2↑𝑘))
26	25	oveq2d 7428	. . . . . 6 ⊢ (𝑥 = 𝑘 → (𝐴 mod (2↑𝑥)) = (𝐴 mod (2↑𝑘)))
27	26	fvoveq1d 7434	. . . . 5 ⊢ (𝑥 = 𝑘 → (bits‘((𝐴 mod (2↑𝑥)) · 𝐵)) = (bits‘((𝐴 mod (2↑𝑘)) · 𝐵)))
28	24, 27	eqeq12d 2779	. . . 4 ⊢ (𝑥 = 𝑘 → ((((bits‘𝐴) ∩ (0..^𝑥)) smul (bits‘𝐵)) = (bits‘((𝐴 mod (2↑𝑥)) · 𝐵)) ↔ (((bits‘𝐴) ∩ (0..^𝑘)) smul (bits‘𝐵)) = (bits‘((𝐴 mod (2↑𝑘)) · 𝐵))))
29	28	imbi2d 343	. . 3 ⊢ (𝑥 = 𝑘 → ((𝜑 → (((bits‘𝐴) ∩ (0..^𝑥)) smul (bits‘𝐵)) = (bits‘((𝐴 mod (2↑𝑥)) · 𝐵))) ↔ (𝜑 → (((bits‘𝐴) ∩ (0..^𝑘)) smul (bits‘𝐵)) = (bits‘((𝐴 mod (2↑𝑘)) · 𝐵)))))
30		oveq2 7420	. . . . . . 7 ⊢ (𝑥 = (𝑘 + 1) → (0..^𝑥) = (0..^(𝑘 + 1)))
31	30	ineq2d 4174	. . . . . 6 ⊢ (𝑥 = (𝑘 + 1) → ((bits‘𝐴) ∩ (0..^𝑥)) = ((bits‘𝐴) ∩ (0..^(𝑘 + 1))))
32	31	oveq1d 7427	. . . . 5 ⊢ (𝑥 = (𝑘 + 1) → (((bits‘𝐴) ∩ (0..^𝑥)) smul (bits‘𝐵)) = (((bits‘𝐴) ∩ (0..^(𝑘 + 1))) smul (bits‘𝐵)))
33		oveq2 7420	. . . . . . 7 ⊢ (𝑥 = (𝑘 + 1) → (2↑𝑥) = (2↑(𝑘 + 1)))
34	33	oveq2d 7428	. . . . . 6 ⊢ (𝑥 = (𝑘 + 1) → (𝐴 mod (2↑𝑥)) = (𝐴 mod (2↑(𝑘 + 1))))
35	34	fvoveq1d 7434	. . . . 5 ⊢ (𝑥 = (𝑘 + 1) → (bits‘((𝐴 mod (2↑𝑥)) · 𝐵)) = (bits‘((𝐴 mod (2↑(𝑘 + 1))) · 𝐵)))
36	32, 35	eqeq12d 2779	. . . 4 ⊢ (𝑥 = (𝑘 + 1) → ((((bits‘𝐴) ∩ (0..^𝑥)) smul (bits‘𝐵)) = (bits‘((𝐴 mod (2↑𝑥)) · 𝐵)) ↔ (((bits‘𝐴) ∩ (0..^(𝑘 + 1))) smul (bits‘𝐵)) = (bits‘((𝐴 mod (2↑(𝑘 + 1))) · 𝐵))))
37	36	imbi2d 343	. . 3 ⊢ (𝑥 = (𝑘 + 1) → ((𝜑 → (((bits‘𝐴) ∩ (0..^𝑥)) smul (bits‘𝐵)) = (bits‘((𝐴 mod (2↑𝑥)) · 𝐵))) ↔ (𝜑 → (((bits‘𝐴) ∩ (0..^(𝑘 + 1))) smul (bits‘𝐵)) = (bits‘((𝐴 mod (2↑(𝑘 + 1))) · 𝐵)))))
38		oveq2 7420	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (0..^𝑥) = (0..^𝑁))
39	38	ineq2d 4174	. . . . . 6 ⊢ (𝑥 = 𝑁 → ((bits‘𝐴) ∩ (0..^𝑥)) = ((bits‘𝐴) ∩ (0..^𝑁)))
40	39	oveq1d 7427	. . . . 5 ⊢ (𝑥 = 𝑁 → (((bits‘𝐴) ∩ (0..^𝑥)) smul (bits‘𝐵)) = (((bits‘𝐴) ∩ (0..^𝑁)) smul (bits‘𝐵)))
41		oveq2 7420	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (2↑𝑥) = (2↑𝑁))
42	41	oveq2d 7428	. . . . . 6 ⊢ (𝑥 = 𝑁 → (𝐴 mod (2↑𝑥)) = (𝐴 mod (2↑𝑁)))
43	42	fvoveq1d 7434	. . . . 5 ⊢ (𝑥 = 𝑁 → (bits‘((𝐴 mod (2↑𝑥)) · 𝐵)) = (bits‘((𝐴 mod (2↑𝑁)) · 𝐵)))
44	40, 43	eqeq12d 2779	. . . 4 ⊢ (𝑥 = 𝑁 → ((((bits‘𝐴) ∩ (0..^𝑥)) smul (bits‘𝐵)) = (bits‘((𝐴 mod (2↑𝑥)) · 𝐵)) ↔ (((bits‘𝐴) ∩ (0..^𝑁)) smul (bits‘𝐵)) = (bits‘((𝐴 mod (2↑𝑁)) · 𝐵))))
45	44	imbi2d 343	. . 3 ⊢ (𝑥 = 𝑁 → ((𝜑 → (((bits‘𝐴) ∩ (0..^𝑥)) smul (bits‘𝐵)) = (bits‘((𝐴 mod (2↑𝑥)) · 𝐵))) ↔ (𝜑 → (((bits‘𝐴) ∩ (0..^𝑁)) smul (bits‘𝐵)) = (bits‘((𝐴 mod (2↑𝑁)) · 𝐵)))))
46		smumullem.a	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℤ)
47		zmod10 13922	. . . . . . . 8 ⊢ (𝐴 ∈ ℤ → (𝐴 mod 1) = 0)
48	46, 47	syl 18	. . . . . . 7 ⊢ (𝜑 → (𝐴 mod 1) = 0)
49	48	oveq1d 7427	. . . . . 6 ⊢ (𝜑 → ((𝐴 mod 1) · 𝐵) = (0 · 𝐵))
50		smumullem.b	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℤ)
51	50	zcnd 12702	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℂ)
52	51	mul02d 11409	. . . . . 6 ⊢ (𝜑 → (0 · 𝐵) = 0)
53	49, 52	eqtrd 2798	. . . . 5 ⊢ (𝜑 → ((𝐴 mod 1) · 𝐵) = 0)
54	53	fveq2d 6887	. . . 4 ⊢ (𝜑 → (bits‘((𝐴 mod 1) · 𝐵)) = (bits‘0))
55		0bits 16498	. . . 4 ⊢ (bits‘0) = ∅
56	54, 55	eqtr2di 2815	. . 3 ⊢ (𝜑 → ∅ = (bits‘((𝐴 mod 1) · 𝐵)))
57		oveq1 7419	. . . . . 6 ⊢ ((((bits‘𝐴) ∩ (0..^𝑘)) smul (bits‘𝐵)) = (bits‘((𝐴 mod (2↑𝑘)) · 𝐵)) → ((((bits‘𝐴) ∩ (0..^𝑘)) smul (bits‘𝐵)) sadd {𝑛 ∈ ℕ0 ∣ (𝑘 ∈ (bits‘𝐴) ∧ (𝑛 − 𝑘) ∈ (bits‘𝐵))}) = ((bits‘((𝐴 mod (2↑𝑘)) · 𝐵)) sadd {𝑛 ∈ ℕ0 ∣ (𝑘 ∈ (bits‘𝐴) ∧ (𝑛 − 𝑘) ∈ (bits‘𝐵))}))
58		bitsss 16485	. . . . . . . . 9 ⊢ (bits‘𝐴) ⊆ ℕ0
59	58	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (bits‘𝐴) ⊆ ℕ0)
60	9	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (bits‘𝐵) ⊆ ℕ0)
61		simpr 489	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0)
62	59, 60, 61	smup1 16548	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (((bits‘𝐴) ∩ (0..^(𝑘 + 1))) smul (bits‘𝐵)) = ((((bits‘𝐴) ∩ (0..^𝑘)) smul (bits‘𝐵)) sadd {𝑛 ∈ ℕ0 ∣ (𝑘 ∈ (bits‘𝐴) ∧ (𝑛 − 𝑘) ∈ (bits‘𝐵))}))
63		bitsinv1lem 16500	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ 𝑘 ∈ ℕ0) → (𝐴 mod (2↑(𝑘 + 1))) = ((𝐴 mod (2↑𝑘)) + if(𝑘 ∈ (bits‘𝐴), (2↑𝑘), 0)))
64	46, 63	sylan 591	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐴 mod (2↑(𝑘 + 1))) = ((𝐴 mod (2↑𝑘)) + if(𝑘 ∈ (bits‘𝐴), (2↑𝑘), 0)))
65	64	oveq1d 7427	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝐴 mod (2↑(𝑘 + 1))) · 𝐵) = (((𝐴 mod (2↑𝑘)) + if(𝑘 ∈ (bits‘𝐴), (2↑𝑘), 0)) · 𝐵))
66	46	adantr 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝐴 ∈ ℤ)
67		2nn 12315	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℕ
68	67	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 2 ∈ ℕ)
69	68, 61	nnexpcld 14283	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (2↑𝑘) ∈ ℕ)
70	66, 69	zmodcld 13927	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐴 mod (2↑𝑘)) ∈ ℕ0)
71	70	nn0cnd 12568	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐴 mod (2↑𝑘)) ∈ ℂ)
72	69	nnnn0d 12566	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (2↑𝑘) ∈ ℕ0)
73		0nn0 12520	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℕ0
74		ifcl 4534	. . . . . . . . . . . . 13 ⊢ (((2↑𝑘) ∈ ℕ0 ∧ 0 ∈ ℕ0) → if(𝑘 ∈ (bits‘𝐴), (2↑𝑘), 0) ∈ ℕ0)
75	72, 73, 74	sylancl 597	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → if(𝑘 ∈ (bits‘𝐴), (2↑𝑘), 0) ∈ ℕ0)
76	75	nn0cnd 12568	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → if(𝑘 ∈ (bits‘𝐴), (2↑𝑘), 0) ∈ ℂ)
77	51	adantr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝐵 ∈ ℂ)
78	71, 76, 77	adddird 11235	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (((𝐴 mod (2↑𝑘)) + if(𝑘 ∈ (bits‘𝐴), (2↑𝑘), 0)) · 𝐵) = (((𝐴 mod (2↑𝑘)) · 𝐵) + (if(𝑘 ∈ (bits‘𝐴), (2↑𝑘), 0) · 𝐵)))
79	76, 77	mulcomd 11231	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (if(𝑘 ∈ (bits‘𝐴), (2↑𝑘), 0) · 𝐵) = (𝐵 · if(𝑘 ∈ (bits‘𝐴), (2↑𝑘), 0)))
80	79	oveq2d 7428	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (((𝐴 mod (2↑𝑘)) · 𝐵) + (if(𝑘 ∈ (bits‘𝐴), (2↑𝑘), 0) · 𝐵)) = (((𝐴 mod (2↑𝑘)) · 𝐵) + (𝐵 · if(𝑘 ∈ (bits‘𝐴), (2↑𝑘), 0))))
81	65, 78, 80	3eqtrd 2802	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝐴 mod (2↑(𝑘 + 1))) · 𝐵) = (((𝐴 mod (2↑𝑘)) · 𝐵) + (𝐵 · if(𝑘 ∈ (bits‘𝐴), (2↑𝑘), 0))))
82	81	fveq2d 6887	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (bits‘((𝐴 mod (2↑(𝑘 + 1))) · 𝐵)) = (bits‘(((𝐴 mod (2↑𝑘)) · 𝐵) + (𝐵 · if(𝑘 ∈ (bits‘𝐴), (2↑𝑘), 0)))))
83	70	nn0zd 12617	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐴 mod (2↑𝑘)) ∈ ℤ)
84	50	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝐵 ∈ ℤ)
85	83, 84	zmulcld 12707	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝐴 mod (2↑𝑘)) · 𝐵) ∈ ℤ)
86	75	nn0zd 12617	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → if(𝑘 ∈ (bits‘𝐴), (2↑𝑘), 0) ∈ ℤ)
87	84, 86	zmulcld 12707	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐵 · if(𝑘 ∈ (bits‘𝐴), (2↑𝑘), 0)) ∈ ℤ)
88		sadadd 16526	. . . . . . . . 9 ⊢ ((((𝐴 mod (2↑𝑘)) · 𝐵) ∈ ℤ ∧ (𝐵 · if(𝑘 ∈ (bits‘𝐴), (2↑𝑘), 0)) ∈ ℤ) → ((bits‘((𝐴 mod (2↑𝑘)) · 𝐵)) sadd (bits‘(𝐵 · if(𝑘 ∈ (bits‘𝐴), (2↑𝑘), 0)))) = (bits‘(((𝐴 mod (2↑𝑘)) · 𝐵) + (𝐵 · if(𝑘 ∈ (bits‘𝐴), (2↑𝑘), 0)))))
89	85, 87, 88	syl2anc 595	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((bits‘((𝐴 mod (2↑𝑘)) · 𝐵)) sadd (bits‘(𝐵 · if(𝑘 ∈ (bits‘𝐴), (2↑𝑘), 0)))) = (bits‘(((𝐴 mod (2↑𝑘)) · 𝐵) + (𝐵 · if(𝑘 ∈ (bits‘𝐴), (2↑𝑘), 0)))))
90		oveq2 7420	. . . . . . . . . . 11 ⊢ ((2↑𝑘) = if(𝑘 ∈ (bits‘𝐴), (2↑𝑘), 0) → (𝐵 · (2↑𝑘)) = (𝐵 · if(𝑘 ∈ (bits‘𝐴), (2↑𝑘), 0)))
91	90	fveqeq2d 6891	. . . . . . . . . 10 ⊢ ((2↑𝑘) = if(𝑘 ∈ (bits‘𝐴), (2↑𝑘), 0) → ((bits‘(𝐵 · (2↑𝑘))) = {𝑛 ∈ ℕ0 ∣ (𝑘 ∈ (bits‘𝐴) ∧ (𝑛 − 𝑘) ∈ (bits‘𝐵))} ↔ (bits‘(𝐵 · if(𝑘 ∈ (bits‘𝐴), (2↑𝑘), 0))) = {𝑛 ∈ ℕ0 ∣ (𝑘 ∈ (bits‘𝐴) ∧ (𝑛 − 𝑘) ∈ (bits‘𝐵))}))
92		oveq2 7420	. . . . . . . . . . 11 ⊢ (0 = if(𝑘 ∈ (bits‘𝐴), (2↑𝑘), 0) → (𝐵 · 0) = (𝐵 · if(𝑘 ∈ (bits‘𝐴), (2↑𝑘), 0)))
93	92	fveqeq2d 6891	. . . . . . . . . 10 ⊢ (0 = if(𝑘 ∈ (bits‘𝐴), (2↑𝑘), 0) → ((bits‘(𝐵 · 0)) = {𝑛 ∈ ℕ0 ∣ (𝑘 ∈ (bits‘𝐴) ∧ (𝑛 − 𝑘) ∈ (bits‘𝐵))} ↔ (bits‘(𝐵 · if(𝑘 ∈ (bits‘𝐴), (2↑𝑘), 0))) = {𝑛 ∈ ℕ0 ∣ (𝑘 ∈ (bits‘𝐴) ∧ (𝑛 − 𝑘) ∈ (bits‘𝐵))}))
94		bitsshft 16534	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ ℤ ∧ 𝑘 ∈ ℕ0) → {𝑛 ∈ ℕ0 ∣ (𝑛 − 𝑘) ∈ (bits‘𝐵)} = (bits‘(𝐵 · (2↑𝑘))))
95	50, 94	sylan 591	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → {𝑛 ∈ ℕ0 ∣ (𝑛 − 𝑘) ∈ (bits‘𝐵)} = (bits‘(𝐵 · (2↑𝑘))))
96		ibar 537	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (bits‘𝐴) → ((𝑛 − 𝑘) ∈ (bits‘𝐵) ↔ (𝑘 ∈ (bits‘𝐴) ∧ (𝑛 − 𝑘) ∈ (bits‘𝐵))))
97	96	rabbidv 3423	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (bits‘𝐴) → {𝑛 ∈ ℕ0 ∣ (𝑛 − 𝑘) ∈ (bits‘𝐵)} = {𝑛 ∈ ℕ0 ∣ (𝑘 ∈ (bits‘𝐴) ∧ (𝑛 − 𝑘) ∈ (bits‘𝐵))})
98	95, 97	sylan9req 2819	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ 𝑘 ∈ (bits‘𝐴)) → (bits‘(𝐵 · (2↑𝑘))) = {𝑛 ∈ ℕ0 ∣ (𝑘 ∈ (bits‘𝐴) ∧ (𝑛 − 𝑘) ∈ (bits‘𝐵))})
99	77	adantr 485	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ∈ (bits‘𝐴)) → 𝐵 ∈ ℂ)
100	99	mul01d 11410	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ∈ (bits‘𝐴)) → (𝐵 · 0) = 0)
101	100	fveq2d 6887	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ∈ (bits‘𝐴)) → (bits‘(𝐵 · 0)) = (bits‘0))
102		simpr 489	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ∈ (bits‘𝐴)) → ¬ 𝑘 ∈ (bits‘𝐴))
103	102	intnanrd 494	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ∈ (bits‘𝐴)) → ¬ (𝑘 ∈ (bits‘𝐴) ∧ (𝑛 − 𝑘) ∈ (bits‘𝐵)))
104	103	ralrimivw 3161	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ∈ (bits‘𝐴)) → ∀𝑛 ∈ ℕ0 ¬ (𝑘 ∈ (bits‘𝐴) ∧ (𝑛 − 𝑘) ∈ (bits‘𝐵)))
105		rabeq0 4346	. . . . . . . . . . . 12 ⊢ ({𝑛 ∈ ℕ0 ∣ (𝑘 ∈ (bits‘𝐴) ∧ (𝑛 − 𝑘) ∈ (bits‘𝐵))} = ∅ ↔ ∀𝑛 ∈ ℕ0 ¬ (𝑘 ∈ (bits‘𝐴) ∧ (𝑛 − 𝑘) ∈ (bits‘𝐵)))
106	104, 105	sylibr 237	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ∈ (bits‘𝐴)) → {𝑛 ∈ ℕ0 ∣ (𝑘 ∈ (bits‘𝐴) ∧ (𝑛 − 𝑘) ∈ (bits‘𝐵))} = ∅)
107	55, 101, 106	3eqtr4a 2824	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ∈ (bits‘𝐴)) → (bits‘(𝐵 · 0)) = {𝑛 ∈ ℕ0 ∣ (𝑘 ∈ (bits‘𝐴) ∧ (𝑛 − 𝑘) ∈ (bits‘𝐵))})
108	91, 93, 98, 107	ifbothda 4527	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (bits‘(𝐵 · if(𝑘 ∈ (bits‘𝐴), (2↑𝑘), 0))) = {𝑛 ∈ ℕ0 ∣ (𝑘 ∈ (bits‘𝐴) ∧ (𝑛 − 𝑘) ∈ (bits‘𝐵))})
109	108	oveq2d 7428	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((bits‘((𝐴 mod (2↑𝑘)) · 𝐵)) sadd (bits‘(𝐵 · if(𝑘 ∈ (bits‘𝐴), (2↑𝑘), 0)))) = ((bits‘((𝐴 mod (2↑𝑘)) · 𝐵)) sadd {𝑛 ∈ ℕ0 ∣ (𝑘 ∈ (bits‘𝐴) ∧ (𝑛 − 𝑘) ∈ (bits‘𝐵))}))
110	82, 89, 109	3eqtr2d 2804	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (bits‘((𝐴 mod (2↑(𝑘 + 1))) · 𝐵)) = ((bits‘((𝐴 mod (2↑𝑘)) · 𝐵)) sadd {𝑛 ∈ ℕ0 ∣ (𝑘 ∈ (bits‘𝐴) ∧ (𝑛 − 𝑘) ∈ (bits‘𝐵))}))
111	62, 110	eqeq12d 2779	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((((bits‘𝐴) ∩ (0..^(𝑘 + 1))) smul (bits‘𝐵)) = (bits‘((𝐴 mod (2↑(𝑘 + 1))) · 𝐵)) ↔ ((((bits‘𝐴) ∩ (0..^𝑘)) smul (bits‘𝐵)) sadd {𝑛 ∈ ℕ0 ∣ (𝑘 ∈ (bits‘𝐴) ∧ (𝑛 − 𝑘) ∈ (bits‘𝐵))}) = ((bits‘((𝐴 mod (2↑𝑘)) · 𝐵)) sadd {𝑛 ∈ ℕ0 ∣ (𝑘 ∈ (bits‘𝐴) ∧ (𝑛 − 𝑘) ∈ (bits‘𝐵))})))
112	57, 111	imbitrrid 249	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((((bits‘𝐴) ∩ (0..^𝑘)) smul (bits‘𝐵)) = (bits‘((𝐴 mod (2↑𝑘)) · 𝐵)) → (((bits‘𝐴) ∩ (0..^(𝑘 + 1))) smul (bits‘𝐵)) = (bits‘((𝐴 mod (2↑(𝑘 + 1))) · 𝐵))))
113	112	expcom 418	. . . 4 ⊢ (𝑘 ∈ ℕ0 → (𝜑 → ((((bits‘𝐴) ∩ (0..^𝑘)) smul (bits‘𝐵)) = (bits‘((𝐴 mod (2↑𝑘)) · 𝐵)) → (((bits‘𝐴) ∩ (0..^(𝑘 + 1))) smul (bits‘𝐵)) = (bits‘((𝐴 mod (2↑(𝑘 + 1))) · 𝐵)))))
114	113	a2d 30	. . 3 ⊢ (𝑘 ∈ ℕ0 → ((𝜑 → (((bits‘𝐴) ∩ (0..^𝑘)) smul (bits‘𝐵)) = (bits‘((𝐴 mod (2↑𝑘)) · 𝐵))) → (𝜑 → (((bits‘𝐴) ∩ (0..^(𝑘 + 1))) smul (bits‘𝐵)) = (bits‘((𝐴 mod (2↑(𝑘 + 1))) · 𝐵)))))
115	21, 29, 37, 45, 56, 114	nn0ind 12692	. 2 ⊢ (𝑁 ∈ ℕ0 → (𝜑 → (((bits‘𝐴) ∩ (0..^𝑁)) smul (bits‘𝐵)) = (bits‘((𝐴 mod (2↑𝑁)) · 𝐵))))
116	1, 115	mpcom 39	1 ⊢ (𝜑 → (((bits‘𝐴) ∩ (0..^𝑁)) smul (bits‘𝐵)) = (bits‘((𝐴 mod (2↑𝑁)) · 𝐵)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ∀wral 3079  {crab 3416   ∩ cin 3905   ⊆ wss 3906  ∅c0 4287  ifcif 4488  ‘cfv 6538  (class class class)co 7412  ℂcc 11099  0cc0 11101  1c1 11102   + caddc 11104   · cmul 11106   − cmin 11442  ℕcn 12234  2c2 12296  ℕ₀cn0 12505  ℤcz 12592  ..^cfzo 13684   mod cmo 13904  ↑cexp 14099  bitscbits 16478   sadd csad 16479   smul csmu 16480

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734  ax-inf2 9611  ax-cnex 11157  ax-resscn 11158  ax-1cn 11159  ax-icn 11160  ax-addcl 11161  ax-addrcl 11162  ax-mulcl 11163  ax-mulrcl 11164  ax-mulcom 11165  ax-addass 11166  ax-mulass 11167  ax-distr 11168  ax-i2m1 11169  ax-1ne0 11170  ax-1rid 11171  ax-rnegex 11172  ax-rrecex 11173  ax-cnre 11174  ax-pre-lttri 11175  ax-pre-lttrn 11176  ax-pre-ltadd 11177  ax-pre-mulgt0 11178  ax-pre-sup 11179

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-xor 1542  df-tru 1573  df-fal 1583  df-had 1624  df-cad 1637  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-int 4914  df-iun 4959  df-disj 5078  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-se 5617  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-isom 6547  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7864  df-1st 7987  df-2nd 7988  df-frecs 8279  df-wrecs 8310  df-recs 8359  df-rdg 8398  df-1o 8454  df-2o 8455  df-oadd 8458  df-er 8695  df-map 8827  df-pm 8828  df-en 8945  df-dom 8946  df-sdom 8947  df-fin 8948  df-sup 9403  df-inf 9404  df-oi 9473  df-dju 9888  df-card 9926  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11444  df-neg 11445  df-div 11873  df-nn 12235  df-2 12304  df-3 12305  df-n0 12506  df-xnn0 12579  df-z 12593  df-uz 12864  df-rp 13018  df-fz 13537  df-fzo 13685  df-fl 13827  df-mod 13905  df-seq 14040  df-exp 14100  df-hash 14369  df-cj 15152  df-re 15153  df-im 15154  df-sqrt 15288  df-abs 15289  df-clim 15541  df-sum 15740  df-dvds 16312  df-bits 16481  df-sad 16510  df-smu 16535

This theorem is referenced by:  smumul  16552