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Theorem lgsdir2 25914

Description: The Legendre symbol is completely multiplicative at 2. (Contributed by Mario Carneiro, 4-Feb-2015.)

**Assertion**
Ref	Expression
lgsdir2	⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴 · 𝐵) /_L 2) = ((𝐴 /_L 2) · (𝐵 /_L 2)))

**Proof of Theorem lgsdir2**
Step	Hyp	Ref	Expression
1		0cn 10622	. . . . . 6 ⊢ 0 ∈ ℂ
2		ax-1cn 10584	. . . . . . 7 ⊢ 1 ∈ ℂ
3		neg1cn 11739	. . . . . . 7 ⊢ -1 ∈ ℂ
4	2, 3	ifcli 4471	. . . . . 6 ⊢ if((𝐵 mod 8) ∈ {1, 7}, 1, -1) ∈ ℂ
5	1, 4	ifcli 4471	. . . . 5 ⊢ if(2 ∥ 𝐵, 0, if((𝐵 mod 8) ∈ {1, 7}, 1, -1)) ∈ ℂ
6	5	mul02i 10818	. . . 4 ⊢ (0 · if(2 ∥ 𝐵, 0, if((𝐵 mod 8) ∈ {1, 7}, 1, -1))) = 0
7		iftrue 4431	. . . . . 6 ⊢ (2 ∥ 𝐴 → if(2 ∥ 𝐴, 0, if((𝐴 mod 8) ∈ {1, 7}, 1, -1)) = 0)
8	7	adantl 485	. . . . 5 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 2 ∥ 𝐴) → if(2 ∥ 𝐴, 0, if((𝐴 mod 8) ∈ {1, 7}, 1, -1)) = 0)
9	8	oveq1d 7150	. . . 4 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 2 ∥ 𝐴) → (if(2 ∥ 𝐴, 0, if((𝐴 mod 8) ∈ {1, 7}, 1, -1)) · if(2 ∥ 𝐵, 0, if((𝐵 mod 8) ∈ {1, 7}, 1, -1))) = (0 · if(2 ∥ 𝐵, 0, if((𝐵 mod 8) ∈ {1, 7}, 1, -1))))
10		2z 12002	. . . . . . 7 ⊢ 2 ∈ ℤ
11		dvdsmultr1 15639	. . . . . . 7 ⊢ ((2 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (2 ∥ 𝐴 → 2 ∥ (𝐴 · 𝐵)))
12	10, 11	mp3an1 1445	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (2 ∥ 𝐴 → 2 ∥ (𝐴 · 𝐵)))
13	12	imp 410	. . . . 5 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 2 ∥ 𝐴) → 2 ∥ (𝐴 · 𝐵))
14	13	iftrued 4433	. . . 4 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 2 ∥ 𝐴) → if(2 ∥ (𝐴 · 𝐵), 0, if(((𝐴 · 𝐵) mod 8) ∈ {1, 7}, 1, -1)) = 0)
15	6, 9, 14	3eqtr4a 2859	. . 3 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 2 ∥ 𝐴) → (if(2 ∥ 𝐴, 0, if((𝐴 mod 8) ∈ {1, 7}, 1, -1)) · if(2 ∥ 𝐵, 0, if((𝐵 mod 8) ∈ {1, 7}, 1, -1))) = if(2 ∥ (𝐴 · 𝐵), 0, if(((𝐴 · 𝐵) mod 8) ∈ {1, 7}, 1, -1)))
16	2, 3	ifcli 4471	. . . . . 6 ⊢ if((𝐴 mod 8) ∈ {1, 7}, 1, -1) ∈ ℂ
17	1, 16	ifcli 4471	. . . . 5 ⊢ if(2 ∥ 𝐴, 0, if((𝐴 mod 8) ∈ {1, 7}, 1, -1)) ∈ ℂ
18	17	mul01i 10819	. . . 4 ⊢ (if(2 ∥ 𝐴, 0, if((𝐴 mod 8) ∈ {1, 7}, 1, -1)) · 0) = 0
19		iftrue 4431	. . . . . 6 ⊢ (2 ∥ 𝐵 → if(2 ∥ 𝐵, 0, if((𝐵 mod 8) ∈ {1, 7}, 1, -1)) = 0)
20	19	adantl 485	. . . . 5 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 2 ∥ 𝐵) → if(2 ∥ 𝐵, 0, if((𝐵 mod 8) ∈ {1, 7}, 1, -1)) = 0)
21	20	oveq2d 7151	. . . 4 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 2 ∥ 𝐵) → (if(2 ∥ 𝐴, 0, if((𝐴 mod 8) ∈ {1, 7}, 1, -1)) · if(2 ∥ 𝐵, 0, if((𝐵 mod 8) ∈ {1, 7}, 1, -1))) = (if(2 ∥ 𝐴, 0, if((𝐴 mod 8) ∈ {1, 7}, 1, -1)) · 0))
22		dvdsmultr2 15641	. . . . . . 7 ⊢ ((2 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (2 ∥ 𝐵 → 2 ∥ (𝐴 · 𝐵)))
23	10, 22	mp3an1 1445	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (2 ∥ 𝐵 → 2 ∥ (𝐴 · 𝐵)))
24	23	imp 410	. . . . 5 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 2 ∥ 𝐵) → 2 ∥ (𝐴 · 𝐵))
25	24	iftrued 4433	. . . 4 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 2 ∥ 𝐵) → if(2 ∥ (𝐴 · 𝐵), 0, if(((𝐴 · 𝐵) mod 8) ∈ {1, 7}, 1, -1)) = 0)
26	18, 21, 25	3eqtr4a 2859	. . 3 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 2 ∥ 𝐵) → (if(2 ∥ 𝐴, 0, if((𝐴 mod 8) ∈ {1, 7}, 1, -1)) · if(2 ∥ 𝐵, 0, if((𝐵 mod 8) ∈ {1, 7}, 1, -1))) = if(2 ∥ (𝐴 · 𝐵), 0, if(((𝐴 · 𝐵) mod 8) ∈ {1, 7}, 1, -1)))
27	4	mulid2i 10635	. . . . . 6 ⊢ (1 · if((𝐵 mod 8) ∈ {1, 7}, 1, -1)) = if((𝐵 mod 8) ∈ {1, 7}, 1, -1)
28		iftrue 4431	. . . . . . . 8 ⊢ ((𝐴 mod 8) ∈ {1, 7} → if((𝐴 mod 8) ∈ {1, 7}, 1, -1) = 1)
29	28	adantl 485	. . . . . . 7 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (¬ 2 ∥ 𝐴 ∧ ¬ 2 ∥ 𝐵)) ∧ (𝐴 mod 8) ∈ {1, 7}) → if((𝐴 mod 8) ∈ {1, 7}, 1, -1) = 1)
30	29	oveq1d 7150	. . . . . 6 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (¬ 2 ∥ 𝐴 ∧ ¬ 2 ∥ 𝐵)) ∧ (𝐴 mod 8) ∈ {1, 7}) → (if((𝐴 mod 8) ∈ {1, 7}, 1, -1) · if((𝐵 mod 8) ∈ {1, 7}, 1, -1)) = (1 · if((𝐵 mod 8) ∈ {1, 7}, 1, -1)))
31		lgsdir2lem4 25912	. . . . . . . 8 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐴 mod 8) ∈ {1, 7}) → (((𝐴 · 𝐵) mod 8) ∈ {1, 7} ↔ (𝐵 mod 8) ∈ {1, 7}))
32	31	adantlr 714	. . . . . . 7 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (¬ 2 ∥ 𝐴 ∧ ¬ 2 ∥ 𝐵)) ∧ (𝐴 mod 8) ∈ {1, 7}) → (((𝐴 · 𝐵) mod 8) ∈ {1, 7} ↔ (𝐵 mod 8) ∈ {1, 7}))
33	32	ifbid 4447	. . . . . 6 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (¬ 2 ∥ 𝐴 ∧ ¬ 2 ∥ 𝐵)) ∧ (𝐴 mod 8) ∈ {1, 7}) → if(((𝐴 · 𝐵) mod 8) ∈ {1, 7}, 1, -1) = if((𝐵 mod 8) ∈ {1, 7}, 1, -1))
34	27, 30, 33	3eqtr4a 2859	. . . . 5 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (¬ 2 ∥ 𝐴 ∧ ¬ 2 ∥ 𝐵)) ∧ (𝐴 mod 8) ∈ {1, 7}) → (if((𝐴 mod 8) ∈ {1, 7}, 1, -1) · if((𝐵 mod 8) ∈ {1, 7}, 1, -1)) = if(((𝐴 · 𝐵) mod 8) ∈ {1, 7}, 1, -1))
35	16	mulid1i 10634	. . . . . 6 ⊢ (if((𝐴 mod 8) ∈ {1, 7}, 1, -1) · 1) = if((𝐴 mod 8) ∈ {1, 7}, 1, -1)
36		iftrue 4431	. . . . . . . 8 ⊢ ((𝐵 mod 8) ∈ {1, 7} → if((𝐵 mod 8) ∈ {1, 7}, 1, -1) = 1)
37	36	adantl 485	. . . . . . 7 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (¬ 2 ∥ 𝐴 ∧ ¬ 2 ∥ 𝐵)) ∧ (𝐵 mod 8) ∈ {1, 7}) → if((𝐵 mod 8) ∈ {1, 7}, 1, -1) = 1)
38	37	oveq2d 7151	. . . . . 6 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (¬ 2 ∥ 𝐴 ∧ ¬ 2 ∥ 𝐵)) ∧ (𝐵 mod 8) ∈ {1, 7}) → (if((𝐴 mod 8) ∈ {1, 7}, 1, -1) · if((𝐵 mod 8) ∈ {1, 7}, 1, -1)) = (if((𝐴 mod 8) ∈ {1, 7}, 1, -1) · 1))
39		zcn 11974	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℂ)
40		zcn 11974	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ ℤ → 𝐵 ∈ ℂ)
41		mulcom 10612	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) = (𝐵 · 𝐴))
42	39, 40, 41	syl2an 598	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 · 𝐵) = (𝐵 · 𝐴))
43	42	ad2antrr 725	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (¬ 2 ∥ 𝐴 ∧ ¬ 2 ∥ 𝐵)) ∧ (𝐵 mod 8) ∈ {1, 7}) → (𝐴 · 𝐵) = (𝐵 · 𝐴))
44	43	oveq1d 7150	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (¬ 2 ∥ 𝐴 ∧ ¬ 2 ∥ 𝐵)) ∧ (𝐵 mod 8) ∈ {1, 7}) → ((𝐴 · 𝐵) mod 8) = ((𝐵 · 𝐴) mod 8))
45	44	eleq1d 2874	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (¬ 2 ∥ 𝐴 ∧ ¬ 2 ∥ 𝐵)) ∧ (𝐵 mod 8) ∈ {1, 7}) → (((𝐴 · 𝐵) mod 8) ∈ {1, 7} ↔ ((𝐵 · 𝐴) mod 8) ∈ {1, 7}))
46		lgsdir2lem4 25912	. . . . . . . . . 10 ⊢ (((𝐵 ∈ ℤ ∧ 𝐴 ∈ ℤ) ∧ (𝐵 mod 8) ∈ {1, 7}) → (((𝐵 · 𝐴) mod 8) ∈ {1, 7} ↔ (𝐴 mod 8) ∈ {1, 7}))
47	46	ancom1s 652	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐵 mod 8) ∈ {1, 7}) → (((𝐵 · 𝐴) mod 8) ∈ {1, 7} ↔ (𝐴 mod 8) ∈ {1, 7}))
48	47	adantlr 714	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (¬ 2 ∥ 𝐴 ∧ ¬ 2 ∥ 𝐵)) ∧ (𝐵 mod 8) ∈ {1, 7}) → (((𝐵 · 𝐴) mod 8) ∈ {1, 7} ↔ (𝐴 mod 8) ∈ {1, 7}))
49	45, 48	bitrd 282	. . . . . . 7 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (¬ 2 ∥ 𝐴 ∧ ¬ 2 ∥ 𝐵)) ∧ (𝐵 mod 8) ∈ {1, 7}) → (((𝐴 · 𝐵) mod 8) ∈ {1, 7} ↔ (𝐴 mod 8) ∈ {1, 7}))
50	49	ifbid 4447	. . . . . 6 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (¬ 2 ∥ 𝐴 ∧ ¬ 2 ∥ 𝐵)) ∧ (𝐵 mod 8) ∈ {1, 7}) → if(((𝐴 · 𝐵) mod 8) ∈ {1, 7}, 1, -1) = if((𝐴 mod 8) ∈ {1, 7}, 1, -1))
51	35, 38, 50	3eqtr4a 2859	. . . . 5 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (¬ 2 ∥ 𝐴 ∧ ¬ 2 ∥ 𝐵)) ∧ (𝐵 mod 8) ∈ {1, 7}) → (if((𝐴 mod 8) ∈ {1, 7}, 1, -1) · if((𝐵 mod 8) ∈ {1, 7}, 1, -1)) = if(((𝐴 · 𝐵) mod 8) ∈ {1, 7}, 1, -1))
52		neg1mulneg1e1 11838	. . . . . 6 ⊢ (-1 · -1) = 1
53		iffalse 4434	. . . . . . . 8 ⊢ (¬ (𝐴 mod 8) ∈ {1, 7} → if((𝐴 mod 8) ∈ {1, 7}, 1, -1) = -1)
54		iffalse 4434	. . . . . . . 8 ⊢ (¬ (𝐵 mod 8) ∈ {1, 7} → if((𝐵 mod 8) ∈ {1, 7}, 1, -1) = -1)
55	53, 54	oveqan12d 7154	. . . . . . 7 ⊢ ((¬ (𝐴 mod 8) ∈ {1, 7} ∧ ¬ (𝐵 mod 8) ∈ {1, 7}) → (if((𝐴 mod 8) ∈ {1, 7}, 1, -1) · if((𝐵 mod 8) ∈ {1, 7}, 1, -1)) = (-1 · -1))
56	55	adantl 485	. . . . . 6 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (¬ 2 ∥ 𝐴 ∧ ¬ 2 ∥ 𝐵)) ∧ (¬ (𝐴 mod 8) ∈ {1, 7} ∧ ¬ (𝐵 mod 8) ∈ {1, 7})) → (if((𝐴 mod 8) ∈ {1, 7}, 1, -1) · if((𝐵 mod 8) ∈ {1, 7}, 1, -1)) = (-1 · -1))
57		lgsdir2lem3 25911	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ ¬ 2 ∥ 𝐴) → (𝐴 mod 8) ∈ ({1, 7} ∪ {3, 5}))
58	57	ad2ant2r 746	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (¬ 2 ∥ 𝐴 ∧ ¬ 2 ∥ 𝐵)) → (𝐴 mod 8) ∈ ({1, 7} ∪ {3, 5}))
59		elun 4076	. . . . . . . . . . 11 ⊢ ((𝐴 mod 8) ∈ ({1, 7} ∪ {3, 5}) ↔ ((𝐴 mod 8) ∈ {1, 7} ∨ (𝐴 mod 8) ∈ {3, 5}))
60	58, 59	sylib 221	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (¬ 2 ∥ 𝐴 ∧ ¬ 2 ∥ 𝐵)) → ((𝐴 mod 8) ∈ {1, 7} ∨ (𝐴 mod 8) ∈ {3, 5}))
61	60	orcanai 1000	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (¬ 2 ∥ 𝐴 ∧ ¬ 2 ∥ 𝐵)) ∧ ¬ (𝐴 mod 8) ∈ {1, 7}) → (𝐴 mod 8) ∈ {3, 5})
62		lgsdir2lem3 25911	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ ℤ ∧ ¬ 2 ∥ 𝐵) → (𝐵 mod 8) ∈ ({1, 7} ∪ {3, 5}))
63	62	ad2ant2l 745	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (¬ 2 ∥ 𝐴 ∧ ¬ 2 ∥ 𝐵)) → (𝐵 mod 8) ∈ ({1, 7} ∪ {3, 5}))
64		elun 4076	. . . . . . . . . . 11 ⊢ ((𝐵 mod 8) ∈ ({1, 7} ∪ {3, 5}) ↔ ((𝐵 mod 8) ∈ {1, 7} ∨ (𝐵 mod 8) ∈ {3, 5}))
65	63, 64	sylib 221	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (¬ 2 ∥ 𝐴 ∧ ¬ 2 ∥ 𝐵)) → ((𝐵 mod 8) ∈ {1, 7} ∨ (𝐵 mod 8) ∈ {3, 5}))
66	65	orcanai 1000	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (¬ 2 ∥ 𝐴 ∧ ¬ 2 ∥ 𝐵)) ∧ ¬ (𝐵 mod 8) ∈ {1, 7}) → (𝐵 mod 8) ∈ {3, 5})
67	61, 66	anim12dan 621	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (¬ 2 ∥ 𝐴 ∧ ¬ 2 ∥ 𝐵)) ∧ (¬ (𝐴 mod 8) ∈ {1, 7} ∧ ¬ (𝐵 mod 8) ∈ {1, 7})) → ((𝐴 mod 8) ∈ {3, 5} ∧ (𝐵 mod 8) ∈ {3, 5}))
68		lgsdir2lem5 25913	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 mod 8) ∈ {3, 5} ∧ (𝐵 mod 8) ∈ {3, 5})) → ((𝐴 · 𝐵) mod 8) ∈ {1, 7})
69	68	adantlr 714	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (¬ 2 ∥ 𝐴 ∧ ¬ 2 ∥ 𝐵)) ∧ ((𝐴 mod 8) ∈ {3, 5} ∧ (𝐵 mod 8) ∈ {3, 5})) → ((𝐴 · 𝐵) mod 8) ∈ {1, 7})
70	67, 69	syldan 594	. . . . . . 7 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (¬ 2 ∥ 𝐴 ∧ ¬ 2 ∥ 𝐵)) ∧ (¬ (𝐴 mod 8) ∈ {1, 7} ∧ ¬ (𝐵 mod 8) ∈ {1, 7})) → ((𝐴 · 𝐵) mod 8) ∈ {1, 7})
71	70	iftrued 4433	. . . . . 6 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (¬ 2 ∥ 𝐴 ∧ ¬ 2 ∥ 𝐵)) ∧ (¬ (𝐴 mod 8) ∈ {1, 7} ∧ ¬ (𝐵 mod 8) ∈ {1, 7})) → if(((𝐴 · 𝐵) mod 8) ∈ {1, 7}, 1, -1) = 1)
72	52, 56, 71	3eqtr4a 2859	. . . . 5 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (¬ 2 ∥ 𝐴 ∧ ¬ 2 ∥ 𝐵)) ∧ (¬ (𝐴 mod 8) ∈ {1, 7} ∧ ¬ (𝐵 mod 8) ∈ {1, 7})) → (if((𝐴 mod 8) ∈ {1, 7}, 1, -1) · if((𝐵 mod 8) ∈ {1, 7}, 1, -1)) = if(((𝐴 · 𝐵) mod 8) ∈ {1, 7}, 1, -1))
73	34, 51, 72	pm2.61ddan 813	. . . 4 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (¬ 2 ∥ 𝐴 ∧ ¬ 2 ∥ 𝐵)) → (if((𝐴 mod 8) ∈ {1, 7}, 1, -1) · if((𝐵 mod 8) ∈ {1, 7}, 1, -1)) = if(((𝐴 · 𝐵) mod 8) ∈ {1, 7}, 1, -1))
74		iffalse 4434	. . . . . 6 ⊢ (¬ 2 ∥ 𝐴 → if(2 ∥ 𝐴, 0, if((𝐴 mod 8) ∈ {1, 7}, 1, -1)) = if((𝐴 mod 8) ∈ {1, 7}, 1, -1))
75		iffalse 4434	. . . . . 6 ⊢ (¬ 2 ∥ 𝐵 → if(2 ∥ 𝐵, 0, if((𝐵 mod 8) ∈ {1, 7}, 1, -1)) = if((𝐵 mod 8) ∈ {1, 7}, 1, -1))
76	74, 75	oveqan12d 7154	. . . . 5 ⊢ ((¬ 2 ∥ 𝐴 ∧ ¬ 2 ∥ 𝐵) → (if(2 ∥ 𝐴, 0, if((𝐴 mod 8) ∈ {1, 7}, 1, -1)) · if(2 ∥ 𝐵, 0, if((𝐵 mod 8) ∈ {1, 7}, 1, -1))) = (if((𝐴 mod 8) ∈ {1, 7}, 1, -1) · if((𝐵 mod 8) ∈ {1, 7}, 1, -1)))
77	76	adantl 485	. . . 4 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (¬ 2 ∥ 𝐴 ∧ ¬ 2 ∥ 𝐵)) → (if(2 ∥ 𝐴, 0, if((𝐴 mod 8) ∈ {1, 7}, 1, -1)) · if(2 ∥ 𝐵, 0, if((𝐵 mod 8) ∈ {1, 7}, 1, -1))) = (if((𝐴 mod 8) ∈ {1, 7}, 1, -1) · if((𝐵 mod 8) ∈ {1, 7}, 1, -1)))
78		ioran 981	. . . . . 6 ⊢ (¬ (2 ∥ 𝐴 ∨ 2 ∥ 𝐵) ↔ (¬ 2 ∥ 𝐴 ∧ ¬ 2 ∥ 𝐵))
79		2prm 16026	. . . . . . . . 9 ⊢ 2 ∈ ℙ
80		euclemma 16047	. . . . . . . . 9 ⊢ ((2 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (2 ∥ (𝐴 · 𝐵) ↔ (2 ∥ 𝐴 ∨ 2 ∥ 𝐵)))
81	79, 80	mp3an1 1445	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (2 ∥ (𝐴 · 𝐵) ↔ (2 ∥ 𝐴 ∨ 2 ∥ 𝐵)))
82	81	notbid 321	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (¬ 2 ∥ (𝐴 · 𝐵) ↔ ¬ (2 ∥ 𝐴 ∨ 2 ∥ 𝐵)))
83	82	biimpar 481	. . . . . 6 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ¬ (2 ∥ 𝐴 ∨ 2 ∥ 𝐵)) → ¬ 2 ∥ (𝐴 · 𝐵))
84	78, 83	sylan2br 597	. . . . 5 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (¬ 2 ∥ 𝐴 ∧ ¬ 2 ∥ 𝐵)) → ¬ 2 ∥ (𝐴 · 𝐵))
85		iffalse 4434	. . . . 5 ⊢ (¬ 2 ∥ (𝐴 · 𝐵) → if(2 ∥ (𝐴 · 𝐵), 0, if(((𝐴 · 𝐵) mod 8) ∈ {1, 7}, 1, -1)) = if(((𝐴 · 𝐵) mod 8) ∈ {1, 7}, 1, -1))
86	84, 85	syl 17	. . . 4 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (¬ 2 ∥ 𝐴 ∧ ¬ 2 ∥ 𝐵)) → if(2 ∥ (𝐴 · 𝐵), 0, if(((𝐴 · 𝐵) mod 8) ∈ {1, 7}, 1, -1)) = if(((𝐴 · 𝐵) mod 8) ∈ {1, 7}, 1, -1))
87	73, 77, 86	3eqtr4d 2843	. . 3 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (¬ 2 ∥ 𝐴 ∧ ¬ 2 ∥ 𝐵)) → (if(2 ∥ 𝐴, 0, if((𝐴 mod 8) ∈ {1, 7}, 1, -1)) · if(2 ∥ 𝐵, 0, if((𝐵 mod 8) ∈ {1, 7}, 1, -1))) = if(2 ∥ (𝐴 · 𝐵), 0, if(((𝐴 · 𝐵) mod 8) ∈ {1, 7}, 1, -1)))
88	15, 26, 87	pm2.61ddan 813	. 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (if(2 ∥ 𝐴, 0, if((𝐴 mod 8) ∈ {1, 7}, 1, -1)) · if(2 ∥ 𝐵, 0, if((𝐵 mod 8) ∈ {1, 7}, 1, -1))) = if(2 ∥ (𝐴 · 𝐵), 0, if(((𝐴 · 𝐵) mod 8) ∈ {1, 7}, 1, -1)))
89		lgs2 25898	. . 3 ⊢ (𝐴 ∈ ℤ → (𝐴 /_L 2) = if(2 ∥ 𝐴, 0, if((𝐴 mod 8) ∈ {1, 7}, 1, -1)))
90		lgs2 25898	. . 3 ⊢ (𝐵 ∈ ℤ → (𝐵 /_L 2) = if(2 ∥ 𝐵, 0, if((𝐵 mod 8) ∈ {1, 7}, 1, -1)))
91	89, 90	oveqan12d 7154	. 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴 /_L 2) · (𝐵 /_L 2)) = (if(2 ∥ 𝐴, 0, if((𝐴 mod 8) ∈ {1, 7}, 1, -1)) · if(2 ∥ 𝐵, 0, if((𝐵 mod 8) ∈ {1, 7}, 1, -1))))
92		zmulcl 12019	. . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 · 𝐵) ∈ ℤ)
93		lgs2 25898	. . 3 ⊢ ((𝐴 · 𝐵) ∈ ℤ → ((𝐴 · 𝐵) /_L 2) = if(2 ∥ (𝐴 · 𝐵), 0, if(((𝐴 · 𝐵) mod 8) ∈ {1, 7}, 1, -1)))
94	92, 93	syl 17	. 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴 · 𝐵) /_L 2) = if(2 ∥ (𝐴 · 𝐵), 0, if(((𝐴 · 𝐵) mod 8) ∈ {1, 7}, 1, -1)))
95	88, 91, 94	3eqtr4rd 2844	1 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴 · 𝐵) /_L 2) = ((𝐴 /_L 2) · (𝐵 /_L 2)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∨ wo 844 = wceq 1538 ∈ wcel 2111 ∪ cun 3879 ifcif 4425 {cpr 4527 class class class wbr 5030 (class class class)co 7135 ℂcc 10524 0cc0 10526 1c1 10527 · cmul 10531 -cneg 10860 2c2 11680 3c3 11681 5c5 11683 7c7 11685 8c8 11686 ℤcz 11969 mod cmo 13232 ∥ cdvds 15599 ℙcprime 16005 /_L clgs 25878

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-sup 8890 df-inf 8891 df-dju 9314 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-xnn0 11956 df-z 11970 df-uz 12232 df-q 12337 df-rp 12378 df-fz 12886 df-fzo 13029 df-fl 13157 df-mod 13233 df-seq 13365 df-exp 13426 df-hash 13687 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-dvds 15600 df-gcd 15834 df-prm 16006 df-phi 16093 df-pc 16164 df-lgs 25879

This theorem is referenced by: lgsdirprm 25915

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