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Theorem lgsdir2 27248

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 11173	. . . . . 6 ⊢ 0 ∈ ℂ
2		ax-1cn 11133	. . . . . . 7 ⊢ 1 ∈ ℂ
3		neg1cn 12178	. . . . . . 7 ⊢ -1 ∈ ℂ
4	2, 3	ifcli 4539	. . . . . 6 ⊢ if((𝐵 mod 8) ∈ {1, 7}, 1, -1) ∈ ℂ
5	1, 4	ifcli 4539	. . . . 5 ⊢ if(2 ∥ 𝐵, 0, if((𝐵 mod 8) ∈ {1, 7}, 1, -1)) ∈ ℂ
6	5	mul02i 11370	. . . 4 ⊢ (0 · if(2 ∥ 𝐵, 0, if((𝐵 mod 8) ∈ {1, 7}, 1, -1))) = 0
7		iftrue 4497	. . . . . 6 ⊢ (2 ∥ 𝐴 → if(2 ∥ 𝐴, 0, if((𝐴 mod 8) ∈ {1, 7}, 1, -1)) = 0)
8	7	adantl 481	. . . . 5 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 2 ∥ 𝐴) → if(2 ∥ 𝐴, 0, if((𝐴 mod 8) ∈ {1, 7}, 1, -1)) = 0)
9	8	oveq1d 7405	. . . 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 12572	. . . . . . 7 ⊢ 2 ∈ ℤ
11		dvdsmultr1 16273	. . . . . . 7 ⊢ ((2 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (2 ∥ 𝐴 → 2 ∥ (𝐴 · 𝐵)))
12	10, 11	mp3an1 1450	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (2 ∥ 𝐴 → 2 ∥ (𝐴 · 𝐵)))
13	12	imp 406	. . . . 5 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 2 ∥ 𝐴) → 2 ∥ (𝐴 · 𝐵))
14	13	iftrued 4499	. . . 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 2791	. . 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 4539	. . . . . 6 ⊢ if((𝐴 mod 8) ∈ {1, 7}, 1, -1) ∈ ℂ
17	1, 16	ifcli 4539	. . . . 5 ⊢ if(2 ∥ 𝐴, 0, if((𝐴 mod 8) ∈ {1, 7}, 1, -1)) ∈ ℂ
18	17	mul01i 11371	. . . 4 ⊢ (if(2 ∥ 𝐴, 0, if((𝐴 mod 8) ∈ {1, 7}, 1, -1)) · 0) = 0
19		iftrue 4497	. . . . . 6 ⊢ (2 ∥ 𝐵 → if(2 ∥ 𝐵, 0, if((𝐵 mod 8) ∈ {1, 7}, 1, -1)) = 0)
20	19	adantl 481	. . . . 5 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 2 ∥ 𝐵) → if(2 ∥ 𝐵, 0, if((𝐵 mod 8) ∈ {1, 7}, 1, -1)) = 0)
21	20	oveq2d 7406	. . . 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 16275	. . . . . . 7 ⊢ ((2 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (2 ∥ 𝐵 → 2 ∥ (𝐴 · 𝐵)))
23	10, 22	mp3an1 1450	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (2 ∥ 𝐵 → 2 ∥ (𝐴 · 𝐵)))
24	23	imp 406	. . . . 5 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 2 ∥ 𝐵) → 2 ∥ (𝐴 · 𝐵))
25	24	iftrued 4499	. . . 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 2791	. . 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	mullidi 11186	. . . . . 6 ⊢ (1 · if((𝐵 mod 8) ∈ {1, 7}, 1, -1)) = if((𝐵 mod 8) ∈ {1, 7}, 1, -1)
28		iftrue 4497	. . . . . . . 8 ⊢ ((𝐴 mod 8) ∈ {1, 7} → if((𝐴 mod 8) ∈ {1, 7}, 1, -1) = 1)
29	28	adantl 481	. . . . . . 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 7405	. . . . . 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 27246	. . . . . . . 8 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐴 mod 8) ∈ {1, 7}) → (((𝐴 · 𝐵) mod 8) ∈ {1, 7} ↔ (𝐵 mod 8) ∈ {1, 7}))
32	31	adantlr 715	. . . . . . 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 4515	. . . . . 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 2791	. . . . 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	mulridi 11185	. . . . . 6 ⊢ (if((𝐴 mod 8) ∈ {1, 7}, 1, -1) · 1) = if((𝐴 mod 8) ∈ {1, 7}, 1, -1)
36		iftrue 4497	. . . . . . . 8 ⊢ ((𝐵 mod 8) ∈ {1, 7} → if((𝐵 mod 8) ∈ {1, 7}, 1, -1) = 1)
37	36	adantl 481	. . . . . . 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 7406	. . . . . 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 12541	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℂ)
40		zcn 12541	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ ℤ → 𝐵 ∈ ℂ)
41		mulcom 11161	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) = (𝐵 · 𝐴))
42	39, 40, 41	syl2an 596	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 · 𝐵) = (𝐵 · 𝐴))
43	42	ad2antrr 726	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (¬ 2 ∥ 𝐴 ∧ ¬ 2 ∥ 𝐵)) ∧ (𝐵 mod 8) ∈ {1, 7}) → (𝐴 · 𝐵) = (𝐵 · 𝐴))
44	43	oveq1d 7405	. . . . . . . . 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 2814	. . . . . . . 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 27246	. . . . . . . . . 10 ⊢ (((𝐵 ∈ ℤ ∧ 𝐴 ∈ ℤ) ∧ (𝐵 mod 8) ∈ {1, 7}) → (((𝐵 · 𝐴) mod 8) ∈ {1, 7} ↔ (𝐴 mod 8) ∈ {1, 7}))
47	46	ancom1s 653	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐵 mod 8) ∈ {1, 7}) → (((𝐵 · 𝐴) mod 8) ∈ {1, 7} ↔ (𝐴 mod 8) ∈ {1, 7}))
48	47	adantlr 715	. . . . . . . 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 279	. . . . . . 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 4515	. . . . . 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 2791	. . . . 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 12401	. . . . . 6 ⊢ (-1 · -1) = 1
53		iffalse 4500	. . . . . . . 8 ⊢ (¬ (𝐴 mod 8) ∈ {1, 7} → if((𝐴 mod 8) ∈ {1, 7}, 1, -1) = -1)
54		iffalse 4500	. . . . . . . 8 ⊢ (¬ (𝐵 mod 8) ∈ {1, 7} → if((𝐵 mod 8) ∈ {1, 7}, 1, -1) = -1)
55	53, 54	oveqan12d 7409	. . . . . . 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 481	. . . . . 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 27245	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ ¬ 2 ∥ 𝐴) → (𝐴 mod 8) ∈ ({1, 7} ∪ {3, 5}))
58	57	ad2ant2r 747	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (¬ 2 ∥ 𝐴 ∧ ¬ 2 ∥ 𝐵)) → (𝐴 mod 8) ∈ ({1, 7} ∪ {3, 5}))
59		elun 4119	. . . . . . . . . . 11 ⊢ ((𝐴 mod 8) ∈ ({1, 7} ∪ {3, 5}) ↔ ((𝐴 mod 8) ∈ {1, 7} ∨ (𝐴 mod 8) ∈ {3, 5}))
60	58, 59	sylib 218	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (¬ 2 ∥ 𝐴 ∧ ¬ 2 ∥ 𝐵)) → ((𝐴 mod 8) ∈ {1, 7} ∨ (𝐴 mod 8) ∈ {3, 5}))
61	60	orcanai 1004	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (¬ 2 ∥ 𝐴 ∧ ¬ 2 ∥ 𝐵)) ∧ ¬ (𝐴 mod 8) ∈ {1, 7}) → (𝐴 mod 8) ∈ {3, 5})
62		lgsdir2lem3 27245	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ ℤ ∧ ¬ 2 ∥ 𝐵) → (𝐵 mod 8) ∈ ({1, 7} ∪ {3, 5}))
63	62	ad2ant2l 746	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (¬ 2 ∥ 𝐴 ∧ ¬ 2 ∥ 𝐵)) → (𝐵 mod 8) ∈ ({1, 7} ∪ {3, 5}))
64		elun 4119	. . . . . . . . . . 11 ⊢ ((𝐵 mod 8) ∈ ({1, 7} ∪ {3, 5}) ↔ ((𝐵 mod 8) ∈ {1, 7} ∨ (𝐵 mod 8) ∈ {3, 5}))
65	63, 64	sylib 218	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (¬ 2 ∥ 𝐴 ∧ ¬ 2 ∥ 𝐵)) → ((𝐵 mod 8) ∈ {1, 7} ∨ (𝐵 mod 8) ∈ {3, 5}))
66	65	orcanai 1004	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (¬ 2 ∥ 𝐴 ∧ ¬ 2 ∥ 𝐵)) ∧ ¬ (𝐵 mod 8) ∈ {1, 7}) → (𝐵 mod 8) ∈ {3, 5})
67	61, 66	anim12dan 619	. . . . . . . 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 27247	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 mod 8) ∈ {3, 5} ∧ (𝐵 mod 8) ∈ {3, 5})) → ((𝐴 · 𝐵) mod 8) ∈ {1, 7})
69	68	adantlr 715	. . . . . . . 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 591	. . . . . . 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 4499	. . . . . 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 2791	. . . . 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 4500	. . . . . 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 4500	. . . . . 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 7409	. . . . 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 481	. . . 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 985	. . . . . 6 ⊢ (¬ (2 ∥ 𝐴 ∨ 2 ∥ 𝐵) ↔ (¬ 2 ∥ 𝐴 ∧ ¬ 2 ∥ 𝐵))
79		2prm 16669	. . . . . . . . 9 ⊢ 2 ∈ ℙ
80		euclemma 16690	. . . . . . . . 9 ⊢ ((2 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (2 ∥ (𝐴 · 𝐵) ↔ (2 ∥ 𝐴 ∨ 2 ∥ 𝐵)))
81	79, 80	mp3an1 1450	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (2 ∥ (𝐴 · 𝐵) ↔ (2 ∥ 𝐴 ∨ 2 ∥ 𝐵)))
82	81	notbid 318	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (¬ 2 ∥ (𝐴 · 𝐵) ↔ ¬ (2 ∥ 𝐴 ∨ 2 ∥ 𝐵)))
83	82	biimpar 477	. . . . . 6 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ¬ (2 ∥ 𝐴 ∨ 2 ∥ 𝐵)) → ¬ 2 ∥ (𝐴 · 𝐵))
84	78, 83	sylan2br 595	. . . . 5 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (¬ 2 ∥ 𝐴 ∧ ¬ 2 ∥ 𝐵)) → ¬ 2 ∥ (𝐴 · 𝐵))
85		iffalse 4500	. . . . 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 2775	. . 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 27232	. . 3 ⊢ (𝐴 ∈ ℤ → (𝐴 /_L 2) = if(2 ∥ 𝐴, 0, if((𝐴 mod 8) ∈ {1, 7}, 1, -1)))
90		lgs2 27232	. . 3 ⊢ (𝐵 ∈ ℤ → (𝐵 /_L 2) = if(2 ∥ 𝐵, 0, if((𝐵 mod 8) ∈ {1, 7}, 1, -1)))
91	89, 90	oveqan12d 7409	. 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 12589	. . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 · 𝐵) ∈ ℤ)
93		lgs2 27232	. . 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 2776	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 206 ∧ wa 395 ∨ wo 847 = wceq 1540 ∈ wcel 2109 ∪ cun 3915 ifcif 4491 {cpr 4594 class class class wbr 5110 (class class class)co 7390 ℂcc 11073 0cc0 11075 1c1 11076 · cmul 11080 -cneg 11413 2c2 12248 3c3 12249 5c5 12251 7c7 12253 8c8 12254 ℤcz 12536 mod cmo 13838 ∥ cdvds 16229 ℙcprime 16648 /_L clgs 27212

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 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-pre-sup 11153

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 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-int 4914 df-iun 4960 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7846 df-1st 7971 df-2nd 7972 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-2o 8438 df-oadd 8441 df-er 8674 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-sup 9400 df-inf 9401 df-dju 9861 df-card 9899 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-div 11843 df-nn 12194 df-2 12256 df-3 12257 df-4 12258 df-5 12259 df-6 12260 df-7 12261 df-8 12262 df-9 12263 df-n0 12450 df-xnn0 12523 df-z 12537 df-uz 12801 df-q 12915 df-rp 12959 df-fz 13476 df-fzo 13623 df-fl 13761 df-mod 13839 df-seq 13974 df-exp 14034 df-hash 14303 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-dvds 16230 df-gcd 16472 df-prm 16649 df-phi 16743 df-pc 16815 df-lgs 27213

This theorem is referenced by: lgsdirprm 27249

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