lgsdir2lem5 - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > lgsdir2lem5		GIF version

Theorem lgsdir2lem5 14100

Description: Lemma for lgsdir2 14101. (Contributed by Mario Carneiro, 4-Feb-2015.)

**Assertion**
Ref	Expression
lgsdir2lem5	⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 mod 8) ∈ {3, 5} ∧ (𝐵 mod 8) ∈ {3, 5})) → ((𝐴 · 𝐵) mod 8) ∈ {1, 7})

**Proof of Theorem lgsdir2lem5**
Step	Hyp	Ref	Expression
1		8nn 9075	. . . . . . . . 9 ⊢ 8 ∈ ℕ
2		zmodcl 10330	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 8 ∈ ℕ) → (𝐴 mod 8) ∈ ℕ₀)
3	1, 2	mpan2 425	. . . . . . . 8 ⊢ (𝐴 ∈ ℤ → (𝐴 mod 8) ∈ ℕ₀)
4	3	adantr 276	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 mod 8) ∈ ℕ₀)
5		elprg 3611	. . . . . . 7 ⊢ ((𝐴 mod 8) ∈ ℕ₀ → ((𝐴 mod 8) ∈ {3, 5} ↔ ((𝐴 mod 8) = 3 ∨ (𝐴 mod 8) = 5)))
6	4, 5	syl 14	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴 mod 8) ∈ {3, 5} ↔ ((𝐴 mod 8) = 3 ∨ (𝐴 mod 8) = 5)))
7		zmodcl 10330	. . . . . . . . 9 ⊢ ((𝐵 ∈ ℤ ∧ 8 ∈ ℕ) → (𝐵 mod 8) ∈ ℕ₀)
8	1, 7	mpan2 425	. . . . . . . 8 ⊢ (𝐵 ∈ ℤ → (𝐵 mod 8) ∈ ℕ₀)
9	8	adantl 277	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐵 mod 8) ∈ ℕ₀)
10		elprg 3611	. . . . . . 7 ⊢ ((𝐵 mod 8) ∈ ℕ₀ → ((𝐵 mod 8) ∈ {3, 5} ↔ ((𝐵 mod 8) = 3 ∨ (𝐵 mod 8) = 5)))
11	9, 10	syl 14	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐵 mod 8) ∈ {3, 5} ↔ ((𝐵 mod 8) = 3 ∨ (𝐵 mod 8) = 5)))
12	6, 11	anbi12d 473	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (((𝐴 mod 8) ∈ {3, 5} ∧ (𝐵 mod 8) ∈ {3, 5}) ↔ (((𝐴 mod 8) = 3 ∨ (𝐴 mod 8) = 5) ∧ ((𝐵 mod 8) = 3 ∨ (𝐵 mod 8) = 5))))
13		simpll 527	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 mod 8) = 3 ∧ (𝐵 mod 8) = 3)) → 𝐴 ∈ ℤ)
14		3z 9271	. . . . . . . . . 10 ⊢ 3 ∈ ℤ
15	14	a1i 9	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 mod 8) = 3 ∧ (𝐵 mod 8) = 3)) → 3 ∈ ℤ)
16		simplr 528	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 mod 8) = 3 ∧ (𝐵 mod 8) = 3)) → 𝐵 ∈ ℤ)
17		nnq 9622	. . . . . . . . . . 11 ⊢ (8 ∈ ℕ → 8 ∈ ℚ)
18	1, 17	ax-mp 5	. . . . . . . . . 10 ⊢ 8 ∈ ℚ
19	18	a1i 9	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 mod 8) = 3 ∧ (𝐵 mod 8) = 3)) → 8 ∈ ℚ)
20		8pos 9011	. . . . . . . . . 10 ⊢ 0 < 8
21	20	a1i 9	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 mod 8) = 3 ∧ (𝐵 mod 8) = 3)) → 0 < 8)
22		simprl 529	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 mod 8) = 3 ∧ (𝐵 mod 8) = 3)) → (𝐴 mod 8) = 3)
23		lgsdir2lem1 14096	. . . . . . . . . . . 12 ⊢ (((1 mod 8) = 1 ∧ (-1 mod 8) = 7) ∧ ((3 mod 8) = 3 ∧ (-3 mod 8) = 5))
24	23	simpri 113	. . . . . . . . . . 11 ⊢ ((3 mod 8) = 3 ∧ (-3 mod 8) = 5)
25	24	simpli 111	. . . . . . . . . 10 ⊢ (3 mod 8) = 3
26	22, 25	eqtr4di 2228	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 mod 8) = 3 ∧ (𝐵 mod 8) = 3)) → (𝐴 mod 8) = (3 mod 8))
27		simprr 531	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 mod 8) = 3 ∧ (𝐵 mod 8) = 3)) → (𝐵 mod 8) = 3)
28	27, 25	eqtr4di 2228	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 mod 8) = 3 ∧ (𝐵 mod 8) = 3)) → (𝐵 mod 8) = (3 mod 8))
29	13, 15, 16, 15, 19, 21, 26, 28	modqmul12d 10364	. . . . . . . 8 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 mod 8) = 3 ∧ (𝐵 mod 8) = 3)) → ((𝐴 · 𝐵) mod 8) = ((3 · 3) mod 8))
30	29	orcd 733	. . . . . . 7 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 mod 8) = 3 ∧ (𝐵 mod 8) = 3)) → (((𝐴 · 𝐵) mod 8) = ((3 · 3) mod 8) ∨ ((𝐴 · 𝐵) mod 8) = (-(3 · 3) mod 8)))
31	30	ex 115	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (((𝐴 mod 8) = 3 ∧ (𝐵 mod 8) = 3) → (((𝐴 · 𝐵) mod 8) = ((3 · 3) mod 8) ∨ ((𝐴 · 𝐵) mod 8) = (-(3 · 3) mod 8))))
32		simpll 527	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 mod 8) = 5 ∧ (𝐵 mod 8) = 3)) → 𝐴 ∈ ℤ)
33		znegcl 9273	. . . . . . . . . . 11 ⊢ (3 ∈ ℤ → -3 ∈ ℤ)
34	14, 33	mp1i 10	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 mod 8) = 5 ∧ (𝐵 mod 8) = 3)) → -3 ∈ ℤ)
35		simplr 528	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 mod 8) = 5 ∧ (𝐵 mod 8) = 3)) → 𝐵 ∈ ℤ)
36	14	a1i 9	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 mod 8) = 5 ∧ (𝐵 mod 8) = 3)) → 3 ∈ ℤ)
37	18	a1i 9	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 mod 8) = 5 ∧ (𝐵 mod 8) = 3)) → 8 ∈ ℚ)
38	20	a1i 9	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 mod 8) = 5 ∧ (𝐵 mod 8) = 3)) → 0 < 8)
39		simprl 529	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 mod 8) = 5 ∧ (𝐵 mod 8) = 3)) → (𝐴 mod 8) = 5)
40	24	simpri 113	. . . . . . . . . . 11 ⊢ (-3 mod 8) = 5
41	39, 40	eqtr4di 2228	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 mod 8) = 5 ∧ (𝐵 mod 8) = 3)) → (𝐴 mod 8) = (-3 mod 8))
42		simprr 531	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 mod 8) = 5 ∧ (𝐵 mod 8) = 3)) → (𝐵 mod 8) = 3)
43	42, 25	eqtr4di 2228	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 mod 8) = 5 ∧ (𝐵 mod 8) = 3)) → (𝐵 mod 8) = (3 mod 8))
44	32, 34, 35, 36, 37, 38, 41, 43	modqmul12d 10364	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 mod 8) = 5 ∧ (𝐵 mod 8) = 3)) → ((𝐴 · 𝐵) mod 8) = ((-3 · 3) mod 8))
45		3cn 8983	. . . . . . . . . . 11 ⊢ 3 ∈ ℂ
46	45, 45	mulneg1i 8351	. . . . . . . . . 10 ⊢ (-3 · 3) = -(3 · 3)
47	46	oveq1i 5879	. . . . . . . . 9 ⊢ ((-3 · 3) mod 8) = (-(3 · 3) mod 8)
48	44, 47	eqtrdi 2226	. . . . . . . 8 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 mod 8) = 5 ∧ (𝐵 mod 8) = 3)) → ((𝐴 · 𝐵) mod 8) = (-(3 · 3) mod 8))
49	48	olcd 734	. . . . . . 7 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 mod 8) = 5 ∧ (𝐵 mod 8) = 3)) → (((𝐴 · 𝐵) mod 8) = ((3 · 3) mod 8) ∨ ((𝐴 · 𝐵) mod 8) = (-(3 · 3) mod 8)))
50	49	ex 115	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (((𝐴 mod 8) = 5 ∧ (𝐵 mod 8) = 3) → (((𝐴 · 𝐵) mod 8) = ((3 · 3) mod 8) ∨ ((𝐴 · 𝐵) mod 8) = (-(3 · 3) mod 8))))
51		simpll 527	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 mod 8) = 3 ∧ (𝐵 mod 8) = 5)) → 𝐴 ∈ ℤ)
52	14	a1i 9	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 mod 8) = 3 ∧ (𝐵 mod 8) = 5)) → 3 ∈ ℤ)
53		simplr 528	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 mod 8) = 3 ∧ (𝐵 mod 8) = 5)) → 𝐵 ∈ ℤ)
54	14, 33	mp1i 10	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 mod 8) = 3 ∧ (𝐵 mod 8) = 5)) → -3 ∈ ℤ)
55	18	a1i 9	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 mod 8) = 3 ∧ (𝐵 mod 8) = 5)) → 8 ∈ ℚ)
56	20	a1i 9	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 mod 8) = 3 ∧ (𝐵 mod 8) = 5)) → 0 < 8)
57		simprl 529	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 mod 8) = 3 ∧ (𝐵 mod 8) = 5)) → (𝐴 mod 8) = 3)
58	57, 25	eqtr4di 2228	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 mod 8) = 3 ∧ (𝐵 mod 8) = 5)) → (𝐴 mod 8) = (3 mod 8))
59		simprr 531	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 mod 8) = 3 ∧ (𝐵 mod 8) = 5)) → (𝐵 mod 8) = 5)
60	59, 40	eqtr4di 2228	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 mod 8) = 3 ∧ (𝐵 mod 8) = 5)) → (𝐵 mod 8) = (-3 mod 8))
61	51, 52, 53, 54, 55, 56, 58, 60	modqmul12d 10364	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 mod 8) = 3 ∧ (𝐵 mod 8) = 5)) → ((𝐴 · 𝐵) mod 8) = ((3 · -3) mod 8))
62	45, 45	mulneg2i 8352	. . . . . . . . . 10 ⊢ (3 · -3) = -(3 · 3)
63	62	oveq1i 5879	. . . . . . . . 9 ⊢ ((3 · -3) mod 8) = (-(3 · 3) mod 8)
64	61, 63	eqtrdi 2226	. . . . . . . 8 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 mod 8) = 3 ∧ (𝐵 mod 8) = 5)) → ((𝐴 · 𝐵) mod 8) = (-(3 · 3) mod 8))
65	64	olcd 734	. . . . . . 7 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 mod 8) = 3 ∧ (𝐵 mod 8) = 5)) → (((𝐴 · 𝐵) mod 8) = ((3 · 3) mod 8) ∨ ((𝐴 · 𝐵) mod 8) = (-(3 · 3) mod 8)))
66	65	ex 115	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (((𝐴 mod 8) = 3 ∧ (𝐵 mod 8) = 5) → (((𝐴 · 𝐵) mod 8) = ((3 · 3) mod 8) ∨ ((𝐴 · 𝐵) mod 8) = (-(3 · 3) mod 8))))
67		simpll 527	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 mod 8) = 5 ∧ (𝐵 mod 8) = 5)) → 𝐴 ∈ ℤ)
68	14, 33	mp1i 10	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 mod 8) = 5 ∧ (𝐵 mod 8) = 5)) → -3 ∈ ℤ)
69		simplr 528	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 mod 8) = 5 ∧ (𝐵 mod 8) = 5)) → 𝐵 ∈ ℤ)
70	18	a1i 9	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 mod 8) = 5 ∧ (𝐵 mod 8) = 5)) → 8 ∈ ℚ)
71	20	a1i 9	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 mod 8) = 5 ∧ (𝐵 mod 8) = 5)) → 0 < 8)
72		simprl 529	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 mod 8) = 5 ∧ (𝐵 mod 8) = 5)) → (𝐴 mod 8) = 5)
73	72, 40	eqtr4di 2228	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 mod 8) = 5 ∧ (𝐵 mod 8) = 5)) → (𝐴 mod 8) = (-3 mod 8))
74		simprr 531	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 mod 8) = 5 ∧ (𝐵 mod 8) = 5)) → (𝐵 mod 8) = 5)
75	74, 40	eqtr4di 2228	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 mod 8) = 5 ∧ (𝐵 mod 8) = 5)) → (𝐵 mod 8) = (-3 mod 8))
76	67, 68, 69, 68, 70, 71, 73, 75	modqmul12d 10364	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 mod 8) = 5 ∧ (𝐵 mod 8) = 5)) → ((𝐴 · 𝐵) mod 8) = ((-3 · -3) mod 8))
77	45, 45	mul2negi 8353	. . . . . . . . . 10 ⊢ (-3 · -3) = (3 · 3)
78	77	oveq1i 5879	. . . . . . . . 9 ⊢ ((-3 · -3) mod 8) = ((3 · 3) mod 8)
79	76, 78	eqtrdi 2226	. . . . . . . 8 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 mod 8) = 5 ∧ (𝐵 mod 8) = 5)) → ((𝐴 · 𝐵) mod 8) = ((3 · 3) mod 8))
80	79	orcd 733	. . . . . . 7 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 mod 8) = 5 ∧ (𝐵 mod 8) = 5)) → (((𝐴 · 𝐵) mod 8) = ((3 · 3) mod 8) ∨ ((𝐴 · 𝐵) mod 8) = (-(3 · 3) mod 8)))
81	80	ex 115	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (((𝐴 mod 8) = 5 ∧ (𝐵 mod 8) = 5) → (((𝐴 · 𝐵) mod 8) = ((3 · 3) mod 8) ∨ ((𝐴 · 𝐵) mod 8) = (-(3 · 3) mod 8))))
82	31, 50, 66, 81	ccased 965	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((((𝐴 mod 8) = 3 ∨ (𝐴 mod 8) = 5) ∧ ((𝐵 mod 8) = 3 ∨ (𝐵 mod 8) = 5)) → (((𝐴 · 𝐵) mod 8) = ((3 · 3) mod 8) ∨ ((𝐴 · 𝐵) mod 8) = (-(3 · 3) mod 8))))
83	12, 82	sylbid 150	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (((𝐴 mod 8) ∈ {3, 5} ∧ (𝐵 mod 8) ∈ {3, 5}) → (((𝐴 · 𝐵) mod 8) = ((3 · 3) mod 8) ∨ ((𝐴 · 𝐵) mod 8) = (-(3 · 3) mod 8))))
84	83	imp 124	. . 3 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 mod 8) ∈ {3, 5} ∧ (𝐵 mod 8) ∈ {3, 5})) → (((𝐴 · 𝐵) mod 8) = ((3 · 3) mod 8) ∨ ((𝐴 · 𝐵) mod 8) = (-(3 · 3) mod 8)))
85		simpll 527	. . . . . 6 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 mod 8) ∈ {3, 5} ∧ (𝐵 mod 8) ∈ {3, 5})) → 𝐴 ∈ ℤ)
86		simplr 528	. . . . . 6 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 mod 8) ∈ {3, 5} ∧ (𝐵 mod 8) ∈ {3, 5})) → 𝐵 ∈ ℤ)
87	85, 86	zmulcld 9370	. . . . 5 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 mod 8) ∈ {3, 5} ∧ (𝐵 mod 8) ∈ {3, 5})) → (𝐴 · 𝐵) ∈ ℤ)
88	1	a1i 9	. . . . 5 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 mod 8) ∈ {3, 5} ∧ (𝐵 mod 8) ∈ {3, 5})) → 8 ∈ ℕ)
89	87, 88	zmodcld 10331	. . . 4 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 mod 8) ∈ {3, 5} ∧ (𝐵 mod 8) ∈ {3, 5})) → ((𝐴 · 𝐵) mod 8) ∈ ℕ₀)
90		elprg 3611	. . . 4 ⊢ (((𝐴 · 𝐵) mod 8) ∈ ℕ₀ → (((𝐴 · 𝐵) mod 8) ∈ {((3 · 3) mod 8), (-(3 · 3) mod 8)} ↔ (((𝐴 · 𝐵) mod 8) = ((3 · 3) mod 8) ∨ ((𝐴 · 𝐵) mod 8) = (-(3 · 3) mod 8))))
91	89, 90	syl 14	. . 3 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 mod 8) ∈ {3, 5} ∧ (𝐵 mod 8) ∈ {3, 5})) → (((𝐴 · 𝐵) mod 8) ∈ {((3 · 3) mod 8), (-(3 · 3) mod 8)} ↔ (((𝐴 · 𝐵) mod 8) = ((3 · 3) mod 8) ∨ ((𝐴 · 𝐵) mod 8) = (-(3 · 3) mod 8))))
92	84, 91	mpbird 167	. 2 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 mod 8) ∈ {3, 5} ∧ (𝐵 mod 8) ∈ {3, 5})) → ((𝐴 · 𝐵) mod 8) ∈ {((3 · 3) mod 8), (-(3 · 3) mod 8)})
93		df-9 8974	. . . . . . . 8 ⊢ 9 = (8 + 1)
94		8cn 8994	. . . . . . . . 9 ⊢ 8 ∈ ℂ
95		ax-1cn 7895	. . . . . . . . 9 ⊢ 1 ∈ ℂ
96	94, 95	addcomi 8091	. . . . . . . 8 ⊢ (8 + 1) = (1 + 8)
97	93, 96	eqtri 2198	. . . . . . 7 ⊢ 9 = (1 + 8)
98		3t3e9 9065	. . . . . . 7 ⊢ (3 · 3) = 9
99	94	mulid2i 7951	. . . . . . . 8 ⊢ (1 · 8) = 8
100	99	oveq2i 5880	. . . . . . 7 ⊢ (1 + (1 · 8)) = (1 + 8)
101	97, 98, 100	3eqtr4i 2208	. . . . . 6 ⊢ (3 · 3) = (1 + (1 · 8))
102	101	oveq1i 5879	. . . . 5 ⊢ ((3 · 3) mod 8) = ((1 + (1 · 8)) mod 8)
103		1nn 8919	. . . . . . 7 ⊢ 1 ∈ ℕ
104		nnq 9622	. . . . . . 7 ⊢ (1 ∈ ℕ → 1 ∈ ℚ)
105	103, 104	ax-mp 5	. . . . . 6 ⊢ 1 ∈ ℚ
106		1z 9268	. . . . . 6 ⊢ 1 ∈ ℤ
107		modqcyc 10345	. . . . . 6 ⊢ (((1 ∈ ℚ ∧ 1 ∈ ℤ) ∧ (8 ∈ ℚ ∧ 0 < 8)) → ((1 + (1 · 8)) mod 8) = (1 mod 8))
108	105, 106, 18, 20, 107	mp4an 427	. . . . 5 ⊢ ((1 + (1 · 8)) mod 8) = (1 mod 8)
109	102, 108	eqtri 2198	. . . 4 ⊢ ((3 · 3) mod 8) = (1 mod 8)
110	23	simpli 111	. . . . 5 ⊢ ((1 mod 8) = 1 ∧ (-1 mod 8) = 7)
111	110	simpli 111	. . . 4 ⊢ (1 mod 8) = 1
112	109, 111	eqtri 2198	. . 3 ⊢ ((3 · 3) mod 8) = 1
113		znegcl 9273	. . . . . . . 8 ⊢ (1 ∈ ℤ → -1 ∈ ℤ)
114	106, 113	mp1i 10	. . . . . . 7 ⊢ (⊤ → -1 ∈ ℤ)
115		3nn 9070	. . . . . . . . . 10 ⊢ 3 ∈ ℕ
116	115, 115	nnmulcli 8930	. . . . . . . . 9 ⊢ (3 · 3) ∈ ℕ
117	116	nnzi 9263	. . . . . . . 8 ⊢ (3 · 3) ∈ ℤ
118	117	a1i 9	. . . . . . 7 ⊢ (⊤ → (3 · 3) ∈ ℤ)
119	106	a1i 9	. . . . . . 7 ⊢ (⊤ → 1 ∈ ℤ)
120	18	a1i 9	. . . . . . 7 ⊢ (⊤ → 8 ∈ ℚ)
121	20	a1i 9	. . . . . . 7 ⊢ (⊤ → 0 < 8)
122		eqidd 2178	. . . . . . 7 ⊢ (⊤ → (-1 mod 8) = (-1 mod 8))
123	109	a1i 9	. . . . . . 7 ⊢ (⊤ → ((3 · 3) mod 8) = (1 mod 8))
124	114, 114, 118, 119, 120, 121, 122, 123	modqmul12d 10364	. . . . . 6 ⊢ (⊤ → ((-1 · (3 · 3)) mod 8) = ((-1 · 1) mod 8))
125	124	mptru 1362	. . . . 5 ⊢ ((-1 · (3 · 3)) mod 8) = ((-1 · 1) mod 8)
126	45, 45	mulcli 7953	. . . . . . 7 ⊢ (3 · 3) ∈ ℂ
127	126	mulm1i 8350	. . . . . 6 ⊢ (-1 · (3 · 3)) = -(3 · 3)
128	127	oveq1i 5879	. . . . 5 ⊢ ((-1 · (3 · 3)) mod 8) = (-(3 · 3) mod 8)
129	95	mulm1i 8350	. . . . . 6 ⊢ (-1 · 1) = -1
130	129	oveq1i 5879	. . . . 5 ⊢ ((-1 · 1) mod 8) = (-1 mod 8)
131	125, 128, 130	3eqtr3i 2206	. . . 4 ⊢ (-(3 · 3) mod 8) = (-1 mod 8)
132	110	simpri 113	. . . 4 ⊢ (-1 mod 8) = 7
133	131, 132	eqtri 2198	. . 3 ⊢ (-(3 · 3) mod 8) = 7
134	112, 133	preq12i 3673	. 2 ⊢ {((3 · 3) mod 8), (-(3 · 3) mod 8)} = {1, 7}
135	92, 134	eleqtrdi 2270	1 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 mod 8) ∈ {3, 5} ∧ (𝐵 mod 8) ∈ {3, 5})) → ((𝐴 · 𝐵) mod 8) ∈ {1, 7})

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 708 = wceq 1353 ⊤wtru 1354 ∈ wcel 2148 {cpr 3592 class class class wbr 4000 (class class class)co 5869 0cc0 7802 1c1 7803 + caddc 7805 · cmul 7807 < clt 7982 -cneg 8119 ℕcn 8908 3c3 8960 5c5 8962 7c7 8964 8c8 8965 9c9 8966 ℕ₀cn0 9165 ℤcz 9242 ℚcq 9608 mod cmo 10308

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-setind 4533 ax-cnex 7893 ax-resscn 7894 ax-1cn 7895 ax-1re 7896 ax-icn 7897 ax-addcl 7898 ax-addrcl 7899 ax-mulcl 7900 ax-mulrcl 7901 ax-addcom 7902 ax-mulcom 7903 ax-addass 7904 ax-mulass 7905 ax-distr 7906 ax-i2m1 7907 ax-0lt1 7908 ax-1rid 7909 ax-0id 7910 ax-rnegex 7911 ax-precex 7912 ax-cnre 7913 ax-pre-ltirr 7914 ax-pre-ltwlin 7915 ax-pre-lttrn 7916 ax-pre-apti 7917 ax-pre-ltadd 7918 ax-pre-mulgt0 7919 ax-pre-mulext 7920 ax-arch 7921

This theorem depends on definitions: df-bi 117 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-id 4290 df-po 4293 df-iso 4294 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-rn 4634 df-res 4635 df-ima 4636 df-iota 5174 df-fun 5214 df-fn 5215 df-f 5216 df-fv 5220 df-riota 5825 df-ov 5872 df-oprab 5873 df-mpo 5874 df-1st 6135 df-2nd 6136 df-pnf 7984 df-mnf 7985 df-xr 7986 df-ltxr 7987 df-le 7988 df-sub 8120 df-neg 8121 df-reap 8522 df-ap 8529 df-div 8619 df-inn 8909 df-2 8967 df-3 8968 df-4 8969 df-5 8970 df-6 8971 df-7 8972 df-8 8973 df-9 8974 df-n0 9166 df-z 9243 df-q 9609 df-rp 9641 df-fl 10256 df-mod 10309

This theorem is referenced by: lgsdir2 14101

W3C validator