lgsdir2lem5 - Intuitionistic Logic Explorer

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

Theorem lgsdir2lem5 15764

Description: Lemma for lgsdir2 15765. (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 9311	. . . . . . . . 9 ⊢ 8 ∈ ℕ
2		zmodcl 10607	. . . . . . . . 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 3689	. . . . . . 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 10607	. . . . . . . . 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 3689	. . . . . . 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 9508	. . . . . . . . . 10 ⊢ 3 ∈ ℤ
15	14	a1i 9	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 mod 8) = 3 ∧ (𝐵 mod 8) = 3)) → 3 ∈ ℤ)
16		simplr 529	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 mod 8) = 3 ∧ (𝐵 mod 8) = 3)) → 𝐵 ∈ ℤ)
17		nnq 9867	. . . . . . . . . . 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 9246	. . . . . . . . . 10 ⊢ 0 < 8
21	20	a1i 9	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 mod 8) = 3 ∧ (𝐵 mod 8) = 3)) → 0 < 8)
22		simprl 531	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 mod 8) = 3 ∧ (𝐵 mod 8) = 3)) → (𝐴 mod 8) = 3)
23		lgsdir2lem1 15760	. . . . . . . . . . . 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 2282	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 mod 8) = 3 ∧ (𝐵 mod 8) = 3)) → (𝐴 mod 8) = (3 mod 8))
27		simprr 533	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 mod 8) = 3 ∧ (𝐵 mod 8) = 3)) → (𝐵 mod 8) = 3)
28	27, 25	eqtr4di 2282	. . . . . . . . 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 10641	. . . . . . . 8 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 mod 8) = 3 ∧ (𝐵 mod 8) = 3)) → ((𝐴 · 𝐵) mod 8) = ((3 · 3) mod 8))
30	29	orcd 740	. . . . . . 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 9510	. . . . . . . . . . 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 529	. . . . . . . . . 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 531	. . . . . . . . . . 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 2282	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 mod 8) = 5 ∧ (𝐵 mod 8) = 3)) → (𝐴 mod 8) = (-3 mod 8))
42		simprr 533	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 mod 8) = 5 ∧ (𝐵 mod 8) = 3)) → (𝐵 mod 8) = 3)
43	42, 25	eqtr4di 2282	. . . . . . . . . 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 10641	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 mod 8) = 5 ∧ (𝐵 mod 8) = 3)) → ((𝐴 · 𝐵) mod 8) = ((-3 · 3) mod 8))
45		3cn 9218	. . . . . . . . . . 11 ⊢ 3 ∈ ℂ
46	45, 45	mulneg1i 8583	. . . . . . . . . 10 ⊢ (-3 · 3) = -(3 · 3)
47	46	oveq1i 6028	. . . . . . . . 9 ⊢ ((-3 · 3) mod 8) = (-(3 · 3) mod 8)
48	44, 47	eqtrdi 2280	. . . . . . . 8 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 mod 8) = 5 ∧ (𝐵 mod 8) = 3)) → ((𝐴 · 𝐵) mod 8) = (-(3 · 3) mod 8))
49	48	olcd 741	. . . . . . 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 529	. . . . . . . . . 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 531	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 mod 8) = 3 ∧ (𝐵 mod 8) = 5)) → (𝐴 mod 8) = 3)
58	57, 25	eqtr4di 2282	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 mod 8) = 3 ∧ (𝐵 mod 8) = 5)) → (𝐴 mod 8) = (3 mod 8))
59		simprr 533	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 mod 8) = 3 ∧ (𝐵 mod 8) = 5)) → (𝐵 mod 8) = 5)
60	59, 40	eqtr4di 2282	. . . . . . . . . 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 10641	. . . . . . . . 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 8584	. . . . . . . . . 10 ⊢ (3 · -3) = -(3 · 3)
63	62	oveq1i 6028	. . . . . . . . 9 ⊢ ((3 · -3) mod 8) = (-(3 · 3) mod 8)
64	61, 63	eqtrdi 2280	. . . . . . . 8 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 mod 8) = 3 ∧ (𝐵 mod 8) = 5)) → ((𝐴 · 𝐵) mod 8) = (-(3 · 3) mod 8))
65	64	olcd 741	. . . . . . 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 529	. . . . . . . . . 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 531	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 mod 8) = 5 ∧ (𝐵 mod 8) = 5)) → (𝐴 mod 8) = 5)
73	72, 40	eqtr4di 2282	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 mod 8) = 5 ∧ (𝐵 mod 8) = 5)) → (𝐴 mod 8) = (-3 mod 8))
74		simprr 533	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 mod 8) = 5 ∧ (𝐵 mod 8) = 5)) → (𝐵 mod 8) = 5)
75	74, 40	eqtr4di 2282	. . . . . . . . . 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 10641	. . . . . . . . 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 8585	. . . . . . . . . 10 ⊢ (-3 · -3) = (3 · 3)
78	77	oveq1i 6028	. . . . . . . . 9 ⊢ ((-3 · -3) mod 8) = ((3 · 3) mod 8)
79	76, 78	eqtrdi 2280	. . . . . . . 8 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 mod 8) = 5 ∧ (𝐵 mod 8) = 5)) → ((𝐴 · 𝐵) mod 8) = ((3 · 3) mod 8))
80	79	orcd 740	. . . . . . 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 973	. . . . 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 529	. . . . . 6 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 mod 8) ∈ {3, 5} ∧ (𝐵 mod 8) ∈ {3, 5})) → 𝐵 ∈ ℤ)
87	85, 86	zmulcld 9608	. . . . 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 10608	. . . 4 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 mod 8) ∈ {3, 5} ∧ (𝐵 mod 8) ∈ {3, 5})) → ((𝐴 · 𝐵) mod 8) ∈ ℕ₀)
90		elprg 3689	. . . 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 9209	. . . . . . . 8 ⊢ 9 = (8 + 1)
94		8cn 9229	. . . . . . . . 9 ⊢ 8 ∈ ℂ
95		ax-1cn 8125	. . . . . . . . 9 ⊢ 1 ∈ ℂ
96	94, 95	addcomi 8323	. . . . . . . 8 ⊢ (8 + 1) = (1 + 8)
97	93, 96	eqtri 2252	. . . . . . 7 ⊢ 9 = (1 + 8)
98		3t3e9 9301	. . . . . . 7 ⊢ (3 · 3) = 9
99	94	mullidi 8182	. . . . . . . 8 ⊢ (1 · 8) = 8
100	99	oveq2i 6029	. . . . . . 7 ⊢ (1 + (1 · 8)) = (1 + 8)
101	97, 98, 100	3eqtr4i 2262	. . . . . 6 ⊢ (3 · 3) = (1 + (1 · 8))
102	101	oveq1i 6028	. . . . 5 ⊢ ((3 · 3) mod 8) = ((1 + (1 · 8)) mod 8)
103		1nn 9154	. . . . . . 7 ⊢ 1 ∈ ℕ
104		nnq 9867	. . . . . . 7 ⊢ (1 ∈ ℕ → 1 ∈ ℚ)
105	103, 104	ax-mp 5	. . . . . 6 ⊢ 1 ∈ ℚ
106		1z 9505	. . . . . 6 ⊢ 1 ∈ ℤ
107		modqcyc 10622	. . . . . 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 2252	. . . 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 2252	. . 3 ⊢ ((3 · 3) mod 8) = 1
113		znegcl 9510	. . . . . . . 8 ⊢ (1 ∈ ℤ → -1 ∈ ℤ)
114	106, 113	mp1i 10	. . . . . . 7 ⊢ (⊤ → -1 ∈ ℤ)
115		3nn 9306	. . . . . . . . . 10 ⊢ 3 ∈ ℕ
116	115, 115	nnmulcli 9165	. . . . . . . . 9 ⊢ (3 · 3) ∈ ℕ
117	116	nnzi 9500	. . . . . . . 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 2232	. . . . . . 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 10641	. . . . . 6 ⊢ (⊤ → ((-1 · (3 · 3)) mod 8) = ((-1 · 1) mod 8))
125	124	mptru 1406	. . . . 5 ⊢ ((-1 · (3 · 3)) mod 8) = ((-1 · 1) mod 8)
126	45, 45	mulcli 8184	. . . . . . 7 ⊢ (3 · 3) ∈ ℂ
127	126	mulm1i 8582	. . . . . 6 ⊢ (-1 · (3 · 3)) = -(3 · 3)
128	127	oveq1i 6028	. . . . 5 ⊢ ((-1 · (3 · 3)) mod 8) = (-(3 · 3) mod 8)
129	95	mulm1i 8582	. . . . . 6 ⊢ (-1 · 1) = -1
130	129	oveq1i 6028	. . . . 5 ⊢ ((-1 · 1) mod 8) = (-1 mod 8)
131	125, 128, 130	3eqtr3i 2260	. . . 4 ⊢ (-(3 · 3) mod 8) = (-1 mod 8)
132	110	simpri 113	. . . 4 ⊢ (-1 mod 8) = 7
133	131, 132	eqtri 2252	. . 3 ⊢ (-(3 · 3) mod 8) = 7
134	112, 133	preq12i 3753	. 2 ⊢ {((3 · 3) mod 8), (-(3 · 3) mod 8)} = {1, 7}
135	92, 134	eleqtrdi 2324	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 715 = wceq 1397 ⊤wtru 1398 ∈ wcel 2202 {cpr 3670 class class class wbr 4088 (class class class)co 6018 0cc0 8032 1c1 8033 + caddc 8035 · cmul 8037 < clt 8214 -cneg 8351 ℕcn 9143 3c3 9195 5c5 9197 7c7 9199 8c8 9200 9c9 9201 ℕ₀cn0 9402 ℤcz 9479 ℚcq 9853 mod cmo 10585

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 619 ax-in2 620 ax-io 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-13 2204 ax-14 2205 ax-ext 2213 ax-sep 4207 ax-pow 4264 ax-pr 4299 ax-un 4530 ax-setind 4635 ax-cnex 8123 ax-resscn 8124 ax-1cn 8125 ax-1re 8126 ax-icn 8127 ax-addcl 8128 ax-addrcl 8129 ax-mulcl 8130 ax-mulrcl 8131 ax-addcom 8132 ax-mulcom 8133 ax-addass 8134 ax-mulass 8135 ax-distr 8136 ax-i2m1 8137 ax-0lt1 8138 ax-1rid 8139 ax-0id 8140 ax-rnegex 8141 ax-precex 8142 ax-cnre 8143 ax-pre-ltirr 8144 ax-pre-ltwlin 8145 ax-pre-lttrn 8146 ax-pre-apti 8147 ax-pre-ltadd 8148 ax-pre-mulgt0 8149 ax-pre-mulext 8150 ax-arch 8151

This theorem depends on definitions: df-bi 117 df-3or 1005 df-3an 1006 df-tru 1400 df-fal 1403 df-nf 1509 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2363 df-ne 2403 df-nel 2498 df-ral 2515 df-rex 2516 df-reu 2517 df-rmo 2518 df-rab 2519 df-v 2804 df-sbc 3032 df-csb 3128 df-dif 3202 df-un 3204 df-in 3206 df-ss 3213 df-pw 3654 df-sn 3675 df-pr 3676 df-op 3678 df-uni 3894 df-int 3929 df-iun 3972 df-br 4089 df-opab 4151 df-mpt 4152 df-id 4390 df-po 4393 df-iso 4394 df-xp 4731 df-rel 4732 df-cnv 4733 df-co 4734 df-dm 4735 df-rn 4736 df-res 4737 df-ima 4738 df-iota 5286 df-fun 5328 df-fn 5329 df-f 5330 df-fv 5334 df-riota 5971 df-ov 6021 df-oprab 6022 df-mpo 6023 df-1st 6303 df-2nd 6304 df-pnf 8216 df-mnf 8217 df-xr 8218 df-ltxr 8219 df-le 8220 df-sub 8352 df-neg 8353 df-reap 8755 df-ap 8762 df-div 8853 df-inn 9144 df-2 9202 df-3 9203 df-4 9204 df-5 9205 df-6 9206 df-7 9207 df-8 9208 df-9 9209 df-n0 9403 df-z 9480 df-q 9854 df-rp 9889 df-fl 10531 df-mod 10586

This theorem is referenced by: lgsdir2 15765

W3C validator