lgsdir2lem5 - Intuitionistic Logic Explorer

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

Theorem lgsdir2lem5 15189

Description: Lemma for lgsdir2 15190. (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 9152	. . . . . . . . 9 ⊢ 8 ∈ ℕ
2		zmodcl 10418	. . . . . . . . 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 3639	. . . . . . 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 10418	. . . . . . . . 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 3639	. . . . . . 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 9349	. . . . . . . . . 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 9701	. . . . . . . . . . 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 9087	. . . . . . . . . 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 15185	. . . . . . . . . . . 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 2244	. . . . . . . . 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 2244	. . . . . . . . 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 10452	. . . . . . . 8 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 mod 8) = 3 ∧ (𝐵 mod 8) = 3)) → ((𝐴 · 𝐵) mod 8) = ((3 · 3) mod 8))
30	29	orcd 734	. . . . . . 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 9351	. . . . . . . . . . 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 2244	. . . . . . . . . 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 2244	. . . . . . . . . 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 10452	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 mod 8) = 5 ∧ (𝐵 mod 8) = 3)) → ((𝐴 · 𝐵) mod 8) = ((-3 · 3) mod 8))
45		3cn 9059	. . . . . . . . . . 11 ⊢ 3 ∈ ℂ
46	45, 45	mulneg1i 8425	. . . . . . . . . 10 ⊢ (-3 · 3) = -(3 · 3)
47	46	oveq1i 5929	. . . . . . . . 9 ⊢ ((-3 · 3) mod 8) = (-(3 · 3) mod 8)
48	44, 47	eqtrdi 2242	. . . . . . . 8 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 mod 8) = 5 ∧ (𝐵 mod 8) = 3)) → ((𝐴 · 𝐵) mod 8) = (-(3 · 3) mod 8))
49	48	olcd 735	. . . . . . 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 2244	. . . . . . . . . 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 2244	. . . . . . . . . 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 10452	. . . . . . . . 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 8426	. . . . . . . . . 10 ⊢ (3 · -3) = -(3 · 3)
63	62	oveq1i 5929	. . . . . . . . 9 ⊢ ((3 · -3) mod 8) = (-(3 · 3) mod 8)
64	61, 63	eqtrdi 2242	. . . . . . . 8 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 mod 8) = 3 ∧ (𝐵 mod 8) = 5)) → ((𝐴 · 𝐵) mod 8) = (-(3 · 3) mod 8))
65	64	olcd 735	. . . . . . 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 2244	. . . . . . . . . 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 2244	. . . . . . . . . 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 10452	. . . . . . . . 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 8427	. . . . . . . . . 10 ⊢ (-3 · -3) = (3 · 3)
78	77	oveq1i 5929	. . . . . . . . 9 ⊢ ((-3 · -3) mod 8) = ((3 · 3) mod 8)
79	76, 78	eqtrdi 2242	. . . . . . . 8 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 mod 8) = 5 ∧ (𝐵 mod 8) = 5)) → ((𝐴 · 𝐵) mod 8) = ((3 · 3) mod 8))
80	79	orcd 734	. . . . . . 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 967	. . . . 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 9448	. . . . 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 10419	. . . 4 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 mod 8) ∈ {3, 5} ∧ (𝐵 mod 8) ∈ {3, 5})) → ((𝐴 · 𝐵) mod 8) ∈ ℕ₀)
90		elprg 3639	. . . 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 9050	. . . . . . . 8 ⊢ 9 = (8 + 1)
94		8cn 9070	. . . . . . . . 9 ⊢ 8 ∈ ℂ
95		ax-1cn 7967	. . . . . . . . 9 ⊢ 1 ∈ ℂ
96	94, 95	addcomi 8165	. . . . . . . 8 ⊢ (8 + 1) = (1 + 8)
97	93, 96	eqtri 2214	. . . . . . 7 ⊢ 9 = (1 + 8)
98		3t3e9 9142	. . . . . . 7 ⊢ (3 · 3) = 9
99	94	mullidi 8024	. . . . . . . 8 ⊢ (1 · 8) = 8
100	99	oveq2i 5930	. . . . . . 7 ⊢ (1 + (1 · 8)) = (1 + 8)
101	97, 98, 100	3eqtr4i 2224	. . . . . 6 ⊢ (3 · 3) = (1 + (1 · 8))
102	101	oveq1i 5929	. . . . 5 ⊢ ((3 · 3) mod 8) = ((1 + (1 · 8)) mod 8)
103		1nn 8995	. . . . . . 7 ⊢ 1 ∈ ℕ
104		nnq 9701	. . . . . . 7 ⊢ (1 ∈ ℕ → 1 ∈ ℚ)
105	103, 104	ax-mp 5	. . . . . 6 ⊢ 1 ∈ ℚ
106		1z 9346	. . . . . 6 ⊢ 1 ∈ ℤ
107		modqcyc 10433	. . . . . 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 2214	. . . 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 2214	. . 3 ⊢ ((3 · 3) mod 8) = 1
113		znegcl 9351	. . . . . . . 8 ⊢ (1 ∈ ℤ → -1 ∈ ℤ)
114	106, 113	mp1i 10	. . . . . . 7 ⊢ (⊤ → -1 ∈ ℤ)
115		3nn 9147	. . . . . . . . . 10 ⊢ 3 ∈ ℕ
116	115, 115	nnmulcli 9006	. . . . . . . . 9 ⊢ (3 · 3) ∈ ℕ
117	116	nnzi 9341	. . . . . . . 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 2194	. . . . . . 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 10452	. . . . . 6 ⊢ (⊤ → ((-1 · (3 · 3)) mod 8) = ((-1 · 1) mod 8))
125	124	mptru 1373	. . . . 5 ⊢ ((-1 · (3 · 3)) mod 8) = ((-1 · 1) mod 8)
126	45, 45	mulcli 8026	. . . . . . 7 ⊢ (3 · 3) ∈ ℂ
127	126	mulm1i 8424	. . . . . 6 ⊢ (-1 · (3 · 3)) = -(3 · 3)
128	127	oveq1i 5929	. . . . 5 ⊢ ((-1 · (3 · 3)) mod 8) = (-(3 · 3) mod 8)
129	95	mulm1i 8424	. . . . . 6 ⊢ (-1 · 1) = -1
130	129	oveq1i 5929	. . . . 5 ⊢ ((-1 · 1) mod 8) = (-1 mod 8)
131	125, 128, 130	3eqtr3i 2222	. . . 4 ⊢ (-(3 · 3) mod 8) = (-1 mod 8)
132	110	simpri 113	. . . 4 ⊢ (-1 mod 8) = 7
133	131, 132	eqtri 2214	. . 3 ⊢ (-(3 · 3) mod 8) = 7
134	112, 133	preq12i 3701	. 2 ⊢ {((3 · 3) mod 8), (-(3 · 3) mod 8)} = {1, 7}
135	92, 134	eleqtrdi 2286	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 709 = wceq 1364 ⊤wtru 1365 ∈ wcel 2164 {cpr 3620 class class class wbr 4030 (class class class)co 5919 0cc0 7874 1c1 7875 + caddc 7877 · cmul 7879 < clt 8056 -cneg 8193 ℕcn 8984 3c3 9036 5c5 9038 7c7 9040 8c8 9041 9c9 9042 ℕ₀cn0 9243 ℤcz 9320 ℚcq 9687 mod cmo 10396

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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-sep 4148 ax-pow 4204 ax-pr 4239 ax-un 4465 ax-setind 4570 ax-cnex 7965 ax-resscn 7966 ax-1cn 7967 ax-1re 7968 ax-icn 7969 ax-addcl 7970 ax-addrcl 7971 ax-mulcl 7972 ax-mulrcl 7973 ax-addcom 7974 ax-mulcom 7975 ax-addass 7976 ax-mulass 7977 ax-distr 7978 ax-i2m1 7979 ax-0lt1 7980 ax-1rid 7981 ax-0id 7982 ax-rnegex 7983 ax-precex 7984 ax-cnre 7985 ax-pre-ltirr 7986 ax-pre-ltwlin 7987 ax-pre-lttrn 7988 ax-pre-apti 7989 ax-pre-ltadd 7990 ax-pre-mulgt0 7991 ax-pre-mulext 7992 ax-arch 7993

This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rmo 2480 df-rab 2481 df-v 2762 df-sbc 2987 df-csb 3082 df-dif 3156 df-un 3158 df-in 3160 df-ss 3167 df-pw 3604 df-sn 3625 df-pr 3626 df-op 3628 df-uni 3837 df-int 3872 df-iun 3915 df-br 4031 df-opab 4092 df-mpt 4093 df-id 4325 df-po 4328 df-iso 4329 df-xp 4666 df-rel 4667 df-cnv 4668 df-co 4669 df-dm 4670 df-rn 4671 df-res 4672 df-ima 4673 df-iota 5216 df-fun 5257 df-fn 5258 df-f 5259 df-fv 5263 df-riota 5874 df-ov 5922 df-oprab 5923 df-mpo 5924 df-1st 6195 df-2nd 6196 df-pnf 8058 df-mnf 8059 df-xr 8060 df-ltxr 8061 df-le 8062 df-sub 8194 df-neg 8195 df-reap 8596 df-ap 8603 df-div 8694 df-inn 8985 df-2 9043 df-3 9044 df-4 9045 df-5 9046 df-6 9047 df-7 9048 df-8 9049 df-9 9050 df-n0 9244 df-z 9321 df-q 9688 df-rp 9723 df-fl 10342 df-mod 10397

This theorem is referenced by: lgsdir2 15190

W3C validator