lgsdir2lem5 - Intuitionistic Logic Explorer

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Theorem lgsdir2lem5 15357

Description: Lemma for lgsdir2 15358. (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 9175	. . . . . . . . 9 ⊢ 8 ∈ ℕ
2		zmodcl 10453	. . . . . . . . 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 3643	. . . . . . 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 10453	. . . . . . . . 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 3643	. . . . . . 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 9372	. . . . . . . . . 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 9724	. . . . . . . . . . 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 9110	. . . . . . . . . 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 15353	. . . . . . . . . . . 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 2247	. . . . . . . . 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 2247	. . . . . . . . 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 10487	. . . . . . . 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 9374	. . . . . . . . . . 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 2247	. . . . . . . . . 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 2247	. . . . . . . . . 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 10487	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 mod 8) = 5 ∧ (𝐵 mod 8) = 3)) → ((𝐴 · 𝐵) mod 8) = ((-3 · 3) mod 8))
45		3cn 9082	. . . . . . . . . . 11 ⊢ 3 ∈ ℂ
46	45, 45	mulneg1i 8447	. . . . . . . . . 10 ⊢ (-3 · 3) = -(3 · 3)
47	46	oveq1i 5935	. . . . . . . . 9 ⊢ ((-3 · 3) mod 8) = (-(3 · 3) mod 8)
48	44, 47	eqtrdi 2245	. . . . . . . 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 2247	. . . . . . . . . 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 2247	. . . . . . . . . 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 10487	. . . . . . . . 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 8448	. . . . . . . . . 10 ⊢ (3 · -3) = -(3 · 3)
63	62	oveq1i 5935	. . . . . . . . 9 ⊢ ((3 · -3) mod 8) = (-(3 · 3) mod 8)
64	61, 63	eqtrdi 2245	. . . . . . . 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 2247	. . . . . . . . . 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 2247	. . . . . . . . . 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 10487	. . . . . . . . 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 8449	. . . . . . . . . 10 ⊢ (-3 · -3) = (3 · 3)
78	77	oveq1i 5935	. . . . . . . . 9 ⊢ ((-3 · -3) mod 8) = ((3 · 3) mod 8)
79	76, 78	eqtrdi 2245	. . . . . . . 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 9471	. . . . 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 10454	. . . 4 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 mod 8) ∈ {3, 5} ∧ (𝐵 mod 8) ∈ {3, 5})) → ((𝐴 · 𝐵) mod 8) ∈ ℕ₀)
90		elprg 3643	. . . 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 9073	. . . . . . . 8 ⊢ 9 = (8 + 1)
94		8cn 9093	. . . . . . . . 9 ⊢ 8 ∈ ℂ
95		ax-1cn 7989	. . . . . . . . 9 ⊢ 1 ∈ ℂ
96	94, 95	addcomi 8187	. . . . . . . 8 ⊢ (8 + 1) = (1 + 8)
97	93, 96	eqtri 2217	. . . . . . 7 ⊢ 9 = (1 + 8)
98		3t3e9 9165	. . . . . . 7 ⊢ (3 · 3) = 9
99	94	mullidi 8046	. . . . . . . 8 ⊢ (1 · 8) = 8
100	99	oveq2i 5936	. . . . . . 7 ⊢ (1 + (1 · 8)) = (1 + 8)
101	97, 98, 100	3eqtr4i 2227	. . . . . 6 ⊢ (3 · 3) = (1 + (1 · 8))
102	101	oveq1i 5935	. . . . 5 ⊢ ((3 · 3) mod 8) = ((1 + (1 · 8)) mod 8)
103		1nn 9018	. . . . . . 7 ⊢ 1 ∈ ℕ
104		nnq 9724	. . . . . . 7 ⊢ (1 ∈ ℕ → 1 ∈ ℚ)
105	103, 104	ax-mp 5	. . . . . 6 ⊢ 1 ∈ ℚ
106		1z 9369	. . . . . 6 ⊢ 1 ∈ ℤ
107		modqcyc 10468	. . . . . 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 2217	. . . 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 2217	. . 3 ⊢ ((3 · 3) mod 8) = 1
113		znegcl 9374	. . . . . . . 8 ⊢ (1 ∈ ℤ → -1 ∈ ℤ)
114	106, 113	mp1i 10	. . . . . . 7 ⊢ (⊤ → -1 ∈ ℤ)
115		3nn 9170	. . . . . . . . . 10 ⊢ 3 ∈ ℕ
116	115, 115	nnmulcli 9029	. . . . . . . . 9 ⊢ (3 · 3) ∈ ℕ
117	116	nnzi 9364	. . . . . . . 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 2197	. . . . . . 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 10487	. . . . . 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 8048	. . . . . . 7 ⊢ (3 · 3) ∈ ℂ
127	126	mulm1i 8446	. . . . . 6 ⊢ (-1 · (3 · 3)) = -(3 · 3)
128	127	oveq1i 5935	. . . . 5 ⊢ ((-1 · (3 · 3)) mod 8) = (-(3 · 3) mod 8)
129	95	mulm1i 8446	. . . . . 6 ⊢ (-1 · 1) = -1
130	129	oveq1i 5935	. . . . 5 ⊢ ((-1 · 1) mod 8) = (-1 mod 8)
131	125, 128, 130	3eqtr3i 2225	. . . 4 ⊢ (-(3 · 3) mod 8) = (-1 mod 8)
132	110	simpri 113	. . . 4 ⊢ (-1 mod 8) = 7
133	131, 132	eqtri 2217	. . 3 ⊢ (-(3 · 3) mod 8) = 7
134	112, 133	preq12i 3705	. 2 ⊢ {((3 · 3) mod 8), (-(3 · 3) mod 8)} = {1, 7}
135	92, 134	eleqtrdi 2289	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 2167 {cpr 3624 class class class wbr 4034 (class class class)co 5925 0cc0 7896 1c1 7897 + caddc 7899 · cmul 7901 < clt 8078 -cneg 8215 ℕcn 9007 3c3 9059 5c5 9061 7c7 9063 8c8 9064 9c9 9065 ℕ₀cn0 9266 ℤcz 9343 ℚcq 9710 mod cmo 10431

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 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-sep 4152 ax-pow 4208 ax-pr 4243 ax-un 4469 ax-setind 4574 ax-cnex 7987 ax-resscn 7988 ax-1cn 7989 ax-1re 7990 ax-icn 7991 ax-addcl 7992 ax-addrcl 7993 ax-mulcl 7994 ax-mulrcl 7995 ax-addcom 7996 ax-mulcom 7997 ax-addass 7998 ax-mulass 7999 ax-distr 8000 ax-i2m1 8001 ax-0lt1 8002 ax-1rid 8003 ax-0id 8004 ax-rnegex 8005 ax-precex 8006 ax-cnre 8007 ax-pre-ltirr 8008 ax-pre-ltwlin 8009 ax-pre-lttrn 8010 ax-pre-apti 8011 ax-pre-ltadd 8012 ax-pre-mulgt0 8013 ax-pre-mulext 8014 ax-arch 8015

This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-nel 2463 df-ral 2480 df-rex 2481 df-reu 2482 df-rmo 2483 df-rab 2484 df-v 2765 df-sbc 2990 df-csb 3085 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-pw 3608 df-sn 3629 df-pr 3630 df-op 3632 df-uni 3841 df-int 3876 df-iun 3919 df-br 4035 df-opab 4096 df-mpt 4097 df-id 4329 df-po 4332 df-iso 4333 df-xp 4670 df-rel 4671 df-cnv 4672 df-co 4673 df-dm 4674 df-rn 4675 df-res 4676 df-ima 4677 df-iota 5220 df-fun 5261 df-fn 5262 df-f 5263 df-fv 5267 df-riota 5880 df-ov 5928 df-oprab 5929 df-mpo 5930 df-1st 6207 df-2nd 6208 df-pnf 8080 df-mnf 8081 df-xr 8082 df-ltxr 8083 df-le 8084 df-sub 8216 df-neg 8217 df-reap 8619 df-ap 8626 df-div 8717 df-inn 9008 df-2 9066 df-3 9067 df-4 9068 df-5 9069 df-6 9070 df-7 9071 df-8 9072 df-9 9073 df-n0 9267 df-z 9344 df-q 9711 df-rp 9746 df-fl 10377 df-mod 10432

This theorem is referenced by: lgsdir2 15358

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