| Step |
Hyp |
Ref |
Expression |
| 1 |
|
ovex |
⊢ ( 𝐴 mod 8 ) ∈ V |
| 2 |
1
|
elpr |
⊢ ( ( 𝐴 mod 8 ) ∈ { 3 , 5 } ↔ ( ( 𝐴 mod 8 ) = 3 ∨ ( 𝐴 mod 8 ) = 5 ) ) |
| 3 |
|
ovex |
⊢ ( 𝐵 mod 8 ) ∈ V |
| 4 |
3
|
elpr |
⊢ ( ( 𝐵 mod 8 ) ∈ { 3 , 5 } ↔ ( ( 𝐵 mod 8 ) = 3 ∨ ( 𝐵 mod 8 ) = 5 ) ) |
| 5 |
2 4
|
anbi12i |
⊢ ( ( ( 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 ) ) ) |
| 6 |
|
simpll |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) ∧ ( ( 𝐴 mod 8 ) = 3 ∧ ( 𝐵 mod 8 ) = 3 ) ) → 𝐴 ∈ ℤ ) |
| 7 |
|
3z |
⊢ 3 ∈ ℤ |
| 8 |
7
|
a1i |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) ∧ ( ( 𝐴 mod 8 ) = 3 ∧ ( 𝐵 mod 8 ) = 3 ) ) → 3 ∈ ℤ ) |
| 9 |
|
simplr |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) ∧ ( ( 𝐴 mod 8 ) = 3 ∧ ( 𝐵 mod 8 ) = 3 ) ) → 𝐵 ∈ ℤ ) |
| 10 |
|
8re |
⊢ 8 ∈ ℝ |
| 11 |
|
8pos |
⊢ 0 < 8 |
| 12 |
10 11
|
elrpii |
⊢ 8 ∈ ℝ+ |
| 13 |
12
|
a1i |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) ∧ ( ( 𝐴 mod 8 ) = 3 ∧ ( 𝐵 mod 8 ) = 3 ) ) → 8 ∈ ℝ+ ) |
| 14 |
|
simprl |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) ∧ ( ( 𝐴 mod 8 ) = 3 ∧ ( 𝐵 mod 8 ) = 3 ) ) → ( 𝐴 mod 8 ) = 3 ) |
| 15 |
|
lgsdir2lem1 |
⊢ ( ( ( 1 mod 8 ) = 1 ∧ ( - 1 mod 8 ) = 7 ) ∧ ( ( 3 mod 8 ) = 3 ∧ ( - 3 mod 8 ) = 5 ) ) |
| 16 |
15
|
simpri |
⊢ ( ( 3 mod 8 ) = 3 ∧ ( - 3 mod 8 ) = 5 ) |
| 17 |
16
|
simpli |
⊢ ( 3 mod 8 ) = 3 |
| 18 |
14 17
|
syl6eqr |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) ∧ ( ( 𝐴 mod 8 ) = 3 ∧ ( 𝐵 mod 8 ) = 3 ) ) → ( 𝐴 mod 8 ) = ( 3 mod 8 ) ) |
| 19 |
|
simprr |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) ∧ ( ( 𝐴 mod 8 ) = 3 ∧ ( 𝐵 mod 8 ) = 3 ) ) → ( 𝐵 mod 8 ) = 3 ) |
| 20 |
19 17
|
syl6eqr |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) ∧ ( ( 𝐴 mod 8 ) = 3 ∧ ( 𝐵 mod 8 ) = 3 ) ) → ( 𝐵 mod 8 ) = ( 3 mod 8 ) ) |
| 21 |
6 8 9 8 13 18 20
|
modmul12d |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) ∧ ( ( 𝐴 mod 8 ) = 3 ∧ ( 𝐵 mod 8 ) = 3 ) ) → ( ( 𝐴 · 𝐵 ) mod 8 ) = ( ( 3 · 3 ) mod 8 ) ) |
| 22 |
21
|
orcd |
⊢ ( ( ( 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 ) ) ) |
| 23 |
22
|
ex |
⊢ ( ( 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 ) ) ) ) |
| 24 |
|
simpll |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) ∧ ( ( 𝐴 mod 8 ) = 5 ∧ ( 𝐵 mod 8 ) = 3 ) ) → 𝐴 ∈ ℤ ) |
| 25 |
|
znegcl |
⊢ ( 3 ∈ ℤ → - 3 ∈ ℤ ) |
| 26 |
7 25
|
mp1i |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) ∧ ( ( 𝐴 mod 8 ) = 5 ∧ ( 𝐵 mod 8 ) = 3 ) ) → - 3 ∈ ℤ ) |
| 27 |
|
simplr |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) ∧ ( ( 𝐴 mod 8 ) = 5 ∧ ( 𝐵 mod 8 ) = 3 ) ) → 𝐵 ∈ ℤ ) |
| 28 |
7
|
a1i |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) ∧ ( ( 𝐴 mod 8 ) = 5 ∧ ( 𝐵 mod 8 ) = 3 ) ) → 3 ∈ ℤ ) |
| 29 |
12
|
a1i |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) ∧ ( ( 𝐴 mod 8 ) = 5 ∧ ( 𝐵 mod 8 ) = 3 ) ) → 8 ∈ ℝ+ ) |
| 30 |
|
simprl |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) ∧ ( ( 𝐴 mod 8 ) = 5 ∧ ( 𝐵 mod 8 ) = 3 ) ) → ( 𝐴 mod 8 ) = 5 ) |
| 31 |
16
|
simpri |
⊢ ( - 3 mod 8 ) = 5 |
| 32 |
30 31
|
syl6eqr |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) ∧ ( ( 𝐴 mod 8 ) = 5 ∧ ( 𝐵 mod 8 ) = 3 ) ) → ( 𝐴 mod 8 ) = ( - 3 mod 8 ) ) |
| 33 |
|
simprr |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) ∧ ( ( 𝐴 mod 8 ) = 5 ∧ ( 𝐵 mod 8 ) = 3 ) ) → ( 𝐵 mod 8 ) = 3 ) |
| 34 |
33 17
|
syl6eqr |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) ∧ ( ( 𝐴 mod 8 ) = 5 ∧ ( 𝐵 mod 8 ) = 3 ) ) → ( 𝐵 mod 8 ) = ( 3 mod 8 ) ) |
| 35 |
24 26 27 28 29 32 34
|
modmul12d |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) ∧ ( ( 𝐴 mod 8 ) = 5 ∧ ( 𝐵 mod 8 ) = 3 ) ) → ( ( 𝐴 · 𝐵 ) mod 8 ) = ( ( - 3 · 3 ) mod 8 ) ) |
| 36 |
|
3cn |
⊢ 3 ∈ ℂ |
| 37 |
36 36
|
mulneg1i |
⊢ ( - 3 · 3 ) = - ( 3 · 3 ) |
| 38 |
37
|
oveq1i |
⊢ ( ( - 3 · 3 ) mod 8 ) = ( - ( 3 · 3 ) mod 8 ) |
| 39 |
35 38
|
syl6eq |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) ∧ ( ( 𝐴 mod 8 ) = 5 ∧ ( 𝐵 mod 8 ) = 3 ) ) → ( ( 𝐴 · 𝐵 ) mod 8 ) = ( - ( 3 · 3 ) mod 8 ) ) |
| 40 |
39
|
olcd |
⊢ ( ( ( 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 ) ) ) |
| 41 |
40
|
ex |
⊢ ( ( 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 ) ) ) ) |
| 42 |
|
simpll |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) ∧ ( ( 𝐴 mod 8 ) = 3 ∧ ( 𝐵 mod 8 ) = 5 ) ) → 𝐴 ∈ ℤ ) |
| 43 |
7
|
a1i |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) ∧ ( ( 𝐴 mod 8 ) = 3 ∧ ( 𝐵 mod 8 ) = 5 ) ) → 3 ∈ ℤ ) |
| 44 |
|
simplr |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) ∧ ( ( 𝐴 mod 8 ) = 3 ∧ ( 𝐵 mod 8 ) = 5 ) ) → 𝐵 ∈ ℤ ) |
| 45 |
7 25
|
mp1i |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) ∧ ( ( 𝐴 mod 8 ) = 3 ∧ ( 𝐵 mod 8 ) = 5 ) ) → - 3 ∈ ℤ ) |
| 46 |
12
|
a1i |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) ∧ ( ( 𝐴 mod 8 ) = 3 ∧ ( 𝐵 mod 8 ) = 5 ) ) → 8 ∈ ℝ+ ) |
| 47 |
|
simprl |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) ∧ ( ( 𝐴 mod 8 ) = 3 ∧ ( 𝐵 mod 8 ) = 5 ) ) → ( 𝐴 mod 8 ) = 3 ) |
| 48 |
47 17
|
syl6eqr |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) ∧ ( ( 𝐴 mod 8 ) = 3 ∧ ( 𝐵 mod 8 ) = 5 ) ) → ( 𝐴 mod 8 ) = ( 3 mod 8 ) ) |
| 49 |
|
simprr |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) ∧ ( ( 𝐴 mod 8 ) = 3 ∧ ( 𝐵 mod 8 ) = 5 ) ) → ( 𝐵 mod 8 ) = 5 ) |
| 50 |
49 31
|
syl6eqr |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) ∧ ( ( 𝐴 mod 8 ) = 3 ∧ ( 𝐵 mod 8 ) = 5 ) ) → ( 𝐵 mod 8 ) = ( - 3 mod 8 ) ) |
| 51 |
42 43 44 45 46 48 50
|
modmul12d |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) ∧ ( ( 𝐴 mod 8 ) = 3 ∧ ( 𝐵 mod 8 ) = 5 ) ) → ( ( 𝐴 · 𝐵 ) mod 8 ) = ( ( 3 · - 3 ) mod 8 ) ) |
| 52 |
36 36
|
mulneg2i |
⊢ ( 3 · - 3 ) = - ( 3 · 3 ) |
| 53 |
52
|
oveq1i |
⊢ ( ( 3 · - 3 ) mod 8 ) = ( - ( 3 · 3 ) mod 8 ) |
| 54 |
51 53
|
syl6eq |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) ∧ ( ( 𝐴 mod 8 ) = 3 ∧ ( 𝐵 mod 8 ) = 5 ) ) → ( ( 𝐴 · 𝐵 ) mod 8 ) = ( - ( 3 · 3 ) mod 8 ) ) |
| 55 |
54
|
olcd |
⊢ ( ( ( 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 ) ) ) |
| 56 |
55
|
ex |
⊢ ( ( 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 ) ) ) ) |
| 57 |
|
simpll |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) ∧ ( ( 𝐴 mod 8 ) = 5 ∧ ( 𝐵 mod 8 ) = 5 ) ) → 𝐴 ∈ ℤ ) |
| 58 |
7 25
|
mp1i |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) ∧ ( ( 𝐴 mod 8 ) = 5 ∧ ( 𝐵 mod 8 ) = 5 ) ) → - 3 ∈ ℤ ) |
| 59 |
|
simplr |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) ∧ ( ( 𝐴 mod 8 ) = 5 ∧ ( 𝐵 mod 8 ) = 5 ) ) → 𝐵 ∈ ℤ ) |
| 60 |
12
|
a1i |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) ∧ ( ( 𝐴 mod 8 ) = 5 ∧ ( 𝐵 mod 8 ) = 5 ) ) → 8 ∈ ℝ+ ) |
| 61 |
|
simprl |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) ∧ ( ( 𝐴 mod 8 ) = 5 ∧ ( 𝐵 mod 8 ) = 5 ) ) → ( 𝐴 mod 8 ) = 5 ) |
| 62 |
61 31
|
syl6eqr |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) ∧ ( ( 𝐴 mod 8 ) = 5 ∧ ( 𝐵 mod 8 ) = 5 ) ) → ( 𝐴 mod 8 ) = ( - 3 mod 8 ) ) |
| 63 |
|
simprr |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) ∧ ( ( 𝐴 mod 8 ) = 5 ∧ ( 𝐵 mod 8 ) = 5 ) ) → ( 𝐵 mod 8 ) = 5 ) |
| 64 |
63 31
|
syl6eqr |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) ∧ ( ( 𝐴 mod 8 ) = 5 ∧ ( 𝐵 mod 8 ) = 5 ) ) → ( 𝐵 mod 8 ) = ( - 3 mod 8 ) ) |
| 65 |
57 58 59 58 60 62 64
|
modmul12d |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) ∧ ( ( 𝐴 mod 8 ) = 5 ∧ ( 𝐵 mod 8 ) = 5 ) ) → ( ( 𝐴 · 𝐵 ) mod 8 ) = ( ( - 3 · - 3 ) mod 8 ) ) |
| 66 |
36 36
|
mul2negi |
⊢ ( - 3 · - 3 ) = ( 3 · 3 ) |
| 67 |
66
|
oveq1i |
⊢ ( ( - 3 · - 3 ) mod 8 ) = ( ( 3 · 3 ) mod 8 ) |
| 68 |
65 67
|
syl6eq |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) ∧ ( ( 𝐴 mod 8 ) = 5 ∧ ( 𝐵 mod 8 ) = 5 ) ) → ( ( 𝐴 · 𝐵 ) mod 8 ) = ( ( 3 · 3 ) mod 8 ) ) |
| 69 |
68
|
orcd |
⊢ ( ( ( 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 ) ) ) |
| 70 |
69
|
ex |
⊢ ( ( 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 ) ) ) ) |
| 71 |
23 41 56 70
|
ccased |
⊢ ( ( 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 ) ) ) ) |
| 72 |
5 71
|
syl5bi |
⊢ ( ( 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 ) ) ) ) |
| 73 |
72
|
imp |
⊢ ( ( ( 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 ) ) ) |
| 74 |
|
ovex |
⊢ ( ( 𝐴 · 𝐵 ) mod 8 ) ∈ V |
| 75 |
74
|
elpr |
⊢ ( ( ( 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 ) ) ) |
| 76 |
73 75
|
sylibr |
⊢ ( ( ( 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 ) } ) |
| 77 |
|
df-9 |
⊢ 9 = ( 8 + 1 ) |
| 78 |
|
8cn |
⊢ 8 ∈ ℂ |
| 79 |
|
ax-1cn |
⊢ 1 ∈ ℂ |
| 80 |
78 79
|
addcomi |
⊢ ( 8 + 1 ) = ( 1 + 8 ) |
| 81 |
77 80
|
eqtri |
⊢ 9 = ( 1 + 8 ) |
| 82 |
|
3t3e9 |
⊢ ( 3 · 3 ) = 9 |
| 83 |
78
|
mulid2i |
⊢ ( 1 · 8 ) = 8 |
| 84 |
83
|
oveq2i |
⊢ ( 1 + ( 1 · 8 ) ) = ( 1 + 8 ) |
| 85 |
81 82 84
|
3eqtr4i |
⊢ ( 3 · 3 ) = ( 1 + ( 1 · 8 ) ) |
| 86 |
85
|
oveq1i |
⊢ ( ( 3 · 3 ) mod 8 ) = ( ( 1 + ( 1 · 8 ) ) mod 8 ) |
| 87 |
|
1re |
⊢ 1 ∈ ℝ |
| 88 |
|
1z |
⊢ 1 ∈ ℤ |
| 89 |
|
modcyc |
⊢ ( ( 1 ∈ ℝ ∧ 8 ∈ ℝ+ ∧ 1 ∈ ℤ ) → ( ( 1 + ( 1 · 8 ) ) mod 8 ) = ( 1 mod 8 ) ) |
| 90 |
87 12 88 89
|
mp3an |
⊢ ( ( 1 + ( 1 · 8 ) ) mod 8 ) = ( 1 mod 8 ) |
| 91 |
86 90
|
eqtri |
⊢ ( ( 3 · 3 ) mod 8 ) = ( 1 mod 8 ) |
| 92 |
15
|
simpli |
⊢ ( ( 1 mod 8 ) = 1 ∧ ( - 1 mod 8 ) = 7 ) |
| 93 |
92
|
simpli |
⊢ ( 1 mod 8 ) = 1 |
| 94 |
91 93
|
eqtri |
⊢ ( ( 3 · 3 ) mod 8 ) = 1 |
| 95 |
|
znegcl |
⊢ ( 1 ∈ ℤ → - 1 ∈ ℤ ) |
| 96 |
88 95
|
mp1i |
⊢ ( ⊤ → - 1 ∈ ℤ ) |
| 97 |
|
3nn |
⊢ 3 ∈ ℕ |
| 98 |
97 97
|
nnmulcli |
⊢ ( 3 · 3 ) ∈ ℕ |
| 99 |
98
|
nnzi |
⊢ ( 3 · 3 ) ∈ ℤ |
| 100 |
99
|
a1i |
⊢ ( ⊤ → ( 3 · 3 ) ∈ ℤ ) |
| 101 |
88
|
a1i |
⊢ ( ⊤ → 1 ∈ ℤ ) |
| 102 |
12
|
a1i |
⊢ ( ⊤ → 8 ∈ ℝ+ ) |
| 103 |
|
eqidd |
⊢ ( ⊤ → ( - 1 mod 8 ) = ( - 1 mod 8 ) ) |
| 104 |
91
|
a1i |
⊢ ( ⊤ → ( ( 3 · 3 ) mod 8 ) = ( 1 mod 8 ) ) |
| 105 |
96 96 100 101 102 103 104
|
modmul12d |
⊢ ( ⊤ → ( ( - 1 · ( 3 · 3 ) ) mod 8 ) = ( ( - 1 · 1 ) mod 8 ) ) |
| 106 |
105
|
mptru |
⊢ ( ( - 1 · ( 3 · 3 ) ) mod 8 ) = ( ( - 1 · 1 ) mod 8 ) |
| 107 |
36 36
|
mulcli |
⊢ ( 3 · 3 ) ∈ ℂ |
| 108 |
107
|
mulm1i |
⊢ ( - 1 · ( 3 · 3 ) ) = - ( 3 · 3 ) |
| 109 |
108
|
oveq1i |
⊢ ( ( - 1 · ( 3 · 3 ) ) mod 8 ) = ( - ( 3 · 3 ) mod 8 ) |
| 110 |
79
|
mulm1i |
⊢ ( - 1 · 1 ) = - 1 |
| 111 |
110
|
oveq1i |
⊢ ( ( - 1 · 1 ) mod 8 ) = ( - 1 mod 8 ) |
| 112 |
106 109 111
|
3eqtr3i |
⊢ ( - ( 3 · 3 ) mod 8 ) = ( - 1 mod 8 ) |
| 113 |
92
|
simpri |
⊢ ( - 1 mod 8 ) = 7 |
| 114 |
112 113
|
eqtri |
⊢ ( - ( 3 · 3 ) mod 8 ) = 7 |
| 115 |
94 114
|
preq12i |
⊢ { ( ( 3 · 3 ) mod 8 ) , ( - ( 3 · 3 ) mod 8 ) } = { 1 , 7 } |
| 116 |
76 115
|
eleqtrdi |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) ∧ ( ( 𝐴 mod 8 ) ∈ { 3 , 5 } ∧ ( 𝐵 mod 8 ) ∈ { 3 , 5 } ) ) → ( ( 𝐴 · 𝐵 ) mod 8 ) ∈ { 1 , 7 } ) |