Step |
Hyp |
Ref |
Expression |
1 |
|
4nn |
|- 4 e. NN |
2 |
1
|
a1i |
|- ( N e. ZZ -> 4 e. NN ) |
3 |
|
3nn0 |
|- 3 e. NN0 |
4 |
3
|
a1i |
|- ( N e. ZZ -> 3 e. NN0 ) |
5 |
|
3lt4 |
|- 3 < 4 |
6 |
4 5
|
jctir |
|- ( N e. ZZ -> ( 3 e. NN0 /\ 3 < 4 ) ) |
7 |
|
modremain |
|- ( ( N e. ZZ /\ 4 e. NN /\ ( 3 e. NN0 /\ 3 < 4 ) ) -> ( ( N mod 4 ) = 3 <-> E. z e. ZZ ( ( z x. 4 ) + 3 ) = N ) ) |
8 |
2 6 7
|
mpd3an23 |
|- ( N e. ZZ -> ( ( N mod 4 ) = 3 <-> E. z e. ZZ ( ( z x. 4 ) + 3 ) = N ) ) |
9 |
|
2cnd |
|- ( ( N e. ZZ /\ z e. ZZ ) -> 2 e. CC ) |
10 |
|
simpr |
|- ( ( N e. ZZ /\ z e. ZZ ) -> z e. ZZ ) |
11 |
|
4z |
|- 4 e. ZZ |
12 |
11
|
a1i |
|- ( ( N e. ZZ /\ z e. ZZ ) -> 4 e. ZZ ) |
13 |
10 12
|
zmulcld |
|- ( ( N e. ZZ /\ z e. ZZ ) -> ( z x. 4 ) e. ZZ ) |
14 |
13
|
zcnd |
|- ( ( N e. ZZ /\ z e. ZZ ) -> ( z x. 4 ) e. CC ) |
15 |
|
3cn |
|- 3 e. CC |
16 |
15
|
a1i |
|- ( ( N e. ZZ /\ z e. ZZ ) -> 3 e. CC ) |
17 |
9 14 16
|
adddid |
|- ( ( N e. ZZ /\ z e. ZZ ) -> ( 2 x. ( ( z x. 4 ) + 3 ) ) = ( ( 2 x. ( z x. 4 ) ) + ( 2 x. 3 ) ) ) |
18 |
10
|
zcnd |
|- ( ( N e. ZZ /\ z e. ZZ ) -> z e. CC ) |
19 |
|
4cn |
|- 4 e. CC |
20 |
19
|
a1i |
|- ( ( N e. ZZ /\ z e. ZZ ) -> 4 e. CC ) |
21 |
9 18 20
|
mul12d |
|- ( ( N e. ZZ /\ z e. ZZ ) -> ( 2 x. ( z x. 4 ) ) = ( z x. ( 2 x. 4 ) ) ) |
22 |
|
2cn |
|- 2 e. CC |
23 |
|
4t2e8 |
|- ( 4 x. 2 ) = 8 |
24 |
19 22 23
|
mulcomli |
|- ( 2 x. 4 ) = 8 |
25 |
24
|
oveq2i |
|- ( z x. ( 2 x. 4 ) ) = ( z x. 8 ) |
26 |
21 25
|
eqtrdi |
|- ( ( N e. ZZ /\ z e. ZZ ) -> ( 2 x. ( z x. 4 ) ) = ( z x. 8 ) ) |
27 |
|
3t2e6 |
|- ( 3 x. 2 ) = 6 |
28 |
15 22 27
|
mulcomli |
|- ( 2 x. 3 ) = 6 |
29 |
28
|
a1i |
|- ( ( N e. ZZ /\ z e. ZZ ) -> ( 2 x. 3 ) = 6 ) |
30 |
26 29
|
oveq12d |
|- ( ( N e. ZZ /\ z e. ZZ ) -> ( ( 2 x. ( z x. 4 ) ) + ( 2 x. 3 ) ) = ( ( z x. 8 ) + 6 ) ) |
31 |
17 30
|
eqtrd |
|- ( ( N e. ZZ /\ z e. ZZ ) -> ( 2 x. ( ( z x. 4 ) + 3 ) ) = ( ( z x. 8 ) + 6 ) ) |
32 |
31
|
oveq1d |
|- ( ( N e. ZZ /\ z e. ZZ ) -> ( ( 2 x. ( ( z x. 4 ) + 3 ) ) + 1 ) = ( ( ( z x. 8 ) + 6 ) + 1 ) ) |
33 |
|
id |
|- ( z e. ZZ -> z e. ZZ ) |
34 |
|
8nn |
|- 8 e. NN |
35 |
34
|
nnzi |
|- 8 e. ZZ |
36 |
35
|
a1i |
|- ( z e. ZZ -> 8 e. ZZ ) |
37 |
33 36
|
zmulcld |
|- ( z e. ZZ -> ( z x. 8 ) e. ZZ ) |
38 |
37
|
zcnd |
|- ( z e. ZZ -> ( z x. 8 ) e. CC ) |
39 |
|
6cn |
|- 6 e. CC |
40 |
39
|
a1i |
|- ( z e. ZZ -> 6 e. CC ) |
41 |
|
1cnd |
|- ( z e. ZZ -> 1 e. CC ) |
42 |
38 40 41
|
addassd |
|- ( z e. ZZ -> ( ( ( z x. 8 ) + 6 ) + 1 ) = ( ( z x. 8 ) + ( 6 + 1 ) ) ) |
43 |
|
6p1e7 |
|- ( 6 + 1 ) = 7 |
44 |
43
|
a1i |
|- ( z e. ZZ -> ( 6 + 1 ) = 7 ) |
45 |
44
|
oveq2d |
|- ( z e. ZZ -> ( ( z x. 8 ) + ( 6 + 1 ) ) = ( ( z x. 8 ) + 7 ) ) |
46 |
42 45
|
eqtrd |
|- ( z e. ZZ -> ( ( ( z x. 8 ) + 6 ) + 1 ) = ( ( z x. 8 ) + 7 ) ) |
47 |
46
|
adantl |
|- ( ( N e. ZZ /\ z e. ZZ ) -> ( ( ( z x. 8 ) + 6 ) + 1 ) = ( ( z x. 8 ) + 7 ) ) |
48 |
32 47
|
eqtrd |
|- ( ( N e. ZZ /\ z e. ZZ ) -> ( ( 2 x. ( ( z x. 4 ) + 3 ) ) + 1 ) = ( ( z x. 8 ) + 7 ) ) |
49 |
48
|
oveq1d |
|- ( ( N e. ZZ /\ z e. ZZ ) -> ( ( ( 2 x. ( ( z x. 4 ) + 3 ) ) + 1 ) mod 8 ) = ( ( ( z x. 8 ) + 7 ) mod 8 ) ) |
50 |
|
nnrp |
|- ( 8 e. NN -> 8 e. RR+ ) |
51 |
34 50
|
mp1i |
|- ( z e. ZZ -> 8 e. RR+ ) |
52 |
|
0xr |
|- 0 e. RR* |
53 |
52
|
a1i |
|- ( z e. ZZ -> 0 e. RR* ) |
54 |
|
8re |
|- 8 e. RR |
55 |
54
|
rexri |
|- 8 e. RR* |
56 |
55
|
a1i |
|- ( z e. ZZ -> 8 e. RR* ) |
57 |
|
7re |
|- 7 e. RR |
58 |
57
|
rexri |
|- 7 e. RR* |
59 |
58
|
a1i |
|- ( z e. ZZ -> 7 e. RR* ) |
60 |
|
0re |
|- 0 e. RR |
61 |
|
7pos |
|- 0 < 7 |
62 |
60 57 61
|
ltleii |
|- 0 <_ 7 |
63 |
62
|
a1i |
|- ( z e. ZZ -> 0 <_ 7 ) |
64 |
|
7lt8 |
|- 7 < 8 |
65 |
64
|
a1i |
|- ( z e. ZZ -> 7 < 8 ) |
66 |
53 56 59 63 65
|
elicod |
|- ( z e. ZZ -> 7 e. ( 0 [,) 8 ) ) |
67 |
|
muladdmodid |
|- ( ( z e. ZZ /\ 8 e. RR+ /\ 7 e. ( 0 [,) 8 ) ) -> ( ( ( z x. 8 ) + 7 ) mod 8 ) = 7 ) |
68 |
51 66 67
|
mpd3an23 |
|- ( z e. ZZ -> ( ( ( z x. 8 ) + 7 ) mod 8 ) = 7 ) |
69 |
68
|
adantl |
|- ( ( N e. ZZ /\ z e. ZZ ) -> ( ( ( z x. 8 ) + 7 ) mod 8 ) = 7 ) |
70 |
49 69
|
eqtrd |
|- ( ( N e. ZZ /\ z e. ZZ ) -> ( ( ( 2 x. ( ( z x. 4 ) + 3 ) ) + 1 ) mod 8 ) = 7 ) |
71 |
|
oveq2 |
|- ( ( ( z x. 4 ) + 3 ) = N -> ( 2 x. ( ( z x. 4 ) + 3 ) ) = ( 2 x. N ) ) |
72 |
71
|
oveq1d |
|- ( ( ( z x. 4 ) + 3 ) = N -> ( ( 2 x. ( ( z x. 4 ) + 3 ) ) + 1 ) = ( ( 2 x. N ) + 1 ) ) |
73 |
72
|
oveq1d |
|- ( ( ( z x. 4 ) + 3 ) = N -> ( ( ( 2 x. ( ( z x. 4 ) + 3 ) ) + 1 ) mod 8 ) = ( ( ( 2 x. N ) + 1 ) mod 8 ) ) |
74 |
73
|
eqeq1d |
|- ( ( ( z x. 4 ) + 3 ) = N -> ( ( ( ( 2 x. ( ( z x. 4 ) + 3 ) ) + 1 ) mod 8 ) = 7 <-> ( ( ( 2 x. N ) + 1 ) mod 8 ) = 7 ) ) |
75 |
70 74
|
syl5ibcom |
|- ( ( N e. ZZ /\ z e. ZZ ) -> ( ( ( z x. 4 ) + 3 ) = N -> ( ( ( 2 x. N ) + 1 ) mod 8 ) = 7 ) ) |
76 |
75
|
rexlimdva |
|- ( N e. ZZ -> ( E. z e. ZZ ( ( z x. 4 ) + 3 ) = N -> ( ( ( 2 x. N ) + 1 ) mod 8 ) = 7 ) ) |
77 |
8 76
|
sylbid |
|- ( N e. ZZ -> ( ( N mod 4 ) = 3 -> ( ( ( 2 x. N ) + 1 ) mod 8 ) = 7 ) ) |
78 |
77
|
imp |
|- ( ( N e. ZZ /\ ( N mod 4 ) = 3 ) -> ( ( ( 2 x. N ) + 1 ) mod 8 ) = 7 ) |