Step |
Hyp |
Ref |
Expression |
1 |
|
numma.1 |
|- T e. NN0 |
2 |
|
numma.2 |
|- A e. NN0 |
3 |
|
numma.3 |
|- B e. NN0 |
4 |
|
numma.4 |
|- C e. NN0 |
5 |
|
numma.5 |
|- D e. NN0 |
6 |
|
numma.6 |
|- M = ( ( T x. A ) + B ) |
7 |
|
numma.7 |
|- N = ( ( T x. C ) + D ) |
8 |
|
numma2c.8 |
|- P e. NN0 |
9 |
|
numma2c.9 |
|- F e. NN0 |
10 |
|
numma2c.10 |
|- G e. NN0 |
11 |
|
numma2c.11 |
|- ( ( P x. A ) + ( C + G ) ) = E |
12 |
|
numma2c.12 |
|- ( ( P x. B ) + D ) = ( ( T x. G ) + F ) |
13 |
8
|
nn0cni |
|- P e. CC |
14 |
1 2 3
|
numcl |
|- ( ( T x. A ) + B ) e. NN0 |
15 |
6 14
|
eqeltri |
|- M e. NN0 |
16 |
15
|
nn0cni |
|- M e. CC |
17 |
13 16
|
mulcomi |
|- ( P x. M ) = ( M x. P ) |
18 |
17
|
oveq1i |
|- ( ( P x. M ) + N ) = ( ( M x. P ) + N ) |
19 |
2
|
nn0cni |
|- A e. CC |
20 |
19 13
|
mulcomi |
|- ( A x. P ) = ( P x. A ) |
21 |
20
|
oveq1i |
|- ( ( A x. P ) + ( C + G ) ) = ( ( P x. A ) + ( C + G ) ) |
22 |
21 11
|
eqtri |
|- ( ( A x. P ) + ( C + G ) ) = E |
23 |
3
|
nn0cni |
|- B e. CC |
24 |
23 13
|
mulcomi |
|- ( B x. P ) = ( P x. B ) |
25 |
24
|
oveq1i |
|- ( ( B x. P ) + D ) = ( ( P x. B ) + D ) |
26 |
25 12
|
eqtri |
|- ( ( B x. P ) + D ) = ( ( T x. G ) + F ) |
27 |
1 2 3 4 5 6 7 8 9 10 22 26
|
nummac |
|- ( ( M x. P ) + N ) = ( ( T x. E ) + F ) |
28 |
18 27
|
eqtri |
|- ( ( P x. M ) + N ) = ( ( T x. E ) + F ) |