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 |
|
numaddc.8 |
|- F e. NN0 |
9 |
|
numaddc.9 |
|- ( ( A + C ) + 1 ) = E |
10 |
|
numaddc.10 |
|- ( B + D ) = ( ( T x. 1 ) + F ) |
11 |
1 2 3
|
numcl |
|- ( ( T x. A ) + B ) e. NN0 |
12 |
6 11
|
eqeltri |
|- M e. NN0 |
13 |
12
|
nn0cni |
|- M e. CC |
14 |
13
|
mulid1i |
|- ( M x. 1 ) = M |
15 |
14
|
oveq1i |
|- ( ( M x. 1 ) + N ) = ( M + N ) |
16 |
|
1nn0 |
|- 1 e. NN0 |
17 |
2
|
nn0cni |
|- A e. CC |
18 |
17
|
mulid1i |
|- ( A x. 1 ) = A |
19 |
18
|
oveq1i |
|- ( ( A x. 1 ) + ( C + 1 ) ) = ( A + ( C + 1 ) ) |
20 |
4
|
nn0cni |
|- C e. CC |
21 |
|
ax-1cn |
|- 1 e. CC |
22 |
17 20 21
|
addassi |
|- ( ( A + C ) + 1 ) = ( A + ( C + 1 ) ) |
23 |
19 22 9
|
3eqtr2i |
|- ( ( A x. 1 ) + ( C + 1 ) ) = E |
24 |
3
|
nn0cni |
|- B e. CC |
25 |
24
|
mulid1i |
|- ( B x. 1 ) = B |
26 |
25
|
oveq1i |
|- ( ( B x. 1 ) + D ) = ( B + D ) |
27 |
26 10
|
eqtri |
|- ( ( B x. 1 ) + D ) = ( ( T x. 1 ) + F ) |
28 |
1 2 3 4 5 6 7 16 8 16 23 27
|
nummac |
|- ( ( M x. 1 ) + N ) = ( ( T x. E ) + F ) |
29 |
15 28
|
eqtr3i |
|- ( M + N ) = ( ( T x. E ) + F ) |