Step |
Hyp |
Ref |
Expression |
1 |
|
problem4.1 |
|- A e. CC |
2 |
|
problem4.2 |
|- B e. CC |
3 |
|
problem4.3 |
|- ( A + B ) = 3 |
4 |
|
problem4.4 |
|- ( ( 3 x. A ) + ( 2 x. B ) ) = 7 |
5 |
|
7re |
|- 7 e. RR |
6 |
5
|
recni |
|- 7 e. CC |
7 |
|
6re |
|- 6 e. RR |
8 |
7
|
recni |
|- 6 e. CC |
9 |
|
ax-1cn |
|- 1 e. CC |
10 |
|
df-7 |
|- 7 = ( 6 + 1 ) |
11 |
10
|
eqcomi |
|- ( 6 + 1 ) = 7 |
12 |
6 8 9 11
|
subaddrii |
|- ( 7 - 6 ) = 1 |
13 |
12
|
eqcomi |
|- 1 = ( 7 - 6 ) |
14 |
|
3cn |
|- 3 e. CC |
15 |
|
2cn |
|- 2 e. CC |
16 |
|
df-3 |
|- 3 = ( 2 + 1 ) |
17 |
16
|
eqcomi |
|- ( 2 + 1 ) = 3 |
18 |
14 15 9 17
|
subaddrii |
|- ( 3 - 2 ) = 1 |
19 |
18
|
oveq1i |
|- ( ( 3 - 2 ) x. A ) = ( 1 x. A ) |
20 |
1
|
mulid2i |
|- ( 1 x. A ) = A |
21 |
19 20
|
eqtri |
|- ( ( 3 - 2 ) x. A ) = A |
22 |
21
|
eqcomi |
|- A = ( ( 3 - 2 ) x. A ) |
23 |
14 15 1
|
subdiri |
|- ( ( 3 - 2 ) x. A ) = ( ( 3 x. A ) - ( 2 x. A ) ) |
24 |
22 23
|
eqtri |
|- A = ( ( 3 x. A ) - ( 2 x. A ) ) |
25 |
24
|
oveq1i |
|- ( A + 6 ) = ( ( ( 3 x. A ) - ( 2 x. A ) ) + 6 ) |
26 |
14 1
|
mulcli |
|- ( 3 x. A ) e. CC |
27 |
15 1
|
mulcli |
|- ( 2 x. A ) e. CC |
28 |
|
subadd23 |
|- ( ( ( 3 x. A ) e. CC /\ ( 2 x. A ) e. CC /\ 6 e. CC ) -> ( ( ( 3 x. A ) - ( 2 x. A ) ) + 6 ) = ( ( 3 x. A ) + ( 6 - ( 2 x. A ) ) ) ) |
29 |
26 27 8 28
|
mp3an |
|- ( ( ( 3 x. A ) - ( 2 x. A ) ) + 6 ) = ( ( 3 x. A ) + ( 6 - ( 2 x. A ) ) ) |
30 |
|
3t2e6 |
|- ( 3 x. 2 ) = 6 |
31 |
1 15
|
mulcomi |
|- ( A x. 2 ) = ( 2 x. A ) |
32 |
30 31
|
oveq12i |
|- ( ( 3 x. 2 ) - ( A x. 2 ) ) = ( 6 - ( 2 x. A ) ) |
33 |
32
|
eqcomi |
|- ( 6 - ( 2 x. A ) ) = ( ( 3 x. 2 ) - ( A x. 2 ) ) |
34 |
14 1 15
|
subdiri |
|- ( ( 3 - A ) x. 2 ) = ( ( 3 x. 2 ) - ( A x. 2 ) ) |
35 |
34
|
eqcomi |
|- ( ( 3 x. 2 ) - ( A x. 2 ) ) = ( ( 3 - A ) x. 2 ) |
36 |
14 1
|
subcli |
|- ( 3 - A ) e. CC |
37 |
15 36
|
mulcomi |
|- ( 2 x. ( 3 - A ) ) = ( ( 3 - A ) x. 2 ) |
38 |
37
|
eqcomi |
|- ( ( 3 - A ) x. 2 ) = ( 2 x. ( 3 - A ) ) |
39 |
14 1 2 3
|
subaddrii |
|- ( 3 - A ) = B |
40 |
39
|
eqcomi |
|- B = ( 3 - A ) |
41 |
40
|
oveq2i |
|- ( 2 x. B ) = ( 2 x. ( 3 - A ) ) |
42 |
41
|
eqcomi |
|- ( 2 x. ( 3 - A ) ) = ( 2 x. B ) |
43 |
38 42
|
eqtri |
|- ( ( 3 - A ) x. 2 ) = ( 2 x. B ) |
44 |
35 43
|
eqtri |
|- ( ( 3 x. 2 ) - ( A x. 2 ) ) = ( 2 x. B ) |
45 |
33 44
|
eqtri |
|- ( 6 - ( 2 x. A ) ) = ( 2 x. B ) |
46 |
45
|
eqcomi |
|- ( 2 x. B ) = ( 6 - ( 2 x. A ) ) |
47 |
46
|
oveq2i |
|- ( ( 3 x. A ) + ( 2 x. B ) ) = ( ( 3 x. A ) + ( 6 - ( 2 x. A ) ) ) |
48 |
47
|
eqcomi |
|- ( ( 3 x. A ) + ( 6 - ( 2 x. A ) ) ) = ( ( 3 x. A ) + ( 2 x. B ) ) |
49 |
29 48
|
eqtri |
|- ( ( ( 3 x. A ) - ( 2 x. A ) ) + 6 ) = ( ( 3 x. A ) + ( 2 x. B ) ) |
50 |
25 49
|
eqtri |
|- ( A + 6 ) = ( ( 3 x. A ) + ( 2 x. B ) ) |
51 |
50 4
|
eqtri |
|- ( A + 6 ) = 7 |
52 |
6 8 1
|
subadd2i |
|- ( ( 7 - 6 ) = A <-> ( A + 6 ) = 7 ) |
53 |
52
|
biimpri |
|- ( ( A + 6 ) = 7 -> ( 7 - 6 ) = A ) |
54 |
51 53
|
ax-mp |
|- ( 7 - 6 ) = A |
55 |
13 54
|
eqtri |
|- 1 = A |
56 |
55
|
eqcomi |
|- A = 1 |
57 |
56
|
oveq2i |
|- ( 3 - A ) = ( 3 - 1 ) |
58 |
14 9 15
|
subadd2i |
|- ( ( 3 - 1 ) = 2 <-> ( 2 + 1 ) = 3 ) |
59 |
58
|
biimpri |
|- ( ( 2 + 1 ) = 3 -> ( 3 - 1 ) = 2 ) |
60 |
17 59
|
ax-mp |
|- ( 3 - 1 ) = 2 |
61 |
57 60
|
eqtri |
|- ( 3 - A ) = 2 |
62 |
40 61
|
eqtri |
|- B = 2 |
63 |
56 62
|
pm3.2i |
|- ( A = 1 /\ B = 2 ) |