Step |
Hyp |
Ref |
Expression |
1 |
|
problem4.1 |
⊢ 𝐴 ∈ ℂ |
2 |
|
problem4.2 |
⊢ 𝐵 ∈ ℂ |
3 |
|
problem4.3 |
⊢ ( 𝐴 + 𝐵 ) = 3 |
4 |
|
problem4.4 |
⊢ ( ( 3 · 𝐴 ) + ( 2 · 𝐵 ) ) = 7 |
5 |
|
7re |
⊢ 7 ∈ ℝ |
6 |
5
|
recni |
⊢ 7 ∈ ℂ |
7 |
|
6re |
⊢ 6 ∈ ℝ |
8 |
7
|
recni |
⊢ 6 ∈ ℂ |
9 |
|
ax-1cn |
⊢ 1 ∈ ℂ |
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 ∈ ℂ |
15 |
|
2cn |
⊢ 2 ∈ ℂ |
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 ) · 𝐴 ) = ( 1 · 𝐴 ) |
20 |
1
|
mulid2i |
⊢ ( 1 · 𝐴 ) = 𝐴 |
21 |
19 20
|
eqtri |
⊢ ( ( 3 − 2 ) · 𝐴 ) = 𝐴 |
22 |
21
|
eqcomi |
⊢ 𝐴 = ( ( 3 − 2 ) · 𝐴 ) |
23 |
14 15 1
|
subdiri |
⊢ ( ( 3 − 2 ) · 𝐴 ) = ( ( 3 · 𝐴 ) − ( 2 · 𝐴 ) ) |
24 |
22 23
|
eqtri |
⊢ 𝐴 = ( ( 3 · 𝐴 ) − ( 2 · 𝐴 ) ) |
25 |
24
|
oveq1i |
⊢ ( 𝐴 + 6 ) = ( ( ( 3 · 𝐴 ) − ( 2 · 𝐴 ) ) + 6 ) |
26 |
14 1
|
mulcli |
⊢ ( 3 · 𝐴 ) ∈ ℂ |
27 |
15 1
|
mulcli |
⊢ ( 2 · 𝐴 ) ∈ ℂ |
28 |
|
subadd23 |
⊢ ( ( ( 3 · 𝐴 ) ∈ ℂ ∧ ( 2 · 𝐴 ) ∈ ℂ ∧ 6 ∈ ℂ ) → ( ( ( 3 · 𝐴 ) − ( 2 · 𝐴 ) ) + 6 ) = ( ( 3 · 𝐴 ) + ( 6 − ( 2 · 𝐴 ) ) ) ) |
29 |
26 27 8 28
|
mp3an |
⊢ ( ( ( 3 · 𝐴 ) − ( 2 · 𝐴 ) ) + 6 ) = ( ( 3 · 𝐴 ) + ( 6 − ( 2 · 𝐴 ) ) ) |
30 |
|
3t2e6 |
⊢ ( 3 · 2 ) = 6 |
31 |
1 15
|
mulcomi |
⊢ ( 𝐴 · 2 ) = ( 2 · 𝐴 ) |
32 |
30 31
|
oveq12i |
⊢ ( ( 3 · 2 ) − ( 𝐴 · 2 ) ) = ( 6 − ( 2 · 𝐴 ) ) |
33 |
32
|
eqcomi |
⊢ ( 6 − ( 2 · 𝐴 ) ) = ( ( 3 · 2 ) − ( 𝐴 · 2 ) ) |
34 |
14 1 15
|
subdiri |
⊢ ( ( 3 − 𝐴 ) · 2 ) = ( ( 3 · 2 ) − ( 𝐴 · 2 ) ) |
35 |
34
|
eqcomi |
⊢ ( ( 3 · 2 ) − ( 𝐴 · 2 ) ) = ( ( 3 − 𝐴 ) · 2 ) |
36 |
14 1
|
subcli |
⊢ ( 3 − 𝐴 ) ∈ ℂ |
37 |
15 36
|
mulcomi |
⊢ ( 2 · ( 3 − 𝐴 ) ) = ( ( 3 − 𝐴 ) · 2 ) |
38 |
37
|
eqcomi |
⊢ ( ( 3 − 𝐴 ) · 2 ) = ( 2 · ( 3 − 𝐴 ) ) |
39 |
14 1 2 3
|
subaddrii |
⊢ ( 3 − 𝐴 ) = 𝐵 |
40 |
39
|
eqcomi |
⊢ 𝐵 = ( 3 − 𝐴 ) |
41 |
40
|
oveq2i |
⊢ ( 2 · 𝐵 ) = ( 2 · ( 3 − 𝐴 ) ) |
42 |
41
|
eqcomi |
⊢ ( 2 · ( 3 − 𝐴 ) ) = ( 2 · 𝐵 ) |
43 |
38 42
|
eqtri |
⊢ ( ( 3 − 𝐴 ) · 2 ) = ( 2 · 𝐵 ) |
44 |
35 43
|
eqtri |
⊢ ( ( 3 · 2 ) − ( 𝐴 · 2 ) ) = ( 2 · 𝐵 ) |
45 |
33 44
|
eqtri |
⊢ ( 6 − ( 2 · 𝐴 ) ) = ( 2 · 𝐵 ) |
46 |
45
|
eqcomi |
⊢ ( 2 · 𝐵 ) = ( 6 − ( 2 · 𝐴 ) ) |
47 |
46
|
oveq2i |
⊢ ( ( 3 · 𝐴 ) + ( 2 · 𝐵 ) ) = ( ( 3 · 𝐴 ) + ( 6 − ( 2 · 𝐴 ) ) ) |
48 |
47
|
eqcomi |
⊢ ( ( 3 · 𝐴 ) + ( 6 − ( 2 · 𝐴 ) ) ) = ( ( 3 · 𝐴 ) + ( 2 · 𝐵 ) ) |
49 |
29 48
|
eqtri |
⊢ ( ( ( 3 · 𝐴 ) − ( 2 · 𝐴 ) ) + 6 ) = ( ( 3 · 𝐴 ) + ( 2 · 𝐵 ) ) |
50 |
25 49
|
eqtri |
⊢ ( 𝐴 + 6 ) = ( ( 3 · 𝐴 ) + ( 2 · 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 ) |