Step |
Hyp |
Ref |
Expression |
1 |
|
oveq2 |
⊢ ( 𝑥 = 1 → ( ( 𝐴 + 𝐵 ) · 𝑥 ) = ( ( 𝐴 + 𝐵 ) · 1 ) ) |
2 |
|
oveq2 |
⊢ ( 𝑥 = 1 → ( 𝐴 · 𝑥 ) = ( 𝐴 · 1 ) ) |
3 |
|
oveq2 |
⊢ ( 𝑥 = 1 → ( 𝐵 · 𝑥 ) = ( 𝐵 · 1 ) ) |
4 |
2 3
|
oveq12d |
⊢ ( 𝑥 = 1 → ( ( 𝐴 · 𝑥 ) + ( 𝐵 · 𝑥 ) ) = ( ( 𝐴 · 1 ) + ( 𝐵 · 1 ) ) ) |
5 |
1 4
|
eqeq12d |
⊢ ( 𝑥 = 1 → ( ( ( 𝐴 + 𝐵 ) · 𝑥 ) = ( ( 𝐴 · 𝑥 ) + ( 𝐵 · 𝑥 ) ) ↔ ( ( 𝐴 + 𝐵 ) · 1 ) = ( ( 𝐴 · 1 ) + ( 𝐵 · 1 ) ) ) ) |
6 |
5
|
imbi2d |
⊢ ( 𝑥 = 1 → ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ) → ( ( 𝐴 + 𝐵 ) · 𝑥 ) = ( ( 𝐴 · 𝑥 ) + ( 𝐵 · 𝑥 ) ) ) ↔ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ) → ( ( 𝐴 + 𝐵 ) · 1 ) = ( ( 𝐴 · 1 ) + ( 𝐵 · 1 ) ) ) ) ) |
7 |
|
oveq2 |
⊢ ( 𝑥 = 𝑦 → ( ( 𝐴 + 𝐵 ) · 𝑥 ) = ( ( 𝐴 + 𝐵 ) · 𝑦 ) ) |
8 |
|
oveq2 |
⊢ ( 𝑥 = 𝑦 → ( 𝐴 · 𝑥 ) = ( 𝐴 · 𝑦 ) ) |
9 |
|
oveq2 |
⊢ ( 𝑥 = 𝑦 → ( 𝐵 · 𝑥 ) = ( 𝐵 · 𝑦 ) ) |
10 |
8 9
|
oveq12d |
⊢ ( 𝑥 = 𝑦 → ( ( 𝐴 · 𝑥 ) + ( 𝐵 · 𝑥 ) ) = ( ( 𝐴 · 𝑦 ) + ( 𝐵 · 𝑦 ) ) ) |
11 |
7 10
|
eqeq12d |
⊢ ( 𝑥 = 𝑦 → ( ( ( 𝐴 + 𝐵 ) · 𝑥 ) = ( ( 𝐴 · 𝑥 ) + ( 𝐵 · 𝑥 ) ) ↔ ( ( 𝐴 + 𝐵 ) · 𝑦 ) = ( ( 𝐴 · 𝑦 ) + ( 𝐵 · 𝑦 ) ) ) ) |
12 |
11
|
imbi2d |
⊢ ( 𝑥 = 𝑦 → ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ) → ( ( 𝐴 + 𝐵 ) · 𝑥 ) = ( ( 𝐴 · 𝑥 ) + ( 𝐵 · 𝑥 ) ) ) ↔ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ) → ( ( 𝐴 + 𝐵 ) · 𝑦 ) = ( ( 𝐴 · 𝑦 ) + ( 𝐵 · 𝑦 ) ) ) ) ) |
13 |
|
oveq2 |
⊢ ( 𝑥 = ( 𝑦 + 1 ) → ( ( 𝐴 + 𝐵 ) · 𝑥 ) = ( ( 𝐴 + 𝐵 ) · ( 𝑦 + 1 ) ) ) |
14 |
|
oveq2 |
⊢ ( 𝑥 = ( 𝑦 + 1 ) → ( 𝐴 · 𝑥 ) = ( 𝐴 · ( 𝑦 + 1 ) ) ) |
15 |
|
oveq2 |
⊢ ( 𝑥 = ( 𝑦 + 1 ) → ( 𝐵 · 𝑥 ) = ( 𝐵 · ( 𝑦 + 1 ) ) ) |
16 |
14 15
|
oveq12d |
⊢ ( 𝑥 = ( 𝑦 + 1 ) → ( ( 𝐴 · 𝑥 ) + ( 𝐵 · 𝑥 ) ) = ( ( 𝐴 · ( 𝑦 + 1 ) ) + ( 𝐵 · ( 𝑦 + 1 ) ) ) ) |
17 |
13 16
|
eqeq12d |
⊢ ( 𝑥 = ( 𝑦 + 1 ) → ( ( ( 𝐴 + 𝐵 ) · 𝑥 ) = ( ( 𝐴 · 𝑥 ) + ( 𝐵 · 𝑥 ) ) ↔ ( ( 𝐴 + 𝐵 ) · ( 𝑦 + 1 ) ) = ( ( 𝐴 · ( 𝑦 + 1 ) ) + ( 𝐵 · ( 𝑦 + 1 ) ) ) ) ) |
18 |
17
|
imbi2d |
⊢ ( 𝑥 = ( 𝑦 + 1 ) → ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ) → ( ( 𝐴 + 𝐵 ) · 𝑥 ) = ( ( 𝐴 · 𝑥 ) + ( 𝐵 · 𝑥 ) ) ) ↔ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ) → ( ( 𝐴 + 𝐵 ) · ( 𝑦 + 1 ) ) = ( ( 𝐴 · ( 𝑦 + 1 ) ) + ( 𝐵 · ( 𝑦 + 1 ) ) ) ) ) ) |
19 |
|
oveq2 |
⊢ ( 𝑥 = 𝐶 → ( ( 𝐴 + 𝐵 ) · 𝑥 ) = ( ( 𝐴 + 𝐵 ) · 𝐶 ) ) |
20 |
|
oveq2 |
⊢ ( 𝑥 = 𝐶 → ( 𝐴 · 𝑥 ) = ( 𝐴 · 𝐶 ) ) |
21 |
|
oveq2 |
⊢ ( 𝑥 = 𝐶 → ( 𝐵 · 𝑥 ) = ( 𝐵 · 𝐶 ) ) |
22 |
20 21
|
oveq12d |
⊢ ( 𝑥 = 𝐶 → ( ( 𝐴 · 𝑥 ) + ( 𝐵 · 𝑥 ) ) = ( ( 𝐴 · 𝐶 ) + ( 𝐵 · 𝐶 ) ) ) |
23 |
19 22
|
eqeq12d |
⊢ ( 𝑥 = 𝐶 → ( ( ( 𝐴 + 𝐵 ) · 𝑥 ) = ( ( 𝐴 · 𝑥 ) + ( 𝐵 · 𝑥 ) ) ↔ ( ( 𝐴 + 𝐵 ) · 𝐶 ) = ( ( 𝐴 · 𝐶 ) + ( 𝐵 · 𝐶 ) ) ) ) |
24 |
23
|
imbi2d |
⊢ ( 𝑥 = 𝐶 → ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ) → ( ( 𝐴 + 𝐵 ) · 𝑥 ) = ( ( 𝐴 · 𝑥 ) + ( 𝐵 · 𝑥 ) ) ) ↔ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ) → ( ( 𝐴 + 𝐵 ) · 𝐶 ) = ( ( 𝐴 · 𝐶 ) + ( 𝐵 · 𝐶 ) ) ) ) ) |
25 |
|
nnaddcl |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ) → ( 𝐴 + 𝐵 ) ∈ ℕ ) |
26 |
25
|
nnred |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ) → ( 𝐴 + 𝐵 ) ∈ ℝ ) |
27 |
|
ax-1rid |
⊢ ( ( 𝐴 + 𝐵 ) ∈ ℝ → ( ( 𝐴 + 𝐵 ) · 1 ) = ( 𝐴 + 𝐵 ) ) |
28 |
26 27
|
syl |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ) → ( ( 𝐴 + 𝐵 ) · 1 ) = ( 𝐴 + 𝐵 ) ) |
29 |
|
nnre |
⊢ ( 𝐴 ∈ ℕ → 𝐴 ∈ ℝ ) |
30 |
|
ax-1rid |
⊢ ( 𝐴 ∈ ℝ → ( 𝐴 · 1 ) = 𝐴 ) |
31 |
29 30
|
syl |
⊢ ( 𝐴 ∈ ℕ → ( 𝐴 · 1 ) = 𝐴 ) |
32 |
|
nnre |
⊢ ( 𝐵 ∈ ℕ → 𝐵 ∈ ℝ ) |
33 |
|
ax-1rid |
⊢ ( 𝐵 ∈ ℝ → ( 𝐵 · 1 ) = 𝐵 ) |
34 |
32 33
|
syl |
⊢ ( 𝐵 ∈ ℕ → ( 𝐵 · 1 ) = 𝐵 ) |
35 |
31 34
|
oveqan12d |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ) → ( ( 𝐴 · 1 ) + ( 𝐵 · 1 ) ) = ( 𝐴 + 𝐵 ) ) |
36 |
28 35
|
eqtr4d |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ) → ( ( 𝐴 + 𝐵 ) · 1 ) = ( ( 𝐴 · 1 ) + ( 𝐵 · 1 ) ) ) |
37 |
|
simp2l |
⊢ ( ( 𝑦 ∈ ℕ ∧ ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ) ∧ ( ( 𝐴 + 𝐵 ) · 𝑦 ) = ( ( 𝐴 · 𝑦 ) + ( 𝐵 · 𝑦 ) ) ) → 𝐴 ∈ ℕ ) |
38 |
|
simp2r |
⊢ ( ( 𝑦 ∈ ℕ ∧ ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ) ∧ ( ( 𝐴 + 𝐵 ) · 𝑦 ) = ( ( 𝐴 · 𝑦 ) + ( 𝐵 · 𝑦 ) ) ) → 𝐵 ∈ ℕ ) |
39 |
37 38
|
nnaddcld |
⊢ ( ( 𝑦 ∈ ℕ ∧ ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ) ∧ ( ( 𝐴 + 𝐵 ) · 𝑦 ) = ( ( 𝐴 · 𝑦 ) + ( 𝐵 · 𝑦 ) ) ) → ( 𝐴 + 𝐵 ) ∈ ℕ ) |
40 |
39
|
nncnd |
⊢ ( ( 𝑦 ∈ ℕ ∧ ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ) ∧ ( ( 𝐴 + 𝐵 ) · 𝑦 ) = ( ( 𝐴 · 𝑦 ) + ( 𝐵 · 𝑦 ) ) ) → ( 𝐴 + 𝐵 ) ∈ ℂ ) |
41 |
|
simp1 |
⊢ ( ( 𝑦 ∈ ℕ ∧ ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ) ∧ ( ( 𝐴 + 𝐵 ) · 𝑦 ) = ( ( 𝐴 · 𝑦 ) + ( 𝐵 · 𝑦 ) ) ) → 𝑦 ∈ ℕ ) |
42 |
41
|
nncnd |
⊢ ( ( 𝑦 ∈ ℕ ∧ ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ) ∧ ( ( 𝐴 + 𝐵 ) · 𝑦 ) = ( ( 𝐴 · 𝑦 ) + ( 𝐵 · 𝑦 ) ) ) → 𝑦 ∈ ℂ ) |
43 |
|
1cnd |
⊢ ( ( 𝑦 ∈ ℕ ∧ ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ) ∧ ( ( 𝐴 + 𝐵 ) · 𝑦 ) = ( ( 𝐴 · 𝑦 ) + ( 𝐵 · 𝑦 ) ) ) → 1 ∈ ℂ ) |
44 |
40 42 43
|
adddid |
⊢ ( ( 𝑦 ∈ ℕ ∧ ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ) ∧ ( ( 𝐴 + 𝐵 ) · 𝑦 ) = ( ( 𝐴 · 𝑦 ) + ( 𝐵 · 𝑦 ) ) ) → ( ( 𝐴 + 𝐵 ) · ( 𝑦 + 1 ) ) = ( ( ( 𝐴 + 𝐵 ) · 𝑦 ) + ( ( 𝐴 + 𝐵 ) · 1 ) ) ) |
45 |
37
|
nnred |
⊢ ( ( 𝑦 ∈ ℕ ∧ ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ) ∧ ( ( 𝐴 + 𝐵 ) · 𝑦 ) = ( ( 𝐴 · 𝑦 ) + ( 𝐵 · 𝑦 ) ) ) → 𝐴 ∈ ℝ ) |
46 |
45 30
|
syl |
⊢ ( ( 𝑦 ∈ ℕ ∧ ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ) ∧ ( ( 𝐴 + 𝐵 ) · 𝑦 ) = ( ( 𝐴 · 𝑦 ) + ( 𝐵 · 𝑦 ) ) ) → ( 𝐴 · 1 ) = 𝐴 ) |
47 |
46
|
oveq2d |
⊢ ( ( 𝑦 ∈ ℕ ∧ ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ) ∧ ( ( 𝐴 + 𝐵 ) · 𝑦 ) = ( ( 𝐴 · 𝑦 ) + ( 𝐵 · 𝑦 ) ) ) → ( ( 𝐴 · 𝑦 ) + ( 𝐴 · 1 ) ) = ( ( 𝐴 · 𝑦 ) + 𝐴 ) ) |
48 |
38
|
nnred |
⊢ ( ( 𝑦 ∈ ℕ ∧ ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ) ∧ ( ( 𝐴 + 𝐵 ) · 𝑦 ) = ( ( 𝐴 · 𝑦 ) + ( 𝐵 · 𝑦 ) ) ) → 𝐵 ∈ ℝ ) |
49 |
48 33
|
syl |
⊢ ( ( 𝑦 ∈ ℕ ∧ ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ) ∧ ( ( 𝐴 + 𝐵 ) · 𝑦 ) = ( ( 𝐴 · 𝑦 ) + ( 𝐵 · 𝑦 ) ) ) → ( 𝐵 · 1 ) = 𝐵 ) |
50 |
49
|
oveq2d |
⊢ ( ( 𝑦 ∈ ℕ ∧ ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ) ∧ ( ( 𝐴 + 𝐵 ) · 𝑦 ) = ( ( 𝐴 · 𝑦 ) + ( 𝐵 · 𝑦 ) ) ) → ( ( 𝐵 · 𝑦 ) + ( 𝐵 · 1 ) ) = ( ( 𝐵 · 𝑦 ) + 𝐵 ) ) |
51 |
47 50
|
oveq12d |
⊢ ( ( 𝑦 ∈ ℕ ∧ ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ) ∧ ( ( 𝐴 + 𝐵 ) · 𝑦 ) = ( ( 𝐴 · 𝑦 ) + ( 𝐵 · 𝑦 ) ) ) → ( ( ( 𝐴 · 𝑦 ) + ( 𝐴 · 1 ) ) + ( ( 𝐵 · 𝑦 ) + ( 𝐵 · 1 ) ) ) = ( ( ( 𝐴 · 𝑦 ) + 𝐴 ) + ( ( 𝐵 · 𝑦 ) + 𝐵 ) ) ) |
52 |
37 41
|
nnmulcld |
⊢ ( ( 𝑦 ∈ ℕ ∧ ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ) ∧ ( ( 𝐴 + 𝐵 ) · 𝑦 ) = ( ( 𝐴 · 𝑦 ) + ( 𝐵 · 𝑦 ) ) ) → ( 𝐴 · 𝑦 ) ∈ ℕ ) |
53 |
52
|
nncnd |
⊢ ( ( 𝑦 ∈ ℕ ∧ ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ) ∧ ( ( 𝐴 + 𝐵 ) · 𝑦 ) = ( ( 𝐴 · 𝑦 ) + ( 𝐵 · 𝑦 ) ) ) → ( 𝐴 · 𝑦 ) ∈ ℂ ) |
54 |
37
|
nncnd |
⊢ ( ( 𝑦 ∈ ℕ ∧ ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ) ∧ ( ( 𝐴 + 𝐵 ) · 𝑦 ) = ( ( 𝐴 · 𝑦 ) + ( 𝐵 · 𝑦 ) ) ) → 𝐴 ∈ ℂ ) |
55 |
38 41
|
nnmulcld |
⊢ ( ( 𝑦 ∈ ℕ ∧ ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ) ∧ ( ( 𝐴 + 𝐵 ) · 𝑦 ) = ( ( 𝐴 · 𝑦 ) + ( 𝐵 · 𝑦 ) ) ) → ( 𝐵 · 𝑦 ) ∈ ℕ ) |
56 |
55 38
|
nnaddcld |
⊢ ( ( 𝑦 ∈ ℕ ∧ ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ) ∧ ( ( 𝐴 + 𝐵 ) · 𝑦 ) = ( ( 𝐴 · 𝑦 ) + ( 𝐵 · 𝑦 ) ) ) → ( ( 𝐵 · 𝑦 ) + 𝐵 ) ∈ ℕ ) |
57 |
56
|
nncnd |
⊢ ( ( 𝑦 ∈ ℕ ∧ ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ) ∧ ( ( 𝐴 + 𝐵 ) · 𝑦 ) = ( ( 𝐴 · 𝑦 ) + ( 𝐵 · 𝑦 ) ) ) → ( ( 𝐵 · 𝑦 ) + 𝐵 ) ∈ ℂ ) |
58 |
53 54 57
|
addassd |
⊢ ( ( 𝑦 ∈ ℕ ∧ ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ) ∧ ( ( 𝐴 + 𝐵 ) · 𝑦 ) = ( ( 𝐴 · 𝑦 ) + ( 𝐵 · 𝑦 ) ) ) → ( ( ( 𝐴 · 𝑦 ) + 𝐴 ) + ( ( 𝐵 · 𝑦 ) + 𝐵 ) ) = ( ( 𝐴 · 𝑦 ) + ( 𝐴 + ( ( 𝐵 · 𝑦 ) + 𝐵 ) ) ) ) |
59 |
55
|
nncnd |
⊢ ( ( 𝑦 ∈ ℕ ∧ ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ) ∧ ( ( 𝐴 + 𝐵 ) · 𝑦 ) = ( ( 𝐴 · 𝑦 ) + ( 𝐵 · 𝑦 ) ) ) → ( 𝐵 · 𝑦 ) ∈ ℂ ) |
60 |
38
|
nncnd |
⊢ ( ( 𝑦 ∈ ℕ ∧ ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ) ∧ ( ( 𝐴 + 𝐵 ) · 𝑦 ) = ( ( 𝐴 · 𝑦 ) + ( 𝐵 · 𝑦 ) ) ) → 𝐵 ∈ ℂ ) |
61 |
54 59 60
|
addassd |
⊢ ( ( 𝑦 ∈ ℕ ∧ ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ) ∧ ( ( 𝐴 + 𝐵 ) · 𝑦 ) = ( ( 𝐴 · 𝑦 ) + ( 𝐵 · 𝑦 ) ) ) → ( ( 𝐴 + ( 𝐵 · 𝑦 ) ) + 𝐵 ) = ( 𝐴 + ( ( 𝐵 · 𝑦 ) + 𝐵 ) ) ) |
62 |
61
|
oveq2d |
⊢ ( ( 𝑦 ∈ ℕ ∧ ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ) ∧ ( ( 𝐴 + 𝐵 ) · 𝑦 ) = ( ( 𝐴 · 𝑦 ) + ( 𝐵 · 𝑦 ) ) ) → ( ( 𝐴 · 𝑦 ) + ( ( 𝐴 + ( 𝐵 · 𝑦 ) ) + 𝐵 ) ) = ( ( 𝐴 · 𝑦 ) + ( 𝐴 + ( ( 𝐵 · 𝑦 ) + 𝐵 ) ) ) ) |
63 |
59 54 60
|
addassd |
⊢ ( ( 𝑦 ∈ ℕ ∧ ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ) ∧ ( ( 𝐴 + 𝐵 ) · 𝑦 ) = ( ( 𝐴 · 𝑦 ) + ( 𝐵 · 𝑦 ) ) ) → ( ( ( 𝐵 · 𝑦 ) + 𝐴 ) + 𝐵 ) = ( ( 𝐵 · 𝑦 ) + ( 𝐴 + 𝐵 ) ) ) |
64 |
63
|
oveq2d |
⊢ ( ( 𝑦 ∈ ℕ ∧ ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ) ∧ ( ( 𝐴 + 𝐵 ) · 𝑦 ) = ( ( 𝐴 · 𝑦 ) + ( 𝐵 · 𝑦 ) ) ) → ( ( 𝐴 · 𝑦 ) + ( ( ( 𝐵 · 𝑦 ) + 𝐴 ) + 𝐵 ) ) = ( ( 𝐴 · 𝑦 ) + ( ( 𝐵 · 𝑦 ) + ( 𝐴 + 𝐵 ) ) ) ) |
65 |
|
nnaddcom |
⊢ ( ( 𝐴 ∈ ℕ ∧ ( 𝐵 · 𝑦 ) ∈ ℕ ) → ( 𝐴 + ( 𝐵 · 𝑦 ) ) = ( ( 𝐵 · 𝑦 ) + 𝐴 ) ) |
66 |
37 55 65
|
syl2anc |
⊢ ( ( 𝑦 ∈ ℕ ∧ ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ) ∧ ( ( 𝐴 + 𝐵 ) · 𝑦 ) = ( ( 𝐴 · 𝑦 ) + ( 𝐵 · 𝑦 ) ) ) → ( 𝐴 + ( 𝐵 · 𝑦 ) ) = ( ( 𝐵 · 𝑦 ) + 𝐴 ) ) |
67 |
66
|
oveq1d |
⊢ ( ( 𝑦 ∈ ℕ ∧ ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ) ∧ ( ( 𝐴 + 𝐵 ) · 𝑦 ) = ( ( 𝐴 · 𝑦 ) + ( 𝐵 · 𝑦 ) ) ) → ( ( 𝐴 + ( 𝐵 · 𝑦 ) ) + 𝐵 ) = ( ( ( 𝐵 · 𝑦 ) + 𝐴 ) + 𝐵 ) ) |
68 |
67
|
oveq2d |
⊢ ( ( 𝑦 ∈ ℕ ∧ ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ) ∧ ( ( 𝐴 + 𝐵 ) · 𝑦 ) = ( ( 𝐴 · 𝑦 ) + ( 𝐵 · 𝑦 ) ) ) → ( ( 𝐴 · 𝑦 ) + ( ( 𝐴 + ( 𝐵 · 𝑦 ) ) + 𝐵 ) ) = ( ( 𝐴 · 𝑦 ) + ( ( ( 𝐵 · 𝑦 ) + 𝐴 ) + 𝐵 ) ) ) |
69 |
53 59 40
|
addassd |
⊢ ( ( 𝑦 ∈ ℕ ∧ ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ) ∧ ( ( 𝐴 + 𝐵 ) · 𝑦 ) = ( ( 𝐴 · 𝑦 ) + ( 𝐵 · 𝑦 ) ) ) → ( ( ( 𝐴 · 𝑦 ) + ( 𝐵 · 𝑦 ) ) + ( 𝐴 + 𝐵 ) ) = ( ( 𝐴 · 𝑦 ) + ( ( 𝐵 · 𝑦 ) + ( 𝐴 + 𝐵 ) ) ) ) |
70 |
64 68 69
|
3eqtr4d |
⊢ ( ( 𝑦 ∈ ℕ ∧ ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ) ∧ ( ( 𝐴 + 𝐵 ) · 𝑦 ) = ( ( 𝐴 · 𝑦 ) + ( 𝐵 · 𝑦 ) ) ) → ( ( 𝐴 · 𝑦 ) + ( ( 𝐴 + ( 𝐵 · 𝑦 ) ) + 𝐵 ) ) = ( ( ( 𝐴 · 𝑦 ) + ( 𝐵 · 𝑦 ) ) + ( 𝐴 + 𝐵 ) ) ) |
71 |
58 62 70
|
3eqtr2d |
⊢ ( ( 𝑦 ∈ ℕ ∧ ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ) ∧ ( ( 𝐴 + 𝐵 ) · 𝑦 ) = ( ( 𝐴 · 𝑦 ) + ( 𝐵 · 𝑦 ) ) ) → ( ( ( 𝐴 · 𝑦 ) + 𝐴 ) + ( ( 𝐵 · 𝑦 ) + 𝐵 ) ) = ( ( ( 𝐴 · 𝑦 ) + ( 𝐵 · 𝑦 ) ) + ( 𝐴 + 𝐵 ) ) ) |
72 |
51 71
|
eqtrd |
⊢ ( ( 𝑦 ∈ ℕ ∧ ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ) ∧ ( ( 𝐴 + 𝐵 ) · 𝑦 ) = ( ( 𝐴 · 𝑦 ) + ( 𝐵 · 𝑦 ) ) ) → ( ( ( 𝐴 · 𝑦 ) + ( 𝐴 · 1 ) ) + ( ( 𝐵 · 𝑦 ) + ( 𝐵 · 1 ) ) ) = ( ( ( 𝐴 · 𝑦 ) + ( 𝐵 · 𝑦 ) ) + ( 𝐴 + 𝐵 ) ) ) |
73 |
54 42 43
|
adddid |
⊢ ( ( 𝑦 ∈ ℕ ∧ ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ) ∧ ( ( 𝐴 + 𝐵 ) · 𝑦 ) = ( ( 𝐴 · 𝑦 ) + ( 𝐵 · 𝑦 ) ) ) → ( 𝐴 · ( 𝑦 + 1 ) ) = ( ( 𝐴 · 𝑦 ) + ( 𝐴 · 1 ) ) ) |
74 |
60 42 43
|
adddid |
⊢ ( ( 𝑦 ∈ ℕ ∧ ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ) ∧ ( ( 𝐴 + 𝐵 ) · 𝑦 ) = ( ( 𝐴 · 𝑦 ) + ( 𝐵 · 𝑦 ) ) ) → ( 𝐵 · ( 𝑦 + 1 ) ) = ( ( 𝐵 · 𝑦 ) + ( 𝐵 · 1 ) ) ) |
75 |
73 74
|
oveq12d |
⊢ ( ( 𝑦 ∈ ℕ ∧ ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ) ∧ ( ( 𝐴 + 𝐵 ) · 𝑦 ) = ( ( 𝐴 · 𝑦 ) + ( 𝐵 · 𝑦 ) ) ) → ( ( 𝐴 · ( 𝑦 + 1 ) ) + ( 𝐵 · ( 𝑦 + 1 ) ) ) = ( ( ( 𝐴 · 𝑦 ) + ( 𝐴 · 1 ) ) + ( ( 𝐵 · 𝑦 ) + ( 𝐵 · 1 ) ) ) ) |
76 |
|
simp3 |
⊢ ( ( 𝑦 ∈ ℕ ∧ ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ) ∧ ( ( 𝐴 + 𝐵 ) · 𝑦 ) = ( ( 𝐴 · 𝑦 ) + ( 𝐵 · 𝑦 ) ) ) → ( ( 𝐴 + 𝐵 ) · 𝑦 ) = ( ( 𝐴 · 𝑦 ) + ( 𝐵 · 𝑦 ) ) ) |
77 |
39
|
nnred |
⊢ ( ( 𝑦 ∈ ℕ ∧ ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ) ∧ ( ( 𝐴 + 𝐵 ) · 𝑦 ) = ( ( 𝐴 · 𝑦 ) + ( 𝐵 · 𝑦 ) ) ) → ( 𝐴 + 𝐵 ) ∈ ℝ ) |
78 |
77 27
|
syl |
⊢ ( ( 𝑦 ∈ ℕ ∧ ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ) ∧ ( ( 𝐴 + 𝐵 ) · 𝑦 ) = ( ( 𝐴 · 𝑦 ) + ( 𝐵 · 𝑦 ) ) ) → ( ( 𝐴 + 𝐵 ) · 1 ) = ( 𝐴 + 𝐵 ) ) |
79 |
76 78
|
oveq12d |
⊢ ( ( 𝑦 ∈ ℕ ∧ ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ) ∧ ( ( 𝐴 + 𝐵 ) · 𝑦 ) = ( ( 𝐴 · 𝑦 ) + ( 𝐵 · 𝑦 ) ) ) → ( ( ( 𝐴 + 𝐵 ) · 𝑦 ) + ( ( 𝐴 + 𝐵 ) · 1 ) ) = ( ( ( 𝐴 · 𝑦 ) + ( 𝐵 · 𝑦 ) ) + ( 𝐴 + 𝐵 ) ) ) |
80 |
72 75 79
|
3eqtr4d |
⊢ ( ( 𝑦 ∈ ℕ ∧ ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ) ∧ ( ( 𝐴 + 𝐵 ) · 𝑦 ) = ( ( 𝐴 · 𝑦 ) + ( 𝐵 · 𝑦 ) ) ) → ( ( 𝐴 · ( 𝑦 + 1 ) ) + ( 𝐵 · ( 𝑦 + 1 ) ) ) = ( ( ( 𝐴 + 𝐵 ) · 𝑦 ) + ( ( 𝐴 + 𝐵 ) · 1 ) ) ) |
81 |
44 80
|
eqtr4d |
⊢ ( ( 𝑦 ∈ ℕ ∧ ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ) ∧ ( ( 𝐴 + 𝐵 ) · 𝑦 ) = ( ( 𝐴 · 𝑦 ) + ( 𝐵 · 𝑦 ) ) ) → ( ( 𝐴 + 𝐵 ) · ( 𝑦 + 1 ) ) = ( ( 𝐴 · ( 𝑦 + 1 ) ) + ( 𝐵 · ( 𝑦 + 1 ) ) ) ) |
82 |
81
|
3exp |
⊢ ( 𝑦 ∈ ℕ → ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ) → ( ( ( 𝐴 + 𝐵 ) · 𝑦 ) = ( ( 𝐴 · 𝑦 ) + ( 𝐵 · 𝑦 ) ) → ( ( 𝐴 + 𝐵 ) · ( 𝑦 + 1 ) ) = ( ( 𝐴 · ( 𝑦 + 1 ) ) + ( 𝐵 · ( 𝑦 + 1 ) ) ) ) ) ) |
83 |
82
|
a2d |
⊢ ( 𝑦 ∈ ℕ → ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ) → ( ( 𝐴 + 𝐵 ) · 𝑦 ) = ( ( 𝐴 · 𝑦 ) + ( 𝐵 · 𝑦 ) ) ) → ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ) → ( ( 𝐴 + 𝐵 ) · ( 𝑦 + 1 ) ) = ( ( 𝐴 · ( 𝑦 + 1 ) ) + ( 𝐵 · ( 𝑦 + 1 ) ) ) ) ) ) |
84 |
6 12 18 24 36 83
|
nnind |
⊢ ( 𝐶 ∈ ℕ → ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ) → ( ( 𝐴 + 𝐵 ) · 𝐶 ) = ( ( 𝐴 · 𝐶 ) + ( 𝐵 · 𝐶 ) ) ) ) |
85 |
84
|
com12 |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ) → ( 𝐶 ∈ ℕ → ( ( 𝐴 + 𝐵 ) · 𝐶 ) = ( ( 𝐴 · 𝐶 ) + ( 𝐵 · 𝐶 ) ) ) ) |
86 |
85
|
3impia |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ ) → ( ( 𝐴 + 𝐵 ) · 𝐶 ) = ( ( 𝐴 · 𝐶 ) + ( 𝐵 · 𝐶 ) ) ) |