Step |
Hyp |
Ref |
Expression |
1 |
|
oveq1 |
⊢ ( 𝑥 = 1 → ( 𝑥 · 𝐵 ) = ( 1 · 𝐵 ) ) |
2 |
|
oveq2 |
⊢ ( 𝑥 = 1 → ( 𝐵 · 𝑥 ) = ( 𝐵 · 1 ) ) |
3 |
1 2
|
eqeq12d |
⊢ ( 𝑥 = 1 → ( ( 𝑥 · 𝐵 ) = ( 𝐵 · 𝑥 ) ↔ ( 1 · 𝐵 ) = ( 𝐵 · 1 ) ) ) |
4 |
3
|
imbi2d |
⊢ ( 𝑥 = 1 → ( ( 𝐵 ∈ ℕ → ( 𝑥 · 𝐵 ) = ( 𝐵 · 𝑥 ) ) ↔ ( 𝐵 ∈ ℕ → ( 1 · 𝐵 ) = ( 𝐵 · 1 ) ) ) ) |
5 |
|
oveq1 |
⊢ ( 𝑥 = 𝑦 → ( 𝑥 · 𝐵 ) = ( 𝑦 · 𝐵 ) ) |
6 |
|
oveq2 |
⊢ ( 𝑥 = 𝑦 → ( 𝐵 · 𝑥 ) = ( 𝐵 · 𝑦 ) ) |
7 |
5 6
|
eqeq12d |
⊢ ( 𝑥 = 𝑦 → ( ( 𝑥 · 𝐵 ) = ( 𝐵 · 𝑥 ) ↔ ( 𝑦 · 𝐵 ) = ( 𝐵 · 𝑦 ) ) ) |
8 |
7
|
imbi2d |
⊢ ( 𝑥 = 𝑦 → ( ( 𝐵 ∈ ℕ → ( 𝑥 · 𝐵 ) = ( 𝐵 · 𝑥 ) ) ↔ ( 𝐵 ∈ ℕ → ( 𝑦 · 𝐵 ) = ( 𝐵 · 𝑦 ) ) ) ) |
9 |
|
oveq1 |
⊢ ( 𝑥 = ( 𝑦 + 1 ) → ( 𝑥 · 𝐵 ) = ( ( 𝑦 + 1 ) · 𝐵 ) ) |
10 |
|
oveq2 |
⊢ ( 𝑥 = ( 𝑦 + 1 ) → ( 𝐵 · 𝑥 ) = ( 𝐵 · ( 𝑦 + 1 ) ) ) |
11 |
9 10
|
eqeq12d |
⊢ ( 𝑥 = ( 𝑦 + 1 ) → ( ( 𝑥 · 𝐵 ) = ( 𝐵 · 𝑥 ) ↔ ( ( 𝑦 + 1 ) · 𝐵 ) = ( 𝐵 · ( 𝑦 + 1 ) ) ) ) |
12 |
11
|
imbi2d |
⊢ ( 𝑥 = ( 𝑦 + 1 ) → ( ( 𝐵 ∈ ℕ → ( 𝑥 · 𝐵 ) = ( 𝐵 · 𝑥 ) ) ↔ ( 𝐵 ∈ ℕ → ( ( 𝑦 + 1 ) · 𝐵 ) = ( 𝐵 · ( 𝑦 + 1 ) ) ) ) ) |
13 |
|
oveq1 |
⊢ ( 𝑥 = 𝐴 → ( 𝑥 · 𝐵 ) = ( 𝐴 · 𝐵 ) ) |
14 |
|
oveq2 |
⊢ ( 𝑥 = 𝐴 → ( 𝐵 · 𝑥 ) = ( 𝐵 · 𝐴 ) ) |
15 |
13 14
|
eqeq12d |
⊢ ( 𝑥 = 𝐴 → ( ( 𝑥 · 𝐵 ) = ( 𝐵 · 𝑥 ) ↔ ( 𝐴 · 𝐵 ) = ( 𝐵 · 𝐴 ) ) ) |
16 |
15
|
imbi2d |
⊢ ( 𝑥 = 𝐴 → ( ( 𝐵 ∈ ℕ → ( 𝑥 · 𝐵 ) = ( 𝐵 · 𝑥 ) ) ↔ ( 𝐵 ∈ ℕ → ( 𝐴 · 𝐵 ) = ( 𝐵 · 𝐴 ) ) ) ) |
17 |
|
nnmul1com |
⊢ ( 𝐵 ∈ ℕ → ( 1 · 𝐵 ) = ( 𝐵 · 1 ) ) |
18 |
|
simp3 |
⊢ ( ( 𝑦 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ ( 𝑦 · 𝐵 ) = ( 𝐵 · 𝑦 ) ) → ( 𝑦 · 𝐵 ) = ( 𝐵 · 𝑦 ) ) |
19 |
17
|
3ad2ant2 |
⊢ ( ( 𝑦 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ ( 𝑦 · 𝐵 ) = ( 𝐵 · 𝑦 ) ) → ( 1 · 𝐵 ) = ( 𝐵 · 1 ) ) |
20 |
18 19
|
oveq12d |
⊢ ( ( 𝑦 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ ( 𝑦 · 𝐵 ) = ( 𝐵 · 𝑦 ) ) → ( ( 𝑦 · 𝐵 ) + ( 1 · 𝐵 ) ) = ( ( 𝐵 · 𝑦 ) + ( 𝐵 · 1 ) ) ) |
21 |
|
simp1 |
⊢ ( ( 𝑦 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ ( 𝑦 · 𝐵 ) = ( 𝐵 · 𝑦 ) ) → 𝑦 ∈ ℕ ) |
22 |
|
1nn |
⊢ 1 ∈ ℕ |
23 |
22
|
a1i |
⊢ ( ( 𝑦 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ ( 𝑦 · 𝐵 ) = ( 𝐵 · 𝑦 ) ) → 1 ∈ ℕ ) |
24 |
|
simp2 |
⊢ ( ( 𝑦 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ ( 𝑦 · 𝐵 ) = ( 𝐵 · 𝑦 ) ) → 𝐵 ∈ ℕ ) |
25 |
|
nnadddir |
⊢ ( ( 𝑦 ∈ ℕ ∧ 1 ∈ ℕ ∧ 𝐵 ∈ ℕ ) → ( ( 𝑦 + 1 ) · 𝐵 ) = ( ( 𝑦 · 𝐵 ) + ( 1 · 𝐵 ) ) ) |
26 |
21 23 24 25
|
syl3anc |
⊢ ( ( 𝑦 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ ( 𝑦 · 𝐵 ) = ( 𝐵 · 𝑦 ) ) → ( ( 𝑦 + 1 ) · 𝐵 ) = ( ( 𝑦 · 𝐵 ) + ( 1 · 𝐵 ) ) ) |
27 |
24
|
nncnd |
⊢ ( ( 𝑦 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ ( 𝑦 · 𝐵 ) = ( 𝐵 · 𝑦 ) ) → 𝐵 ∈ ℂ ) |
28 |
21
|
nncnd |
⊢ ( ( 𝑦 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ ( 𝑦 · 𝐵 ) = ( 𝐵 · 𝑦 ) ) → 𝑦 ∈ ℂ ) |
29 |
|
1cnd |
⊢ ( ( 𝑦 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ ( 𝑦 · 𝐵 ) = ( 𝐵 · 𝑦 ) ) → 1 ∈ ℂ ) |
30 |
27 28 29
|
adddid |
⊢ ( ( 𝑦 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ ( 𝑦 · 𝐵 ) = ( 𝐵 · 𝑦 ) ) → ( 𝐵 · ( 𝑦 + 1 ) ) = ( ( 𝐵 · 𝑦 ) + ( 𝐵 · 1 ) ) ) |
31 |
20 26 30
|
3eqtr4d |
⊢ ( ( 𝑦 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ ( 𝑦 · 𝐵 ) = ( 𝐵 · 𝑦 ) ) → ( ( 𝑦 + 1 ) · 𝐵 ) = ( 𝐵 · ( 𝑦 + 1 ) ) ) |
32 |
31
|
3exp |
⊢ ( 𝑦 ∈ ℕ → ( 𝐵 ∈ ℕ → ( ( 𝑦 · 𝐵 ) = ( 𝐵 · 𝑦 ) → ( ( 𝑦 + 1 ) · 𝐵 ) = ( 𝐵 · ( 𝑦 + 1 ) ) ) ) ) |
33 |
32
|
a2d |
⊢ ( 𝑦 ∈ ℕ → ( ( 𝐵 ∈ ℕ → ( 𝑦 · 𝐵 ) = ( 𝐵 · 𝑦 ) ) → ( 𝐵 ∈ ℕ → ( ( 𝑦 + 1 ) · 𝐵 ) = ( 𝐵 · ( 𝑦 + 1 ) ) ) ) ) |
34 |
4 8 12 16 17 33
|
nnind |
⊢ ( 𝐴 ∈ ℕ → ( 𝐵 ∈ ℕ → ( 𝐴 · 𝐵 ) = ( 𝐵 · 𝐴 ) ) ) |
35 |
34
|
imp |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ) → ( 𝐴 · 𝐵 ) = ( 𝐵 · 𝐴 ) ) |