Step |
Hyp |
Ref |
Expression |
1 |
|
elnn0 |
⊢ ( 𝐵 ∈ ℕ0 ↔ ( 𝐵 ∈ ℕ ∨ 𝐵 = 0 ) ) |
2 |
|
elnn0 |
⊢ ( 𝐴 ∈ ℕ0 ↔ ( 𝐴 ∈ ℕ ∨ 𝐴 = 0 ) ) |
3 |
|
nnmulcom |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ) → ( 𝐴 · 𝐵 ) = ( 𝐵 · 𝐴 ) ) |
4 |
|
nnre |
⊢ ( 𝐵 ∈ ℕ → 𝐵 ∈ ℝ ) |
5 |
|
remul02 |
⊢ ( 𝐵 ∈ ℝ → ( 0 · 𝐵 ) = 0 ) |
6 |
|
remul01 |
⊢ ( 𝐵 ∈ ℝ → ( 𝐵 · 0 ) = 0 ) |
7 |
5 6
|
eqtr4d |
⊢ ( 𝐵 ∈ ℝ → ( 0 · 𝐵 ) = ( 𝐵 · 0 ) ) |
8 |
4 7
|
syl |
⊢ ( 𝐵 ∈ ℕ → ( 0 · 𝐵 ) = ( 𝐵 · 0 ) ) |
9 |
|
oveq1 |
⊢ ( 𝐴 = 0 → ( 𝐴 · 𝐵 ) = ( 0 · 𝐵 ) ) |
10 |
|
oveq2 |
⊢ ( 𝐴 = 0 → ( 𝐵 · 𝐴 ) = ( 𝐵 · 0 ) ) |
11 |
9 10
|
eqeq12d |
⊢ ( 𝐴 = 0 → ( ( 𝐴 · 𝐵 ) = ( 𝐵 · 𝐴 ) ↔ ( 0 · 𝐵 ) = ( 𝐵 · 0 ) ) ) |
12 |
8 11
|
syl5ibrcom |
⊢ ( 𝐵 ∈ ℕ → ( 𝐴 = 0 → ( 𝐴 · 𝐵 ) = ( 𝐵 · 𝐴 ) ) ) |
13 |
12
|
impcom |
⊢ ( ( 𝐴 = 0 ∧ 𝐵 ∈ ℕ ) → ( 𝐴 · 𝐵 ) = ( 𝐵 · 𝐴 ) ) |
14 |
3 13
|
jaoian |
⊢ ( ( ( 𝐴 ∈ ℕ ∨ 𝐴 = 0 ) ∧ 𝐵 ∈ ℕ ) → ( 𝐴 · 𝐵 ) = ( 𝐵 · 𝐴 ) ) |
15 |
2 14
|
sylanb |
⊢ ( ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ) → ( 𝐴 · 𝐵 ) = ( 𝐵 · 𝐴 ) ) |
16 |
|
nn0re |
⊢ ( 𝐴 ∈ ℕ0 → 𝐴 ∈ ℝ ) |
17 |
|
remul01 |
⊢ ( 𝐴 ∈ ℝ → ( 𝐴 · 0 ) = 0 ) |
18 |
|
remul02 |
⊢ ( 𝐴 ∈ ℝ → ( 0 · 𝐴 ) = 0 ) |
19 |
17 18
|
eqtr4d |
⊢ ( 𝐴 ∈ ℝ → ( 𝐴 · 0 ) = ( 0 · 𝐴 ) ) |
20 |
16 19
|
syl |
⊢ ( 𝐴 ∈ ℕ0 → ( 𝐴 · 0 ) = ( 0 · 𝐴 ) ) |
21 |
|
oveq2 |
⊢ ( 𝐵 = 0 → ( 𝐴 · 𝐵 ) = ( 𝐴 · 0 ) ) |
22 |
|
oveq1 |
⊢ ( 𝐵 = 0 → ( 𝐵 · 𝐴 ) = ( 0 · 𝐴 ) ) |
23 |
21 22
|
eqeq12d |
⊢ ( 𝐵 = 0 → ( ( 𝐴 · 𝐵 ) = ( 𝐵 · 𝐴 ) ↔ ( 𝐴 · 0 ) = ( 0 · 𝐴 ) ) ) |
24 |
20 23
|
syl5ibrcom |
⊢ ( 𝐴 ∈ ℕ0 → ( 𝐵 = 0 → ( 𝐴 · 𝐵 ) = ( 𝐵 · 𝐴 ) ) ) |
25 |
24
|
imp |
⊢ ( ( 𝐴 ∈ ℕ0 ∧ 𝐵 = 0 ) → ( 𝐴 · 𝐵 ) = ( 𝐵 · 𝐴 ) ) |
26 |
15 25
|
jaodan |
⊢ ( ( 𝐴 ∈ ℕ0 ∧ ( 𝐵 ∈ ℕ ∨ 𝐵 = 0 ) ) → ( 𝐴 · 𝐵 ) = ( 𝐵 · 𝐴 ) ) |
27 |
1 26
|
sylan2b |
⊢ ( ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) → ( 𝐴 · 𝐵 ) = ( 𝐵 · 𝐴 ) ) |