Step |
Hyp |
Ref |
Expression |
1 |
|
elnn0 |
⊢ ( 𝐵 ∈ ℕ0 ↔ ( 𝐵 ∈ ℕ ∨ 𝐵 = 0 ) ) |
2 |
|
renegneg |
⊢ ( 𝐴 ∈ ℝ → ( 0 −ℝ ( 0 −ℝ 𝐴 ) ) = 𝐴 ) |
3 |
2
|
oveq1d |
⊢ ( 𝐴 ∈ ℝ → ( ( 0 −ℝ ( 0 −ℝ 𝐴 ) ) · 𝐵 ) = ( 𝐴 · 𝐵 ) ) |
4 |
3
|
ad2antrr |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ ( 0 −ℝ 𝐴 ) ∈ ℕ ) ∧ 𝐵 ∈ ℕ ) → ( ( 0 −ℝ ( 0 −ℝ 𝐴 ) ) · 𝐵 ) = ( 𝐴 · 𝐵 ) ) |
5 |
|
rernegcl |
⊢ ( 𝐴 ∈ ℝ → ( 0 −ℝ 𝐴 ) ∈ ℝ ) |
6 |
5
|
ad2antrr |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ ( 0 −ℝ 𝐴 ) ∈ ℕ ) ∧ 𝐵 ∈ ℕ ) → ( 0 −ℝ 𝐴 ) ∈ ℝ ) |
7 |
|
simpr |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ ( 0 −ℝ 𝐴 ) ∈ ℕ ) ∧ 𝐵 ∈ ℕ ) → 𝐵 ∈ ℕ ) |
8 |
6 7
|
renegmulnnass |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ ( 0 −ℝ 𝐴 ) ∈ ℕ ) ∧ 𝐵 ∈ ℕ ) → ( ( 0 −ℝ ( 0 −ℝ 𝐴 ) ) · 𝐵 ) = ( 0 −ℝ ( ( 0 −ℝ 𝐴 ) · 𝐵 ) ) ) |
9 |
|
nnmulcom |
⊢ ( ( ( 0 −ℝ 𝐴 ) ∈ ℕ ∧ 𝐵 ∈ ℕ ) → ( ( 0 −ℝ 𝐴 ) · 𝐵 ) = ( 𝐵 · ( 0 −ℝ 𝐴 ) ) ) |
10 |
9
|
adantll |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ ( 0 −ℝ 𝐴 ) ∈ ℕ ) ∧ 𝐵 ∈ ℕ ) → ( ( 0 −ℝ 𝐴 ) · 𝐵 ) = ( 𝐵 · ( 0 −ℝ 𝐴 ) ) ) |
11 |
10
|
oveq2d |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ ( 0 −ℝ 𝐴 ) ∈ ℕ ) ∧ 𝐵 ∈ ℕ ) → ( 0 −ℝ ( ( 0 −ℝ 𝐴 ) · 𝐵 ) ) = ( 0 −ℝ ( 𝐵 · ( 0 −ℝ 𝐴 ) ) ) ) |
12 |
|
nnre |
⊢ ( 𝐵 ∈ ℕ → 𝐵 ∈ ℝ ) |
13 |
12
|
adantl |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ ( 0 −ℝ 𝐴 ) ∈ ℕ ) ∧ 𝐵 ∈ ℕ ) → 𝐵 ∈ ℝ ) |
14 |
|
0red |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ ( 0 −ℝ 𝐴 ) ∈ ℕ ) ∧ 𝐵 ∈ ℕ ) → 0 ∈ ℝ ) |
15 |
|
resubdi |
⊢ ( ( 𝐵 ∈ ℝ ∧ 0 ∈ ℝ ∧ ( 0 −ℝ 𝐴 ) ∈ ℝ ) → ( 𝐵 · ( 0 −ℝ ( 0 −ℝ 𝐴 ) ) ) = ( ( 𝐵 · 0 ) −ℝ ( 𝐵 · ( 0 −ℝ 𝐴 ) ) ) ) |
16 |
13 14 6 15
|
syl3anc |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ ( 0 −ℝ 𝐴 ) ∈ ℕ ) ∧ 𝐵 ∈ ℕ ) → ( 𝐵 · ( 0 −ℝ ( 0 −ℝ 𝐴 ) ) ) = ( ( 𝐵 · 0 ) −ℝ ( 𝐵 · ( 0 −ℝ 𝐴 ) ) ) ) |
17 |
|
remul01 |
⊢ ( 𝐵 ∈ ℝ → ( 𝐵 · 0 ) = 0 ) |
18 |
12 17
|
syl |
⊢ ( 𝐵 ∈ ℕ → ( 𝐵 · 0 ) = 0 ) |
19 |
18
|
adantl |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ ( 0 −ℝ 𝐴 ) ∈ ℕ ) ∧ 𝐵 ∈ ℕ ) → ( 𝐵 · 0 ) = 0 ) |
20 |
19
|
oveq1d |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ ( 0 −ℝ 𝐴 ) ∈ ℕ ) ∧ 𝐵 ∈ ℕ ) → ( ( 𝐵 · 0 ) −ℝ ( 𝐵 · ( 0 −ℝ 𝐴 ) ) ) = ( 0 −ℝ ( 𝐵 · ( 0 −ℝ 𝐴 ) ) ) ) |
21 |
16 20
|
eqtrd |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ ( 0 −ℝ 𝐴 ) ∈ ℕ ) ∧ 𝐵 ∈ ℕ ) → ( 𝐵 · ( 0 −ℝ ( 0 −ℝ 𝐴 ) ) ) = ( 0 −ℝ ( 𝐵 · ( 0 −ℝ 𝐴 ) ) ) ) |
22 |
2
|
ad2antrr |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ ( 0 −ℝ 𝐴 ) ∈ ℕ ) ∧ 𝐵 ∈ ℕ ) → ( 0 −ℝ ( 0 −ℝ 𝐴 ) ) = 𝐴 ) |
23 |
22
|
oveq2d |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ ( 0 −ℝ 𝐴 ) ∈ ℕ ) ∧ 𝐵 ∈ ℕ ) → ( 𝐵 · ( 0 −ℝ ( 0 −ℝ 𝐴 ) ) ) = ( 𝐵 · 𝐴 ) ) |
24 |
11 21 23
|
3eqtr2d |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ ( 0 −ℝ 𝐴 ) ∈ ℕ ) ∧ 𝐵 ∈ ℕ ) → ( 0 −ℝ ( ( 0 −ℝ 𝐴 ) · 𝐵 ) ) = ( 𝐵 · 𝐴 ) ) |
25 |
8 4 24
|
3eqtr3d |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ ( 0 −ℝ 𝐴 ) ∈ ℕ ) ∧ 𝐵 ∈ ℕ ) → ( 𝐴 · 𝐵 ) = ( 𝐵 · 𝐴 ) ) |
26 |
4 4 25
|
3eqtr3d |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ ( 0 −ℝ 𝐴 ) ∈ ℕ ) ∧ 𝐵 ∈ ℕ ) → ( 𝐴 · 𝐵 ) = ( 𝐵 · 𝐴 ) ) |
27 |
|
remul01 |
⊢ ( 𝐴 ∈ ℝ → ( 𝐴 · 0 ) = 0 ) |
28 |
|
remul02 |
⊢ ( 𝐴 ∈ ℝ → ( 0 · 𝐴 ) = 0 ) |
29 |
27 28
|
eqtr4d |
⊢ ( 𝐴 ∈ ℝ → ( 𝐴 · 0 ) = ( 0 · 𝐴 ) ) |
30 |
29
|
adantr |
⊢ ( ( 𝐴 ∈ ℝ ∧ ( 0 −ℝ 𝐴 ) ∈ ℕ ) → ( 𝐴 · 0 ) = ( 0 · 𝐴 ) ) |
31 |
|
oveq2 |
⊢ ( 𝐵 = 0 → ( 𝐴 · 𝐵 ) = ( 𝐴 · 0 ) ) |
32 |
|
oveq1 |
⊢ ( 𝐵 = 0 → ( 𝐵 · 𝐴 ) = ( 0 · 𝐴 ) ) |
33 |
31 32
|
eqeq12d |
⊢ ( 𝐵 = 0 → ( ( 𝐴 · 𝐵 ) = ( 𝐵 · 𝐴 ) ↔ ( 𝐴 · 0 ) = ( 0 · 𝐴 ) ) ) |
34 |
30 33
|
syl5ibrcom |
⊢ ( ( 𝐴 ∈ ℝ ∧ ( 0 −ℝ 𝐴 ) ∈ ℕ ) → ( 𝐵 = 0 → ( 𝐴 · 𝐵 ) = ( 𝐵 · 𝐴 ) ) ) |
35 |
34
|
imp |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ ( 0 −ℝ 𝐴 ) ∈ ℕ ) ∧ 𝐵 = 0 ) → ( 𝐴 · 𝐵 ) = ( 𝐵 · 𝐴 ) ) |
36 |
26 35
|
jaodan |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ ( 0 −ℝ 𝐴 ) ∈ ℕ ) ∧ ( 𝐵 ∈ ℕ ∨ 𝐵 = 0 ) ) → ( 𝐴 · 𝐵 ) = ( 𝐵 · 𝐴 ) ) |
37 |
1 36
|
sylan2b |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ ( 0 −ℝ 𝐴 ) ∈ ℕ ) ∧ 𝐵 ∈ ℕ0 ) → ( 𝐴 · 𝐵 ) = ( 𝐵 · 𝐴 ) ) |