Step |
Hyp |
Ref |
Expression |
1 |
|
simpr |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ ( 0 −ℝ 𝐴 ) ∈ ℕ ) ∧ 𝐵 ∈ ℕ0 ) → 𝐵 ∈ ℕ0 ) |
2 |
1
|
nn0cnd |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ ( 0 −ℝ 𝐴 ) ∈ ℕ ) ∧ 𝐵 ∈ ℕ0 ) → 𝐵 ∈ ℂ ) |
3 |
|
rernegcl |
⊢ ( 𝐴 ∈ ℝ → ( 0 −ℝ 𝐴 ) ∈ ℝ ) |
4 |
3
|
ad2antrr |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ ( 0 −ℝ 𝐴 ) ∈ ℕ ) ∧ 𝐵 ∈ ℕ0 ) → ( 0 −ℝ 𝐴 ) ∈ ℝ ) |
5 |
4
|
recnd |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ ( 0 −ℝ 𝐴 ) ∈ ℕ ) ∧ 𝐵 ∈ ℕ0 ) → ( 0 −ℝ 𝐴 ) ∈ ℂ ) |
6 |
|
simpll |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ ( 0 −ℝ 𝐴 ) ∈ ℕ ) ∧ 𝐵 ∈ ℕ0 ) → 𝐴 ∈ ℝ ) |
7 |
6
|
recnd |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ ( 0 −ℝ 𝐴 ) ∈ ℕ ) ∧ 𝐵 ∈ ℕ0 ) → 𝐴 ∈ ℂ ) |
8 |
2 5 7
|
addassd |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ ( 0 −ℝ 𝐴 ) ∈ ℕ ) ∧ 𝐵 ∈ ℕ0 ) → ( ( 𝐵 + ( 0 −ℝ 𝐴 ) ) + 𝐴 ) = ( 𝐵 + ( ( 0 −ℝ 𝐴 ) + 𝐴 ) ) ) |
9 |
|
renegid2 |
⊢ ( 𝐴 ∈ ℝ → ( ( 0 −ℝ 𝐴 ) + 𝐴 ) = 0 ) |
10 |
9
|
ad2antrr |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ ( 0 −ℝ 𝐴 ) ∈ ℕ ) ∧ 𝐵 ∈ ℕ0 ) → ( ( 0 −ℝ 𝐴 ) + 𝐴 ) = 0 ) |
11 |
10
|
oveq2d |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ ( 0 −ℝ 𝐴 ) ∈ ℕ ) ∧ 𝐵 ∈ ℕ0 ) → ( 𝐵 + ( ( 0 −ℝ 𝐴 ) + 𝐴 ) ) = ( 𝐵 + 0 ) ) |
12 |
|
nn0re |
⊢ ( 𝐵 ∈ ℕ0 → 𝐵 ∈ ℝ ) |
13 |
|
readdrid |
⊢ ( 𝐵 ∈ ℝ → ( 𝐵 + 0 ) = 𝐵 ) |
14 |
12 13
|
syl |
⊢ ( 𝐵 ∈ ℕ0 → ( 𝐵 + 0 ) = 𝐵 ) |
15 |
14
|
adantl |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ ( 0 −ℝ 𝐴 ) ∈ ℕ ) ∧ 𝐵 ∈ ℕ0 ) → ( 𝐵 + 0 ) = 𝐵 ) |
16 |
8 11 15
|
3eqtrrd |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ ( 0 −ℝ 𝐴 ) ∈ ℕ ) ∧ 𝐵 ∈ ℕ0 ) → 𝐵 = ( ( 𝐵 + ( 0 −ℝ 𝐴 ) ) + 𝐴 ) ) |
17 |
9
|
oveq1d |
⊢ ( 𝐴 ∈ ℝ → ( ( ( 0 −ℝ 𝐴 ) + 𝐴 ) + 𝐵 ) = ( 0 + 𝐵 ) ) |
18 |
17
|
adantr |
⊢ ( ( 𝐴 ∈ ℝ ∧ ( 0 −ℝ 𝐴 ) ∈ ℕ ) → ( ( ( 0 −ℝ 𝐴 ) + 𝐴 ) + 𝐵 ) = ( 0 + 𝐵 ) ) |
19 |
|
readdlid |
⊢ ( 𝐵 ∈ ℝ → ( 0 + 𝐵 ) = 𝐵 ) |
20 |
12 19
|
syl |
⊢ ( 𝐵 ∈ ℕ0 → ( 0 + 𝐵 ) = 𝐵 ) |
21 |
18 20
|
sylan9eq |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ ( 0 −ℝ 𝐴 ) ∈ ℕ ) ∧ 𝐵 ∈ ℕ0 ) → ( ( ( 0 −ℝ 𝐴 ) + 𝐴 ) + 𝐵 ) = 𝐵 ) |
22 |
|
nnnn0 |
⊢ ( ( 0 −ℝ 𝐴 ) ∈ ℕ → ( 0 −ℝ 𝐴 ) ∈ ℕ0 ) |
23 |
|
nn0addcom |
⊢ ( ( ( 0 −ℝ 𝐴 ) ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) → ( ( 0 −ℝ 𝐴 ) + 𝐵 ) = ( 𝐵 + ( 0 −ℝ 𝐴 ) ) ) |
24 |
22 23
|
sylan |
⊢ ( ( ( 0 −ℝ 𝐴 ) ∈ ℕ ∧ 𝐵 ∈ ℕ0 ) → ( ( 0 −ℝ 𝐴 ) + 𝐵 ) = ( 𝐵 + ( 0 −ℝ 𝐴 ) ) ) |
25 |
24
|
adantll |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ ( 0 −ℝ 𝐴 ) ∈ ℕ ) ∧ 𝐵 ∈ ℕ0 ) → ( ( 0 −ℝ 𝐴 ) + 𝐵 ) = ( 𝐵 + ( 0 −ℝ 𝐴 ) ) ) |
26 |
25
|
oveq1d |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ ( 0 −ℝ 𝐴 ) ∈ ℕ ) ∧ 𝐵 ∈ ℕ0 ) → ( ( ( 0 −ℝ 𝐴 ) + 𝐵 ) + 𝐴 ) = ( ( 𝐵 + ( 0 −ℝ 𝐴 ) ) + 𝐴 ) ) |
27 |
16 21 26
|
3eqtr4d |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ ( 0 −ℝ 𝐴 ) ∈ ℕ ) ∧ 𝐵 ∈ ℕ0 ) → ( ( ( 0 −ℝ 𝐴 ) + 𝐴 ) + 𝐵 ) = ( ( ( 0 −ℝ 𝐴 ) + 𝐵 ) + 𝐴 ) ) |
28 |
5 7 2
|
addassd |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ ( 0 −ℝ 𝐴 ) ∈ ℕ ) ∧ 𝐵 ∈ ℕ0 ) → ( ( ( 0 −ℝ 𝐴 ) + 𝐴 ) + 𝐵 ) = ( ( 0 −ℝ 𝐴 ) + ( 𝐴 + 𝐵 ) ) ) |
29 |
5 2 7
|
addassd |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ ( 0 −ℝ 𝐴 ) ∈ ℕ ) ∧ 𝐵 ∈ ℕ0 ) → ( ( ( 0 −ℝ 𝐴 ) + 𝐵 ) + 𝐴 ) = ( ( 0 −ℝ 𝐴 ) + ( 𝐵 + 𝐴 ) ) ) |
30 |
27 28 29
|
3eqtr3d |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ ( 0 −ℝ 𝐴 ) ∈ ℕ ) ∧ 𝐵 ∈ ℕ0 ) → ( ( 0 −ℝ 𝐴 ) + ( 𝐴 + 𝐵 ) ) = ( ( 0 −ℝ 𝐴 ) + ( 𝐵 + 𝐴 ) ) ) |
31 |
7 2
|
addcld |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ ( 0 −ℝ 𝐴 ) ∈ ℕ ) ∧ 𝐵 ∈ ℕ0 ) → ( 𝐴 + 𝐵 ) ∈ ℂ ) |
32 |
2 7
|
addcld |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ ( 0 −ℝ 𝐴 ) ∈ ℕ ) ∧ 𝐵 ∈ ℕ0 ) → ( 𝐵 + 𝐴 ) ∈ ℂ ) |
33 |
5 31 32
|
sn-addcand |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ ( 0 −ℝ 𝐴 ) ∈ ℕ ) ∧ 𝐵 ∈ ℕ0 ) → ( ( ( 0 −ℝ 𝐴 ) + ( 𝐴 + 𝐵 ) ) = ( ( 0 −ℝ 𝐴 ) + ( 𝐵 + 𝐴 ) ) ↔ ( 𝐴 + 𝐵 ) = ( 𝐵 + 𝐴 ) ) ) |
34 |
30 33
|
mpbid |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ ( 0 −ℝ 𝐴 ) ∈ ℕ ) ∧ 𝐵 ∈ ℕ0 ) → ( 𝐴 + 𝐵 ) = ( 𝐵 + 𝐴 ) ) |