Step |
Hyp |
Ref |
Expression |
1 |
|
xadd0ge.a |
⊢ ( 𝜑 → 𝐴 ∈ ℝ* ) |
2 |
|
xadd0ge.b |
⊢ ( 𝜑 → 𝐵 ∈ ( 0 [,] +∞ ) ) |
3 |
|
xaddid1 |
⊢ ( 𝐴 ∈ ℝ* → ( 𝐴 +𝑒 0 ) = 𝐴 ) |
4 |
1 3
|
syl |
⊢ ( 𝜑 → ( 𝐴 +𝑒 0 ) = 𝐴 ) |
5 |
4
|
eqcomd |
⊢ ( 𝜑 → 𝐴 = ( 𝐴 +𝑒 0 ) ) |
6 |
|
0xr |
⊢ 0 ∈ ℝ* |
7 |
6
|
a1i |
⊢ ( 𝜑 → 0 ∈ ℝ* ) |
8 |
1 7
|
jca |
⊢ ( 𝜑 → ( 𝐴 ∈ ℝ* ∧ 0 ∈ ℝ* ) ) |
9 |
|
iccssxr |
⊢ ( 0 [,] +∞ ) ⊆ ℝ* |
10 |
9 2
|
sseldi |
⊢ ( 𝜑 → 𝐵 ∈ ℝ* ) |
11 |
1 10
|
jca |
⊢ ( 𝜑 → ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) ) |
12 |
8 11
|
jca |
⊢ ( 𝜑 → ( ( 𝐴 ∈ ℝ* ∧ 0 ∈ ℝ* ) ∧ ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) ) ) |
13 |
1
|
xrleidd |
⊢ ( 𝜑 → 𝐴 ≤ 𝐴 ) |
14 |
|
pnfxr |
⊢ +∞ ∈ ℝ* |
15 |
14
|
a1i |
⊢ ( 𝜑 → +∞ ∈ ℝ* ) |
16 |
|
iccgelb |
⊢ ( ( 0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝐵 ∈ ( 0 [,] +∞ ) ) → 0 ≤ 𝐵 ) |
17 |
7 15 2 16
|
syl3anc |
⊢ ( 𝜑 → 0 ≤ 𝐵 ) |
18 |
13 17
|
jca |
⊢ ( 𝜑 → ( 𝐴 ≤ 𝐴 ∧ 0 ≤ 𝐵 ) ) |
19 |
|
xle2add |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 0 ∈ ℝ* ) ∧ ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) ) → ( ( 𝐴 ≤ 𝐴 ∧ 0 ≤ 𝐵 ) → ( 𝐴 +𝑒 0 ) ≤ ( 𝐴 +𝑒 𝐵 ) ) ) |
20 |
12 18 19
|
sylc |
⊢ ( 𝜑 → ( 𝐴 +𝑒 0 ) ≤ ( 𝐴 +𝑒 𝐵 ) ) |
21 |
5 20
|
eqbrtrd |
⊢ ( 𝜑 → 𝐴 ≤ ( 𝐴 +𝑒 𝐵 ) ) |