Metamath Proof Explorer
Description: The sum of positive numbers is positive. Proof of addgt0d without
ax-mulcom . (Contributed by SN, 25-Jan-2025)
|
|
Ref |
Expression |
|
Hypotheses |
sn-addgt0d.a |
⊢ ( 𝜑 → 𝐴 ∈ ℝ ) |
|
|
sn-addgt0d.b |
⊢ ( 𝜑 → 𝐵 ∈ ℝ ) |
|
|
sn-addgt0d.1 |
⊢ ( 𝜑 → 0 < 𝐴 ) |
|
|
sn-addgt0d.2 |
⊢ ( 𝜑 → 0 < 𝐵 ) |
|
Assertion |
sn-addgt0d |
⊢ ( 𝜑 → 0 < ( 𝐴 + 𝐵 ) ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
sn-addgt0d.a |
⊢ ( 𝜑 → 𝐴 ∈ ℝ ) |
2 |
|
sn-addgt0d.b |
⊢ ( 𝜑 → 𝐵 ∈ ℝ ) |
3 |
|
sn-addgt0d.1 |
⊢ ( 𝜑 → 0 < 𝐴 ) |
4 |
|
sn-addgt0d.2 |
⊢ ( 𝜑 → 0 < 𝐵 ) |
5 |
|
0red |
⊢ ( 𝜑 → 0 ∈ ℝ ) |
6 |
1 2
|
readdcld |
⊢ ( 𝜑 → ( 𝐴 + 𝐵 ) ∈ ℝ ) |
7 |
|
sn-ltaddpos |
⊢ ( ( 𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ ) → ( 0 < 𝐵 ↔ 𝐴 < ( 𝐴 + 𝐵 ) ) ) |
8 |
2 1 7
|
syl2anc |
⊢ ( 𝜑 → ( 0 < 𝐵 ↔ 𝐴 < ( 𝐴 + 𝐵 ) ) ) |
9 |
4 8
|
mpbid |
⊢ ( 𝜑 → 𝐴 < ( 𝐴 + 𝐵 ) ) |
10 |
5 1 6 3 9
|
lttrd |
⊢ ( 𝜑 → 0 < ( 𝐴 + 𝐵 ) ) |