Step |
Hyp |
Ref |
Expression |
1 |
|
iftrue |
⊢ ( 𝐴 ≤ 𝐵 → if ( 𝐴 ≤ 𝐵 , 𝐴 , 𝐵 ) = 𝐴 ) |
2 |
1
|
adantl |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) ∧ 𝐴 ≤ 𝐵 ) → if ( 𝐴 ≤ 𝐵 , 𝐴 , 𝐵 ) = 𝐴 ) |
3 |
|
xrleid |
⊢ ( 𝐴 ∈ ℝ* → 𝐴 ≤ 𝐴 ) |
4 |
3
|
ad2antrr |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) ∧ 𝐴 ≤ 𝐵 ) → 𝐴 ≤ 𝐴 ) |
5 |
2 4
|
eqbrtrd |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) ∧ 𝐴 ≤ 𝐵 ) → if ( 𝐴 ≤ 𝐵 , 𝐴 , 𝐵 ) ≤ 𝐴 ) |
6 |
|
iffalse |
⊢ ( ¬ 𝐴 ≤ 𝐵 → if ( 𝐴 ≤ 𝐵 , 𝐴 , 𝐵 ) = 𝐵 ) |
7 |
6
|
adantl |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) ∧ ¬ 𝐴 ≤ 𝐵 ) → if ( 𝐴 ≤ 𝐵 , 𝐴 , 𝐵 ) = 𝐵 ) |
8 |
|
xrletri |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( 𝐴 ≤ 𝐵 ∨ 𝐵 ≤ 𝐴 ) ) |
9 |
8
|
orcanai |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) ∧ ¬ 𝐴 ≤ 𝐵 ) → 𝐵 ≤ 𝐴 ) |
10 |
7 9
|
eqbrtrd |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) ∧ ¬ 𝐴 ≤ 𝐵 ) → if ( 𝐴 ≤ 𝐵 , 𝐴 , 𝐵 ) ≤ 𝐴 ) |
11 |
5 10
|
pm2.61dan |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → if ( 𝐴 ≤ 𝐵 , 𝐴 , 𝐵 ) ≤ 𝐴 ) |