Description: The minimum of two extended reals is less than or equal to one of them. (Contributed by NM, 7-Feb-2007)
Ref | Expression | ||
---|---|---|---|
Assertion | xrmin2 | ⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → if ( 𝐴 ≤ 𝐵 , 𝐴 , 𝐵 ) ≤ 𝐵 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xrleid | ⊢ ( 𝐵 ∈ ℝ* → 𝐵 ≤ 𝐵 ) | |
2 | iffalse | ⊢ ( ¬ 𝐴 ≤ 𝐵 → if ( 𝐴 ≤ 𝐵 , 𝐴 , 𝐵 ) = 𝐵 ) | |
3 | 2 | breq1d | ⊢ ( ¬ 𝐴 ≤ 𝐵 → ( if ( 𝐴 ≤ 𝐵 , 𝐴 , 𝐵 ) ≤ 𝐵 ↔ 𝐵 ≤ 𝐵 ) ) |
4 | 1 3 | syl5ibrcom | ⊢ ( 𝐵 ∈ ℝ* → ( ¬ 𝐴 ≤ 𝐵 → if ( 𝐴 ≤ 𝐵 , 𝐴 , 𝐵 ) ≤ 𝐵 ) ) |
5 | iftrue | ⊢ ( 𝐴 ≤ 𝐵 → if ( 𝐴 ≤ 𝐵 , 𝐴 , 𝐵 ) = 𝐴 ) | |
6 | id | ⊢ ( 𝐴 ≤ 𝐵 → 𝐴 ≤ 𝐵 ) | |
7 | 5 6 | eqbrtrd | ⊢ ( 𝐴 ≤ 𝐵 → if ( 𝐴 ≤ 𝐵 , 𝐴 , 𝐵 ) ≤ 𝐵 ) |
8 | 4 7 | pm2.61d2 | ⊢ ( 𝐵 ∈ ℝ* → if ( 𝐴 ≤ 𝐵 , 𝐴 , 𝐵 ) ≤ 𝐵 ) |
9 | 8 | adantl | ⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → if ( 𝐴 ≤ 𝐵 , 𝐴 , 𝐵 ) ≤ 𝐵 ) |