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Metamath Proof Explorer

Theorem somincom

Description: Commutativity of minimum in a total order. (Contributed by Stefan O'Rear, 17-Jan-2015)

Ref Expression
Assertion somincom ( ( 𝑅 Or 𝑋 ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ) → if ( 𝐴 𝑅 𝐵 , 𝐴 , 𝐵 ) = if ( 𝐵 𝑅 𝐴 , 𝐵 , 𝐴 ) )

Proof

Step Hyp Ref Expression
1 so2nr ⊢ ( ( 𝑅 Or 𝑋 ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ) → ¬ ( 𝐴 𝑅 𝐵 ∧ 𝐵 𝑅 𝐴 ) )
2 nan ⊢ ( ( ( 𝑅 Or 𝑋 ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ) → ¬ ( 𝐴 𝑅 𝐵 ∧ 𝐵 𝑅 𝐴 ) ) ↔ ( ( ( 𝑅 Or 𝑋 ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ) ∧ 𝐴 𝑅 𝐵 ) → ¬ 𝐵 𝑅 𝐴 ) )
3 1 2 mpbi ⊢ ( ( ( 𝑅 Or 𝑋 ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ) ∧ 𝐴 𝑅 𝐵 ) → ¬ 𝐵 𝑅 𝐴 )
4 3 iffalsed ⊢ ( ( ( 𝑅 Or 𝑋 ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ) ∧ 𝐴 𝑅 𝐵 ) → if ( 𝐵 𝑅 𝐴 , 𝐵 , 𝐴 ) = 𝐴 )
5 4 eqcomd ⊢ ( ( ( 𝑅 Or 𝑋 ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ) ∧ 𝐴 𝑅 𝐵 ) → 𝐴 = if ( 𝐵 𝑅 𝐴 , 𝐵 , 𝐴 ) )
6 sotric ⊢ ( ( 𝑅 Or 𝑋 ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ) → ( 𝐴 𝑅 𝐵 ↔ ¬ ( 𝐴 = 𝐵 ∨ 𝐵 𝑅 𝐴 ) ) )
7 6 con2bid ⊢ ( ( 𝑅 Or 𝑋 ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ) → ( ( 𝐴 = 𝐵 ∨ 𝐵 𝑅 𝐴 ) ↔ ¬ 𝐴 𝑅 𝐵 ) )
8 ifeq2 ⊢ ( 𝐴 = 𝐵 → if ( 𝐵 𝑅 𝐴 , 𝐵 , 𝐴 ) = if ( 𝐵 𝑅 𝐴 , 𝐵 , 𝐵 ) )
9 ifid ⊢ if ( 𝐵 𝑅 𝐴 , 𝐵 , 𝐵 ) = 𝐵
10 8 9 eqtr2di ⊢ ( 𝐴 = 𝐵 → 𝐵 = if ( 𝐵 𝑅 𝐴 , 𝐵 , 𝐴 ) )
11 iftrue ⊢ ( 𝐵 𝑅 𝐴 → if ( 𝐵 𝑅 𝐴 , 𝐵 , 𝐴 ) = 𝐵 )
12 11 eqcomd ⊢ ( 𝐵 𝑅 𝐴 → 𝐵 = if ( 𝐵 𝑅 𝐴 , 𝐵 , 𝐴 ) )
13 10 12 jaoi ⊢ ( ( 𝐴 = 𝐵 ∨ 𝐵 𝑅 𝐴 ) → 𝐵 = if ( 𝐵 𝑅 𝐴 , 𝐵 , 𝐴 ) )
14 7 13 biimtrrdi ⊢ ( ( 𝑅 Or 𝑋 ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ) → ( ¬ 𝐴 𝑅 𝐵 → 𝐵 = if ( 𝐵 𝑅 𝐴 , 𝐵 , 𝐴 ) ) )
15 14 imp ⊢ ( ( ( 𝑅 Or 𝑋 ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ) ∧ ¬ 𝐴 𝑅 𝐵 ) → 𝐵 = if ( 𝐵 𝑅 𝐴 , 𝐵 , 𝐴 ) )
16 5 15 ifeqda ⊢ ( ( 𝑅 Or 𝑋 ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ) → if ( 𝐴 𝑅 𝐵 , 𝐴 , 𝐵 ) = if ( 𝐵 𝑅 𝐴 , 𝐵 , 𝐴 ) )