This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem addcomnq

Description: Addition of positive fractions is commutative. (Contributed by NM, 30-Aug-1995) (Revised by Mario Carneiro, 28-Apr-2013) (New usage is discouraged.)

Ref Expression
Assertion addcomnq ( 𝐴 +Q 𝐵 ) = ( 𝐵 +Q 𝐴 )

Proof

Step Hyp Ref Expression
1 addcompq ⊢ ( 𝐴 +pQ 𝐵 ) = ( 𝐵 +pQ 𝐴 )
2 1 fveq2i ⊢ ( [Q] ‘ ( 𝐴 +pQ 𝐵 ) ) = ( [Q] ‘ ( 𝐵 +pQ 𝐴 ) )
3 addpqnq ⊢ ( ( 𝐴 ∈ Q ∧ 𝐵 ∈ Q ) → ( 𝐴 +Q 𝐵 ) = ( [Q] ‘ ( 𝐴 +pQ 𝐵 ) ) )
4 addpqnq ⊢ ( ( 𝐵 ∈ Q ∧ 𝐴 ∈ Q ) → ( 𝐵 +Q 𝐴 ) = ( [Q] ‘ ( 𝐵 +pQ 𝐴 ) ) )
5 4 ancoms ⊢ ( ( 𝐴 ∈ Q ∧ 𝐵 ∈ Q ) → ( 𝐵 +Q 𝐴 ) = ( [Q] ‘ ( 𝐵 +pQ 𝐴 ) ) )
6 2 3 5 3eqtr4a ⊢ ( ( 𝐴 ∈ Q ∧ 𝐵 ∈ Q ) → ( 𝐴 +Q 𝐵 ) = ( 𝐵 +Q 𝐴 ) )
7 addnqf ⊢ +Q : ( Q × Q ) ⟶ Q
8 7 fdmi ⊢ dom +Q = ( Q × Q )
9 8 ndmovcom ⊢ ( ¬ ( 𝐴 ∈ Q ∧ 𝐵 ∈ Q ) → ( 𝐴 +Q 𝐵 ) = ( 𝐵 +Q 𝐴 ) )
10 6 9 pm2.61i ⊢ ( 𝐴 +Q 𝐵 ) = ( 𝐵 +Q 𝐴 )