Theorem addasspr

Description: Addition of positive reals is associative. Proposition 9-3.5(i) of Gleason p. 123. (Contributed by NM, 18-Mar-1996) (New usage is discouraged.)

	Ref	Expression
Assertion	addasspr	⊢ ( ( 𝐴 +P 𝐵 ) +P 𝐶 ) = ( 𝐴 +P ( 𝐵 +P 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		df-plp	⊢ +P = ( 𝑤 ∈ P , 𝑣 ∈ P ↦ { 𝑥 ∣ ∃ 𝑦 ∈ 𝑤 ∃ 𝑧 ∈ 𝑣 𝑥 = ( 𝑦 +Q 𝑧 ) } )
2		addclnq	⊢ ( ( 𝑦 ∈ Q ∧ 𝑧 ∈ Q ) → ( 𝑦 +Q 𝑧 ) ∈ Q )
3		dmplp	⊢ dom +P = ( P × P )
4		addclpr	⊢ ( ( 𝑓 ∈ P ∧ 𝑔 ∈ P ) → ( 𝑓 +P 𝑔 ) ∈ P )
5		addassnq	⊢ ( ( 𝑓 +Q 𝑔 ) +Q ℎ ) = ( 𝑓 +Q ( 𝑔 +Q ℎ ) )
6	1 2 3 4 5	genpass	⊢ ( ( 𝐴 +P 𝐵 ) +P 𝐶 ) = ( 𝐴 +P ( 𝐵 +P 𝐶 ) )