Theorem hvadd4i

Description: Hilbert vector space addition law. (Contributed by NM, 3-Sep-1999) (New usage is discouraged.)

	Ref	Expression
Hypotheses	hvass.1	⊢ 𝐴 ∈ ℋ
	hvass.2	⊢ 𝐵 ∈ ℋ
	hvass.3	⊢ 𝐶 ∈ ℋ
	hvadd4.4	⊢ 𝐷 ∈ ℋ
Assertion	hvadd4i	⊢ ( ( 𝐴 +ℎ 𝐵 ) +ℎ ( 𝐶 +ℎ 𝐷 ) ) = ( ( 𝐴 +ℎ 𝐶 ) +ℎ ( 𝐵 +ℎ 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		hvass.1	⊢ 𝐴 ∈ ℋ
2		hvass.2	⊢ 𝐵 ∈ ℋ
3		hvass.3	⊢ 𝐶 ∈ ℋ
4		hvadd4.4	⊢ 𝐷 ∈ ℋ
5		hvadd4	⊢ ( ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) ∧ ( 𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ ) ) → ( ( 𝐴 +ℎ 𝐵 ) +ℎ ( 𝐶 +ℎ 𝐷 ) ) = ( ( 𝐴 +ℎ 𝐶 ) +ℎ ( 𝐵 +ℎ 𝐷 ) ) )
6	1 2 3 4 5	mp4an	⊢ ( ( 𝐴 +ℎ 𝐵 ) +ℎ ( 𝐶 +ℎ 𝐷 ) ) = ( ( 𝐴 +ℎ 𝐶 ) +ℎ ( 𝐵 +ℎ 𝐷 ) )