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Metamath Proof Explorer

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 ) )