Theorem hiassdi

Description: Distributive/associative law for inner product, useful for linearity proofs. (Contributed by NM, 10-May-2005) (New usage is discouraged.)

	Ref	Expression
Assertion	hiassdi	⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ) ∧ ( 𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ ) ) → ( ( ( 𝐴 ·ℎ 𝐵 ) +ℎ 𝐶 ) ·ih 𝐷 ) = ( ( 𝐴 · ( 𝐵 ·ih 𝐷 ) ) + ( 𝐶 ·ih 𝐷 ) ) )

Proof

Step	Hyp	Ref	Expression
1		hvmulcl	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ) → ( 𝐴 ·ℎ 𝐵 ) ∈ ℋ )
2		ax-his2	⊢ ( ( ( 𝐴 ·ℎ 𝐵 ) ∈ ℋ ∧ 𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ ) → ( ( ( 𝐴 ·ℎ 𝐵 ) +ℎ 𝐶 ) ·ih 𝐷 ) = ( ( ( 𝐴 ·ℎ 𝐵 ) ·ih 𝐷 ) + ( 𝐶 ·ih 𝐷 ) ) )
3	2	3expb	⊢ ( ( ( 𝐴 ·ℎ 𝐵 ) ∈ ℋ ∧ ( 𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ ) ) → ( ( ( 𝐴 ·ℎ 𝐵 ) +ℎ 𝐶 ) ·ih 𝐷 ) = ( ( ( 𝐴 ·ℎ 𝐵 ) ·ih 𝐷 ) + ( 𝐶 ·ih 𝐷 ) ) )
4	1 3	sylan	⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ) ∧ ( 𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ ) ) → ( ( ( 𝐴 ·ℎ 𝐵 ) +ℎ 𝐶 ) ·ih 𝐷 ) = ( ( ( 𝐴 ·ℎ 𝐵 ) ·ih 𝐷 ) + ( 𝐶 ·ih 𝐷 ) ) )
5		ax-his3	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐷 ∈ ℋ ) → ( ( 𝐴 ·ℎ 𝐵 ) ·ih 𝐷 ) = ( 𝐴 · ( 𝐵 ·ih 𝐷 ) ) )
6	5	3expa	⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ) ∧ 𝐷 ∈ ℋ ) → ( ( 𝐴 ·ℎ 𝐵 ) ·ih 𝐷 ) = ( 𝐴 · ( 𝐵 ·ih 𝐷 ) ) )
7	6	adantrl	⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ) ∧ ( 𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ ) ) → ( ( 𝐴 ·ℎ 𝐵 ) ·ih 𝐷 ) = ( 𝐴 · ( 𝐵 ·ih 𝐷 ) ) )
8	7	oveq1d	⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ) ∧ ( 𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ ) ) → ( ( ( 𝐴 ·ℎ 𝐵 ) ·ih 𝐷 ) + ( 𝐶 ·ih 𝐷 ) ) = ( ( 𝐴 · ( 𝐵 ·ih 𝐷 ) ) + ( 𝐶 ·ih 𝐷 ) ) )
9	4 8	eqtrd	⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ) ∧ ( 𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ ) ) → ( ( ( 𝐴 ·ℎ 𝐵 ) +ℎ 𝐶 ) ·ih 𝐷 ) = ( ( 𝐴 · ( 𝐵 ·ih 𝐷 ) ) + ( 𝐶 ·ih 𝐷 ) ) )