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Metamath Proof Explorer

Theorem his35i

Description: Move scalar multiplication to outside of inner product. (Contributed by NM, 1-Jul-2005) (Revised by Mario Carneiro, 15-May-2014) (New usage is discouraged.)

	Ref	Expression
Hypotheses	his35.1	⊢ 𝐴 ∈ ℂ
	his35.2	⊢ 𝐵 ∈ ℂ
	his35.3	⊢ 𝐶 ∈ ℋ
	his35.4	⊢ 𝐷 ∈ ℋ
Assertion	his35i	⊢ ( ( 𝐴 ·_ℎ 𝐶 ) ·_ih ( 𝐵 ·_ℎ 𝐷 ) ) = ( ( 𝐴 · ( ∗ ‘ 𝐵 ) ) · ( 𝐶 ·_ih 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		his35.1	⊢ 𝐴 ∈ ℂ
2		his35.2	⊢ 𝐵 ∈ ℂ
3		his35.3	⊢ 𝐶 ∈ ℋ
4		his35.4	⊢ 𝐷 ∈ ℋ
5		his35	⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) ∧ ( 𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ ) ) → ( ( 𝐴 ·_ℎ 𝐶 ) ·_ih ( 𝐵 ·_ℎ 𝐷 ) ) = ( ( 𝐴 · ( ∗ ‘ 𝐵 ) ) · ( 𝐶 ·_ih 𝐷 ) ) )
6	1 2 3 4 5	mp4an	⊢ ( ( 𝐴 ·_ℎ 𝐶 ) ·_ih ( 𝐵 ·_ℎ 𝐷 ) ) = ( ( 𝐴 · ( ∗ ‘ 𝐵 ) ) · ( 𝐶 ·_ih 𝐷 ) )