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Metamath Proof Explorer

Theorem bramul

Description: Linearity property of bra for multiplication. (Contributed by NM, 23-May-2006) (New usage is discouraged.)

Ref Expression
Assertion bramul ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ ) → ( ( bra ‘ 𝐴 ) ‘ ( 𝐵 ·ℎ 𝐶 ) ) = ( 𝐵 · ( ( bra ‘ 𝐴 ) ‘ 𝐶 ) ) )

Proof

Step Hyp Ref Expression
1 ax-his3 ⊢ ( ( 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ ∧ 𝐴 ∈ ℋ ) → ( ( 𝐵 ·ℎ 𝐶 ) ·ih 𝐴 ) = ( 𝐵 · ( 𝐶 ·ih 𝐴 ) ) )
2 1 3comr ⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ ) → ( ( 𝐵 ·ℎ 𝐶 ) ·ih 𝐴 ) = ( 𝐵 · ( 𝐶 ·ih 𝐴 ) ) )
3 hvmulcl ⊢ ( ( 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ ) → ( 𝐵 ·ℎ 𝐶 ) ∈ ℋ )
4 braval ⊢ ( ( 𝐴 ∈ ℋ ∧ ( 𝐵 ·ℎ 𝐶 ) ∈ ℋ ) → ( ( bra ‘ 𝐴 ) ‘ ( 𝐵 ·ℎ 𝐶 ) ) = ( ( 𝐵 ·ℎ 𝐶 ) ·ih 𝐴 ) )
5 3 4 sylan2 ⊢ ( ( 𝐴 ∈ ℋ ∧ ( 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ ) ) → ( ( bra ‘ 𝐴 ) ‘ ( 𝐵 ·ℎ 𝐶 ) ) = ( ( 𝐵 ·ℎ 𝐶 ) ·ih 𝐴 ) )
6 5 3impb ⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ ) → ( ( bra ‘ 𝐴 ) ‘ ( 𝐵 ·ℎ 𝐶 ) ) = ( ( 𝐵 ·ℎ 𝐶 ) ·ih 𝐴 ) )
7 braval ⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐶 ∈ ℋ ) → ( ( bra ‘ 𝐴 ) ‘ 𝐶 ) = ( 𝐶 ·ih 𝐴 ) )
8 7 3adant2 ⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ ) → ( ( bra ‘ 𝐴 ) ‘ 𝐶 ) = ( 𝐶 ·ih 𝐴 ) )
9 8 oveq2d ⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ ) → ( 𝐵 · ( ( bra ‘ 𝐴 ) ‘ 𝐶 ) ) = ( 𝐵 · ( 𝐶 ·ih 𝐴 ) ) )
10 2 6 9 3eqtr4d ⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ ) → ( ( bra ‘ 𝐴 ) ‘ ( 𝐵 ·ℎ 𝐶 ) ) = ( 𝐵 · ( ( bra ‘ 𝐴 ) ‘ 𝐶 ) ) )