Theorem braadd

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

	Ref	Expression
Assertion	braadd	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ) → ( ( bra ‘ 𝐴 ) ‘ ( 𝐵 +ℎ 𝐶 ) ) = ( ( ( bra ‘ 𝐴 ) ‘ 𝐵 ) + ( ( bra ‘ 𝐴 ) ‘ 𝐶 ) ) )

Proof

Step	Hyp	Ref	Expression
1		ax-his2	⊢ ( ( 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ∧ 𝐴 ∈ ℋ ) → ( ( 𝐵 +ℎ 𝐶 ) ·ih 𝐴 ) = ( ( 𝐵 ·ih 𝐴 ) + ( 𝐶 ·ih 𝐴 ) ) )
2	1	3comr	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ) → ( ( 𝐵 +ℎ 𝐶 ) ·ih 𝐴 ) = ( ( 𝐵 ·ih 𝐴 ) + ( 𝐶 ·ih 𝐴 ) ) )
3		hvaddcl	⊢ ( ( 𝐵 ∈ ℋ ∧ 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 ∈ ℋ ∧ 𝐶 ∈ ℋ ) ) → ( ( bra ‘ 𝐴 ) ‘ ( 𝐵 +ℎ 𝐶 ) ) = ( ( 𝐵 +ℎ 𝐶 ) ·ih 𝐴 ) )
6	5	3impb	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ) → ( ( bra ‘ 𝐴 ) ‘ ( 𝐵 +ℎ 𝐶 ) ) = ( ( 𝐵 +ℎ 𝐶 ) ·ih 𝐴 ) )
7		braval	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( ( bra ‘ 𝐴 ) ‘ 𝐵 ) = ( 𝐵 ·ih 𝐴 ) )
8	7	3adant3	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ) → ( ( bra ‘ 𝐴 ) ‘ 𝐵 ) = ( 𝐵 ·ih 𝐴 ) )
9		braval	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐶 ∈ ℋ ) → ( ( bra ‘ 𝐴 ) ‘ 𝐶 ) = ( 𝐶 ·ih 𝐴 ) )
10	9	3adant2	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ) → ( ( bra ‘ 𝐴 ) ‘ 𝐶 ) = ( 𝐶 ·ih 𝐴 ) )
11	8 10	oveq12d	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ) → ( ( ( bra ‘ 𝐴 ) ‘ 𝐵 ) + ( ( bra ‘ 𝐴 ) ‘ 𝐶 ) ) = ( ( 𝐵 ·ih 𝐴 ) + ( 𝐶 ·ih 𝐴 ) ) )
12	2 6 11	3eqtr4d	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ) → ( ( bra ‘ 𝐴 ) ‘ ( 𝐵 +ℎ 𝐶 ) ) = ( ( ( bra ‘ 𝐴 ) ‘ 𝐵 ) + ( ( bra ‘ 𝐴 ) ‘ 𝐶 ) ) )