# Metamath Proof Explorer

## Theorem his7

Description: Distributive law for inner product. Lemma 3.1(S7) of Beran p. 95. (Contributed by NM, 31-Jul-1999) (New usage is discouraged.)

Ref Expression
Assertion his7 ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ) → ( 𝐴 ·ih ( 𝐵 + 𝐶 ) ) = ( ( 𝐴 ·ih 𝐵 ) + ( 𝐴 ·ih 𝐶 ) ) )

### Proof

Step Hyp Ref Expression
1 ax-his2 ( ( 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ∧ 𝐴 ∈ ℋ ) → ( ( 𝐵 + 𝐶 ) ·ih 𝐴 ) = ( ( 𝐵 ·ih 𝐴 ) + ( 𝐶 ·ih 𝐴 ) ) )
2 1 fveq2d ( ( 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ∧ 𝐴 ∈ ℋ ) → ( ∗ ‘ ( ( 𝐵 + 𝐶 ) ·ih 𝐴 ) ) = ( ∗ ‘ ( ( 𝐵 ·ih 𝐴 ) + ( 𝐶 ·ih 𝐴 ) ) ) )
3 hicl ( ( 𝐵 ∈ ℋ ∧ 𝐴 ∈ ℋ ) → ( 𝐵 ·ih 𝐴 ) ∈ ℂ )
4 hicl ( ( 𝐶 ∈ ℋ ∧ 𝐴 ∈ ℋ ) → ( 𝐶 ·ih 𝐴 ) ∈ ℂ )
5 cjadd ( ( ( 𝐵 ·ih 𝐴 ) ∈ ℂ ∧ ( 𝐶 ·ih 𝐴 ) ∈ ℂ ) → ( ∗ ‘ ( ( 𝐵 ·ih 𝐴 ) + ( 𝐶 ·ih 𝐴 ) ) ) = ( ( ∗ ‘ ( 𝐵 ·ih 𝐴 ) ) + ( ∗ ‘ ( 𝐶 ·ih 𝐴 ) ) ) )
6 3 4 5 syl2an ( ( ( 𝐵 ∈ ℋ ∧ 𝐴 ∈ ℋ ) ∧ ( 𝐶 ∈ ℋ ∧ 𝐴 ∈ ℋ ) ) → ( ∗ ‘ ( ( 𝐵 ·ih 𝐴 ) + ( 𝐶 ·ih 𝐴 ) ) ) = ( ( ∗ ‘ ( 𝐵 ·ih 𝐴 ) ) + ( ∗ ‘ ( 𝐶 ·ih 𝐴 ) ) ) )
7 6 3impdir ( ( 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ∧ 𝐴 ∈ ℋ ) → ( ∗ ‘ ( ( 𝐵 ·ih 𝐴 ) + ( 𝐶 ·ih 𝐴 ) ) ) = ( ( ∗ ‘ ( 𝐵 ·ih 𝐴 ) ) + ( ∗ ‘ ( 𝐶 ·ih 𝐴 ) ) ) )
8 2 7 eqtrd ( ( 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ∧ 𝐴 ∈ ℋ ) → ( ∗ ‘ ( ( 𝐵 + 𝐶 ) ·ih 𝐴 ) ) = ( ( ∗ ‘ ( 𝐵 ·ih 𝐴 ) ) + ( ∗ ‘ ( 𝐶 ·ih 𝐴 ) ) ) )
9 8 3comr ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ) → ( ∗ ‘ ( ( 𝐵 + 𝐶 ) ·ih 𝐴 ) ) = ( ( ∗ ‘ ( 𝐵 ·ih 𝐴 ) ) + ( ∗ ‘ ( 𝐶 ·ih 𝐴 ) ) ) )
10 hvaddcl ( ( 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ) → ( 𝐵 + 𝐶 ) ∈ ℋ )
11 ax-his1 ( ( 𝐴 ∈ ℋ ∧ ( 𝐵 + 𝐶 ) ∈ ℋ ) → ( 𝐴 ·ih ( 𝐵 + 𝐶 ) ) = ( ∗ ‘ ( ( 𝐵 + 𝐶 ) ·ih 𝐴 ) ) )
12 10 11 sylan2 ( ( 𝐴 ∈ ℋ ∧ ( 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ) ) → ( 𝐴 ·ih ( 𝐵 + 𝐶 ) ) = ( ∗ ‘ ( ( 𝐵 + 𝐶 ) ·ih 𝐴 ) ) )
13 12 3impb ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ) → ( 𝐴 ·ih ( 𝐵 + 𝐶 ) ) = ( ∗ ‘ ( ( 𝐵 + 𝐶 ) ·ih 𝐴 ) ) )
14 ax-his1 ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( 𝐴 ·ih 𝐵 ) = ( ∗ ‘ ( 𝐵 ·ih 𝐴 ) ) )
15 14 3adant3 ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ) → ( 𝐴 ·ih 𝐵 ) = ( ∗ ‘ ( 𝐵 ·ih 𝐴 ) ) )
16 ax-his1 ( ( 𝐴 ∈ ℋ ∧ 𝐶 ∈ ℋ ) → ( 𝐴 ·ih 𝐶 ) = ( ∗ ‘ ( 𝐶 ·ih 𝐴 ) ) )
17 16 3adant2 ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ) → ( 𝐴 ·ih 𝐶 ) = ( ∗ ‘ ( 𝐶 ·ih 𝐴 ) ) )
18 15 17 oveq12d ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ) → ( ( 𝐴 ·ih 𝐵 ) + ( 𝐴 ·ih 𝐶 ) ) = ( ( ∗ ‘ ( 𝐵 ·ih 𝐴 ) ) + ( ∗ ‘ ( 𝐶 ·ih 𝐴 ) ) ) )
19 9 13 18 3eqtr4d ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ) → ( 𝐴 ·ih ( 𝐵 + 𝐶 ) ) = ( ( 𝐴 ·ih 𝐵 ) + ( 𝐴 ·ih 𝐶 ) ) )