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Metamath Proof Explorer

Axiom ax-his2

Description: Distributive law for inner product. Postulate (S2) of Beran p. 95. (Contributed by NM, 31-Jul-1999) (New usage is discouraged.)

Ref Expression
Assertion ax-his2 ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ) → ( ( 𝐴 +ℎ 𝐵 ) ·ih 𝐶 ) = ( ( 𝐴 ·ih 𝐶 ) + ( 𝐵 ·ih 𝐶 ) ) )

Detailed syntax breakdown

Step Hyp Ref Expression
0 cA ⊢ 𝐴
1 chba ⊢ ℋ
2 0 1 wcel ⊢ 𝐴 ∈ ℋ
3 cB ⊢ 𝐵
4 3 1 wcel ⊢ 𝐵 ∈ ℋ
5 cC ⊢ 𝐶
6 5 1 wcel ⊢ 𝐶 ∈ ℋ
7 2 4 6 w3a ⊢ ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ )
8 cva ⊢ +ℎ
9 0 3 8 co ⊢ ( 𝐴 +ℎ 𝐵 )
10 csp ⊢ ·ih
11 9 5 10 co ⊢ ( ( 𝐴 +ℎ 𝐵 ) ·ih 𝐶 )
12 0 5 10 co ⊢ ( 𝐴 ·ih 𝐶 )
13 caddc ⊢ +
14 3 5 10 co ⊢ ( 𝐵 ·ih 𝐶 )
15 12 14 13 co ⊢ ( ( 𝐴 ·ih 𝐶 ) + ( 𝐵 ·ih 𝐶 ) )
16 11 15 wceq ⊢ ( ( 𝐴 +ℎ 𝐵 ) ·ih 𝐶 ) = ( ( 𝐴 ·ih 𝐶 ) + ( 𝐵 ·ih 𝐶 ) )
17 7 16 wi ⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ) → ( ( 𝐴 +ℎ 𝐵 ) ·ih 𝐶 ) = ( ( 𝐴 ·ih 𝐶 ) + ( 𝐵 ·ih 𝐶 ) ) )