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 ) ) )