Axiom ax-hvdistr2

Description: Scalar multiplication distributive law. (Contributed by NM, 30-May-1999) (New usage is discouraged.)

	Ref	Expression
Assertion	ax-hvdistr2	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ ) → ( ( 𝐴 + 𝐵 ) ·ℎ 𝐶 ) = ( ( 𝐴 ·ℎ 𝐶 ) +ℎ ( 𝐵 ·ℎ 𝐶 ) ) )

Detailed syntax breakdown

Step	Hyp	Ref	Expression
0		cA	⊢ 𝐴
1		cc	⊢ ℂ
2	0 1	wcel	⊢ 𝐴 ∈ ℂ
3		cB	⊢ 𝐵
4	3 1	wcel	⊢ 𝐵 ∈ ℂ
5		cC	⊢ 𝐶
6		chba	⊢ ℋ
7	5 6	wcel	⊢ 𝐶 ∈ ℋ
8	2 4 7	w3a	⊢ ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ )
9		caddc	⊢ +
10	0 3 9	co	⊢ ( 𝐴 + 𝐵 )
11		csm	⊢ ·ℎ
12	10 5 11	co	⊢ ( ( 𝐴 + 𝐵 ) ·ℎ 𝐶 )
13	0 5 11	co	⊢ ( 𝐴 ·ℎ 𝐶 )
14		cva	⊢ +ℎ
15	3 5 11	co	⊢ ( 𝐵 ·ℎ 𝐶 )
16	13 15 14	co	⊢ ( ( 𝐴 ·ℎ 𝐶 ) +ℎ ( 𝐵 ·ℎ 𝐶 ) )
17	12 16	wceq	⊢ ( ( 𝐴 + 𝐵 ) ·ℎ 𝐶 ) = ( ( 𝐴 ·ℎ 𝐶 ) +ℎ ( 𝐵 ·ℎ 𝐶 ) )
18	8 17	wi	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ ) → ( ( 𝐴 + 𝐵 ) ·ℎ 𝐶 ) = ( ( 𝐴 ·ℎ 𝐶 ) +ℎ ( 𝐵 ·ℎ 𝐶 ) ) )