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Metamath Proof Explorer

Axiom ax-hvdistr1

Description: Scalar multiplication distributive law. (Contributed by NM, 3-Sep-1999) (New usage is discouraged.)

Ref Expression
Assertion ax-hvdistr1 ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ) → ( 𝐴 ·ℎ ( 𝐵 +ℎ 𝐶 ) ) = ( ( 𝐴 ·ℎ 𝐵 ) +ℎ ( 𝐴 ·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 chba ⊢ ℋ
5 3 4 wcel ⊢ 𝐵 ∈ ℋ
6 cC ⊢ 𝐶
7 6 4 wcel ⊢ 𝐶 ∈ ℋ
8 2 5 7 w3a ⊢ ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ )
9 csm ⊢ ·ℎ
10 cva ⊢ +ℎ
11 3 6 10 co ⊢ ( 𝐵 +ℎ 𝐶 )
12 0 11 9 co ⊢ ( 𝐴 ·ℎ ( 𝐵 +ℎ 𝐶 ) )
13 0 3 9 co ⊢ ( 𝐴 ·ℎ 𝐵 )
14 0 6 9 co ⊢ ( 𝐴 ·ℎ 𝐶 )
15 13 14 10 co ⊢ ( ( 𝐴 ·ℎ 𝐵 ) +ℎ ( 𝐴 ·ℎ 𝐶 ) )
16 12 15 wceq ⊢ ( 𝐴 ·ℎ ( 𝐵 +ℎ 𝐶 ) ) = ( ( 𝐴 ·ℎ 𝐵 ) +ℎ ( 𝐴 ·ℎ 𝐶 ) )
17 8 16 wi ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ) → ( 𝐴 ·ℎ ( 𝐵 +ℎ 𝐶 ) ) = ( ( 𝐴 ·ℎ 𝐵 ) +ℎ ( 𝐴 ·ℎ 𝐶 ) ) )