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Mirrors > Home > HSE Home > Th. List > hvdistr1i | Structured version Visualization version GIF version |
Description: Scalar multiplication distributive law. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hvdistr1.1 | ⊢ 𝐴 ∈ ℂ |
hvdistr1.2 | ⊢ 𝐵 ∈ ℋ |
hvdistr1.3 | ⊢ 𝐶 ∈ ℋ |
Ref | Expression |
---|---|
hvdistr1i | ⊢ (𝐴 ·ℎ (𝐵 +ℎ 𝐶)) = ((𝐴 ·ℎ 𝐵) +ℎ (𝐴 ·ℎ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hvdistr1.1 | . 2 ⊢ 𝐴 ∈ ℂ | |
2 | hvdistr1.2 | . 2 ⊢ 𝐵 ∈ ℋ | |
3 | hvdistr1.3 | . 2 ⊢ 𝐶 ∈ ℋ | |
4 | ax-hvdistr1 28791 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 ·ℎ (𝐵 +ℎ 𝐶)) = ((𝐴 ·ℎ 𝐵) +ℎ (𝐴 ·ℎ 𝐶))) | |
5 | 1, 2, 3, 4 | mp3an 1458 | 1 ⊢ (𝐴 ·ℎ (𝐵 +ℎ 𝐶)) = ((𝐴 ·ℎ 𝐵) +ℎ (𝐴 ·ℎ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 ∈ wcel 2111 (class class class)co 7135 ℂcc 10524 ℋchba 28702 +ℎ cva 28703 ·ℎ csm 28704 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-hvdistr1 28791 |
This theorem depends on definitions: df-bi 210 df-an 400 df-3an 1086 |
This theorem is referenced by: hvsubsub4i 28842 hvnegdii 28845 pjmulii 29460 lnophmlem2 29800 |
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