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| Mirrors > Home > HSE Home > Th. List > hvmulassi | Structured version Visualization version GIF version | ||
| Description: Scalar multiplication associative law. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| hvmulcom.1 | ⊢ 𝐴 ∈ ℂ | 
| hvmulcom.2 | ⊢ 𝐵 ∈ ℂ | 
| hvmulcom.3 | ⊢ 𝐶 ∈ ℋ | 
| Ref | Expression | 
|---|---|
| hvmulassi | ⊢ ((𝐴 · 𝐵) ·ℎ 𝐶) = (𝐴 ·ℎ (𝐵 ·ℎ 𝐶)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | hvmulcom.1 | . 2 ⊢ 𝐴 ∈ ℂ | |
| 2 | hvmulcom.2 | . 2 ⊢ 𝐵 ∈ ℂ | |
| 3 | hvmulcom.3 | . 2 ⊢ 𝐶 ∈ ℋ | |
| 4 | ax-hvmulass 31026 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝐴 · 𝐵) ·ℎ 𝐶) = (𝐴 ·ℎ (𝐵 ·ℎ 𝐶))) | |
| 5 | 1, 2, 3, 4 | mp3an 1463 | 1 ⊢ ((𝐴 · 𝐵) ·ℎ 𝐶) = (𝐴 ·ℎ (𝐵 ·ℎ 𝐶)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1540 ∈ wcel 2108 (class class class)co 7431 ℂcc 11153 · cmul 11160 ℋchba 30938 ·ℎ csm 30940 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-hvmulass 31026 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-3an 1089 | 
| This theorem is referenced by: hvmul2negi 31067 hvnegdii 31081 normlem0 31128 lnophmlem2 32036 | 
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