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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 31036 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝐴 · 𝐵) ·ℎ 𝐶) = (𝐴 ·ℎ (𝐵 ·ℎ 𝐶))) | |
5 | 1, 2, 3, 4 | mp3an 1460 | 1 ⊢ ((𝐴 · 𝐵) ·ℎ 𝐶) = (𝐴 ·ℎ (𝐵 ·ℎ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2106 (class class class)co 7431 ℂcc 11151 · cmul 11158 ℋchba 30948 ·ℎ csm 30950 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-hvmulass 31036 |
This theorem depends on definitions: df-bi 207 df-an 396 df-3an 1088 |
This theorem is referenced by: hvmul2negi 31077 hvnegdii 31091 normlem0 31138 lnophmlem2 32046 |
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