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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 28381 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝐴 · 𝐵) ·ℎ 𝐶) = (𝐴 ·ℎ (𝐵 ·ℎ 𝐶))) | |
5 | 1, 2, 3, 4 | mp3an 1586 | 1 ⊢ ((𝐴 · 𝐵) ·ℎ 𝐶) = (𝐴 ·ℎ (𝐵 ·ℎ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1653 ∈ wcel 2157 (class class class)co 6876 ℂcc 10220 · cmul 10227 ℋchba 28293 ·ℎ csm 28295 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-hvmulass 28381 |
This theorem depends on definitions: df-bi 199 df-an 386 df-3an 1110 |
This theorem is referenced by: hvmul2negi 28422 hvnegdii 28436 normlem0 28483 lnophmlem2 29393 |
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