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| Mirrors > Home > HSE Home > Th. List > hvmulcomi | Structured version Visualization version GIF version | ||
| Description: Scalar multiplication commutative 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 | 
|---|---|
| hvmulcomi | ⊢ (𝐴 ·ℎ (𝐵 ·ℎ 𝐶)) = (𝐵 ·ℎ (𝐴 ·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 | hvmulcom 31062 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (𝐴 ·ℎ (𝐵 ·ℎ 𝐶)) = (𝐵 ·ℎ (𝐴 ·ℎ 𝐶))) | |
| 5 | 1, 2, 3, 4 | mp3an 1463 | 1 ⊢ (𝐴 ·ℎ (𝐵 ·ℎ 𝐶)) = (𝐵 ·ℎ (𝐴 ·ℎ 𝐶)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1540 ∈ wcel 2108 (class class class)co 7431 ℂcc 11153 ℋchba 30938 ·ℎ csm 30940 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-mulcom 11219 ax-hvmulass 31026 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-ov 7434 | 
| This theorem is referenced by: (None) | 
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