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| Description: The ring of scalars 𝐹 of a subcomplex module is the restriction of the field of complex numbers to the base set of 𝐹. (Contributed by Mario Carneiro, 16-Oct-2015.) | 
| Ref | Expression | 
|---|---|
| isclm.f | ⊢ 𝐹 = (Scalar‘𝑊) | 
| isclm.k | ⊢ 𝐾 = (Base‘𝐹) | 
| Ref | Expression | 
|---|---|
| clmsca | ⊢ (𝑊 ∈ ℂMod → 𝐹 = (ℂfld ↾s 𝐾)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | isclm.f | . . 3 ⊢ 𝐹 = (Scalar‘𝑊) | |
| 2 | isclm.k | . . 3 ⊢ 𝐾 = (Base‘𝐹) | |
| 3 | 1, 2 | isclm 25098 | . 2 ⊢ (𝑊 ∈ ℂMod ↔ (𝑊 ∈ LMod ∧ 𝐹 = (ℂfld ↾s 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld))) | 
| 4 | 3 | simp2bi 1146 | 1 ⊢ (𝑊 ∈ ℂMod → 𝐹 = (ℂfld ↾s 𝐾)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2107 ‘cfv 6560 (class class class)co 7432 Basecbs 17248 ↾s cress 17275 Scalarcsca 17301 SubRingcsubrg 20570 LModclmod 20859 ℂfldccnfld 21365 ℂModcclm 25096 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-nul 5305 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-rab 3436 df-v 3481 df-sbc 3788 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-iota 6513 df-fv 6568 df-ov 7435 df-clm 25097 | 
| This theorem is referenced by: clm0 25106 clm1 25107 clmadd 25108 clmmul 25109 clmcj 25110 clmsub 25114 clmneg 25115 clmabs 25117 cvsdiv 25166 isncvsngp 25184 | 
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