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Mirrors > Home > MPE Home > Th. List > clmsca | Structured version Visualization version GIF version |
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 25111 | . 2 ⊢ (𝑊 ∈ ℂMod ↔ (𝑊 ∈ LMod ∧ 𝐹 = (ℂfld ↾s 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld))) |
4 | 3 | simp2bi 1145 | 1 ⊢ (𝑊 ∈ ℂMod → 𝐹 = (ℂfld ↾s 𝐾)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 ↾s cress 17274 Scalarcsca 17301 SubRingcsubrg 20586 LModclmod 20875 ℂfldccnfld 21382 ℂModcclm 25109 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-nul 5312 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-iota 6516 df-fv 6571 df-ov 7434 df-clm 25110 |
This theorem is referenced by: clm0 25119 clm1 25120 clmadd 25121 clmmul 25122 clmcj 25123 clmsub 25127 clmneg 25128 clmabs 25130 cvsdiv 25179 isncvsngp 25197 |
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