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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 22864 | . 2 ⊢ (𝑊 ∈ ℂMod ↔ (𝑊 ∈ LMod ∧ 𝐹 = (ℂfld ↾s 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld))) |
4 | 3 | simp2bi 1077 | 1 ⊢ (𝑊 ∈ ℂMod → 𝐹 = (ℂfld ↾s 𝐾)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 ↾s cress 15858 Scalarcsca 15944 SubRingcsubrg 18776 LModclmod 18863 ℂfldccnfld 19746 ℂModcclm 22862 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-nul 4789 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-iota 5851 df-fv 5896 df-ov 6653 df-clm 22863 |
This theorem is referenced by: clm0 22872 clm1 22873 clmadd 22874 clmmul 22875 clmcj 22876 clmsub 22880 clmneg 22881 clmabs 22883 cvsdiv 22932 isncvsngp 22949 |
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