Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 24237 | . 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 1539 ∈ wcel 2106 ‘cfv 6426 (class class class)co 7267 Basecbs 16922 ↾s cress 16951 Scalarcsca 16975 SubRingcsubrg 20030 LModclmod 20133 ℂfldccnfld 20607 ℂModcclm 24235 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-nul 5228 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3431 df-sbc 3716 df-dif 3889 df-un 3891 df-in 3893 df-ss 3903 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5074 df-iota 6384 df-fv 6434 df-ov 7270 df-clm 24236 |
This theorem is referenced by: clm0 24245 clm1 24246 clmadd 24247 clmmul 24248 clmcj 24249 clmsub 24253 clmneg 24254 clmabs 24256 cvsdiv 24305 isncvsngp 24323 |
Copyright terms: Public domain | W3C validator |