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Mirrors > Home > MPE Home > Th. List > clmsubrg | Structured version Visualization version GIF version |
Description: The base set of the ring of scalars of a subcomplex module is the base set of a subring of the field of complex numbers. (Contributed by Mario Carneiro, 16-Oct-2015.) |
Ref | Expression |
---|---|
isclm.f | ⊢ 𝐹 = (Scalar‘𝑊) |
isclm.k | ⊢ 𝐾 = (Base‘𝐹) |
Ref | Expression |
---|---|
clmsubrg | ⊢ (𝑊 ∈ ℂMod → 𝐾 ∈ (SubRing‘ℂfld)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isclm.f | . . 3 ⊢ 𝐹 = (Scalar‘𝑊) | |
2 | isclm.k | . . 3 ⊢ 𝐾 = (Base‘𝐹) | |
3 | 1, 2 | isclm 23660 | . 2 ⊢ (𝑊 ∈ ℂMod ↔ (𝑊 ∈ LMod ∧ 𝐹 = (ℂfld ↾s 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld))) |
4 | 3 | simp3bi 1142 | 1 ⊢ (𝑊 ∈ ℂMod → 𝐾 ∈ (SubRing‘ℂfld)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1531 ∈ wcel 2108 ‘cfv 6348 (class class class)co 7148 Basecbs 16475 ↾s cress 16476 Scalarcsca 16560 SubRingcsubrg 19523 LModclmod 19626 ℂfldccnfld 20537 ℂModcclm 23658 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-nul 5201 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ral 3141 df-rex 3142 df-rab 3145 df-v 3495 df-sbc 3771 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4831 df-br 5058 df-iota 6307 df-fv 6356 df-ov 7151 df-clm 23659 |
This theorem is referenced by: clm0 23668 clm1 23669 clmzss 23674 clmsscn 23675 clmsub 23676 clmneg 23677 clmabs 23679 clmacl 23680 clmmcl 23681 clmsubcl 23682 cmodscexp 23717 cvsdiv 23728 isncvsngp 23745 |
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