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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 23669 | . 2 ⊢ (𝑊 ∈ ℂMod ↔ (𝑊 ∈ LMod ∧ 𝐹 = (ℂfld ↾s 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld))) |
4 | 3 | simp3bi 1144 | 1 ⊢ (𝑊 ∈ ℂMod → 𝐾 ∈ (SubRing‘ℂfld)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 ↾s cress 16476 Scalarcsca 16560 SubRingcsubrg 19524 LModclmod 19627 ℂfldccnfld 20091 ℂModcclm 23667 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-nul 5174 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-iota 6283 df-fv 6332 df-ov 7138 df-clm 23668 |
This theorem is referenced by: clm0 23677 clm1 23678 clmzss 23683 clmsscn 23684 clmsub 23685 clmneg 23686 clmabs 23688 clmacl 23689 clmmcl 23690 clmsubcl 23691 cmodscexp 23726 cvsdiv 23737 isncvsngp 23754 |
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