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Mirrors > Home > MPE Home > Th. List > clmlmod | Structured version Visualization version GIF version |
Description: A subcomplex module is a left module. (Contributed by Mario Carneiro, 16-Oct-2015.) |
Ref | Expression |
---|---|
clmlmod | ⊢ (𝑊 ∈ ℂMod → 𝑊 ∈ LMod) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2798 | . . 3 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
2 | eqid 2798 | . . 3 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) | |
3 | 1, 2 | isclm 23669 | . 2 ⊢ (𝑊 ∈ ℂMod ↔ (𝑊 ∈ LMod ∧ (Scalar‘𝑊) = (ℂfld ↾s (Base‘(Scalar‘𝑊))) ∧ (Base‘(Scalar‘𝑊)) ∈ (SubRing‘ℂfld))) |
4 | 3 | simp1bi 1142 | 1 ⊢ (𝑊 ∈ ℂMod → 𝑊 ∈ LMod) |
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: clmgrp 23673 clmabl 23674 clmring 23675 clmfgrp 23676 clmvscl 23693 clmvsass 23694 clmvsdir 23696 clmvsdi 23697 clmvs1 23698 clmvs2 23699 clm0vs 23700 clmopfne 23701 clmvneg1 23704 clmvsneg 23705 clmsubdir 23707 clmvsubval 23714 zlmclm 23717 cmodscmulexp 23727 iscvs 23732 cvsi 23735 isncvsngp 23754 ttgbtwnid 26678 ttgcontlem1 26679 |
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