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| 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 2737 | . . 3 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
| 2 | eqid 2737 | . . 3 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) | |
| 3 | 1, 2 | isclm 25097 | . 2 ⊢ (𝑊 ∈ ℂMod ↔ (𝑊 ∈ LMod ∧ (Scalar‘𝑊) = (ℂfld ↾s (Base‘(Scalar‘𝑊))) ∧ (Base‘(Scalar‘𝑊)) ∈ (SubRing‘ℂfld))) | 
| 4 | 3 | simp1bi 1146 | 1 ⊢ (𝑊 ∈ ℂMod → 𝑊 ∈ LMod) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 ↾s cress 17274 Scalarcsca 17300 SubRingcsubrg 20569 LModclmod 20858 ℂfldccnfld 21364 ℂModcclm 25095 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-nul 5306 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-ov 7434 df-clm 25096 | 
| This theorem is referenced by: clmgrp 25101 clmabl 25102 clmring 25103 clmfgrp 25104 clmvscl 25121 clmvsass 25122 clmvsdir 25124 clmvsdi 25125 clmvs1 25126 clmvs2 25127 clm0vs 25128 clmopfne 25129 clmvneg1 25132 clmvsneg 25133 clmsubdir 25135 clmvsubval 25142 zlmclm 25145 cmodscmulexp 25155 iscvs 25160 cvsi 25163 isncvsngp 25183 ttgbtwnid 28898 ttgcontlem1 28899 | 
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