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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 2735 | . . 3 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
2 | eqid 2735 | . . 3 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) | |
3 | 1, 2 | isclm 25111 | . 2 ⊢ (𝑊 ∈ ℂMod ↔ (𝑊 ∈ LMod ∧ (Scalar‘𝑊) = (ℂfld ↾s (Base‘(Scalar‘𝑊))) ∧ (Base‘(Scalar‘𝑊)) ∈ (SubRing‘ℂfld))) |
4 | 3 | simp1bi 1144 | 1 ⊢ (𝑊 ∈ ℂMod → 𝑊 ∈ LMod) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 ↾s cress 17274 Scalarcsca 17301 SubRingcsubrg 20586 LModclmod 20875 ℂfldccnfld 21382 ℂModcclm 25109 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-nul 5312 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-iota 6516 df-fv 6571 df-ov 7434 df-clm 25110 |
This theorem is referenced by: clmgrp 25115 clmabl 25116 clmring 25117 clmfgrp 25118 clmvscl 25135 clmvsass 25136 clmvsdir 25138 clmvsdi 25139 clmvs1 25140 clmvs2 25141 clm0vs 25142 clmopfne 25143 clmvneg1 25146 clmvsneg 25147 clmsubdir 25149 clmvsubval 25156 zlmclm 25159 cmodscmulexp 25169 iscvs 25174 cvsi 25177 isncvsngp 25197 ttgbtwnid 28913 ttgcontlem1 28914 |
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