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Mirrors > Home > MPE Home > Th. List > clmlmod | Structured version Visualization version GIF version |
Description: A complex 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 2610 | . . 3 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
2 | eqid 2610 | . . 3 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) | |
3 | 1, 2 | isclm 22672 | . 2 ⊢ (𝑊 ∈ ℂMod ↔ (𝑊 ∈ LMod ∧ (Scalar‘𝑊) = (ℂfld ↾s (Base‘(Scalar‘𝑊))) ∧ (Base‘(Scalar‘𝑊)) ∈ (SubRing‘ℂfld))) |
4 | 3 | simp1bi 1069 | 1 ⊢ (𝑊 ∈ ℂMod → 𝑊 ∈ LMod) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 ↾s cress 15696 Scalarcsca 15771 SubRingcsubrg 18599 LModclmod 18686 ℂfldccnfld 19567 ℂModcclm 22670 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-nul 4717 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-iota 5768 df-fv 5812 df-ov 6552 df-clm 22671 |
This theorem is referenced by: clmgrp 22676 clmabl 22677 clmring 22678 clmfgrp 22679 clmvscl 22696 clmvsass 22697 clmvsdir 22699 clmvsdi 22700 clmvs1 22701 clmvs2 22702 clm0vs 22703 clmopfne 22704 clmvneg1 22707 clmvsneg 22708 clmsubdir 22710 clmvsubval 22717 zlmclm 22720 cmodscmulexp 22730 iscvs 22735 cvsi 22738 isncvsngp 22757 ttgbtwnid 25564 ttgcontlem1 25565 |
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