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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 2821 | . . 3 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
2 | eqid 2821 | . . 3 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) | |
3 | 1, 2 | isclm 23668 | . 2 ⊢ (𝑊 ∈ ℂMod ↔ (𝑊 ∈ LMod ∧ (Scalar‘𝑊) = (ℂfld ↾s (Base‘(Scalar‘𝑊))) ∧ (Base‘(Scalar‘𝑊)) ∈ (SubRing‘ℂfld))) |
4 | 3 | simp1bi 1141 | 1 ⊢ (𝑊 ∈ ℂMod → 𝑊 ∈ LMod) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ‘cfv 6355 (class class class)co 7156 Basecbs 16483 ↾s cress 16484 Scalarcsca 16568 SubRingcsubrg 19531 LModclmod 19634 ℂfldccnfld 20545 ℂModcclm 23666 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-nul 5210 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-iota 6314 df-fv 6363 df-ov 7159 df-clm 23667 |
This theorem is referenced by: clmgrp 23672 clmabl 23673 clmring 23674 clmfgrp 23675 clmvscl 23692 clmvsass 23693 clmvsdir 23695 clmvsdi 23696 clmvs1 23697 clmvs2 23698 clm0vs 23699 clmopfne 23700 clmvneg1 23703 clmvsneg 23704 clmsubdir 23706 clmvsubval 23713 zlmclm 23716 cmodscmulexp 23726 iscvs 23731 cvsi 23734 isncvsngp 23753 ttgbtwnid 26670 ttgcontlem1 26671 |
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