Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  lmhmclm Structured version   Unicode version

Theorem lmhmclm 19104
 Description: The domain of a linear operator is a complex module iff the range is. (Contributed by Mario Carneiro, 21-Oct-2015.)
Assertion
Ref Expression
lmhmclm LMHom CMod CMod

Proof of Theorem lmhmclm
StepHypRef Expression
1 lmhmlmod1 16102 . . . 4 LMHom
2 lmhmlmod2 16101 . . . 4 LMHom
31, 22thd 232 . . 3 LMHom
4 eqid 2436 . . . . . 6 Scalar Scalar
5 eqid 2436 . . . . . 6 Scalar Scalar
64, 5lmhmsca 16099 . . . . 5 LMHom Scalar Scalar
76eqcomd 2441 . . . 4 LMHom Scalar Scalar
87fveq2d 5725 . . . . 5 LMHom Scalar Scalar
98oveq2d 6090 . . . 4 LMHom flds Scalar flds Scalar
107, 9eqeq12d 2450 . . 3 LMHom Scalar flds Scalar Scalar flds Scalar
118eleq1d 2502 . . 3 LMHom Scalar SubRingfld Scalar SubRingfld
123, 10, 113anbi123d 1254 . 2 LMHom Scalar flds Scalar Scalar SubRingfld Scalar flds Scalar Scalar SubRingfld
13 eqid 2436 . . 3 Scalar Scalar
144, 13isclm 19082 . 2 CMod Scalar flds Scalar Scalar SubRingfld
15 eqid 2436 . . 3 Scalar Scalar
165, 15isclm 19082 . 2 CMod Scalar flds Scalar Scalar SubRingfld
1712, 14, 163bitr4g 280 1 LMHom CMod CMod
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   w3a 936   wceq 1652   wcel 1725  cfv 5447  (class class class)co 6074  cbs 13462   ↾s cress 13463  Scalarcsca 13525  SubRingcsubrg 15857  clmod 15943   LMHom clmhm 16088  ℂfldccnfld 16696  CModcclm 19080 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2703  df-rex 2704  df-rab 2707  df-v 2951  df-sbc 3155  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-sn 3813  df-pr 3814  df-op 3816  df-uni 4009  df-br 4206  df-opab 4260  df-id 4491  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-iota 5411  df-fun 5449  df-fv 5455  df-ov 6077  df-oprab 6078  df-mpt2 6079  df-lmhm 16091  df-clm 19081
 Copyright terms: Public domain W3C validator