Theorem lmhmclm 22887

Description: The domain of a linear operator is a subcomplex module iff the range is. (Contributed by Mario Carneiro, 21-Oct-2015.)

Assertion

Ref	Expression
lmhmclm	|- ( F e. ( S LMHom T ) -> ( S e. CMod <-> T e. CMod ) )

Proof of Theorem lmhmclm

Step	Hyp	Ref	Expression
1		lmhmlmod1 19033	. . . 4 |- ( F e. ( S LMHom T ) -> S e. LMod )
2		lmhmlmod2 19032	. . . 4 |- ( F e. ( S LMHom T ) -> T e. LMod )
3	1, 2	2thd 255	. . 3 |- ( F e. ( S LMHom T ) -> ( S e. LMod <-> T e. LMod ) )
4		eqid 2622	. . . . . 6 |- (Scalar ` S ) = (Scalar ` S )
5		eqid 2622	. . . . . 6 |- (Scalar ` T ) = (Scalar ` T )
6	4, 5	lmhmsca 19030	. . . . 5 |- ( F e. ( S LMHom T ) -> (Scalar ` T ) = (Scalar ` S ) )
7	6	eqcomd 2628	. . . 4 |- ( F e. ( S LMHom T ) -> (Scalar ` S ) = (Scalar ` T ) )
8	7	fveq2d 6195	. . . . 5 |- ( F e. ( S LMHom T ) -> ( Base ` (Scalar ` S ) ) = ( Base ` (Scalar ` T ) ) )
9	8	oveq2d 6666	. . . 4 |- ( F e. ( S LMHom T ) -> (ℂfld ↾s ( Base ` (Scalar ` S ) ) ) = (ℂfld ↾s ( Base ` (Scalar ` T ) ) ) )
10	7, 9	eqeq12d 2637	. . 3 |- ( F e. ( S LMHom T ) -> ( (Scalar ` S ) = (ℂfld ↾s ( Base ` (Scalar ` S ) ) ) <-> (Scalar ` T ) = (ℂfld ↾s ( Base ` (Scalar ` T ) ) ) ) )
11	8	eleq1d 2686	. . 3 |- ( F e. ( S LMHom T ) -> ( ( Base ` (Scalar ` S ) ) e. (SubRing ` ℂfld ) <-> ( Base ` (Scalar ` T ) ) e. (SubRing ` ℂfld ) ) )
12	3, 10, 11	3anbi123d 1399	. 2 |- ( F e. ( S LMHom T ) -> ( ( S e. LMod /\ (Scalar ` S ) = (ℂfld ↾s ( Base ` (Scalar ` S ) ) ) /\ ( Base ` (Scalar ` S ) ) e. (SubRing ` ℂfld ) ) <-> ( T e. LMod /\ (Scalar ` T ) = (ℂfld ↾s ( Base ` (Scalar ` T ) ) ) /\ ( Base ` (Scalar ` T ) ) e. (SubRing ` ℂfld ) ) ) )
13		eqid 2622	. . 3 |- ( Base ` (Scalar ` S ) ) = ( Base ` (Scalar ` S ) )
14	4, 13	isclm 22864	. 2 |- ( S e. CMod <-> ( S e. LMod /\ (Scalar ` S ) = (ℂfld ↾s ( Base ` (Scalar ` S ) ) ) /\ ( Base ` (Scalar ` S ) ) e. (SubRing ` ℂfld ) ) )
15		eqid 2622	. . 3 |- ( Base ` (Scalar ` T ) ) = ( Base ` (Scalar ` T ) )
16	5, 15	isclm 22864	. 2 |- ( T e. CMod <-> ( T e. LMod /\ (Scalar ` T ) = (ℂfld ↾s ( Base ` (Scalar ` T ) ) ) /\ ( Base ` (Scalar ` T ) ) e. (SubRing ` ℂfld ) ) )
17	12, 14, 16	3bitr4g 303	1 |- ( F e. ( S LMHom T ) -> ( S e. CMod <-> T e. CMod ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ w3a 1037   = wceq 1483   e. wcel 1990   ` cfv 5888  (class class class)co 6650   Basecbs 15857   ↾_s cress 15858  Scalarcsca 15944  SubRingcsubrg 18776   LModclmod 18863   LMHom clmhm 19019  ℂ_fldccnfld 19746  CModcclm 22862

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-lmhm 19022  df-clm 22863

This theorem is referenced by: (None)