Theorem isclm 22864

Description: A subcomplex module is a left module over a subring of the field of complex numbers. (Contributed by Mario Carneiro, 16-Oct-2015.)

Hypotheses

Ref	Expression
isclm.f	|- F = (Scalar ` W )
isclm.k	|- K = ( Base ` F )

Assertion

Ref	Expression
isclm	|- ( W e. CMod <-> ( W e. LMod /\ F = (ℂfld ↾s K ) /\ K e. (SubRing ` ℂfld ) ) )

Proof of Theorem isclm

Dummy variables f k w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvexd 6203	. . . 4 |- ( w = W -> (Scalar ` w ) e. _V )
2		fvexd 6203	. . . . 5 |- ( ( w = W /\ f = (Scalar ` w ) ) -> ( Base ` f ) e. _V )
3		id 22	. . . . . . . . 9 |- ( f = (Scalar ` w ) -> f = (Scalar ` w ) )
4		fveq2 6191	. . . . . . . . . 10 |- ( w = W -> (Scalar ` w ) = (Scalar ` W ) )
5		isclm.f	. . . . . . . . . 10 |- F = (Scalar ` W )
6	4, 5	syl6eqr 2674	. . . . . . . . 9 |- ( w = W -> (Scalar ` w ) = F )
7	3, 6	sylan9eqr 2678	. . . . . . . 8 |- ( ( w = W /\ f = (Scalar ` w ) ) -> f = F )
8	7	adantr 481	. . . . . . 7 |- ( ( ( w = W /\ f = (Scalar ` w ) ) /\ k = ( Base ` f ) ) -> f = F )
9		id 22	. . . . . . . . 9 |- ( k = ( Base ` f ) -> k = ( Base ` f ) )
10	7	fveq2d 6195	. . . . . . . . . 10 |- ( ( w = W /\ f = (Scalar ` w ) ) -> ( Base ` f ) = ( Base ` F ) )
11		isclm.k	. . . . . . . . . 10 |- K = ( Base ` F )
12	10, 11	syl6eqr 2674	. . . . . . . . 9 |- ( ( w = W /\ f = (Scalar ` w ) ) -> ( Base ` f ) = K )
13	9, 12	sylan9eqr 2678	. . . . . . . 8 |- ( ( ( w = W /\ f = (Scalar ` w ) ) /\ k = ( Base ` f ) ) -> k = K )
14	13	oveq2d 6666	. . . . . . 7 |- ( ( ( w = W /\ f = (Scalar ` w ) ) /\ k = ( Base ` f ) ) -> (ℂfld ↾s k ) = (ℂfld ↾s K ) )
15	8, 14	eqeq12d 2637	. . . . . 6 |- ( ( ( w = W /\ f = (Scalar ` w ) ) /\ k = ( Base ` f ) ) -> ( f = (ℂfld ↾s k ) <-> F = (ℂfld ↾s K ) ) )
16	13	eleq1d 2686	. . . . . 6 |- ( ( ( w = W /\ f = (Scalar ` w ) ) /\ k = ( Base ` f ) ) -> ( k e. (SubRing ` ℂfld ) <-> K e. (SubRing ` ℂfld ) ) )
17	15, 16	anbi12d 747	. . . . 5 |- ( ( ( w = W /\ f = (Scalar ` w ) ) /\ k = ( Base ` f ) ) -> ( ( f = (ℂfld ↾s k ) /\ k e. (SubRing ` ℂfld ) ) <-> ( F = (ℂfld ↾s K ) /\ K e. (SubRing ` ℂfld ) ) ) )
18	2, 17	sbcied 3472	. . . 4 |- ( ( w = W /\ f = (Scalar ` w ) ) -> ( [. ( Base ` f ) / k ]. ( f = (ℂfld ↾s k ) /\ k e. (SubRing ` ℂfld ) ) <-> ( F = (ℂfld ↾s K ) /\ K e. (SubRing ` ℂfld ) ) ) )
19	1, 18	sbcied 3472	. . 3 |- ( w = W -> ( [. (Scalar ` w ) / f ]. [. ( Base ` f ) / k ]. ( f = (ℂfld ↾s k ) /\ k e. (SubRing ` ℂfld ) ) <-> ( F = (ℂfld ↾s K ) /\ K e. (SubRing ` ℂfld ) ) ) )
20		df-clm 22863	. . 3 |- CMod = { w e. LMod | [. (Scalar ` w ) / f ]. [. ( Base ` f ) / k ]. ( f = (ℂfld ↾s k ) /\ k e. (SubRing ` ℂfld ) ) }
21	19, 20	elrab2 3366	. 2 |- ( W e. CMod <-> ( W e. LMod /\ ( F = (ℂfld ↾s K ) /\ K e. (SubRing ` ℂfld ) ) ) )
22		3anass 1042	. 2 |- ( ( W e. LMod /\ F = (ℂfld ↾s K ) /\ K e. (SubRing ` ℂfld ) ) <-> ( W e. LMod /\ ( F = (ℂfld ↾s K ) /\ K e. (SubRing ` ℂfld ) ) ) )
23	21, 22	bitr4i 267	1 |- ( W e. CMod <-> ( W e. LMod /\ F = (ℂfld ↾s K ) /\ K e. (SubRing ` ℂfld ) ) )

Syntax hints:   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   [.wsbc 3435   ` cfv 5888  (class class class)co 6650   Basecbs 15857   ↾_s cress 15858  Scalarcsca 15944  SubRingcsubrg 18776   LModclmod 18863  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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789

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-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  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-iota 5851  df-fv 5896  df-ov 6653  df-clm 22863

This theorem is referenced by:  clmsca  22865  clmsubrg  22866  clmlmod  22867  isclmi  22877  lmhmclm  22887  isclmp  22897  cphclm  22989  tchclm  23031