Theorem mapdvalc 36918

Description: Value of projectivity from vector space H to dual space. (Contributed by NM, 27-Jan-2015.)

Hypotheses

Ref	Expression
mapdval.h	|- H = ( LHyp ` K )
mapdval.u	|- U = ( ( DVecH ` K ) ` W )
mapdval.s	|- S = ( LSubSp ` U )
mapdval.f	|- F = (LFnl ` U )
mapdval.l	|- L = (LKer ` U )
mapdval.o	|- O = ( ( ocH ` K ) ` W )
mapdval.m	|- M = ( (mapd ` K ) ` W )
mapdval.k	|- ( ph -> ( K e. X /\ W e. H ) )
mapdval.t	|- ( ph -> T e. S )
mapdvalc.c	|- C = { g e. F | ( O ` ( O ` ( L ` g ) ) ) = ( L ` g ) }

Assertion

Ref	Expression
mapdvalc	|- ( ph -> ( M ` T ) = { f e. C | ( O ` ( L ` f ) ) C_ T } )

Distinct variable groups:   f, K   f, F   f, W   f, g, F   g, L   g, O   T, f   ph, f

Allowed substitution hints:   ph( g)   C( f, g)   S( f, g)   T( g)   U( f, g)   H( f, g)   K( g)   L( f)   M( f, g)   O( f)   W( g)   X( f, g)

Proof of Theorem mapdvalc

Step	Hyp	Ref	Expression
1		mapdval.h	. . 3 |- H = ( LHyp ` K )
2		mapdval.u	. . 3 |- U = ( ( DVecH ` K ) ` W )
3		mapdval.s	. . 3 |- S = ( LSubSp ` U )
4		mapdval.f	. . 3 |- F = (LFnl ` U )
5		mapdval.l	. . 3 |- L = (LKer ` U )
6		mapdval.o	. . 3 |- O = ( ( ocH ` K ) ` W )
7		mapdval.m	. . 3 |- M = ( (mapd ` K ) ` W )
8		mapdval.k	. . 3 |- ( ph -> ( K e. X /\ W e. H ) )
9		mapdval.t	. . 3 |- ( ph -> T e. S )
10	1, 2, 3, 4, 5, 6, 7, 8, 9	mapdval 36917	. 2 |- ( ph -> ( M ` T ) = { f e. F | ( ( O ` ( O ` ( L ` f ) ) ) = ( L ` f ) /\ ( O ` ( L ` f ) ) C_ T ) } )
11		anass 681	. . . 4 |- ( ( ( f e. F /\ ( O ` ( O ` ( L ` f ) ) ) = ( L ` f ) ) /\ ( O ` ( L ` f ) ) C_ T ) <-> ( f e. F /\ ( ( O ` ( O ` ( L ` f ) ) ) = ( L ` f ) /\ ( O ` ( L ` f ) ) C_ T ) ) )
12		mapdvalc.c	. . . . . . . 8 |- C = { g e. F | ( O ` ( O ` ( L ` g ) ) ) = ( L ` g ) }
13	12	lcfl1lem 36780	. . . . . . 7 |- ( f e. C <-> ( f e. F /\ ( O ` ( O ` ( L ` f ) ) ) = ( L ` f ) ) )
14	13	anbi1i 731	. . . . . 6 |- ( ( f e. C /\ ( O ` ( L ` f ) ) C_ T ) <-> ( ( f e. F /\ ( O ` ( O ` ( L ` f ) ) ) = ( L ` f ) ) /\ ( O ` ( L ` f ) ) C_ T ) )
15	14	bicomi 214	. . . . 5 |- ( ( ( f e. F /\ ( O ` ( O ` ( L ` f ) ) ) = ( L ` f ) ) /\ ( O ` ( L ` f ) ) C_ T ) <-> ( f e. C /\ ( O ` ( L ` f ) ) C_ T ) )
16	15	a1i 11	. . . 4 |- ( ph -> ( ( ( f e. F /\ ( O ` ( O ` ( L ` f ) ) ) = ( L ` f ) ) /\ ( O ` ( L ` f ) ) C_ T ) <-> ( f e. C /\ ( O ` ( L ` f ) ) C_ T ) ) )
17	11, 16	syl5bbr 274	. . 3 |- ( ph -> ( ( f e. F /\ ( ( O ` ( O ` ( L ` f ) ) ) = ( L ` f ) /\ ( O ` ( L ` f ) ) C_ T ) ) <-> ( f e. C /\ ( O ` ( L ` f ) ) C_ T ) ) )
18	17	rabbidva2 3186	. 2 |- ( ph -> { f e. F | ( ( O ` ( O ` ( L ` f ) ) ) = ( L ` f ) /\ ( O ` ( L ` f ) ) C_ T ) } = { f e. C | ( O ` ( L ` f ) ) C_ T } )
19	10, 18	eqtrd 2656	1 |- ( ph -> ( M ` T ) = { f e. C | ( O ` ( L ` f ) ) C_ T } )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   C_ wss 3574   ` cfv 5888   LSubSpclss 18932  LFnlclfn 34344  LKerclk 34372   LHypclh 35270   DVecHcdvh 36367   ocHcoch 36636  mapdcmpd 36913

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-rep 4771  ax-sep 4781  ax-nul 4789  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-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  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-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-mapd 36914

This theorem is referenced by:  mapdval2N  36919  mapdordlem2  36926  mapdrval  36936