Theorem mapd1o 36937

Description: The map defined by df-mapd 36914 is one-to-one and onto the set of dual subspaces of functionals with closed kernels. This shows M satisfies part of the definition of projectivity of [Baer] p. 40. TODO: change theorems leading to this (lcfr 36874, mapdrval 36936, lclkrs 36828, lclkr 36822,...) to use T i^i ~P C? TODO: maybe get rid of $d's for g vs. K U W,. propagate to mapdrn 36938 and any others. (Contributed by NM, 12-Mar-2015.)

Hypotheses

Ref	Expression
mapd1o.h	|- H = ( LHyp ` K )
mapd1o.o	|- O = ( ( ocH ` K ) ` W )
mapd1o.m	|- M = ( (mapd ` K ) ` W )
mapd1o.u	|- U = ( ( DVecH ` K ) ` W )
mapd1o.s	|- S = ( LSubSp ` U )
mapd1o.f	|- F = (LFnl ` U )
mapd1o.l	|- L = (LKer ` U )
mapd1o.d	|- D = (LDual ` U )
mapd1o.t	|- T = ( LSubSp ` D )
mapd1o.c	|- C = { g e. F | ( O ` ( O ` ( L ` g ) ) ) = ( L ` g ) }
mapd1o.k	|- ( ph -> ( K e. HL /\ W e. H ) )

Assertion

Ref	Expression
mapd1o	|- ( ph -> M : S -1-1-onto-> ( T i^i ~P C ) )

Distinct variable groups:   g, F   g, K   g, L   g, O   U, g   g, W

Allowed substitution hints:   ph( g)   C( g)   D( g)   S( g)   T( g)   H( g)   M( g)

Proof of Theorem mapd1o

Dummy variables f c t u are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mapd1o.f	. . . . . 6 |- F = (LFnl ` U )
2		fvex 6201	. . . . . 6 |- (LFnl ` U ) e. _V
3	1, 2	eqeltri 2697	. . . . 5 |- F e. _V
4	3	rabex 4813	. . . 4 |- { f e. F | ( ( O ` ( O ` ( L ` f ) ) ) = ( L ` f ) /\ ( O ` ( L ` f ) ) C_ t ) } e. _V
5		eqid 2622	. . . 4 |- ( t e. S |-> { f e. F | ( ( O ` ( O ` ( L ` f ) ) ) = ( L ` f ) /\ ( O ` ( L ` f ) ) C_ t ) } ) = ( t e. S |-> { f e. F | ( ( O ` ( O ` ( L ` f ) ) ) = ( L ` f ) /\ ( O ` ( L ` f ) ) C_ t ) } )
6	4, 5	fnmpti 6022	. . 3 |- ( t e. S |-> { f e. F | ( ( O ` ( O ` ( L ` f ) ) ) = ( L ` f ) /\ ( O ` ( L ` f ) ) C_ t ) } ) Fn S
7		mapd1o.k	. . . . 5 |- ( ph -> ( K e. HL /\ W e. H ) )
8		mapd1o.h	. . . . . 6 |- H = ( LHyp ` K )
9		mapd1o.u	. . . . . 6 |- U = ( ( DVecH ` K ) ` W )
10		mapd1o.s	. . . . . 6 |- S = ( LSubSp ` U )
11		mapd1o.l	. . . . . 6 |- L = (LKer ` U )
12		mapd1o.o	. . . . . 6 |- O = ( ( ocH ` K ) ` W )
13		mapd1o.m	. . . . . 6 |- M = ( (mapd ` K ) ` W )
14	8, 9, 10, 1, 11, 12, 13	mapdfval 36916	. . . . 5 |- ( ( K e. HL /\ W e. H ) -> M = ( t e. S |-> { f e. F | ( ( O ` ( O ` ( L ` f ) ) ) = ( L ` f ) /\ ( O ` ( L ` f ) ) C_ t ) } ) )
15	7, 14	syl 17	. . . 4 |- ( ph -> M = ( t e. S |-> { f e. F | ( ( O ` ( O ` ( L ` f ) ) ) = ( L ` f ) /\ ( O ` ( L ` f ) ) C_ t ) } ) )
16	15	fneq1d 5981	. . 3 |- ( ph -> ( M Fn S <-> ( t e. S |-> { f e. F | ( ( O ` ( O ` ( L ` f ) ) ) = ( L ` f ) /\ ( O ` ( L ` f ) ) C_ t ) } ) Fn S ) )
17	6, 16	mpbiri 248	. 2 |- ( ph -> M Fn S )
18	3	rabex 4813	. . . . . . 7 |- { g e. F | ( ( O ` ( O ` ( L ` g ) ) ) = ( L ` g ) /\ ( O ` ( L ` g ) ) C_ t ) } e. _V
19		eqid 2622	. . . . . . 7 |- ( t e. S |-> { g e. F | ( ( O ` ( O ` ( L ` g ) ) ) = ( L ` g ) /\ ( O ` ( L ` g ) ) C_ t ) } ) = ( t e. S |-> { g e. F | ( ( O ` ( O ` ( L ` g ) ) ) = ( L ` g ) /\ ( O ` ( L ` g ) ) C_ t ) } )
20	18, 19	fnmpti 6022	. . . . . 6 |- ( t e. S |-> { g e. F | ( ( O ` ( O ` ( L ` g ) ) ) = ( L ` g ) /\ ( O ` ( L ` g ) ) C_ t ) } ) Fn S
21	8, 9, 10, 1, 11, 12, 13	mapdfval 36916	. . . . . . . 8 |- ( ( K e. HL /\ W e. H ) -> M = ( t e. S |-> { g e. F | ( ( O ` ( O ` ( L ` g ) ) ) = ( L ` g ) /\ ( O ` ( L ` g ) ) C_ t ) } ) )
22	7, 21	syl 17	. . . . . . 7 |- ( ph -> M = ( t e. S |-> { g e. F | ( ( O ` ( O ` ( L ` g ) ) ) = ( L ` g ) /\ ( O ` ( L ` g ) ) C_ t ) } ) )
23	22	fneq1d 5981	. . . . . 6 |- ( ph -> ( M Fn S <-> ( t e. S |-> { g e. F | ( ( O ` ( O ` ( L ` g ) ) ) = ( L ` g ) /\ ( O ` ( L ` g ) ) C_ t ) } ) Fn S ) )
24	20, 23	mpbiri 248	. . . . 5 |- ( ph -> M Fn S )
25		fvelrnb 6243	. . . . 5 |- ( M Fn S -> ( t e. ran M <-> E. c e. S ( M ` c ) = t ) )
26	24, 25	syl 17	. . . 4 |- ( ph -> ( t e. ran M <-> E. c e. S ( M ` c ) = t ) )
27	7	adantr 481	. . . . . . . . 9 |- ( ( ph /\ c e. S ) -> ( K e. HL /\ W e. H ) )
28		simpr 477	. . . . . . . . 9 |- ( ( ph /\ c e. S ) -> c e. S )
29	8, 9, 10, 1, 11, 12, 13, 27, 28	mapdval 36917	. . . . . . . 8 |- ( ( ph /\ c e. S ) -> ( M ` c ) = { f e. F | ( ( O ` ( O ` ( L ` f ) ) ) = ( L ` f ) /\ ( O ` ( L ` f ) ) C_ c ) } )
30		mapd1o.d	. . . . . . . . . 10 |- D = (LDual ` U )
31		mapd1o.t	. . . . . . . . . 10 |- T = ( LSubSp ` D )
32		mapd1o.c	. . . . . . . . . 10 |- C = { g e. F | ( O ` ( O ` ( L ` g ) ) ) = ( L ` g ) }
33		eqid 2622	. . . . . . . . . 10 |- { f e. F | ( ( O ` ( O ` ( L ` f ) ) ) = ( L ` f ) /\ ( O ` ( L ` f ) ) C_ c ) } = { f e. F | ( ( O ` ( O ` ( L ` f ) ) ) = ( L ` f ) /\ ( O ` ( L ` f ) ) C_ c ) }
34	8, 12, 9, 10, 1, 11, 30, 31, 32, 33, 27, 28	lclkrs2 36829	. . . . . . . . 9 |- ( ( ph /\ c e. S ) -> ( { f e. F | ( ( O ` ( O ` ( L ` f ) ) ) = ( L ` f ) /\ ( O ` ( L ` f ) ) C_ c ) } e. T /\ { f e. F | ( ( O ` ( O ` ( L ` f ) ) ) = ( L ` f ) /\ ( O ` ( L ` f ) ) C_ c ) } C_ C ) )
35		elin 3796	. . . . . . . . . 10 |- ( { f e. F | ( ( O ` ( O ` ( L ` f ) ) ) = ( L ` f ) /\ ( O ` ( L ` f ) ) C_ c ) } e. ( T i^i ~P C ) <-> ( { f e. F | ( ( O ` ( O ` ( L ` f ) ) ) = ( L ` f ) /\ ( O ` ( L ` f ) ) C_ c ) } e. T /\ { f e. F | ( ( O ` ( O ` ( L ` f ) ) ) = ( L ` f ) /\ ( O ` ( L ` f ) ) C_ c ) } e. ~P C ) )
36	3	rabex 4813	. . . . . . . . . . . 12 |- { f e. F | ( ( O ` ( O ` ( L ` f ) ) ) = ( L ` f ) /\ ( O ` ( L ` f ) ) C_ c ) } e. _V
37	36	elpw 4164	. . . . . . . . . . 11 |- ( { f e. F | ( ( O ` ( O ` ( L ` f ) ) ) = ( L ` f ) /\ ( O ` ( L ` f ) ) C_ c ) } e. ~P C <-> { f e. F | ( ( O ` ( O ` ( L ` f ) ) ) = ( L ` f ) /\ ( O ` ( L ` f ) ) C_ c ) } C_ C )
38	37	anbi2i 730	. . . . . . . . . 10 |- ( ( { f e. F | ( ( O ` ( O ` ( L ` f ) ) ) = ( L ` f ) /\ ( O ` ( L ` f ) ) C_ c ) } e. T /\ { f e. F | ( ( O ` ( O ` ( L ` f ) ) ) = ( L ` f ) /\ ( O ` ( L ` f ) ) C_ c ) } e. ~P C ) <-> ( { f e. F | ( ( O ` ( O ` ( L ` f ) ) ) = ( L ` f ) /\ ( O ` ( L ` f ) ) C_ c ) } e. T /\ { f e. F | ( ( O ` ( O ` ( L ` f ) ) ) = ( L ` f ) /\ ( O ` ( L ` f ) ) C_ c ) } C_ C ) )
39	35, 38	bitr2i 265	. . . . . . . . 9 |- ( ( { f e. F | ( ( O ` ( O ` ( L ` f ) ) ) = ( L ` f ) /\ ( O ` ( L ` f ) ) C_ c ) } e. T /\ { f e. F | ( ( O ` ( O ` ( L ` f ) ) ) = ( L ` f ) /\ ( O ` ( L ` f ) ) C_ c ) } C_ C ) <-> { f e. F | ( ( O ` ( O ` ( L ` f ) ) ) = ( L ` f ) /\ ( O ` ( L ` f ) ) C_ c ) } e. ( T i^i ~P C ) )
40	34, 39	sylib 208	. . . . . . . 8 |- ( ( ph /\ c e. S ) -> { f e. F | ( ( O ` ( O ` ( L ` f ) ) ) = ( L ` f ) /\ ( O ` ( L ` f ) ) C_ c ) } e. ( T i^i ~P C ) )
41	29, 40	eqeltrd 2701	. . . . . . 7 |- ( ( ph /\ c e. S ) -> ( M ` c ) e. ( T i^i ~P C ) )
42		eleq1 2689	. . . . . . 7 |- ( ( M ` c ) = t -> ( ( M ` c ) e. ( T i^i ~P C ) <-> t e. ( T i^i ~P C ) ) )
43	41, 42	syl5ibcom 235	. . . . . 6 |- ( ( ph /\ c e. S ) -> ( ( M ` c ) = t -> t e. ( T i^i ~P C ) ) )
44	43	rexlimdva 3031	. . . . 5 |- ( ph -> ( E. c e. S ( M ` c ) = t -> t e. ( T i^i ~P C ) ) )
45		eqid 2622	. . . . . . . 8 |- U_ f e. t ( O ` ( L ` f ) ) = U_ f e. t ( O ` ( L ` f ) )
46	7	adantr 481	. . . . . . . 8 |- ( ( ph /\ t e. ( T i^i ~P C ) ) -> ( K e. HL /\ W e. H ) )
47		inss1 3833	. . . . . . . . . 10 |- ( T i^i ~P C ) C_ T
48	47	sseli 3599	. . . . . . . . 9 |- ( t e. ( T i^i ~P C ) -> t e. T )
49	48	adantl 482	. . . . . . . 8 |- ( ( ph /\ t e. ( T i^i ~P C ) ) -> t e. T )
50		inss2 3834	. . . . . . . . . . 11 |- ( T i^i ~P C ) C_ ~P C
51	50	sseli 3599	. . . . . . . . . 10 |- ( t e. ( T i^i ~P C ) -> t e. ~P C )
52	51	elpwid 4170	. . . . . . . . 9 |- ( t e. ( T i^i ~P C ) -> t C_ C )
53	52	adantl 482	. . . . . . . 8 |- ( ( ph /\ t e. ( T i^i ~P C ) ) -> t C_ C )
54	8, 12, 9, 10, 1, 11, 30, 31, 32, 45, 46, 49, 53	lcfr 36874	. . . . . . 7 |- ( ( ph /\ t e. ( T i^i ~P C ) ) -> U_ f e. t ( O ` ( L ` f ) ) e. S )
55	8, 12, 13, 9, 10, 1, 11, 30, 31, 32, 46, 49, 53, 45	mapdrval 36936	. . . . . . 7 |- ( ( ph /\ t e. ( T i^i ~P C ) ) -> ( M ` U_ f e. t ( O ` ( L ` f ) ) ) = t )
56		fveq2 6191	. . . . . . . . 9 |- ( c = U_ f e. t ( O ` ( L ` f ) ) -> ( M ` c ) = ( M ` U_ f e. t ( O ` ( L ` f ) ) ) )
57	56	eqeq1d 2624	. . . . . . . 8 |- ( c = U_ f e. t ( O ` ( L ` f ) ) -> ( ( M ` c ) = t <-> ( M ` U_ f e. t ( O ` ( L ` f ) ) ) = t ) )
58	57	rspcev 3309	. . . . . . 7 |- ( ( U_ f e. t ( O ` ( L ` f ) ) e. S /\ ( M ` U_ f e. t ( O ` ( L ` f ) ) ) = t ) -> E. c e. S ( M ` c ) = t )
59	54, 55, 58	syl2anc 693	. . . . . 6 |- ( ( ph /\ t e. ( T i^i ~P C ) ) -> E. c e. S ( M ` c ) = t )
60	59	ex 450	. . . . 5 |- ( ph -> ( t e. ( T i^i ~P C ) -> E. c e. S ( M ` c ) = t ) )
61	44, 60	impbid 202	. . . 4 |- ( ph -> ( E. c e. S ( M ` c ) = t <-> t e. ( T i^i ~P C ) ) )
62	26, 61	bitrd 268	. . 3 |- ( ph -> ( t e. ran M <-> t e. ( T i^i ~P C ) ) )
63	62	eqrdv 2620	. 2 |- ( ph -> ran M = ( T i^i ~P C ) )
64	7	adantr 481	. . . . 5 |- ( ( ph /\ ( t e. S /\ u e. S ) ) -> ( K e. HL /\ W e. H ) )
65		simprl 794	. . . . 5 |- ( ( ph /\ ( t e. S /\ u e. S ) ) -> t e. S )
66		simprr 796	. . . . 5 |- ( ( ph /\ ( t e. S /\ u e. S ) ) -> u e. S )
67	8, 9, 10, 13, 64, 65, 66	mapd11 36928	. . . 4 |- ( ( ph /\ ( t e. S /\ u e. S ) ) -> ( ( M ` t ) = ( M ` u ) <-> t = u ) )
68	67	biimpd 219	. . 3 |- ( ( ph /\ ( t e. S /\ u e. S ) ) -> ( ( M ` t ) = ( M ` u ) -> t = u ) )
69	68	ralrimivva 2971	. 2 |- ( ph -> A. t e. S A. u e. S ( ( M ` t ) = ( M ` u ) -> t = u ) )
70		dff1o6 6531	. 2 |- ( M : S -1-1-onto-> ( T i^i ~P C ) <-> ( M Fn S /\ ran M = ( T i^i ~P C ) /\ A. t e. S A. u e. S ( ( M ` t ) = ( M ` u ) -> t = u ) ) )
71	17, 63, 69, 70	syl3anbrc 1246	1 |- ( ph -> M : S -1-1-onto-> ( T i^i ~P C ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   {crab 2916   _Vcvv 3200   i^i cin 3573   C_ wss 3574   ~Pcpw 4158   U_ciun 4520   |-> cmpt 4729   ran crn 5115   Fn wfn 5883   -1-1-onto->wf1o 5887   ` cfv 5888   LSubSpclss 18932  LFnlclfn 34344  LKerclk 34372  LDualcld 34410   HLchlt 34637   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-8 1992  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-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-riotaBAD 34239

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  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-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-tpos 7352  df-undef 7399  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-sca 15957  df-vsca 15958  df-0g 16102  df-mre 16246  df-mrc 16247  df-acs 16249  df-preset 16928  df-poset 16946  df-plt 16958  df-lub 16974  df-glb 16975  df-join 16976  df-meet 16977  df-p0 17039  df-p1 17040  df-lat 17046  df-clat 17108  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-grp 17425  df-minusg 17426  df-sbg 17427  df-subg 17591  df-cntz 17750  df-oppg 17776  df-lsm 18051  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-ring 18549  df-oppr 18623  df-dvdsr 18641  df-unit 18642  df-invr 18672  df-dvr 18683  df-drng 18749  df-lmod 18865  df-lss 18933  df-lsp 18972  df-lvec 19103  df-lsatoms 34263  df-lshyp 34264  df-lcv 34306  df-lfl 34345  df-lkr 34373  df-ldual 34411  df-oposet 34463  df-ol 34465  df-oml 34466  df-covers 34553  df-ats 34554  df-atl 34585  df-cvlat 34609  df-hlat 34638  df-llines 34784  df-lplanes 34785  df-lvols 34786  df-lines 34787  df-psubsp 34789  df-pmap 34790  df-padd 35082  df-lhyp 35274  df-laut 35275  df-ldil 35390  df-ltrn 35391  df-trl 35446  df-tgrp 36031  df-tendo 36043  df-edring 36045  df-dveca 36291  df-disoa 36318  df-dvech 36368  df-dib 36428  df-dic 36462  df-dih 36518  df-doch 36637  df-djh 36684  df-mapd 36914

This theorem is referenced by:  mapdrn  36938  mapdcnvcl  36941  mapdcl  36942  mapdcnvid1N  36943  mapdcnvid2  36946