Theorem mapdval4N 36921

Description: Value of projectivity from vector space H to dual space. TODO: 1. This is shorter than others - make it the official def? (but is not as obvious that it is C_ C) 2. The unneeded direction of lcfl8a 36792 has awkward E.- add another thm with only one direction of it? 3. Swap O ` { v } and L ` f? (Contributed by NM, 31-Jan-2015.) (New usage is discouraged.)

Hypotheses

Ref	Expression
mapdval4.h	|- H = ( LHyp ` K )
mapdval4.u	|- U = ( ( DVecH ` K ) ` W )
mapdval4.s	|- S = ( LSubSp ` U )
mapdval4.f	|- F = (LFnl ` U )
mapdval4.l	|- L = (LKer ` U )
mapdval4.o	|- O = ( ( ocH ` K ) ` W )
mapdval4.m	|- M = ( (mapd ` K ) ` W )
mapdval4.k	|- ( ph -> ( K e. HL /\ W e. H ) )
mapdval4.t	|- ( ph -> T e. S )

Assertion

Ref	Expression
mapdval4N	|- ( ph -> ( M ` T ) = { f e. F | E. v e. T ( O ` { v } ) = ( L ` f ) } )

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

Allowed substitution hints:   S( v, f)   U( f)   H( v, f)   K( v)   L( f)   M( v, f)   O( f)   W( v)

Proof of Theorem mapdval4N

Dummy variables g w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mapdval4.h	. . 3 |- H = ( LHyp ` K )
2		mapdval4.u	. . 3 |- U = ( ( DVecH ` K ) ` W )
3		mapdval4.s	. . 3 |- S = ( LSubSp ` U )
4		eqid 2622	. . 3 |- ( LSpan ` U ) = ( LSpan ` U )
5		mapdval4.f	. . 3 |- F = (LFnl ` U )
6		mapdval4.l	. . 3 |- L = (LKer ` U )
7		mapdval4.o	. . 3 |- O = ( ( ocH ` K ) ` W )
8		mapdval4.m	. . 3 |- M = ( (mapd ` K ) ` W )
9		mapdval4.k	. . 3 |- ( ph -> ( K e. HL /\ W e. H ) )
10		mapdval4.t	. . 3 |- ( ph -> T e. S )
11		eqid 2622	. . 3 |- { g e. F | ( O ` ( O ` ( L ` g ) ) ) = ( L ` g ) } = { g e. F | ( O ` ( O ` ( L ` g ) ) ) = ( L ` g ) }
12	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11	mapdval2N 36919	. 2 |- ( ph -> ( M ` T ) = { f e. { g e. F | ( O ` ( O ` ( L ` g ) ) ) = ( L ` g ) } | E. v e. T ( O ` ( L ` f ) ) = ( ( LSpan ` U ) ` { v } ) } )
13	11	lcfl1lem 36780	. . . . . 6 |- ( f e. { g e. F | ( O ` ( O ` ( L ` g ) ) ) = ( L ` g ) } <-> ( f e. F /\ ( O ` ( O ` ( L ` f ) ) ) = ( L ` f ) ) )
14	13	anbi1i 731	. . . . 5 |- ( ( f e. { g e. F | ( O ` ( O ` ( L ` g ) ) ) = ( L ` g ) } /\ E. v e. T ( O ` ( L ` f ) ) = ( ( LSpan ` U ) ` { v } ) ) <-> ( ( f e. F /\ ( O ` ( O ` ( L ` f ) ) ) = ( L ` f ) ) /\ E. v e. T ( O ` ( L ` f ) ) = ( ( LSpan ` U ) ` { v } ) ) )
15		anass 681	. . . . 5 |- ( ( ( f e. F /\ ( O ` ( O ` ( L ` f ) ) ) = ( L ` f ) ) /\ E. v e. T ( O ` ( L ` f ) ) = ( ( LSpan ` U ) ` { v } ) ) <-> ( f e. F /\ ( ( O ` ( O ` ( L ` f ) ) ) = ( L ` f ) /\ E. v e. T ( O ` ( L ` f ) ) = ( ( LSpan ` U ) ` { v } ) ) ) )
16	14, 15	bitri 264	. . . 4 |- ( ( f e. { g e. F | ( O ` ( O ` ( L ` g ) ) ) = ( L ` g ) } /\ E. v e. T ( O ` ( L ` f ) ) = ( ( LSpan ` U ) ` { v } ) ) <-> ( f e. F /\ ( ( O ` ( O ` ( L ` f ) ) ) = ( L ` f ) /\ E. v e. T ( O ` ( L ` f ) ) = ( ( LSpan ` U ) ` { v } ) ) ) )
17		r19.42v 3092	. . . . . 6 |- ( E. v e. T ( ( O ` ( O ` ( L ` f ) ) ) = ( L ` f ) /\ ( O ` ( L ` f ) ) = ( ( LSpan ` U ) ` { v } ) ) <-> ( ( O ` ( O ` ( L ` f ) ) ) = ( L ` f ) /\ E. v e. T ( O ` ( L ` f ) ) = ( ( LSpan ` U ) ` { v } ) ) )
18		simprr 796	. . . . . . . . . 10 |- ( ( ( ( ph /\ f e. F ) /\ v e. T ) /\ ( ( O ` ( O ` ( L ` f ) ) ) = ( L ` f ) /\ ( O ` ( L ` f ) ) = ( ( LSpan ` U ) ` { v } ) ) ) -> ( O ` ( L ` f ) ) = ( ( LSpan ` U ) ` { v } ) )
19	18	fveq2d 6195	. . . . . . . . 9 |- ( ( ( ( ph /\ f e. F ) /\ v e. T ) /\ ( ( O ` ( O ` ( L ` f ) ) ) = ( L ` f ) /\ ( O ` ( L ` f ) ) = ( ( LSpan ` U ) ` { v } ) ) ) -> ( O ` ( O ` ( L ` f ) ) ) = ( O ` ( ( LSpan ` U ) ` { v } ) ) )
20		simprl 794	. . . . . . . . 9 |- ( ( ( ( ph /\ f e. F ) /\ v e. T ) /\ ( ( O ` ( O ` ( L ` f ) ) ) = ( L ` f ) /\ ( O ` ( L ` f ) ) = ( ( LSpan ` U ) ` { v } ) ) ) -> ( O ` ( O ` ( L ` f ) ) ) = ( L ` f ) )
21		eqid 2622	. . . . . . . . . 10 |- ( Base ` U ) = ( Base ` U )
22	9	adantr 481	. . . . . . . . . . . 12 |- ( ( ph /\ f e. F ) -> ( K e. HL /\ W e. H ) )
23	22	adantr 481	. . . . . . . . . . 11 |- ( ( ( ph /\ f e. F ) /\ v e. T ) -> ( K e. HL /\ W e. H ) )
24	23	adantr 481	. . . . . . . . . 10 |- ( ( ( ( ph /\ f e. F ) /\ v e. T ) /\ ( ( O ` ( O ` ( L ` f ) ) ) = ( L ` f ) /\ ( O ` ( L ` f ) ) = ( ( LSpan ` U ) ` { v } ) ) ) -> ( K e. HL /\ W e. H ) )
25	10	adantr 481	. . . . . . . . . . . . 13 |- ( ( ph /\ f e. F ) -> T e. S )
26	21, 3	lssel 18938	. . . . . . . . . . . . 13 |- ( ( T e. S /\ v e. T ) -> v e. ( Base ` U ) )
27	25, 26	sylan 488	. . . . . . . . . . . 12 |- ( ( ( ph /\ f e. F ) /\ v e. T ) -> v e. ( Base ` U ) )
28	27	snssd 4340	. . . . . . . . . . 11 |- ( ( ( ph /\ f e. F ) /\ v e. T ) -> { v } C_ ( Base ` U ) )
29	28	adantr 481	. . . . . . . . . 10 |- ( ( ( ( ph /\ f e. F ) /\ v e. T ) /\ ( ( O ` ( O ` ( L ` f ) ) ) = ( L ` f ) /\ ( O ` ( L ` f ) ) = ( ( LSpan ` U ) ` { v } ) ) ) -> { v } C_ ( Base ` U ) )
30	1, 2, 7, 21, 4, 24, 29	dochocsp 36668	. . . . . . . . 9 |- ( ( ( ( ph /\ f e. F ) /\ v e. T ) /\ ( ( O ` ( O ` ( L ` f ) ) ) = ( L ` f ) /\ ( O ` ( L ` f ) ) = ( ( LSpan ` U ) ` { v } ) ) ) -> ( O ` ( ( LSpan ` U ) ` { v } ) ) = ( O ` { v } ) )
31	19, 20, 30	3eqtr3rd 2665	. . . . . . . 8 |- ( ( ( ( ph /\ f e. F ) /\ v e. T ) /\ ( ( O ` ( O ` ( L ` f ) ) ) = ( L ` f ) /\ ( O ` ( L ` f ) ) = ( ( LSpan ` U ) ` { v } ) ) ) -> ( O ` { v } ) = ( L ` f ) )
32	27	adantr 481	. . . . . . . . . . 11 |- ( ( ( ( ph /\ f e. F ) /\ v e. T ) /\ ( O ` { v } ) = ( L ` f ) ) -> v e. ( Base ` U ) )
33		simpr 477	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ f e. F ) /\ v e. T ) /\ ( O ` { v } ) = ( L ` f ) ) -> ( O ` { v } ) = ( L ` f ) )
34	33	eqcomd 2628	. . . . . . . . . . 11 |- ( ( ( ( ph /\ f e. F ) /\ v e. T ) /\ ( O ` { v } ) = ( L ` f ) ) -> ( L ` f ) = ( O ` { v } ) )
35		sneq 4187	. . . . . . . . . . . . . 14 |- ( w = v -> { w } = { v } )
36	35	fveq2d 6195	. . . . . . . . . . . . 13 |- ( w = v -> ( O ` { w } ) = ( O ` { v } ) )
37	36	eqeq2d 2632	. . . . . . . . . . . 12 |- ( w = v -> ( ( L ` f ) = ( O ` { w } ) <-> ( L ` f ) = ( O ` { v } ) ) )
38	37	rspcev 3309	. . . . . . . . . . 11 |- ( ( v e. ( Base ` U ) /\ ( L ` f ) = ( O ` { v } ) ) -> E. w e. ( Base ` U ) ( L ` f ) = ( O ` { w } ) )
39	32, 34, 38	syl2anc 693	. . . . . . . . . 10 |- ( ( ( ( ph /\ f e. F ) /\ v e. T ) /\ ( O ` { v } ) = ( L ` f ) ) -> E. w e. ( Base ` U ) ( L ` f ) = ( O ` { w } ) )
40	23	adantr 481	. . . . . . . . . . 11 |- ( ( ( ( ph /\ f e. F ) /\ v e. T ) /\ ( O ` { v } ) = ( L ` f ) ) -> ( K e. HL /\ W e. H ) )
41		simpllr 799	. . . . . . . . . . 11 |- ( ( ( ( ph /\ f e. F ) /\ v e. T ) /\ ( O ` { v } ) = ( L ` f ) ) -> f e. F )
42	1, 7, 2, 21, 5, 6, 40, 41	lcfl8a 36792	. . . . . . . . . 10 |- ( ( ( ( ph /\ f e. F ) /\ v e. T ) /\ ( O ` { v } ) = ( L ` f ) ) -> ( ( O ` ( O ` ( L ` f ) ) ) = ( L ` f ) <-> E. w e. ( Base ` U ) ( L ` f ) = ( O ` { w } ) ) )
43	39, 42	mpbird 247	. . . . . . . . 9 |- ( ( ( ( ph /\ f e. F ) /\ v e. T ) /\ ( O ` { v } ) = ( L ` f ) ) -> ( O ` ( O ` ( L ` f ) ) ) = ( L ` f ) )
44	1, 2, 7, 21, 4, 23, 27	dochocsn 36670	. . . . . . . . . . 11 |- ( ( ( ph /\ f e. F ) /\ v e. T ) -> ( O ` ( O ` { v } ) ) = ( ( LSpan ` U ) ` { v } ) )
45		fveq2 6191	. . . . . . . . . . 11 |- ( ( O ` { v } ) = ( L ` f ) -> ( O ` ( O ` { v } ) ) = ( O ` ( L ` f ) ) )
46	44, 45	sylan9req 2677	. . . . . . . . . 10 |- ( ( ( ( ph /\ f e. F ) /\ v e. T ) /\ ( O ` { v } ) = ( L ` f ) ) -> ( ( LSpan ` U ) ` { v } ) = ( O ` ( L ` f ) ) )
47	46	eqcomd 2628	. . . . . . . . 9 |- ( ( ( ( ph /\ f e. F ) /\ v e. T ) /\ ( O ` { v } ) = ( L ` f ) ) -> ( O ` ( L ` f ) ) = ( ( LSpan ` U ) ` { v } ) )
48	43, 47	jca 554	. . . . . . . 8 |- ( ( ( ( ph /\ f e. F ) /\ v e. T ) /\ ( O ` { v } ) = ( L ` f ) ) -> ( ( O ` ( O ` ( L ` f ) ) ) = ( L ` f ) /\ ( O ` ( L ` f ) ) = ( ( LSpan ` U ) ` { v } ) ) )
49	31, 48	impbida 877	. . . . . . 7 |- ( ( ( ph /\ f e. F ) /\ v e. T ) -> ( ( ( O ` ( O ` ( L ` f ) ) ) = ( L ` f ) /\ ( O ` ( L ` f ) ) = ( ( LSpan ` U ) ` { v } ) ) <-> ( O ` { v } ) = ( L ` f ) ) )
50	49	rexbidva 3049	. . . . . 6 |- ( ( ph /\ f e. F ) -> ( E. v e. T ( ( O ` ( O ` ( L ` f ) ) ) = ( L ` f ) /\ ( O ` ( L ` f ) ) = ( ( LSpan ` U ) ` { v } ) ) <-> E. v e. T ( O ` { v } ) = ( L ` f ) ) )
51	17, 50	syl5bbr 274	. . . . 5 |- ( ( ph /\ f e. F ) -> ( ( ( O ` ( O ` ( L ` f ) ) ) = ( L ` f ) /\ E. v e. T ( O ` ( L ` f ) ) = ( ( LSpan ` U ) ` { v } ) ) <-> E. v e. T ( O ` { v } ) = ( L ` f ) ) )
52	51	pm5.32da 673	. . . 4 |- ( ph -> ( ( f e. F /\ ( ( O ` ( O ` ( L ` f ) ) ) = ( L ` f ) /\ E. v e. T ( O ` ( L ` f ) ) = ( ( LSpan ` U ) ` { v } ) ) ) <-> ( f e. F /\ E. v e. T ( O ` { v } ) = ( L ` f ) ) ) )
53	16, 52	syl5bb 272	. . 3 |- ( ph -> ( ( f e. { g e. F | ( O ` ( O ` ( L ` g ) ) ) = ( L ` g ) } /\ E. v e. T ( O ` ( L ` f ) ) = ( ( LSpan ` U ) ` { v } ) ) <-> ( f e. F /\ E. v e. T ( O ` { v } ) = ( L ` f ) ) ) )
54	53	rabbidva2 3186	. 2 |- ( ph -> { f e. { g e. F | ( O ` ( O ` ( L ` g ) ) ) = ( L ` g ) } | E. v e. T ( O ` ( L ` f ) ) = ( ( LSpan ` U ) ` { v } ) } = { f e. F | E. v e. T ( O ` { v } ) = ( L ` f ) } )
55	12, 54	eqtrd 2656	1 |- ( ph -> ( M ` T ) = { f e. F | E. v e. T ( O ` { v } ) = ( L ` f ) } )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   {crab 2916   C_ wss 3574   {csn 4177   ` cfv 5888   Basecbs 15857   LSubSpclss 18932   LSpanclspn 18971  LFnlclfn 34344  LKerclk 34372   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-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-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-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-lfl 34345  df-lkr 34373  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:  mapdval5N  36922  mapd1dim2lem1N  36933