Theorem docavalN 36412

Description: Subspace orthocomplement for DVecA partial vector space. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
docaval.j	|- .\/ = ( join ` K )
docaval.m	|- ./\ = ( meet ` K )
docaval.o	|- ._|_ = ( oc ` K )
docaval.h	|- H = ( LHyp ` K )
docaval.t	|- T = ( ( LTrn ` K ) ` W )
docaval.i	|- I = ( ( DIsoA ` K ) ` W )
docaval.n	|- N = ( ( ocA ` K ) ` W )

Assertion

Ref	Expression
docavalN	|- ( ( ( K e. HL /\ W e. H ) /\ X C_ T ) -> ( N ` X ) = ( I ` ( ( ( ._|_ ` ( `' I ` |^| { z e. ran I | X C_ z } ) ) .\/ ( ._|_ ` W ) ) ./\ W ) ) )

Distinct variable groups:   z, K   z, I   z, W   z, T   z, X

Allowed substitution hints:   H( z)   .\/ ( z)   ./\ ( z)   N( z)   ._|_ ( z)

Proof of Theorem docavalN

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		docaval.j	. . . . 5 |- .\/ = ( join ` K )
2		docaval.m	. . . . 5 |- ./\ = ( meet ` K )
3		docaval.o	. . . . 5 |- ._|_ = ( oc ` K )
4		docaval.h	. . . . 5 |- H = ( LHyp ` K )
5		docaval.t	. . . . 5 |- T = ( ( LTrn ` K ) ` W )
6		docaval.i	. . . . 5 |- I = ( ( DIsoA ` K ) ` W )
7		docaval.n	. . . . 5 |- N = ( ( ocA ` K ) ` W )
8	1, 2, 3, 4, 5, 6, 7	docafvalN 36411	. . . 4 |- ( ( K e. HL /\ W e. H ) -> N = ( x e. ~P T |-> ( I ` ( ( ( ._|_ ` ( `' I ` |^| { z e. ran I | x C_ z } ) ) .\/ ( ._|_ ` W ) ) ./\ W ) ) ) )
9	8	adantr 481	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ X C_ T ) -> N = ( x e. ~P T |-> ( I ` ( ( ( ._|_ ` ( `' I ` |^| { z e. ran I | x C_ z } ) ) .\/ ( ._|_ ` W ) ) ./\ W ) ) ) )
10	9	fveq1d 6193	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ X C_ T ) -> ( N ` X ) = ( ( x e. ~P T |-> ( I ` ( ( ( ._|_ ` ( `' I ` |^| { z e. ran I | x C_ z } ) ) .\/ ( ._|_ ` W ) ) ./\ W ) ) ) ` X ) )
11		fvex 6201	. . . . . . 7 |- ( ( LTrn ` K ) ` W ) e. _V
12	5, 11	eqeltri 2697	. . . . . 6 |- T e. _V
13	12	elpw2 4828	. . . . 5 |- ( X e. ~P T <-> X C_ T )
14	13	biimpri 218	. . . 4 |- ( X C_ T -> X e. ~P T )
15	14	adantl 482	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ X C_ T ) -> X e. ~P T )
16		fvex 6201	. . 3 |- ( I ` ( ( ( ._|_ ` ( `' I ` |^| { z e. ran I | X C_ z } ) ) .\/ ( ._|_ ` W ) ) ./\ W ) ) e. _V
17		sseq1 3626	. . . . . . . . . . 11 |- ( x = X -> ( x C_ z <-> X C_ z ) )
18	17	rabbidv 3189	. . . . . . . . . 10 |- ( x = X -> { z e. ran I | x C_ z } = { z e. ran I | X C_ z } )
19	18	inteqd 4480	. . . . . . . . 9 |- ( x = X -> |^| { z e. ran I | x C_ z } = |^| { z e. ran I | X C_ z } )
20	19	fveq2d 6195	. . . . . . . 8 |- ( x = X -> ( `' I ` |^| { z e. ran I | x C_ z } ) = ( `' I ` |^| { z e. ran I | X C_ z } ) )
21	20	fveq2d 6195	. . . . . . 7 |- ( x = X -> ( ._|_ ` ( `' I ` |^| { z e. ran I | x C_ z } ) ) = ( ._|_ ` ( `' I ` |^| { z e. ran I | X C_ z } ) ) )
22	21	oveq1d 6665	. . . . . 6 |- ( x = X -> ( ( ._|_ ` ( `' I ` |^| { z e. ran I | x C_ z } ) ) .\/ ( ._|_ ` W ) ) = ( ( ._|_ ` ( `' I ` |^| { z e. ran I | X C_ z } ) ) .\/ ( ._|_ ` W ) ) )
23	22	oveq1d 6665	. . . . 5 |- ( x = X -> ( ( ( ._|_ ` ( `' I ` |^| { z e. ran I | x C_ z } ) ) .\/ ( ._|_ ` W ) ) ./\ W ) = ( ( ( ._|_ ` ( `' I ` |^| { z e. ran I | X C_ z } ) ) .\/ ( ._|_ ` W ) ) ./\ W ) )
24	23	fveq2d 6195	. . . 4 |- ( x = X -> ( I ` ( ( ( ._|_ ` ( `' I ` |^| { z e. ran I | x C_ z } ) ) .\/ ( ._|_ ` W ) ) ./\ W ) ) = ( I ` ( ( ( ._|_ ` ( `' I ` |^| { z e. ran I | X C_ z } ) ) .\/ ( ._|_ ` W ) ) ./\ W ) ) )
25		eqid 2622	. . . 4 |- ( x e. ~P T |-> ( I ` ( ( ( ._|_ ` ( `' I ` |^| { z e. ran I | x C_ z } ) ) .\/ ( ._|_ ` W ) ) ./\ W ) ) ) = ( x e. ~P T |-> ( I ` ( ( ( ._|_ ` ( `' I ` |^| { z e. ran I | x C_ z } ) ) .\/ ( ._|_ ` W ) ) ./\ W ) ) )
26	24, 25	fvmptg 6280	. . 3 |- ( ( X e. ~P T /\ ( I ` ( ( ( ._|_ ` ( `' I ` |^| { z e. ran I | X C_ z } ) ) .\/ ( ._|_ ` W ) ) ./\ W ) ) e. _V ) -> ( ( x e. ~P T |-> ( I ` ( ( ( ._|_ ` ( `' I ` |^| { z e. ran I | x C_ z } ) ) .\/ ( ._|_ ` W ) ) ./\ W ) ) ) ` X ) = ( I ` ( ( ( ._|_ ` ( `' I ` |^| { z e. ran I | X C_ z } ) ) .\/ ( ._|_ ` W ) ) ./\ W ) ) )
27	15, 16, 26	sylancl 694	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ X C_ T ) -> ( ( x e. ~P T |-> ( I ` ( ( ( ._|_ ` ( `' I ` |^| { z e. ran I | x C_ z } ) ) .\/ ( ._|_ ` W ) ) ./\ W ) ) ) ` X ) = ( I ` ( ( ( ._|_ ` ( `' I ` |^| { z e. ran I | X C_ z } ) ) .\/ ( ._|_ ` W ) ) ./\ W ) ) )
28	10, 27	eqtrd 2656	1 |- ( ( ( K e. HL /\ W e. H ) /\ X C_ T ) -> ( N ` X ) = ( I ` ( ( ( ._|_ ` ( `' I ` |^| { z e. ran I | X C_ z } ) ) .\/ ( ._|_ ` W ) ) ./\ W ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   _Vcvv 3200   C_ wss 3574   ~Pcpw 4158   |^|cint 4475   |-> cmpt 4729   `'ccnv 5113   ran crn 5115   ` cfv 5888  (class class class)co 6650   occoc 15949   joincjn 16944   meetcmee 16945   HLchlt 34637   LHypclh 35270   LTrncltrn 35387   DIsoAcdia 36317   ocAcocaN 36408

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-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-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-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-int 4476  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-ov 6653  df-docaN 36409

This theorem is referenced by:  docaclN  36413  diaocN  36414