Theorem docafvalN 36411

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
docafvalN	|- ( ( K e. V /\ W e. H ) -> N = ( x e. ~P T |-> ( I ` ( ( ( ._|_ ` ( `' I ` |^| { z e. ran I | x C_ z } ) ) .\/ ( ._|_ ` W ) ) ./\ W ) ) ) )

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

Allowed substitution hints:   T( z)   H( x, z)   .\/ ( x, z)   ./\ ( x, z)   N( x, z)   ._|_ ( x, z)   V( x, z)

Proof of Theorem docafvalN

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		docaval.n	. . 3 |- N = ( ( ocA ` K ) ` W )
2		docaval.j	. . . . 5 |- .\/ = ( join ` K )
3		docaval.m	. . . . 5 |- ./\ = ( meet ` K )
4		docaval.o	. . . . 5 |- ._|_ = ( oc ` K )
5		docaval.h	. . . . 5 |- H = ( LHyp ` K )
6	2, 3, 4, 5	docaffvalN 36410	. . . 4 |- ( K e. V -> ( ocA ` K ) = ( w e. H |-> ( x e. ~P ( ( LTrn ` K ) ` w ) |-> ( ( ( DIsoA ` K ) ` w ) ` ( ( ( ._|_ ` ( `' ( ( DIsoA ` K ) ` w ) ` |^| { z e. ran ( ( DIsoA ` K ) ` w ) | x C_ z } ) ) .\/ ( ._|_ ` w ) ) ./\ w ) ) ) ) )
7	6	fveq1d 6193	. . 3 |- ( K e. V -> ( ( ocA ` K ) ` W ) = ( ( w e. H |-> ( x e. ~P ( ( LTrn ` K ) ` w ) |-> ( ( ( DIsoA ` K ) ` w ) ` ( ( ( ._|_ ` ( `' ( ( DIsoA ` K ) ` w ) ` |^| { z e. ran ( ( DIsoA ` K ) ` w ) | x C_ z } ) ) .\/ ( ._|_ ` w ) ) ./\ w ) ) ) ) ` W ) )
8	1, 7	syl5eq 2668	. 2 |- ( K e. V -> N = ( ( w e. H |-> ( x e. ~P ( ( LTrn ` K ) ` w ) |-> ( ( ( DIsoA ` K ) ` w ) ` ( ( ( ._|_ ` ( `' ( ( DIsoA ` K ) ` w ) ` |^| { z e. ran ( ( DIsoA ` K ) ` w ) | x C_ z } ) ) .\/ ( ._|_ ` w ) ) ./\ w ) ) ) ) ` W ) )
9		fveq2 6191	. . . . . 6 |- ( w = W -> ( ( LTrn ` K ) ` w ) = ( ( LTrn ` K ) ` W ) )
10		docaval.t	. . . . . 6 |- T = ( ( LTrn ` K ) ` W )
11	9, 10	syl6eqr 2674	. . . . 5 |- ( w = W -> ( ( LTrn ` K ) ` w ) = T )
12	11	pweqd 4163	. . . 4 |- ( w = W -> ~P ( ( LTrn ` K ) ` w ) = ~P T )
13		fveq2 6191	. . . . . 6 |- ( w = W -> ( ( DIsoA ` K ) ` w ) = ( ( DIsoA ` K ) ` W ) )
14		docaval.i	. . . . . 6 |- I = ( ( DIsoA ` K ) ` W )
15	13, 14	syl6eqr 2674	. . . . 5 |- ( w = W -> ( ( DIsoA ` K ) ` w ) = I )
16	15	cnveqd 5298	. . . . . . . . 9 |- ( w = W -> `' ( ( DIsoA ` K ) ` w ) = `' I )
17	15	rneqd 5353	. . . . . . . . . . 11 |- ( w = W -> ran ( ( DIsoA ` K ) ` w ) = ran I )
18		rabeq 3192	. . . . . . . . . . 11 |- ( ran ( ( DIsoA ` K ) ` w ) = ran I -> { z e. ran ( ( DIsoA ` K ) ` w ) | x C_ z } = { z e. ran I | x C_ z } )
19	17, 18	syl 17	. . . . . . . . . 10 |- ( w = W -> { z e. ran ( ( DIsoA ` K ) ` w ) | x C_ z } = { z e. ran I | x C_ z } )
20	19	inteqd 4480	. . . . . . . . 9 |- ( w = W -> |^| { z e. ran ( ( DIsoA ` K ) ` w ) | x C_ z } = |^| { z e. ran I | x C_ z } )
21	16, 20	fveq12d 6197	. . . . . . . 8 |- ( w = W -> ( `' ( ( DIsoA ` K ) ` w ) ` |^| { z e. ran ( ( DIsoA ` K ) ` w ) | x C_ z } ) = ( `' I ` |^| { z e. ran I | x C_ z } ) )
22	21	fveq2d 6195	. . . . . . 7 |- ( w = W -> ( ._|_ ` ( `' ( ( DIsoA ` K ) ` w ) ` |^| { z e. ran ( ( DIsoA ` K ) ` w ) | x C_ z } ) ) = ( ._|_ ` ( `' I ` |^| { z e. ran I | x C_ z } ) ) )
23		fveq2 6191	. . . . . . 7 |- ( w = W -> ( ._|_ ` w ) = ( ._|_ ` W ) )
24	22, 23	oveq12d 6668	. . . . . 6 |- ( w = W -> ( ( ._|_ ` ( `' ( ( DIsoA ` K ) ` w ) ` |^| { z e. ran ( ( DIsoA ` K ) ` w ) | x C_ z } ) ) .\/ ( ._|_ ` w ) ) = ( ( ._|_ ` ( `' I ` |^| { z e. ran I | x C_ z } ) ) .\/ ( ._|_ ` W ) ) )
25		id 22	. . . . . 6 |- ( w = W -> w = W )
26	24, 25	oveq12d 6668	. . . . 5 |- ( w = W -> ( ( ( ._|_ ` ( `' ( ( DIsoA ` K ) ` w ) ` |^| { z e. ran ( ( DIsoA ` K ) ` w ) | x C_ z } ) ) .\/ ( ._|_ ` w ) ) ./\ w ) = ( ( ( ._|_ ` ( `' I ` |^| { z e. ran I | x C_ z } ) ) .\/ ( ._|_ ` W ) ) ./\ W ) )
27	15, 26	fveq12d 6197	. . . 4 |- ( w = W -> ( ( ( DIsoA ` K ) ` w ) ` ( ( ( ._|_ ` ( `' ( ( DIsoA ` K ) ` w ) ` |^| { z e. ran ( ( DIsoA ` K ) ` w ) | x C_ z } ) ) .\/ ( ._|_ ` w ) ) ./\ w ) ) = ( I ` ( ( ( ._|_ ` ( `' I ` |^| { z e. ran I | x C_ z } ) ) .\/ ( ._|_ ` W ) ) ./\ W ) ) )
28	12, 27	mpteq12dv 4733	. . 3 |- ( w = W -> ( x e. ~P ( ( LTrn ` K ) ` w ) |-> ( ( ( DIsoA ` K ) ` w ) ` ( ( ( ._|_ ` ( `' ( ( DIsoA ` K ) ` w ) ` |^| { z e. ran ( ( DIsoA ` K ) ` w ) | x C_ z } ) ) .\/ ( ._|_ ` w ) ) ./\ w ) ) ) = ( x e. ~P T |-> ( I ` ( ( ( ._|_ ` ( `' I ` |^| { z e. ran I | x C_ z } ) ) .\/ ( ._|_ ` W ) ) ./\ W ) ) ) )
29		eqid 2622	. . 3 |- ( w e. H |-> ( x e. ~P ( ( LTrn ` K ) ` w ) |-> ( ( ( DIsoA ` K ) ` w ) ` ( ( ( ._|_ ` ( `' ( ( DIsoA ` K ) ` w ) ` |^| { z e. ran ( ( DIsoA ` K ) ` w ) | x C_ z } ) ) .\/ ( ._|_ ` w ) ) ./\ w ) ) ) ) = ( w e. H |-> ( x e. ~P ( ( LTrn ` K ) ` w ) |-> ( ( ( DIsoA ` K ) ` w ) ` ( ( ( ._|_ ` ( `' ( ( DIsoA ` K ) ` w ) ` |^| { z e. ran ( ( DIsoA ` K ) ` w ) | x C_ z } ) ) .\/ ( ._|_ ` w ) ) ./\ w ) ) ) )
30		fvex 6201	. . . . . 6 |- ( ( LTrn ` K ) ` W ) e. _V
31	10, 30	eqeltri 2697	. . . . 5 |- T e. _V
32	31	pwex 4848	. . . 4 |- ~P T e. _V
33	32	mptex 6486	. . 3 |- ( x e. ~P T |-> ( I ` ( ( ( ._|_ ` ( `' I ` |^| { z e. ran I | x C_ z } ) ) .\/ ( ._|_ ` W ) ) ./\ W ) ) ) e. _V
34	28, 29, 33	fvmpt 6282	. 2 |- ( W e. H -> ( ( w e. H |-> ( x e. ~P ( ( LTrn ` K ) ` w ) |-> ( ( ( DIsoA ` K ) ` w ) ` ( ( ( ._|_ ` ( `' ( ( DIsoA ` K ) ` w ) ` |^| { z e. ran ( ( DIsoA ` K ) ` w ) | x C_ z } ) ) .\/ ( ._|_ ` w ) ) ./\ w ) ) ) ) ` W ) = ( x e. ~P T |-> ( I ` ( ( ( ._|_ ` ( `' I ` |^| { z e. ran I | x C_ z } ) ) .\/ ( ._|_ ` W ) ) ./\ W ) ) ) )
35	8, 34	sylan9eq 2676	1 |- ( ( K e. V /\ W e. H ) -> N = ( x e. ~P T |-> ( 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   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:  docavalN  36412