Theorem docaffvalN 36410

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 )

Assertion

Ref	Expression
docaffvalN	|- ( 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 ) ) ) ) )

Distinct variable groups:   w, H   x, w, z, K

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

Proof of Theorem docaffvalN

Dummy variable k is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3212	. 2 |- ( K e. V -> K e. _V )
2		fveq2 6191	. . . . 5 |- ( k = K -> ( LHyp ` k ) = ( LHyp ` K ) )
3		docaval.h	. . . . 5 |- H = ( LHyp ` K )
4	2, 3	syl6eqr 2674	. . . 4 |- ( k = K -> ( LHyp ` k ) = H )
5		fveq2 6191	. . . . . . 7 |- ( k = K -> ( LTrn ` k ) = ( LTrn ` K ) )
6	5	fveq1d 6193	. . . . . 6 |- ( k = K -> ( ( LTrn ` k ) ` w ) = ( ( LTrn ` K ) ` w ) )
7	6	pweqd 4163	. . . . 5 |- ( k = K -> ~P ( ( LTrn ` k ) ` w ) = ~P ( ( LTrn ` K ) ` w ) )
8		fveq2 6191	. . . . . . 7 |- ( k = K -> ( DIsoA ` k ) = ( DIsoA ` K ) )
9	8	fveq1d 6193	. . . . . 6 |- ( k = K -> ( ( DIsoA ` k ) ` w ) = ( ( DIsoA ` K ) ` w ) )
10		fveq2 6191	. . . . . . . 8 |- ( k = K -> ( meet ` k ) = ( meet ` K ) )
11		docaval.m	. . . . . . . 8 |- ./\ = ( meet ` K )
12	10, 11	syl6eqr 2674	. . . . . . 7 |- ( k = K -> ( meet ` k ) = ./\ )
13		fveq2 6191	. . . . . . . . 9 |- ( k = K -> ( join ` k ) = ( join ` K ) )
14		docaval.j	. . . . . . . . 9 |- .\/ = ( join ` K )
15	13, 14	syl6eqr 2674	. . . . . . . 8 |- ( k = K -> ( join ` k ) = .\/ )
16		fveq2 6191	. . . . . . . . . 10 |- ( k = K -> ( oc ` k ) = ( oc ` K ) )
17		docaval.o	. . . . . . . . . 10 |- ._|_ = ( oc ` K )
18	16, 17	syl6eqr 2674	. . . . . . . . 9 |- ( k = K -> ( oc ` k ) = ._|_ )
19	9	cnveqd 5298	. . . . . . . . . 10 |- ( k = K -> `' ( ( DIsoA ` k ) ` w ) = `' ( ( DIsoA ` K ) ` w ) )
20	9	rneqd 5353	. . . . . . . . . . . 12 |- ( k = K -> ran ( ( DIsoA ` k ) ` w ) = ran ( ( DIsoA ` K ) ` w ) )
21		rabeq 3192	. . . . . . . . . . . 12 |- ( ran ( ( DIsoA ` k ) ` w ) = ran ( ( DIsoA ` K ) ` w ) -> { z e. ran ( ( DIsoA ` k ) ` w ) | x C_ z } = { z e. ran ( ( DIsoA ` K ) ` w ) | x C_ z } )
22	20, 21	syl 17	. . . . . . . . . . 11 |- ( k = K -> { z e. ran ( ( DIsoA ` k ) ` w ) | x C_ z } = { z e. ran ( ( DIsoA ` K ) ` w ) | x C_ z } )
23	22	inteqd 4480	. . . . . . . . . 10 |- ( k = K -> |^| { z e. ran ( ( DIsoA ` k ) ` w ) | x C_ z } = |^| { z e. ran ( ( DIsoA ` K ) ` w ) | x C_ z } )
24	19, 23	fveq12d 6197	. . . . . . . . 9 |- ( k = K -> ( `' ( ( DIsoA ` k ) ` w ) ` |^| { z e. ran ( ( DIsoA ` k ) ` w ) | x C_ z } ) = ( `' ( ( DIsoA ` K ) ` w ) ` |^| { z e. ran ( ( DIsoA ` K ) ` w ) | x C_ z } ) )
25	18, 24	fveq12d 6197	. . . . . . . 8 |- ( k = K -> ( ( oc ` k ) ` ( `' ( ( DIsoA ` k ) ` w ) ` |^| { z e. ran ( ( DIsoA ` k ) ` w ) | x C_ z } ) ) = ( ._|_ ` ( `' ( ( DIsoA ` K ) ` w ) ` |^| { z e. ran ( ( DIsoA ` K ) ` w ) | x C_ z } ) ) )
26	18	fveq1d 6193	. . . . . . . 8 |- ( k = K -> ( ( oc ` k ) ` w ) = ( ._|_ ` w ) )
27	15, 25, 26	oveq123d 6671	. . . . . . 7 |- ( k = K -> ( ( ( oc ` k ) ` ( `' ( ( DIsoA ` k ) ` w ) ` |^| { z e. ran ( ( DIsoA ` k ) ` w ) | x C_ z } ) ) ( join ` k ) ( ( oc ` k ) ` w ) ) = ( ( ._|_ ` ( `' ( ( DIsoA ` K ) ` w ) ` |^| { z e. ran ( ( DIsoA ` K ) ` w ) | x C_ z } ) ) .\/ ( ._|_ ` w ) ) )
28		eqidd 2623	. . . . . . 7 |- ( k = K -> w = w )
29	12, 27, 28	oveq123d 6671	. . . . . 6 |- ( k = K -> ( ( ( ( oc ` k ) ` ( `' ( ( DIsoA ` k ) ` w ) ` |^| { z e. ran ( ( DIsoA ` k ) ` w ) | x C_ z } ) ) ( join ` k ) ( ( oc ` k ) ` w ) ) ( meet ` k ) w ) = ( ( ( ._|_ ` ( `' ( ( DIsoA ` K ) ` w ) ` |^| { z e. ran ( ( DIsoA ` K ) ` w ) | x C_ z } ) ) .\/ ( ._|_ ` w ) ) ./\ w ) )
30	9, 29	fveq12d 6197	. . . . 5 |- ( k = K -> ( ( ( DIsoA ` k ) ` w ) ` ( ( ( ( oc ` k ) ` ( `' ( ( DIsoA ` k ) ` w ) ` |^| { z e. ran ( ( DIsoA ` k ) ` w ) | x C_ z } ) ) ( join ` k ) ( ( oc ` k ) ` w ) ) ( meet ` k ) w ) ) = ( ( ( DIsoA ` K ) ` w ) ` ( ( ( ._|_ ` ( `' ( ( DIsoA ` K ) ` w ) ` |^| { z e. ran ( ( DIsoA ` K ) ` w ) | x C_ z } ) ) .\/ ( ._|_ ` w ) ) ./\ w ) ) )
31	7, 30	mpteq12dv 4733	. . . 4 |- ( k = K -> ( x e. ~P ( ( LTrn ` k ) ` w ) |-> ( ( ( DIsoA ` k ) ` w ) ` ( ( ( ( oc ` k ) ` ( `' ( ( DIsoA ` k ) ` w ) ` |^| { z e. ran ( ( DIsoA ` k ) ` w ) | x C_ z } ) ) ( join ` k ) ( ( oc ` k ) ` w ) ) ( meet ` k ) w ) ) ) = ( x e. ~P ( ( LTrn ` K ) ` w ) |-> ( ( ( DIsoA ` K ) ` w ) ` ( ( ( ._|_ ` ( `' ( ( DIsoA ` K ) ` w ) ` |^| { z e. ran ( ( DIsoA ` K ) ` w ) | x C_ z } ) ) .\/ ( ._|_ ` w ) ) ./\ w ) ) ) )
32	4, 31	mpteq12dv 4733	. . 3 |- ( k = K -> ( w e. ( LHyp ` k ) |-> ( x e. ~P ( ( LTrn ` k ) ` w ) |-> ( ( ( DIsoA ` k ) ` w ) ` ( ( ( ( oc ` k ) ` ( `' ( ( DIsoA ` k ) ` w ) ` |^| { z e. ran ( ( DIsoA ` k ) ` w ) | x C_ z } ) ) ( join ` k ) ( ( oc ` k ) ` w ) ) ( meet ` 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 ) ) ) ) )
33		df-docaN 36409	. . 3 |- ocA = ( k e. _V |-> ( w e. ( LHyp ` k ) |-> ( x e. ~P ( ( LTrn ` k ) ` w ) |-> ( ( ( DIsoA ` k ) ` w ) ` ( ( ( ( oc ` k ) ` ( `' ( ( DIsoA ` k ) ` w ) ` |^| { z e. ran ( ( DIsoA ` k ) ` w ) | x C_ z } ) ) ( join ` k ) ( ( oc ` k ) ` w ) ) ( meet ` k ) w ) ) ) ) )
34		fvex 6201	. . . . 5 |- ( LHyp ` K ) e. _V
35	3, 34	eqeltri 2697	. . . 4 |- H e. _V
36	35	mptex 6486	. . 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 ) ) ) ) e. _V
37	32, 33, 36	fvmpt 6282	. 2 |- ( 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 ) ) ) ) )
38	1, 37	syl 17	1 |- ( 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 ) ) ) ) )

Syntax hints:   -> wi 4   = 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-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:  docafvalN  36411