Theorem djafvalN 36423

Description: Subspace join for DVecA partial vector space. TODO: take out hypothesis .i, no longer used. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
djaval.h	|- H = ( LHyp ` K )
djaval.t	|- T = ( ( LTrn ` K ) ` W )
djaval.i	|- I = ( ( DIsoA ` K ) ` W )
djaval.n	|- ._|_ = ( ( ocA ` K ) ` W )
djaval.j	|- J = ( ( vA ` K ) ` W )

Assertion

Ref	Expression
djafvalN	|- ( ( K e. V /\ W e. H ) -> J = ( x e. ~P T , y e. ~P T |-> ( ._|_ ` ( ( ._|_ ` x ) i^i ( ._|_ ` y ) ) ) ) )

Distinct variable groups:   x, y, K   x, T, y   x, W, y

Allowed substitution hints:   H( x, y)   I( x, y)   J( x, y)   ._|_ ( x, y)   V( x, y)

Proof of Theorem djafvalN

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		djaval.j	. . 3 |- J = ( ( vA ` K ) ` W )
2		djaval.h	. . . . 5 |- H = ( LHyp ` K )
3	2	djaffvalN 36422	. . . 4 |- ( K e. V -> ( vA ` K ) = ( w e. H |-> ( x e. ~P ( ( LTrn ` K ) ` w ) , y e. ~P ( ( LTrn ` K ) ` w ) |-> ( ( ( ocA ` K ) ` w ) ` ( ( ( ( ocA ` K ) ` w ) ` x ) i^i ( ( ( ocA ` K ) ` w ) ` y ) ) ) ) ) )
4	3	fveq1d 6193	. . 3 |- ( K e. V -> ( ( vA ` K ) ` W ) = ( ( w e. H |-> ( x e. ~P ( ( LTrn ` K ) ` w ) , y e. ~P ( ( LTrn ` K ) ` w ) |-> ( ( ( ocA ` K ) ` w ) ` ( ( ( ( ocA ` K ) ` w ) ` x ) i^i ( ( ( ocA ` K ) ` w ) ` y ) ) ) ) ) ` W ) )
5	1, 4	syl5eq 2668	. 2 |- ( K e. V -> J = ( ( w e. H |-> ( x e. ~P ( ( LTrn ` K ) ` w ) , y e. ~P ( ( LTrn ` K ) ` w ) |-> ( ( ( ocA ` K ) ` w ) ` ( ( ( ( ocA ` K ) ` w ) ` x ) i^i ( ( ( ocA ` K ) ` w ) ` y ) ) ) ) ) ` W ) )
6		fveq2 6191	. . . . . 6 |- ( w = W -> ( ( LTrn ` K ) ` w ) = ( ( LTrn ` K ) ` W ) )
7		djaval.t	. . . . . 6 |- T = ( ( LTrn ` K ) ` W )
8	6, 7	syl6eqr 2674	. . . . 5 |- ( w = W -> ( ( LTrn ` K ) ` w ) = T )
9	8	pweqd 4163	. . . 4 |- ( w = W -> ~P ( ( LTrn ` K ) ` w ) = ~P T )
10		fveq2 6191	. . . . . 6 |- ( w = W -> ( ( ocA ` K ) ` w ) = ( ( ocA ` K ) ` W ) )
11		djaval.n	. . . . . 6 |- ._|_ = ( ( ocA ` K ) ` W )
12	10, 11	syl6eqr 2674	. . . . 5 |- ( w = W -> ( ( ocA ` K ) ` w ) = ._|_ )
13	12	fveq1d 6193	. . . . . 6 |- ( w = W -> ( ( ( ocA ` K ) ` w ) ` x ) = ( ._|_ ` x ) )
14	12	fveq1d 6193	. . . . . 6 |- ( w = W -> ( ( ( ocA ` K ) ` w ) ` y ) = ( ._|_ ` y ) )
15	13, 14	ineq12d 3815	. . . . 5 |- ( w = W -> ( ( ( ( ocA ` K ) ` w ) ` x ) i^i ( ( ( ocA ` K ) ` w ) ` y ) ) = ( ( ._|_ ` x ) i^i ( ._|_ ` y ) ) )
16	12, 15	fveq12d 6197	. . . 4 |- ( w = W -> ( ( ( ocA ` K ) ` w ) ` ( ( ( ( ocA ` K ) ` w ) ` x ) i^i ( ( ( ocA ` K ) ` w ) ` y ) ) ) = ( ._|_ ` ( ( ._|_ ` x ) i^i ( ._|_ ` y ) ) ) )
17	9, 9, 16	mpt2eq123dv 6717	. . 3 |- ( w = W -> ( x e. ~P ( ( LTrn ` K ) ` w ) , y e. ~P ( ( LTrn ` K ) ` w ) |-> ( ( ( ocA ` K ) ` w ) ` ( ( ( ( ocA ` K ) ` w ) ` x ) i^i ( ( ( ocA ` K ) ` w ) ` y ) ) ) ) = ( x e. ~P T , y e. ~P T |-> ( ._|_ ` ( ( ._|_ ` x ) i^i ( ._|_ ` y ) ) ) ) )
18		eqid 2622	. . 3 |- ( w e. H |-> ( x e. ~P ( ( LTrn ` K ) ` w ) , y e. ~P ( ( LTrn ` K ) ` w ) |-> ( ( ( ocA ` K ) ` w ) ` ( ( ( ( ocA ` K ) ` w ) ` x ) i^i ( ( ( ocA ` K ) ` w ) ` y ) ) ) ) ) = ( w e. H |-> ( x e. ~P ( ( LTrn ` K ) ` w ) , y e. ~P ( ( LTrn ` K ) ` w ) |-> ( ( ( ocA ` K ) ` w ) ` ( ( ( ( ocA ` K ) ` w ) ` x ) i^i ( ( ( ocA ` K ) ` w ) ` y ) ) ) ) )
19		fvex 6201	. . . . . 6 |- ( ( LTrn ` K ) ` W ) e. _V
20	7, 19	eqeltri 2697	. . . . 5 |- T e. _V
21	20	pwex 4848	. . . 4 |- ~P T e. _V
22	21, 21	mpt2ex 7247	. . 3 |- ( x e. ~P T , y e. ~P T |-> ( ._|_ ` ( ( ._|_ ` x ) i^i ( ._|_ ` y ) ) ) ) e. _V
23	17, 18, 22	fvmpt 6282	. 2 |- ( W e. H -> ( ( w e. H |-> ( x e. ~P ( ( LTrn ` K ) ` w ) , y e. ~P ( ( LTrn ` K ) ` w ) |-> ( ( ( ocA ` K ) ` w ) ` ( ( ( ( ocA ` K ) ` w ) ` x ) i^i ( ( ( ocA ` K ) ` w ) ` y ) ) ) ) ) ` W ) = ( x e. ~P T , y e. ~P T |-> ( ._|_ ` ( ( ._|_ ` x ) i^i ( ._|_ ` y ) ) ) ) )
24	5, 23	sylan9eq 2676	1 |- ( ( K e. V /\ W e. H ) -> J = ( x e. ~P T , y e. ~P T |-> ( ._|_ ` ( ( ._|_ ` x ) i^i ( ._|_ ` y ) ) ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   i^i cin 3573   ~Pcpw 4158   |-> cmpt 4729   ` cfv 5888   |-> cmpt2 6652   LHypclh 35270   LTrncltrn 35387   DIsoAcdia 36317   ocAcocaN 36408   vAcdjaN 36420

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

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-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-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-djaN 36421

This theorem is referenced by:  djavalN  36424