Theorem hlhilset 37226

Description: The final Hilbert space constructed from a Hilbert lattice K and an arbitrary hyperplane W in K. (Contributed by NM, 21-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.)

Hypotheses

Ref	Expression
hlhilset.h	|- H = ( LHyp ` K )
hlhilset.l	|- L = ( (HLHil ` K ) ` W )
hlhilset.u	|- U = ( ( DVecH ` K ) ` W )
hlhilset.v	|- V = ( Base ` U )
hlhilset.p	|- .+ = ( +g ` U )
hlhilset.e	|- E = ( ( EDRing ` K ) ` W )
hlhilset.g	|- G = ( (HGMap ` K ) ` W )
hlhilset.r	|- R = ( E sSet <. ( *r ` ndx ) , G >. )
hlhilset.t	|- .x. = ( .s ` U )
hlhilset.s	|- S = ( (HDMap ` K ) ` W )
hlhilset.i	|- ., = ( x e. V , y e. V |-> ( ( S ` y ) ` x ) )
hlhilset.k	|- ( ph -> ( K e. HL /\ W e. H ) )

Assertion

Ref	Expression
hlhilset	|- ( ph -> L = ( { <. ( Base ` ndx ) , V >. , <. ( +g ` ndx ) , .+ >. , <. (Scalar ` ndx ) , R >. } u. { <. ( .s ` ndx ) , .x. >. , <. ( .i ` ndx ) , ., >. } ) )

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

Allowed substitution hints:   .+ ( x, y)   R( x, y)   S( x, y)   .x. ( x, y)   U( x, y)   E( x, y)   G( x, y)   H( x, y)   ., ( x, y)   L( x, y)   V( x, y)

Proof of Theorem hlhilset

Dummy variables w k u v are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		hlhilset.l	. 2 |- L = ( (HLHil ` K ) ` W )
2		hlhilset.k	. . . . 5 |- ( ph -> ( K e. HL /\ W e. H ) )
3		elex 3212	. . . . . 6 |- ( K e. HL -> K e. _V )
4	3	adantr 481	. . . . 5 |- ( ( K e. HL /\ W e. H ) -> K e. _V )
5	2, 4	syl 17	. . . 4 |- ( ph -> K e. _V )
6		hlhilset.h	. . . . . 6 |- H = ( LHyp ` K )
7		fvex 6201	. . . . . 6 |- ( LHyp ` K ) e. _V
8	6, 7	eqeltri 2697	. . . . 5 |- H e. _V
9	8	mptex 6486	. . . 4 |- ( w e. H |-> [_ K / k ]_ [_ ( ( DVecH ` k ) ` w ) / u ]_ [_ ( Base ` u ) / v ]_ ( { <. ( Base ` ndx ) , v >. , <. ( +g ` ndx ) , ( +g ` u ) >. , <. (Scalar ` ndx ) , ( ( ( EDRing ` k ) ` w ) sSet <. ( *r ` ndx ) , ( (HGMap ` k ) ` w ) >. ) >. } u. { <. ( .s ` ndx ) , ( .s ` u ) >. , <. ( .i ` ndx ) , ( x e. v , y e. v |-> ( ( ( (HDMap ` k ) ` w ) ` y ) ` x ) ) >. } ) ) e. _V
10		nfcv 2764	. . . . 5 |- F/_ k K
11		nfcv 2764	. . . . . 6 |- F/_ k H
12		nfcsb1v 3549	. . . . . 6 |- F/_ k [_ K / k ]_ [_ ( ( DVecH ` k ) ` w ) / u ]_ [_ ( Base ` u ) / v ]_ ( { <. ( Base ` ndx ) , v >. , <. ( +g ` ndx ) , ( +g ` u ) >. , <. (Scalar ` ndx ) , ( ( ( EDRing ` k ) ` w ) sSet <. ( *r ` ndx ) , ( (HGMap ` k ) ` w ) >. ) >. } u. { <. ( .s ` ndx ) , ( .s ` u ) >. , <. ( .i ` ndx ) , ( x e. v , y e. v |-> ( ( ( (HDMap ` k ) ` w ) ` y ) ` x ) ) >. } )
13	11, 12	nfmpt 4746	. . . . 5 |- F/_ k ( w e. H |-> [_ K / k ]_ [_ ( ( DVecH ` k ) ` w ) / u ]_ [_ ( Base ` u ) / v ]_ ( { <. ( Base ` ndx ) , v >. , <. ( +g ` ndx ) , ( +g ` u ) >. , <. (Scalar ` ndx ) , ( ( ( EDRing ` k ) ` w ) sSet <. ( *r ` ndx ) , ( (HGMap ` k ) ` w ) >. ) >. } u. { <. ( .s ` ndx ) , ( .s ` u ) >. , <. ( .i ` ndx ) , ( x e. v , y e. v |-> ( ( ( (HDMap ` k ) ` w ) ` y ) ` x ) ) >. } ) )
14		fveq2 6191	. . . . . . 7 |- ( k = K -> ( LHyp ` k ) = ( LHyp ` K ) )
15	14, 6	syl6eqr 2674	. . . . . 6 |- ( k = K -> ( LHyp ` k ) = H )
16		csbeq1a 3542	. . . . . 6 |- ( k = K -> [_ ( ( DVecH ` k ) ` w ) / u ]_ [_ ( Base ` u ) / v ]_ ( { <. ( Base ` ndx ) , v >. , <. ( +g ` ndx ) , ( +g ` u ) >. , <. (Scalar ` ndx ) , ( ( ( EDRing ` k ) ` w ) sSet <. ( *r ` ndx ) , ( (HGMap ` k ) ` w ) >. ) >. } u. { <. ( .s ` ndx ) , ( .s ` u ) >. , <. ( .i ` ndx ) , ( x e. v , y e. v |-> ( ( ( (HDMap ` k ) ` w ) ` y ) ` x ) ) >. } ) = [_ K / k ]_ [_ ( ( DVecH ` k ) ` w ) / u ]_ [_ ( Base ` u ) / v ]_ ( { <. ( Base ` ndx ) , v >. , <. ( +g ` ndx ) , ( +g ` u ) >. , <. (Scalar ` ndx ) , ( ( ( EDRing ` k ) ` w ) sSet <. ( *r ` ndx ) , ( (HGMap ` k ) ` w ) >. ) >. } u. { <. ( .s ` ndx ) , ( .s ` u ) >. , <. ( .i ` ndx ) , ( x e. v , y e. v |-> ( ( ( (HDMap ` k ) ` w ) ` y ) ` x ) ) >. } ) )
17	15, 16	mpteq12dv 4733	. . . . 5 |- ( k = K -> ( w e. ( LHyp ` k ) |-> [_ ( ( DVecH ` k ) ` w ) / u ]_ [_ ( Base ` u ) / v ]_ ( { <. ( Base ` ndx ) , v >. , <. ( +g ` ndx ) , ( +g ` u ) >. , <. (Scalar ` ndx ) , ( ( ( EDRing ` k ) ` w ) sSet <. ( *r ` ndx ) , ( (HGMap ` k ) ` w ) >. ) >. } u. { <. ( .s ` ndx ) , ( .s ` u ) >. , <. ( .i ` ndx ) , ( x e. v , y e. v |-> ( ( ( (HDMap ` k ) ` w ) ` y ) ` x ) ) >. } ) ) = ( w e. H |-> [_ K / k ]_ [_ ( ( DVecH ` k ) ` w ) / u ]_ [_ ( Base ` u ) / v ]_ ( { <. ( Base ` ndx ) , v >. , <. ( +g ` ndx ) , ( +g ` u ) >. , <. (Scalar ` ndx ) , ( ( ( EDRing ` k ) ` w ) sSet <. ( *r ` ndx ) , ( (HGMap ` k ) ` w ) >. ) >. } u. { <. ( .s ` ndx ) , ( .s ` u ) >. , <. ( .i ` ndx ) , ( x e. v , y e. v |-> ( ( ( (HDMap ` k ) ` w ) ` y ) ` x ) ) >. } ) ) )
18		df-hlhil 37225	. . . . 5 |- HLHil = ( k e. _V |-> ( w e. ( LHyp ` k ) |-> [_ ( ( DVecH ` k ) ` w ) / u ]_ [_ ( Base ` u ) / v ]_ ( { <. ( Base ` ndx ) , v >. , <. ( +g ` ndx ) , ( +g ` u ) >. , <. (Scalar ` ndx ) , ( ( ( EDRing ` k ) ` w ) sSet <. ( *r ` ndx ) , ( (HGMap ` k ) ` w ) >. ) >. } u. { <. ( .s ` ndx ) , ( .s ` u ) >. , <. ( .i ` ndx ) , ( x e. v , y e. v |-> ( ( ( (HDMap ` k ) ` w ) ` y ) ` x ) ) >. } ) ) )
19	10, 13, 17, 18	fvmptf 6301	. . . 4 |- ( ( K e. _V /\ ( w e. H |-> [_ K / k ]_ [_ ( ( DVecH ` k ) ` w ) / u ]_ [_ ( Base ` u ) / v ]_ ( { <. ( Base ` ndx ) , v >. , <. ( +g ` ndx ) , ( +g ` u ) >. , <. (Scalar ` ndx ) , ( ( ( EDRing ` k ) ` w ) sSet <. ( *r ` ndx ) , ( (HGMap ` k ) ` w ) >. ) >. } u. { <. ( .s ` ndx ) , ( .s ` u ) >. , <. ( .i ` ndx ) , ( x e. v , y e. v |-> ( ( ( (HDMap ` k ) ` w ) ` y ) ` x ) ) >. } ) ) e. _V ) -> (HLHil ` K ) = ( w e. H |-> [_ K / k ]_ [_ ( ( DVecH ` k ) ` w ) / u ]_ [_ ( Base ` u ) / v ]_ ( { <. ( Base ` ndx ) , v >. , <. ( +g ` ndx ) , ( +g ` u ) >. , <. (Scalar ` ndx ) , ( ( ( EDRing ` k ) ` w ) sSet <. ( *r ` ndx ) , ( (HGMap ` k ) ` w ) >. ) >. } u. { <. ( .s ` ndx ) , ( .s ` u ) >. , <. ( .i ` ndx ) , ( x e. v , y e. v |-> ( ( ( (HDMap ` k ) ` w ) ` y ) ` x ) ) >. } ) ) )
20	5, 9, 19	sylancl 694	. . 3 |- ( ph -> (HLHil ` K ) = ( w e. H |-> [_ K / k ]_ [_ ( ( DVecH ` k ) ` w ) / u ]_ [_ ( Base ` u ) / v ]_ ( { <. ( Base ` ndx ) , v >. , <. ( +g ` ndx ) , ( +g ` u ) >. , <. (Scalar ` ndx ) , ( ( ( EDRing ` k ) ` w ) sSet <. ( *r ` ndx ) , ( (HGMap ` k ) ` w ) >. ) >. } u. { <. ( .s ` ndx ) , ( .s ` u ) >. , <. ( .i ` ndx ) , ( x e. v , y e. v |-> ( ( ( (HDMap ` k ) ` w ) ` y ) ` x ) ) >. } ) ) )
21	5	adantr 481	. . . 4 |- ( ( ph /\ w = W ) -> K e. _V )
22		fvexd 6203	. . . . 5 |- ( ( ( ph /\ w = W ) /\ k = K ) -> ( ( DVecH ` k ) ` w ) e. _V )
23		fvexd 6203	. . . . . 6 |- ( ( ( ( ph /\ w = W ) /\ k = K ) /\ u = ( ( DVecH ` k ) ` w ) ) -> ( Base ` u ) e. _V )
24		id 22	. . . . . . . . . 10 |- ( v = ( Base ` u ) -> v = ( Base ` u ) )
25		id 22	. . . . . . . . . . . . 13 |- ( u = ( ( DVecH ` k ) ` w ) -> u = ( ( DVecH ` k ) ` w ) )
26		simpr 477	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ w = W ) /\ k = K ) -> k = K )
27	26	fveq2d 6195	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ w = W ) /\ k = K ) -> ( DVecH ` k ) = ( DVecH ` K ) )
28		simplr 792	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ w = W ) /\ k = K ) -> w = W )
29	27, 28	fveq12d 6197	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ w = W ) /\ k = K ) -> ( ( DVecH ` k ) ` w ) = ( ( DVecH ` K ) ` W ) )
30		hlhilset.u	. . . . . . . . . . . . . 14 |- U = ( ( DVecH ` K ) ` W )
31	29, 30	syl6eqr 2674	. . . . . . . . . . . . 13 |- ( ( ( ph /\ w = W ) /\ k = K ) -> ( ( DVecH ` k ) ` w ) = U )
32	25, 31	sylan9eqr 2678	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ w = W ) /\ k = K ) /\ u = ( ( DVecH ` k ) ` w ) ) -> u = U )
33	32	fveq2d 6195	. . . . . . . . . . 11 |- ( ( ( ( ph /\ w = W ) /\ k = K ) /\ u = ( ( DVecH ` k ) ` w ) ) -> ( Base ` u ) = ( Base ` U ) )
34		hlhilset.v	. . . . . . . . . . 11 |- V = ( Base ` U )
35	33, 34	syl6eqr 2674	. . . . . . . . . 10 |- ( ( ( ( ph /\ w = W ) /\ k = K ) /\ u = ( ( DVecH ` k ) ` w ) ) -> ( Base ` u ) = V )
36	24, 35	sylan9eqr 2678	. . . . . . . . 9 |- ( ( ( ( ( ph /\ w = W ) /\ k = K ) /\ u = ( ( DVecH ` k ) ` w ) ) /\ v = ( Base ` u ) ) -> v = V )
37	36	opeq2d 4409	. . . . . . . 8 |- ( ( ( ( ( ph /\ w = W ) /\ k = K ) /\ u = ( ( DVecH ` k ) ` w ) ) /\ v = ( Base ` u ) ) -> <. ( Base ` ndx ) , v >. = <. ( Base ` ndx ) , V >. )
38	32	adantr 481	. . . . . . . . . . 11 |- ( ( ( ( ( ph /\ w = W ) /\ k = K ) /\ u = ( ( DVecH ` k ) ` w ) ) /\ v = ( Base ` u ) ) -> u = U )
39	38	fveq2d 6195	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ w = W ) /\ k = K ) /\ u = ( ( DVecH ` k ) ` w ) ) /\ v = ( Base ` u ) ) -> ( +g ` u ) = ( +g ` U ) )
40		hlhilset.p	. . . . . . . . . 10 |- .+ = ( +g ` U )
41	39, 40	syl6eqr 2674	. . . . . . . . 9 |- ( ( ( ( ( ph /\ w = W ) /\ k = K ) /\ u = ( ( DVecH ` k ) ` w ) ) /\ v = ( Base ` u ) ) -> ( +g ` u ) = .+ )
42	41	opeq2d 4409	. . . . . . . 8 |- ( ( ( ( ( ph /\ w = W ) /\ k = K ) /\ u = ( ( DVecH ` k ) ` w ) ) /\ v = ( Base ` u ) ) -> <. ( +g ` ndx ) , ( +g ` u ) >. = <. ( +g ` ndx ) , .+ >. )
43	26	fveq2d 6195	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ w = W ) /\ k = K ) -> ( EDRing ` k ) = ( EDRing ` K ) )
44	43, 28	fveq12d 6197	. . . . . . . . . . . . 13 |- ( ( ( ph /\ w = W ) /\ k = K ) -> ( ( EDRing ` k ) ` w ) = ( ( EDRing ` K ) ` W ) )
45		hlhilset.e	. . . . . . . . . . . . 13 |- E = ( ( EDRing ` K ) ` W )
46	44, 45	syl6eqr 2674	. . . . . . . . . . . 12 |- ( ( ( ph /\ w = W ) /\ k = K ) -> ( ( EDRing ` k ) ` w ) = E )
47	26	fveq2d 6195	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ w = W ) /\ k = K ) -> (HGMap ` k ) = (HGMap ` K ) )
48	47, 28	fveq12d 6197	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ w = W ) /\ k = K ) -> ( (HGMap ` k ) ` w ) = ( (HGMap ` K ) ` W ) )
49		hlhilset.g	. . . . . . . . . . . . . 14 |- G = ( (HGMap ` K ) ` W )
50	48, 49	syl6eqr 2674	. . . . . . . . . . . . 13 |- ( ( ( ph /\ w = W ) /\ k = K ) -> ( (HGMap ` k ) ` w ) = G )
51	50	opeq2d 4409	. . . . . . . . . . . 12 |- ( ( ( ph /\ w = W ) /\ k = K ) -> <. ( *r ` ndx ) , ( (HGMap ` k ) ` w ) >. = <. ( *r ` ndx ) , G >. )
52	46, 51	oveq12d 6668	. . . . . . . . . . 11 |- ( ( ( ph /\ w = W ) /\ k = K ) -> ( ( ( EDRing ` k ) ` w ) sSet <. ( *r ` ndx ) , ( (HGMap ` k ) ` w ) >. ) = ( E sSet <. ( *r ` ndx ) , G >. ) )
53		hlhilset.r	. . . . . . . . . . 11 |- R = ( E sSet <. ( *r ` ndx ) , G >. )
54	52, 53	syl6eqr 2674	. . . . . . . . . 10 |- ( ( ( ph /\ w = W ) /\ k = K ) -> ( ( ( EDRing ` k ) ` w ) sSet <. ( *r ` ndx ) , ( (HGMap ` k ) ` w ) >. ) = R )
55	54	opeq2d 4409	. . . . . . . . 9 |- ( ( ( ph /\ w = W ) /\ k = K ) -> <. (Scalar ` ndx ) , ( ( ( EDRing ` k ) ` w ) sSet <. ( *r ` ndx ) , ( (HGMap ` k ) ` w ) >. ) >. = <. (Scalar ` ndx ) , R >. )
56	55	ad2antrr 762	. . . . . . . 8 |- ( ( ( ( ( ph /\ w = W ) /\ k = K ) /\ u = ( ( DVecH ` k ) ` w ) ) /\ v = ( Base ` u ) ) -> <. (Scalar ` ndx ) , ( ( ( EDRing ` k ) ` w ) sSet <. ( *r ` ndx ) , ( (HGMap ` k ) ` w ) >. ) >. = <. (Scalar ` ndx ) , R >. )
57	37, 42, 56	tpeq123d 4283	. . . . . . 7 |- ( ( ( ( ( ph /\ w = W ) /\ k = K ) /\ u = ( ( DVecH ` k ) ` w ) ) /\ v = ( Base ` u ) ) -> { <. ( Base ` ndx ) , v >. , <. ( +g ` ndx ) , ( +g ` u ) >. , <. (Scalar ` ndx ) , ( ( ( EDRing ` k ) ` w ) sSet <. ( *r ` ndx ) , ( (HGMap ` k ) ` w ) >. ) >. } = { <. ( Base ` ndx ) , V >. , <. ( +g ` ndx ) , .+ >. , <. (Scalar ` ndx ) , R >. } )
58	38	fveq2d 6195	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ w = W ) /\ k = K ) /\ u = ( ( DVecH ` k ) ` w ) ) /\ v = ( Base ` u ) ) -> ( .s ` u ) = ( .s ` U ) )
59		hlhilset.t	. . . . . . . . . 10 |- .x. = ( .s ` U )
60	58, 59	syl6eqr 2674	. . . . . . . . 9 |- ( ( ( ( ( ph /\ w = W ) /\ k = K ) /\ u = ( ( DVecH ` k ) ` w ) ) /\ v = ( Base ` u ) ) -> ( .s ` u ) = .x. )
61	60	opeq2d 4409	. . . . . . . 8 |- ( ( ( ( ( ph /\ w = W ) /\ k = K ) /\ u = ( ( DVecH ` k ) ` w ) ) /\ v = ( Base ` u ) ) -> <. ( .s ` ndx ) , ( .s ` u ) >. = <. ( .s ` ndx ) , .x. >. )
62	26	fveq2d 6195	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ w = W ) /\ k = K ) -> (HDMap ` k ) = (HDMap ` K ) )
63	62, 28	fveq12d 6197	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ w = W ) /\ k = K ) -> ( (HDMap ` k ) ` w ) = ( (HDMap ` K ) ` W ) )
64		hlhilset.s	. . . . . . . . . . . . . . 15 |- S = ( (HDMap ` K ) ` W )
65	63, 64	syl6eqr 2674	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ w = W ) /\ k = K ) -> ( (HDMap ` k ) ` w ) = S )
66	65	ad2antrr 762	. . . . . . . . . . . . 13 |- ( ( ( ( ( ph /\ w = W ) /\ k = K ) /\ u = ( ( DVecH ` k ) ` w ) ) /\ v = ( Base ` u ) ) -> ( (HDMap ` k ) ` w ) = S )
67	66	fveq1d 6193	. . . . . . . . . . . 12 |- ( ( ( ( ( ph /\ w = W ) /\ k = K ) /\ u = ( ( DVecH ` k ) ` w ) ) /\ v = ( Base ` u ) ) -> ( ( (HDMap ` k ) ` w ) ` y ) = ( S ` y ) )
68	67	fveq1d 6193	. . . . . . . . . . 11 |- ( ( ( ( ( ph /\ w = W ) /\ k = K ) /\ u = ( ( DVecH ` k ) ` w ) ) /\ v = ( Base ` u ) ) -> ( ( ( (HDMap ` k ) ` w ) ` y ) ` x ) = ( ( S ` y ) ` x ) )
69	36, 36, 68	mpt2eq123dv 6717	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ w = W ) /\ k = K ) /\ u = ( ( DVecH ` k ) ` w ) ) /\ v = ( Base ` u ) ) -> ( x e. v , y e. v |-> ( ( ( (HDMap ` k ) ` w ) ` y ) ` x ) ) = ( x e. V , y e. V |-> ( ( S ` y ) ` x ) ) )
70		hlhilset.i	. . . . . . . . . 10 |- ., = ( x e. V , y e. V |-> ( ( S ` y ) ` x ) )
71	69, 70	syl6eqr 2674	. . . . . . . . 9 |- ( ( ( ( ( ph /\ w = W ) /\ k = K ) /\ u = ( ( DVecH ` k ) ` w ) ) /\ v = ( Base ` u ) ) -> ( x e. v , y e. v |-> ( ( ( (HDMap ` k ) ` w ) ` y ) ` x ) ) = ., )
72	71	opeq2d 4409	. . . . . . . 8 |- ( ( ( ( ( ph /\ w = W ) /\ k = K ) /\ u = ( ( DVecH ` k ) ` w ) ) /\ v = ( Base ` u ) ) -> <. ( .i ` ndx ) , ( x e. v , y e. v |-> ( ( ( (HDMap ` k ) ` w ) ` y ) ` x ) ) >. = <. ( .i ` ndx ) , ., >. )
73	61, 72	preq12d 4276	. . . . . . 7 |- ( ( ( ( ( ph /\ w = W ) /\ k = K ) /\ u = ( ( DVecH ` k ) ` w ) ) /\ v = ( Base ` u ) ) -> { <. ( .s ` ndx ) , ( .s ` u ) >. , <. ( .i ` ndx ) , ( x e. v , y e. v |-> ( ( ( (HDMap ` k ) ` w ) ` y ) ` x ) ) >. } = { <. ( .s ` ndx ) , .x. >. , <. ( .i ` ndx ) , ., >. } )
74	57, 73	uneq12d 3768	. . . . . 6 |- ( ( ( ( ( ph /\ w = W ) /\ k = K ) /\ u = ( ( DVecH ` k ) ` w ) ) /\ v = ( Base ` u ) ) -> ( { <. ( Base ` ndx ) , v >. , <. ( +g ` ndx ) , ( +g ` u ) >. , <. (Scalar ` ndx ) , ( ( ( EDRing ` k ) ` w ) sSet <. ( *r ` ndx ) , ( (HGMap ` k ) ` w ) >. ) >. } u. { <. ( .s ` ndx ) , ( .s ` u ) >. , <. ( .i ` ndx ) , ( x e. v , y e. v |-> ( ( ( (HDMap ` k ) ` w ) ` y ) ` x ) ) >. } ) = ( { <. ( Base ` ndx ) , V >. , <. ( +g ` ndx ) , .+ >. , <. (Scalar ` ndx ) , R >. } u. { <. ( .s ` ndx ) , .x. >. , <. ( .i ` ndx ) , ., >. } ) )
75	23, 74	csbied 3560	. . . . 5 |- ( ( ( ( ph /\ w = W ) /\ k = K ) /\ u = ( ( DVecH ` k ) ` w ) ) -> [_ ( Base ` u ) / v ]_ ( { <. ( Base ` ndx ) , v >. , <. ( +g ` ndx ) , ( +g ` u ) >. , <. (Scalar ` ndx ) , ( ( ( EDRing ` k ) ` w ) sSet <. ( *r ` ndx ) , ( (HGMap ` k ) ` w ) >. ) >. } u. { <. ( .s ` ndx ) , ( .s ` u ) >. , <. ( .i ` ndx ) , ( x e. v , y e. v |-> ( ( ( (HDMap ` k ) ` w ) ` y ) ` x ) ) >. } ) = ( { <. ( Base ` ndx ) , V >. , <. ( +g ` ndx ) , .+ >. , <. (Scalar ` ndx ) , R >. } u. { <. ( .s ` ndx ) , .x. >. , <. ( .i ` ndx ) , ., >. } ) )
76	22, 75	csbied 3560	. . . 4 |- ( ( ( ph /\ w = W ) /\ k = K ) -> [_ ( ( DVecH ` k ) ` w ) / u ]_ [_ ( Base ` u ) / v ]_ ( { <. ( Base ` ndx ) , v >. , <. ( +g ` ndx ) , ( +g ` u ) >. , <. (Scalar ` ndx ) , ( ( ( EDRing ` k ) ` w ) sSet <. ( *r ` ndx ) , ( (HGMap ` k ) ` w ) >. ) >. } u. { <. ( .s ` ndx ) , ( .s ` u ) >. , <. ( .i ` ndx ) , ( x e. v , y e. v |-> ( ( ( (HDMap ` k ) ` w ) ` y ) ` x ) ) >. } ) = ( { <. ( Base ` ndx ) , V >. , <. ( +g ` ndx ) , .+ >. , <. (Scalar ` ndx ) , R >. } u. { <. ( .s ` ndx ) , .x. >. , <. ( .i ` ndx ) , ., >. } ) )
77	21, 76	csbied 3560	. . 3 |- ( ( ph /\ w = W ) -> [_ K / k ]_ [_ ( ( DVecH ` k ) ` w ) / u ]_ [_ ( Base ` u ) / v ]_ ( { <. ( Base ` ndx ) , v >. , <. ( +g ` ndx ) , ( +g ` u ) >. , <. (Scalar ` ndx ) , ( ( ( EDRing ` k ) ` w ) sSet <. ( *r ` ndx ) , ( (HGMap ` k ) ` w ) >. ) >. } u. { <. ( .s ` ndx ) , ( .s ` u ) >. , <. ( .i ` ndx ) , ( x e. v , y e. v |-> ( ( ( (HDMap ` k ) ` w ) ` y ) ` x ) ) >. } ) = ( { <. ( Base ` ndx ) , V >. , <. ( +g ` ndx ) , .+ >. , <. (Scalar ` ndx ) , R >. } u. { <. ( .s ` ndx ) , .x. >. , <. ( .i ` ndx ) , ., >. } ) )
78	2	simprd 479	. . 3 |- ( ph -> W e. H )
79		tpex 6957	. . . . 5 |- { <. ( Base ` ndx ) , V >. , <. ( +g ` ndx ) , .+ >. , <. (Scalar ` ndx ) , R >. } e. _V
80		prex 4909	. . . . 5 |- { <. ( .s ` ndx ) , .x. >. , <. ( .i ` ndx ) , ., >. } e. _V
81	79, 80	unex 6956	. . . 4 |- ( { <. ( Base ` ndx ) , V >. , <. ( +g ` ndx ) , .+ >. , <. (Scalar ` ndx ) , R >. } u. { <. ( .s ` ndx ) , .x. >. , <. ( .i ` ndx ) , ., >. } ) e. _V
82	81	a1i 11	. . 3 |- ( ph -> ( { <. ( Base ` ndx ) , V >. , <. ( +g ` ndx ) , .+ >. , <. (Scalar ` ndx ) , R >. } u. { <. ( .s ` ndx ) , .x. >. , <. ( .i ` ndx ) , ., >. } ) e. _V )
83	20, 77, 78, 82	fvmptd 6288	. 2 |- ( ph -> ( (HLHil ` K ) ` W ) = ( { <. ( Base ` ndx ) , V >. , <. ( +g ` ndx ) , .+ >. , <. (Scalar ` ndx ) , R >. } u. { <. ( .s ` ndx ) , .x. >. , <. ( .i ` ndx ) , ., >. } ) )
84	1, 83	syl5eq 2668	1 |- ( ph -> L = ( { <. ( Base ` ndx ) , V >. , <. ( +g ` ndx ) , .+ >. , <. (Scalar ` ndx ) , R >. } u. { <. ( .s ` ndx ) , .x. >. , <. ( .i ` ndx ) , ., >. } ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   [_csb 3533   u. cun 3572   {cpr 4179   {ctp 4181   <.cop 4183   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   ndxcnx 15854   sSet csts 15855   Basecbs 15857   +g cplusg 15941   *rcstv 15943  Scalarcsca 15944   .scvsca 15945   .icip 15946   HLchlt 34637   LHypclh 35270   EDRingcedring 36041   DVecHcdvh 36367  HDMapchdma 37082  HGMapchg 37175  HLHilchlh 37224

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-sn 4178  df-pr 4180  df-tp 4182  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-ov 6653  df-oprab 6654  df-mpt2 6655  df-hlhil 37225

This theorem is referenced by:  hlhilsca  37227  hlhilbase  37228  hlhilplus  37229  hlhilvsca  37239  hlhilip  37240