Theorem hlhilset 35582

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 2981	. . . . . 6 |- ( K e. HL -> K e. _V )
4	3	adantr 465	. . . . 5 |- ( ( K e. HL /\ W e. H ) -> K e. _V )
5	2, 4	syl 16	. . . 4 |- ( ph -> K e. _V )
6		hlhilset.h	. . . . . 6 |- H = ( LHyp ` K )
7		fvex 5701	. . . . . 6 |- ( LHyp ` K ) e. _V
8	6, 7	eqeltri 2513	. . . . 5 |- H e. _V
9	8	mptex 5948	. . . 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 2579	. . . . 5 |- F/_ k K
11		nfcv 2579	. . . . . 6 |- F/_ k H
12		nfcsb1v 3304	. . . . . 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 4380	. . . . 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 5691	. . . . . . 7 |- ( k = K -> ( LHyp ` k ) = ( LHyp ` K ) )
15	14, 6	syl6eqr 2493	. . . . . 6 |- ( k = K -> ( LHyp ` k ) = H )
16		csbeq1a 3297	. . . . . 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 4370	. . . . 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 35581	. . . . 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 5790	. . . 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 662	. . 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 465	. . . 4 |- ( ( ph /\ w = W ) -> K e. _V )
22		fvex 5701	. . . . . 6 |- ( ( DVecH ` k ) ` w ) e. _V
23	22	a1i 11	. . . . 5 |- ( ( ( ph /\ w = W ) /\ k = K ) -> ( ( DVecH ` k ) ` w ) e. _V )
24		fvex 5701	. . . . . . 7 |- ( Base ` u ) e. _V
25	24	a1i 11	. . . . . 6 |- ( ( ( ( ph /\ w = W ) /\ k = K ) /\ u = ( ( DVecH ` k ) ` w ) ) -> ( Base ` u ) e. _V )
26		id 22	. . . . . . . . . 10 |- ( v = ( Base ` u ) -> v = ( Base ` u ) )
27		id 22	. . . . . . . . . . . . 13 |- ( u = ( ( DVecH ` k ) ` w ) -> u = ( ( DVecH ` k ) ` w ) )
28		simpr 461	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ w = W ) /\ k = K ) -> k = K )
29	28	fveq2d 5695	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ w = W ) /\ k = K ) -> ( DVecH ` k ) = ( DVecH ` K ) )
30		simplr 754	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ w = W ) /\ k = K ) -> w = W )
31	29, 30	fveq12d 5697	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ w = W ) /\ k = K ) -> ( ( DVecH ` k ) ` w ) = ( ( DVecH ` K ) ` W ) )
32		hlhilset.u	. . . . . . . . . . . . . 14 |- U = ( ( DVecH ` K ) ` W )
33	31, 32	syl6eqr 2493	. . . . . . . . . . . . 13 |- ( ( ( ph /\ w = W ) /\ k = K ) -> ( ( DVecH ` k ) ` w ) = U )
34	27, 33	sylan9eqr 2497	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ w = W ) /\ k = K ) /\ u = ( ( DVecH ` k ) ` w ) ) -> u = U )
35	34	fveq2d 5695	. . . . . . . . . . 11 |- ( ( ( ( ph /\ w = W ) /\ k = K ) /\ u = ( ( DVecH ` k ) ` w ) ) -> ( Base ` u ) = ( Base ` U ) )
36		hlhilset.v	. . . . . . . . . . 11 |- V = ( Base ` U )
37	35, 36	syl6eqr 2493	. . . . . . . . . 10 |- ( ( ( ( ph /\ w = W ) /\ k = K ) /\ u = ( ( DVecH ` k ) ` w ) ) -> ( Base ` u ) = V )
38	26, 37	sylan9eqr 2497	. . . . . . . . 9 |- ( ( ( ( ( ph /\ w = W ) /\ k = K ) /\ u = ( ( DVecH ` k ) ` w ) ) /\ v = ( Base ` u ) ) -> v = V )
39	38	opeq2d 4066	. . . . . . . 8 |- ( ( ( ( ( ph /\ w = W ) /\ k = K ) /\ u = ( ( DVecH ` k ) ` w ) ) /\ v = ( Base ` u ) ) -> <. ( Base ` ndx ) , v >. = <. ( Base ` ndx ) , V >. )
40	34	adantr 465	. . . . . . . . . . 11 |- ( ( ( ( ( ph /\ w = W ) /\ k = K ) /\ u = ( ( DVecH ` k ) ` w ) ) /\ v = ( Base ` u ) ) -> u = U )
41	40	fveq2d 5695	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ w = W ) /\ k = K ) /\ u = ( ( DVecH ` k ) ` w ) ) /\ v = ( Base ` u ) ) -> ( +g ` u ) = ( +g ` U ) )
42		hlhilset.p	. . . . . . . . . 10 |- .+ = ( +g ` U )
43	41, 42	syl6eqr 2493	. . . . . . . . 9 |- ( ( ( ( ( ph /\ w = W ) /\ k = K ) /\ u = ( ( DVecH ` k ) ` w ) ) /\ v = ( Base ` u ) ) -> ( +g ` u ) = .+ )
44	43	opeq2d 4066	. . . . . . . 8 |- ( ( ( ( ( ph /\ w = W ) /\ k = K ) /\ u = ( ( DVecH ` k ) ` w ) ) /\ v = ( Base ` u ) ) -> <. ( +g ` ndx ) , ( +g ` u ) >. = <. ( +g ` ndx ) , .+ >. )
45	28	fveq2d 5695	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ w = W ) /\ k = K ) -> ( EDRing ` k ) = ( EDRing ` K ) )
46	45, 30	fveq12d 5697	. . . . . . . . . . . . 13 |- ( ( ( ph /\ w = W ) /\ k = K ) -> ( ( EDRing ` k ) ` w ) = ( ( EDRing ` K ) ` W ) )
47		hlhilset.e	. . . . . . . . . . . . 13 |- E = ( ( EDRing ` K ) ` W )
48	46, 47	syl6eqr 2493	. . . . . . . . . . . 12 |- ( ( ( ph /\ w = W ) /\ k = K ) -> ( ( EDRing ` k ) ` w ) = E )
49	28	fveq2d 5695	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ w = W ) /\ k = K ) -> (HGMap ` k ) = (HGMap ` K ) )
50	49, 30	fveq12d 5697	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ w = W ) /\ k = K ) -> ( (HGMap ` k ) ` w ) = ( (HGMap ` K ) ` W ) )
51		hlhilset.g	. . . . . . . . . . . . . 14 |- G = ( (HGMap ` K ) ` W )
52	50, 51	syl6eqr 2493	. . . . . . . . . . . . 13 |- ( ( ( ph /\ w = W ) /\ k = K ) -> ( (HGMap ` k ) ` w ) = G )
53	52	opeq2d 4066	. . . . . . . . . . . 12 |- ( ( ( ph /\ w = W ) /\ k = K ) -> <. ( *r ` ndx ) , ( (HGMap ` k ) ` w ) >. = <. ( *r ` ndx ) , G >. )
54	48, 53	oveq12d 6109	. . . . . . . . . . 11 |- ( ( ( ph /\ w = W ) /\ k = K ) -> ( ( ( EDRing ` k ) ` w ) sSet <. ( *r ` ndx ) , ( (HGMap ` k ) ` w ) >. ) = ( E sSet <. ( *r ` ndx ) , G >. ) )
55		hlhilset.r	. . . . . . . . . . 11 |- R = ( E sSet <. ( *r ` ndx ) , G >. )
56	54, 55	syl6eqr 2493	. . . . . . . . . 10 |- ( ( ( ph /\ w = W ) /\ k = K ) -> ( ( ( EDRing ` k ) ` w ) sSet <. ( *r ` ndx ) , ( (HGMap ` k ) ` w ) >. ) = R )
57	56	opeq2d 4066	. . . . . . . . 9 |- ( ( ( ph /\ w = W ) /\ k = K ) -> <. (Scalar ` ndx ) , ( ( ( EDRing ` k ) ` w ) sSet <. ( *r ` ndx ) , ( (HGMap ` k ) ` w ) >. ) >. = <. (Scalar ` ndx ) , R >. )
58	57	ad2antrr 725	. . . . . . . 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 >. )
59	39, 44, 58	tpeq123d 3969	. . . . . . 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 >. } )
60	40	fveq2d 5695	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ w = W ) /\ k = K ) /\ u = ( ( DVecH ` k ) ` w ) ) /\ v = ( Base ` u ) ) -> ( .s ` u ) = ( .s ` U ) )
61		hlhilset.t	. . . . . . . . . 10 |- .x. = ( .s ` U )
62	60, 61	syl6eqr 2493	. . . . . . . . 9 |- ( ( ( ( ( ph /\ w = W ) /\ k = K ) /\ u = ( ( DVecH ` k ) ` w ) ) /\ v = ( Base ` u ) ) -> ( .s ` u ) = .x. )
63	62	opeq2d 4066	. . . . . . . 8 |- ( ( ( ( ( ph /\ w = W ) /\ k = K ) /\ u = ( ( DVecH ` k ) ` w ) ) /\ v = ( Base ` u ) ) -> <. ( .s ` ndx ) , ( .s ` u ) >. = <. ( .s ` ndx ) , .x. >. )
64	28	fveq2d 5695	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ w = W ) /\ k = K ) -> (HDMap ` k ) = (HDMap ` K ) )
65	64, 30	fveq12d 5697	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ w = W ) /\ k = K ) -> ( (HDMap ` k ) ` w ) = ( (HDMap ` K ) ` W ) )
66		hlhilset.s	. . . . . . . . . . . . . . 15 |- S = ( (HDMap ` K ) ` W )
67	65, 66	syl6eqr 2493	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ w = W ) /\ k = K ) -> ( (HDMap ` k ) ` w ) = S )
68	67	ad2antrr 725	. . . . . . . . . . . . 13 |- ( ( ( ( ( ph /\ w = W ) /\ k = K ) /\ u = ( ( DVecH ` k ) ` w ) ) /\ v = ( Base ` u ) ) -> ( (HDMap ` k ) ` w ) = S )
69	68	fveq1d 5693	. . . . . . . . . . . 12 |- ( ( ( ( ( ph /\ w = W ) /\ k = K ) /\ u = ( ( DVecH ` k ) ` w ) ) /\ v = ( Base ` u ) ) -> ( ( (HDMap ` k ) ` w ) ` y ) = ( S ` y ) )
70	69	fveq1d 5693	. . . . . . . . . . 11 |- ( ( ( ( ( ph /\ w = W ) /\ k = K ) /\ u = ( ( DVecH ` k ) ` w ) ) /\ v = ( Base ` u ) ) -> ( ( ( (HDMap ` k ) ` w ) ` y ) ` x ) = ( ( S ` y ) ` x ) )
71	38, 38, 70	mpt2eq123dv 6148	. . . . . . . . . 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 ) ) )
72		hlhilset.i	. . . . . . . . . 10 |- ., = ( x e. V , y e. V |-> ( ( S ` y ) ` x ) )
73	71, 72	syl6eqr 2493	. . . . . . . . 9 |- ( ( ( ( ( ph /\ w = W ) /\ k = K ) /\ u = ( ( DVecH ` k ) ` w ) ) /\ v = ( Base ` u ) ) -> ( x e. v , y e. v |-> ( ( ( (HDMap ` k ) ` w ) ` y ) ` x ) ) = ., )
74	73	opeq2d 4066	. . . . . . . 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 ) , ., >. )
75	63, 74	preq12d 3962	. . . . . . 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 ) , ., >. } )
76	59, 75	uneq12d 3511	. . . . . 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 ) , ., >. } ) )
77	25, 76	csbied 3314	. . . . 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 ) , ., >. } ) )
78	23, 77	csbied 3314	. . . 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 ) , ., >. } ) )
79	21, 78	csbied 3314	. . 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 ) , ., >. } ) )
80	2	simprd 463	. . 3 |- ( ph -> W e. H )
81		tpex 6379	. . . . 5 |- { <. ( Base ` ndx ) , V >. , <. ( +g ` ndx ) , .+ >. , <. (Scalar ` ndx ) , R >. } e. _V
82		prex 4534	. . . . 5 |- { <. ( .s ` ndx ) , .x. >. , <. ( .i ` ndx ) , ., >. } e. _V
83	81, 82	unex 6378	. . . 4 |- ( { <. ( Base ` ndx ) , V >. , <. ( +g ` ndx ) , .+ >. , <. (Scalar ` ndx ) , R >. } u. { <. ( .s ` ndx ) , .x. >. , <. ( .i ` ndx ) , ., >. } ) e. _V
84	83	a1i 11	. . 3 |- ( ph -> ( { <. ( Base ` ndx ) , V >. , <. ( +g ` ndx ) , .+ >. , <. (Scalar ` ndx ) , R >. } u. { <. ( .s ` ndx ) , .x. >. , <. ( .i ` ndx ) , ., >. } ) e. _V )
85	20, 79, 80, 84	fvmptd 5779	. 2 |- ( ph -> ( (HLHil ` K ) ` W ) = ( { <. ( Base ` ndx ) , V >. , <. ( +g ` ndx ) , .+ >. , <. (Scalar ` ndx ) , R >. } u. { <. ( .s ` ndx ) , .x. >. , <. ( .i ` ndx ) , ., >. } ) )
86	1, 85	syl5eq 2487	1 |- ( ph -> L = ( { <. ( Base ` ndx ) , V >. , <. ( +g ` ndx ) , .+ >. , <. (Scalar ` ndx ) , R >. } u. { <. ( .s ` ndx ) , .x. >. , <. ( .i ` ndx ) , ., >. } ) )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1369   e. wcel 1756   _Vcvv 2972   [_csb 3288   u. cun 3326   {cpr 3879   {ctp 3881   <.cop 3883   e. cmpt 4350   ` cfv 5418  (class class class)co 6091   e. cmpt2 6093   ndxcnx 14171   sSet csts 14172   Basecbs 14174   +g cplusg 14238   *rcstv 14240  Scalarcsca 14241   .scvsca 14242   .icip 14243   HLchlt 32995   LHypclh 33628   EDRingcedring 34397   DVecHcdvh 34723  HDMapchdma 35438  HGMapchg 35531  HLHilchlh 35580

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-hlhil 35581

This theorem is referenced by:  hlhilsca  35583  hlhilbase  35584  hlhilplus  35585  hlhilvsca  35595  hlhilip  35596