Theorem hlhilset 35304

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 2979	. . . . . 6 |- ( K e. HL -> K e. _V )
4	3	adantr 462	. . . . 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 5698	. . . . . 6 |- ( LHyp ` K ) e. _V
8	6, 7	eqeltri 2511	. . . . 5 |- H e. _V
9	8	mptex 5945	. . . 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 2577	. . . . 5 |- F/_ k K
11		nfcv 2577	. . . . . 6 |- F/_ k H
12		nfcsb1v 3301	. . . . . 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 4377	. . . . 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 5688	. . . . . . 7 |- ( k = K -> ( LHyp ` k ) = ( LHyp ` K ) )
15	14, 6	syl6eqr 2491	. . . . . 6 |- ( k = K -> ( LHyp ` k ) = H )
16		csbeq1a 3294	. . . . . 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 4367	. . . . 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 35303	. . . . 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 5787	. . . 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 657	. . 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 462	. . . 4 |- ( ( ph /\ w = W ) -> K e. _V )
22		fvex 5698	. . . . . 6 |- ( ( DVecH ` k ) ` w ) e. _V
23	22	a1i 11	. . . . 5 |- ( ( ( ph /\ w = W ) /\ k = K ) -> ( ( DVecH ` k ) ` w ) e. _V )
24		fvex 5698	. . . . . . 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 458	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ w = W ) /\ k = K ) -> k = K )
29	28	fveq2d 5692	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ w = W ) /\ k = K ) -> ( DVecH ` k ) = ( DVecH ` K ) )
30		simplr 749	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ w = W ) /\ k = K ) -> w = W )
31	29, 30	fveq12d 5694	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ w = W ) /\ k = K ) -> ( ( DVecH ` k ) ` w ) = ( ( DVecH ` K ) ` W ) )
32		hlhilset.u	. . . . . . . . . . . . . 14 |- U = ( ( DVecH ` K ) ` W )
33	31, 32	syl6eqr 2491	. . . . . . . . . . . . 13 |- ( ( ( ph /\ w = W ) /\ k = K ) -> ( ( DVecH ` k ) ` w ) = U )
34	27, 33	sylan9eqr 2495	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ w = W ) /\ k = K ) /\ u = ( ( DVecH ` k ) ` w ) ) -> u = U )
35	34	fveq2d 5692	. . . . . . . . . . 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 2491	. . . . . . . . . 10 |- ( ( ( ( ph /\ w = W ) /\ k = K ) /\ u = ( ( DVecH ` k ) ` w ) ) -> ( Base ` u ) = V )
38	26, 37	sylan9eqr 2495	. . . . . . . . 9 |- ( ( ( ( ( ph /\ w = W ) /\ k = K ) /\ u = ( ( DVecH ` k ) ` w ) ) /\ v = ( Base ` u ) ) -> v = V )
39	38	opeq2d 4063	. . . . . . . 8 |- ( ( ( ( ( ph /\ w = W ) /\ k = K ) /\ u = ( ( DVecH ` k ) ` w ) ) /\ v = ( Base ` u ) ) -> <. ( Base ` ndx ) , v >. = <. ( Base ` ndx ) , V >. )
40	34	adantr 462	. . . . . . . . . . 11 |- ( ( ( ( ( ph /\ w = W ) /\ k = K ) /\ u = ( ( DVecH ` k ) ` w ) ) /\ v = ( Base ` u ) ) -> u = U )
41	40	fveq2d 5692	. . . . . . . . . 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 2491	. . . . . . . . 9 |- ( ( ( ( ( ph /\ w = W ) /\ k = K ) /\ u = ( ( DVecH ` k ) ` w ) ) /\ v = ( Base ` u ) ) -> ( +g ` u ) = .+ )
44	43	opeq2d 4063	. . . . . . . 8 |- ( ( ( ( ( ph /\ w = W ) /\ k = K ) /\ u = ( ( DVecH ` k ) ` w ) ) /\ v = ( Base ` u ) ) -> <. ( +g ` ndx ) , ( +g ` u ) >. = <. ( +g ` ndx ) , .+ >. )
45	28	fveq2d 5692	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ w = W ) /\ k = K ) -> ( EDRing ` k ) = ( EDRing ` K ) )
46	45, 30	fveq12d 5694	. . . . . . . . . . . . 13 |- ( ( ( ph /\ w = W ) /\ k = K ) -> ( ( EDRing ` k ) ` w ) = ( ( EDRing ` K ) ` W ) )
47		hlhilset.e	. . . . . . . . . . . . 13 |- E = ( ( EDRing ` K ) ` W )
48	46, 47	syl6eqr 2491	. . . . . . . . . . . 12 |- ( ( ( ph /\ w = W ) /\ k = K ) -> ( ( EDRing ` k ) ` w ) = E )
49	28	fveq2d 5692	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ w = W ) /\ k = K ) -> (HGMap ` k ) = (HGMap ` K ) )
50	49, 30	fveq12d 5694	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ w = W ) /\ k = K ) -> ( (HGMap ` k ) ` w ) = ( (HGMap ` K ) ` W ) )
51		hlhilset.g	. . . . . . . . . . . . . 14 |- G = ( (HGMap ` K ) ` W )
52	50, 51	syl6eqr 2491	. . . . . . . . . . . . 13 |- ( ( ( ph /\ w = W ) /\ k = K ) -> ( (HGMap ` k ) ` w ) = G )
53	52	opeq2d 4063	. . . . . . . . . . . 12 |- ( ( ( ph /\ w = W ) /\ k = K ) -> <. ( *r ` ndx ) , ( (HGMap ` k ) ` w ) >. = <. ( *r ` ndx ) , G >. )
54	48, 53	oveq12d 6108	. . . . . . . . . . 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 2491	. . . . . . . . . 10 |- ( ( ( ph /\ w = W ) /\ k = K ) -> ( ( ( EDRing ` k ) ` w ) sSet <. ( *r ` ndx ) , ( (HGMap ` k ) ` w ) >. ) = R )
57	56	opeq2d 4063	. . . . . . . . 9 |- ( ( ( ph /\ w = W ) /\ k = K ) -> <. (Scalar ` ndx ) , ( ( ( EDRing ` k ) ` w ) sSet <. ( *r ` ndx ) , ( (HGMap ` k ) ` w ) >. ) >. = <. (Scalar ` ndx ) , R >. )
58	57	ad2antrr 720	. . . . . . . 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 3966	. . . . . . 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 5692	. . . . . . . . . 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 2491	. . . . . . . . 9 |- ( ( ( ( ( ph /\ w = W ) /\ k = K ) /\ u = ( ( DVecH ` k ) ` w ) ) /\ v = ( Base ` u ) ) -> ( .s ` u ) = .x. )
63	62	opeq2d 4063	. . . . . . . 8 |- ( ( ( ( ( ph /\ w = W ) /\ k = K ) /\ u = ( ( DVecH ` k ) ` w ) ) /\ v = ( Base ` u ) ) -> <. ( .s ` ndx ) , ( .s ` u ) >. = <. ( .s ` ndx ) , .x. >. )
64	28	fveq2d 5692	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ w = W ) /\ k = K ) -> (HDMap ` k ) = (HDMap ` K ) )
65	64, 30	fveq12d 5694	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ w = W ) /\ k = K ) -> ( (HDMap ` k ) ` w ) = ( (HDMap ` K ) ` W ) )
66		hlhilset.s	. . . . . . . . . . . . . . 15 |- S = ( (HDMap ` K ) ` W )
67	65, 66	syl6eqr 2491	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ w = W ) /\ k = K ) -> ( (HDMap ` k ) ` w ) = S )
68	67	ad2antrr 720	. . . . . . . . . . . . 13 |- ( ( ( ( ( ph /\ w = W ) /\ k = K ) /\ u = ( ( DVecH ` k ) ` w ) ) /\ v = ( Base ` u ) ) -> ( (HDMap ` k ) ` w ) = S )
69	68	fveq1d 5690	. . . . . . . . . . . 12 |- ( ( ( ( ( ph /\ w = W ) /\ k = K ) /\ u = ( ( DVecH ` k ) ` w ) ) /\ v = ( Base ` u ) ) -> ( ( (HDMap ` k ) ` w ) ` y ) = ( S ` y ) )
70	69	fveq1d 5690	. . . . . . . . . . 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 6147	. . . . . . . . . 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 2491	. . . . . . . . 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 4063	. . . . . . . 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 3959	. . . . . . 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 3508	. . . . . 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 3311	. . . . 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 3311	. . . 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 3311	. . 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 460	. . 3 |- ( ph -> W e. H )
81		tpex 6378	. . . . 5 |- { <. ( Base ` ndx ) , V >. , <. ( +g ` ndx ) , .+ >. , <. (Scalar ` ndx ) , R >. } e. _V
82		prex 4531	. . . . 5 |- { <. ( .s ` ndx ) , .x. >. , <. ( .i ` ndx ) , ., >. } e. _V
83	81, 82	unex 6377	. . . 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 5776	. 2 |- ( ph -> ( (HLHil ` K ) ` W ) = ( { <. ( Base ` ndx ) , V >. , <. ( +g ` ndx ) , .+ >. , <. (Scalar ` ndx ) , R >. } u. { <. ( .s ` ndx ) , .x. >. , <. ( .i ` ndx ) , ., >. } ) )
86	1, 85	syl5eq 2485	1 |- ( ph -> L = ( { <. ( Base ` ndx ) , V >. , <. ( +g ` ndx ) , .+ >. , <. (Scalar ` ndx ) , R >. } u. { <. ( .s ` ndx ) , .x. >. , <. ( .i ` ndx ) , ., >. } ) )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1364   e. wcel 1761   _Vcvv 2970   [_csb 3285   u. cun 3323   {cpr 3876   {ctp 3878   <.cop 3880   e. cmpt 4347   ` cfv 5415  (class class class)co 6090   e. cmpt2 6092   ndxcnx 14167   sSet csts 14168   Basecbs 14170   +g cplusg 14234   *rcstv 14236  Scalarcsca 14237   .scvsca 14238   .icip 14239   HLchlt 32717   LHypclh 33350   EDRingcedring 34119   DVecHcdvh 34445  HDMapchdma 35160  HGMapchg 35253  HLHilchlh 35302

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-hlhil 35303

This theorem is referenced by:  hlhilsca  35305  hlhilbase  35306  hlhilplus  35307  hlhilvsca  35317  hlhilip  35318