Theorem hdmapfval 37119

Description: Map from vectors to functionals in the closed kernel dual space. (Contributed by NM, 15-May-2015.)

Hypotheses

Ref	Expression
hdmapval.h	|- H = ( LHyp ` K )
hdmapfval.e	|- E = <. ( _I |` ( Base ` K ) ) , ( _I |` ( ( LTrn ` K ) ` W ) ) >.
hdmapfval.u	|- U = ( ( DVecH ` K ) ` W )
hdmapfval.v	|- V = ( Base ` U )
hdmapfval.n	|- N = ( LSpan ` U )
hdmapfval.c	|- C = ( (LCDual ` K ) ` W )
hdmapfval.d	|- D = ( Base ` C )
hdmapfval.j	|- J = ( (HVMap ` K ) ` W )
hdmapfval.i	|- I = ( (HDMap1 ` K ) ` W )
hdmapfval.s	|- S = ( (HDMap ` K ) ` W )
hdmapfval.k	|- ( ph -> ( K e. A /\ W e. H ) )

Assertion

Ref	Expression
hdmapfval	|- ( ph -> S = ( t e. V |-> ( iota_ y e. D A. z e. V ( -. z e. ( ( N ` { E } ) u. ( N ` { t } ) ) -> y = ( I ` <. z , ( I ` <. E , ( J ` E ) , z >. ) , t >. ) ) ) ) )

Distinct variable groups:   y, t, z, K   y, D   t, E, y, z   t, I, y, z   t, U, y, z   t, V, y, z   t, W, y, z

Allowed substitution hints:   ph( y, z, t)   A( y, z, t)   C( y, z, t)   D( z, t)   S( y, z, t)   H( y, z, t)   J( y, z, t)   N( y, z, t)

Proof of Theorem hdmapfval

Dummy variables w e a i u v are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		hdmapfval.k	. 2 |- ( ph -> ( K e. A /\ W e. H ) )
2		hdmapfval.s	. . . 4 |- S = ( (HDMap ` K ) ` W )
3		hdmapval.h	. . . . . 6 |- H = ( LHyp ` K )
4	3	hdmapffval 37118	. . . . 5 |- ( K e. A -> (HDMap ` K ) = ( w e. H |-> { a | [. <. ( _I |` ( Base ` K ) ) , ( _I |` ( ( LTrn ` K ) ` w ) ) >. / e ]. [. ( ( DVecH ` K ) ` w ) / u ]. [. ( Base ` u ) / v ]. [. ( (HDMap1 ` K ) ` w ) / i ]. a e. ( t e. v |-> ( iota_ y e. ( Base ` ( (LCDual ` K ) ` w ) ) A. z e. v ( -. z e. ( ( ( LSpan ` u ) ` { e } ) u. ( ( LSpan ` u ) ` { t } ) ) -> y = ( i ` <. z , ( i ` <. e , ( ( (HVMap ` K ) ` w ) ` e ) , z >. ) , t >. ) ) ) ) } ) )
5	4	fveq1d 6193	. . . 4 |- ( K e. A -> ( (HDMap ` K ) ` W ) = ( ( w e. H |-> { a | [. <. ( _I |` ( Base ` K ) ) , ( _I |` ( ( LTrn ` K ) ` w ) ) >. / e ]. [. ( ( DVecH ` K ) ` w ) / u ]. [. ( Base ` u ) / v ]. [. ( (HDMap1 ` K ) ` w ) / i ]. a e. ( t e. v |-> ( iota_ y e. ( Base ` ( (LCDual ` K ) ` w ) ) A. z e. v ( -. z e. ( ( ( LSpan ` u ) ` { e } ) u. ( ( LSpan ` u ) ` { t } ) ) -> y = ( i ` <. z , ( i ` <. e , ( ( (HVMap ` K ) ` w ) ` e ) , z >. ) , t >. ) ) ) ) } ) ` W ) )
6	2, 5	syl5eq 2668	. . 3 |- ( K e. A -> S = ( ( w e. H |-> { a | [. <. ( _I |` ( Base ` K ) ) , ( _I |` ( ( LTrn ` K ) ` w ) ) >. / e ]. [. ( ( DVecH ` K ) ` w ) / u ]. [. ( Base ` u ) / v ]. [. ( (HDMap1 ` K ) ` w ) / i ]. a e. ( t e. v |-> ( iota_ y e. ( Base ` ( (LCDual ` K ) ` w ) ) A. z e. v ( -. z e. ( ( ( LSpan ` u ) ` { e } ) u. ( ( LSpan ` u ) ` { t } ) ) -> y = ( i ` <. z , ( i ` <. e , ( ( (HVMap ` K ) ` w ) ` e ) , z >. ) , t >. ) ) ) ) } ) ` W ) )
7		fveq2 6191	. . . . . . . . 9 |- ( w = W -> ( ( LTrn ` K ) ` w ) = ( ( LTrn ` K ) ` W ) )
8	7	reseq2d 5396	. . . . . . . 8 |- ( w = W -> ( _I |` ( ( LTrn ` K ) ` w ) ) = ( _I |` ( ( LTrn ` K ) ` W ) ) )
9	8	opeq2d 4409	. . . . . . 7 |- ( w = W -> <. ( _I |` ( Base ` K ) ) , ( _I |` ( ( LTrn ` K ) ` w ) ) >. = <. ( _I |` ( Base ` K ) ) , ( _I |` ( ( LTrn ` K ) ` W ) ) >. )
10		fveq2 6191	. . . . . . . 8 |- ( w = W -> ( ( DVecH ` K ) ` w ) = ( ( DVecH ` K ) ` W ) )
11		fveq2 6191	. . . . . . . . . 10 |- ( w = W -> ( (HDMap1 ` K ) ` w ) = ( (HDMap1 ` K ) ` W ) )
12		fveq2 6191	. . . . . . . . . . . . . 14 |- ( w = W -> ( (LCDual ` K ) ` w ) = ( (LCDual ` K ) ` W ) )
13	12	fveq2d 6195	. . . . . . . . . . . . 13 |- ( w = W -> ( Base ` ( (LCDual ` K ) ` w ) ) = ( Base ` ( (LCDual ` K ) ` W ) ) )
14		fveq2 6191	. . . . . . . . . . . . . . . . . . . . 21 |- ( w = W -> ( (HVMap ` K ) ` w ) = ( (HVMap ` K ) ` W ) )
15	14	fveq1d 6193	. . . . . . . . . . . . . . . . . . . 20 |- ( w = W -> ( ( (HVMap ` K ) ` w ) ` e ) = ( ( (HVMap ` K ) ` W ) ` e ) )
16	15	oteq2d 4415	. . . . . . . . . . . . . . . . . . 19 |- ( w = W -> <. e , ( ( (HVMap ` K ) ` w ) ` e ) , z >. = <. e , ( ( (HVMap ` K ) ` W ) ` e ) , z >. )
17	16	fveq2d 6195	. . . . . . . . . . . . . . . . . 18 |- ( w = W -> ( i ` <. e , ( ( (HVMap ` K ) ` w ) ` e ) , z >. ) = ( i ` <. e , ( ( (HVMap ` K ) ` W ) ` e ) , z >. ) )
18	17	oteq2d 4415	. . . . . . . . . . . . . . . . 17 |- ( w = W -> <. z , ( i ` <. e , ( ( (HVMap ` K ) ` w ) ` e ) , z >. ) , t >. = <. z , ( i ` <. e , ( ( (HVMap ` K ) ` W ) ` e ) , z >. ) , t >. )
19	18	fveq2d 6195	. . . . . . . . . . . . . . . 16 |- ( w = W -> ( i ` <. z , ( i ` <. e , ( ( (HVMap ` K ) ` w ) ` e ) , z >. ) , t >. ) = ( i ` <. z , ( i ` <. e , ( ( (HVMap ` K ) ` W ) ` e ) , z >. ) , t >. ) )
20	19	eqeq2d 2632	. . . . . . . . . . . . . . 15 |- ( w = W -> ( y = ( i ` <. z , ( i ` <. e , ( ( (HVMap ` K ) ` w ) ` e ) , z >. ) , t >. ) <-> y = ( i ` <. z , ( i ` <. e , ( ( (HVMap ` K ) ` W ) ` e ) , z >. ) , t >. ) ) )
21	20	imbi2d 330	. . . . . . . . . . . . . 14 |- ( w = W -> ( ( -. z e. ( ( ( LSpan ` u ) ` { e } ) u. ( ( LSpan ` u ) ` { t } ) ) -> y = ( i ` <. z , ( i ` <. e , ( ( (HVMap ` K ) ` w ) ` e ) , z >. ) , t >. ) ) <-> ( -. z e. ( ( ( LSpan ` u ) ` { e } ) u. ( ( LSpan ` u ) ` { t } ) ) -> y = ( i ` <. z , ( i ` <. e , ( ( (HVMap ` K ) ` W ) ` e ) , z >. ) , t >. ) ) ) )
22	21	ralbidv 2986	. . . . . . . . . . . . 13 |- ( w = W -> ( A. z e. v ( -. z e. ( ( ( LSpan ` u ) ` { e } ) u. ( ( LSpan ` u ) ` { t } ) ) -> y = ( i ` <. z , ( i ` <. e , ( ( (HVMap ` K ) ` w ) ` e ) , z >. ) , t >. ) ) <-> A. z e. v ( -. z e. ( ( ( LSpan ` u ) ` { e } ) u. ( ( LSpan ` u ) ` { t } ) ) -> y = ( i ` <. z , ( i ` <. e , ( ( (HVMap ` K ) ` W ) ` e ) , z >. ) , t >. ) ) ) )
23	13, 22	riotaeqbidv 6614	. . . . . . . . . . . 12 |- ( w = W -> ( iota_ y e. ( Base ` ( (LCDual ` K ) ` w ) ) A. z e. v ( -. z e. ( ( ( LSpan ` u ) ` { e } ) u. ( ( LSpan ` u ) ` { t } ) ) -> y = ( i ` <. z , ( i ` <. e , ( ( (HVMap ` K ) ` w ) ` e ) , z >. ) , t >. ) ) ) = ( iota_ y e. ( Base ` ( (LCDual ` K ) ` W ) ) A. z e. v ( -. z e. ( ( ( LSpan ` u ) ` { e } ) u. ( ( LSpan ` u ) ` { t } ) ) -> y = ( i ` <. z , ( i ` <. e , ( ( (HVMap ` K ) ` W ) ` e ) , z >. ) , t >. ) ) ) )
24	23	mpteq2dv 4745	. . . . . . . . . . 11 |- ( w = W -> ( t e. v |-> ( iota_ y e. ( Base ` ( (LCDual ` K ) ` w ) ) A. z e. v ( -. z e. ( ( ( LSpan ` u ) ` { e } ) u. ( ( LSpan ` u ) ` { t } ) ) -> y = ( i ` <. z , ( i ` <. e , ( ( (HVMap ` K ) ` w ) ` e ) , z >. ) , t >. ) ) ) ) = ( t e. v |-> ( iota_ y e. ( Base ` ( (LCDual ` K ) ` W ) ) A. z e. v ( -. z e. ( ( ( LSpan ` u ) ` { e } ) u. ( ( LSpan ` u ) ` { t } ) ) -> y = ( i ` <. z , ( i ` <. e , ( ( (HVMap ` K ) ` W ) ` e ) , z >. ) , t >. ) ) ) ) )
25	24	eleq2d 2687	. . . . . . . . . 10 |- ( w = W -> ( a e. ( t e. v |-> ( iota_ y e. ( Base ` ( (LCDual ` K ) ` w ) ) A. z e. v ( -. z e. ( ( ( LSpan ` u ) ` { e } ) u. ( ( LSpan ` u ) ` { t } ) ) -> y = ( i ` <. z , ( i ` <. e , ( ( (HVMap ` K ) ` w ) ` e ) , z >. ) , t >. ) ) ) ) <-> a e. ( t e. v |-> ( iota_ y e. ( Base ` ( (LCDual ` K ) ` W ) ) A. z e. v ( -. z e. ( ( ( LSpan ` u ) ` { e } ) u. ( ( LSpan ` u ) ` { t } ) ) -> y = ( i ` <. z , ( i ` <. e , ( ( (HVMap ` K ) ` W ) ` e ) , z >. ) , t >. ) ) ) ) ) )
26	11, 25	sbceqbid 3442	. . . . . . . . 9 |- ( w = W -> ( [. ( (HDMap1 ` K ) ` w ) / i ]. a e. ( t e. v |-> ( iota_ y e. ( Base ` ( (LCDual ` K ) ` w ) ) A. z e. v ( -. z e. ( ( ( LSpan ` u ) ` { e } ) u. ( ( LSpan ` u ) ` { t } ) ) -> y = ( i ` <. z , ( i ` <. e , ( ( (HVMap ` K ) ` w ) ` e ) , z >. ) , t >. ) ) ) ) <-> [. ( (HDMap1 ` K ) ` W ) / i ]. a e. ( t e. v |-> ( iota_ y e. ( Base ` ( (LCDual ` K ) ` W ) ) A. z e. v ( -. z e. ( ( ( LSpan ` u ) ` { e } ) u. ( ( LSpan ` u ) ` { t } ) ) -> y = ( i ` <. z , ( i ` <. e , ( ( (HVMap ` K ) ` W ) ` e ) , z >. ) , t >. ) ) ) ) ) )
27	26	sbcbidv 3490	. . . . . . . 8 |- ( w = W -> ( [. ( Base ` u ) / v ]. [. ( (HDMap1 ` K ) ` w ) / i ]. a e. ( t e. v |-> ( iota_ y e. ( Base ` ( (LCDual ` K ) ` w ) ) A. z e. v ( -. z e. ( ( ( LSpan ` u ) ` { e } ) u. ( ( LSpan ` u ) ` { t } ) ) -> y = ( i ` <. z , ( i ` <. e , ( ( (HVMap ` K ) ` w ) ` e ) , z >. ) , t >. ) ) ) ) <-> [. ( Base ` u ) / v ]. [. ( (HDMap1 ` K ) ` W ) / i ]. a e. ( t e. v |-> ( iota_ y e. ( Base ` ( (LCDual ` K ) ` W ) ) A. z e. v ( -. z e. ( ( ( LSpan ` u ) ` { e } ) u. ( ( LSpan ` u ) ` { t } ) ) -> y = ( i ` <. z , ( i ` <. e , ( ( (HVMap ` K ) ` W ) ` e ) , z >. ) , t >. ) ) ) ) ) )
28	10, 27	sbceqbid 3442	. . . . . . 7 |- ( w = W -> ( [. ( ( DVecH ` K ) ` w ) / u ]. [. ( Base ` u ) / v ]. [. ( (HDMap1 ` K ) ` w ) / i ]. a e. ( t e. v |-> ( iota_ y e. ( Base ` ( (LCDual ` K ) ` w ) ) A. z e. v ( -. z e. ( ( ( LSpan ` u ) ` { e } ) u. ( ( LSpan ` u ) ` { t } ) ) -> y = ( i ` <. z , ( i ` <. e , ( ( (HVMap ` K ) ` w ) ` e ) , z >. ) , t >. ) ) ) ) <-> [. ( ( DVecH ` K ) ` W ) / u ]. [. ( Base ` u ) / v ]. [. ( (HDMap1 ` K ) ` W ) / i ]. a e. ( t e. v |-> ( iota_ y e. ( Base ` ( (LCDual ` K ) ` W ) ) A. z e. v ( -. z e. ( ( ( LSpan ` u ) ` { e } ) u. ( ( LSpan ` u ) ` { t } ) ) -> y = ( i ` <. z , ( i ` <. e , ( ( (HVMap ` K ) ` W ) ` e ) , z >. ) , t >. ) ) ) ) ) )
29	9, 28	sbceqbid 3442	. . . . . 6 |- ( w = W -> ( [. <. ( _I |` ( Base ` K ) ) , ( _I |` ( ( LTrn ` K ) ` w ) ) >. / e ]. [. ( ( DVecH ` K ) ` w ) / u ]. [. ( Base ` u ) / v ]. [. ( (HDMap1 ` K ) ` w ) / i ]. a e. ( t e. v |-> ( iota_ y e. ( Base ` ( (LCDual ` K ) ` w ) ) A. z e. v ( -. z e. ( ( ( LSpan ` u ) ` { e } ) u. ( ( LSpan ` u ) ` { t } ) ) -> y = ( i ` <. z , ( i ` <. e , ( ( (HVMap ` K ) ` w ) ` e ) , z >. ) , t >. ) ) ) ) <-> [. <. ( _I |` ( Base ` K ) ) , ( _I |` ( ( LTrn ` K ) ` W ) ) >. / e ]. [. ( ( DVecH ` K ) ` W ) / u ]. [. ( Base ` u ) / v ]. [. ( (HDMap1 ` K ) ` W ) / i ]. a e. ( t e. v |-> ( iota_ y e. ( Base ` ( (LCDual ` K ) ` W ) ) A. z e. v ( -. z e. ( ( ( LSpan ` u ) ` { e } ) u. ( ( LSpan ` u ) ` { t } ) ) -> y = ( i ` <. z , ( i ` <. e , ( ( (HVMap ` K ) ` W ) ` e ) , z >. ) , t >. ) ) ) ) ) )
30		opex 4932	. . . . . . 7 |- <. ( _I |` ( Base ` K ) ) , ( _I |` ( ( LTrn ` K ) ` W ) ) >. e. _V
31		fvex 6201	. . . . . . 7 |- ( ( DVecH ` K ) ` W ) e. _V
32		fvex 6201	. . . . . . 7 |- ( Base ` u ) e. _V
33		simp1 1061	. . . . . . . . 9 |- ( ( e = <. ( _I |` ( Base ` K ) ) , ( _I |` ( ( LTrn ` K ) ` W ) ) >. /\ u = ( ( DVecH ` K ) ` W ) /\ v = ( Base ` u ) ) -> e = <. ( _I |` ( Base ` K ) ) , ( _I |` ( ( LTrn ` K ) ` W ) ) >. )
34		hdmapfval.e	. . . . . . . . 9 |- E = <. ( _I |` ( Base ` K ) ) , ( _I |` ( ( LTrn ` K ) ` W ) ) >.
35	33, 34	syl6eqr 2674	. . . . . . . 8 |- ( ( e = <. ( _I |` ( Base ` K ) ) , ( _I |` ( ( LTrn ` K ) ` W ) ) >. /\ u = ( ( DVecH ` K ) ` W ) /\ v = ( Base ` u ) ) -> e = E )
36		simp2 1062	. . . . . . . . 9 |- ( ( e = <. ( _I |` ( Base ` K ) ) , ( _I |` ( ( LTrn ` K ) ` W ) ) >. /\ u = ( ( DVecH ` K ) ` W ) /\ v = ( Base ` u ) ) -> u = ( ( DVecH ` K ) ` W ) )
37		hdmapfval.u	. . . . . . . . 9 |- U = ( ( DVecH ` K ) ` W )
38	36, 37	syl6eqr 2674	. . . . . . . 8 |- ( ( e = <. ( _I |` ( Base ` K ) ) , ( _I |` ( ( LTrn ` K ) ` W ) ) >. /\ u = ( ( DVecH ` K ) ` W ) /\ v = ( Base ` u ) ) -> u = U )
39		simp3 1063	. . . . . . . . . 10 |- ( ( e = <. ( _I |` ( Base ` K ) ) , ( _I |` ( ( LTrn ` K ) ` W ) ) >. /\ u = ( ( DVecH ` K ) ` W ) /\ v = ( Base ` u ) ) -> v = ( Base ` u ) )
40	38	fveq2d 6195	. . . . . . . . . 10 |- ( ( e = <. ( _I |` ( Base ` K ) ) , ( _I |` ( ( LTrn ` K ) ` W ) ) >. /\ u = ( ( DVecH ` K ) ` W ) /\ v = ( Base ` u ) ) -> ( Base ` u ) = ( Base ` U ) )
41	39, 40	eqtrd 2656	. . . . . . . . 9 |- ( ( e = <. ( _I |` ( Base ` K ) ) , ( _I |` ( ( LTrn ` K ) ` W ) ) >. /\ u = ( ( DVecH ` K ) ` W ) /\ v = ( Base ` u ) ) -> v = ( Base ` U ) )
42		hdmapfval.v	. . . . . . . . 9 |- V = ( Base ` U )
43	41, 42	syl6eqr 2674	. . . . . . . 8 |- ( ( e = <. ( _I |` ( Base ` K ) ) , ( _I |` ( ( LTrn ` K ) ` W ) ) >. /\ u = ( ( DVecH ` K ) ` W ) /\ v = ( Base ` u ) ) -> v = V )
44		fvex 6201	. . . . . . . . . 10 |- ( (HDMap1 ` K ) ` W ) e. _V
45		id 22	. . . . . . . . . . . 12 |- ( i = ( (HDMap1 ` K ) ` W ) -> i = ( (HDMap1 ` K ) ` W ) )
46		hdmapfval.i	. . . . . . . . . . . 12 |- I = ( (HDMap1 ` K ) ` W )
47	45, 46	syl6eqr 2674	. . . . . . . . . . 11 |- ( i = ( (HDMap1 ` K ) ` W ) -> i = I )
48		fveq1 6190	. . . . . . . . . . . . . . . . . 18 |- ( i = I -> ( i ` <. z , ( i ` <. e , ( ( (HVMap ` K ) ` W ) ` e ) , z >. ) , t >. ) = ( I ` <. z , ( i ` <. e , ( ( (HVMap ` K ) ` W ) ` e ) , z >. ) , t >. ) )
49		fveq1 6190	. . . . . . . . . . . . . . . . . . . 20 |- ( i = I -> ( i ` <. e , ( ( (HVMap ` K ) ` W ) ` e ) , z >. ) = ( I ` <. e , ( ( (HVMap ` K ) ` W ) ` e ) , z >. ) )
50	49	oteq2d 4415	. . . . . . . . . . . . . . . . . . 19 |- ( i = I -> <. z , ( i ` <. e , ( ( (HVMap ` K ) ` W ) ` e ) , z >. ) , t >. = <. z , ( I ` <. e , ( ( (HVMap ` K ) ` W ) ` e ) , z >. ) , t >. )
51	50	fveq2d 6195	. . . . . . . . . . . . . . . . . 18 |- ( i = I -> ( I ` <. z , ( i ` <. e , ( ( (HVMap ` K ) ` W ) ` e ) , z >. ) , t >. ) = ( I ` <. z , ( I ` <. e , ( ( (HVMap ` K ) ` W ) ` e ) , z >. ) , t >. ) )
52	48, 51	eqtrd 2656	. . . . . . . . . . . . . . . . 17 |- ( i = I -> ( i ` <. z , ( i ` <. e , ( ( (HVMap ` K ) ` W ) ` e ) , z >. ) , t >. ) = ( I ` <. z , ( I ` <. e , ( ( (HVMap ` K ) ` W ) ` e ) , z >. ) , t >. ) )
53	52	eqeq2d 2632	. . . . . . . . . . . . . . . 16 |- ( i = I -> ( y = ( i ` <. z , ( i ` <. e , ( ( (HVMap ` K ) ` W ) ` e ) , z >. ) , t >. ) <-> y = ( I ` <. z , ( I ` <. e , ( ( (HVMap ` K ) ` W ) ` e ) , z >. ) , t >. ) ) )
54	53	imbi2d 330	. . . . . . . . . . . . . . 15 |- ( i = I -> ( ( -. z e. ( ( ( LSpan ` u ) ` { e } ) u. ( ( LSpan ` u ) ` { t } ) ) -> y = ( i ` <. z , ( i ` <. e , ( ( (HVMap ` K ) ` W ) ` e ) , z >. ) , t >. ) ) <-> ( -. z e. ( ( ( LSpan ` u ) ` { e } ) u. ( ( LSpan ` u ) ` { t } ) ) -> y = ( I ` <. z , ( I ` <. e , ( ( (HVMap ` K ) ` W ) ` e ) , z >. ) , t >. ) ) ) )
55	54	ralbidv 2986	. . . . . . . . . . . . . 14 |- ( i = I -> ( A. z e. v ( -. z e. ( ( ( LSpan ` u ) ` { e } ) u. ( ( LSpan ` u ) ` { t } ) ) -> y = ( i ` <. z , ( i ` <. e , ( ( (HVMap ` K ) ` W ) ` e ) , z >. ) , t >. ) ) <-> A. z e. v ( -. z e. ( ( ( LSpan ` u ) ` { e } ) u. ( ( LSpan ` u ) ` { t } ) ) -> y = ( I ` <. z , ( I ` <. e , ( ( (HVMap ` K ) ` W ) ` e ) , z >. ) , t >. ) ) ) )
56	55	riotabidv 6613	. . . . . . . . . . . . 13 |- ( i = I -> ( iota_ y e. ( Base ` ( (LCDual ` K ) ` W ) ) A. z e. v ( -. z e. ( ( ( LSpan ` u ) ` { e } ) u. ( ( LSpan ` u ) ` { t } ) ) -> y = ( i ` <. z , ( i ` <. e , ( ( (HVMap ` K ) ` W ) ` e ) , z >. ) , t >. ) ) ) = ( iota_ y e. ( Base ` ( (LCDual ` K ) ` W ) ) A. z e. v ( -. z e. ( ( ( LSpan ` u ) ` { e } ) u. ( ( LSpan ` u ) ` { t } ) ) -> y = ( I ` <. z , ( I ` <. e , ( ( (HVMap ` K ) ` W ) ` e ) , z >. ) , t >. ) ) ) )
57	56	mpteq2dv 4745	. . . . . . . . . . . 12 |- ( i = I -> ( t e. v |-> ( iota_ y e. ( Base ` ( (LCDual ` K ) ` W ) ) A. z e. v ( -. z e. ( ( ( LSpan ` u ) ` { e } ) u. ( ( LSpan ` u ) ` { t } ) ) -> y = ( i ` <. z , ( i ` <. e , ( ( (HVMap ` K ) ` W ) ` e ) , z >. ) , t >. ) ) ) ) = ( t e. v |-> ( iota_ y e. ( Base ` ( (LCDual ` K ) ` W ) ) A. z e. v ( -. z e. ( ( ( LSpan ` u ) ` { e } ) u. ( ( LSpan ` u ) ` { t } ) ) -> y = ( I ` <. z , ( I ` <. e , ( ( (HVMap ` K ) ` W ) ` e ) , z >. ) , t >. ) ) ) ) )
58	57	eleq2d 2687	. . . . . . . . . . 11 |- ( i = I -> ( a e. ( t e. v |-> ( iota_ y e. ( Base ` ( (LCDual ` K ) ` W ) ) A. z e. v ( -. z e. ( ( ( LSpan ` u ) ` { e } ) u. ( ( LSpan ` u ) ` { t } ) ) -> y = ( i ` <. z , ( i ` <. e , ( ( (HVMap ` K ) ` W ) ` e ) , z >. ) , t >. ) ) ) ) <-> a e. ( t e. v |-> ( iota_ y e. ( Base ` ( (LCDual ` K ) ` W ) ) A. z e. v ( -. z e. ( ( ( LSpan ` u ) ` { e } ) u. ( ( LSpan ` u ) ` { t } ) ) -> y = ( I ` <. z , ( I ` <. e , ( ( (HVMap ` K ) ` W ) ` e ) , z >. ) , t >. ) ) ) ) ) )
59	47, 58	syl 17	. . . . . . . . . 10 |- ( i = ( (HDMap1 ` K ) ` W ) -> ( a e. ( t e. v |-> ( iota_ y e. ( Base ` ( (LCDual ` K ) ` W ) ) A. z e. v ( -. z e. ( ( ( LSpan ` u ) ` { e } ) u. ( ( LSpan ` u ) ` { t } ) ) -> y = ( i ` <. z , ( i ` <. e , ( ( (HVMap ` K ) ` W ) ` e ) , z >. ) , t >. ) ) ) ) <-> a e. ( t e. v |-> ( iota_ y e. ( Base ` ( (LCDual ` K ) ` W ) ) A. z e. v ( -. z e. ( ( ( LSpan ` u ) ` { e } ) u. ( ( LSpan ` u ) ` { t } ) ) -> y = ( I ` <. z , ( I ` <. e , ( ( (HVMap ` K ) ` W ) ` e ) , z >. ) , t >. ) ) ) ) ) )
60	44, 59	sbcie 3470	. . . . . . . . 9 |- ( [. ( (HDMap1 ` K ) ` W ) / i ]. a e. ( t e. v |-> ( iota_ y e. ( Base ` ( (LCDual ` K ) ` W ) ) A. z e. v ( -. z e. ( ( ( LSpan ` u ) ` { e } ) u. ( ( LSpan ` u ) ` { t } ) ) -> y = ( i ` <. z , ( i ` <. e , ( ( (HVMap ` K ) ` W ) ` e ) , z >. ) , t >. ) ) ) ) <-> a e. ( t e. v |-> ( iota_ y e. ( Base ` ( (LCDual ` K ) ` W ) ) A. z e. v ( -. z e. ( ( ( LSpan ` u ) ` { e } ) u. ( ( LSpan ` u ) ` { t } ) ) -> y = ( I ` <. z , ( I ` <. e , ( ( (HVMap ` K ) ` W ) ` e ) , z >. ) , t >. ) ) ) ) )
61		simp3 1063	. . . . . . . . . . 11 |- ( ( e = E /\ u = U /\ v = V ) -> v = V )
62		hdmapfval.d	. . . . . . . . . . . . . 14 |- D = ( Base ` C )
63		hdmapfval.c	. . . . . . . . . . . . . . 15 |- C = ( (LCDual ` K ) ` W )
64	63	fveq2i 6194	. . . . . . . . . . . . . 14 |- ( Base ` C ) = ( Base ` ( (LCDual ` K ) ` W ) )
65	62, 64	eqtr2i 2645	. . . . . . . . . . . . 13 |- ( Base ` ( (LCDual ` K ) ` W ) ) = D
66	65	a1i 11	. . . . . . . . . . . 12 |- ( ( e = E /\ u = U /\ v = V ) -> ( Base ` ( (LCDual ` K ) ` W ) ) = D )
67		simp2 1062	. . . . . . . . . . . . . . . . . . . 20 |- ( ( e = E /\ u = U /\ v = V ) -> u = U )
68	67	fveq2d 6195	. . . . . . . . . . . . . . . . . . 19 |- ( ( e = E /\ u = U /\ v = V ) -> ( LSpan ` u ) = ( LSpan ` U ) )
69		hdmapfval.n	. . . . . . . . . . . . . . . . . . 19 |- N = ( LSpan ` U )
70	68, 69	syl6eqr 2674	. . . . . . . . . . . . . . . . . 18 |- ( ( e = E /\ u = U /\ v = V ) -> ( LSpan ` u ) = N )
71		simp1 1061	. . . . . . . . . . . . . . . . . . 19 |- ( ( e = E /\ u = U /\ v = V ) -> e = E )
72	71	sneqd 4189	. . . . . . . . . . . . . . . . . 18 |- ( ( e = E /\ u = U /\ v = V ) -> { e } = { E } )
73	70, 72	fveq12d 6197	. . . . . . . . . . . . . . . . 17 |- ( ( e = E /\ u = U /\ v = V ) -> ( ( LSpan ` u ) ` { e } ) = ( N ` { E } ) )
74	70	fveq1d 6193	. . . . . . . . . . . . . . . . 17 |- ( ( e = E /\ u = U /\ v = V ) -> ( ( LSpan ` u ) ` { t } ) = ( N ` { t } ) )
75	73, 74	uneq12d 3768	. . . . . . . . . . . . . . . 16 |- ( ( e = E /\ u = U /\ v = V ) -> ( ( ( LSpan ` u ) ` { e } ) u. ( ( LSpan ` u ) ` { t } ) ) = ( ( N ` { E } ) u. ( N ` { t } ) ) )
76	75	eleq2d 2687	. . . . . . . . . . . . . . 15 |- ( ( e = E /\ u = U /\ v = V ) -> ( z e. ( ( ( LSpan ` u ) ` { e } ) u. ( ( LSpan ` u ) ` { t } ) ) <-> z e. ( ( N ` { E } ) u. ( N ` { t } ) ) ) )
77	76	notbid 308	. . . . . . . . . . . . . 14 |- ( ( e = E /\ u = U /\ v = V ) -> ( -. z e. ( ( ( LSpan ` u ) ` { e } ) u. ( ( LSpan ` u ) ` { t } ) ) <-> -. z e. ( ( N ` { E } ) u. ( N ` { t } ) ) ) )
78	71	oteq1d 4414	. . . . . . . . . . . . . . . . . . 19 |- ( ( e = E /\ u = U /\ v = V ) -> <. e , ( ( (HVMap ` K ) ` W ) ` e ) , z >. = <. E , ( ( (HVMap ` K ) ` W ) ` e ) , z >. )
79	71	fveq2d 6195	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( e = E /\ u = U /\ v = V ) -> ( ( (HVMap ` K ) ` W ) ` e ) = ( ( (HVMap ` K ) ` W ) ` E ) )
80		hdmapfval.j	. . . . . . . . . . . . . . . . . . . . . 22 |- J = ( (HVMap ` K ) ` W )
81	80	fveq1i 6192	. . . . . . . . . . . . . . . . . . . . 21 |- ( J ` E ) = ( ( (HVMap ` K ) ` W ) ` E )
82	79, 81	syl6eqr 2674	. . . . . . . . . . . . . . . . . . . 20 |- ( ( e = E /\ u = U /\ v = V ) -> ( ( (HVMap ` K ) ` W ) ` e ) = ( J ` E ) )
83	82	oteq2d 4415	. . . . . . . . . . . . . . . . . . 19 |- ( ( e = E /\ u = U /\ v = V ) -> <. E , ( ( (HVMap ` K ) ` W ) ` e ) , z >. = <. E , ( J ` E ) , z >. )
84	78, 83	eqtrd 2656	. . . . . . . . . . . . . . . . . 18 |- ( ( e = E /\ u = U /\ v = V ) -> <. e , ( ( (HVMap ` K ) ` W ) ` e ) , z >. = <. E , ( J ` E ) , z >. )
85	84	fveq2d 6195	. . . . . . . . . . . . . . . . 17 |- ( ( e = E /\ u = U /\ v = V ) -> ( I ` <. e , ( ( (HVMap ` K ) ` W ) ` e ) , z >. ) = ( I ` <. E , ( J ` E ) , z >. ) )
86	85	oteq2d 4415	. . . . . . . . . . . . . . . 16 |- ( ( e = E /\ u = U /\ v = V ) -> <. z , ( I ` <. e , ( ( (HVMap ` K ) ` W ) ` e ) , z >. ) , t >. = <. z , ( I ` <. E , ( J ` E ) , z >. ) , t >. )
87	86	fveq2d 6195	. . . . . . . . . . . . . . 15 |- ( ( e = E /\ u = U /\ v = V ) -> ( I ` <. z , ( I ` <. e , ( ( (HVMap ` K ) ` W ) ` e ) , z >. ) , t >. ) = ( I ` <. z , ( I ` <. E , ( J ` E ) , z >. ) , t >. ) )
88	87	eqeq2d 2632	. . . . . . . . . . . . . 14 |- ( ( e = E /\ u = U /\ v = V ) -> ( y = ( I ` <. z , ( I ` <. e , ( ( (HVMap ` K ) ` W ) ` e ) , z >. ) , t >. ) <-> y = ( I ` <. z , ( I ` <. E , ( J ` E ) , z >. ) , t >. ) ) )
89	77, 88	imbi12d 334	. . . . . . . . . . . . 13 |- ( ( e = E /\ u = U /\ v = V ) -> ( ( -. z e. ( ( ( LSpan ` u ) ` { e } ) u. ( ( LSpan ` u ) ` { t } ) ) -> y = ( I ` <. z , ( I ` <. e , ( ( (HVMap ` K ) ` W ) ` e ) , z >. ) , t >. ) ) <-> ( -. z e. ( ( N ` { E } ) u. ( N ` { t } ) ) -> y = ( I ` <. z , ( I ` <. E , ( J ` E ) , z >. ) , t >. ) ) ) )
90	61, 89	raleqbidv 3152	. . . . . . . . . . . 12 |- ( ( e = E /\ u = U /\ v = V ) -> ( A. z e. v ( -. z e. ( ( ( LSpan ` u ) ` { e } ) u. ( ( LSpan ` u ) ` { t } ) ) -> y = ( I ` <. z , ( I ` <. e , ( ( (HVMap ` K ) ` W ) ` e ) , z >. ) , t >. ) ) <-> A. z e. V ( -. z e. ( ( N ` { E } ) u. ( N ` { t } ) ) -> y = ( I ` <. z , ( I ` <. E , ( J ` E ) , z >. ) , t >. ) ) ) )
91	66, 90	riotaeqbidv 6614	. . . . . . . . . . 11 |- ( ( e = E /\ u = U /\ v = V ) -> ( iota_ y e. ( Base ` ( (LCDual ` K ) ` W ) ) A. z e. v ( -. z e. ( ( ( LSpan ` u ) ` { e } ) u. ( ( LSpan ` u ) ` { t } ) ) -> y = ( I ` <. z , ( I ` <. e , ( ( (HVMap ` K ) ` W ) ` e ) , z >. ) , t >. ) ) ) = ( iota_ y e. D A. z e. V ( -. z e. ( ( N ` { E } ) u. ( N ` { t } ) ) -> y = ( I ` <. z , ( I ` <. E , ( J ` E ) , z >. ) , t >. ) ) ) )
92	61, 91	mpteq12dv 4733	. . . . . . . . . 10 |- ( ( e = E /\ u = U /\ v = V ) -> ( t e. v |-> ( iota_ y e. ( Base ` ( (LCDual ` K ) ` W ) ) A. z e. v ( -. z e. ( ( ( LSpan ` u ) ` { e } ) u. ( ( LSpan ` u ) ` { t } ) ) -> y = ( I ` <. z , ( I ` <. e , ( ( (HVMap ` K ) ` W ) ` e ) , z >. ) , t >. ) ) ) ) = ( t e. V |-> ( iota_ y e. D A. z e. V ( -. z e. ( ( N ` { E } ) u. ( N ` { t } ) ) -> y = ( I ` <. z , ( I ` <. E , ( J ` E ) , z >. ) , t >. ) ) ) ) )
93	92	eleq2d 2687	. . . . . . . . 9 |- ( ( e = E /\ u = U /\ v = V ) -> ( a e. ( t e. v |-> ( iota_ y e. ( Base ` ( (LCDual ` K ) ` W ) ) A. z e. v ( -. z e. ( ( ( LSpan ` u ) ` { e } ) u. ( ( LSpan ` u ) ` { t } ) ) -> y = ( I ` <. z , ( I ` <. e , ( ( (HVMap ` K ) ` W ) ` e ) , z >. ) , t >. ) ) ) ) <-> a e. ( t e. V |-> ( iota_ y e. D A. z e. V ( -. z e. ( ( N ` { E } ) u. ( N ` { t } ) ) -> y = ( I ` <. z , ( I ` <. E , ( J ` E ) , z >. ) , t >. ) ) ) ) ) )
94	60, 93	syl5bb 272	. . . . . . . 8 |- ( ( e = E /\ u = U /\ v = V ) -> ( [. ( (HDMap1 ` K ) ` W ) / i ]. a e. ( t e. v |-> ( iota_ y e. ( Base ` ( (LCDual ` K ) ` W ) ) A. z e. v ( -. z e. ( ( ( LSpan ` u ) ` { e } ) u. ( ( LSpan ` u ) ` { t } ) ) -> y = ( i ` <. z , ( i ` <. e , ( ( (HVMap ` K ) ` W ) ` e ) , z >. ) , t >. ) ) ) ) <-> a e. ( t e. V |-> ( iota_ y e. D A. z e. V ( -. z e. ( ( N ` { E } ) u. ( N ` { t } ) ) -> y = ( I ` <. z , ( I ` <. E , ( J ` E ) , z >. ) , t >. ) ) ) ) ) )
95	35, 38, 43, 94	syl3anc 1326	. . . . . . 7 |- ( ( e = <. ( _I |` ( Base ` K ) ) , ( _I |` ( ( LTrn ` K ) ` W ) ) >. /\ u = ( ( DVecH ` K ) ` W ) /\ v = ( Base ` u ) ) -> ( [. ( (HDMap1 ` K ) ` W ) / i ]. a e. ( t e. v |-> ( iota_ y e. ( Base ` ( (LCDual ` K ) ` W ) ) A. z e. v ( -. z e. ( ( ( LSpan ` u ) ` { e } ) u. ( ( LSpan ` u ) ` { t } ) ) -> y = ( i ` <. z , ( i ` <. e , ( ( (HVMap ` K ) ` W ) ` e ) , z >. ) , t >. ) ) ) ) <-> a e. ( t e. V |-> ( iota_ y e. D A. z e. V ( -. z e. ( ( N ` { E } ) u. ( N ` { t } ) ) -> y = ( I ` <. z , ( I ` <. E , ( J ` E ) , z >. ) , t >. ) ) ) ) ) )
96	30, 31, 32, 95	sbc3ie 3507	. . . . . 6 |- ( [. <. ( _I |` ( Base ` K ) ) , ( _I |` ( ( LTrn ` K ) ` W ) ) >. / e ]. [. ( ( DVecH ` K ) ` W ) / u ]. [. ( Base ` u ) / v ]. [. ( (HDMap1 ` K ) ` W ) / i ]. a e. ( t e. v |-> ( iota_ y e. ( Base ` ( (LCDual ` K ) ` W ) ) A. z e. v ( -. z e. ( ( ( LSpan ` u ) ` { e } ) u. ( ( LSpan ` u ) ` { t } ) ) -> y = ( i ` <. z , ( i ` <. e , ( ( (HVMap ` K ) ` W ) ` e ) , z >. ) , t >. ) ) ) ) <-> a e. ( t e. V |-> ( iota_ y e. D A. z e. V ( -. z e. ( ( N ` { E } ) u. ( N ` { t } ) ) -> y = ( I ` <. z , ( I ` <. E , ( J ` E ) , z >. ) , t >. ) ) ) ) )
97	29, 96	syl6bb 276	. . . . 5 |- ( w = W -> ( [. <. ( _I |` ( Base ` K ) ) , ( _I |` ( ( LTrn ` K ) ` w ) ) >. / e ]. [. ( ( DVecH ` K ) ` w ) / u ]. [. ( Base ` u ) / v ]. [. ( (HDMap1 ` K ) ` w ) / i ]. a e. ( t e. v |-> ( iota_ y e. ( Base ` ( (LCDual ` K ) ` w ) ) A. z e. v ( -. z e. ( ( ( LSpan ` u ) ` { e } ) u. ( ( LSpan ` u ) ` { t } ) ) -> y = ( i ` <. z , ( i ` <. e , ( ( (HVMap ` K ) ` w ) ` e ) , z >. ) , t >. ) ) ) ) <-> a e. ( t e. V |-> ( iota_ y e. D A. z e. V ( -. z e. ( ( N ` { E } ) u. ( N ` { t } ) ) -> y = ( I ` <. z , ( I ` <. E , ( J ` E ) , z >. ) , t >. ) ) ) ) ) )
98	97	abbi1dv 2743	. . . 4 |- ( w = W -> { a | [. <. ( _I |` ( Base ` K ) ) , ( _I |` ( ( LTrn ` K ) ` w ) ) >. / e ]. [. ( ( DVecH ` K ) ` w ) / u ]. [. ( Base ` u ) / v ]. [. ( (HDMap1 ` K ) ` w ) / i ]. a e. ( t e. v |-> ( iota_ y e. ( Base ` ( (LCDual ` K ) ` w ) ) A. z e. v ( -. z e. ( ( ( LSpan ` u ) ` { e } ) u. ( ( LSpan ` u ) ` { t } ) ) -> y = ( i ` <. z , ( i ` <. e , ( ( (HVMap ` K ) ` w ) ` e ) , z >. ) , t >. ) ) ) ) } = ( t e. V |-> ( iota_ y e. D A. z e. V ( -. z e. ( ( N ` { E } ) u. ( N ` { t } ) ) -> y = ( I ` <. z , ( I ` <. E , ( J ` E ) , z >. ) , t >. ) ) ) ) )
99		eqid 2622	. . . 4 |- ( w e. H |-> { a | [. <. ( _I |` ( Base ` K ) ) , ( _I |` ( ( LTrn ` K ) ` w ) ) >. / e ]. [. ( ( DVecH ` K ) ` w ) / u ]. [. ( Base ` u ) / v ]. [. ( (HDMap1 ` K ) ` w ) / i ]. a e. ( t e. v |-> ( iota_ y e. ( Base ` ( (LCDual ` K ) ` w ) ) A. z e. v ( -. z e. ( ( ( LSpan ` u ) ` { e } ) u. ( ( LSpan ` u ) ` { t } ) ) -> y = ( i ` <. z , ( i ` <. e , ( ( (HVMap ` K ) ` w ) ` e ) , z >. ) , t >. ) ) ) ) } ) = ( w e. H |-> { a | [. <. ( _I |` ( Base ` K ) ) , ( _I |` ( ( LTrn ` K ) ` w ) ) >. / e ]. [. ( ( DVecH ` K ) ` w ) / u ]. [. ( Base ` u ) / v ]. [. ( (HDMap1 ` K ) ` w ) / i ]. a e. ( t e. v |-> ( iota_ y e. ( Base ` ( (LCDual ` K ) ` w ) ) A. z e. v ( -. z e. ( ( ( LSpan ` u ) ` { e } ) u. ( ( LSpan ` u ) ` { t } ) ) -> y = ( i ` <. z , ( i ` <. e , ( ( (HVMap ` K ) ` w ) ` e ) , z >. ) , t >. ) ) ) ) } )
100		fvex 6201	. . . . . 6 |- ( Base ` U ) e. _V
101	42, 100	eqeltri 2697	. . . . 5 |- V e. _V
102	101	mptex 6486	. . . 4 |- ( t e. V |-> ( iota_ y e. D A. z e. V ( -. z e. ( ( N ` { E } ) u. ( N ` { t } ) ) -> y = ( I ` <. z , ( I ` <. E , ( J ` E ) , z >. ) , t >. ) ) ) ) e. _V
103	98, 99, 102	fvmpt 6282	. . 3 |- ( W e. H -> ( ( w e. H |-> { a | [. <. ( _I |` ( Base ` K ) ) , ( _I |` ( ( LTrn ` K ) ` w ) ) >. / e ]. [. ( ( DVecH ` K ) ` w ) / u ]. [. ( Base ` u ) / v ]. [. ( (HDMap1 ` K ) ` w ) / i ]. a e. ( t e. v |-> ( iota_ y e. ( Base ` ( (LCDual ` K ) ` w ) ) A. z e. v ( -. z e. ( ( ( LSpan ` u ) ` { e } ) u. ( ( LSpan ` u ) ` { t } ) ) -> y = ( i ` <. z , ( i ` <. e , ( ( (HVMap ` K ) ` w ) ` e ) , z >. ) , t >. ) ) ) ) } ) ` W ) = ( t e. V |-> ( iota_ y e. D A. z e. V ( -. z e. ( ( N ` { E } ) u. ( N ` { t } ) ) -> y = ( I ` <. z , ( I ` <. E , ( J ` E ) , z >. ) , t >. ) ) ) ) )
104	6, 103	sylan9eq 2676	. 2 |- ( ( K e. A /\ W e. H ) -> S = ( t e. V |-> ( iota_ y e. D A. z e. V ( -. z e. ( ( N ` { E } ) u. ( N ` { t } ) ) -> y = ( I ` <. z , ( I ` <. E , ( J ` E ) , z >. ) , t >. ) ) ) ) )
105	1, 104	syl 17	1 |- ( ph -> S = ( t e. V |-> ( iota_ y e. D A. z e. V ( -. z e. ( ( N ` { E } ) u. ( N ` { t } ) ) -> y = ( I ` <. z , ( I ` <. E , ( J ` E ) , z >. ) , t >. ) ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   {cab 2608   A.wral 2912   _Vcvv 3200   [.wsbc 3435   u. cun 3572   {csn 4177   <.cop 4183   <.cotp 4185   |-> cmpt 4729   _I cid 5023   |` cres 5116   ` cfv 5888   iota_crio 6610   Basecbs 15857   LSpanclspn 18971   LHypclh 35270   LTrncltrn 35387   DVecHcdvh 36367  LCDualclcd 36875  HVMapchvm 37045  HDMap1chdma1 37081  HDMapchdma 37082

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-sn 4178  df-pr 4180  df-op 4184  df-ot 4186  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-riota 6611  df-hdmap 37084

This theorem is referenced by:  hdmapval  37120  hdmapfnN  37121