Theorem hgmapffval 37177

Description: Map from the scalar division ring of the vector space to the scalar division ring of its closed kernel dual. (Contributed by NM, 25-Mar-2015.)

Hypothesis

Ref	Expression
hgmapval.h	|- H = ( LHyp ` K )

Assertion

Ref	Expression
hgmapffval	|- ( K e. X -> (HGMap ` K ) = ( w e. H |-> { a | [. ( ( DVecH ` K ) ` w ) / u ]. [. ( Base ` (Scalar ` u ) ) / b ]. [. ( (HDMap ` K ) ` w ) / m ]. a e. ( x e. b |-> ( iota_ y e. b A. v e. ( Base ` u ) ( m ` ( x ( .s ` u ) v ) ) = ( y ( .s ` ( (LCDual ` K ) ` w ) ) ( m ` v ) ) ) ) } ) )

Distinct variable groups:   w, H   a, b, m, u, v, w, x, y, K

Allowed substitution hints:   H( x, y, v, u, m, a, b)   X( x, y, w, v, u, m, a, b)

Proof of Theorem hgmapffval

Dummy variable k is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3212	. 2 |- ( K e. X -> K e. _V )
2		fveq2 6191	. . . . 5 |- ( k = K -> ( LHyp ` k ) = ( LHyp ` K ) )
3		hgmapval.h	. . . . 5 |- H = ( LHyp ` K )
4	2, 3	syl6eqr 2674	. . . 4 |- ( k = K -> ( LHyp ` k ) = H )
5		fveq2 6191	. . . . . . 7 |- ( k = K -> ( DVecH ` k ) = ( DVecH ` K ) )
6	5	fveq1d 6193	. . . . . 6 |- ( k = K -> ( ( DVecH ` k ) ` w ) = ( ( DVecH ` K ) ` w ) )
7		fveq2 6191	. . . . . . . . 9 |- ( k = K -> (HDMap ` k ) = (HDMap ` K ) )
8	7	fveq1d 6193	. . . . . . . 8 |- ( k = K -> ( (HDMap ` k ) ` w ) = ( (HDMap ` K ) ` w ) )
9		fveq2 6191	. . . . . . . . . . . . . . . 16 |- ( k = K -> (LCDual ` k ) = (LCDual ` K ) )
10	9	fveq1d 6193	. . . . . . . . . . . . . . 15 |- ( k = K -> ( (LCDual ` k ) ` w ) = ( (LCDual ` K ) ` w ) )
11	10	fveq2d 6195	. . . . . . . . . . . . . 14 |- ( k = K -> ( .s ` ( (LCDual ` k ) ` w ) ) = ( .s ` ( (LCDual ` K ) ` w ) ) )
12	11	oveqd 6667	. . . . . . . . . . . . 13 |- ( k = K -> ( y ( .s ` ( (LCDual ` k ) ` w ) ) ( m ` v ) ) = ( y ( .s ` ( (LCDual ` K ) ` w ) ) ( m ` v ) ) )
13	12	eqeq2d 2632	. . . . . . . . . . . 12 |- ( k = K -> ( ( m ` ( x ( .s ` u ) v ) ) = ( y ( .s ` ( (LCDual ` k ) ` w ) ) ( m ` v ) ) <-> ( m ` ( x ( .s ` u ) v ) ) = ( y ( .s ` ( (LCDual ` K ) ` w ) ) ( m ` v ) ) ) )
14	13	ralbidv 2986	. . . . . . . . . . 11 |- ( k = K -> ( A. v e. ( Base ` u ) ( m ` ( x ( .s ` u ) v ) ) = ( y ( .s ` ( (LCDual ` k ) ` w ) ) ( m ` v ) ) <-> A. v e. ( Base ` u ) ( m ` ( x ( .s ` u ) v ) ) = ( y ( .s ` ( (LCDual ` K ) ` w ) ) ( m ` v ) ) ) )
15	14	riotabidv 6613	. . . . . . . . . 10 |- ( k = K -> ( iota_ y e. b A. v e. ( Base ` u ) ( m ` ( x ( .s ` u ) v ) ) = ( y ( .s ` ( (LCDual ` k ) ` w ) ) ( m ` v ) ) ) = ( iota_ y e. b A. v e. ( Base ` u ) ( m ` ( x ( .s ` u ) v ) ) = ( y ( .s ` ( (LCDual ` K ) ` w ) ) ( m ` v ) ) ) )
16	15	mpteq2dv 4745	. . . . . . . . 9 |- ( k = K -> ( x e. b |-> ( iota_ y e. b A. v e. ( Base ` u ) ( m ` ( x ( .s ` u ) v ) ) = ( y ( .s ` ( (LCDual ` k ) ` w ) ) ( m ` v ) ) ) ) = ( x e. b |-> ( iota_ y e. b A. v e. ( Base ` u ) ( m ` ( x ( .s ` u ) v ) ) = ( y ( .s ` ( (LCDual ` K ) ` w ) ) ( m ` v ) ) ) ) )
17	16	eleq2d 2687	. . . . . . . 8 |- ( k = K -> ( a e. ( x e. b |-> ( iota_ y e. b A. v e. ( Base ` u ) ( m ` ( x ( .s ` u ) v ) ) = ( y ( .s ` ( (LCDual ` k ) ` w ) ) ( m ` v ) ) ) ) <-> a e. ( x e. b |-> ( iota_ y e. b A. v e. ( Base ` u ) ( m ` ( x ( .s ` u ) v ) ) = ( y ( .s ` ( (LCDual ` K ) ` w ) ) ( m ` v ) ) ) ) ) )
18	8, 17	sbceqbid 3442	. . . . . . 7 |- ( k = K -> ( [. ( (HDMap ` k ) ` w ) / m ]. a e. ( x e. b |-> ( iota_ y e. b A. v e. ( Base ` u ) ( m ` ( x ( .s ` u ) v ) ) = ( y ( .s ` ( (LCDual ` k ) ` w ) ) ( m ` v ) ) ) ) <-> [. ( (HDMap ` K ) ` w ) / m ]. a e. ( x e. b |-> ( iota_ y e. b A. v e. ( Base ` u ) ( m ` ( x ( .s ` u ) v ) ) = ( y ( .s ` ( (LCDual ` K ) ` w ) ) ( m ` v ) ) ) ) ) )
19	18	sbcbidv 3490	. . . . . 6 |- ( k = K -> ( [. ( Base ` (Scalar ` u ) ) / b ]. [. ( (HDMap ` k ) ` w ) / m ]. a e. ( x e. b |-> ( iota_ y e. b A. v e. ( Base ` u ) ( m ` ( x ( .s ` u ) v ) ) = ( y ( .s ` ( (LCDual ` k ) ` w ) ) ( m ` v ) ) ) ) <-> [. ( Base ` (Scalar ` u ) ) / b ]. [. ( (HDMap ` K ) ` w ) / m ]. a e. ( x e. b |-> ( iota_ y e. b A. v e. ( Base ` u ) ( m ` ( x ( .s ` u ) v ) ) = ( y ( .s ` ( (LCDual ` K ) ` w ) ) ( m ` v ) ) ) ) ) )
20	6, 19	sbceqbid 3442	. . . . 5 |- ( k = K -> ( [. ( ( DVecH ` k ) ` w ) / u ]. [. ( Base ` (Scalar ` u ) ) / b ]. [. ( (HDMap ` k ) ` w ) / m ]. a e. ( x e. b |-> ( iota_ y e. b A. v e. ( Base ` u ) ( m ` ( x ( .s ` u ) v ) ) = ( y ( .s ` ( (LCDual ` k ) ` w ) ) ( m ` v ) ) ) ) <-> [. ( ( DVecH ` K ) ` w ) / u ]. [. ( Base ` (Scalar ` u ) ) / b ]. [. ( (HDMap ` K ) ` w ) / m ]. a e. ( x e. b |-> ( iota_ y e. b A. v e. ( Base ` u ) ( m ` ( x ( .s ` u ) v ) ) = ( y ( .s ` ( (LCDual ` K ) ` w ) ) ( m ` v ) ) ) ) ) )
21	20	abbidv 2741	. . . 4 |- ( k = K -> { a | [. ( ( DVecH ` k ) ` w ) / u ]. [. ( Base ` (Scalar ` u ) ) / b ]. [. ( (HDMap ` k ) ` w ) / m ]. a e. ( x e. b |-> ( iota_ y e. b A. v e. ( Base ` u ) ( m ` ( x ( .s ` u ) v ) ) = ( y ( .s ` ( (LCDual ` k ) ` w ) ) ( m ` v ) ) ) ) } = { a | [. ( ( DVecH ` K ) ` w ) / u ]. [. ( Base ` (Scalar ` u ) ) / b ]. [. ( (HDMap ` K ) ` w ) / m ]. a e. ( x e. b |-> ( iota_ y e. b A. v e. ( Base ` u ) ( m ` ( x ( .s ` u ) v ) ) = ( y ( .s ` ( (LCDual ` K ) ` w ) ) ( m ` v ) ) ) ) } )
22	4, 21	mpteq12dv 4733	. . 3 |- ( k = K -> ( w e. ( LHyp ` k ) |-> { a | [. ( ( DVecH ` k ) ` w ) / u ]. [. ( Base ` (Scalar ` u ) ) / b ]. [. ( (HDMap ` k ) ` w ) / m ]. a e. ( x e. b |-> ( iota_ y e. b A. v e. ( Base ` u ) ( m ` ( x ( .s ` u ) v ) ) = ( y ( .s ` ( (LCDual ` k ) ` w ) ) ( m ` v ) ) ) ) } ) = ( w e. H |-> { a | [. ( ( DVecH ` K ) ` w ) / u ]. [. ( Base ` (Scalar ` u ) ) / b ]. [. ( (HDMap ` K ) ` w ) / m ]. a e. ( x e. b |-> ( iota_ y e. b A. v e. ( Base ` u ) ( m ` ( x ( .s ` u ) v ) ) = ( y ( .s ` ( (LCDual ` K ) ` w ) ) ( m ` v ) ) ) ) } ) )
23		df-hgmap 37176	. . 3 |- HGMap = ( k e. _V |-> ( w e. ( LHyp ` k ) |-> { a | [. ( ( DVecH ` k ) ` w ) / u ]. [. ( Base ` (Scalar ` u ) ) / b ]. [. ( (HDMap ` k ) ` w ) / m ]. a e. ( x e. b |-> ( iota_ y e. b A. v e. ( Base ` u ) ( m ` ( x ( .s ` u ) v ) ) = ( y ( .s ` ( (LCDual ` k ) ` w ) ) ( m ` v ) ) ) ) } ) )
24		fvex 6201	. . . . 5 |- ( LHyp ` K ) e. _V
25	3, 24	eqeltri 2697	. . . 4 |- H e. _V
26	25	mptex 6486	. . 3 |- ( w e. H |-> { a | [. ( ( DVecH ` K ) ` w ) / u ]. [. ( Base ` (Scalar ` u ) ) / b ]. [. ( (HDMap ` K ) ` w ) / m ]. a e. ( x e. b |-> ( iota_ y e. b A. v e. ( Base ` u ) ( m ` ( x ( .s ` u ) v ) ) = ( y ( .s ` ( (LCDual ` K ) ` w ) ) ( m ` v ) ) ) ) } ) e. _V
27	22, 23, 26	fvmpt 6282	. 2 |- ( K e. _V -> (HGMap ` K ) = ( w e. H |-> { a | [. ( ( DVecH ` K ) ` w ) / u ]. [. ( Base ` (Scalar ` u ) ) / b ]. [. ( (HDMap ` K ) ` w ) / m ]. a e. ( x e. b |-> ( iota_ y e. b A. v e. ( Base ` u ) ( m ` ( x ( .s ` u ) v ) ) = ( y ( .s ` ( (LCDual ` K ) ` w ) ) ( m ` v ) ) ) ) } ) )
28	1, 27	syl 17	1 |- ( K e. X -> (HGMap ` K ) = ( w e. H |-> { a | [. ( ( DVecH ` K ) ` w ) / u ]. [. ( Base ` (Scalar ` u ) ) / b ]. [. ( (HDMap ` K ) ` w ) / m ]. a e. ( x e. b |-> ( iota_ y e. b A. v e. ( Base ` u ) ( m ` ( x ( .s ` u ) v ) ) = ( y ( .s ` ( (LCDual ` K ) ` w ) ) ( m ` v ) ) ) ) } ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   {cab 2608   A.wral 2912   _Vcvv 3200   [.wsbc 3435   |-> cmpt 4729   ` cfv 5888   iota_crio 6610  (class class class)co 6650   Basecbs 15857  Scalarcsca 15944   .scvsca 15945   LHypclh 35270   DVecHcdvh 36367  LCDualclcd 36875  HDMapchdma 37082  HGMapchg 37175

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-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-ov 6653  df-hgmap 37176

This theorem is referenced by:  hgmapfval  37178