Theorem dvhb1dimN 36274

Description: Two expressions for the 1-dimensional subspaces of vector space H, in the isomorphism B case where the 2nd vector component is zero. (Contributed by NM, 23-Feb-2014.) (New usage is discouraged.)

Hypotheses

Ref	Expression
dvhb1dim.l	|- .<_ = ( le ` K )
dvhb1dim.h	|- H = ( LHyp ` K )
dvhb1dim.t	|- T = ( ( LTrn ` K ) ` W )
dvhb1dim.r	|- R = ( ( trL ` K ) ` W )
dvhb1dim.e	|- E = ( ( TEndo ` K ) ` W )
dvhb1dim.o	|- .0. = ( h e. T |-> ( _I |` B ) )

Assertion

Ref	Expression
dvhb1dimN	|- ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> { g e. ( T X. E ) | E. s e. E g = <. ( s ` F ) , .0. >. } = { g e. ( T X. E ) | ( ( R ` ( 1st ` g ) ) .<_ ( R ` F ) /\ ( 2nd ` g ) = .0. ) } )

Distinct variable groups:   .<_ , s   E, s   g, s, F   g, H, s   g, K, s   .0. , s   R, s   g, h, T, s   g, W, s

Allowed substitution hints:   B( g, h, s)   R( g, h)   E( g, h)   F( h)   H( h)   K( h)   .<_ ( g, h)   W( h)   .0. ( g, h)

Proof of Theorem dvhb1dimN

Dummy variable f is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqop 7208	. . . . 5 |- ( g e. ( T X. E ) -> ( g = <. ( s ` F ) , .0. >. <-> ( ( 1st ` g ) = ( s ` F ) /\ ( 2nd ` g ) = .0. ) ) )
2	1	adantl 482	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T ) /\ g e. ( T X. E ) ) -> ( g = <. ( s ` F ) , .0. >. <-> ( ( 1st ` g ) = ( s ` F ) /\ ( 2nd ` g ) = .0. ) ) )
3	2	rexbidv 3052	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T ) /\ g e. ( T X. E ) ) -> ( E. s e. E g = <. ( s ` F ) , .0. >. <-> E. s e. E ( ( 1st ` g ) = ( s ` F ) /\ ( 2nd ` g ) = .0. ) ) )
4		r19.41v 3089	. . . 4 |- ( E. s e. E ( ( 1st ` g ) = ( s ` F ) /\ ( 2nd ` g ) = .0. ) <-> ( E. s e. E ( 1st ` g ) = ( s ` F ) /\ ( 2nd ` g ) = .0. ) )
5		fvex 6201	. . . . . . . 8 |- ( 1st ` g ) e. _V
6		eqeq1 2626	. . . . . . . . 9 |- ( f = ( 1st ` g ) -> ( f = ( s ` F ) <-> ( 1st ` g ) = ( s ` F ) ) )
7	6	rexbidv 3052	. . . . . . . 8 |- ( f = ( 1st ` g ) -> ( E. s e. E f = ( s ` F ) <-> E. s e. E ( 1st ` g ) = ( s ` F ) ) )
8	5, 7	elab 3350	. . . . . . 7 |- ( ( 1st ` g ) e. { f | E. s e. E f = ( s ` F ) } <-> E. s e. E ( 1st ` g ) = ( s ` F ) )
9		dvhb1dim.l	. . . . . . . . . 10 |- .<_ = ( le ` K )
10		dvhb1dim.h	. . . . . . . . . 10 |- H = ( LHyp ` K )
11		dvhb1dim.t	. . . . . . . . . 10 |- T = ( ( LTrn ` K ) ` W )
12		dvhb1dim.r	. . . . . . . . . 10 |- R = ( ( trL ` K ) ` W )
13		dvhb1dim.e	. . . . . . . . . 10 |- E = ( ( TEndo ` K ) ` W )
14	9, 10, 11, 12, 13	dva1dim 36273	. . . . . . . . 9 |- ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> { f | E. s e. E f = ( s ` F ) } = { f e. T | ( R ` f ) .<_ ( R ` F ) } )
15	14	adantr 481	. . . . . . . 8 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T ) /\ g e. ( T X. E ) ) -> { f | E. s e. E f = ( s ` F ) } = { f e. T | ( R ` f ) .<_ ( R ` F ) } )
16	15	eleq2d 2687	. . . . . . 7 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T ) /\ g e. ( T X. E ) ) -> ( ( 1st ` g ) e. { f | E. s e. E f = ( s ` F ) } <-> ( 1st ` g ) e. { f e. T | ( R ` f ) .<_ ( R ` F ) } ) )
17	8, 16	syl5bbr 274	. . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T ) /\ g e. ( T X. E ) ) -> ( E. s e. E ( 1st ` g ) = ( s ` F ) <-> ( 1st ` g ) e. { f e. T | ( R ` f ) .<_ ( R ` F ) } ) )
18		xp1st 7198	. . . . . . . 8 |- ( g e. ( T X. E ) -> ( 1st ` g ) e. T )
19	18	adantl 482	. . . . . . 7 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T ) /\ g e. ( T X. E ) ) -> ( 1st ` g ) e. T )
20		fveq2 6191	. . . . . . . . 9 |- ( f = ( 1st ` g ) -> ( R ` f ) = ( R ` ( 1st ` g ) ) )
21	20	breq1d 4663	. . . . . . . 8 |- ( f = ( 1st ` g ) -> ( ( R ` f ) .<_ ( R ` F ) <-> ( R ` ( 1st ` g ) ) .<_ ( R ` F ) ) )
22	21	elrab3 3364	. . . . . . 7 |- ( ( 1st ` g ) e. T -> ( ( 1st ` g ) e. { f e. T | ( R ` f ) .<_ ( R ` F ) } <-> ( R ` ( 1st ` g ) ) .<_ ( R ` F ) ) )
23	19, 22	syl 17	. . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T ) /\ g e. ( T X. E ) ) -> ( ( 1st ` g ) e. { f e. T | ( R ` f ) .<_ ( R ` F ) } <-> ( R ` ( 1st ` g ) ) .<_ ( R ` F ) ) )
24	17, 23	bitrd 268	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T ) /\ g e. ( T X. E ) ) -> ( E. s e. E ( 1st ` g ) = ( s ` F ) <-> ( R ` ( 1st ` g ) ) .<_ ( R ` F ) ) )
25	24	anbi1d 741	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T ) /\ g e. ( T X. E ) ) -> ( ( E. s e. E ( 1st ` g ) = ( s ` F ) /\ ( 2nd ` g ) = .0. ) <-> ( ( R ` ( 1st ` g ) ) .<_ ( R ` F ) /\ ( 2nd ` g ) = .0. ) ) )
26	4, 25	syl5bb 272	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T ) /\ g e. ( T X. E ) ) -> ( E. s e. E ( ( 1st ` g ) = ( s ` F ) /\ ( 2nd ` g ) = .0. ) <-> ( ( R ` ( 1st ` g ) ) .<_ ( R ` F ) /\ ( 2nd ` g ) = .0. ) ) )
27	3, 26	bitrd 268	. 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T ) /\ g e. ( T X. E ) ) -> ( E. s e. E g = <. ( s ` F ) , .0. >. <-> ( ( R ` ( 1st ` g ) ) .<_ ( R ` F ) /\ ( 2nd ` g ) = .0. ) ) )
28	27	rabbidva 3188	1 |- ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> { g e. ( T X. E ) | E. s e. E g = <. ( s ` F ) , .0. >. } = { g e. ( T X. E ) | ( ( R ` ( 1st ` g ) ) .<_ ( R ` F ) /\ ( 2nd ` g ) = .0. ) } )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   E.wrex 2913   {crab 2916   <.cop 4183   class class class wbr 4653   |-> cmpt 4729   _I cid 5023   X. cxp 5112   |` cres 5116   ` cfv 5888   1stc1st 7166   2ndc2nd 7167   lecple 15948   HLchlt 34637   LHypclh 35270   LTrncltrn 35387   trLctrl 35445   TEndoctendo 36040

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  ax-riotaBAD 34239

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  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-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  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-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-iin 4523  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-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-undef 7399  df-map 7859  df-preset 16928  df-poset 16946  df-plt 16958  df-lub 16974  df-glb 16975  df-join 16976  df-meet 16977  df-p0 17039  df-p1 17040  df-lat 17046  df-clat 17108  df-oposet 34463  df-ol 34465  df-oml 34466  df-covers 34553  df-ats 34554  df-atl 34585  df-cvlat 34609  df-hlat 34638  df-llines 34784  df-lplanes 34785  df-lvols 34786  df-lines 34787  df-psubsp 34789  df-pmap 34790  df-padd 35082  df-lhyp 35274  df-laut 35275  df-ldil 35390  df-ltrn 35391  df-trl 35446  df-tendo 36043

This theorem is referenced by: (None)