Theorem dia1dim 36350

Description: Two expressions for the 1-dimensional subspaces of partial vector space A (when F is a nonzero vector i.e. non-identity translation). Remark after Lemma L in [Crawley] p. 120 line 21. (Contributed by NM, 15-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)

Hypotheses

Ref	Expression
dia1dim.h	|- H = ( LHyp ` K )
dia1dim.t	|- T = ( ( LTrn ` K ) ` W )
dia1dim.r	|- R = ( ( trL ` K ) ` W )
dia1dim.e	|- E = ( ( TEndo ` K ) ` W )
dia1dim.i	|- I = ( ( DIsoA ` K ) ` W )

Assertion

Ref	Expression
dia1dim	|- ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> ( I ` ( R ` F ) ) = { g | E. s e. E g = ( s ` F ) } )

Distinct variable groups:   E, s   g, s, F   g, H, s   g, K, s   R, g, s   T, g, s   g, W, s

Allowed substitution hints:   E( g)   I( g, s)

Proof of Theorem dia1dim

Step	Hyp	Ref	Expression
1		simpl 473	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> ( K e. HL /\ W e. H ) )
2		eqid 2622	. . . 4 |- ( Base ` K ) = ( Base ` K )
3		dia1dim.h	. . . 4 |- H = ( LHyp ` K )
4		dia1dim.t	. . . 4 |- T = ( ( LTrn ` K ) ` W )
5		dia1dim.r	. . . 4 |- R = ( ( trL ` K ) ` W )
6	2, 3, 4, 5	trlcl 35451	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> ( R ` F ) e. ( Base ` K ) )
7		eqid 2622	. . . 4 |- ( le ` K ) = ( le ` K )
8	7, 3, 4, 5	trlle 35471	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> ( R ` F ) ( le ` K ) W )
9		dia1dim.i	. . . 4 |- I = ( ( DIsoA ` K ) ` W )
10	2, 7, 3, 4, 5, 9	diaval 36321	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( R ` F ) e. ( Base ` K ) /\ ( R ` F ) ( le ` K ) W ) ) -> ( I ` ( R ` F ) ) = { g e. T | ( R ` g ) ( le ` K ) ( R ` F ) } )
11	1, 6, 8, 10	syl12anc 1324	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> ( I ` ( R ` F ) ) = { g e. T | ( R ` g ) ( le ` K ) ( R ` F ) } )
12		dia1dim.e	. . 3 |- E = ( ( TEndo ` K ) ` W )
13	7, 3, 4, 5, 12	dva1dim 36273	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> { g | E. s e. E g = ( s ` F ) } = { g e. T | ( R ` g ) ( le ` K ) ( R ` F ) } )
14	11, 13	eqtr4d 2659	1 |- ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> ( I ` ( R ` F ) ) = { g | E. s e. E g = ( s ` F ) } )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   E.wrex 2913   {crab 2916   class class class wbr 4653   ` cfv 5888   Basecbs 15857   lecple 15948   HLchlt 34637   LHypclh 35270   LTrncltrn 35387   trLctrl 35445   TEndoctendo 36040   DIsoAcdia 36317

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  df-disoa 36318

This theorem is referenced by:  dia1dim2  36351  dib1dim  36454