Theorem diclss 34526

Description: The value of partial isomorphism C is a subspace of partial vector space H. (Contributed by NM, 16-Feb-2014.)

Hypotheses

Ref	Expression
diclss.l	|- .<_ = ( le ` K )
diclss.a	|- A = ( Atoms ` K )
diclss.h	|- H = ( LHyp ` K )
diclss.u	|- U = ( ( DVecH ` K ) ` W )
diclss.i	|- I = ( ( DIsoC ` K ) ` W )
diclss.s	|- S = ( LSubSp ` U )

Assertion

Ref	Expression
diclss	|- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( I ` Q ) e. S )

Proof of Theorem diclss

Dummy variables a b x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqidd 2442	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> (Scalar ` U ) = (Scalar ` U ) )
2		diclss.h	. . . . 5 |- H = ( LHyp ` K )
3		eqid 2441	. . . . 5 |- ( ( TEndo ` K ) ` W ) = ( ( TEndo ` K ) ` W )
4		diclss.u	. . . . 5 |- U = ( ( DVecH ` K ) ` W )
5		eqid 2441	. . . . 5 |- (Scalar ` U ) = (Scalar ` U )
6		eqid 2441	. . . . 5 |- ( Base ` (Scalar ` U ) ) = ( Base ` (Scalar ` U ) )
7	2, 3, 4, 5, 6	dvhbase 34416	. . . 4 |- ( ( K e. HL /\ W e. H ) -> ( Base ` (Scalar ` U ) ) = ( ( TEndo ` K ) ` W ) )
8	7	eqcomd 2446	. . 3 |- ( ( K e. HL /\ W e. H ) -> ( ( TEndo ` K ) ` W ) = ( Base ` (Scalar ` U ) ) )
9	8	adantr 462	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( ( TEndo ` K ) ` W ) = ( Base ` (Scalar ` U ) ) )
10		eqid 2441	. . . . 5 |- ( ( LTrn ` K ) ` W ) = ( ( LTrn ` K ) ` W )
11		eqid 2441	. . . . 5 |- ( Base ` U ) = ( Base ` U )
12	2, 10, 3, 4, 11	dvhvbase 34420	. . . 4 |- ( ( K e. HL /\ W e. H ) -> ( Base ` U ) = ( ( ( LTrn ` K ) ` W ) X. ( ( TEndo ` K ) ` W ) ) )
13	12	eqcomd 2446	. . 3 |- ( ( K e. HL /\ W e. H ) -> ( ( ( LTrn ` K ) ` W ) X. ( ( TEndo ` K ) ` W ) ) = ( Base ` U ) )
14	13	adantr 462	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( ( ( LTrn ` K ) ` W ) X. ( ( TEndo ` K ) ` W ) ) = ( Base ` U ) )
15		eqidd 2442	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( +g ` U ) = ( +g ` U ) )
16		eqidd 2442	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( .s ` U ) = ( .s ` U ) )
17		diclss.s	. . 3 |- S = ( LSubSp ` U )
18	17	a1i 11	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> S = ( LSubSp ` U ) )
19		diclss.l	. . . 4 |- .<_ = ( le ` K )
20		diclss.a	. . . 4 |- A = ( Atoms ` K )
21		diclss.i	. . . 4 |- I = ( ( DIsoC ` K ) ` W )
22	19, 20, 2, 21, 4, 11	dicssdvh 34519	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( I ` Q ) C_ ( Base ` U ) )
23	22, 14	sseqtr4d 3390	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( I ` Q ) C_ ( ( ( LTrn ` K ) ` W ) X. ( ( TEndo ` K ) ` W ) ) )
24	19, 20, 2, 21	dicn0 34525	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( I ` Q ) =/= (/) )
25		simpll 748	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( x e. ( ( TEndo ` K ) ` W ) /\ a e. ( I ` Q ) /\ b e. ( I ` Q ) ) ) -> ( K e. HL /\ W e. H ) )
26		simplr 749	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( x e. ( ( TEndo ` K ) ` W ) /\ a e. ( I ` Q ) /\ b e. ( I ` Q ) ) ) -> ( Q e. A /\ -. Q .<_ W ) )
27		simpr1 989	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( x e. ( ( TEndo ` K ) ` W ) /\ a e. ( I ` Q ) /\ b e. ( I ` Q ) ) ) -> x e. ( ( TEndo ` K ) ` W ) )
28		simpr2 990	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( x e. ( ( TEndo ` K ) ` W ) /\ a e. ( I ` Q ) /\ b e. ( I ` Q ) ) ) -> a e. ( I ` Q ) )
29		eqid 2441	. . . . 5 |- ( .s ` U ) = ( .s ` U )
30	19, 20, 2, 3, 4, 21, 29	dicvscacl 34524	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( x e. ( ( TEndo ` K ) ` W ) /\ a e. ( I ` Q ) ) ) -> ( x ( .s ` U ) a ) e. ( I ` Q ) )
31	25, 26, 27, 28, 30	syl112anc 1217	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( x e. ( ( TEndo ` K ) ` W ) /\ a e. ( I ` Q ) /\ b e. ( I ` Q ) ) ) -> ( x ( .s ` U ) a ) e. ( I ` Q ) )
32		simpr3 991	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( x e. ( ( TEndo ` K ) ` W ) /\ a e. ( I ` Q ) /\ b e. ( I ` Q ) ) ) -> b e. ( I ` Q ) )
33		eqid 2441	. . . 4 |- ( +g ` U ) = ( +g ` U )
34	19, 20, 2, 4, 21, 33	dicvaddcl 34523	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( ( x ( .s ` U ) a ) e. ( I ` Q ) /\ b e. ( I ` Q ) ) ) -> ( ( x ( .s ` U ) a ) ( +g ` U ) b ) e. ( I ` Q ) )
35	25, 26, 31, 32, 34	syl112anc 1217	. 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( x e. ( ( TEndo ` K ) ` W ) /\ a e. ( I ` Q ) /\ b e. ( I ` Q ) ) ) -> ( ( x ( .s ` U ) a ) ( +g ` U ) b ) e. ( I ` Q ) )
36	1, 9, 14, 15, 16, 18, 23, 24, 35	islssd 16995	1 |- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( I ` Q ) e. S )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 369   /\ w3a 960   = wceq 1364   e. wcel 1761   class class class wbr 4289   X. cxp 4834   ` cfv 5415  (class class class)co 6090   Basecbs 14170   +g cplusg 14234  Scalarcsca 14237   .scvsca 14238   lecple 14241   LSubSpclss 16991   Atomscatm 32596   HLchlt 32683   LHypclh 33316   LTrncltrn 33433   TEndoctendo 34084   DVecHcdvh 34411   DIsoCcdic 34505

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  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-riotaBAD 32292

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  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-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-iin 4171  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  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-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-undef 6788  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-map 7212  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-n0 10576  df-z 10643  df-uz 10858  df-fz 11434  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-plusg 14247  df-mulr 14248  df-sca 14250  df-vsca 14251  df-poset 15112  df-plt 15124  df-lub 15140  df-glb 15141  df-join 15142  df-meet 15143  df-p0 15205  df-p1 15206  df-lat 15212  df-clat 15274  df-lss 16992  df-oposet 32509  df-ol 32511  df-oml 32512  df-covers 32599  df-ats 32600  df-atl 32631  df-cvlat 32655  df-hlat 32684  df-llines 32830  df-lplanes 32831  df-lvols 32832  df-lines 32833  df-psubsp 32835  df-pmap 32836  df-padd 33128  df-lhyp 33320  df-laut 33321  df-ldil 33436  df-ltrn 33437  df-trl 33491  df-tendo 34087  df-edring 34089  df-dvech 34412  df-dic 34506

This theorem is referenced by:  cdlemn5pre  34533  cdlemn11c  34542  dihjustlem  34549  dihord1  34551  dihord2a  34552  dihord2b  34553  dihord11c  34557  dihlsscpre  34567  dihvalcqat  34572  dihopelvalcpre  34581  dihord6apre  34589  dihord5b  34592  dihord5apre  34595