Theorem diclss 34193

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 2403	. 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 2402	. . . . 5 |- ( ( TEndo ` K ) ` W ) = ( ( TEndo ` K ) ` W )
4		diclss.u	. . . . 5 |- U = ( ( DVecH ` K ) ` W )
5		eqid 2402	. . . . 5 |- (Scalar ` U ) = (Scalar ` U )
6		eqid 2402	. . . . 5 |- ( Base ` (Scalar ` U ) ) = ( Base ` (Scalar ` U ) )
7	2, 3, 4, 5, 6	dvhbase 34083	. . . 4 |- ( ( K e. HL /\ W e. H ) -> ( Base ` (Scalar ` U ) ) = ( ( TEndo ` K ) ` W ) )
8	7	eqcomd 2410	. . 3 |- ( ( K e. HL /\ W e. H ) -> ( ( TEndo ` K ) ` W ) = ( Base ` (Scalar ` U ) ) )
9	8	adantr 463	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( ( TEndo ` K ) ` W ) = ( Base ` (Scalar ` U ) ) )
10		eqid 2402	. . . . 5 |- ( ( LTrn ` K ) ` W ) = ( ( LTrn ` K ) ` W )
11		eqid 2402	. . . . 5 |- ( Base ` U ) = ( Base ` U )
12	2, 10, 3, 4, 11	dvhvbase 34087	. . . 4 |- ( ( K e. HL /\ W e. H ) -> ( Base ` U ) = ( ( ( LTrn ` K ) ` W ) X. ( ( TEndo ` K ) ` W ) ) )
13	12	eqcomd 2410	. . 3 |- ( ( K e. HL /\ W e. H ) -> ( ( ( LTrn ` K ) ` W ) X. ( ( TEndo ` K ) ` W ) ) = ( Base ` U ) )
14	13	adantr 463	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( ( ( LTrn ` K ) ` W ) X. ( ( TEndo ` K ) ` W ) ) = ( Base ` U ) )
15		eqidd 2403	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( +g ` U ) = ( +g ` U ) )
16		eqidd 2403	. 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 34186	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( I ` Q ) C_ ( Base ` U ) )
23	22, 14	sseqtr4d 3478	. 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 34192	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( I ` Q ) =/= (/) )
25		simpll 752	. . 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 754	. . 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 1003	. . . 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 1004	. . . 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 2402	. . . . 5 |- ( .s ` U ) = ( .s ` U )
30	19, 20, 2, 3, 4, 21, 29	dicvscacl 34191	. . . 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 1234	. . 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 1005	. . 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 2402	. . . 4 |- ( +g ` U ) = ( +g ` U )
34	19, 20, 2, 4, 21, 33	dicvaddcl 34190	. . 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 1234	. 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 17900	1 |- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( I ` Q ) e. S )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 367   /\ w3a 974   = wceq 1405   e. wcel 1842   class class class wbr 4394   X. cxp 4820   ` cfv 5568  (class class class)co 6277   Basecbs 14839   +g cplusg 14907  Scalarcsca 14910   .scvsca 14911   lecple 14914   LSubSpclss 17896   Atomscatm 32261   HLchlt 32348   LHypclh 32981   LTrncltrn 33098   TEndoctendo 33751   DVecHcdvh 34078   DIsoCcdic 34172

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597  ax-pre-mulgt0 9598  ax-riotaBAD 31957

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-iin 4273  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6683  df-1st 6783  df-2nd 6784  df-undef 7004  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-1o 7166  df-oadd 7170  df-er 7347  df-map 7458  df-en 7554  df-dom 7555  df-sdom 7556  df-fin 7557  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-le 9663  df-sub 9842  df-neg 9843  df-nn 10576  df-2 10634  df-3 10635  df-4 10636  df-5 10637  df-6 10638  df-n0 10836  df-z 10905  df-uz 11127  df-fz 11725  df-struct 14841  df-ndx 14842  df-slot 14843  df-base 14844  df-plusg 14920  df-mulr 14921  df-sca 14923  df-vsca 14924  df-preset 15879  df-poset 15897  df-plt 15910  df-lub 15926  df-glb 15927  df-join 15928  df-meet 15929  df-p0 15991  df-p1 15992  df-lat 15998  df-clat 16060  df-lss 17897  df-oposet 32174  df-ol 32176  df-oml 32177  df-covers 32264  df-ats 32265  df-atl 32296  df-cvlat 32320  df-hlat 32349  df-llines 32495  df-lplanes 32496  df-lvols 32497  df-lines 32498  df-psubsp 32500  df-pmap 32501  df-padd 32793  df-lhyp 32985  df-laut 32986  df-ldil 33101  df-ltrn 33102  df-trl 33157  df-tendo 33754  df-edring 33756  df-dvech 34079  df-dic 34173

This theorem is referenced by:  cdlemn5pre  34200  cdlemn11c  34209  dihjustlem  34216  dihord1  34218  dihord2a  34219  dihord2b  34220  dihord11c  34224  dihlsscpre  34234  dihvalcqat  34239  dihopelvalcpre  34248  dihord6apre  34256  dihord5b  34259  dihord5apre  34262