Theorem diclss 31676

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 2405	. 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 2404	. . . . 5 |- ( ( TEndo ` K ) ` W ) = ( ( TEndo ` K ) ` W )
4		diclss.u	. . . . 5 |- U = ( ( DVecH ` K ) ` W )
5		eqid 2404	. . . . 5 |- (Scalar ` U ) = (Scalar ` U )
6		eqid 2404	. . . . 5 |- ( Base ` (Scalar ` U ) ) = ( Base ` (Scalar ` U ) )
7	2, 3, 4, 5, 6	dvhbase 31566	. . . 4 |- ( ( K e. HL /\ W e. H ) -> ( Base ` (Scalar ` U ) ) = ( ( TEndo ` K ) ` W ) )
8	7	eqcomd 2409	. . 3 |- ( ( K e. HL /\ W e. H ) -> ( ( TEndo ` K ) ` W ) = ( Base ` (Scalar ` U ) ) )
9	8	adantr 452	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( ( TEndo ` K ) ` W ) = ( Base ` (Scalar ` U ) ) )
10		eqid 2404	. . . . 5 |- ( ( LTrn ` K ) ` W ) = ( ( LTrn ` K ) ` W )
11		eqid 2404	. . . . 5 |- ( Base ` U ) = ( Base ` U )
12	2, 10, 3, 4, 11	dvhvbase 31570	. . . 4 |- ( ( K e. HL /\ W e. H ) -> ( Base ` U ) = ( ( ( LTrn ` K ) ` W ) X. ( ( TEndo ` K ) ` W ) ) )
13	12	eqcomd 2409	. . 3 |- ( ( K e. HL /\ W e. H ) -> ( ( ( LTrn ` K ) ` W ) X. ( ( TEndo ` K ) ` W ) ) = ( Base ` U ) )
14	13	adantr 452	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( ( ( LTrn ` K ) ` W ) X. ( ( TEndo ` K ) ` W ) ) = ( Base ` U ) )
15		eqidd 2405	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( +g ` U ) = ( +g ` U ) )
16		eqidd 2405	. 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 31669	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( I ` Q ) C_ ( Base ` U ) )
23	22, 14	sseqtr4d 3345	. 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 31675	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( I ` Q ) =/= (/) )
25		simpll 731	. . 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 732	. . 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 963	. . . 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 964	. . . 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 2404	. . . . 5 |- ( .s ` U ) = ( .s ` U )
30	19, 20, 2, 3, 4, 21, 29	dicvscacl 31674	. . . 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 1188	. . 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 965	. . 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 2404	. . . 4 |- ( +g ` U ) = ( +g ` U )
34	19, 20, 2, 4, 21, 33	dicvaddcl 31673	. . 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 1188	. 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 15967	1 |- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( I ` Q ) e. S )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 359   /\ w3a 936   = wceq 1649   e. wcel 1721   class class class wbr 4172   X. cxp 4835   ` cfv 5413  (class class class)co 6040   Basecbs 13424   +g cplusg 13484  Scalarcsca 13487   .scvsca 13488   lecple 13491   LSubSpclss 15963   Atomscatm 29746   HLchlt 29833   LHypclh 30466   LTrncltrn 30583   TEndoctendo 31234   DVecHcdvh 31561   DIsoCcdic 31655

This theorem is referenced by:  cdlemn5pre  31683  cdlemn11c  31692  dihjustlem  31699  dihord1  31701  dihord2a  31702  dihord2b  31703  dihord11c  31707  dihlsscpre  31717  dihvalcqat  31722  dihopelvalcpre  31731  dihord6apre  31739  dihord5b  31742  dihord5apre  31745

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-undef 6502  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-n0 10178  df-z 10239  df-uz 10445  df-fz 11000  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-plusg 13497  df-mulr 13498  df-sca 13500  df-vsca 13501  df-poset 14358  df-plt 14370  df-lub 14386  df-glb 14387  df-join 14388  df-meet 14389  df-p0 14423  df-p1 14424  df-lat 14430  df-clat 14492  df-lss 15964  df-oposet 29659  df-ol 29661  df-oml 29662  df-covers 29749  df-ats 29750  df-atl 29781  df-cvlat 29805  df-hlat 29834  df-llines 29980  df-lplanes 29981  df-lvols 29982  df-lines 29983  df-psubsp 29985  df-pmap 29986  df-padd 30278  df-lhyp 30470  df-laut 30471  df-ldil 30586  df-ltrn 30587  df-trl 30641  df-tendo 31237  df-edring 31239  df-dvech 31562  df-dic 31656