Theorem diclss 35201

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 2455	. 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 2454	. . . . 5 |- ( ( TEndo ` K ) ` W ) = ( ( TEndo ` K ) ` W )
4		diclss.u	. . . . 5 |- U = ( ( DVecH ` K ) ` W )
5		eqid 2454	. . . . 5 |- (Scalar ` U ) = (Scalar ` U )
6		eqid 2454	. . . . 5 |- ( Base ` (Scalar ` U ) ) = ( Base ` (Scalar ` U ) )
7	2, 3, 4, 5, 6	dvhbase 35091	. . . 4 |- ( ( K e. HL /\ W e. H ) -> ( Base ` (Scalar ` U ) ) = ( ( TEndo ` K ) ` W ) )
8	7	eqcomd 2462	. . 3 |- ( ( K e. HL /\ W e. H ) -> ( ( TEndo ` K ) ` W ) = ( Base ` (Scalar ` U ) ) )
9	8	adantr 465	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( ( TEndo ` K ) ` W ) = ( Base ` (Scalar ` U ) ) )
10		eqid 2454	. . . . 5 |- ( ( LTrn ` K ) ` W ) = ( ( LTrn ` K ) ` W )
11		eqid 2454	. . . . 5 |- ( Base ` U ) = ( Base ` U )
12	2, 10, 3, 4, 11	dvhvbase 35095	. . . 4 |- ( ( K e. HL /\ W e. H ) -> ( Base ` U ) = ( ( ( LTrn ` K ) ` W ) X. ( ( TEndo ` K ) ` W ) ) )
13	12	eqcomd 2462	. . 3 |- ( ( K e. HL /\ W e. H ) -> ( ( ( LTrn ` K ) ` W ) X. ( ( TEndo ` K ) ` W ) ) = ( Base ` U ) )
14	13	adantr 465	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( ( ( LTrn ` K ) ` W ) X. ( ( TEndo ` K ) ` W ) ) = ( Base ` U ) )
15		eqidd 2455	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( +g ` U ) = ( +g ` U ) )
16		eqidd 2455	. 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 35194	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( I ` Q ) C_ ( Base ` U ) )
23	22, 14	sseqtr4d 3504	. 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 35200	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( I ` Q ) =/= (/) )
25		simpll 753	. . 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 994	. . . 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 995	. . . 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 2454	. . . . 5 |- ( .s ` U ) = ( .s ` U )
30	19, 20, 2, 3, 4, 21, 29	dicvscacl 35199	. . . 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 1223	. . 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 996	. . 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 2454	. . . 4 |- ( +g ` U ) = ( +g ` U )
34	19, 20, 2, 4, 21, 33	dicvaddcl 35198	. . 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 1223	. 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 17150	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 965   = wceq 1370   e. wcel 1758   class class class wbr 4403   X. cxp 4949   ` cfv 5529  (class class class)co 6203   Basecbs 14296   +g cplusg 14361  Scalarcsca 14364   .scvsca 14365   lecple 14368   LSubSpclss 17146   Atomscatm 33271   HLchlt 33358   LHypclh 33991   LTrncltrn 34108   TEndoctendo 34759   DVecHcdvh 35086   DIsoCcdic 35180

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9453  ax-resscn 9454  ax-1cn 9455  ax-icn 9456  ax-addcl 9457  ax-addrcl 9458  ax-mulcl 9459  ax-mulrcl 9460  ax-mulcom 9461  ax-addass 9462  ax-mulass 9463  ax-distr 9464  ax-i2m1 9465  ax-1ne0 9466  ax-1rid 9467  ax-rnegex 9468  ax-rrecex 9469  ax-cnre 9470  ax-pre-lttri 9471  ax-pre-lttrn 9472  ax-pre-ltadd 9473  ax-pre-mulgt0 9474  ax-riotaBAD 32967

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-iin 4285  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-undef 6905  df-recs 6945  df-rdg 6979  df-1o 7033  df-oadd 7037  df-er 7214  df-map 7329  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-pnf 9535  df-mnf 9536  df-xr 9537  df-ltxr 9538  df-le 9539  df-sub 9712  df-neg 9713  df-nn 10438  df-2 10495  df-3 10496  df-4 10497  df-5 10498  df-6 10499  df-n0 10695  df-z 10762  df-uz 10977  df-fz 11559  df-struct 14298  df-ndx 14299  df-slot 14300  df-base 14301  df-plusg 14374  df-mulr 14375  df-sca 14377  df-vsca 14378  df-poset 15239  df-plt 15251  df-lub 15267  df-glb 15268  df-join 15269  df-meet 15270  df-p0 15332  df-p1 15333  df-lat 15339  df-clat 15401  df-lss 17147  df-oposet 33184  df-ol 33186  df-oml 33187  df-covers 33274  df-ats 33275  df-atl 33306  df-cvlat 33330  df-hlat 33359  df-llines 33505  df-lplanes 33506  df-lvols 33507  df-lines 33508  df-psubsp 33510  df-pmap 33511  df-padd 33803  df-lhyp 33995  df-laut 33996  df-ldil 34111  df-ltrn 34112  df-trl 34166  df-tendo 34762  df-edring 34764  df-dvech 35087  df-dic 35181

This theorem is referenced by:  cdlemn5pre  35208  cdlemn11c  35217  dihjustlem  35224  dihord1  35226  dihord2a  35227  dihord2b  35228  dihord11c  35232  dihlsscpre  35242  dihvalcqat  35247  dihopelvalcpre  35256  dihord6apre  35264  dihord5b  35267  dihord5apre  35270