Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dia2dimlem5 Unicode version

Theorem dia2dimlem5 30059
Description: Lemma for dia2dim 30068. The sum of vectors  G and  D belongs to the sum of the subspaces generated by them. Thus  F  =  ( G  o.  D ) belongs to the subspace sum. Part of proof of Lemma M in [Crawley] p. 121 line 5. (Contributed by NM, 8-Sep-2014.)
Hypotheses
Ref Expression
dia2dimlem5.l  |-  .<_  =  ( le `  K )
dia2dimlem5.j  |-  .\/  =  ( join `  K )
dia2dimlem5.m  |-  ./\  =  ( meet `  K )
dia2dimlem5.a  |-  A  =  ( Atoms `  K )
dia2dimlem5.h  |-  H  =  ( LHyp `  K
)
dia2dimlem5.t  |-  T  =  ( ( LTrn `  K
) `  W )
dia2dimlem5.r  |-  R  =  ( ( trL `  K
) `  W )
dia2dimlem5.y  |-  Y  =  ( ( DVecA `  K
) `  W )
dia2dimlem5.s  |-  S  =  ( LSubSp `  Y )
dia2dimlem5.pl  |-  .(+)  =  (
LSSum `  Y )
dia2dimlem5.n  |-  N  =  ( LSpan `  Y )
dia2dimlem5.i  |-  I  =  ( ( DIsoA `  K
) `  W )
dia2dimlem5.q  |-  Q  =  ( ( P  .\/  U )  ./\  ( ( F `  P )  .\/  V ) )
dia2dimlem5.k  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
dia2dimlem5.u  |-  ( ph  ->  ( U  e.  A  /\  U  .<_  W ) )
dia2dimlem5.v  |-  ( ph  ->  ( V  e.  A  /\  V  .<_  W ) )
dia2dimlem5.p  |-  ( ph  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
dia2dimlem5.f  |-  ( ph  ->  ( F  e.  T  /\  ( F `  P
)  =/=  P ) )
dia2dimlem5.rf  |-  ( ph  ->  ( R `  F
)  .<_  ( U  .\/  V ) )
dia2dimlem5.uv  |-  ( ph  ->  U  =/=  V )
dia2dimlem5.ru  |-  ( ph  ->  ( R `  F
)  =/=  U )
dia2dimlem5.rv  |-  ( ph  ->  ( R `  F
)  =/=  V )
dia2dimlem5.g  |-  ( ph  ->  G  e.  T )
dia2dimlem5.gv  |-  ( ph  ->  ( G `  P
)  =  Q )
dia2dimlem5.d  |-  ( ph  ->  D  e.  T )
dia2dimlem5.dv  |-  ( ph  ->  ( D `  Q
)  =  ( F `
 P ) )
Assertion
Ref Expression
dia2dimlem5  |-  ( ph  ->  F  e.  ( ( I `  U ) 
.(+)  ( I `  V ) ) )

Proof of Theorem dia2dimlem5
StepHypRef Expression
1 dia2dimlem5.k . . . . 5  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
2 dia2dimlem5.d . . . . 5  |-  ( ph  ->  D  e.  T )
3 dia2dimlem5.g . . . . 5  |-  ( ph  ->  G  e.  T )
4 dia2dimlem5.h . . . . . 6  |-  H  =  ( LHyp `  K
)
5 dia2dimlem5.t . . . . . 6  |-  T  =  ( ( LTrn `  K
) `  W )
6 dia2dimlem5.y . . . . . 6  |-  Y  =  ( ( DVecA `  K
) `  W )
7 eqid 2253 . . . . . 6  |-  ( +g  `  Y )  =  ( +g  `  Y )
84, 5, 6, 7dvavadd 30005 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( D  e.  T  /\  G  e.  T ) )  -> 
( D ( +g  `  Y ) G )  =  ( D  o.  G ) )
91, 2, 3, 8syl12anc 1185 . . . 4  |-  ( ph  ->  ( D ( +g  `  Y ) G )  =  ( D  o.  G ) )
10 dia2dimlem5.l . . . . 5  |-  .<_  =  ( le `  K )
11 dia2dimlem5.a . . . . 5  |-  A  =  ( Atoms `  K )
12 dia2dimlem5.p . . . . 5  |-  ( ph  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
13 dia2dimlem5.f . . . . . 6  |-  ( ph  ->  ( F  e.  T  /\  ( F `  P
)  =/=  P ) )
1413simpld 447 . . . . 5  |-  ( ph  ->  F  e.  T )
15 dia2dimlem5.gv . . . . 5  |-  ( ph  ->  ( G `  P
)  =  Q )
16 dia2dimlem5.dv . . . . 5  |-  ( ph  ->  ( D `  Q
)  =  ( F `
 P ) )
1710, 11, 4, 5, 1, 12, 14, 3, 15, 2, 16dia2dimlem4 30058 . . . 4  |-  ( ph  ->  ( D  o.  G
)  =  F )
189, 17eqtr2d 2286 . . 3  |-  ( ph  ->  F  =  ( D ( +g  `  Y
) G ) )
194, 6dvalvec 30017 . . . . . . 7  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  Y  e.  LVec )
20 lveclmod 15694 . . . . . . 7  |-  ( Y  e.  LVec  ->  Y  e. 
LMod )
211, 19, 203syl 20 . . . . . 6  |-  ( ph  ->  Y  e.  LMod )
22 dia2dimlem5.s . . . . . . 7  |-  S  =  ( LSubSp `  Y )
2322lsssssubg 15550 . . . . . 6  |-  ( Y  e.  LMod  ->  S  C_  (SubGrp `  Y ) )
2421, 23syl 17 . . . . 5  |-  ( ph  ->  S  C_  (SubGrp `  Y
) )
25 dia2dimlem5.v . . . . . . . 8  |-  ( ph  ->  ( V  e.  A  /\  V  .<_  W ) )
2625simpld 447 . . . . . . 7  |-  ( ph  ->  V  e.  A )
27 eqid 2253 . . . . . . . 8  |-  ( Base `  K )  =  (
Base `  K )
2827, 11atbase 28280 . . . . . . 7  |-  ( V  e.  A  ->  V  e.  ( Base `  K
) )
2926, 28syl 17 . . . . . 6  |-  ( ph  ->  V  e.  ( Base `  K ) )
3025simprd 451 . . . . . 6  |-  ( ph  ->  V  .<_  W )
31 dia2dimlem5.i . . . . . . 7  |-  I  =  ( ( DIsoA `  K
) `  W )
3227, 10, 4, 6, 31, 22dialss 30037 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( V  e.  ( Base `  K
)  /\  V  .<_  W ) )  ->  (
I `  V )  e.  S )
331, 29, 30, 32syl12anc 1185 . . . . 5  |-  ( ph  ->  ( I `  V
)  e.  S )
3424, 33sseldd 3104 . . . 4  |-  ( ph  ->  ( I `  V
)  e.  (SubGrp `  Y ) )
35 dia2dimlem5.u . . . . . . . 8  |-  ( ph  ->  ( U  e.  A  /\  U  .<_  W ) )
3635simpld 447 . . . . . . 7  |-  ( ph  ->  U  e.  A )
3727, 11atbase 28280 . . . . . . 7  |-  ( U  e.  A  ->  U  e.  ( Base `  K
) )
3836, 37syl 17 . . . . . 6  |-  ( ph  ->  U  e.  ( Base `  K ) )
3935simprd 451 . . . . . 6  |-  ( ph  ->  U  .<_  W )
4027, 10, 4, 6, 31, 22dialss 30037 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  ( Base `  K
)  /\  U  .<_  W ) )  ->  (
I `  U )  e.  S )
411, 38, 39, 40syl12anc 1185 . . . . 5  |-  ( ph  ->  ( I `  U
)  e.  S )
4224, 41sseldd 3104 . . . 4  |-  ( ph  ->  ( I `  U
)  e.  (SubGrp `  Y ) )
43 dia2dimlem5.r . . . . . . . 8  |-  R  =  ( ( trL `  K
) `  W )
44 dia2dimlem5.n . . . . . . . 8  |-  N  =  ( LSpan `  Y )
454, 5, 43, 6, 31, 44dia1dim2 30053 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  D  e.  T
)  ->  ( I `  ( R `  D
) )  =  ( N `  { D } ) )
461, 2, 45syl2anc 645 . . . . . 6  |-  ( ph  ->  ( I `  ( R `  D )
)  =  ( N `
 { D }
) )
47 dia2dimlem5.j . . . . . . . . . 10  |-  .\/  =  ( join `  K )
48 dia2dimlem5.m . . . . . . . . . 10  |-  ./\  =  ( meet `  K )
49 dia2dimlem5.q . . . . . . . . . 10  |-  Q  =  ( ( P  .\/  U )  ./\  ( ( F `  P )  .\/  V ) )
50 dia2dimlem5.rf . . . . . . . . . 10  |-  ( ph  ->  ( R `  F
)  .<_  ( U  .\/  V ) )
51 dia2dimlem5.uv . . . . . . . . . 10  |-  ( ph  ->  U  =/=  V )
52 dia2dimlem5.ru . . . . . . . . . 10  |-  ( ph  ->  ( R `  F
)  =/=  U )
53 dia2dimlem5.rv . . . . . . . . . 10  |-  ( ph  ->  ( R `  F
)  =/=  V )
5410, 47, 48, 11, 4, 5, 43, 49, 1, 35, 25, 12, 13, 50, 51, 52, 53, 2, 16dia2dimlem3 30057 . . . . . . . . 9  |-  ( ph  ->  ( R `  D
)  =  V )
5554fveq2d 5381 . . . . . . . 8  |-  ( ph  ->  ( I `  ( R `  D )
)  =  ( I `
 V ) )
56 eqss 3115 . . . . . . . 8  |-  ( ( I `  ( R `
 D ) )  =  ( I `  V )  <->  ( (
I `  ( R `  D ) )  C_  ( I `  V
)  /\  ( I `  V )  C_  (
I `  ( R `  D ) ) ) )
5755, 56sylib 190 . . . . . . 7  |-  ( ph  ->  ( ( I `  ( R `  D ) )  C_  ( I `  V )  /\  (
I `  V )  C_  ( I `  ( R `  D )
) ) )
5857simpld 447 . . . . . 6  |-  ( ph  ->  ( I `  ( R `  D )
)  C_  ( I `  V ) )
5946, 58eqsstr3d 3134 . . . . 5  |-  ( ph  ->  ( N `  { D } )  C_  (
I `  V )
)
60 eqid 2253 . . . . . 6  |-  ( Base `  Y )  =  (
Base `  Y )
614, 5, 6, 60dvavbase 30003 . . . . . . . 8  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( Base `  Y
)  =  T )
621, 61syl 17 . . . . . . 7  |-  ( ph  ->  ( Base `  Y
)  =  T )
632, 62eleqtrrd 2330 . . . . . 6  |-  ( ph  ->  D  e.  ( Base `  Y ) )
6460, 22, 44, 21, 33, 63lspsnel5 15587 . . . . 5  |-  ( ph  ->  ( D  e.  ( I `  V )  <-> 
( N `  { D } )  C_  (
I `  V )
) )
6559, 64mpbird 225 . . . 4  |-  ( ph  ->  D  e.  ( I `
 V ) )
664, 5, 43, 6, 31, 44dia1dim2 30053 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T
)  ->  ( I `  ( R `  G
) )  =  ( N `  { G } ) )
671, 3, 66syl2anc 645 . . . . . 6  |-  ( ph  ->  ( I `  ( R `  G )
)  =  ( N `
 { G }
) )
6810, 47, 48, 11, 4, 5, 43, 49, 1, 35, 25, 12, 13, 50, 53, 3, 15dia2dimlem2 30056 . . . . . . . . 9  |-  ( ph  ->  ( R `  G
)  =  U )
6968fveq2d 5381 . . . . . . . 8  |-  ( ph  ->  ( I `  ( R `  G )
)  =  ( I `
 U ) )
70 eqss 3115 . . . . . . . 8  |-  ( ( I `  ( R `
 G ) )  =  ( I `  U )  <->  ( (
I `  ( R `  G ) )  C_  ( I `  U
)  /\  ( I `  U )  C_  (
I `  ( R `  G ) ) ) )
7169, 70sylib 190 . . . . . . 7  |-  ( ph  ->  ( ( I `  ( R `  G ) )  C_  ( I `  U )  /\  (
I `  U )  C_  ( I `  ( R `  G )
) ) )
7271simpld 447 . . . . . 6  |-  ( ph  ->  ( I `  ( R `  G )
)  C_  ( I `  U ) )
7367, 72eqsstr3d 3134 . . . . 5  |-  ( ph  ->  ( N `  { G } )  C_  (
I `  U )
)
743, 62eleqtrrd 2330 . . . . . 6  |-  ( ph  ->  G  e.  ( Base `  Y ) )
7560, 22, 44, 21, 41, 74lspsnel5 15587 . . . . 5  |-  ( ph  ->  ( G  e.  ( I `  U )  <-> 
( N `  { G } )  C_  (
I `  U )
) )
7673, 75mpbird 225 . . . 4  |-  ( ph  ->  G  e.  ( I `
 U ) )
77 dia2dimlem5.pl . . . . 5  |-  .(+)  =  (
LSSum `  Y )
787, 77lsmelvali 14796 . . . 4  |-  ( ( ( ( I `  V )  e.  (SubGrp `  Y )  /\  (
I `  U )  e.  (SubGrp `  Y )
)  /\  ( D  e.  ( I `  V
)  /\  G  e.  ( I `  U
) ) )  -> 
( D ( +g  `  Y ) G )  e.  ( ( I `
 V )  .(+)  ( I `  U ) ) )
7934, 42, 65, 76, 78syl22anc 1188 . . 3  |-  ( ph  ->  ( D ( +g  `  Y ) G )  e.  ( ( I `
 V )  .(+)  ( I `  U ) ) )
8018, 79eqeltrd 2327 . 2  |-  ( ph  ->  F  e.  ( ( I `  V ) 
.(+)  ( I `  U ) ) )
81 lmodabl 15507 . . . 4  |-  ( Y  e.  LMod  ->  Y  e. 
Abel )
8221, 81syl 17 . . 3  |-  ( ph  ->  Y  e.  Abel )
8377lsmcom 14985 . . 3  |-  ( ( Y  e.  Abel  /\  (
I `  V )  e.  (SubGrp `  Y )  /\  ( I `  U
)  e.  (SubGrp `  Y ) )  -> 
( ( I `  V )  .(+)  ( I `
 U ) )  =  ( ( I `
 U )  .(+)  ( I `  V ) ) )
8482, 34, 42, 83syl3anc 1187 . 2  |-  ( ph  ->  ( ( I `  V )  .(+)  ( I `
 U ) )  =  ( ( I `
 U )  .(+)  ( I `  V ) ) )
8580, 84eleqtrd 2329 1  |-  ( ph  ->  F  e.  ( ( I `  U ) 
.(+)  ( I `  V ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360    = wceq 1619    e. wcel 1621    =/= wne 2412    C_ wss 3078   {csn 3544   class class class wbr 3920    o. ccom 4584   ` cfv 4592  (class class class)co 5710   Basecbs 13022   +g cplusg 13082   lecple 13089   joincjn 13922   meetcmee 13923  SubGrpcsubg 14450   LSSumclsm 14780   Abelcabel 14925   LModclmod 15462   LSubSpclss 15524   LSpanclspn 15563   LVecclvec 15690   Atomscatm 28254   HLchlt 28341   LHypclh 28974   LTrncltrn 29091   trLctrl 29148   DVecAcdveca 29992   DIsoAcdia 30019
This theorem is referenced by:  dia2dimlem6  30060
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-cnex 8673  ax-resscn 8674  ax-1cn 8675  ax-icn 8676  ax-addcl 8677  ax-addrcl 8678  ax-mulcl 8679  ax-mulrcl 8680  ax-mulcom 8681  ax-addass 8682  ax-mulass 8683  ax-distr 8684  ax-i2m1 8685  ax-1ne0 8686  ax-1rid 8687  ax-rnegex 8688  ax-rrecex 8689  ax-cnre 8690  ax-pre-lttri 8691  ax-pre-lttrn 8692  ax-pre-ltadd 8693  ax-pre-mulgt0 8694
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-fal 1316  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-int 3761  df-iun 3805  df-iin 3806  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-tpos 6086  df-iota 6143  df-undef 6182  df-riota 6190  df-recs 6274  df-rdg 6309  df-1o 6365  df-oadd 6369  df-er 6546  df-map 6660  df-en 6750  df-dom 6751  df-sdom 6752  df-fin 6753  df-pnf 8749  df-mnf 8750  df-xr 8751  df-ltxr 8752  df-le 8753  df-sub 8919  df-neg 8920  df-n 9627  df-2 9684  df-3 9685  df-4 9686  df-5 9687  df-6 9688  df-n0 9845  df-z 9904  df-uz 10110  df-fz 10661  df-struct 13024  df-ndx 13025  df-slot 13026  df-base 13027  df-sets 13028  df-ress 13029  df-plusg 13095  df-mulr 13096  df-sca 13098  df-vsca 13099  df-0g 13278  df-poset 13924  df-plt 13936  df-lub 13952  df-glb 13953  df-join 13954  df-meet 13955  df-p0 13989  df-p1 13990  df-lat 13996  df-clat 14058  df-mnd 14202  df-grp 14324  df-minusg 14325  df-sbg 14326  df-subg 14453  df-lsm 14782  df-cmn 14926  df-abl 14927  df-mgp 15161  df-ring 15175  df-ur 15177  df-oppr 15240  df-dvdsr 15258  df-unit 15259  df-invr 15289  df-dvr 15300  df-drng 15349  df-lmod 15464  df-lss 15525  df-lsp 15564  df-lvec 15691  df-oposet 28167  df-ol 28169  df-oml 28170  df-covers 28257  df-ats 28258  df-atl 28289  df-cvlat 28313  df-hlat 28342  df-llines 28488  df-lplanes 28489  df-lvols 28490  df-lines 28491  df-psubsp 28493  df-pmap 28494  df-padd 28786  df-lhyp 28978  df-laut 28979  df-ldil 29094  df-ltrn 29095  df-trl 29149  df-tgrp 29733  df-tendo 29745  df-edring 29747  df-dveca 29993  df-disoa 30020
  Copyright terms: Public domain W3C validator