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Theorem dia2dimlem5 31804
Description: Lemma for dia2dim 31813. 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 2436 . . . . . 6  |-  ( +g  `  Y )  =  ( +g  `  Y )
84, 5, 6, 7dvavadd 31750 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( D  e.  T  /\  G  e.  T ) )  -> 
( D ( +g  `  Y ) G )  =  ( D  o.  G ) )
91, 2, 3, 8syl12anc 1182 . . . 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 446 . . . . 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 31803 . . . 4  |-  ( ph  ->  ( D  o.  G
)  =  F )
189, 17eqtr2d 2469 . . 3  |-  ( ph  ->  F  =  ( D ( +g  `  Y
) G ) )
194, 6dvalvec 31762 . . . . . . 7  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  Y  e.  LVec )
20 lveclmod 16171 . . . . . . 7  |-  ( Y  e.  LVec  ->  Y  e. 
LMod )
211, 19, 203syl 19 . . . . . 6  |-  ( ph  ->  Y  e.  LMod )
22 dia2dimlem5.s . . . . . . 7  |-  S  =  ( LSubSp `  Y )
2322lsssssubg 16027 . . . . . 6  |-  ( Y  e.  LMod  ->  S  C_  (SubGrp `  Y ) )
2421, 23syl 16 . . . . 5  |-  ( ph  ->  S  C_  (SubGrp `  Y
) )
25 dia2dimlem5.v . . . . . . . 8  |-  ( ph  ->  ( V  e.  A  /\  V  .<_  W ) )
2625simpld 446 . . . . . . 7  |-  ( ph  ->  V  e.  A )
27 eqid 2436 . . . . . . . 8  |-  ( Base `  K )  =  (
Base `  K )
2827, 11atbase 30025 . . . . . . 7  |-  ( V  e.  A  ->  V  e.  ( Base `  K
) )
2926, 28syl 16 . . . . . 6  |-  ( ph  ->  V  e.  ( Base `  K ) )
3025simprd 450 . . . . . 6  |-  ( ph  ->  V  .<_  W )
31 dia2dimlem5.i . . . . . . 7  |-  I  =  ( ( DIsoA `  K
) `  W )
3227, 10, 4, 6, 31, 22dialss 31782 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( V  e.  ( Base `  K
)  /\  V  .<_  W ) )  ->  (
I `  V )  e.  S )
331, 29, 30, 32syl12anc 1182 . . . . 5  |-  ( ph  ->  ( I `  V
)  e.  S )
3424, 33sseldd 3342 . . . 4  |-  ( ph  ->  ( I `  V
)  e.  (SubGrp `  Y ) )
35 dia2dimlem5.u . . . . . . . 8  |-  ( ph  ->  ( U  e.  A  /\  U  .<_  W ) )
3635simpld 446 . . . . . . 7  |-  ( ph  ->  U  e.  A )
3727, 11atbase 30025 . . . . . . 7  |-  ( U  e.  A  ->  U  e.  ( Base `  K
) )
3836, 37syl 16 . . . . . 6  |-  ( ph  ->  U  e.  ( Base `  K ) )
3935simprd 450 . . . . . 6  |-  ( ph  ->  U  .<_  W )
4027, 10, 4, 6, 31, 22dialss 31782 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  ( Base `  K
)  /\  U  .<_  W ) )  ->  (
I `  U )  e.  S )
411, 38, 39, 40syl12anc 1182 . . . . 5  |-  ( ph  ->  ( I `  U
)  e.  S )
4224, 41sseldd 3342 . . . 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 31798 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  D  e.  T
)  ->  ( I `  ( R `  D
) )  =  ( N `  { D } ) )
461, 2, 45syl2anc 643 . . . . . 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 31802 . . . . . . . . 9  |-  ( ph  ->  ( R `  D
)  =  V )
5554fveq2d 5725 . . . . . . . 8  |-  ( ph  ->  ( I `  ( R `  D )
)  =  ( I `
 V ) )
56 eqss 3356 . . . . . . . 8  |-  ( ( I `  ( R `
 D ) )  =  ( I `  V )  <->  ( (
I `  ( R `  D ) )  C_  ( I `  V
)  /\  ( I `  V )  C_  (
I `  ( R `  D ) ) ) )
5755, 56sylib 189 . . . . . . 7  |-  ( ph  ->  ( ( I `  ( R `  D ) )  C_  ( I `  V )  /\  (
I `  V )  C_  ( I `  ( R `  D )
) ) )
5857simpld 446 . . . . . 6  |-  ( ph  ->  ( I `  ( R `  D )
)  C_  ( I `  V ) )
5946, 58eqsstr3d 3376 . . . . 5  |-  ( ph  ->  ( N `  { D } )  C_  (
I `  V )
)
60 eqid 2436 . . . . . 6  |-  ( Base `  Y )  =  (
Base `  Y )
614, 5, 6, 60dvavbase 31748 . . . . . . . 8  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( Base `  Y
)  =  T )
621, 61syl 16 . . . . . . 7  |-  ( ph  ->  ( Base `  Y
)  =  T )
632, 62eleqtrrd 2513 . . . . . 6  |-  ( ph  ->  D  e.  ( Base `  Y ) )
6460, 22, 44, 21, 33, 63lspsnel5 16064 . . . . 5  |-  ( ph  ->  ( D  e.  ( I `  V )  <-> 
( N `  { D } )  C_  (
I `  V )
) )
6559, 64mpbird 224 . . . 4  |-  ( ph  ->  D  e.  ( I `
 V ) )
664, 5, 43, 6, 31, 44dia1dim2 31798 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T
)  ->  ( I `  ( R `  G
) )  =  ( N `  { G } ) )
671, 3, 66syl2anc 643 . . . . . 6  |-  ( ph  ->  ( I `  ( R `  G )
)  =  ( N `
 { G }
) )
6810, 47, 48, 11, 4, 5, 43, 49, 1, 35, 25, 12, 13, 50, 53, 3, 15dia2dimlem2 31801 . . . . . . . . 9  |-  ( ph  ->  ( R `  G
)  =  U )
6968fveq2d 5725 . . . . . . . 8  |-  ( ph  ->  ( I `  ( R `  G )
)  =  ( I `
 U ) )
70 eqss 3356 . . . . . . . 8  |-  ( ( I `  ( R `
 G ) )  =  ( I `  U )  <->  ( (
I `  ( R `  G ) )  C_  ( I `  U
)  /\  ( I `  U )  C_  (
I `  ( R `  G ) ) ) )
7169, 70sylib 189 . . . . . . 7  |-  ( ph  ->  ( ( I `  ( R `  G ) )  C_  ( I `  U )  /\  (
I `  U )  C_  ( I `  ( R `  G )
) ) )
7271simpld 446 . . . . . 6  |-  ( ph  ->  ( I `  ( R `  G )
)  C_  ( I `  U ) )
7367, 72eqsstr3d 3376 . . . . 5  |-  ( ph  ->  ( N `  { G } )  C_  (
I `  U )
)
743, 62eleqtrrd 2513 . . . . . 6  |-  ( ph  ->  G  e.  ( Base `  Y ) )
7560, 22, 44, 21, 41, 74lspsnel5 16064 . . . . 5  |-  ( ph  ->  ( G  e.  ( I `  U )  <-> 
( N `  { G } )  C_  (
I `  U )
) )
7673, 75mpbird 224 . . . 4  |-  ( ph  ->  G  e.  ( I `
 U ) )
77 dia2dimlem5.pl . . . . 5  |-  .(+)  =  (
LSSum `  Y )
787, 77lsmelvali 15277 . . . 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 1185 . . 3  |-  ( ph  ->  ( D ( +g  `  Y ) G )  e.  ( ( I `
 V )  .(+)  ( I `  U ) ) )
8018, 79eqeltrd 2510 . 2  |-  ( ph  ->  F  e.  ( ( I `  V ) 
.(+)  ( I `  U ) ) )
81 lmodabl 15984 . . . 4  |-  ( Y  e.  LMod  ->  Y  e. 
Abel )
8221, 81syl 16 . . 3  |-  ( ph  ->  Y  e.  Abel )
8377lsmcom 15466 . . 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 1184 . 2  |-  ( ph  ->  ( ( I `  V )  .(+)  ( I `
 U ) )  =  ( ( I `
 U )  .(+)  ( I `  V ) ) )
8580, 84eleqtrd 2512 1  |-  ( ph  ->  F  e.  ( ( I `  U ) 
.(+)  ( I `  V ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2599    C_ wss 3313   {csn 3807   class class class wbr 4205    o. ccom 4875   ` cfv 5447  (class class class)co 6074   Basecbs 13462   +g cplusg 13522   lecple 13529   joincjn 14394   meetcmee 14395  SubGrpcsubg 14931   LSSumclsm 15261   Abelcabel 15406   LModclmod 15943   LSubSpclss 16001   LSpanclspn 16040   LVecclvec 16167   Atomscatm 29999   HLchlt 30086   LHypclh 30719   LTrncltrn 30836   trLctrl 30893   DVecAcdveca 31737   DIsoAcdia 31764
This theorem is referenced by:  dia2dimlem6  31805
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4313  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396  ax-un 4694  ax-cnex 9039  ax-resscn 9040  ax-1cn 9041  ax-icn 9042  ax-addcl 9043  ax-addrcl 9044  ax-mulcl 9045  ax-mulrcl 9046  ax-mulcom 9047  ax-addass 9048  ax-mulass 9049  ax-distr 9050  ax-i2m1 9051  ax-1ne0 9052  ax-1rid 9053  ax-rnegex 9054  ax-rrecex 9055  ax-cnre 9056  ax-pre-lttri 9057  ax-pre-lttrn 9058  ax-pre-ltadd 9059  ax-pre-mulgt0 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-fal 1329  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2703  df-rex 2704  df-reu 2705  df-rmo 2706  df-rab 2707  df-v 2951  df-sbc 3155  df-csb 3245  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-pss 3329  df-nul 3622  df-if 3733  df-pw 3794  df-sn 3813  df-pr 3814  df-tp 3815  df-op 3816  df-uni 4009  df-int 4044  df-iun 4088  df-iin 4089  df-br 4206  df-opab 4260  df-mpt 4261  df-tr 4296  df-eprel 4487  df-id 4491  df-po 4496  df-so 4497  df-fr 4534  df-we 4536  df-ord 4577  df-on 4578  df-lim 4579  df-suc 4580  df-om 4839  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-res 4883  df-ima 4884  df-iota 5411  df-fun 5449  df-fn 5450  df-f 5451  df-f1 5452  df-fo 5453  df-f1o 5454  df-fv 5455  df-ov 6077  df-oprab 6078  df-mpt2 6079  df-1st 6342  df-2nd 6343  df-tpos 6472  df-undef 6536  df-riota 6542  df-recs 6626  df-rdg 6661  df-1o 6717  df-oadd 6721  df-er 6898  df-map 7013  df-en 7103  df-dom 7104  df-sdom 7105  df-fin 7106  df-pnf 9115  df-mnf 9116  df-xr 9117  df-ltxr 9118  df-le 9119  df-sub 9286  df-neg 9287  df-nn 9994  df-2 10051  df-3 10052  df-4 10053  df-5 10054  df-6 10055  df-n0 10215  df-z 10276  df-uz 10482  df-fz 11037  df-struct 13464  df-ndx 13465  df-slot 13466  df-base 13467  df-sets 13468  df-ress 13469  df-plusg 13535  df-mulr 13536  df-sca 13538  df-vsca 13539  df-0g 13720  df-poset 14396  df-plt 14408  df-lub 14424  df-glb 14425  df-join 14426  df-meet 14427  df-p0 14461  df-p1 14462  df-lat 14468  df-clat 14530  df-mnd 14683  df-grp 14805  df-minusg 14806  df-sbg 14807  df-subg 14934  df-lsm 15263  df-cmn 15407  df-abl 15408  df-mgp 15642  df-rng 15656  df-ur 15658  df-oppr 15721  df-dvdsr 15739  df-unit 15740  df-invr 15770  df-dvr 15781  df-drng 15830  df-lmod 15945  df-lss 16002  df-lsp 16041  df-lvec 16168  df-oposet 29912  df-ol 29914  df-oml 29915  df-covers 30002  df-ats 30003  df-atl 30034  df-cvlat 30058  df-hlat 30087  df-llines 30233  df-lplanes 30234  df-lvols 30235  df-lines 30236  df-psubsp 30238  df-pmap 30239  df-padd 30531  df-lhyp 30723  df-laut 30724  df-ldil 30839  df-ltrn 30840  df-trl 30894  df-tgrp 31478  df-tendo 31490  df-edring 31492  df-dveca 31738  df-disoa 31765
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