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Theorem dia2dimlem5 36896
Description: Lemma for dia2dim 36905. 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 2457 . . . . . 6  |-  ( +g  `  Y )  =  ( +g  `  Y )
84, 5, 6, 7dvavadd 36842 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( D  e.  T  /\  G  e.  T ) )  -> 
( D ( +g  `  Y ) G )  =  ( D  o.  G ) )
91, 2, 3, 8syl12anc 1226 . . . 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 459 . . . . 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 36895 . . . 4  |-  ( ph  ->  ( D  o.  G
)  =  F )
189, 17eqtr2d 2499 . . 3  |-  ( ph  ->  F  =  ( D ( +g  `  Y
) G ) )
194, 6dvalvec 36854 . . . . . . 7  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  Y  e.  LVec )
20 lveclmod 17878 . . . . . . 7  |-  ( Y  e.  LVec  ->  Y  e. 
LMod )
211, 19, 203syl 20 . . . . . 6  |-  ( ph  ->  Y  e.  LMod )
22 dia2dimlem5.s . . . . . . 7  |-  S  =  ( LSubSp `  Y )
2322lsssssubg 17730 . . . . . 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 459 . . . . . . 7  |-  ( ph  ->  V  e.  A )
27 eqid 2457 . . . . . . . 8  |-  ( Base `  K )  =  (
Base `  K )
2827, 11atbase 35115 . . . . . . 7  |-  ( V  e.  A  ->  V  e.  ( Base `  K
) )
2926, 28syl 16 . . . . . 6  |-  ( ph  ->  V  e.  ( Base `  K ) )
3025simprd 463 . . . . . 6  |-  ( ph  ->  V  .<_  W )
31 dia2dimlem5.i . . . . . . 7  |-  I  =  ( ( DIsoA `  K
) `  W )
3227, 10, 4, 6, 31, 22dialss 36874 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( V  e.  ( Base `  K
)  /\  V  .<_  W ) )  ->  (
I `  V )  e.  S )
331, 29, 30, 32syl12anc 1226 . . . . 5  |-  ( ph  ->  ( I `  V
)  e.  S )
3424, 33sseldd 3500 . . . 4  |-  ( ph  ->  ( I `  V
)  e.  (SubGrp `  Y ) )
35 dia2dimlem5.u . . . . . . . 8  |-  ( ph  ->  ( U  e.  A  /\  U  .<_  W ) )
3635simpld 459 . . . . . . 7  |-  ( ph  ->  U  e.  A )
3727, 11atbase 35115 . . . . . . 7  |-  ( U  e.  A  ->  U  e.  ( Base `  K
) )
3836, 37syl 16 . . . . . 6  |-  ( ph  ->  U  e.  ( Base `  K ) )
3935simprd 463 . . . . . 6  |-  ( ph  ->  U  .<_  W )
4027, 10, 4, 6, 31, 22dialss 36874 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  ( Base `  K
)  /\  U  .<_  W ) )  ->  (
I `  U )  e.  S )
411, 38, 39, 40syl12anc 1226 . . . . 5  |-  ( ph  ->  ( I `  U
)  e.  S )
4224, 41sseldd 3500 . . . 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 36890 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  D  e.  T
)  ->  ( I `  ( R `  D
) )  =  ( N `  { D } ) )
461, 2, 45syl2anc 661 . . . . . 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 36894 . . . . . . . . 9  |-  ( ph  ->  ( R `  D
)  =  V )
5554fveq2d 5876 . . . . . . . 8  |-  ( ph  ->  ( I `  ( R `  D )
)  =  ( I `
 V ) )
56 eqss 3514 . . . . . . . 8  |-  ( ( I `  ( R `
 D ) )  =  ( I `  V )  <->  ( (
I `  ( R `  D ) )  C_  ( I `  V
)  /\  ( I `  V )  C_  (
I `  ( R `  D ) ) ) )
5755, 56sylib 196 . . . . . . 7  |-  ( ph  ->  ( ( I `  ( R `  D ) )  C_  ( I `  V )  /\  (
I `  V )  C_  ( I `  ( R `  D )
) ) )
5857simpld 459 . . . . . 6  |-  ( ph  ->  ( I `  ( R `  D )
)  C_  ( I `  V ) )
5946, 58eqsstr3d 3534 . . . . 5  |-  ( ph  ->  ( N `  { D } )  C_  (
I `  V )
)
60 eqid 2457 . . . . . 6  |-  ( Base `  Y )  =  (
Base `  Y )
614, 5, 6, 60dvavbase 36840 . . . . . . . 8  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( Base `  Y
)  =  T )
621, 61syl 16 . . . . . . 7  |-  ( ph  ->  ( Base `  Y
)  =  T )
632, 62eleqtrrd 2548 . . . . . 6  |-  ( ph  ->  D  e.  ( Base `  Y ) )
6460, 22, 44, 21, 33, 63lspsnel5 17767 . . . . 5  |-  ( ph  ->  ( D  e.  ( I `  V )  <-> 
( N `  { D } )  C_  (
I `  V )
) )
6559, 64mpbird 232 . . . 4  |-  ( ph  ->  D  e.  ( I `
 V ) )
664, 5, 43, 6, 31, 44dia1dim2 36890 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T
)  ->  ( I `  ( R `  G
) )  =  ( N `  { G } ) )
671, 3, 66syl2anc 661 . . . . . 6  |-  ( ph  ->  ( I `  ( R `  G )
)  =  ( N `
 { G }
) )
6810, 47, 48, 11, 4, 5, 43, 49, 1, 35, 25, 12, 13, 50, 53, 3, 15dia2dimlem2 36893 . . . . . . . . 9  |-  ( ph  ->  ( R `  G
)  =  U )
6968fveq2d 5876 . . . . . . . 8  |-  ( ph  ->  ( I `  ( R `  G )
)  =  ( I `
 U ) )
70 eqss 3514 . . . . . . . 8  |-  ( ( I `  ( R `
 G ) )  =  ( I `  U )  <->  ( (
I `  ( R `  G ) )  C_  ( I `  U
)  /\  ( I `  U )  C_  (
I `  ( R `  G ) ) ) )
7169, 70sylib 196 . . . . . . 7  |-  ( ph  ->  ( ( I `  ( R `  G ) )  C_  ( I `  U )  /\  (
I `  U )  C_  ( I `  ( R `  G )
) ) )
7271simpld 459 . . . . . 6  |-  ( ph  ->  ( I `  ( R `  G )
)  C_  ( I `  U ) )
7367, 72eqsstr3d 3534 . . . . 5  |-  ( ph  ->  ( N `  { G } )  C_  (
I `  U )
)
743, 62eleqtrrd 2548 . . . . . 6  |-  ( ph  ->  G  e.  ( Base `  Y ) )
7560, 22, 44, 21, 41, 74lspsnel5 17767 . . . . 5  |-  ( ph  ->  ( G  e.  ( I `  U )  <-> 
( N `  { G } )  C_  (
I `  U )
) )
7673, 75mpbird 232 . . . 4  |-  ( ph  ->  G  e.  ( I `
 U ) )
77 dia2dimlem5.pl . . . . 5  |-  .(+)  =  (
LSSum `  Y )
787, 77lsmelvali 16796 . . . 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 1229 . . 3  |-  ( ph  ->  ( D ( +g  `  Y ) G )  e.  ( ( I `
 V )  .(+)  ( I `  U ) ) )
8018, 79eqeltrd 2545 . 2  |-  ( ph  ->  F  e.  ( ( I `  V ) 
.(+)  ( I `  U ) ) )
81 lmodabl 17683 . . . 4  |-  ( Y  e.  LMod  ->  Y  e. 
Abel )
8221, 81syl 16 . . 3  |-  ( ph  ->  Y  e.  Abel )
8377lsmcom 16990 . . 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 1228 . 2  |-  ( ph  ->  ( ( I `  V )  .(+)  ( I `
 U ) )  =  ( ( I `
 U )  .(+)  ( I `  V ) ) )
8580, 84eleqtrd 2547 1  |-  ( ph  ->  F  e.  ( ( I `  U ) 
.(+)  ( I `  V ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819    =/= wne 2652    C_ wss 3471   {csn 4032   class class class wbr 4456    o. ccom 5012   ` cfv 5594  (class class class)co 6296   Basecbs 14643   +g cplusg 14711   lecple 14718   joincjn 15699   meetcmee 15700  SubGrpcsubg 16321   LSSumclsm 16780   Abelcabl 16925   LModclmod 17638   LSubSpclss 17704   LSpanclspn 17743   LVecclvec 17874   Atomscatm 35089   HLchlt 35176   LHypclh 35809   LTrncltrn 35926   trLctrl 35984   DVecAcdveca 36829   DIsoAcdia 36856
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-riotaBAD 34785
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-iin 4335  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-tpos 6973  df-undef 7020  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-struct 14645  df-ndx 14646  df-slot 14647  df-base 14648  df-sets 14649  df-ress 14650  df-plusg 14724  df-mulr 14725  df-sca 14727  df-vsca 14728  df-0g 14858  df-preset 15683  df-poset 15701  df-plt 15714  df-lub 15730  df-glb 15731  df-join 15732  df-meet 15733  df-p0 15795  df-p1 15796  df-lat 15802  df-clat 15864  df-mgm 15998  df-sgrp 16037  df-mnd 16047  df-grp 16183  df-minusg 16184  df-sbg 16185  df-subg 16324  df-lsm 16782  df-cmn 16926  df-abl 16927  df-mgp 17268  df-ur 17280  df-ring 17326  df-oppr 17398  df-dvdsr 17416  df-unit 17417  df-invr 17447  df-dvr 17458  df-drng 17524  df-lmod 17640  df-lss 17705  df-lsp 17744  df-lvec 17875  df-oposet 35002  df-ol 35004  df-oml 35005  df-covers 35092  df-ats 35093  df-atl 35124  df-cvlat 35148  df-hlat 35177  df-llines 35323  df-lplanes 35324  df-lvols 35325  df-lines 35326  df-psubsp 35328  df-pmap 35329  df-padd 35621  df-lhyp 35813  df-laut 35814  df-ldil 35929  df-ltrn 35930  df-trl 35985  df-tgrp 36570  df-tendo 36582  df-edring 36584  df-dveca 36830  df-disoa 36857
This theorem is referenced by:  dia2dimlem6  36897
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