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Theorem dia2dimlem5 31234
Description: Lemma for dia2dim 31243. 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 2380 . . . . . 6  |-  ( +g  `  Y )  =  ( +g  `  Y )
84, 5, 6, 7dvavadd 31180 . . . . 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 31233 . . . 4  |-  ( ph  ->  ( D  o.  G
)  =  F )
189, 17eqtr2d 2413 . . 3  |-  ( ph  ->  F  =  ( D ( +g  `  Y
) G ) )
194, 6dvalvec 31192 . . . . . . 7  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  Y  e.  LVec )
20 lveclmod 16098 . . . . . . 7  |-  ( Y  e.  LVec  ->  Y  e. 
LMod )
211, 19, 203syl 19 . . . . . 6  |-  ( ph  ->  Y  e.  LMod )
22 dia2dimlem5.s . . . . . . 7  |-  S  =  ( LSubSp `  Y )
2322lsssssubg 15954 . . . . . 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 2380 . . . . . . . 8  |-  ( Base `  K )  =  (
Base `  K )
2827, 11atbase 29455 . . . . . . 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 31212 . . . . . 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 3285 . . . 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 29455 . . . . . . 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 31212 . . . . . 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 3285 . . . 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 31228 . . . . . . 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 31232 . . . . . . . . 9  |-  ( ph  ->  ( R `  D
)  =  V )
5554fveq2d 5665 . . . . . . . 8  |-  ( ph  ->  ( I `  ( R `  D )
)  =  ( I `
 V ) )
56 eqss 3299 . . . . . . . 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 3319 . . . . 5  |-  ( ph  ->  ( N `  { D } )  C_  (
I `  V )
)
60 eqid 2380 . . . . . 6  |-  ( Base `  Y )  =  (
Base `  Y )
614, 5, 6, 60dvavbase 31178 . . . . . . . 8  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( Base `  Y
)  =  T )
621, 61syl 16 . . . . . . 7  |-  ( ph  ->  ( Base `  Y
)  =  T )
632, 62eleqtrrd 2457 . . . . . 6  |-  ( ph  ->  D  e.  ( Base `  Y ) )
6460, 22, 44, 21, 33, 63lspsnel5 15991 . . . . 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 31228 . . . . . . 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 31231 . . . . . . . . 9  |-  ( ph  ->  ( R `  G
)  =  U )
6968fveq2d 5665 . . . . . . . 8  |-  ( ph  ->  ( I `  ( R `  G )
)  =  ( I `
 U ) )
70 eqss 3299 . . . . . . . 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 3319 . . . . 5  |-  ( ph  ->  ( N `  { G } )  C_  (
I `  U )
)
743, 62eleqtrrd 2457 . . . . . 6  |-  ( ph  ->  G  e.  ( Base `  Y ) )
7560, 22, 44, 21, 41, 74lspsnel5 15991 . . . . 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 15204 . . . 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 2454 . 2  |-  ( ph  ->  F  e.  ( ( I `  V ) 
.(+)  ( I `  U ) ) )
81 lmodabl 15911 . . . 4  |-  ( Y  e.  LMod  ->  Y  e. 
Abel )
8221, 81syl 16 . . 3  |-  ( ph  ->  Y  e.  Abel )
8377lsmcom 15393 . . 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 2456 1  |-  ( ph  ->  F  e.  ( ( I `  U ) 
.(+)  ( I `  V ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717    =/= wne 2543    C_ wss 3256   {csn 3750   class class class wbr 4146    o. ccom 4815   ` cfv 5387  (class class class)co 6013   Basecbs 13389   +g cplusg 13449   lecple 13456   joincjn 14321   meetcmee 14322  SubGrpcsubg 14858   LSSumclsm 15188   Abelcabel 15333   LModclmod 15870   LSubSpclss 15928   LSpanclspn 15967   LVecclvec 16094   Atomscatm 29429   HLchlt 29516   LHypclh 30149   LTrncltrn 30266   trLctrl 30323   DVecAcdveca 31167   DIsoAcdia 31194
This theorem is referenced by:  dia2dimlem6  31235
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-fal 1326  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-int 3986  df-iun 4030  df-iin 4031  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-tpos 6408  df-undef 6472  df-riota 6478  df-recs 6562  df-rdg 6597  df-1o 6653  df-oadd 6657  df-er 6834  df-map 6949  df-en 7039  df-dom 7040  df-sdom 7041  df-fin 7042  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-nn 9926  df-2 9983  df-3 9984  df-4 9985  df-5 9986  df-6 9987  df-n0 10147  df-z 10208  df-uz 10414  df-fz 10969  df-struct 13391  df-ndx 13392  df-slot 13393  df-base 13394  df-sets 13395  df-ress 13396  df-plusg 13462  df-mulr 13463  df-sca 13465  df-vsca 13466  df-0g 13647  df-poset 14323  df-plt 14335  df-lub 14351  df-glb 14352  df-join 14353  df-meet 14354  df-p0 14388  df-p1 14389  df-lat 14395  df-clat 14457  df-mnd 14610  df-grp 14732  df-minusg 14733  df-sbg 14734  df-subg 14861  df-lsm 15190  df-cmn 15334  df-abl 15335  df-mgp 15569  df-rng 15583  df-ur 15585  df-oppr 15648  df-dvdsr 15666  df-unit 15667  df-invr 15697  df-dvr 15708  df-drng 15757  df-lmod 15872  df-lss 15929  df-lsp 15968  df-lvec 16095  df-oposet 29342  df-ol 29344  df-oml 29345  df-covers 29432  df-ats 29433  df-atl 29464  df-cvlat 29488  df-hlat 29517  df-llines 29663  df-lplanes 29664  df-lvols 29665  df-lines 29666  df-psubsp 29668  df-pmap 29669  df-padd 29961  df-lhyp 30153  df-laut 30154  df-ldil 30269  df-ltrn 30270  df-trl 30324  df-tgrp 30908  df-tendo 30920  df-edring 30922  df-dveca 31168  df-disoa 31195
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