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Theorem dia2dimlem5 31331
Description: Lemma for dia2dim 31340. 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 2285 . . . . . 6  |-  ( +g  `  Y )  =  ( +g  `  Y )
84, 5, 6, 7dvavadd 31277 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( D  e.  T  /\  G  e.  T ) )  -> 
( D ( +g  `  Y ) G )  =  ( D  o.  G ) )
91, 2, 3, 8syl12anc 1180 . . . 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 445 . . . . 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 31330 . . . 4  |-  ( ph  ->  ( D  o.  G
)  =  F )
189, 17eqtr2d 2318 . . 3  |-  ( ph  ->  F  =  ( D ( +g  `  Y
) G ) )
194, 6dvalvec 31289 . . . . . . 7  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  Y  e.  LVec )
20 lveclmod 15861 . . . . . . 7  |-  ( Y  e.  LVec  ->  Y  e. 
LMod )
211, 19, 203syl 18 . . . . . 6  |-  ( ph  ->  Y  e.  LMod )
22 dia2dimlem5.s . . . . . . 7  |-  S  =  ( LSubSp `  Y )
2322lsssssubg 15717 . . . . . 6  |-  ( Y  e.  LMod  ->  S  C_  (SubGrp `  Y ) )
2421, 23syl 15 . . . . 5  |-  ( ph  ->  S  C_  (SubGrp `  Y
) )
25 dia2dimlem5.v . . . . . . . 8  |-  ( ph  ->  ( V  e.  A  /\  V  .<_  W ) )
2625simpld 445 . . . . . . 7  |-  ( ph  ->  V  e.  A )
27 eqid 2285 . . . . . . . 8  |-  ( Base `  K )  =  (
Base `  K )
2827, 11atbase 29552 . . . . . . 7  |-  ( V  e.  A  ->  V  e.  ( Base `  K
) )
2926, 28syl 15 . . . . . 6  |-  ( ph  ->  V  e.  ( Base `  K ) )
3025simprd 449 . . . . . 6  |-  ( ph  ->  V  .<_  W )
31 dia2dimlem5.i . . . . . . 7  |-  I  =  ( ( DIsoA `  K
) `  W )
3227, 10, 4, 6, 31, 22dialss 31309 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( V  e.  ( Base `  K
)  /\  V  .<_  W ) )  ->  (
I `  V )  e.  S )
331, 29, 30, 32syl12anc 1180 . . . . 5  |-  ( ph  ->  ( I `  V
)  e.  S )
3424, 33sseldd 3183 . . . 4  |-  ( ph  ->  ( I `  V
)  e.  (SubGrp `  Y ) )
35 dia2dimlem5.u . . . . . . . 8  |-  ( ph  ->  ( U  e.  A  /\  U  .<_  W ) )
3635simpld 445 . . . . . . 7  |-  ( ph  ->  U  e.  A )
3727, 11atbase 29552 . . . . . . 7  |-  ( U  e.  A  ->  U  e.  ( Base `  K
) )
3836, 37syl 15 . . . . . 6  |-  ( ph  ->  U  e.  ( Base `  K ) )
3935simprd 449 . . . . . 6  |-  ( ph  ->  U  .<_  W )
4027, 10, 4, 6, 31, 22dialss 31309 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  ( Base `  K
)  /\  U  .<_  W ) )  ->  (
I `  U )  e.  S )
411, 38, 39, 40syl12anc 1180 . . . . 5  |-  ( ph  ->  ( I `  U
)  e.  S )
4224, 41sseldd 3183 . . . 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 31325 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  D  e.  T
)  ->  ( I `  ( R `  D
) )  =  ( N `  { D } ) )
461, 2, 45syl2anc 642 . . . . . 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 31329 . . . . . . . . 9  |-  ( ph  ->  ( R `  D
)  =  V )
5554fveq2d 5531 . . . . . . . 8  |-  ( ph  ->  ( I `  ( R `  D )
)  =  ( I `
 V ) )
56 eqss 3196 . . . . . . . 8  |-  ( ( I `  ( R `
 D ) )  =  ( I `  V )  <->  ( (
I `  ( R `  D ) )  C_  ( I `  V
)  /\  ( I `  V )  C_  (
I `  ( R `  D ) ) ) )
5755, 56sylib 188 . . . . . . 7  |-  ( ph  ->  ( ( I `  ( R `  D ) )  C_  ( I `  V )  /\  (
I `  V )  C_  ( I `  ( R `  D )
) ) )
5857simpld 445 . . . . . 6  |-  ( ph  ->  ( I `  ( R `  D )
)  C_  ( I `  V ) )
5946, 58eqsstr3d 3215 . . . . 5  |-  ( ph  ->  ( N `  { D } )  C_  (
I `  V )
)
60 eqid 2285 . . . . . 6  |-  ( Base `  Y )  =  (
Base `  Y )
614, 5, 6, 60dvavbase 31275 . . . . . . . 8  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( Base `  Y
)  =  T )
621, 61syl 15 . . . . . . 7  |-  ( ph  ->  ( Base `  Y
)  =  T )
632, 62eleqtrrd 2362 . . . . . 6  |-  ( ph  ->  D  e.  ( Base `  Y ) )
6460, 22, 44, 21, 33, 63lspsnel5 15754 . . . . 5  |-  ( ph  ->  ( D  e.  ( I `  V )  <-> 
( N `  { D } )  C_  (
I `  V )
) )
6559, 64mpbird 223 . . . 4  |-  ( ph  ->  D  e.  ( I `
 V ) )
664, 5, 43, 6, 31, 44dia1dim2 31325 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T
)  ->  ( I `  ( R `  G
) )  =  ( N `  { G } ) )
671, 3, 66syl2anc 642 . . . . . 6  |-  ( ph  ->  ( I `  ( R `  G )
)  =  ( N `
 { G }
) )
6810, 47, 48, 11, 4, 5, 43, 49, 1, 35, 25, 12, 13, 50, 53, 3, 15dia2dimlem2 31328 . . . . . . . . 9  |-  ( ph  ->  ( R `  G
)  =  U )
6968fveq2d 5531 . . . . . . . 8  |-  ( ph  ->  ( I `  ( R `  G )
)  =  ( I `
 U ) )
70 eqss 3196 . . . . . . . 8  |-  ( ( I `  ( R `
 G ) )  =  ( I `  U )  <->  ( (
I `  ( R `  G ) )  C_  ( I `  U
)  /\  ( I `  U )  C_  (
I `  ( R `  G ) ) ) )
7169, 70sylib 188 . . . . . . 7  |-  ( ph  ->  ( ( I `  ( R `  G ) )  C_  ( I `  U )  /\  (
I `  U )  C_  ( I `  ( R `  G )
) ) )
7271simpld 445 . . . . . 6  |-  ( ph  ->  ( I `  ( R `  G )
)  C_  ( I `  U ) )
7367, 72eqsstr3d 3215 . . . . 5  |-  ( ph  ->  ( N `  { G } )  C_  (
I `  U )
)
743, 62eleqtrrd 2362 . . . . . 6  |-  ( ph  ->  G  e.  ( Base `  Y ) )
7560, 22, 44, 21, 41, 74lspsnel5 15754 . . . . 5  |-  ( ph  ->  ( G  e.  ( I `  U )  <-> 
( N `  { G } )  C_  (
I `  U )
) )
7673, 75mpbird 223 . . . 4  |-  ( ph  ->  G  e.  ( I `
 U ) )
77 dia2dimlem5.pl . . . . 5  |-  .(+)  =  (
LSSum `  Y )
787, 77lsmelvali 14963 . . . 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 1183 . . 3  |-  ( ph  ->  ( D ( +g  `  Y ) G )  e.  ( ( I `
 V )  .(+)  ( I `  U ) ) )
8018, 79eqeltrd 2359 . 2  |-  ( ph  ->  F  e.  ( ( I `  V ) 
.(+)  ( I `  U ) ) )
81 lmodabl 15674 . . . 4  |-  ( Y  e.  LMod  ->  Y  e. 
Abel )
8221, 81syl 15 . . 3  |-  ( ph  ->  Y  e.  Abel )
8377lsmcom 15152 . . 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 1182 . 2  |-  ( ph  ->  ( ( I `  V )  .(+)  ( I `
 U ) )  =  ( ( I `
 U )  .(+)  ( I `  V ) ) )
8580, 84eleqtrd 2361 1  |-  ( ph  ->  F  e.  ( ( I `  U ) 
.(+)  ( I `  V ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1625    e. wcel 1686    =/= wne 2448    C_ wss 3154   {csn 3642   class class class wbr 4025    o. ccom 4695   ` cfv 5257  (class class class)co 5860   Basecbs 13150   +g cplusg 13210   lecple 13217   joincjn 14080   meetcmee 14081  SubGrpcsubg 14617   LSSumclsm 14947   Abelcabel 15092   LModclmod 15629   LSubSpclss 15691   LSpanclspn 15730   LVecclvec 15857   Atomscatm 29526   HLchlt 29613   LHypclh 30246   LTrncltrn 30363   trLctrl 30420   DVecAcdveca 31264   DIsoAcdia 31291
This theorem is referenced by:  dia2dimlem6  31332
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-cnex 8795  ax-resscn 8796  ax-1cn 8797  ax-icn 8798  ax-addcl 8799  ax-addrcl 8800  ax-mulcl 8801  ax-mulrcl 8802  ax-mulcom 8803  ax-addass 8804  ax-mulass 8805  ax-distr 8806  ax-i2m1 8807  ax-1ne0 8808  ax-1rid 8809  ax-rnegex 8810  ax-rrecex 8811  ax-cnre 8812  ax-pre-lttri 8813  ax-pre-lttrn 8814  ax-pre-ltadd 8815  ax-pre-mulgt0 8816
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-fal 1311  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-int 3865  df-iun 3909  df-iin 3910  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-1st 6124  df-2nd 6125  df-tpos 6236  df-undef 6300  df-riota 6306  df-recs 6390  df-rdg 6425  df-1o 6481  df-oadd 6485  df-er 6662  df-map 6776  df-en 6866  df-dom 6867  df-sdom 6868  df-fin 6869  df-pnf 8871  df-mnf 8872  df-xr 8873  df-ltxr 8874  df-le 8875  df-sub 9041  df-neg 9042  df-nn 9749  df-2 9806  df-3 9807  df-4 9808  df-5 9809  df-6 9810  df-n0 9968  df-z 10027  df-uz 10233  df-fz 10785  df-struct 13152  df-ndx 13153  df-slot 13154  df-base 13155  df-sets 13156  df-ress 13157  df-plusg 13223  df-mulr 13224  df-sca 13226  df-vsca 13227  df-0g 13406  df-poset 14082  df-plt 14094  df-lub 14110  df-glb 14111  df-join 14112  df-meet 14113  df-p0 14147  df-p1 14148  df-lat 14154  df-clat 14216  df-mnd 14369  df-grp 14491  df-minusg 14492  df-sbg 14493  df-subg 14620  df-lsm 14949  df-cmn 15093  df-abl 15094  df-mgp 15328  df-rng 15342  df-ur 15344  df-oppr 15407  df-dvdsr 15425  df-unit 15426  df-invr 15456  df-dvr 15467  df-drng 15516  df-lmod 15631  df-lss 15692  df-lsp 15731  df-lvec 15858  df-oposet 29439  df-ol 29441  df-oml 29442  df-covers 29529  df-ats 29530  df-atl 29561  df-cvlat 29585  df-hlat 29614  df-llines 29760  df-lplanes 29761  df-lvols 29762  df-lines 29763  df-psubsp 29765  df-pmap 29766  df-padd 30058  df-lhyp 30250  df-laut 30251  df-ldil 30366  df-ltrn 30367  df-trl 30421  df-tgrp 31005  df-tendo 31017  df-edring 31019  df-dveca 31265  df-disoa 31292
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