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Theorem lcfrlem3 32356
Description: Lemma for lcfr 32397. (Contributed by NM, 27-Feb-2015.)
Hypotheses
Ref Expression
lcfrlem1.v  |-  V  =  ( Base `  U
)
lcfrlem1.s  |-  S  =  (Scalar `  U )
lcfrlem1.q  |-  .X.  =  ( .r `  S )
lcfrlem1.z  |-  .0.  =  ( 0g `  S )
lcfrlem1.i  |-  I  =  ( invr `  S
)
lcfrlem1.f  |-  F  =  (LFnl `  U )
lcfrlem1.d  |-  D  =  (LDual `  U )
lcfrlem1.t  |-  .x.  =  ( .s `  D )
lcfrlem1.m  |-  .-  =  ( -g `  D )
lcfrlem1.u  |-  ( ph  ->  U  e.  LVec )
lcfrlem1.e  |-  ( ph  ->  E  e.  F )
lcfrlem1.g  |-  ( ph  ->  G  e.  F )
lcfrlem1.x  |-  ( ph  ->  X  e.  V )
lcfrlem1.n  |-  ( ph  ->  ( G `  X
)  =/=  .0.  )
lcfrlem1.h  |-  H  =  ( E  .-  (
( ( I `  ( G `  X ) )  .X.  ( E `  X ) )  .x.  G ) )
lcfrlem2.l  |-  L  =  (LKer `  U )
Assertion
Ref Expression
lcfrlem3  |-  ( ph  ->  X  e.  ( L `
 H ) )

Proof of Theorem lcfrlem3
StepHypRef Expression
1 lcfrlem1.v . . 3  |-  V  =  ( Base `  U
)
2 lcfrlem1.s . . 3  |-  S  =  (Scalar `  U )
3 lcfrlem1.q . . 3  |-  .X.  =  ( .r `  S )
4 lcfrlem1.z . . 3  |-  .0.  =  ( 0g `  S )
5 lcfrlem1.i . . 3  |-  I  =  ( invr `  S
)
6 lcfrlem1.f . . 3  |-  F  =  (LFnl `  U )
7 lcfrlem1.d . . 3  |-  D  =  (LDual `  U )
8 lcfrlem1.t . . 3  |-  .x.  =  ( .s `  D )
9 lcfrlem1.m . . 3  |-  .-  =  ( -g `  D )
10 lcfrlem1.u . . 3  |-  ( ph  ->  U  e.  LVec )
11 lcfrlem1.e . . 3  |-  ( ph  ->  E  e.  F )
12 lcfrlem1.g . . 3  |-  ( ph  ->  G  e.  F )
13 lcfrlem1.x . . 3  |-  ( ph  ->  X  e.  V )
14 lcfrlem1.n . . 3  |-  ( ph  ->  ( G `  X
)  =/=  .0.  )
15 lcfrlem1.h . . 3  |-  H  =  ( E  .-  (
( ( I `  ( G `  X ) )  .X.  ( E `  X ) )  .x.  G ) )
161, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15lcfrlem1 32354 . 2  |-  ( ph  ->  ( H `  X
)  =  .0.  )
17 lcfrlem2.l . . 3  |-  L  =  (LKer `  U )
18 lveclmod 15875 . . . . . 6  |-  ( U  e.  LVec  ->  U  e. 
LMod )
1910, 18syl 15 . . . . 5  |-  ( ph  ->  U  e.  LMod )
20 eqid 2296 . . . . . 6  |-  ( Base `  S )  =  (
Base `  S )
212lmodrng 15651 . . . . . . . 8  |-  ( U  e.  LMod  ->  S  e. 
Ring )
2219, 21syl 15 . . . . . . 7  |-  ( ph  ->  S  e.  Ring )
232lvecdrng 15874 . . . . . . . . 9  |-  ( U  e.  LVec  ->  S  e.  DivRing )
2410, 23syl 15 . . . . . . . 8  |-  ( ph  ->  S  e.  DivRing )
252, 20, 1, 6lflcl 29876 . . . . . . . . 9  |-  ( ( U  e.  LVec  /\  G  e.  F  /\  X  e.  V )  ->  ( G `  X )  e.  ( Base `  S
) )
2610, 12, 13, 25syl3anc 1182 . . . . . . . 8  |-  ( ph  ->  ( G `  X
)  e.  ( Base `  S ) )
2720, 4, 5drnginvrcl 15545 . . . . . . . 8  |-  ( ( S  e.  DivRing  /\  ( G `  X )  e.  ( Base `  S
)  /\  ( G `  X )  =/=  .0.  )  ->  ( I `  ( G `  X ) )  e.  ( Base `  S ) )
2824, 26, 14, 27syl3anc 1182 . . . . . . 7  |-  ( ph  ->  ( I `  ( G `  X )
)  e.  ( Base `  S ) )
292, 20, 1, 6lflcl 29876 . . . . . . . 8  |-  ( ( U  e.  LVec  /\  E  e.  F  /\  X  e.  V )  ->  ( E `  X )  e.  ( Base `  S
) )
3010, 11, 13, 29syl3anc 1182 . . . . . . 7  |-  ( ph  ->  ( E `  X
)  e.  ( Base `  S ) )
3120, 3rngcl 15370 . . . . . . 7  |-  ( ( S  e.  Ring  /\  (
I `  ( G `  X ) )  e.  ( Base `  S
)  /\  ( E `  X )  e.  (
Base `  S )
)  ->  ( (
I `  ( G `  X ) )  .X.  ( E `  X ) )  e.  ( Base `  S ) )
3222, 28, 30, 31syl3anc 1182 . . . . . 6  |-  ( ph  ->  ( ( I `  ( G `  X ) )  .X.  ( E `  X ) )  e.  ( Base `  S
) )
336, 2, 20, 7, 8, 19, 32, 12ldualvscl 29951 . . . . 5  |-  ( ph  ->  ( ( ( I `
 ( G `  X ) )  .X.  ( E `  X ) )  .x.  G )  e.  F )
346, 7, 9, 19, 11, 33ldualvsubcl 29968 . . . 4  |-  ( ph  ->  ( E  .-  (
( ( I `  ( G `  X ) )  .X.  ( E `  X ) )  .x.  G ) )  e.  F )
3515, 34syl5eqel 2380 . . 3  |-  ( ph  ->  H  e.  F )
361, 2, 4, 6, 17, 10, 35, 13ellkr2 29903 . 2  |-  ( ph  ->  ( X  e.  ( L `  H )  <-> 
( H `  X
)  =  .0.  )
)
3716, 36mpbird 223 1  |-  ( ph  ->  X  e.  ( L `
 H ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696    =/= wne 2459   ` cfv 5271  (class class class)co 5874   Basecbs 13164   .rcmulr 13225  Scalarcsca 13227   .scvsca 13228   0gc0g 13416   -gcsg 14381   Ringcrg 15353   invrcinvr 15469   DivRingcdr 15528   LModclmod 15643   LVecclvec 15871  LFnlclfn 29869  LKerclk 29897  LDualcld 29935
This theorem is referenced by:  lcfrlem35  32389
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-tpos 6250  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-sca 13240  df-vsca 13241  df-0g 13420  df-mnd 14383  df-grp 14505  df-minusg 14506  df-sbg 14507  df-cmn 15107  df-abl 15108  df-mgp 15342  df-rng 15356  df-ur 15358  df-oppr 15421  df-dvdsr 15439  df-unit 15440  df-invr 15470  df-drng 15530  df-lmod 15645  df-lvec 15872  df-lfl 29870  df-lkr 29898  df-ldual 29936
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