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Theorem lcfrlem3 31661
Description: Lemma for lcfr 31702. (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 31659 . 2  |-  ( ph  ->  ( H `  X
)  =  .0.  )
17 lcfrlem2.l . . 3  |-  L  =  (LKer `  U )
18 lveclmod 16107 . . . . . 6  |-  ( U  e.  LVec  ->  U  e. 
LMod )
1910, 18syl 16 . . . . 5  |-  ( ph  ->  U  e.  LMod )
20 eqid 2389 . . . . . 6  |-  ( Base `  S )  =  (
Base `  S )
212lmodrng 15887 . . . . . . . 8  |-  ( U  e.  LMod  ->  S  e. 
Ring )
2219, 21syl 16 . . . . . . 7  |-  ( ph  ->  S  e.  Ring )
232lvecdrng 16106 . . . . . . . . 9  |-  ( U  e.  LVec  ->  S  e.  DivRing )
2410, 23syl 16 . . . . . . . 8  |-  ( ph  ->  S  e.  DivRing )
252, 20, 1, 6lflcl 29181 . . . . . . . . 9  |-  ( ( U  e.  LVec  /\  G  e.  F  /\  X  e.  V )  ->  ( G `  X )  e.  ( Base `  S
) )
2610, 12, 13, 25syl3anc 1184 . . . . . . . 8  |-  ( ph  ->  ( G `  X
)  e.  ( Base `  S ) )
2720, 4, 5drnginvrcl 15781 . . . . . . . 8  |-  ( ( S  e.  DivRing  /\  ( G `  X )  e.  ( Base `  S
)  /\  ( G `  X )  =/=  .0.  )  ->  ( I `  ( G `  X ) )  e.  ( Base `  S ) )
2824, 26, 14, 27syl3anc 1184 . . . . . . 7  |-  ( ph  ->  ( I `  ( G `  X )
)  e.  ( Base `  S ) )
292, 20, 1, 6lflcl 29181 . . . . . . . 8  |-  ( ( U  e.  LVec  /\  E  e.  F  /\  X  e.  V )  ->  ( E `  X )  e.  ( Base `  S
) )
3010, 11, 13, 29syl3anc 1184 . . . . . . 7  |-  ( ph  ->  ( E `  X
)  e.  ( Base `  S ) )
3120, 3rngcl 15606 . . . . . . 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 1184 . . . . . 6  |-  ( ph  ->  ( ( I `  ( G `  X ) )  .X.  ( E `  X ) )  e.  ( Base `  S
) )
336, 2, 20, 7, 8, 19, 32, 12ldualvscl 29256 . . . . 5  |-  ( ph  ->  ( ( ( I `
 ( G `  X ) )  .X.  ( E `  X ) )  .x.  G )  e.  F )
346, 7, 9, 19, 11, 33ldualvsubcl 29273 . . . 4  |-  ( ph  ->  ( E  .-  (
( ( I `  ( G `  X ) )  .X.  ( E `  X ) )  .x.  G ) )  e.  F )
3515, 34syl5eqel 2473 . . 3  |-  ( ph  ->  H  e.  F )
361, 2, 4, 6, 17, 10, 35, 13ellkr2 29208 . 2  |-  ( ph  ->  ( X  e.  ( L `  H )  <-> 
( H `  X
)  =  .0.  )
)
3716, 36mpbird 224 1  |-  ( ph  ->  X  e.  ( L `
 H ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717    =/= wne 2552   ` cfv 5396  (class class class)co 6022   Basecbs 13398   .rcmulr 13459  Scalarcsca 13461   .scvsca 13462   0gc0g 13652   -gcsg 14617   Ringcrg 15589   invrcinvr 15705   DivRingcdr 15764   LModclmod 15879   LVecclvec 16103  LFnlclfn 29174  LKerclk 29202  LDualcld 29240
This theorem is referenced by:  lcfrlem35  31694
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 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-cnex 8981  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-int 3995  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-of 6246  df-1st 6290  df-2nd 6291  df-tpos 6417  df-riota 6487  df-recs 6571  df-rdg 6606  df-1o 6662  df-oadd 6666  df-er 6843  df-map 6958  df-en 7048  df-dom 7049  df-sdom 7050  df-fin 7051  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-nn 9935  df-2 9992  df-3 9993  df-4 9994  df-5 9995  df-6 9996  df-n0 10156  df-z 10217  df-uz 10423  df-fz 10978  df-struct 13400  df-ndx 13401  df-slot 13402  df-base 13403  df-sets 13404  df-ress 13405  df-plusg 13471  df-mulr 13472  df-sca 13474  df-vsca 13475  df-0g 13656  df-mnd 14619  df-grp 14741  df-minusg 14742  df-sbg 14743  df-cmn 15343  df-abl 15344  df-mgp 15578  df-rng 15592  df-ur 15594  df-oppr 15657  df-dvdsr 15675  df-unit 15676  df-invr 15706  df-drng 15766  df-lmod 15881  df-lvec 16104  df-lfl 29175  df-lkr 29203  df-ldual 29241
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