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Theorem lcfrlem1 35156
Description: Lemma for lcfr 35199. Note that  X is z in Mario's notes. (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 ) )
Assertion
Ref Expression
lcfrlem1  |-  ( ph  ->  ( H `  X
)  =  .0.  )

Proof of Theorem lcfrlem1
StepHypRef Expression
1 lcfrlem1.h . . 3  |-  H  =  ( E  .-  (
( ( I `  ( G `  X ) )  .X.  ( E `  X ) )  .x.  G ) )
21fveq1i 5893 . 2  |-  ( H `
 X )  =  ( ( E  .-  ( ( ( I `
 ( G `  X ) )  .X.  ( E `  X ) )  .x.  G ) ) `  X )
3 lcfrlem1.v . . . 4  |-  V  =  ( Base `  U
)
4 lcfrlem1.s . . . 4  |-  S  =  (Scalar `  U )
5 eqid 2462 . . . 4  |-  ( -g `  S )  =  (
-g `  S )
6 lcfrlem1.f . . . 4  |-  F  =  (LFnl `  U )
7 lcfrlem1.d . . . 4  |-  D  =  (LDual `  U )
8 lcfrlem1.m . . . 4  |-  .-  =  ( -g `  D )
9 lcfrlem1.u . . . . 5  |-  ( ph  ->  U  e.  LVec )
10 lveclmod 18384 . . . . 5  |-  ( U  e.  LVec  ->  U  e. 
LMod )
119, 10syl 17 . . . 4  |-  ( ph  ->  U  e.  LMod )
12 lcfrlem1.e . . . 4  |-  ( ph  ->  E  e.  F )
13 eqid 2462 . . . . 5  |-  ( Base `  S )  =  (
Base `  S )
14 lcfrlem1.t . . . . 5  |-  .x.  =  ( .s `  D )
154lvecdrng 18383 . . . . . . . 8  |-  ( U  e.  LVec  ->  S  e.  DivRing )
169, 15syl 17 . . . . . . 7  |-  ( ph  ->  S  e.  DivRing )
17 lcfrlem1.g . . . . . . . 8  |-  ( ph  ->  G  e.  F )
18 lcfrlem1.x . . . . . . . 8  |-  ( ph  ->  X  e.  V )
194, 13, 3, 6lflcl 32676 . . . . . . . 8  |-  ( ( U  e.  LVec  /\  G  e.  F  /\  X  e.  V )  ->  ( G `  X )  e.  ( Base `  S
) )
209, 17, 18, 19syl3anc 1276 . . . . . . 7  |-  ( ph  ->  ( G `  X
)  e.  ( Base `  S ) )
21 lcfrlem1.n . . . . . . 7  |-  ( ph  ->  ( G `  X
)  =/=  .0.  )
22 lcfrlem1.z . . . . . . . 8  |-  .0.  =  ( 0g `  S )
23 lcfrlem1.i . . . . . . . 8  |-  I  =  ( invr `  S
)
2413, 22, 23drnginvrcl 18047 . . . . . . 7  |-  ( ( S  e.  DivRing  /\  ( G `  X )  e.  ( Base `  S
)  /\  ( G `  X )  =/=  .0.  )  ->  ( I `  ( G `  X ) )  e.  ( Base `  S ) )
2516, 20, 21, 24syl3anc 1276 . . . . . 6  |-  ( ph  ->  ( I `  ( G `  X )
)  e.  ( Base `  S ) )
264, 13, 3, 6lflcl 32676 . . . . . . 7  |-  ( ( U  e.  LVec  /\  E  e.  F  /\  X  e.  V )  ->  ( E `  X )  e.  ( Base `  S
) )
279, 12, 18, 26syl3anc 1276 . . . . . 6  |-  ( ph  ->  ( E `  X
)  e.  ( Base `  S ) )
28 lcfrlem1.q . . . . . . 7  |-  .X.  =  ( .r `  S )
294, 13, 28lmodmcl 18158 . . . . . 6  |-  ( ( U  e.  LMod  /\  (
I `  ( G `  X ) )  e.  ( Base `  S
)  /\  ( E `  X )  e.  (
Base `  S )
)  ->  ( (
I `  ( G `  X ) )  .X.  ( E `  X ) )  e.  ( Base `  S ) )
3011, 25, 27, 29syl3anc 1276 . . . . 5  |-  ( ph  ->  ( ( I `  ( G `  X ) )  .X.  ( E `  X ) )  e.  ( Base `  S
) )
316, 4, 13, 7, 14, 11, 30, 17ldualvscl 32751 . . . 4  |-  ( ph  ->  ( ( ( I `
 ( G `  X ) )  .X.  ( E `  X ) )  .x.  G )  e.  F )
323, 4, 5, 6, 7, 8, 11, 12, 31, 18ldualvsubval 32769 . . 3  |-  ( ph  ->  ( ( E  .-  ( ( ( I `
 ( G `  X ) )  .X.  ( E `  X ) )  .x.  G ) ) `  X )  =  ( ( E `
 X ) (
-g `  S )
( ( ( ( I `  ( G `
 X ) ) 
.X.  ( E `  X ) )  .x.  G ) `  X
) ) )
336, 3, 4, 13, 28, 7, 14, 9, 30, 17, 18ldualvsval 32750 . . . . 5  |-  ( ph  ->  ( ( ( ( I `  ( G `
 X ) ) 
.X.  ( E `  X ) )  .x.  G ) `  X
)  =  ( ( G `  X ) 
.X.  ( ( I `
 ( G `  X ) )  .X.  ( E `  X ) ) ) )
34 eqid 2462 . . . . . . . . 9  |-  ( 1r
`  S )  =  ( 1r `  S
)
3513, 22, 28, 34, 23drnginvrr 18050 . . . . . . . 8  |-  ( ( S  e.  DivRing  /\  ( G `  X )  e.  ( Base `  S
)  /\  ( G `  X )  =/=  .0.  )  ->  ( ( G `
 X )  .X.  ( I `  ( G `  X )
) )  =  ( 1r `  S ) )
3616, 20, 21, 35syl3anc 1276 . . . . . . 7  |-  ( ph  ->  ( ( G `  X )  .X.  (
I `  ( G `  X ) ) )  =  ( 1r `  S ) )
3736oveq1d 6335 . . . . . 6  |-  ( ph  ->  ( ( ( G `
 X )  .X.  ( I `  ( G `  X )
) )  .X.  ( E `  X )
)  =  ( ( 1r `  S ) 
.X.  ( E `  X ) ) )
384lmodring 18154 . . . . . . . 8  |-  ( U  e.  LMod  ->  S  e. 
Ring )
3911, 38syl 17 . . . . . . 7  |-  ( ph  ->  S  e.  Ring )
4013, 28ringass 17852 . . . . . . 7  |-  ( ( S  e.  Ring  /\  (
( G `  X
)  e.  ( Base `  S )  /\  (
I `  ( G `  X ) )  e.  ( Base `  S
)  /\  ( E `  X )  e.  (
Base `  S )
) )  ->  (
( ( G `  X )  .X.  (
I `  ( G `  X ) ) ) 
.X.  ( E `  X ) )  =  ( ( G `  X )  .X.  (
( I `  ( G `  X )
)  .X.  ( E `  X ) ) ) )
4139, 20, 25, 27, 40syl13anc 1278 . . . . . 6  |-  ( ph  ->  ( ( ( G `
 X )  .X.  ( I `  ( G `  X )
) )  .X.  ( E `  X )
)  =  ( ( G `  X ) 
.X.  ( ( I `
 ( G `  X ) )  .X.  ( E `  X ) ) ) )
4213, 28, 34ringlidm 17859 . . . . . . 7  |-  ( ( S  e.  Ring  /\  ( E `  X )  e.  ( Base `  S
) )  ->  (
( 1r `  S
)  .X.  ( E `  X ) )  =  ( E `  X
) )
4339, 27, 42syl2anc 671 . . . . . 6  |-  ( ph  ->  ( ( 1r `  S )  .X.  ( E `  X )
)  =  ( E `
 X ) )
4437, 41, 433eqtr3d 2504 . . . . 5  |-  ( ph  ->  ( ( G `  X )  .X.  (
( I `  ( G `  X )
)  .X.  ( E `  X ) ) )  =  ( E `  X ) )
4533, 44eqtrd 2496 . . . 4  |-  ( ph  ->  ( ( ( ( I `  ( G `
 X ) ) 
.X.  ( E `  X ) )  .x.  G ) `  X
)  =  ( E `
 X ) )
4645oveq2d 6336 . . 3  |-  ( ph  ->  ( ( E `  X ) ( -g `  S ) ( ( ( ( I `  ( G `  X ) )  .X.  ( E `  X ) )  .x.  G ) `  X
) )  =  ( ( E `  X
) ( -g `  S
) ( E `  X ) ) )
474lmodfgrp 18155 . . . . 5  |-  ( U  e.  LMod  ->  S  e. 
Grp )
4811, 47syl 17 . . . 4  |-  ( ph  ->  S  e.  Grp )
4913, 22, 5grpsubid 16793 . . . 4  |-  ( ( S  e.  Grp  /\  ( E `  X )  e.  ( Base `  S
) )  ->  (
( E `  X
) ( -g `  S
) ( E `  X ) )  =  .0.  )
5048, 27, 49syl2anc 671 . . 3  |-  ( ph  ->  ( ( E `  X ) ( -g `  S ) ( E `
 X ) )  =  .0.  )
5132, 46, 503eqtrd 2500 . 2  |-  ( ph  ->  ( ( E  .-  ( ( ( I `
 ( G `  X ) )  .X.  ( E `  X ) )  .x.  G ) ) `  X )  =  .0.  )
522, 51syl5eq 2508 1  |-  ( ph  ->  ( H `  X
)  =  .0.  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1455    e. wcel 1898    =/= wne 2633   ` cfv 5605  (class class class)co 6320   Basecbs 15176   .rcmulr 15246  Scalarcsca 15248   .scvsca 15249   0gc0g 15393   Grpcgrp 16724   -gcsg 16726   1rcur 17790   Ringcrg 17835   invrcinvr 17954   DivRingcdr 18030   LModclmod 18146   LVecclvec 18380  LFnlclfn 32669  LDualcld 32735
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4531  ax-sep 4541  ax-nul 4550  ax-pow 4598  ax-pr 4656  ax-un 6615  ax-cnex 9626  ax-resscn 9627  ax-1cn 9628  ax-icn 9629  ax-addcl 9630  ax-addrcl 9631  ax-mulcl 9632  ax-mulrcl 9633  ax-mulcom 9634  ax-addass 9635  ax-mulass 9636  ax-distr 9637  ax-i2m1 9638  ax-1ne0 9639  ax-1rid 9640  ax-rnegex 9641  ax-rrecex 9642  ax-cnre 9643  ax-pre-lttri 9644  ax-pre-lttrn 9645  ax-pre-ltadd 9646  ax-pre-mulgt0 9647
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-nel 2636  df-ral 2754  df-rex 2755  df-reu 2756  df-rmo 2757  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-int 4249  df-iun 4294  df-br 4419  df-opab 4478  df-mpt 4479  df-tr 4514  df-eprel 4767  df-id 4771  df-po 4777  df-so 4778  df-fr 4815  df-we 4817  df-xp 4862  df-rel 4863  df-cnv 4864  df-co 4865  df-dm 4866  df-rn 4867  df-res 4868  df-ima 4869  df-pred 5403  df-ord 5449  df-on 5450  df-lim 5451  df-suc 5452  df-iota 5569  df-fun 5607  df-fn 5608  df-f 5609  df-f1 5610  df-fo 5611  df-f1o 5612  df-fv 5613  df-riota 6282  df-ov 6323  df-oprab 6324  df-mpt2 6325  df-of 6563  df-om 6725  df-1st 6825  df-2nd 6826  df-tpos 7004  df-wrecs 7059  df-recs 7121  df-rdg 7159  df-1o 7213  df-oadd 7217  df-er 7394  df-map 7505  df-en 7601  df-dom 7602  df-sdom 7603  df-fin 7604  df-pnf 9708  df-mnf 9709  df-xr 9710  df-ltxr 9711  df-le 9712  df-sub 9893  df-neg 9894  df-nn 10643  df-2 10701  df-3 10702  df-4 10703  df-5 10704  df-6 10705  df-n0 10904  df-z 10972  df-uz 11194  df-fz 11820  df-struct 15178  df-ndx 15179  df-slot 15180  df-base 15181  df-sets 15182  df-ress 15183  df-plusg 15258  df-mulr 15259  df-sca 15261  df-vsca 15262  df-0g 15395  df-mgm 16543  df-sgrp 16582  df-mnd 16592  df-grp 16728  df-minusg 16729  df-sbg 16730  df-cmn 17487  df-abl 17488  df-mgp 17779  df-ur 17791  df-ring 17837  df-oppr 17906  df-dvdsr 17924  df-unit 17925  df-invr 17955  df-drng 18032  df-lmod 18148  df-lvec 18381  df-lfl 32670  df-ldual 32736
This theorem is referenced by:  lcfrlem3  35158
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