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Theorem lcfrlem1 32438
Description: Lemma for lcfr 32481. 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 5758 . 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 2442 . . . 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 16209 . . . . 5  |-  ( U  e.  LVec  ->  U  e. 
LMod )
119, 10syl 16 . . . 4  |-  ( ph  ->  U  e.  LMod )
12 lcfrlem1.e . . . 4  |-  ( ph  ->  E  e.  F )
13 eqid 2442 . . . . 5  |-  ( Base `  S )  =  (
Base `  S )
14 lcfrlem1.t . . . . 5  |-  .x.  =  ( .s `  D )
154lvecdrng 16208 . . . . . . . 8  |-  ( U  e.  LVec  ->  S  e.  DivRing )
169, 15syl 16 . . . . . . 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 29960 . . . . . . . 8  |-  ( ( U  e.  LVec  /\  G  e.  F  /\  X  e.  V )  ->  ( G `  X )  e.  ( Base `  S
) )
209, 17, 18, 19syl3anc 1185 . . . . . . 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 15883 . . . . . . 7  |-  ( ( S  e.  DivRing  /\  ( G `  X )  e.  ( Base `  S
)  /\  ( G `  X )  =/=  .0.  )  ->  ( I `  ( G `  X ) )  e.  ( Base `  S ) )
2516, 20, 21, 24syl3anc 1185 . . . . . 6  |-  ( ph  ->  ( I `  ( G `  X )
)  e.  ( Base `  S ) )
264, 13, 3, 6lflcl 29960 . . . . . . 7  |-  ( ( U  e.  LVec  /\  E  e.  F  /\  X  e.  V )  ->  ( E `  X )  e.  ( Base `  S
) )
279, 12, 18, 26syl3anc 1185 . . . . . 6  |-  ( ph  ->  ( E `  X
)  e.  ( Base `  S ) )
28 lcfrlem1.q . . . . . . 7  |-  .X.  =  ( .r `  S )
294, 13, 28lmodmcl 15993 . . . . . 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 1185 . . . . 5  |-  ( ph  ->  ( ( I `  ( G `  X ) )  .X.  ( E `  X ) )  e.  ( Base `  S
) )
316, 4, 13, 7, 14, 11, 30, 17ldualvscl 30035 . . . 4  |-  ( ph  ->  ( ( ( I `
 ( G `  X ) )  .X.  ( E `  X ) )  .x.  G )  e.  F )
323, 4, 5, 6, 7, 8, 11, 12, 31, 18ldualvsubval 30053 . . 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 30034 . . . . 5  |-  ( ph  ->  ( ( ( ( I `  ( G `
 X ) ) 
.X.  ( E `  X ) )  .x.  G ) `  X
)  =  ( ( G `  X ) 
.X.  ( ( I `
 ( G `  X ) )  .X.  ( E `  X ) ) ) )
34 eqid 2442 . . . . . . . . 9  |-  ( 1r
`  S )  =  ( 1r `  S
)
3513, 22, 28, 34, 23drnginvrr 15886 . . . . . . . 8  |-  ( ( S  e.  DivRing  /\  ( G `  X )  e.  ( Base `  S
)  /\  ( G `  X )  =/=  .0.  )  ->  ( ( G `
 X )  .X.  ( I `  ( G `  X )
) )  =  ( 1r `  S ) )
3616, 20, 21, 35syl3anc 1185 . . . . . . 7  |-  ( ph  ->  ( ( G `  X )  .X.  (
I `  ( G `  X ) ) )  =  ( 1r `  S ) )
3736oveq1d 6125 . . . . . 6  |-  ( ph  ->  ( ( ( G `
 X )  .X.  ( I `  ( G `  X )
) )  .X.  ( E `  X )
)  =  ( ( 1r `  S ) 
.X.  ( E `  X ) ) )
384lmodrng 15989 . . . . . . . 8  |-  ( U  e.  LMod  ->  S  e. 
Ring )
3911, 38syl 16 . . . . . . 7  |-  ( ph  ->  S  e.  Ring )
4013, 28rngass 15711 . . . . . . 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 1187 . . . . . 6  |-  ( ph  ->  ( ( ( G `
 X )  .X.  ( I `  ( G `  X )
) )  .X.  ( E `  X )
)  =  ( ( G `  X ) 
.X.  ( ( I `
 ( G `  X ) )  .X.  ( E `  X ) ) ) )
4213, 28, 34rnglidm 15718 . . . . . . 7  |-  ( ( S  e.  Ring  /\  ( E `  X )  e.  ( Base `  S
) )  ->  (
( 1r `  S
)  .X.  ( E `  X ) )  =  ( E `  X
) )
4339, 27, 42syl2anc 644 . . . . . 6  |-  ( ph  ->  ( ( 1r `  S )  .X.  ( E `  X )
)  =  ( E `
 X ) )
4437, 41, 433eqtr3d 2482 . . . . 5  |-  ( ph  ->  ( ( G `  X )  .X.  (
( I `  ( G `  X )
)  .X.  ( E `  X ) ) )  =  ( E `  X ) )
4533, 44eqtrd 2474 . . . 4  |-  ( ph  ->  ( ( ( ( I `  ( G `
 X ) ) 
.X.  ( E `  X ) )  .x.  G ) `  X
)  =  ( E `
 X ) )
4645oveq2d 6126 . . 3  |-  ( ph  ->  ( ( E `  X ) ( -g `  S ) ( ( ( ( I `  ( G `  X ) )  .X.  ( E `  X ) )  .x.  G ) `  X
) )  =  ( ( E `  X
) ( -g `  S
) ( E `  X ) ) )
474lmodfgrp 15990 . . . . 5  |-  ( U  e.  LMod  ->  S  e. 
Grp )
4811, 47syl 16 . . . 4  |-  ( ph  ->  S  e.  Grp )
4913, 22, 5grpsubid 14904 . . . 4  |-  ( ( S  e.  Grp  /\  ( E `  X )  e.  ( Base `  S
) )  ->  (
( E `  X
) ( -g `  S
) ( E `  X ) )  =  .0.  )
5048, 27, 49syl2anc 644 . . 3  |-  ( ph  ->  ( ( E `  X ) ( -g `  S ) ( E `
 X ) )  =  .0.  )
5132, 46, 503eqtrd 2478 . 2  |-  ( ph  ->  ( ( E  .-  ( ( ( I `
 ( G `  X ) )  .X.  ( E `  X ) )  .x.  G ) ) `  X )  =  .0.  )
522, 51syl5eq 2486 1  |-  ( ph  ->  ( H `  X
)  =  .0.  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1727    =/= wne 2605   ` cfv 5483  (class class class)co 6110   Basecbs 13500   .rcmulr 13561  Scalarcsca 13563   .scvsca 13564   0gc0g 13754   Grpcgrp 14716   -gcsg 14719   Ringcrg 15691   1rcur 15693   invrcinvr 15807   DivRingcdr 15866   LModclmod 15981   LVecclvec 16205  LFnlclfn 29953  LDualcld 30019
This theorem is referenced by:  lcfrlem3  32440
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-rep 4345  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730  ax-cnex 9077  ax-resscn 9078  ax-1cn 9079  ax-icn 9080  ax-addcl 9081  ax-addrcl 9082  ax-mulcl 9083  ax-mulrcl 9084  ax-mulcom 9085  ax-addass 9086  ax-mulass 9087  ax-distr 9088  ax-i2m1 9089  ax-1ne0 9090  ax-1rid 9091  ax-rnegex 9092  ax-rrecex 9093  ax-cnre 9094  ax-pre-lttri 9095  ax-pre-lttrn 9096  ax-pre-ltadd 9097  ax-pre-mulgt0 9098
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2716  df-rex 2717  df-reu 2718  df-rmo 2719  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-pss 3322  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-tp 3846  df-op 3847  df-uni 4040  df-int 4075  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-tr 4328  df-eprel 4523  df-id 4527  df-po 4532  df-so 4533  df-fr 4570  df-we 4572  df-ord 4613  df-on 4614  df-lim 4615  df-suc 4616  df-om 4875  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-of 6334  df-1st 6378  df-2nd 6379  df-tpos 6508  df-riota 6578  df-recs 6662  df-rdg 6697  df-1o 6753  df-oadd 6757  df-er 6934  df-map 7049  df-en 7139  df-dom 7140  df-sdom 7141  df-fin 7142  df-pnf 9153  df-mnf 9154  df-xr 9155  df-ltxr 9156  df-le 9157  df-sub 9324  df-neg 9325  df-nn 10032  df-2 10089  df-3 10090  df-4 10091  df-5 10092  df-6 10093  df-n0 10253  df-z 10314  df-uz 10520  df-fz 11075  df-struct 13502  df-ndx 13503  df-slot 13504  df-base 13505  df-sets 13506  df-ress 13507  df-plusg 13573  df-mulr 13574  df-sca 13576  df-vsca 13577  df-0g 13758  df-mnd 14721  df-grp 14843  df-minusg 14844  df-sbg 14845  df-cmn 15445  df-abl 15446  df-mgp 15680  df-rng 15694  df-ur 15696  df-oppr 15759  df-dvdsr 15777  df-unit 15778  df-invr 15808  df-drng 15868  df-lmod 15983  df-lvec 16206  df-lfl 29954  df-ldual 30020
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