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Theorem lcfrlem1 34526
Description: Lemma for lcfr 34569. 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 5804 . 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 2400 . . . 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 17962 . . . . 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 2400 . . . . 5  |-  ( Base `  S )  =  (
Base `  S )
14 lcfrlem1.t . . . . 5  |-  .x.  =  ( .s `  D )
154lvecdrng 17961 . . . . . . . 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 32046 . . . . . . . 8  |-  ( ( U  e.  LVec  /\  G  e.  F  /\  X  e.  V )  ->  ( G `  X )  e.  ( Base `  S
) )
209, 17, 18, 19syl3anc 1228 . . . . . . 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 17623 . . . . . . 7  |-  ( ( S  e.  DivRing  /\  ( G `  X )  e.  ( Base `  S
)  /\  ( G `  X )  =/=  .0.  )  ->  ( I `  ( G `  X ) )  e.  ( Base `  S ) )
2516, 20, 21, 24syl3anc 1228 . . . . . 6  |-  ( ph  ->  ( I `  ( G `  X )
)  e.  ( Base `  S ) )
264, 13, 3, 6lflcl 32046 . . . . . . 7  |-  ( ( U  e.  LVec  /\  E  e.  F  /\  X  e.  V )  ->  ( E `  X )  e.  ( Base `  S
) )
279, 12, 18, 26syl3anc 1228 . . . . . 6  |-  ( ph  ->  ( E `  X
)  e.  ( Base `  S ) )
28 lcfrlem1.q . . . . . . 7  |-  .X.  =  ( .r `  S )
294, 13, 28lmodmcl 17734 . . . . . 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 1228 . . . . 5  |-  ( ph  ->  ( ( I `  ( G `  X ) )  .X.  ( E `  X ) )  e.  ( Base `  S
) )
316, 4, 13, 7, 14, 11, 30, 17ldualvscl 32121 . . . 4  |-  ( ph  ->  ( ( ( I `
 ( G `  X ) )  .X.  ( E `  X ) )  .x.  G )  e.  F )
323, 4, 5, 6, 7, 8, 11, 12, 31, 18ldualvsubval 32139 . . 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 32120 . . . . 5  |-  ( ph  ->  ( ( ( ( I `  ( G `
 X ) ) 
.X.  ( E `  X ) )  .x.  G ) `  X
)  =  ( ( G `  X ) 
.X.  ( ( I `
 ( G `  X ) )  .X.  ( E `  X ) ) ) )
34 eqid 2400 . . . . . . . . 9  |-  ( 1r
`  S )  =  ( 1r `  S
)
3513, 22, 28, 34, 23drnginvrr 17626 . . . . . . . 8  |-  ( ( S  e.  DivRing  /\  ( G `  X )  e.  ( Base `  S
)  /\  ( G `  X )  =/=  .0.  )  ->  ( ( G `
 X )  .X.  ( I `  ( G `  X )
) )  =  ( 1r `  S ) )
3616, 20, 21, 35syl3anc 1228 . . . . . . 7  |-  ( ph  ->  ( ( G `  X )  .X.  (
I `  ( G `  X ) ) )  =  ( 1r `  S ) )
3736oveq1d 6247 . . . . . 6  |-  ( ph  ->  ( ( ( G `
 X )  .X.  ( I `  ( G `  X )
) )  .X.  ( E `  X )
)  =  ( ( 1r `  S ) 
.X.  ( E `  X ) ) )
384lmodring 17730 . . . . . . . 8  |-  ( U  e.  LMod  ->  S  e. 
Ring )
3911, 38syl 17 . . . . . . 7  |-  ( ph  ->  S  e.  Ring )
4013, 28ringass 17425 . . . . . . 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 1230 . . . . . 6  |-  ( ph  ->  ( ( ( G `
 X )  .X.  ( I `  ( G `  X )
) )  .X.  ( E `  X )
)  =  ( ( G `  X ) 
.X.  ( ( I `
 ( G `  X ) )  .X.  ( E `  X ) ) ) )
4213, 28, 34ringlidm 17432 . . . . . . 7  |-  ( ( S  e.  Ring  /\  ( E `  X )  e.  ( Base `  S
) )  ->  (
( 1r `  S
)  .X.  ( E `  X ) )  =  ( E `  X
) )
4339, 27, 42syl2anc 659 . . . . . 6  |-  ( ph  ->  ( ( 1r `  S )  .X.  ( E `  X )
)  =  ( E `
 X ) )
4437, 41, 433eqtr3d 2449 . . . . 5  |-  ( ph  ->  ( ( G `  X )  .X.  (
( I `  ( G `  X )
)  .X.  ( E `  X ) ) )  =  ( E `  X ) )
4533, 44eqtrd 2441 . . . 4  |-  ( ph  ->  ( ( ( ( I `  ( G `
 X ) ) 
.X.  ( E `  X ) )  .x.  G ) `  X
)  =  ( E `
 X ) )
4645oveq2d 6248 . . 3  |-  ( ph  ->  ( ( E `  X ) ( -g `  S ) ( ( ( ( I `  ( G `  X ) )  .X.  ( E `  X ) )  .x.  G ) `  X
) )  =  ( ( E `  X
) ( -g `  S
) ( E `  X ) ) )
474lmodfgrp 17731 . . . . 5  |-  ( U  e.  LMod  ->  S  e. 
Grp )
4811, 47syl 17 . . . 4  |-  ( ph  ->  S  e.  Grp )
4913, 22, 5grpsubid 16336 . . . 4  |-  ( ( S  e.  Grp  /\  ( E `  X )  e.  ( Base `  S
) )  ->  (
( E `  X
) ( -g `  S
) ( E `  X ) )  =  .0.  )
5048, 27, 49syl2anc 659 . . 3  |-  ( ph  ->  ( ( E `  X ) ( -g `  S ) ( E `
 X ) )  =  .0.  )
5132, 46, 503eqtrd 2445 . 2  |-  ( ph  ->  ( ( E  .-  ( ( ( I `
 ( G `  X ) )  .X.  ( E `  X ) )  .x.  G ) ) `  X )  =  .0.  )
522, 51syl5eq 2453 1  |-  ( ph  ->  ( H `  X
)  =  .0.  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1403    e. wcel 1840    =/= wne 2596   ` cfv 5523  (class class class)co 6232   Basecbs 14731   .rcmulr 14800  Scalarcsca 14802   .scvsca 14803   0gc0g 14944   Grpcgrp 16267   -gcsg 16269   1rcur 17363   Ringcrg 17408   invrcinvr 17530   DivRingcdr 17606   LModclmod 17722   LVecclvec 17958  LFnlclfn 32039  LDualcld 32105
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528  ax-cnex 9496  ax-resscn 9497  ax-1cn 9498  ax-icn 9499  ax-addcl 9500  ax-addrcl 9501  ax-mulcl 9502  ax-mulrcl 9503  ax-mulcom 9504  ax-addass 9505  ax-mulass 9506  ax-distr 9507  ax-i2m1 9508  ax-1ne0 9509  ax-1rid 9510  ax-rnegex 9511  ax-rrecex 9512  ax-cnre 9513  ax-pre-lttri 9514  ax-pre-lttrn 9515  ax-pre-ltadd 9516  ax-pre-mulgt0 9517
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 973  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-nel 2599  df-ral 2756  df-rex 2757  df-reu 2758  df-rmo 2759  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-pss 3427  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-tp 3974  df-op 3976  df-uni 4189  df-int 4225  df-iun 4270  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4487  df-eprel 4731  df-id 4735  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-riota 6194  df-ov 6235  df-oprab 6236  df-mpt2 6237  df-of 6475  df-om 6637  df-1st 6736  df-2nd 6737  df-tpos 6910  df-recs 6997  df-rdg 7031  df-1o 7085  df-oadd 7089  df-er 7266  df-map 7377  df-en 7473  df-dom 7474  df-sdom 7475  df-fin 7476  df-pnf 9578  df-mnf 9579  df-xr 9580  df-ltxr 9581  df-le 9582  df-sub 9761  df-neg 9762  df-nn 10495  df-2 10553  df-3 10554  df-4 10555  df-5 10556  df-6 10557  df-n0 10755  df-z 10824  df-uz 11044  df-fz 11642  df-struct 14733  df-ndx 14734  df-slot 14735  df-base 14736  df-sets 14737  df-ress 14738  df-plusg 14812  df-mulr 14813  df-sca 14815  df-vsca 14816  df-0g 14946  df-mgm 16086  df-sgrp 16125  df-mnd 16135  df-grp 16271  df-minusg 16272  df-sbg 16273  df-cmn 17014  df-abl 17015  df-mgp 17352  df-ur 17364  df-ring 17410  df-oppr 17482  df-dvdsr 17500  df-unit 17501  df-invr 17531  df-drng 17608  df-lmod 17724  df-lvec 17959  df-lfl 32040  df-ldual 32106
This theorem is referenced by:  lcfrlem3  34528
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