Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  lcfrlem1 Unicode version

Theorem lcfrlem1 31801
Description: Lemma for lcfr 31844. 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 5609 . 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 2358 . . . 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 15958 . . . . 5  |-  ( U  e.  LVec  ->  U  e. 
LMod )
119, 10syl 15 . . . 4  |-  ( ph  ->  U  e.  LMod )
12 lcfrlem1.e . . . 4  |-  ( ph  ->  E  e.  F )
13 eqid 2358 . . . . 5  |-  ( Base `  S )  =  (
Base `  S )
14 lcfrlem1.t . . . . 5  |-  .x.  =  ( .s `  D )
154lvecdrng 15957 . . . . . . . 8  |-  ( U  e.  LVec  ->  S  e.  DivRing )
169, 15syl 15 . . . . . . 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 29323 . . . . . . . 8  |-  ( ( U  e.  LVec  /\  G  e.  F  /\  X  e.  V )  ->  ( G `  X )  e.  ( Base `  S
) )
209, 17, 18, 19syl3anc 1182 . . . . . . 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 15628 . . . . . . 7  |-  ( ( S  e.  DivRing  /\  ( G `  X )  e.  ( Base `  S
)  /\  ( G `  X )  =/=  .0.  )  ->  ( I `  ( G `  X ) )  e.  ( Base `  S ) )
2516, 20, 21, 24syl3anc 1182 . . . . . 6  |-  ( ph  ->  ( I `  ( G `  X )
)  e.  ( Base `  S ) )
264, 13, 3, 6lflcl 29323 . . . . . . 7  |-  ( ( U  e.  LVec  /\  E  e.  F  /\  X  e.  V )  ->  ( E `  X )  e.  ( Base `  S
) )
279, 12, 18, 26syl3anc 1182 . . . . . 6  |-  ( ph  ->  ( E `  X
)  e.  ( Base `  S ) )
28 lcfrlem1.q . . . . . . 7  |-  .X.  =  ( .r `  S )
294, 13, 28lmodmcl 15738 . . . . . 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 1182 . . . . 5  |-  ( ph  ->  ( ( I `  ( G `  X ) )  .X.  ( E `  X ) )  e.  ( Base `  S
) )
316, 4, 13, 7, 14, 11, 30, 17ldualvscl 29398 . . . 4  |-  ( ph  ->  ( ( ( I `
 ( G `  X ) )  .X.  ( E `  X ) )  .x.  G )  e.  F )
323, 4, 5, 6, 7, 8, 11, 12, 31, 18ldualvsubval 29416 . . 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 29397 . . . . 5  |-  ( ph  ->  ( ( ( ( I `  ( G `
 X ) ) 
.X.  ( E `  X ) )  .x.  G ) `  X
)  =  ( ( G `  X ) 
.X.  ( ( I `
 ( G `  X ) )  .X.  ( E `  X ) ) ) )
34 eqid 2358 . . . . . . . . 9  |-  ( 1r
`  S )  =  ( 1r `  S
)
3513, 22, 28, 34, 23drnginvrr 15631 . . . . . . . 8  |-  ( ( S  e.  DivRing  /\  ( G `  X )  e.  ( Base `  S
)  /\  ( G `  X )  =/=  .0.  )  ->  ( ( G `
 X )  .X.  ( I `  ( G `  X )
) )  =  ( 1r `  S ) )
3616, 20, 21, 35syl3anc 1182 . . . . . . 7  |-  ( ph  ->  ( ( G `  X )  .X.  (
I `  ( G `  X ) ) )  =  ( 1r `  S ) )
3736oveq1d 5960 . . . . . 6  |-  ( ph  ->  ( ( ( G `
 X )  .X.  ( I `  ( G `  X )
) )  .X.  ( E `  X )
)  =  ( ( 1r `  S ) 
.X.  ( E `  X ) ) )
384lmodrng 15734 . . . . . . . 8  |-  ( U  e.  LMod  ->  S  e. 
Ring )
3911, 38syl 15 . . . . . . 7  |-  ( ph  ->  S  e.  Ring )
4013, 28rngass 15456 . . . . . . 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 1184 . . . . . 6  |-  ( ph  ->  ( ( ( G `
 X )  .X.  ( I `  ( G `  X )
) )  .X.  ( E `  X )
)  =  ( ( G `  X ) 
.X.  ( ( I `
 ( G `  X ) )  .X.  ( E `  X ) ) ) )
4213, 28, 34rnglidm 15463 . . . . . . 7  |-  ( ( S  e.  Ring  /\  ( E `  X )  e.  ( Base `  S
) )  ->  (
( 1r `  S
)  .X.  ( E `  X ) )  =  ( E `  X
) )
4339, 27, 42syl2anc 642 . . . . . 6  |-  ( ph  ->  ( ( 1r `  S )  .X.  ( E `  X )
)  =  ( E `
 X ) )
4437, 41, 433eqtr3d 2398 . . . . 5  |-  ( ph  ->  ( ( G `  X )  .X.  (
( I `  ( G `  X )
)  .X.  ( E `  X ) ) )  =  ( E `  X ) )
4533, 44eqtrd 2390 . . . 4  |-  ( ph  ->  ( ( ( ( I `  ( G `
 X ) ) 
.X.  ( E `  X ) )  .x.  G ) `  X
)  =  ( E `
 X ) )
4645oveq2d 5961 . . 3  |-  ( ph  ->  ( ( E `  X ) ( -g `  S ) ( ( ( ( I `  ( G `  X ) )  .X.  ( E `  X ) )  .x.  G ) `  X
) )  =  ( ( E `  X
) ( -g `  S
) ( E `  X ) ) )
474lmodfgrp 15735 . . . . 5  |-  ( U  e.  LMod  ->  S  e. 
Grp )
4811, 47syl 15 . . . 4  |-  ( ph  ->  S  e.  Grp )
4913, 22, 5grpsubid 14649 . . . 4  |-  ( ( S  e.  Grp  /\  ( E `  X )  e.  ( Base `  S
) )  ->  (
( E `  X
) ( -g `  S
) ( E `  X ) )  =  .0.  )
5048, 27, 49syl2anc 642 . . 3  |-  ( ph  ->  ( ( E `  X ) ( -g `  S ) ( E `
 X ) )  =  .0.  )
5132, 46, 503eqtrd 2394 . 2  |-  ( ph  ->  ( ( E  .-  ( ( ( I `
 ( G `  X ) )  .X.  ( E `  X ) )  .x.  G ) ) `  X )  =  .0.  )
522, 51syl5eq 2402 1  |-  ( ph  ->  ( H `  X
)  =  .0.  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1642    e. wcel 1710    =/= wne 2521   ` cfv 5337  (class class class)co 5945   Basecbs 13245   .rcmulr 13306  Scalarcsca 13308   .scvsca 13309   0gc0g 13499   Grpcgrp 14461   -gcsg 14464   Ringcrg 15436   1rcur 15438   invrcinvr 15552   DivRingcdr 15611   LModclmod 15726   LVecclvec 15954  LFnlclfn 29316  LDualcld 29382
This theorem is referenced by:  lcfrlem3  31803
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-cnex 8883  ax-resscn 8884  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-addrcl 8888  ax-mulcl 8889  ax-mulrcl 8890  ax-mulcom 8891  ax-addass 8892  ax-mulass 8893  ax-distr 8894  ax-i2m1 8895  ax-1ne0 8896  ax-1rid 8897  ax-rnegex 8898  ax-rrecex 8899  ax-cnre 8900  ax-pre-lttri 8901  ax-pre-lttrn 8902  ax-pre-ltadd 8903  ax-pre-mulgt0 8904
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-int 3944  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-of 6165  df-1st 6209  df-2nd 6210  df-tpos 6321  df-riota 6391  df-recs 6475  df-rdg 6510  df-1o 6566  df-oadd 6570  df-er 6747  df-map 6862  df-en 6952  df-dom 6953  df-sdom 6954  df-fin 6955  df-pnf 8959  df-mnf 8960  df-xr 8961  df-ltxr 8962  df-le 8963  df-sub 9129  df-neg 9130  df-nn 9837  df-2 9894  df-3 9895  df-4 9896  df-5 9897  df-6 9898  df-n0 10058  df-z 10117  df-uz 10323  df-fz 10875  df-struct 13247  df-ndx 13248  df-slot 13249  df-base 13250  df-sets 13251  df-ress 13252  df-plusg 13318  df-mulr 13319  df-sca 13321  df-vsca 13322  df-0g 13503  df-mnd 14466  df-grp 14588  df-minusg 14589  df-sbg 14590  df-cmn 15190  df-abl 15191  df-mgp 15425  df-rng 15439  df-ur 15441  df-oppr 15504  df-dvdsr 15522  df-unit 15523  df-invr 15553  df-drng 15613  df-lmod 15728  df-lvec 15955  df-lfl 29317  df-ldual 29383
  Copyright terms: Public domain W3C validator