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Theorem lcfrlem2 34544
Description: Lemma for lcfr 34586. (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
lcfrlem2  |-  ( ph  ->  ( ( L `  E )  i^i  ( L `  G )
)  C_  ( L `  H ) )

Proof of Theorem lcfrlem2
StepHypRef Expression
1 lcfrlem1.s . . . . . 6  |-  S  =  (Scalar `  U )
2 eqid 2402 . . . . . 6  |-  ( Base `  S )  =  (
Base `  S )
3 lcfrlem1.f . . . . . 6  |-  F  =  (LFnl `  U )
4 lcfrlem2.l . . . . . 6  |-  L  =  (LKer `  U )
5 lcfrlem1.d . . . . . 6  |-  D  =  (LDual `  U )
6 lcfrlem1.t . . . . . 6  |-  .x.  =  ( .s `  D )
7 lcfrlem1.u . . . . . 6  |-  ( ph  ->  U  e.  LVec )
8 lcfrlem1.g . . . . . 6  |-  ( ph  ->  G  e.  F )
9 lveclmod 17964 . . . . . . . . 9  |-  ( U  e.  LVec  ->  U  e. 
LMod )
107, 9syl 17 . . . . . . . 8  |-  ( ph  ->  U  e.  LMod )
111lmodring 17732 . . . . . . . 8  |-  ( U  e.  LMod  ->  S  e. 
Ring )
1210, 11syl 17 . . . . . . 7  |-  ( ph  ->  S  e.  Ring )
131lvecdrng 17963 . . . . . . . . 9  |-  ( U  e.  LVec  ->  S  e.  DivRing )
147, 13syl 17 . . . . . . . 8  |-  ( ph  ->  S  e.  DivRing )
15 lcfrlem1.x . . . . . . . . 9  |-  ( ph  ->  X  e.  V )
16 lcfrlem1.v . . . . . . . . . 10  |-  V  =  ( Base `  U
)
171, 2, 16, 3lflcl 32063 . . . . . . . . 9  |-  ( ( U  e.  LVec  /\  G  e.  F  /\  X  e.  V )  ->  ( G `  X )  e.  ( Base `  S
) )
187, 8, 15, 17syl3anc 1230 . . . . . . . 8  |-  ( ph  ->  ( G `  X
)  e.  ( Base `  S ) )
19 lcfrlem1.n . . . . . . . 8  |-  ( ph  ->  ( G `  X
)  =/=  .0.  )
20 lcfrlem1.z . . . . . . . . 9  |-  .0.  =  ( 0g `  S )
21 lcfrlem1.i . . . . . . . . 9  |-  I  =  ( invr `  S
)
222, 20, 21drnginvrcl 17625 . . . . . . . 8  |-  ( ( S  e.  DivRing  /\  ( G `  X )  e.  ( Base `  S
)  /\  ( G `  X )  =/=  .0.  )  ->  ( I `  ( G `  X ) )  e.  ( Base `  S ) )
2314, 18, 19, 22syl3anc 1230 . . . . . . 7  |-  ( ph  ->  ( I `  ( G `  X )
)  e.  ( Base `  S ) )
24 lcfrlem1.e . . . . . . . 8  |-  ( ph  ->  E  e.  F )
251, 2, 16, 3lflcl 32063 . . . . . . . 8  |-  ( ( U  e.  LVec  /\  E  e.  F  /\  X  e.  V )  ->  ( E `  X )  e.  ( Base `  S
) )
267, 24, 15, 25syl3anc 1230 . . . . . . 7  |-  ( ph  ->  ( E `  X
)  e.  ( Base `  S ) )
27 lcfrlem1.q . . . . . . . 8  |-  .X.  =  ( .r `  S )
282, 27ringcl 17424 . . . . . . 7  |-  ( ( S  e.  Ring  /\  (
I `  ( G `  X ) )  e.  ( Base `  S
)  /\  ( E `  X )  e.  (
Base `  S )
)  ->  ( (
I `  ( G `  X ) )  .X.  ( E `  X ) )  e.  ( Base `  S ) )
2912, 23, 26, 28syl3anc 1230 . . . . . 6  |-  ( ph  ->  ( ( I `  ( G `  X ) )  .X.  ( E `  X ) )  e.  ( Base `  S
) )
301, 2, 3, 4, 5, 6, 7, 8, 29lkrss 32167 . . . . 5  |-  ( ph  ->  ( L `  G
)  C_  ( L `  ( ( ( I `
 ( G `  X ) )  .X.  ( E `  X ) )  .x.  G ) ) )
313, 1, 2, 5, 6, 10, 29, 8ldualvscl 32138 . . . . . 6  |-  ( ph  ->  ( ( ( I `
 ( G `  X ) )  .X.  ( E `  X ) )  .x.  G )  e.  F )
32 ringgrp 17415 . . . . . . . 8  |-  ( S  e.  Ring  ->  S  e. 
Grp )
3312, 32syl 17 . . . . . . 7  |-  ( ph  ->  S  e.  Grp )
34 eqid 2402 . . . . . . . . 9  |-  ( 1r
`  S )  =  ( 1r `  S
)
352, 34ringidcl 17431 . . . . . . . 8  |-  ( S  e.  Ring  ->  ( 1r
`  S )  e.  ( Base `  S
) )
3612, 35syl 17 . . . . . . 7  |-  ( ph  ->  ( 1r `  S
)  e.  ( Base `  S ) )
37 eqid 2402 . . . . . . . 8  |-  ( invg `  S )  =  ( invg `  S )
382, 37grpinvcl 16311 . . . . . . 7  |-  ( ( S  e.  Grp  /\  ( 1r `  S )  e.  ( Base `  S
) )  ->  (
( invg `  S ) `  ( 1r `  S ) )  e.  ( Base `  S
) )
3933, 36, 38syl2anc 659 . . . . . 6  |-  ( ph  ->  ( ( invg `  S ) `  ( 1r `  S ) )  e.  ( Base `  S
) )
401, 2, 3, 4, 5, 6, 7, 31, 39lkrss 32167 . . . . 5  |-  ( ph  ->  ( L `  (
( ( I `  ( G `  X ) )  .X.  ( E `  X ) )  .x.  G ) )  C_  ( L `  ( ( ( invg `  S ) `  ( 1r `  S ) ) 
.x.  ( ( ( I `  ( G `
 X ) ) 
.X.  ( E `  X ) )  .x.  G ) ) ) )
4130, 40sstrd 3451 . . . 4  |-  ( ph  ->  ( L `  G
)  C_  ( L `  ( ( ( invg `  S ) `
 ( 1r `  S ) )  .x.  ( ( ( I `
 ( G `  X ) )  .X.  ( E `  X ) )  .x.  G ) ) ) )
42 sslin 3664 . . . 4  |-  ( ( L `  G ) 
C_  ( L `  ( ( ( invg `  S ) `
 ( 1r `  S ) )  .x.  ( ( ( I `
 ( G `  X ) )  .X.  ( E `  X ) )  .x.  G ) ) )  ->  (
( L `  E
)  i^i  ( L `  G ) )  C_  ( ( L `  E )  i^i  ( L `  ( (
( invg `  S ) `  ( 1r `  S ) ) 
.x.  ( ( ( I `  ( G `
 X ) ) 
.X.  ( E `  X ) )  .x.  G ) ) ) ) )
4341, 42syl 17 . . 3  |-  ( ph  ->  ( ( L `  E )  i^i  ( L `  G )
)  C_  ( ( L `  E )  i^i  ( L `  (
( ( invg `  S ) `  ( 1r `  S ) ) 
.x.  ( ( ( I `  ( G `
 X ) ) 
.X.  ( E `  X ) )  .x.  G ) ) ) ) )
44 eqid 2402 . . . 4  |-  ( +g  `  D )  =  ( +g  `  D )
453, 1, 2, 5, 6, 10, 39, 31ldualvscl 32138 . . . 4  |-  ( ph  ->  ( ( ( invg `  S ) `
 ( 1r `  S ) )  .x.  ( ( ( I `
 ( G `  X ) )  .X.  ( E `  X ) )  .x.  G ) )  e.  F )
463, 4, 5, 44, 10, 24, 45lkrin 32163 . . 3  |-  ( ph  ->  ( ( L `  E )  i^i  ( L `  ( (
( invg `  S ) `  ( 1r `  S ) ) 
.x.  ( ( ( I `  ( G `
 X ) ) 
.X.  ( E `  X ) )  .x.  G ) ) ) )  C_  ( L `  ( E ( +g  `  D ) ( ( ( invg `  S ) `  ( 1r `  S ) ) 
.x.  ( ( ( I `  ( G `
 X ) ) 
.X.  ( E `  X ) )  .x.  G ) ) ) ) )
4743, 46sstrd 3451 . 2  |-  ( ph  ->  ( ( L `  E )  i^i  ( L `  G )
)  C_  ( L `  ( E ( +g  `  D ) ( ( ( invg `  S ) `  ( 1r `  S ) ) 
.x.  ( ( ( I `  ( G `
 X ) ) 
.X.  ( E `  X ) )  .x.  G ) ) ) ) )
48 lcfrlem1.h . . . 4  |-  H  =  ( E  .-  (
( ( I `  ( G `  X ) )  .X.  ( E `  X ) )  .x.  G ) )
4948fveq2i 5808 . . 3  |-  ( L `
 H )  =  ( L `  ( E  .-  ( ( ( I `  ( G `
 X ) ) 
.X.  ( E `  X ) )  .x.  G ) ) )
50 lcfrlem1.m . . . . 5  |-  .-  =  ( -g `  D )
511, 37, 34, 3, 5, 44, 6, 50, 10, 24, 31ldualvsub 32154 . . . 4  |-  ( ph  ->  ( E  .-  (
( ( I `  ( G `  X ) )  .X.  ( E `  X ) )  .x.  G ) )  =  ( E ( +g  `  D ) ( ( ( invg `  S ) `  ( 1r `  S ) ) 
.x.  ( ( ( I `  ( G `
 X ) ) 
.X.  ( E `  X ) )  .x.  G ) ) ) )
5251fveq2d 5809 . . 3  |-  ( ph  ->  ( L `  ( E  .-  ( ( ( I `  ( G `
 X ) ) 
.X.  ( E `  X ) )  .x.  G ) ) )  =  ( L `  ( E ( +g  `  D
) ( ( ( invg `  S
) `  ( 1r `  S ) )  .x.  ( ( ( I `
 ( G `  X ) )  .X.  ( E `  X ) )  .x.  G ) ) ) ) )
5349, 52syl5req 2456 . 2  |-  ( ph  ->  ( L `  ( E ( +g  `  D
) ( ( ( invg `  S
) `  ( 1r `  S ) )  .x.  ( ( ( I `
 ( G `  X ) )  .X.  ( E `  X ) )  .x.  G ) ) ) )  =  ( L `  H
) )
5447, 53sseqtrd 3477 1  |-  ( ph  ->  ( ( L `  E )  i^i  ( L `  G )
)  C_  ( L `  H ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1405    e. wcel 1842    =/= wne 2598    i^i cin 3412    C_ wss 3413   ` cfv 5525  (class class class)co 6234   Basecbs 14733   +g cplusg 14801   .rcmulr 14802  Scalarcsca 14804   .scvsca 14805   0gc0g 14946   Grpcgrp 16269   invgcminusg 16270   -gcsg 16271   1rcur 17365   Ringcrg 17410   invrcinvr 17532   DivRingcdr 17608   LModclmod 17724   LVecclvec 17960  LFnlclfn 32056  LKerclk 32084  LDualcld 32122
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530  ax-cnex 9498  ax-resscn 9499  ax-1cn 9500  ax-icn 9501  ax-addcl 9502  ax-addrcl 9503  ax-mulcl 9504  ax-mulrcl 9505  ax-mulcom 9506  ax-addass 9507  ax-mulass 9508  ax-distr 9509  ax-i2m1 9510  ax-1ne0 9511  ax-1rid 9512  ax-rnegex 9513  ax-rrecex 9514  ax-cnre 9515  ax-pre-lttri 9516  ax-pre-lttrn 9517  ax-pre-ltadd 9518  ax-pre-mulgt0 9519
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-ord 4824  df-on 4825  df-lim 4826  df-suc 4827  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6196  df-ov 6237  df-oprab 6238  df-mpt2 6239  df-of 6477  df-om 6639  df-1st 6738  df-2nd 6739  df-tpos 6912  df-recs 6999  df-rdg 7033  df-1o 7087  df-oadd 7091  df-er 7268  df-map 7379  df-en 7475  df-dom 7476  df-sdom 7477  df-fin 7478  df-pnf 9580  df-mnf 9581  df-xr 9582  df-ltxr 9583  df-le 9584  df-sub 9763  df-neg 9764  df-nn 10497  df-2 10555  df-3 10556  df-4 10557  df-5 10558  df-6 10559  df-n0 10757  df-z 10826  df-uz 11046  df-fz 11644  df-struct 14735  df-ndx 14736  df-slot 14737  df-base 14738  df-sets 14739  df-ress 14740  df-plusg 14814  df-mulr 14815  df-sca 14817  df-vsca 14818  df-0g 14948  df-mgm 16088  df-sgrp 16127  df-mnd 16137  df-grp 16273  df-minusg 16274  df-sbg 16275  df-cmn 17016  df-abl 17017  df-mgp 17354  df-ur 17366  df-ring 17412  df-oppr 17484  df-dvdsr 17502  df-unit 17503  df-invr 17533  df-drng 17610  df-lmod 17726  df-lss 17791  df-lvec 17961  df-lfl 32057  df-lkr 32085  df-ldual 32123
This theorem is referenced by:  lcfrlem35  34578
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