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Theorem lcfrlem2 31733
Description: Lemma for lcfr 31775. (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 2283 . . . . . 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 15859 . . . . . . . . 9  |-  ( U  e.  LVec  ->  U  e. 
LMod )
107, 9syl 15 . . . . . . . 8  |-  ( ph  ->  U  e.  LMod )
111lmodrng 15635 . . . . . . . 8  |-  ( U  e.  LMod  ->  S  e. 
Ring )
1210, 11syl 15 . . . . . . 7  |-  ( ph  ->  S  e.  Ring )
131lvecdrng 15858 . . . . . . . . 9  |-  ( U  e.  LVec  ->  S  e.  DivRing )
147, 13syl 15 . . . . . . . 8  |-  ( ph  ->  S  e.  DivRing )
15 lcfrlem1.x . . . . . . . . 9  |-  ( ph  ->  X  e.  V )
16 lcfrlem1.v . . . . . . . . . 10  |-  V  =  ( Base `  U
)
171, 2, 16, 3lflcl 29254 . . . . . . . . 9  |-  ( ( U  e.  LVec  /\  G  e.  F  /\  X  e.  V )  ->  ( G `  X )  e.  ( Base `  S
) )
187, 8, 15, 17syl3anc 1182 . . . . . . . 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 15529 . . . . . . . 8  |-  ( ( S  e.  DivRing  /\  ( G `  X )  e.  ( Base `  S
)  /\  ( G `  X )  =/=  .0.  )  ->  ( I `  ( G `  X ) )  e.  ( Base `  S ) )
2314, 18, 19, 22syl3anc 1182 . . . . . . 7  |-  ( ph  ->  ( I `  ( G `  X )
)  e.  ( Base `  S ) )
24 lcfrlem1.e . . . . . . . 8  |-  ( ph  ->  E  e.  F )
251, 2, 16, 3lflcl 29254 . . . . . . . 8  |-  ( ( U  e.  LVec  /\  E  e.  F  /\  X  e.  V )  ->  ( E `  X )  e.  ( Base `  S
) )
267, 24, 15, 25syl3anc 1182 . . . . . . 7  |-  ( ph  ->  ( E `  X
)  e.  ( Base `  S ) )
27 lcfrlem1.q . . . . . . . 8  |-  .X.  =  ( .r `  S )
282, 27rngcl 15354 . . . . . . 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 1182 . . . . . 6  |-  ( ph  ->  ( ( I `  ( G `  X ) )  .X.  ( E `  X ) )  e.  ( Base `  S
) )
301, 2, 3, 4, 5, 6, 7, 8, 29lkrss 29358 . . . . 5  |-  ( ph  ->  ( L `  G
)  C_  ( L `  ( ( ( I `
 ( G `  X ) )  .X.  ( E `  X ) )  .x.  G ) ) )
313, 1, 2, 5, 6, 10, 29, 8ldualvscl 29329 . . . . . 6  |-  ( ph  ->  ( ( ( I `
 ( G `  X ) )  .X.  ( E `  X ) )  .x.  G )  e.  F )
32 rnggrp 15346 . . . . . . . 8  |-  ( S  e.  Ring  ->  S  e. 
Grp )
3312, 32syl 15 . . . . . . 7  |-  ( ph  ->  S  e.  Grp )
34 eqid 2283 . . . . . . . . 9  |-  ( 1r
`  S )  =  ( 1r `  S
)
352, 34rngidcl 15361 . . . . . . . 8  |-  ( S  e.  Ring  ->  ( 1r
`  S )  e.  ( Base `  S
) )
3612, 35syl 15 . . . . . . 7  |-  ( ph  ->  ( 1r `  S
)  e.  ( Base `  S ) )
37 eqid 2283 . . . . . . . 8  |-  ( inv g `  S )  =  ( inv g `  S )
382, 37grpinvcl 14527 . . . . . . 7  |-  ( ( S  e.  Grp  /\  ( 1r `  S )  e.  ( Base `  S
) )  ->  (
( inv g `  S ) `  ( 1r `  S ) )  e.  ( Base `  S
) )
3933, 36, 38syl2anc 642 . . . . . 6  |-  ( ph  ->  ( ( inv g `  S ) `  ( 1r `  S ) )  e.  ( Base `  S
) )
401, 2, 3, 4, 5, 6, 7, 31, 39lkrss 29358 . . . . 5  |-  ( ph  ->  ( L `  (
( ( I `  ( G `  X ) )  .X.  ( E `  X ) )  .x.  G ) )  C_  ( L `  ( ( ( inv g `  S ) `  ( 1r `  S ) ) 
.x.  ( ( ( I `  ( G `
 X ) ) 
.X.  ( E `  X ) )  .x.  G ) ) ) )
4130, 40sstrd 3189 . . . 4  |-  ( ph  ->  ( L `  G
)  C_  ( L `  ( ( ( inv g `  S ) `
 ( 1r `  S ) )  .x.  ( ( ( I `
 ( G `  X ) )  .X.  ( E `  X ) )  .x.  G ) ) ) )
42 sslin 3395 . . . 4  |-  ( ( L `  G ) 
C_  ( L `  ( ( ( inv g `  S ) `
 ( 1r `  S ) )  .x.  ( ( ( I `
 ( G `  X ) )  .X.  ( E `  X ) )  .x.  G ) ) )  ->  (
( L `  E
)  i^i  ( L `  G ) )  C_  ( ( L `  E )  i^i  ( L `  ( (
( inv g `  S ) `  ( 1r `  S ) ) 
.x.  ( ( ( I `  ( G `
 X ) ) 
.X.  ( E `  X ) )  .x.  G ) ) ) ) )
4341, 42syl 15 . . 3  |-  ( ph  ->  ( ( L `  E )  i^i  ( L `  G )
)  C_  ( ( L `  E )  i^i  ( L `  (
( ( inv g `  S ) `  ( 1r `  S ) ) 
.x.  ( ( ( I `  ( G `
 X ) ) 
.X.  ( E `  X ) )  .x.  G ) ) ) ) )
44 eqid 2283 . . . 4  |-  ( +g  `  D )  =  ( +g  `  D )
453, 1, 2, 5, 6, 10, 39, 31ldualvscl 29329 . . . 4  |-  ( ph  ->  ( ( ( inv g `  S ) `
 ( 1r `  S ) )  .x.  ( ( ( I `
 ( G `  X ) )  .X.  ( E `  X ) )  .x.  G ) )  e.  F )
463, 4, 5, 44, 10, 24, 45lkrin 29354 . . 3  |-  ( ph  ->  ( ( L `  E )  i^i  ( L `  ( (
( inv g `  S ) `  ( 1r `  S ) ) 
.x.  ( ( ( I `  ( G `
 X ) ) 
.X.  ( E `  X ) )  .x.  G ) ) ) )  C_  ( L `  ( E ( +g  `  D ) ( ( ( inv g `  S ) `  ( 1r `  S ) ) 
.x.  ( ( ( I `  ( G `
 X ) ) 
.X.  ( E `  X ) )  .x.  G ) ) ) ) )
4743, 46sstrd 3189 . 2  |-  ( ph  ->  ( ( L `  E )  i^i  ( L `  G )
)  C_  ( L `  ( E ( +g  `  D ) ( ( ( inv g `  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 5528 . . 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 29345 . . . 4  |-  ( ph  ->  ( E  .-  (
( ( I `  ( G `  X ) )  .X.  ( E `  X ) )  .x.  G ) )  =  ( E ( +g  `  D ) ( ( ( inv g `  S ) `  ( 1r `  S ) ) 
.x.  ( ( ( I `  ( G `
 X ) ) 
.X.  ( E `  X ) )  .x.  G ) ) ) )
5251fveq2d 5529 . . 3  |-  ( ph  ->  ( L `  ( E  .-  ( ( ( I `  ( G `
 X ) ) 
.X.  ( E `  X ) )  .x.  G ) ) )  =  ( L `  ( E ( +g  `  D
) ( ( ( inv g `  S
) `  ( 1r `  S ) )  .x.  ( ( ( I `
 ( G `  X ) )  .X.  ( E `  X ) )  .x.  G ) ) ) ) )
5349, 52syl5req 2328 . 2  |-  ( ph  ->  ( L `  ( E ( +g  `  D
) ( ( ( inv g `  S
) `  ( 1r `  S ) )  .x.  ( ( ( I `
 ( G `  X ) )  .X.  ( E `  X ) )  .x.  G ) ) ) )  =  ( L `  H
) )
5447, 53sseqtrd 3214 1  |-  ( ph  ->  ( ( L `  E )  i^i  ( L `  G )
)  C_  ( L `  H ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684    =/= wne 2446    i^i cin 3151    C_ wss 3152   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208   .rcmulr 13209  Scalarcsca 13211   .scvsca 13212   0gc0g 13400   Grpcgrp 14362   inv gcminusg 14363   -gcsg 14365   Ringcrg 15337   1rcur 15339   invrcinvr 15453   DivRingcdr 15512   LModclmod 15627   LVecclvec 15855  LFnlclfn 29247  LKerclk 29275  LDualcld 29313
This theorem is referenced by:  lcfrlem35  31767
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-tpos 6234  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-sca 13224  df-vsca 13225  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490  df-sbg 14491  df-cmn 15091  df-abl 15092  df-mgp 15326  df-rng 15340  df-ur 15342  df-oppr 15405  df-dvdsr 15423  df-unit 15424  df-invr 15454  df-drng 15514  df-lmod 15629  df-lss 15690  df-lvec 15856  df-lfl 29248  df-lkr 29276  df-ldual 29314
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