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Theorem lcfrlem2 35497
Description: Lemma for lcfr 35539. (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 2451 . . . . . 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 17302 . . . . . . . . 9  |-  ( U  e.  LVec  ->  U  e. 
LMod )
107, 9syl 16 . . . . . . . 8  |-  ( ph  ->  U  e.  LMod )
111lmodrng 17071 . . . . . . . 8  |-  ( U  e.  LMod  ->  S  e. 
Ring )
1210, 11syl 16 . . . . . . 7  |-  ( ph  ->  S  e.  Ring )
131lvecdrng 17301 . . . . . . . . 9  |-  ( U  e.  LVec  ->  S  e.  DivRing )
147, 13syl 16 . . . . . . . 8  |-  ( ph  ->  S  e.  DivRing )
15 lcfrlem1.x . . . . . . . . 9  |-  ( ph  ->  X  e.  V )
16 lcfrlem1.v . . . . . . . . . 10  |-  V  =  ( Base `  U
)
171, 2, 16, 3lflcl 33018 . . . . . . . . 9  |-  ( ( U  e.  LVec  /\  G  e.  F  /\  X  e.  V )  ->  ( G `  X )  e.  ( Base `  S
) )
187, 8, 15, 17syl3anc 1219 . . . . . . . 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 16964 . . . . . . . 8  |-  ( ( S  e.  DivRing  /\  ( G `  X )  e.  ( Base `  S
)  /\  ( G `  X )  =/=  .0.  )  ->  ( I `  ( G `  X ) )  e.  ( Base `  S ) )
2314, 18, 19, 22syl3anc 1219 . . . . . . 7  |-  ( ph  ->  ( I `  ( G `  X )
)  e.  ( Base `  S ) )
24 lcfrlem1.e . . . . . . . 8  |-  ( ph  ->  E  e.  F )
251, 2, 16, 3lflcl 33018 . . . . . . . 8  |-  ( ( U  e.  LVec  /\  E  e.  F  /\  X  e.  V )  ->  ( E `  X )  e.  ( Base `  S
) )
267, 24, 15, 25syl3anc 1219 . . . . . . 7  |-  ( ph  ->  ( E `  X
)  e.  ( Base `  S ) )
27 lcfrlem1.q . . . . . . . 8  |-  .X.  =  ( .r `  S )
282, 27rngcl 16773 . . . . . . 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 1219 . . . . . 6  |-  ( ph  ->  ( ( I `  ( G `  X ) )  .X.  ( E `  X ) )  e.  ( Base `  S
) )
301, 2, 3, 4, 5, 6, 7, 8, 29lkrss 33122 . . . . 5  |-  ( ph  ->  ( L `  G
)  C_  ( L `  ( ( ( I `
 ( G `  X ) )  .X.  ( E `  X ) )  .x.  G ) ) )
313, 1, 2, 5, 6, 10, 29, 8ldualvscl 33093 . . . . . 6  |-  ( ph  ->  ( ( ( I `
 ( G `  X ) )  .X.  ( E `  X ) )  .x.  G )  e.  F )
32 rnggrp 16765 . . . . . . . 8  |-  ( S  e.  Ring  ->  S  e. 
Grp )
3312, 32syl 16 . . . . . . 7  |-  ( ph  ->  S  e.  Grp )
34 eqid 2451 . . . . . . . . 9  |-  ( 1r
`  S )  =  ( 1r `  S
)
352, 34rngidcl 16780 . . . . . . . 8  |-  ( S  e.  Ring  ->  ( 1r
`  S )  e.  ( Base `  S
) )
3612, 35syl 16 . . . . . . 7  |-  ( ph  ->  ( 1r `  S
)  e.  ( Base `  S ) )
37 eqid 2451 . . . . . . . 8  |-  ( invg `  S )  =  ( invg `  S )
382, 37grpinvcl 15694 . . . . . . 7  |-  ( ( S  e.  Grp  /\  ( 1r `  S )  e.  ( Base `  S
) )  ->  (
( invg `  S ) `  ( 1r `  S ) )  e.  ( Base `  S
) )
3933, 36, 38syl2anc 661 . . . . . 6  |-  ( ph  ->  ( ( invg `  S ) `  ( 1r `  S ) )  e.  ( Base `  S
) )
401, 2, 3, 4, 5, 6, 7, 31, 39lkrss 33122 . . . . 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 3467 . . . 4  |-  ( ph  ->  ( L `  G
)  C_  ( L `  ( ( ( invg `  S ) `
 ( 1r `  S ) )  .x.  ( ( ( I `
 ( G `  X ) )  .X.  ( E `  X ) )  .x.  G ) ) ) )
42 sslin 3677 . . . 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 16 . . 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 2451 . . . 4  |-  ( +g  `  D )  =  ( +g  `  D )
453, 1, 2, 5, 6, 10, 39, 31ldualvscl 33093 . . . 4  |-  ( ph  ->  ( ( ( invg `  S ) `
 ( 1r `  S ) )  .x.  ( ( ( I `
 ( G `  X ) )  .X.  ( E `  X ) )  .x.  G ) )  e.  F )
463, 4, 5, 44, 10, 24, 45lkrin 33118 . . 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 3467 . 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 5795 . . 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 33109 . . . 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 5796 . . 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 2505 . 2  |-  ( ph  ->  ( L `  ( E ( +g  `  D
) ( ( ( invg `  S
) `  ( 1r `  S ) )  .x.  ( ( ( I `
 ( G `  X ) )  .X.  ( E `  X ) )  .x.  G ) ) ) )  =  ( L `  H
) )
5447, 53sseqtrd 3493 1  |-  ( ph  ->  ( ( L `  E )  i^i  ( L `  G )
)  C_  ( L `  H ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758    =/= wne 2644    i^i cin 3428    C_ wss 3429   ` cfv 5519  (class class class)co 6193   Basecbs 14285   +g cplusg 14349   .rcmulr 14350  Scalarcsca 14352   .scvsca 14353   0gc0g 14489   Grpcgrp 15521   invgcminusg 15522   -gcsg 15524   1rcur 16717   Ringcrg 16760   invrcinvr 16878   DivRingcdr 16947   LModclmod 17063   LVecclvec 17298  LFnlclfn 33011  LKerclk 33039  LDualcld 33077
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-int 4230  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-of 6423  df-om 6580  df-1st 6680  df-2nd 6681  df-tpos 6848  df-recs 6935  df-rdg 6969  df-1o 7023  df-oadd 7027  df-er 7204  df-map 7319  df-en 7414  df-dom 7415  df-sdom 7416  df-fin 7417  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-nn 10427  df-2 10484  df-3 10485  df-4 10486  df-5 10487  df-6 10488  df-n0 10684  df-z 10751  df-uz 10966  df-fz 11548  df-struct 14287  df-ndx 14288  df-slot 14289  df-base 14290  df-sets 14291  df-ress 14292  df-plusg 14362  df-mulr 14363  df-sca 14365  df-vsca 14366  df-0g 14491  df-mnd 15526  df-grp 15656  df-minusg 15657  df-sbg 15658  df-cmn 16392  df-abl 16393  df-mgp 16706  df-ur 16718  df-rng 16762  df-oppr 16830  df-dvdsr 16848  df-unit 16849  df-invr 16879  df-drng 16949  df-lmod 17065  df-lss 17129  df-lvec 17299  df-lfl 33012  df-lkr 33040  df-ldual 33078
This theorem is referenced by:  lcfrlem35  35531
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