Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  lcfrlem2 Structured version   Unicode version

Theorem lcfrlem2 36340
Description: Lemma for lcfr 36382. (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 2467 . . . . . 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 17535 . . . . . . . . 9  |-  ( U  e.  LVec  ->  U  e. 
LMod )
107, 9syl 16 . . . . . . . 8  |-  ( ph  ->  U  e.  LMod )
111lmodrng 17303 . . . . . . . 8  |-  ( U  e.  LMod  ->  S  e. 
Ring )
1210, 11syl 16 . . . . . . 7  |-  ( ph  ->  S  e.  Ring )
131lvecdrng 17534 . . . . . . . . 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 33861 . . . . . . . . 9  |-  ( ( U  e.  LVec  /\  G  e.  F  /\  X  e.  V )  ->  ( G `  X )  e.  ( Base `  S
) )
187, 8, 15, 17syl3anc 1228 . . . . . . . 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 17196 . . . . . . . 8  |-  ( ( S  e.  DivRing  /\  ( G `  X )  e.  ( Base `  S
)  /\  ( G `  X )  =/=  .0.  )  ->  ( I `  ( G `  X ) )  e.  ( Base `  S ) )
2314, 18, 19, 22syl3anc 1228 . . . . . . 7  |-  ( ph  ->  ( I `  ( G `  X )
)  e.  ( Base `  S ) )
24 lcfrlem1.e . . . . . . . 8  |-  ( ph  ->  E  e.  F )
251, 2, 16, 3lflcl 33861 . . . . . . . 8  |-  ( ( U  e.  LVec  /\  E  e.  F  /\  X  e.  V )  ->  ( E `  X )  e.  ( Base `  S
) )
267, 24, 15, 25syl3anc 1228 . . . . . . 7  |-  ( ph  ->  ( E `  X
)  e.  ( Base `  S ) )
27 lcfrlem1.q . . . . . . . 8  |-  .X.  =  ( .r `  S )
282, 27rngcl 16999 . . . . . . 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 1228 . . . . . 6  |-  ( ph  ->  ( ( I `  ( G `  X ) )  .X.  ( E `  X ) )  e.  ( Base `  S
) )
301, 2, 3, 4, 5, 6, 7, 8, 29lkrss 33965 . . . . 5  |-  ( ph  ->  ( L `  G
)  C_  ( L `  ( ( ( I `
 ( G `  X ) )  .X.  ( E `  X ) )  .x.  G ) ) )
313, 1, 2, 5, 6, 10, 29, 8ldualvscl 33936 . . . . . 6  |-  ( ph  ->  ( ( ( I `
 ( G `  X ) )  .X.  ( E `  X ) )  .x.  G )  e.  F )
32 rnggrp 16991 . . . . . . . 8  |-  ( S  e.  Ring  ->  S  e. 
Grp )
3312, 32syl 16 . . . . . . 7  |-  ( ph  ->  S  e.  Grp )
34 eqid 2467 . . . . . . . . 9  |-  ( 1r
`  S )  =  ( 1r `  S
)
352, 34rngidcl 17006 . . . . . . . 8  |-  ( S  e.  Ring  ->  ( 1r
`  S )  e.  ( Base `  S
) )
3612, 35syl 16 . . . . . . 7  |-  ( ph  ->  ( 1r `  S
)  e.  ( Base `  S ) )
37 eqid 2467 . . . . . . . 8  |-  ( invg `  S )  =  ( invg `  S )
382, 37grpinvcl 15896 . . . . . . 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 33965 . . . . 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 3514 . . . 4  |-  ( ph  ->  ( L `  G
)  C_  ( L `  ( ( ( invg `  S ) `
 ( 1r `  S ) )  .x.  ( ( ( I `
 ( G `  X ) )  .X.  ( E `  X ) )  .x.  G ) ) ) )
42 sslin 3724 . . . 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 2467 . . . 4  |-  ( +g  `  D )  =  ( +g  `  D )
453, 1, 2, 5, 6, 10, 39, 31ldualvscl 33936 . . . 4  |-  ( ph  ->  ( ( ( invg `  S ) `
 ( 1r `  S ) )  .x.  ( ( ( I `
 ( G `  X ) )  .X.  ( E `  X ) )  .x.  G ) )  e.  F )
463, 4, 5, 44, 10, 24, 45lkrin 33961 . . 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 3514 . 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 5867 . . 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 33952 . . . 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 5868 . . 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 2521 . 2  |-  ( ph  ->  ( L `  ( E ( +g  `  D
) ( ( ( invg `  S
) `  ( 1r `  S ) )  .x.  ( ( ( I `
 ( G `  X ) )  .X.  ( E `  X ) )  .x.  G ) ) ) )  =  ( L `  H
) )
5447, 53sseqtrd 3540 1  |-  ( ph  ->  ( ( L `  E )  i^i  ( L `  G )
)  C_  ( L `  H ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767    =/= wne 2662    i^i cin 3475    C_ wss 3476   ` cfv 5586  (class class class)co 6282   Basecbs 14486   +g cplusg 14551   .rcmulr 14552  Scalarcsca 14554   .scvsca 14555   0gc0g 14691   Grpcgrp 15723   invgcminusg 15724   -gcsg 15726   1rcur 16943   Ringcrg 16986   invrcinvr 17104   DivRingcdr 17179   LModclmod 17295   LVecclvec 17531  LFnlclfn 33854  LKerclk 33882  LDualcld 33920
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-of 6522  df-om 6679  df-1st 6781  df-2nd 6782  df-tpos 6952  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-map 7419  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-n0 10792  df-z 10861  df-uz 11079  df-fz 11669  df-struct 14488  df-ndx 14489  df-slot 14490  df-base 14491  df-sets 14492  df-ress 14493  df-plusg 14564  df-mulr 14565  df-sca 14567  df-vsca 14568  df-0g 14693  df-mnd 15728  df-grp 15858  df-minusg 15859  df-sbg 15860  df-cmn 16596  df-abl 16597  df-mgp 16932  df-ur 16944  df-rng 16988  df-oppr 17056  df-dvdsr 17074  df-unit 17075  df-invr 17105  df-drng 17181  df-lmod 17297  df-lss 17362  df-lvec 17532  df-lfl 33855  df-lkr 33883  df-ldual 33921
This theorem is referenced by:  lcfrlem35  36374
  Copyright terms: Public domain W3C validator