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Theorem lclkrlem2m 31709
Description: Lemma for lclkr 31723. Construct a vector  B that makes the sum of functionals zero. Combine with  B  e.  V to shorten overall proof. (Contributed by NM, 17-Jan-2015.)
Hypotheses
Ref Expression
lclkrlem2m.v  |-  V  =  ( Base `  U
)
lclkrlem2m.t  |-  .x.  =  ( .s `  U )
lclkrlem2m.s  |-  S  =  (Scalar `  U )
lclkrlem2m.q  |-  .X.  =  ( .r `  S )
lclkrlem2m.z  |-  .0.  =  ( 0g `  S )
lclkrlem2m.i  |-  I  =  ( invr `  S
)
lclkrlem2m.m  |-  .-  =  ( -g `  U )
lclkrlem2m.f  |-  F  =  (LFnl `  U )
lclkrlem2m.d  |-  D  =  (LDual `  U )
lclkrlem2m.p  |-  .+  =  ( +g  `  D )
lclkrlem2m.x  |-  ( ph  ->  X  e.  V )
lclkrlem2m.y  |-  ( ph  ->  Y  e.  V )
lclkrlem2m.e  |-  ( ph  ->  E  e.  F )
lclkrlem2m.g  |-  ( ph  ->  G  e.  F )
lclkrlem2m.w  |-  ( ph  ->  U  e.  LVec )
lclkrlem2m.b  |-  B  =  ( X  .-  (
( ( ( E 
.+  G ) `  X )  .X.  (
I `  ( ( E  .+  G ) `  Y ) ) ) 
.x.  Y ) )
lclkrlem2m.n  |-  ( ph  ->  ( ( E  .+  G ) `  Y
)  =/=  .0.  )
Assertion
Ref Expression
lclkrlem2m  |-  ( ph  ->  ( B  e.  V  /\  ( ( E  .+  G ) `  B
)  =  .0.  )
)

Proof of Theorem lclkrlem2m
StepHypRef Expression
1 lclkrlem2m.b . . 3  |-  B  =  ( X  .-  (
( ( ( E 
.+  G ) `  X )  .X.  (
I `  ( ( E  .+  G ) `  Y ) ) ) 
.x.  Y ) )
2 lclkrlem2m.w . . . . . 6  |-  ( ph  ->  U  e.  LVec )
3 lveclmod 15859 . . . . . 6  |-  ( U  e.  LVec  ->  U  e. 
LMod )
42, 3syl 15 . . . . 5  |-  ( ph  ->  U  e.  LMod )
5 lmodgrp 15634 . . . . 5  |-  ( U  e.  LMod  ->  U  e. 
Grp )
64, 5syl 15 . . . 4  |-  ( ph  ->  U  e.  Grp )
7 lclkrlem2m.x . . . 4  |-  ( ph  ->  X  e.  V )
8 lclkrlem2m.s . . . . . . . 8  |-  S  =  (Scalar `  U )
98lmodrng 15635 . . . . . . 7  |-  ( U  e.  LMod  ->  S  e. 
Ring )
104, 9syl 15 . . . . . 6  |-  ( ph  ->  S  e.  Ring )
11 lclkrlem2m.f . . . . . . . 8  |-  F  =  (LFnl `  U )
12 lclkrlem2m.d . . . . . . . 8  |-  D  =  (LDual `  U )
13 lclkrlem2m.p . . . . . . . 8  |-  .+  =  ( +g  `  D )
14 lclkrlem2m.e . . . . . . . 8  |-  ( ph  ->  E  e.  F )
15 lclkrlem2m.g . . . . . . . 8  |-  ( ph  ->  G  e.  F )
1611, 12, 13, 4, 14, 15ldualvaddcl 29320 . . . . . . 7  |-  ( ph  ->  ( E  .+  G
)  e.  F )
17 eqid 2283 . . . . . . . 8  |-  ( Base `  S )  =  (
Base `  S )
18 lclkrlem2m.v . . . . . . . 8  |-  V  =  ( Base `  U
)
198, 17, 18, 11lflcl 29254 . . . . . . 7  |-  ( ( U  e.  LVec  /\  ( E  .+  G )  e.  F  /\  X  e.  V )  ->  (
( E  .+  G
) `  X )  e.  ( Base `  S
) )
202, 16, 7, 19syl3anc 1182 . . . . . 6  |-  ( ph  ->  ( ( E  .+  G ) `  X
)  e.  ( Base `  S ) )
218lvecdrng 15858 . . . . . . . 8  |-  ( U  e.  LVec  ->  S  e.  DivRing )
222, 21syl 15 . . . . . . 7  |-  ( ph  ->  S  e.  DivRing )
23 lclkrlem2m.y . . . . . . . 8  |-  ( ph  ->  Y  e.  V )
248, 17, 18, 11lflcl 29254 . . . . . . . 8  |-  ( ( U  e.  LVec  /\  ( E  .+  G )  e.  F  /\  Y  e.  V )  ->  (
( E  .+  G
) `  Y )  e.  ( Base `  S
) )
252, 16, 23, 24syl3anc 1182 . . . . . . 7  |-  ( ph  ->  ( ( E  .+  G ) `  Y
)  e.  ( Base `  S ) )
26 lclkrlem2m.n . . . . . . 7  |-  ( ph  ->  ( ( E  .+  G ) `  Y
)  =/=  .0.  )
27 lclkrlem2m.z . . . . . . . 8  |-  .0.  =  ( 0g `  S )
28 lclkrlem2m.i . . . . . . . 8  |-  I  =  ( invr `  S
)
2917, 27, 28drnginvrcl 15529 . . . . . . 7  |-  ( ( S  e.  DivRing  /\  (
( E  .+  G
) `  Y )  e.  ( Base `  S
)  /\  ( ( E  .+  G ) `  Y )  =/=  .0.  )  ->  ( I `  ( ( E  .+  G ) `  Y
) )  e.  (
Base `  S )
)
3022, 25, 26, 29syl3anc 1182 . . . . . 6  |-  ( ph  ->  ( I `  (
( E  .+  G
) `  Y )
)  e.  ( Base `  S ) )
31 lclkrlem2m.q . . . . . . 7  |-  .X.  =  ( .r `  S )
3217, 31rngcl 15354 . . . . . 6  |-  ( ( S  e.  Ring  /\  (
( E  .+  G
) `  X )  e.  ( Base `  S
)  /\  ( I `  ( ( E  .+  G ) `  Y
) )  e.  (
Base `  S )
)  ->  ( (
( E  .+  G
) `  X )  .X.  ( I `  (
( E  .+  G
) `  Y )
) )  e.  (
Base `  S )
)
3310, 20, 30, 32syl3anc 1182 . . . . 5  |-  ( ph  ->  ( ( ( E 
.+  G ) `  X )  .X.  (
I `  ( ( E  .+  G ) `  Y ) ) )  e.  ( Base `  S
) )
34 lclkrlem2m.t . . . . . 6  |-  .x.  =  ( .s `  U )
3518, 8, 34, 17lmodvscl 15644 . . . . 5  |-  ( ( U  e.  LMod  /\  (
( ( E  .+  G ) `  X
)  .X.  ( I `  ( ( E  .+  G ) `  Y
) ) )  e.  ( Base `  S
)  /\  Y  e.  V )  ->  (
( ( ( E 
.+  G ) `  X )  .X.  (
I `  ( ( E  .+  G ) `  Y ) ) ) 
.x.  Y )  e.  V )
364, 33, 23, 35syl3anc 1182 . . . 4  |-  ( ph  ->  ( ( ( ( E  .+  G ) `
 X )  .X.  ( I `  (
( E  .+  G
) `  Y )
) )  .x.  Y
)  e.  V )
37 lclkrlem2m.m . . . . 5  |-  .-  =  ( -g `  U )
3818, 37grpsubcl 14546 . . . 4  |-  ( ( U  e.  Grp  /\  X  e.  V  /\  ( ( ( ( E  .+  G ) `
 X )  .X.  ( I `  (
( E  .+  G
) `  Y )
) )  .x.  Y
)  e.  V )  ->  ( X  .-  ( ( ( ( E  .+  G ) `
 X )  .X.  ( I `  (
( E  .+  G
) `  Y )
) )  .x.  Y
) )  e.  V
)
396, 7, 36, 38syl3anc 1182 . . 3  |-  ( ph  ->  ( X  .-  (
( ( ( E 
.+  G ) `  X )  .X.  (
I `  ( ( E  .+  G ) `  Y ) ) ) 
.x.  Y ) )  e.  V )
401, 39syl5eqel 2367 . 2  |-  ( ph  ->  B  e.  V )
411fveq2i 5528 . . 3  |-  ( ( E  .+  G ) `
 B )  =  ( ( E  .+  G ) `  ( X  .-  ( ( ( ( E  .+  G
) `  X )  .X.  ( I `  (
( E  .+  G
) `  Y )
) )  .x.  Y
) ) )
42 eqid 2283 . . . . . 6  |-  ( -g `  S )  =  (
-g `  S )
438, 42, 18, 37, 11lflsub 29257 . . . . 5  |-  ( ( U  e.  LMod  /\  ( E  .+  G )  e.  F  /\  ( X  e.  V  /\  (
( ( ( E 
.+  G ) `  X )  .X.  (
I `  ( ( E  .+  G ) `  Y ) ) ) 
.x.  Y )  e.  V ) )  -> 
( ( E  .+  G ) `  ( X  .-  ( ( ( ( E  .+  G
) `  X )  .X.  ( I `  (
( E  .+  G
) `  Y )
) )  .x.  Y
) ) )  =  ( ( ( E 
.+  G ) `  X ) ( -g `  S ) ( ( E  .+  G ) `
 ( ( ( ( E  .+  G
) `  X )  .X.  ( I `  (
( E  .+  G
) `  Y )
) )  .x.  Y
) ) ) )
444, 16, 7, 36, 43syl112anc 1186 . . . 4  |-  ( ph  ->  ( ( E  .+  G ) `  ( X  .-  ( ( ( ( E  .+  G
) `  X )  .X.  ( I `  (
( E  .+  G
) `  Y )
) )  .x.  Y
) ) )  =  ( ( ( E 
.+  G ) `  X ) ( -g `  S ) ( ( E  .+  G ) `
 ( ( ( ( E  .+  G
) `  X )  .X.  ( I `  (
( E  .+  G
) `  Y )
) )  .x.  Y
) ) ) )
458, 17, 31, 18, 34, 11lflmul 29258 . . . . . . 7  |-  ( ( U  e.  LMod  /\  ( E  .+  G )  e.  F  /\  ( ( ( ( E  .+  G ) `  X
)  .X.  ( I `  ( ( E  .+  G ) `  Y
) ) )  e.  ( Base `  S
)  /\  Y  e.  V ) )  -> 
( ( E  .+  G ) `  (
( ( ( E 
.+  G ) `  X )  .X.  (
I `  ( ( E  .+  G ) `  Y ) ) ) 
.x.  Y ) )  =  ( ( ( ( E  .+  G
) `  X )  .X.  ( I `  (
( E  .+  G
) `  Y )
) )  .X.  (
( E  .+  G
) `  Y )
) )
464, 16, 33, 23, 45syl112anc 1186 . . . . . 6  |-  ( ph  ->  ( ( E  .+  G ) `  (
( ( ( E 
.+  G ) `  X )  .X.  (
I `  ( ( E  .+  G ) `  Y ) ) ) 
.x.  Y ) )  =  ( ( ( ( E  .+  G
) `  X )  .X.  ( I `  (
( E  .+  G
) `  Y )
) )  .X.  (
( E  .+  G
) `  Y )
) )
4717, 31rngass 15357 . . . . . . . 8  |-  ( ( S  e.  Ring  /\  (
( ( E  .+  G ) `  X
)  e.  ( Base `  S )  /\  (
I `  ( ( E  .+  G ) `  Y ) )  e.  ( Base `  S
)  /\  ( ( E  .+  G ) `  Y )  e.  (
Base `  S )
) )  ->  (
( ( ( E 
.+  G ) `  X )  .X.  (
I `  ( ( E  .+  G ) `  Y ) ) ) 
.X.  ( ( E 
.+  G ) `  Y ) )  =  ( ( ( E 
.+  G ) `  X )  .X.  (
( I `  (
( E  .+  G
) `  Y )
)  .X.  ( ( E  .+  G ) `  Y ) ) ) )
4810, 20, 30, 25, 47syl13anc 1184 . . . . . . 7  |-  ( ph  ->  ( ( ( ( E  .+  G ) `
 X )  .X.  ( I `  (
( E  .+  G
) `  Y )
) )  .X.  (
( E  .+  G
) `  Y )
)  =  ( ( ( E  .+  G
) `  X )  .X.  ( ( I `  ( ( E  .+  G ) `  Y
) )  .X.  (
( E  .+  G
) `  Y )
) ) )
49 eqid 2283 . . . . . . . . . 10  |-  ( 1r
`  S )  =  ( 1r `  S
)
5017, 27, 31, 49, 28drnginvrl 15531 . . . . . . . . 9  |-  ( ( S  e.  DivRing  /\  (
( E  .+  G
) `  Y )  e.  ( Base `  S
)  /\  ( ( E  .+  G ) `  Y )  =/=  .0.  )  ->  ( ( I `
 ( ( E 
.+  G ) `  Y ) )  .X.  ( ( E  .+  G ) `  Y
) )  =  ( 1r `  S ) )
5122, 25, 26, 50syl3anc 1182 . . . . . . . 8  |-  ( ph  ->  ( ( I `  ( ( E  .+  G ) `  Y
) )  .X.  (
( E  .+  G
) `  Y )
)  =  ( 1r
`  S ) )
5251oveq2d 5874 . . . . . . 7  |-  ( ph  ->  ( ( ( E 
.+  G ) `  X )  .X.  (
( I `  (
( E  .+  G
) `  Y )
)  .X.  ( ( E  .+  G ) `  Y ) ) )  =  ( ( ( E  .+  G ) `
 X )  .X.  ( 1r `  S ) ) )
5348, 52eqtrd 2315 . . . . . 6  |-  ( ph  ->  ( ( ( ( E  .+  G ) `
 X )  .X.  ( I `  (
( E  .+  G
) `  Y )
) )  .X.  (
( E  .+  G
) `  Y )
)  =  ( ( ( E  .+  G
) `  X )  .X.  ( 1r `  S
) ) )
5417, 31, 49rngridm 15365 . . . . . . 7  |-  ( ( S  e.  Ring  /\  (
( E  .+  G
) `  X )  e.  ( Base `  S
) )  ->  (
( ( E  .+  G ) `  X
)  .X.  ( 1r `  S ) )  =  ( ( E  .+  G ) `  X
) )
5510, 20, 54syl2anc 642 . . . . . 6  |-  ( ph  ->  ( ( ( E 
.+  G ) `  X )  .X.  ( 1r `  S ) )  =  ( ( E 
.+  G ) `  X ) )
5646, 53, 553eqtrd 2319 . . . . 5  |-  ( ph  ->  ( ( E  .+  G ) `  (
( ( ( E 
.+  G ) `  X )  .X.  (
I `  ( ( E  .+  G ) `  Y ) ) ) 
.x.  Y ) )  =  ( ( E 
.+  G ) `  X ) )
5756oveq2d 5874 . . . 4  |-  ( ph  ->  ( ( ( E 
.+  G ) `  X ) ( -g `  S ) ( ( E  .+  G ) `
 ( ( ( ( E  .+  G
) `  X )  .X.  ( I `  (
( E  .+  G
) `  Y )
) )  .x.  Y
) ) )  =  ( ( ( E 
.+  G ) `  X ) ( -g `  S ) ( ( E  .+  G ) `
 X ) ) )
58 rnggrp 15346 . . . . . 6  |-  ( S  e.  Ring  ->  S  e. 
Grp )
5910, 58syl 15 . . . . 5  |-  ( ph  ->  S  e.  Grp )
6017, 27, 42grpsubid 14550 . . . . 5  |-  ( ( S  e.  Grp  /\  ( ( E  .+  G ) `  X
)  e.  ( Base `  S ) )  -> 
( ( ( E 
.+  G ) `  X ) ( -g `  S ) ( ( E  .+  G ) `
 X ) )  =  .0.  )
6159, 20, 60syl2anc 642 . . . 4  |-  ( ph  ->  ( ( ( E 
.+  G ) `  X ) ( -g `  S ) ( ( E  .+  G ) `
 X ) )  =  .0.  )
6244, 57, 613eqtrd 2319 . . 3  |-  ( ph  ->  ( ( E  .+  G ) `  ( X  .-  ( ( ( ( E  .+  G
) `  X )  .X.  ( I `  (
( E  .+  G
) `  Y )
) )  .x.  Y
) ) )  =  .0.  )
6341, 62syl5eq 2327 . 2  |-  ( ph  ->  ( ( E  .+  G ) `  B
)  =  .0.  )
6440, 63jca 518 1  |-  ( ph  ->  ( B  e.  V  /\  ( ( E  .+  G ) `  B
)  =  .0.  )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208   .rcmulr 13209  Scalarcsca 13211   .scvsca 13212   0gc0g 13400   Grpcgrp 14362   -gcsg 14365   Ringcrg 15337   1rcur 15339   invrcinvr 15453   DivRingcdr 15512   LModclmod 15627   LVecclvec 15855  LFnlclfn 29247  LDualcld 29313
This theorem is referenced by:  lclkrlem2o  31711  lclkrlem2q  31713
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-lvec 15856  df-lfl 29248  df-ldual 29314
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