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Theorem baerlem5amN 31831
Description: An equality that holds when  X,  Y,  Z are independent (non-colinear) vectors. Subtraction version of first equation of part (5) in [Baer] p. 46. TODO: This is the subtraction version, may not be needed. TODO: delete if baerlem5abmN 31833 is used. (Contributed by NM, 24-May-2015.) (New usage is discouraged.)
Hypotheses
Ref Expression
baerlem3.v  |-  V  =  ( Base `  W
)
baerlem3.m  |-  .-  =  ( -g `  W )
baerlem3.o  |-  .0.  =  ( 0g `  W )
baerlem3.s  |-  .(+)  =  (
LSSum `  W )
baerlem3.n  |-  N  =  ( LSpan `  W )
baerlem3.w  |-  ( ph  ->  W  e.  LVec )
baerlem3.x  |-  ( ph  ->  X  e.  V )
baerlem3.c  |-  ( ph  ->  -.  X  e.  ( N `  { Y ,  Z } ) )
baerlem3.d  |-  ( ph  ->  ( N `  { Y } )  =/=  ( N `  { Z } ) )
baerlem3.y  |-  ( ph  ->  Y  e.  ( V 
\  {  .0.  }
) )
baerlem3.z  |-  ( ph  ->  Z  e.  ( V 
\  {  .0.  }
) )
baerlem5a.p  |-  .+  =  ( +g  `  W )
Assertion
Ref Expression
baerlem5amN  |-  ( ph  ->  ( N `  {
( X  .-  ( Y  .-  Z ) ) } )  =  ( ( ( N `  { ( X  .-  Y ) } ) 
.(+)  ( N `  { Z } ) )  i^i  ( ( N `
 { ( X 
.+  Z ) } )  .(+)  ( N `  { Y } ) ) ) )

Proof of Theorem baerlem5amN
StepHypRef Expression
1 baerlem3.y . . . . . . 7  |-  ( ph  ->  Y  e.  ( V 
\  {  .0.  }
) )
21eldifad 3275 . . . . . 6  |-  ( ph  ->  Y  e.  V )
3 baerlem3.z . . . . . . 7  |-  ( ph  ->  Z  e.  ( V 
\  {  .0.  }
) )
43eldifad 3275 . . . . . 6  |-  ( ph  ->  Z  e.  V )
5 baerlem3.v . . . . . . 7  |-  V  =  ( Base `  W
)
6 baerlem5a.p . . . . . . 7  |-  .+  =  ( +g  `  W )
7 eqid 2387 . . . . . . 7  |-  ( inv g `  W )  =  ( inv g `  W )
8 baerlem3.m . . . . . . 7  |-  .-  =  ( -g `  W )
95, 6, 7, 8grpsubval 14775 . . . . . 6  |-  ( ( Y  e.  V  /\  Z  e.  V )  ->  ( Y  .-  Z
)  =  ( Y 
.+  ( ( inv g `  W ) `
 Z ) ) )
102, 4, 9syl2anc 643 . . . . 5  |-  ( ph  ->  ( Y  .-  Z
)  =  ( Y 
.+  ( ( inv g `  W ) `
 Z ) ) )
1110oveq2d 6036 . . . 4  |-  ( ph  ->  ( X  .-  ( Y  .-  Z ) )  =  ( X  .-  ( Y  .+  ( ( inv g `  W
) `  Z )
) ) )
1211sneqd 3770 . . 3  |-  ( ph  ->  { ( X  .-  ( Y  .-  Z ) ) }  =  {
( X  .-  ( Y  .+  ( ( inv g `  W ) `
 Z ) ) ) } )
1312fveq2d 5672 . 2  |-  ( ph  ->  ( N `  {
( X  .-  ( Y  .-  Z ) ) } )  =  ( N `  { ( X  .-  ( Y 
.+  ( ( inv g `  W ) `
 Z ) ) ) } ) )
14 baerlem3.o . . 3  |-  .0.  =  ( 0g `  W )
15 baerlem3.s . . 3  |-  .(+)  =  (
LSSum `  W )
16 baerlem3.n . . 3  |-  N  =  ( LSpan `  W )
17 baerlem3.w . . 3  |-  ( ph  ->  W  e.  LVec )
18 baerlem3.x . . 3  |-  ( ph  ->  X  e.  V )
19 lveclmod 16105 . . . . . 6  |-  ( W  e.  LVec  ->  W  e. 
LMod )
2017, 19syl 16 . . . . 5  |-  ( ph  ->  W  e.  LMod )
215, 7lmodvnegcl 15912 . . . . 5  |-  ( ( W  e.  LMod  /\  Z  e.  V )  ->  (
( inv g `  W ) `  Z
)  e.  V )
2220, 4, 21syl2anc 643 . . . 4  |-  ( ph  ->  ( ( inv g `  W ) `  Z
)  e.  V )
23 eqid 2387 . . . . . 6  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
245, 23, 16, 20, 2, 4lspprcl 15981 . . . . . 6  |-  ( ph  ->  ( N `  { Y ,  Z }
)  e.  ( LSubSp `  W ) )
25 baerlem3.c . . . . . 6  |-  ( ph  ->  -.  X  e.  ( N `  { Y ,  Z } ) )
265, 14, 23, 20, 24, 18, 25lssneln0 15955 . . . . 5  |-  ( ph  ->  X  e.  ( V 
\  {  .0.  }
) )
275, 16, 17, 18, 2, 4, 25lspindpi 16131 . . . . . 6  |-  ( ph  ->  ( ( N `  { X } )  =/=  ( N `  { Y } )  /\  ( N `  { X } )  =/=  ( N `  { Z } ) ) )
2827simpld 446 . . . . 5  |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y } ) )
295, 14, 16, 17, 26, 2, 28lspsnne1 16116 . . . 4  |-  ( ph  ->  -.  X  e.  ( N `  { Y } ) )
30 baerlem3.d . . . . . . . 8  |-  ( ph  ->  ( N `  { Y } )  =/=  ( N `  { Z } ) )
3130necomd 2633 . . . . . . 7  |-  ( ph  ->  ( N `  { Z } )  =/=  ( N `  { Y } ) )
325, 14, 16, 17, 3, 2, 31lspsnne1 16116 . . . . . 6  |-  ( ph  ->  -.  Z  e.  ( N `  { Y } ) )
335, 16, 17, 18, 4, 2, 32, 25lspexchn2 16130 . . . . 5  |-  ( ph  ->  -.  Z  e.  ( N `  { Y ,  X } ) )
34 lmodgrp 15884 . . . . . . . . 9  |-  ( W  e.  LMod  ->  W  e. 
Grp )
3517, 19, 343syl 19 . . . . . . . 8  |-  ( ph  ->  W  e.  Grp )
3635adantr 452 . . . . . . 7  |-  ( (
ph  /\  ( ( inv g `  W ) `
 Z )  e.  ( N `  { Y ,  X }
) )  ->  W  e.  Grp )
374adantr 452 . . . . . . 7  |-  ( (
ph  /\  ( ( inv g `  W ) `
 Z )  e.  ( N `  { Y ,  X }
) )  ->  Z  e.  V )
385, 7grpinvinv 14785 . . . . . . 7  |-  ( ( W  e.  Grp  /\  Z  e.  V )  ->  ( ( inv g `  W ) `  (
( inv g `  W ) `  Z
) )  =  Z )
3936, 37, 38syl2anc 643 . . . . . 6  |-  ( (
ph  /\  ( ( inv g `  W ) `
 Z )  e.  ( N `  { Y ,  X }
) )  ->  (
( inv g `  W ) `  (
( inv g `  W ) `  Z
) )  =  Z )
4020adantr 452 . . . . . . 7  |-  ( (
ph  /\  ( ( inv g `  W ) `
 Z )  e.  ( N `  { Y ,  X }
) )  ->  W  e.  LMod )
415, 23, 16, 20, 2, 18lspprcl 15981 . . . . . . . 8  |-  ( ph  ->  ( N `  { Y ,  X }
)  e.  ( LSubSp `  W ) )
4241adantr 452 . . . . . . 7  |-  ( (
ph  /\  ( ( inv g `  W ) `
 Z )  e.  ( N `  { Y ,  X }
) )  ->  ( N `  { Y ,  X } )  e.  ( LSubSp `  W )
)
43 simpr 448 . . . . . . 7  |-  ( (
ph  /\  ( ( inv g `  W ) `
 Z )  e.  ( N `  { Y ,  X }
) )  ->  (
( inv g `  W ) `  Z
)  e.  ( N `
 { Y ,  X } ) )
4423, 7lssvnegcl 15959 . . . . . . 7  |-  ( ( W  e.  LMod  /\  ( N `  { Y ,  X } )  e.  ( LSubSp `  W )  /\  ( ( inv g `  W ) `  Z
)  e.  ( N `
 { Y ,  X } ) )  -> 
( ( inv g `  W ) `  (
( inv g `  W ) `  Z
) )  e.  ( N `  { Y ,  X } ) )
4540, 42, 43, 44syl3anc 1184 . . . . . 6  |-  ( (
ph  /\  ( ( inv g `  W ) `
 Z )  e.  ( N `  { Y ,  X }
) )  ->  (
( inv g `  W ) `  (
( inv g `  W ) `  Z
) )  e.  ( N `  { Y ,  X } ) )
4639, 45eqeltrrd 2462 . . . . 5  |-  ( (
ph  /\  ( ( inv g `  W ) `
 Z )  e.  ( N `  { Y ,  X }
) )  ->  Z  e.  ( N `  { Y ,  X }
) )
4733, 46mtand 641 . . . 4  |-  ( ph  ->  -.  ( ( inv g `  W ) `
 Z )  e.  ( N `  { Y ,  X }
) )
485, 16, 17, 22, 18, 2, 29, 47lspexchn2 16130 . . 3  |-  ( ph  ->  -.  X  e.  ( N `  { Y ,  ( ( inv g `  W ) `
 Z ) } ) )
495, 7, 16lspsnneg 16009 . . . . 5  |-  ( ( W  e.  LMod  /\  Z  e.  V )  ->  ( N `  { (
( inv g `  W ) `  Z
) } )  =  ( N `  { Z } ) )
5020, 4, 49syl2anc 643 . . . 4  |-  ( ph  ->  ( N `  {
( ( inv g `  W ) `  Z
) } )  =  ( N `  { Z } ) )
5130, 50neeqtrrd 2574 . . 3  |-  ( ph  ->  ( N `  { Y } )  =/=  ( N `  { (
( inv g `  W ) `  Z
) } ) )
525, 14, 7grpinvnzcl 14790 . . . 4  |-  ( ( W  e.  Grp  /\  Z  e.  ( V  \  {  .0.  } ) )  ->  ( ( inv g `  W ) `
 Z )  e.  ( V  \  {  .0.  } ) )
5335, 3, 52syl2anc 643 . . 3  |-  ( ph  ->  ( ( inv g `  W ) `  Z
)  e.  ( V 
\  {  .0.  }
) )
545, 8, 14, 15, 16, 17, 18, 48, 51, 1, 53, 6baerlem5a 31829 . 2  |-  ( ph  ->  ( N `  {
( X  .-  ( Y  .+  ( ( inv g `  W ) `
 Z ) ) ) } )  =  ( ( ( N `
 { ( X 
.-  Y ) } )  .(+)  ( N `  { ( ( inv g `  W ) `
 Z ) } ) )  i^i  (
( N `  {
( X  .-  (
( inv g `  W ) `  Z
) ) } ) 
.(+)  ( N `  { Y } ) ) ) )
5550oveq2d 6036 . . 3  |-  ( ph  ->  ( ( N `  { ( X  .-  Y ) } ) 
.(+)  ( N `  { ( ( inv g `  W ) `
 Z ) } ) )  =  ( ( N `  {
( X  .-  Y
) } )  .(+)  ( N `  { Z } ) ) )
565, 6, 8, 7, 35, 18, 4grpsubinv 14791 . . . . . 6  |-  ( ph  ->  ( X  .-  (
( inv g `  W ) `  Z
) )  =  ( X  .+  Z ) )
5756sneqd 3770 . . . . 5  |-  ( ph  ->  { ( X  .-  ( ( inv g `  W ) `  Z
) ) }  =  { ( X  .+  Z ) } )
5857fveq2d 5672 . . . 4  |-  ( ph  ->  ( N `  {
( X  .-  (
( inv g `  W ) `  Z
) ) } )  =  ( N `  { ( X  .+  Z ) } ) )
5958oveq1d 6035 . . 3  |-  ( ph  ->  ( ( N `  { ( X  .-  ( ( inv g `  W ) `  Z
) ) } ) 
.(+)  ( N `  { Y } ) )  =  ( ( N `
 { ( X 
.+  Z ) } )  .(+)  ( N `  { Y } ) ) )
6055, 59ineq12d 3486 . 2  |-  ( ph  ->  ( ( ( N `
 { ( X 
.-  Y ) } )  .(+)  ( N `  { ( ( inv g `  W ) `
 Z ) } ) )  i^i  (
( N `  {
( X  .-  (
( inv g `  W ) `  Z
) ) } ) 
.(+)  ( N `  { Y } ) ) )  =  ( ( ( N `  {
( X  .-  Y
) } )  .(+)  ( N `  { Z } ) )  i^i  ( ( N `  { ( X  .+  Z ) } ) 
.(+)  ( N `  { Y } ) ) ) )
6113, 54, 603eqtrd 2423 1  |-  ( ph  ->  ( N `  {
( X  .-  ( Y  .-  Z ) ) } )  =  ( ( ( N `  { ( X  .-  Y ) } ) 
.(+)  ( N `  { Z } ) )  i^i  ( ( N `
 { ( X 
.+  Z ) } )  .(+)  ( N `  { Y } ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717    =/= wne 2550    \ cdif 3260    i^i cin 3262   {csn 3757   {cpr 3758   ` cfv 5394  (class class class)co 6020   Basecbs 13396   +g cplusg 13456   0gc0g 13650   Grpcgrp 14612   inv gcminusg 14613   -gcsg 14615   LSSumclsm 15195   LModclmod 15877   LSubSpclss 15935   LSpanclspn 15974   LVecclvec 16101
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-tpos 6415  df-riota 6485  df-recs 6569  df-rdg 6604  df-er 6841  df-en 7046  df-dom 7047  df-sdom 7048  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-nn 9933  df-2 9990  df-3 9991  df-ndx 13399  df-slot 13400  df-base 13401  df-sets 13402  df-ress 13403  df-plusg 13469  df-mulr 13470  df-0g 13654  df-mnd 14617  df-submnd 14666  df-grp 14739  df-minusg 14740  df-sbg 14741  df-subg 14868  df-cntz 15043  df-lsm 15197  df-cmn 15341  df-abl 15342  df-mgp 15576  df-rng 15590  df-ur 15592  df-oppr 15655  df-dvdsr 15673  df-unit 15674  df-invr 15704  df-drng 15764  df-lmod 15879  df-lss 15936  df-lsp 15975  df-lvec 16102
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