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Theorem baerlem5bmN 31175
Description: An equality that holds when  X,  Y,  Z are independent (non-colinear) vectors. Subtraction version of second equation of part (5) in [Baer] p. 46. TODO: This is the subtraction version, may not be needed. TODO: delete if baerlem5abmN 31176 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
baerlem5bmN  |-  ( ph  ->  ( N `  {
( Y  .-  Z
) } )  =  ( ( ( N `
 { Y }
)  .(+)  ( N `  { Z } ) )  i^i  ( ( N `
 { ( X 
.-  ( Y  .-  Z ) ) } )  .(+)  ( N `  { X } ) ) ) )

Proof of Theorem baerlem5bmN
StepHypRef Expression
1 baerlem3.y . . . . . 6  |-  ( ph  ->  Y  e.  ( V 
\  {  .0.  }
) )
2 eldifi 3300 . . . . . 6  |-  ( Y  e.  ( V  \  {  .0.  } )  ->  Y  e.  V )
31, 2syl 17 . . . . 5  |-  ( ph  ->  Y  e.  V )
4 baerlem3.z . . . . . 6  |-  ( ph  ->  Z  e.  ( V 
\  {  .0.  }
) )
5 eldifi 3300 . . . . . 6  |-  ( Z  e.  ( V  \  {  .0.  } )  ->  Z  e.  V )
64, 5syl 17 . . . . 5  |-  ( ph  ->  Z  e.  V )
7 baerlem3.v . . . . . 6  |-  V  =  ( Base `  W
)
8 baerlem5a.p . . . . . 6  |-  .+  =  ( +g  `  W )
9 eqid 2285 . . . . . 6  |-  ( inv g `  W )  =  ( inv g `  W )
10 baerlem3.m . . . . . 6  |-  .-  =  ( -g `  W )
117, 8, 9, 10grpsubval 14520 . . . . 5  |-  ( ( Y  e.  V  /\  Z  e.  V )  ->  ( Y  .-  Z
)  =  ( Y 
.+  ( ( inv g `  W ) `
 Z ) ) )
123, 6, 11syl2anc 644 . . . 4  |-  ( ph  ->  ( Y  .-  Z
)  =  ( Y 
.+  ( ( inv g `  W ) `
 Z ) ) )
1312sneqd 3655 . . 3  |-  ( ph  ->  { ( Y  .-  Z ) }  =  { ( Y  .+  ( ( inv g `  W ) `  Z
) ) } )
1413fveq2d 5490 . 2  |-  ( ph  ->  ( N `  {
( Y  .-  Z
) } )  =  ( N `  {
( Y  .+  (
( inv g `  W ) `  Z
) ) } ) )
15 baerlem3.o . . 3  |-  .0.  =  ( 0g `  W )
16 baerlem3.s . . 3  |-  .(+)  =  (
LSSum `  W )
17 baerlem3.n . . 3  |-  N  =  ( LSpan `  W )
18 baerlem3.w . . 3  |-  ( ph  ->  W  e.  LVec )
19 baerlem3.x . . 3  |-  ( ph  ->  X  e.  V )
20 lveclmod 15854 . . . . . 6  |-  ( W  e.  LVec  ->  W  e. 
LMod )
2118, 20syl 17 . . . . 5  |-  ( ph  ->  W  e.  LMod )
227, 9lmodvnegcl 15660 . . . . 5  |-  ( ( W  e.  LMod  /\  Z  e.  V )  ->  (
( inv g `  W ) `  Z
)  e.  V )
2321, 6, 22syl2anc 644 . . . 4  |-  ( ph  ->  ( ( inv g `  W ) `  Z
)  e.  V )
24 eqid 2285 . . . . . 6  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
257, 24, 17, 21, 3, 6lspprcl 15730 . . . . . 6  |-  ( ph  ->  ( N `  { Y ,  Z }
)  e.  ( LSubSp `  W ) )
26 baerlem3.c . . . . . 6  |-  ( ph  ->  -.  X  e.  ( N `  { Y ,  Z } ) )
277, 15, 24, 21, 25, 19, 26lssneln0 15704 . . . . 5  |-  ( ph  ->  X  e.  ( V 
\  {  .0.  }
) )
287, 17, 18, 19, 3, 6, 26lspindpi 15880 . . . . . 6  |-  ( ph  ->  ( ( N `  { X } )  =/=  ( N `  { Y } )  /\  ( N `  { X } )  =/=  ( N `  { Z } ) ) )
2928simpld 447 . . . . 5  |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y } ) )
307, 15, 17, 18, 27, 3, 29lspsnne1 15865 . . . 4  |-  ( ph  ->  -.  X  e.  ( N `  { Y } ) )
31 baerlem3.d . . . . . . . 8  |-  ( ph  ->  ( N `  { Y } )  =/=  ( N `  { Z } ) )
3231necomd 2531 . . . . . . 7  |-  ( ph  ->  ( N `  { Z } )  =/=  ( N `  { Y } ) )
337, 15, 17, 18, 4, 3, 32lspsnne1 15865 . . . . . 6  |-  ( ph  ->  -.  Z  e.  ( N `  { Y } ) )
347, 17, 18, 19, 6, 3, 33, 26lspexchn2 15879 . . . . 5  |-  ( ph  ->  -.  Z  e.  ( N `  { Y ,  X } ) )
35 lmodgrp 15629 . . . . . . . . 9  |-  ( W  e.  LMod  ->  W  e. 
Grp )
3621, 35syl 17 . . . . . . . 8  |-  ( ph  ->  W  e.  Grp )
3736adantr 453 . . . . . . 7  |-  ( (
ph  /\  ( ( inv g `  W ) `
 Z )  e.  ( N `  { Y ,  X }
) )  ->  W  e.  Grp )
386adantr 453 . . . . . . 7  |-  ( (
ph  /\  ( ( inv g `  W ) `
 Z )  e.  ( N `  { Y ,  X }
) )  ->  Z  e.  V )
397, 9grpinvinv 14530 . . . . . . 7  |-  ( ( W  e.  Grp  /\  Z  e.  V )  ->  ( ( inv g `  W ) `  (
( inv g `  W ) `  Z
) )  =  Z )
4037, 38, 39syl2anc 644 . . . . . 6  |-  ( (
ph  /\  ( ( inv g `  W ) `
 Z )  e.  ( N `  { Y ,  X }
) )  ->  (
( inv g `  W ) `  (
( inv g `  W ) `  Z
) )  =  Z )
4121adantr 453 . . . . . . 7  |-  ( (
ph  /\  ( ( inv g `  W ) `
 Z )  e.  ( N `  { Y ,  X }
) )  ->  W  e.  LMod )
427, 24, 17, 21, 3, 19lspprcl 15730 . . . . . . . 8  |-  ( ph  ->  ( N `  { Y ,  X }
)  e.  ( LSubSp `  W ) )
4342adantr 453 . . . . . . 7  |-  ( (
ph  /\  ( ( inv g `  W ) `
 Z )  e.  ( N `  { Y ,  X }
) )  ->  ( N `  { Y ,  X } )  e.  ( LSubSp `  W )
)
44 simpr 449 . . . . . . 7  |-  ( (
ph  /\  ( ( inv g `  W ) `
 Z )  e.  ( N `  { Y ,  X }
) )  ->  (
( inv g `  W ) `  Z
)  e.  ( N `
 { Y ,  X } ) )
4524, 9lssvnegcl 15708 . . . . . . 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 } ) )
4641, 43, 44, 45syl3anc 1184 . . . . . 6  |-  ( (
ph  /\  ( ( inv g `  W ) `
 Z )  e.  ( N `  { Y ,  X }
) )  ->  (
( inv g `  W ) `  (
( inv g `  W ) `  Z
) )  e.  ( N `  { Y ,  X } ) )
4740, 46eqeltrrd 2360 . . . . 5  |-  ( (
ph  /\  ( ( inv g `  W ) `
 Z )  e.  ( N `  { Y ,  X }
) )  ->  Z  e.  ( N `  { Y ,  X }
) )
4834, 47mtand 642 . . . 4  |-  ( ph  ->  -.  ( ( inv g `  W ) `
 Z )  e.  ( N `  { Y ,  X }
) )
497, 17, 18, 23, 19, 3, 30, 48lspexchn2 15879 . . 3  |-  ( ph  ->  -.  X  e.  ( N `  { Y ,  ( ( inv g `  W ) `
 Z ) } ) )
507, 9, 17lspsnneg 15758 . . . . 5  |-  ( ( W  e.  LMod  /\  Z  e.  V )  ->  ( N `  { (
( inv g `  W ) `  Z
) } )  =  ( N `  { Z } ) )
5121, 6, 50syl2anc 644 . . . 4  |-  ( ph  ->  ( N `  {
( ( inv g `  W ) `  Z
) } )  =  ( N `  { Z } ) )
5231, 51neeqtrrd 2472 . . 3  |-  ( ph  ->  ( N `  { Y } )  =/=  ( N `  { (
( inv g `  W ) `  Z
) } ) )
537, 15, 9grpinvnzcl 14535 . . . 4  |-  ( ( W  e.  Grp  /\  Z  e.  ( V  \  {  .0.  } ) )  ->  ( ( inv g `  W ) `
 Z )  e.  ( V  \  {  .0.  } ) )
5436, 4, 53syl2anc 644 . . 3  |-  ( ph  ->  ( ( inv g `  W ) `  Z
)  e.  ( V 
\  {  .0.  }
) )
557, 10, 15, 16, 17, 18, 19, 49, 52, 1, 54, 8baerlem5b 31173 . 2  |-  ( ph  ->  ( N `  {
( Y  .+  (
( inv g `  W ) `  Z
) ) } )  =  ( ( ( N `  { Y } )  .(+)  ( N `
 { ( ( inv g `  W
) `  Z ) } ) )  i^i  ( ( N `  { ( X  .-  ( Y  .+  ( ( inv g `  W
) `  Z )
) ) } ) 
.(+)  ( N `  { X } ) ) ) )
5651oveq2d 5836 . . 3  |-  ( ph  ->  ( ( N `  { Y } )  .(+)  ( N `  { ( ( inv g `  W ) `  Z
) } ) )  =  ( ( N `
 { Y }
)  .(+)  ( N `  { Z } ) ) )
5712eqcomd 2290 . . . . . . 7  |-  ( ph  ->  ( Y  .+  (
( inv g `  W ) `  Z
) )  =  ( Y  .-  Z ) )
5857oveq2d 5836 . . . . . 6  |-  ( ph  ->  ( X  .-  ( Y  .+  ( ( inv g `  W ) `
 Z ) ) )  =  ( X 
.-  ( Y  .-  Z ) ) )
5958sneqd 3655 . . . . 5  |-  ( ph  ->  { ( X  .-  ( Y  .+  ( ( inv g `  W
) `  Z )
) ) }  =  { ( X  .-  ( Y  .-  Z ) ) } )
6059fveq2d 5490 . . . 4  |-  ( ph  ->  ( N `  {
( X  .-  ( Y  .+  ( ( inv g `  W ) `
 Z ) ) ) } )  =  ( N `  {
( X  .-  ( Y  .-  Z ) ) } ) )
6160oveq1d 5835 . . 3  |-  ( ph  ->  ( ( N `  { ( X  .-  ( Y  .+  ( ( inv g `  W
) `  Z )
) ) } ) 
.(+)  ( N `  { X } ) )  =  ( ( N `
 { ( X 
.-  ( Y  .-  Z ) ) } )  .(+)  ( N `  { X } ) ) )
6256, 61ineq12d 3373 . 2  |-  ( ph  ->  ( ( ( N `
 { Y }
)  .(+)  ( N `  { ( ( inv g `  W ) `
 Z ) } ) )  i^i  (
( N `  {
( X  .-  ( Y  .+  ( ( inv g `  W ) `
 Z ) ) ) } )  .(+)  ( N `  { X } ) ) )  =  ( ( ( N `  { Y } )  .(+)  ( N `
 { Z }
) )  i^i  (
( N `  {
( X  .-  ( Y  .-  Z ) ) } )  .(+)  ( N `
 { X }
) ) ) )
6314, 55, 623eqtrd 2321 1  |-  ( ph  ->  ( N `  {
( Y  .-  Z
) } )  =  ( ( ( N `
 { Y }
)  .(+)  ( N `  { Z } ) )  i^i  ( ( N `
 { ( X 
.-  ( Y  .-  Z ) ) } )  .(+)  ( N `  { X } ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360    = wceq 1624    e. wcel 1685    =/= wne 2448    \ cdif 3151    i^i cin 3153   {csn 3642   {cpr 3643   ` cfv 5222  (class class class)co 5820   Basecbs 13143   +g cplusg 13203   0gc0g 13395   Grpcgrp 14357   inv gcminusg 14358   -gcsg 14360   LSSumclsm 14940   LModclmod 15622   LSubSpclss 15684   LSpanclspn 15723   LVecclvec 15850
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8789  ax-resscn 8790  ax-1cn 8791  ax-icn 8792  ax-addcl 8793  ax-addrcl 8794  ax-mulcl 8795  ax-mulrcl 8796  ax-mulcom 8797  ax-addass 8798  ax-mulass 8799  ax-distr 8800  ax-i2m1 8801  ax-1ne0 8802  ax-1rid 8803  ax-rnegex 8804  ax-rrecex 8805  ax-cnre 8806  ax-pre-lttri 8807  ax-pre-lttrn 8808  ax-pre-ltadd 8809  ax-pre-mulgt0 8810
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-int 3865  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  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-fun 5224  df-fn 5225  df-f 5226  df-f1 5227  df-fo 5228  df-f1o 5229  df-fv 5230  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-1st 6084  df-2nd 6085  df-tpos 6196  df-iota 6253  df-riota 6300  df-recs 6384  df-rdg 6419  df-er 6656  df-en 6860  df-dom 6861  df-sdom 6862  df-pnf 8865  df-mnf 8866  df-xr 8867  df-ltxr 8868  df-le 8869  df-sub 9035  df-neg 9036  df-nn 9743  df-2 9800  df-3 9801  df-ndx 13146  df-slot 13147  df-base 13148  df-sets 13149  df-ress 13150  df-plusg 13216  df-mulr 13217  df-0g 13399  df-mnd 14362  df-submnd 14411  df-grp 14484  df-minusg 14485  df-sbg 14486  df-subg 14613  df-cntz 14788  df-lsm 14942  df-cmn 15086  df-abl 15087  df-mgp 15321  df-rng 15335  df-ur 15337  df-oppr 15400  df-dvdsr 15418  df-unit 15419  df-invr 15449  df-drng 15509  df-lmod 15624  df-lss 15685  df-lsp 15724  df-lvec 15851
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