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Theorem baerlem5bmN 30596
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 30597 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 3215 . . . . . 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 3215 . . . . . 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 2253 . . . . . 6  |-  ( inv g `  W )  =  ( inv g `  W )
10 baerlem3.m . . . . . 6  |-  .-  =  ( -g `  W )
117, 8, 9, 10grpsubval 14360 . . . . 5  |-  ( ( Y  e.  V  /\  Z  e.  V )  ->  ( Y  .-  Z
)  =  ( Y 
.+  ( ( inv g `  W ) `
 Z ) ) )
123, 6, 11syl2anc 645 . . . 4  |-  ( ph  ->  ( Y  .-  Z
)  =  ( Y 
.+  ( ( inv g `  W ) `
 Z ) ) )
1312sneqd 3557 . . 3  |-  ( ph  ->  { ( Y  .-  Z ) }  =  { ( Y  .+  ( ( inv g `  W ) `  Z
) ) } )
1413fveq2d 5381 . 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 15694 . . . . . 6  |-  ( W  e.  LVec  ->  W  e. 
LMod )
2118, 20syl 17 . . . . 5  |-  ( ph  ->  W  e.  LMod )
227, 9lmodvnegcl 15500 . . . . 5  |-  ( ( W  e.  LMod  /\  Z  e.  V )  ->  (
( inv g `  W ) `  Z
)  e.  V )
2321, 6, 22syl2anc 645 . . . 4  |-  ( ph  ->  ( ( inv g `  W ) `  Z
)  e.  V )
24 eqid 2253 . . . . . 6  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
257, 24, 17, 21, 3, 6lspprcl 15570 . . . . . 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 15544 . . . . 5  |-  ( ph  ->  X  e.  ( V 
\  {  .0.  }
) )
287, 17, 18, 19, 3, 6, 26lspindpi 15720 . . . . . 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 15705 . . . 4  |-  ( ph  ->  -.  X  e.  ( N `  { Y } ) )
31 baerlem3.d . . . . . . . 8  |-  ( ph  ->  ( N `  { Y } )  =/=  ( N `  { Z } ) )
3231necomd 2495 . . . . . . 7  |-  ( ph  ->  ( N `  { Z } )  =/=  ( N `  { Y } ) )
337, 15, 17, 18, 4, 3, 32lspsnne1 15705 . . . . . 6  |-  ( ph  ->  -.  Z  e.  ( N `  { Y } ) )
347, 17, 18, 19, 6, 3, 33, 26lspexchn2 15719 . . . . 5  |-  ( ph  ->  -.  Z  e.  ( N `  { Y ,  X } ) )
35 lmodgrp 15469 . . . . . . . . 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 14370 . . . . . . 7  |-  ( ( W  e.  Grp  /\  Z  e.  V )  ->  ( ( inv g `  W ) `  (
( inv g `  W ) `  Z
) )  =  Z )
4037, 38, 39syl2anc 645 . . . . . 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 15570 . . . . . . . 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 15548 . . . . . . 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 1187 . . . . . 6  |-  ( (
ph  /\  ( ( inv g `  W ) `
 Z )  e.  ( N `  { Y ,  X }
) )  ->  (
( inv g `  W ) `  (
( inv g `  W ) `  Z
) )  e.  ( N `  { Y ,  X } ) )
4740, 46eqeltrrd 2328 . . . . 5  |-  ( (
ph  /\  ( ( inv g `  W ) `
 Z )  e.  ( N `  { Y ,  X }
) )  ->  Z  e.  ( N `  { Y ,  X }
) )
4834, 47mtand 643 . . . 4  |-  ( ph  ->  -.  ( ( inv g `  W ) `
 Z )  e.  ( N `  { Y ,  X }
) )
497, 17, 18, 23, 19, 3, 30, 48lspexchn2 15719 . . 3  |-  ( ph  ->  -.  X  e.  ( N `  { Y ,  ( ( inv g `  W ) `
 Z ) } ) )
507, 9, 17lspsnneg 15598 . . . . 5  |-  ( ( W  e.  LMod  /\  Z  e.  V )  ->  ( N `  { (
( inv g `  W ) `  Z
) } )  =  ( N `  { Z } ) )
5121, 6, 50syl2anc 645 . . . 4  |-  ( ph  ->  ( N `  {
( ( inv g `  W ) `  Z
) } )  =  ( N `  { Z } ) )
5231, 51neeqtrrd 2436 . . 3  |-  ( ph  ->  ( N `  { Y } )  =/=  ( N `  { (
( inv g `  W ) `  Z
) } ) )
537, 15, 9grpinvnzcl 14375 . . . 4  |-  ( ( W  e.  Grp  /\  Z  e.  ( V  \  {  .0.  } ) )  ->  ( ( inv g `  W ) `
 Z )  e.  ( V  \  {  .0.  } ) )
5436, 4, 53syl2anc 645 . . 3  |-  ( ph  ->  ( ( inv g `  W ) `  Z
)  e.  ( V 
\  {  .0.  }
) )
557, 10, 15, 16, 17, 18, 19, 49, 52, 1, 54, 8baerlem5b 30594 . 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 5726 . . 3  |-  ( ph  ->  ( ( N `  { Y } )  .(+)  ( N `  { ( ( inv g `  W ) `  Z
) } ) )  =  ( ( N `
 { Y }
)  .(+)  ( N `  { Z } ) ) )
5712eqcomd 2258 . . . . . . 7  |-  ( ph  ->  ( Y  .+  (
( inv g `  W ) `  Z
) )  =  ( Y  .-  Z ) )
5857oveq2d 5726 . . . . . 6  |-  ( ph  ->  ( X  .-  ( Y  .+  ( ( inv g `  W ) `
 Z ) ) )  =  ( X 
.-  ( Y  .-  Z ) ) )
5958sneqd 3557 . . . . 5  |-  ( ph  ->  { ( X  .-  ( Y  .+  ( ( inv g `  W
) `  Z )
) ) }  =  { ( X  .-  ( Y  .-  Z ) ) } )
6059fveq2d 5381 . . . 4  |-  ( ph  ->  ( N `  {
( X  .-  ( Y  .+  ( ( inv g `  W ) `
 Z ) ) ) } )  =  ( N `  {
( X  .-  ( Y  .-  Z ) ) } ) )
6160oveq1d 5725 . . 3  |-  ( ph  ->  ( ( N `  { ( X  .-  ( Y  .+  ( ( inv g `  W
) `  Z )
) ) } ) 
.(+)  ( N `  { X } ) )  =  ( ( N `
 { ( X 
.-  ( Y  .-  Z ) ) } )  .(+)  ( N `  { X } ) ) )
6256, 61ineq12d 3279 . 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 2289 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 1619    e. wcel 1621    =/= wne 2412    \ cdif 3075    i^i cin 3077   {csn 3544   {cpr 3545   ` cfv 4592  (class class class)co 5710   Basecbs 13022   +g cplusg 13082   0gc0g 13274   Grpcgrp 14197   inv gcminusg 14198   -gcsg 14200   LSSumclsm 14780   LModclmod 15462   LSubSpclss 15524   LSpanclspn 15563   LVecclvec 15690
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-cnex 8673  ax-resscn 8674  ax-1cn 8675  ax-icn 8676  ax-addcl 8677  ax-addrcl 8678  ax-mulcl 8679  ax-mulrcl 8680  ax-mulcom 8681  ax-addass 8682  ax-mulass 8683  ax-distr 8684  ax-i2m1 8685  ax-1ne0 8686  ax-1rid 8687  ax-rnegex 8688  ax-rrecex 8689  ax-cnre 8690  ax-pre-lttri 8691  ax-pre-lttrn 8692  ax-pre-ltadd 8693  ax-pre-mulgt0 8694
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-int 3761  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-tpos 6086  df-iota 6143  df-riota 6190  df-recs 6274  df-rdg 6309  df-er 6546  df-en 6750  df-dom 6751  df-sdom 6752  df-pnf 8749  df-mnf 8750  df-xr 8751  df-ltxr 8752  df-le 8753  df-sub 8919  df-neg 8920  df-n 9627  df-2 9684  df-3 9685  df-ndx 13025  df-slot 13026  df-base 13027  df-sets 13028  df-ress 13029  df-plusg 13095  df-mulr 13096  df-0g 13278  df-mnd 14202  df-submnd 14251  df-grp 14324  df-minusg 14325  df-sbg 14326  df-subg 14453  df-cntz 14628  df-lsm 14782  df-cmn 14926  df-abl 14927  df-mgp 15161  df-ring 15175  df-ur 15177  df-oppr 15240  df-dvdsr 15258  df-unit 15259  df-invr 15289  df-drng 15349  df-lmod 15464  df-lss 15525  df-lsp 15564  df-lvec 15691
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