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Theorem baerlem5amN 32528
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 32530 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.  }
) )
2 eldifi 3311 . . . . . . 7  |-  ( Y  e.  ( V  \  {  .0.  } )  ->  Y  e.  V )
31, 2syl 15 . . . . . 6  |-  ( ph  ->  Y  e.  V )
4 baerlem3.z . . . . . . 7  |-  ( ph  ->  Z  e.  ( V 
\  {  .0.  }
) )
5 eldifi 3311 . . . . . . 7  |-  ( Z  e.  ( V  \  {  .0.  } )  ->  Z  e.  V )
64, 5syl 15 . . . . . 6  |-  ( ph  ->  Z  e.  V )
7 baerlem3.v . . . . . . 7  |-  V  =  ( Base `  W
)
8 baerlem5a.p . . . . . . 7  |-  .+  =  ( +g  `  W )
9 eqid 2296 . . . . . . 7  |-  ( inv g `  W )  =  ( inv g `  W )
10 baerlem3.m . . . . . . 7  |-  .-  =  ( -g `  W )
117, 8, 9, 10grpsubval 14541 . . . . . 6  |-  ( ( Y  e.  V  /\  Z  e.  V )  ->  ( Y  .-  Z
)  =  ( Y 
.+  ( ( inv g `  W ) `
 Z ) ) )
123, 6, 11syl2anc 642 . . . . 5  |-  ( ph  ->  ( Y  .-  Z
)  =  ( Y 
.+  ( ( inv g `  W ) `
 Z ) ) )
1312oveq2d 5890 . . . 4  |-  ( ph  ->  ( X  .-  ( Y  .-  Z ) )  =  ( X  .-  ( Y  .+  ( ( inv g `  W
) `  Z )
) ) )
1413sneqd 3666 . . 3  |-  ( ph  ->  { ( X  .-  ( Y  .-  Z ) ) }  =  {
( X  .-  ( Y  .+  ( ( inv g `  W ) `
 Z ) ) ) } )
1514fveq2d 5545 . 2  |-  ( ph  ->  ( N `  {
( X  .-  ( Y  .-  Z ) ) } )  =  ( N `  { ( X  .-  ( Y 
.+  ( ( inv g `  W ) `
 Z ) ) ) } ) )
16 baerlem3.o . . 3  |-  .0.  =  ( 0g `  W )
17 baerlem3.s . . 3  |-  .(+)  =  (
LSSum `  W )
18 baerlem3.n . . 3  |-  N  =  ( LSpan `  W )
19 baerlem3.w . . 3  |-  ( ph  ->  W  e.  LVec )
20 baerlem3.x . . 3  |-  ( ph  ->  X  e.  V )
21 lveclmod 15875 . . . . . 6  |-  ( W  e.  LVec  ->  W  e. 
LMod )
2219, 21syl 15 . . . . 5  |-  ( ph  ->  W  e.  LMod )
237, 9lmodvnegcl 15681 . . . . 5  |-  ( ( W  e.  LMod  /\  Z  e.  V )  ->  (
( inv g `  W ) `  Z
)  e.  V )
2422, 6, 23syl2anc 642 . . . 4  |-  ( ph  ->  ( ( inv g `  W ) `  Z
)  e.  V )
25 eqid 2296 . . . . . 6  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
267, 25, 18, 22, 3, 6lspprcl 15751 . . . . . 6  |-  ( ph  ->  ( N `  { Y ,  Z }
)  e.  ( LSubSp `  W ) )
27 baerlem3.c . . . . . 6  |-  ( ph  ->  -.  X  e.  ( N `  { Y ,  Z } ) )
287, 16, 25, 22, 26, 20, 27lssneln0 15725 . . . . 5  |-  ( ph  ->  X  e.  ( V 
\  {  .0.  }
) )
297, 18, 19, 20, 3, 6, 27lspindpi 15901 . . . . . 6  |-  ( ph  ->  ( ( N `  { X } )  =/=  ( N `  { Y } )  /\  ( N `  { X } )  =/=  ( N `  { Z } ) ) )
3029simpld 445 . . . . 5  |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y } ) )
317, 16, 18, 19, 28, 3, 30lspsnne1 15886 . . . 4  |-  ( ph  ->  -.  X  e.  ( N `  { Y } ) )
32 baerlem3.d . . . . . . . 8  |-  ( ph  ->  ( N `  { Y } )  =/=  ( N `  { Z } ) )
3332necomd 2542 . . . . . . 7  |-  ( ph  ->  ( N `  { Z } )  =/=  ( N `  { Y } ) )
347, 16, 18, 19, 4, 3, 33lspsnne1 15886 . . . . . 6  |-  ( ph  ->  -.  Z  e.  ( N `  { Y } ) )
357, 18, 19, 20, 6, 3, 34, 27lspexchn2 15900 . . . . 5  |-  ( ph  ->  -.  Z  e.  ( N `  { Y ,  X } ) )
36 lmodgrp 15650 . . . . . . . . 9  |-  ( W  e.  LMod  ->  W  e. 
Grp )
3719, 21, 363syl 18 . . . . . . . 8  |-  ( ph  ->  W  e.  Grp )
3837adantr 451 . . . . . . 7  |-  ( (
ph  /\  ( ( inv g `  W ) `
 Z )  e.  ( N `  { Y ,  X }
) )  ->  W  e.  Grp )
396adantr 451 . . . . . . 7  |-  ( (
ph  /\  ( ( inv g `  W ) `
 Z )  e.  ( N `  { Y ,  X }
) )  ->  Z  e.  V )
407, 9grpinvinv 14551 . . . . . . 7  |-  ( ( W  e.  Grp  /\  Z  e.  V )  ->  ( ( inv g `  W ) `  (
( inv g `  W ) `  Z
) )  =  Z )
4138, 39, 40syl2anc 642 . . . . . 6  |-  ( (
ph  /\  ( ( inv g `  W ) `
 Z )  e.  ( N `  { Y ,  X }
) )  ->  (
( inv g `  W ) `  (
( inv g `  W ) `  Z
) )  =  Z )
4222adantr 451 . . . . . . 7  |-  ( (
ph  /\  ( ( inv g `  W ) `
 Z )  e.  ( N `  { Y ,  X }
) )  ->  W  e.  LMod )
437, 25, 18, 22, 3, 20lspprcl 15751 . . . . . . . 8  |-  ( ph  ->  ( N `  { Y ,  X }
)  e.  ( LSubSp `  W ) )
4443adantr 451 . . . . . . 7  |-  ( (
ph  /\  ( ( inv g `  W ) `
 Z )  e.  ( N `  { Y ,  X }
) )  ->  ( N `  { Y ,  X } )  e.  ( LSubSp `  W )
)
45 simpr 447 . . . . . . 7  |-  ( (
ph  /\  ( ( inv g `  W ) `
 Z )  e.  ( N `  { Y ,  X }
) )  ->  (
( inv g `  W ) `  Z
)  e.  ( N `
 { Y ,  X } ) )
4625, 9lssvnegcl 15729 . . . . . . 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 } ) )
4742, 44, 45, 46syl3anc 1182 . . . . . 6  |-  ( (
ph  /\  ( ( inv g `  W ) `
 Z )  e.  ( N `  { Y ,  X }
) )  ->  (
( inv g `  W ) `  (
( inv g `  W ) `  Z
) )  e.  ( N `  { Y ,  X } ) )
4841, 47eqeltrrd 2371 . . . . 5  |-  ( (
ph  /\  ( ( inv g `  W ) `
 Z )  e.  ( N `  { Y ,  X }
) )  ->  Z  e.  ( N `  { Y ,  X }
) )
4935, 48mtand 640 . . . 4  |-  ( ph  ->  -.  ( ( inv g `  W ) `
 Z )  e.  ( N `  { Y ,  X }
) )
507, 18, 19, 24, 20, 3, 31, 49lspexchn2 15900 . . 3  |-  ( ph  ->  -.  X  e.  ( N `  { Y ,  ( ( inv g `  W ) `
 Z ) } ) )
517, 9, 18lspsnneg 15779 . . . . 5  |-  ( ( W  e.  LMod  /\  Z  e.  V )  ->  ( N `  { (
( inv g `  W ) `  Z
) } )  =  ( N `  { Z } ) )
5222, 6, 51syl2anc 642 . . . 4  |-  ( ph  ->  ( N `  {
( ( inv g `  W ) `  Z
) } )  =  ( N `  { Z } ) )
5332, 52neeqtrrd 2483 . . 3  |-  ( ph  ->  ( N `  { Y } )  =/=  ( N `  { (
( inv g `  W ) `  Z
) } ) )
547, 16, 9grpinvnzcl 14556 . . . 4  |-  ( ( W  e.  Grp  /\  Z  e.  ( V  \  {  .0.  } ) )  ->  ( ( inv g `  W ) `
 Z )  e.  ( V  \  {  .0.  } ) )
5537, 4, 54syl2anc 642 . . 3  |-  ( ph  ->  ( ( inv g `  W ) `  Z
)  e.  ( V 
\  {  .0.  }
) )
567, 10, 16, 17, 18, 19, 20, 50, 53, 1, 55, 8baerlem5a 32526 . 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 } ) ) ) )
5752oveq2d 5890 . . 3  |-  ( ph  ->  ( ( N `  { ( X  .-  Y ) } ) 
.(+)  ( N `  { ( ( inv g `  W ) `
 Z ) } ) )  =  ( ( N `  {
( X  .-  Y
) } )  .(+)  ( N `  { Z } ) ) )
587, 8, 10, 9, 37, 20, 6grpsubinv 14557 . . . . . 6  |-  ( ph  ->  ( X  .-  (
( inv g `  W ) `  Z
) )  =  ( X  .+  Z ) )
5958sneqd 3666 . . . . 5  |-  ( ph  ->  { ( X  .-  ( ( inv g `  W ) `  Z
) ) }  =  { ( X  .+  Z ) } )
6059fveq2d 5545 . . . 4  |-  ( ph  ->  ( N `  {
( X  .-  (
( inv g `  W ) `  Z
) ) } )  =  ( N `  { ( X  .+  Z ) } ) )
6160oveq1d 5889 . . 3  |-  ( ph  ->  ( ( N `  { ( X  .-  ( ( inv g `  W ) `  Z
) ) } ) 
.(+)  ( N `  { Y } ) )  =  ( ( N `
 { ( X 
.+  Z ) } )  .(+)  ( N `  { Y } ) ) )
6257, 61ineq12d 3384 . 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 } ) ) ) )
6315, 56, 623eqtrd 2332 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 358    = wceq 1632    e. wcel 1696    =/= wne 2459    \ cdif 3162    i^i cin 3164   {csn 3653   {cpr 3654   ` cfv 5271  (class class class)co 5874   Basecbs 13164   +g cplusg 13224   0gc0g 13416   Grpcgrp 14378   inv gcminusg 14379   -gcsg 14381   LSSumclsm 14961   LModclmod 15643   LSubSpclss 15705   LSpanclspn 15744   LVecclvec 15871
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-tpos 6250  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-0g 13420  df-mnd 14383  df-submnd 14432  df-grp 14505  df-minusg 14506  df-sbg 14507  df-subg 14634  df-cntz 14809  df-lsm 14963  df-cmn 15107  df-abl 15108  df-mgp 15342  df-rng 15356  df-ur 15358  df-oppr 15421  df-dvdsr 15439  df-unit 15440  df-invr 15470  df-drng 15530  df-lmod 15645  df-lss 15706  df-lsp 15745  df-lvec 15872
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