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Theorem baerlem5bmN 35367
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 35368 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.  }
) )
21eldifad 3345 . . . . 5  |-  ( ph  ->  Y  e.  V )
3 baerlem3.z . . . . . 6  |-  ( ph  ->  Z  e.  ( V 
\  {  .0.  }
) )
43eldifad 3345 . . . . 5  |-  ( ph  ->  Z  e.  V )
5 baerlem3.v . . . . . 6  |-  V  =  ( Base `  W
)
6 baerlem5a.p . . . . . 6  |-  .+  =  ( +g  `  W )
7 eqid 2443 . . . . . 6  |-  ( invg `  W )  =  ( invg `  W )
8 baerlem3.m . . . . . 6  |-  .-  =  ( -g `  W )
95, 6, 7, 8grpsubval 15586 . . . . 5  |-  ( ( Y  e.  V  /\  Z  e.  V )  ->  ( Y  .-  Z
)  =  ( Y 
.+  ( ( invg `  W ) `
 Z ) ) )
102, 4, 9syl2anc 661 . . . 4  |-  ( ph  ->  ( Y  .-  Z
)  =  ( Y 
.+  ( ( invg `  W ) `
 Z ) ) )
1110sneqd 3894 . . 3  |-  ( ph  ->  { ( Y  .-  Z ) }  =  { ( Y  .+  ( ( invg `  W ) `  Z
) ) } )
1211fveq2d 5700 . 2  |-  ( ph  ->  ( N `  {
( Y  .-  Z
) } )  =  ( N `  {
( Y  .+  (
( invg `  W ) `  Z
) ) } ) )
13 baerlem3.o . . 3  |-  .0.  =  ( 0g `  W )
14 baerlem3.s . . 3  |-  .(+)  =  (
LSSum `  W )
15 baerlem3.n . . 3  |-  N  =  ( LSpan `  W )
16 baerlem3.w . . 3  |-  ( ph  ->  W  e.  LVec )
17 baerlem3.x . . 3  |-  ( ph  ->  X  e.  V )
18 lveclmod 17192 . . . . . 6  |-  ( W  e.  LVec  ->  W  e. 
LMod )
1916, 18syl 16 . . . . 5  |-  ( ph  ->  W  e.  LMod )
205, 7lmodvnegcl 16991 . . . . 5  |-  ( ( W  e.  LMod  /\  Z  e.  V )  ->  (
( invg `  W ) `  Z
)  e.  V )
2119, 4, 20syl2anc 661 . . . 4  |-  ( ph  ->  ( ( invg `  W ) `  Z
)  e.  V )
22 eqid 2443 . . . . . 6  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
235, 22, 15, 19, 2, 4lspprcl 17064 . . . . . 6  |-  ( ph  ->  ( N `  { Y ,  Z }
)  e.  ( LSubSp `  W ) )
24 baerlem3.c . . . . . 6  |-  ( ph  ->  -.  X  e.  ( N `  { Y ,  Z } ) )
255, 13, 22, 19, 23, 17, 24lssneln0 17038 . . . . 5  |-  ( ph  ->  X  e.  ( V 
\  {  .0.  }
) )
265, 15, 16, 17, 2, 4, 24lspindpi 17218 . . . . . 6  |-  ( ph  ->  ( ( N `  { X } )  =/=  ( N `  { Y } )  /\  ( N `  { X } )  =/=  ( N `  { Z } ) ) )
2726simpld 459 . . . . 5  |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y } ) )
285, 13, 15, 16, 25, 2, 27lspsnne1 17203 . . . 4  |-  ( ph  ->  -.  X  e.  ( N `  { Y } ) )
29 baerlem3.d . . . . . . . 8  |-  ( ph  ->  ( N `  { Y } )  =/=  ( N `  { Z } ) )
3029necomd 2700 . . . . . . 7  |-  ( ph  ->  ( N `  { Z } )  =/=  ( N `  { Y } ) )
315, 13, 15, 16, 3, 2, 30lspsnne1 17203 . . . . . 6  |-  ( ph  ->  -.  Z  e.  ( N `  { Y } ) )
325, 15, 16, 17, 4, 2, 31, 24lspexchn2 17217 . . . . 5  |-  ( ph  ->  -.  Z  e.  ( N `  { Y ,  X } ) )
33 lmodgrp 16960 . . . . . . . . 9  |-  ( W  e.  LMod  ->  W  e. 
Grp )
3419, 33syl 16 . . . . . . . 8  |-  ( ph  ->  W  e.  Grp )
3534adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( ( invg `  W ) `
 Z )  e.  ( N `  { Y ,  X }
) )  ->  W  e.  Grp )
364adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( ( invg `  W ) `
 Z )  e.  ( N `  { Y ,  X }
) )  ->  Z  e.  V )
375, 7grpinvinv 15598 . . . . . . 7  |-  ( ( W  e.  Grp  /\  Z  e.  V )  ->  ( ( invg `  W ) `  (
( invg `  W ) `  Z
) )  =  Z )
3835, 36, 37syl2anc 661 . . . . . 6  |-  ( (
ph  /\  ( ( invg `  W ) `
 Z )  e.  ( N `  { Y ,  X }
) )  ->  (
( invg `  W ) `  (
( invg `  W ) `  Z
) )  =  Z )
3919adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( ( invg `  W ) `
 Z )  e.  ( N `  { Y ,  X }
) )  ->  W  e.  LMod )
405, 22, 15, 19, 2, 17lspprcl 17064 . . . . . . . 8  |-  ( ph  ->  ( N `  { Y ,  X }
)  e.  ( LSubSp `  W ) )
4140adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( ( invg `  W ) `
 Z )  e.  ( N `  { Y ,  X }
) )  ->  ( N `  { Y ,  X } )  e.  ( LSubSp `  W )
)
42 simpr 461 . . . . . . 7  |-  ( (
ph  /\  ( ( invg `  W ) `
 Z )  e.  ( N `  { Y ,  X }
) )  ->  (
( invg `  W ) `  Z
)  e.  ( N `
 { Y ,  X } ) )
4322, 7lssvnegcl 17042 . . . . . . 7  |-  ( ( W  e.  LMod  /\  ( N `  { Y ,  X } )  e.  ( LSubSp `  W )  /\  ( ( invg `  W ) `  Z
)  e.  ( N `
 { Y ,  X } ) )  -> 
( ( invg `  W ) `  (
( invg `  W ) `  Z
) )  e.  ( N `  { Y ,  X } ) )
4439, 41, 42, 43syl3anc 1218 . . . . . 6  |-  ( (
ph  /\  ( ( invg `  W ) `
 Z )  e.  ( N `  { Y ,  X }
) )  ->  (
( invg `  W ) `  (
( invg `  W ) `  Z
) )  e.  ( N `  { Y ,  X } ) )
4538, 44eqeltrrd 2518 . . . . 5  |-  ( (
ph  /\  ( ( invg `  W ) `
 Z )  e.  ( N `  { Y ,  X }
) )  ->  Z  e.  ( N `  { Y ,  X }
) )
4632, 45mtand 659 . . . 4  |-  ( ph  ->  -.  ( ( invg `  W ) `
 Z )  e.  ( N `  { Y ,  X }
) )
475, 15, 16, 21, 17, 2, 28, 46lspexchn2 17217 . . 3  |-  ( ph  ->  -.  X  e.  ( N `  { Y ,  ( ( invg `  W ) `
 Z ) } ) )
485, 7, 15lspsnneg 17092 . . . . 5  |-  ( ( W  e.  LMod  /\  Z  e.  V )  ->  ( N `  { (
( invg `  W ) `  Z
) } )  =  ( N `  { Z } ) )
4919, 4, 48syl2anc 661 . . . 4  |-  ( ph  ->  ( N `  {
( ( invg `  W ) `  Z
) } )  =  ( N `  { Z } ) )
5029, 49neeqtrrd 2637 . . 3  |-  ( ph  ->  ( N `  { Y } )  =/=  ( N `  { (
( invg `  W ) `  Z
) } ) )
515, 13, 7grpinvnzcl 15603 . . . 4  |-  ( ( W  e.  Grp  /\  Z  e.  ( V  \  {  .0.  } ) )  ->  ( ( invg `  W ) `
 Z )  e.  ( V  \  {  .0.  } ) )
5234, 3, 51syl2anc 661 . . 3  |-  ( ph  ->  ( ( invg `  W ) `  Z
)  e.  ( V 
\  {  .0.  }
) )
535, 8, 13, 14, 15, 16, 17, 47, 50, 1, 52, 6baerlem5b 35365 . 2  |-  ( ph  ->  ( N `  {
( Y  .+  (
( invg `  W ) `  Z
) ) } )  =  ( ( ( N `  { Y } )  .(+)  ( N `
 { ( ( invg `  W
) `  Z ) } ) )  i^i  ( ( N `  { ( X  .-  ( Y  .+  ( ( invg `  W
) `  Z )
) ) } ) 
.(+)  ( N `  { X } ) ) ) )
5449oveq2d 6112 . . 3  |-  ( ph  ->  ( ( N `  { Y } )  .(+)  ( N `  { ( ( invg `  W ) `  Z
) } ) )  =  ( ( N `
 { Y }
)  .(+)  ( N `  { Z } ) ) )
5510eqcomd 2448 . . . . . . 7  |-  ( ph  ->  ( Y  .+  (
( invg `  W ) `  Z
) )  =  ( Y  .-  Z ) )
5655oveq2d 6112 . . . . . 6  |-  ( ph  ->  ( X  .-  ( Y  .+  ( ( invg `  W ) `
 Z ) ) )  =  ( X 
.-  ( Y  .-  Z ) ) )
5756sneqd 3894 . . . . 5  |-  ( ph  ->  { ( X  .-  ( Y  .+  ( ( invg `  W
) `  Z )
) ) }  =  { ( X  .-  ( Y  .-  Z ) ) } )
5857fveq2d 5700 . . . 4  |-  ( ph  ->  ( N `  {
( X  .-  ( Y  .+  ( ( invg `  W ) `
 Z ) ) ) } )  =  ( N `  {
( X  .-  ( Y  .-  Z ) ) } ) )
5958oveq1d 6111 . . 3  |-  ( ph  ->  ( ( N `  { ( X  .-  ( Y  .+  ( ( invg `  W
) `  Z )
) ) } ) 
.(+)  ( N `  { X } ) )  =  ( ( N `
 { ( X 
.-  ( Y  .-  Z ) ) } )  .(+)  ( N `  { X } ) ) )
6054, 59ineq12d 3558 . 2  |-  ( ph  ->  ( ( ( N `
 { Y }
)  .(+)  ( N `  { ( ( invg `  W ) `
 Z ) } ) )  i^i  (
( N `  {
( X  .-  ( Y  .+  ( ( invg `  W ) `
 Z ) ) ) } )  .(+)  ( N `  { X } ) ) )  =  ( ( ( N `  { Y } )  .(+)  ( N `
 { Z }
) )  i^i  (
( N `  {
( X  .-  ( Y  .-  Z ) ) } )  .(+)  ( N `
 { X }
) ) ) )
6112, 53, 603eqtrd 2479 1  |-  ( ph  ->  ( N `  {
( Y  .-  Z
) } )  =  ( ( ( N `
 { Y }
)  .(+)  ( N `  { Z } ) )  i^i  ( ( N `
 { ( X 
.-  ( Y  .-  Z ) ) } )  .(+)  ( N `  { X } ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2611    \ cdif 3330    i^i cin 3332   {csn 3882   {cpr 3884   ` cfv 5423  (class class class)co 6096   Basecbs 14179   +g cplusg 14243   0gc0g 14383   Grpcgrp 15415   invgcminusg 15416   -gcsg 15418   LSSumclsm 16138   LModclmod 16953   LSubSpclss 17018   LSpanclspn 17057   LVecclvec 17188
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-tpos 6750  df-recs 6837  df-rdg 6871  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-nn 10328  df-2 10385  df-3 10386  df-ndx 14182  df-slot 14183  df-base 14184  df-sets 14185  df-ress 14186  df-plusg 14256  df-mulr 14257  df-0g 14385  df-mnd 15420  df-submnd 15470  df-grp 15550  df-minusg 15551  df-sbg 15552  df-subg 15683  df-cntz 15840  df-lsm 16140  df-cmn 16284  df-abl 16285  df-mgp 16597  df-ur 16609  df-rng 16652  df-oppr 16720  df-dvdsr 16738  df-unit 16739  df-invr 16769  df-drng 16839  df-lmod 16955  df-lss 17019  df-lsp 17058  df-lvec 17189
This theorem is referenced by: (None)
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