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Theorem baerlem5amN 35373
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 35375 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 3352 . . . . . 6  |-  ( ph  ->  Y  e.  V )
3 baerlem3.z . . . . . . 7  |-  ( ph  ->  Z  e.  ( V 
\  {  .0.  }
) )
43eldifad 3352 . . . . . 6  |-  ( ph  ->  Z  e.  V )
5 baerlem3.v . . . . . . 7  |-  V  =  ( Base `  W
)
6 baerlem5a.p . . . . . . 7  |-  .+  =  ( +g  `  W )
7 eqid 2443 . . . . . . 7  |-  ( invg `  W )  =  ( invg `  W )
8 baerlem3.m . . . . . . 7  |-  .-  =  ( -g `  W )
95, 6, 7, 8grpsubval 15593 . . . . . 6  |-  ( ( Y  e.  V  /\  Z  e.  V )  ->  ( Y  .-  Z
)  =  ( Y 
.+  ( ( invg `  W ) `
 Z ) ) )
102, 4, 9syl2anc 661 . . . . 5  |-  ( ph  ->  ( Y  .-  Z
)  =  ( Y 
.+  ( ( invg `  W ) `
 Z ) ) )
1110oveq2d 6119 . . . 4  |-  ( ph  ->  ( X  .-  ( Y  .-  Z ) )  =  ( X  .-  ( Y  .+  ( ( invg `  W
) `  Z )
) ) )
1211sneqd 3901 . . 3  |-  ( ph  ->  { ( X  .-  ( Y  .-  Z ) ) }  =  {
( X  .-  ( Y  .+  ( ( invg `  W ) `
 Z ) ) ) } )
1312fveq2d 5707 . 2  |-  ( ph  ->  ( N `  {
( X  .-  ( Y  .-  Z ) ) } )  =  ( N `  { ( X  .-  ( Y 
.+  ( ( invg `  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 17199 . . . . . 6  |-  ( W  e.  LVec  ->  W  e. 
LMod )
2017, 19syl 16 . . . . 5  |-  ( ph  ->  W  e.  LMod )
215, 7lmodvnegcl 16998 . . . . 5  |-  ( ( W  e.  LMod  /\  Z  e.  V )  ->  (
( invg `  W ) `  Z
)  e.  V )
2220, 4, 21syl2anc 661 . . . 4  |-  ( ph  ->  ( ( invg `  W ) `  Z
)  e.  V )
23 eqid 2443 . . . . . 6  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
245, 23, 16, 20, 2, 4lspprcl 17071 . . . . . 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 17045 . . . . 5  |-  ( ph  ->  X  e.  ( V 
\  {  .0.  }
) )
275, 16, 17, 18, 2, 4, 25lspindpi 17225 . . . . . 6  |-  ( ph  ->  ( ( N `  { X } )  =/=  ( N `  { Y } )  /\  ( N `  { X } )  =/=  ( N `  { Z } ) ) )
2827simpld 459 . . . . 5  |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y } ) )
295, 14, 16, 17, 26, 2, 28lspsnne1 17210 . . . 4  |-  ( ph  ->  -.  X  e.  ( N `  { Y } ) )
30 baerlem3.d . . . . . . . 8  |-  ( ph  ->  ( N `  { Y } )  =/=  ( N `  { Z } ) )
3130necomd 2707 . . . . . . 7  |-  ( ph  ->  ( N `  { Z } )  =/=  ( N `  { Y } ) )
325, 14, 16, 17, 3, 2, 31lspsnne1 17210 . . . . . 6  |-  ( ph  ->  -.  Z  e.  ( N `  { Y } ) )
335, 16, 17, 18, 4, 2, 32, 25lspexchn2 17224 . . . . 5  |-  ( ph  ->  -.  Z  e.  ( N `  { Y ,  X } ) )
34 lmodgrp 16967 . . . . . . . . 9  |-  ( W  e.  LMod  ->  W  e. 
Grp )
3517, 19, 343syl 20 . . . . . . . 8  |-  ( ph  ->  W  e.  Grp )
3635adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( ( invg `  W ) `
 Z )  e.  ( N `  { Y ,  X }
) )  ->  W  e.  Grp )
374adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( ( invg `  W ) `
 Z )  e.  ( N `  { Y ,  X }
) )  ->  Z  e.  V )
385, 7grpinvinv 15605 . . . . . . 7  |-  ( ( W  e.  Grp  /\  Z  e.  V )  ->  ( ( invg `  W ) `  (
( invg `  W ) `  Z
) )  =  Z )
3936, 37, 38syl2anc 661 . . . . . 6  |-  ( (
ph  /\  ( ( invg `  W ) `
 Z )  e.  ( N `  { Y ,  X }
) )  ->  (
( invg `  W ) `  (
( invg `  W ) `  Z
) )  =  Z )
4020adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( ( invg `  W ) `
 Z )  e.  ( N `  { Y ,  X }
) )  ->  W  e.  LMod )
415, 23, 16, 20, 2, 18lspprcl 17071 . . . . . . . 8  |-  ( ph  ->  ( N `  { Y ,  X }
)  e.  ( LSubSp `  W ) )
4241adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( ( invg `  W ) `
 Z )  e.  ( N `  { Y ,  X }
) )  ->  ( N `  { Y ,  X } )  e.  ( LSubSp `  W )
)
43 simpr 461 . . . . . . 7  |-  ( (
ph  /\  ( ( invg `  W ) `
 Z )  e.  ( N `  { Y ,  X }
) )  ->  (
( invg `  W ) `  Z
)  e.  ( N `
 { Y ,  X } ) )
4423, 7lssvnegcl 17049 . . . . . . 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 } ) )
4540, 42, 43, 44syl3anc 1218 . . . . . 6  |-  ( (
ph  /\  ( ( invg `  W ) `
 Z )  e.  ( N `  { Y ,  X }
) )  ->  (
( invg `  W ) `  (
( invg `  W ) `  Z
) )  e.  ( N `  { Y ,  X } ) )
4639, 45eqeltrrd 2518 . . . . 5  |-  ( (
ph  /\  ( ( invg `  W ) `
 Z )  e.  ( N `  { Y ,  X }
) )  ->  Z  e.  ( N `  { Y ,  X }
) )
4733, 46mtand 659 . . . 4  |-  ( ph  ->  -.  ( ( invg `  W ) `
 Z )  e.  ( N `  { Y ,  X }
) )
485, 16, 17, 22, 18, 2, 29, 47lspexchn2 17224 . . 3  |-  ( ph  ->  -.  X  e.  ( N `  { Y ,  ( ( invg `  W ) `
 Z ) } ) )
495, 7, 16lspsnneg 17099 . . . . 5  |-  ( ( W  e.  LMod  /\  Z  e.  V )  ->  ( N `  { (
( invg `  W ) `  Z
) } )  =  ( N `  { Z } ) )
5020, 4, 49syl2anc 661 . . . 4  |-  ( ph  ->  ( N `  {
( ( invg `  W ) `  Z
) } )  =  ( N `  { Z } ) )
5130, 50neeqtrrd 2644 . . 3  |-  ( ph  ->  ( N `  { Y } )  =/=  ( N `  { (
( invg `  W ) `  Z
) } ) )
525, 14, 7grpinvnzcl 15610 . . . 4  |-  ( ( W  e.  Grp  /\  Z  e.  ( V  \  {  .0.  } ) )  ->  ( ( invg `  W ) `
 Z )  e.  ( V  \  {  .0.  } ) )
5335, 3, 52syl2anc 661 . . 3  |-  ( ph  ->  ( ( invg `  W ) `  Z
)  e.  ( V 
\  {  .0.  }
) )
545, 8, 14, 15, 16, 17, 18, 48, 51, 1, 53, 6baerlem5a 35371 . 2  |-  ( ph  ->  ( N `  {
( X  .-  ( Y  .+  ( ( invg `  W ) `
 Z ) ) ) } )  =  ( ( ( N `
 { ( X 
.-  Y ) } )  .(+)  ( N `  { ( ( invg `  W ) `
 Z ) } ) )  i^i  (
( N `  {
( X  .-  (
( invg `  W ) `  Z
) ) } ) 
.(+)  ( N `  { Y } ) ) ) )
5550oveq2d 6119 . . 3  |-  ( ph  ->  ( ( N `  { ( X  .-  Y ) } ) 
.(+)  ( N `  { ( ( invg `  W ) `
 Z ) } ) )  =  ( ( N `  {
( X  .-  Y
) } )  .(+)  ( N `  { Z } ) ) )
565, 6, 8, 7, 35, 18, 4grpsubinv 15611 . . . . . 6  |-  ( ph  ->  ( X  .-  (
( invg `  W ) `  Z
) )  =  ( X  .+  Z ) )
5756sneqd 3901 . . . . 5  |-  ( ph  ->  { ( X  .-  ( ( invg `  W ) `  Z
) ) }  =  { ( X  .+  Z ) } )
5857fveq2d 5707 . . . 4  |-  ( ph  ->  ( N `  {
( X  .-  (
( invg `  W ) `  Z
) ) } )  =  ( N `  { ( X  .+  Z ) } ) )
5958oveq1d 6118 . . 3  |-  ( ph  ->  ( ( N `  { ( X  .-  ( ( invg `  W ) `  Z
) ) } ) 
.(+)  ( N `  { Y } ) )  =  ( ( N `
 { ( X 
.+  Z ) } )  .(+)  ( N `  { Y } ) ) )
6055, 59ineq12d 3565 . 2  |-  ( ph  ->  ( ( ( N `
 { ( X 
.-  Y ) } )  .(+)  ( N `  { ( ( invg `  W ) `
 Z ) } ) )  i^i  (
( N `  {
( X  .-  (
( invg `  W ) `  Z
) ) } ) 
.(+)  ( N `  { Y } ) ) )  =  ( ( ( N `  {
( X  .-  Y
) } )  .(+)  ( N `  { Z } ) )  i^i  ( ( N `  { ( X  .+  Z ) } ) 
.(+)  ( N `  { Y } ) ) ) )
6113, 54, 603eqtrd 2479 1  |-  ( ph  ->  ( N `  {
( X  .-  ( Y  .-  Z ) ) } )  =  ( ( ( N `  { ( X  .-  Y ) } ) 
.(+)  ( N `  { Z } ) )  i^i  ( ( N `
 { ( X 
.+  Z ) } )  .(+)  ( N `  { Y } ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2618    \ cdif 3337    i^i cin 3339   {csn 3889   {cpr 3891   ` cfv 5430  (class class class)co 6103   Basecbs 14186   +g cplusg 14250   0gc0g 14390   Grpcgrp 15422   invgcminusg 15423   -gcsg 15425   LSSumclsm 16145   LModclmod 16960   LSubSpclss 17025   LSpanclspn 17064   LVecclvec 17195
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 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-cnex 9350  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-mulcom 9358  ax-addass 9359  ax-mulass 9360  ax-distr 9361  ax-i2m1 9362  ax-1ne0 9363  ax-1rid 9364  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367  ax-pre-lttri 9368  ax-pre-lttrn 9369  ax-pre-ltadd 9370  ax-pre-mulgt0 9371
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 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rmo 2735  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-int 4141  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-om 6489  df-1st 6589  df-2nd 6590  df-tpos 6757  df-recs 6844  df-rdg 6878  df-er 7113  df-en 7323  df-dom 7324  df-sdom 7325  df-pnf 9432  df-mnf 9433  df-xr 9434  df-ltxr 9435  df-le 9436  df-sub 9609  df-neg 9610  df-nn 10335  df-2 10392  df-3 10393  df-ndx 14189  df-slot 14190  df-base 14191  df-sets 14192  df-ress 14193  df-plusg 14263  df-mulr 14264  df-0g 14392  df-mnd 15427  df-submnd 15477  df-grp 15557  df-minusg 15558  df-sbg 15559  df-subg 15690  df-cntz 15847  df-lsm 16147  df-cmn 16291  df-abl 16292  df-mgp 16604  df-ur 16616  df-rng 16659  df-oppr 16727  df-dvdsr 16745  df-unit 16746  df-invr 16776  df-drng 16846  df-lmod 16962  df-lss 17026  df-lsp 17065  df-lvec 17196
This theorem is referenced by: (None)
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