Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  baerlem5amN Structured version   Unicode version

Theorem baerlem5amN 37586
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 37588 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 3483 . . . . . 6  |-  ( ph  ->  Y  e.  V )
3 baerlem3.z . . . . . . 7  |-  ( ph  ->  Z  e.  ( V 
\  {  .0.  }
) )
43eldifad 3483 . . . . . 6  |-  ( ph  ->  Z  e.  V )
5 baerlem3.v . . . . . . 7  |-  V  =  ( Base `  W
)
6 baerlem5a.p . . . . . . 7  |-  .+  =  ( +g  `  W )
7 eqid 2457 . . . . . . 7  |-  ( invg `  W )  =  ( invg `  W )
8 baerlem3.m . . . . . . 7  |-  .-  =  ( -g `  W )
95, 6, 7, 8grpsubval 16220 . . . . . 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 6312 . . . 4  |-  ( ph  ->  ( X  .-  ( Y  .-  Z ) )  =  ( X  .-  ( Y  .+  ( ( invg `  W
) `  Z )
) ) )
1211sneqd 4044 . . 3  |-  ( ph  ->  { ( X  .-  ( Y  .-  Z ) ) }  =  {
( X  .-  ( Y  .+  ( ( invg `  W ) `
 Z ) ) ) } )
1312fveq2d 5876 . 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 17879 . . . . . 6  |-  ( W  e.  LVec  ->  W  e. 
LMod )
2017, 19syl 16 . . . . 5  |-  ( ph  ->  W  e.  LMod )
215, 7lmodvnegcl 17678 . . . . 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 2457 . . . . . 6  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
245, 23, 16, 20, 2, 4lspprcl 17751 . . . . . 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 17725 . . . . 5  |-  ( ph  ->  X  e.  ( V 
\  {  .0.  }
) )
275, 16, 17, 18, 2, 4, 25lspindpi 17905 . . . . . 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 17890 . . . 4  |-  ( ph  ->  -.  X  e.  ( N `  { Y } ) )
30 baerlem3.d . . . . . . . 8  |-  ( ph  ->  ( N `  { Y } )  =/=  ( N `  { Z } ) )
3130necomd 2728 . . . . . . 7  |-  ( ph  ->  ( N `  { Z } )  =/=  ( N `  { Y } ) )
325, 14, 16, 17, 3, 2, 31lspsnne1 17890 . . . . . 6  |-  ( ph  ->  -.  Z  e.  ( N `  { Y } ) )
335, 16, 17, 18, 4, 2, 32, 25lspexchn2 17904 . . . . 5  |-  ( ph  ->  -.  Z  e.  ( N `  { Y ,  X } ) )
34 lmodgrp 17646 . . . . . . . . 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 16232 . . . . . . 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 17751 . . . . . . . 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 17729 . . . . . . 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 1228 . . . . . 6  |-  ( (
ph  /\  ( ( invg `  W ) `
 Z )  e.  ( N `  { Y ,  X }
) )  ->  (
( invg `  W ) `  (
( invg `  W ) `  Z
) )  e.  ( N `  { Y ,  X } ) )
4639, 45eqeltrrd 2546 . . . . 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 17904 . . 3  |-  ( ph  ->  -.  X  e.  ( N `  { Y ,  ( ( invg `  W ) `
 Z ) } ) )
495, 7, 16lspsnneg 17779 . . . . 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 2757 . . 3  |-  ( ph  ->  ( N `  { Y } )  =/=  ( N `  { (
( invg `  W ) `  Z
) } ) )
525, 14, 7grpinvnzcl 16237 . . . 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 37584 . 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 6312 . . 3  |-  ( ph  ->  ( ( N `  { ( X  .-  Y ) } ) 
.(+)  ( N `  { ( ( invg `  W ) `
 Z ) } ) )  =  ( ( N `  {
( X  .-  Y
) } )  .(+)  ( N `  { Z } ) ) )
565, 6, 8, 7, 35, 18, 4grpsubinv 16238 . . . . . 6  |-  ( ph  ->  ( X  .-  (
( invg `  W ) `  Z
) )  =  ( X  .+  Z ) )
5756sneqd 4044 . . . . 5  |-  ( ph  ->  { ( X  .-  ( ( invg `  W ) `  Z
) ) }  =  { ( X  .+  Z ) } )
5857fveq2d 5876 . . . 4  |-  ( ph  ->  ( N `  {
( X  .-  (
( invg `  W ) `  Z
) ) } )  =  ( N `  { ( X  .+  Z ) } ) )
5958oveq1d 6311 . . 3  |-  ( ph  ->  ( ( N `  { ( X  .-  ( ( invg `  W ) `  Z
) ) } ) 
.(+)  ( N `  { Y } ) )  =  ( ( N `
 { ( X 
.+  Z ) } )  .(+)  ( N `  { Y } ) ) )
6055, 59ineq12d 3697 . 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 2502 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 1395    e. wcel 1819    =/= wne 2652    \ cdif 3468    i^i cin 3470   {csn 4032   {cpr 4034   ` cfv 5594  (class class class)co 6296   Basecbs 14644   +g cplusg 14712   0gc0g 14857   Grpcgrp 16180   invgcminusg 16181   -gcsg 16182   LSSumclsm 16781   LModclmod 17639   LSubSpclss 17705   LSpanclspn 17744   LVecclvec 17875
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-tpos 6973  df-recs 7060  df-rdg 7094  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-3 10616  df-ndx 14647  df-slot 14648  df-base 14649  df-sets 14650  df-ress 14651  df-plusg 14725  df-mulr 14726  df-0g 14859  df-mgm 15999  df-sgrp 16038  df-mnd 16048  df-submnd 16094  df-grp 16184  df-minusg 16185  df-sbg 16186  df-subg 16325  df-cntz 16482  df-lsm 16783  df-cmn 16927  df-abl 16928  df-mgp 17269  df-ur 17281  df-ring 17327  df-oppr 17399  df-dvdsr 17417  df-unit 17418  df-invr 17448  df-drng 17525  df-lmod 17641  df-lss 17706  df-lsp 17745  df-lvec 17876
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator