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Theorem baerlem5amN 31036
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 31038 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 3240 . . . . . . 7  |-  ( Y  e.  ( V  \  {  .0.  } )  ->  Y  e.  V )
31, 2syl 17 . . . . . 6  |-  ( ph  ->  Y  e.  V )
4 baerlem3.z . . . . . . 7  |-  ( ph  ->  Z  e.  ( V 
\  {  .0.  }
) )
5 eldifi 3240 . . . . . . 7  |-  ( Z  e.  ( V  \  {  .0.  } )  ->  Z  e.  V )
64, 5syl 17 . . . . . 6  |-  ( ph  ->  Z  e.  V )
7 baerlem3.v . . . . . . 7  |-  V  =  ( Base `  W
)
8 baerlem5a.p . . . . . . 7  |-  .+  =  ( +g  `  W )
9 eqid 2256 . . . . . . 7  |-  ( inv g `  W )  =  ( inv g `  W )
10 baerlem3.m . . . . . . 7  |-  .-  =  ( -g `  W )
117, 8, 9, 10grpsubval 14452 . . . . . 6  |-  ( ( Y  e.  V  /\  Z  e.  V )  ->  ( Y  .-  Z
)  =  ( Y 
.+  ( ( inv g `  W ) `
 Z ) ) )
123, 6, 11syl2anc 645 . . . . 5  |-  ( ph  ->  ( Y  .-  Z
)  =  ( Y 
.+  ( ( inv g `  W ) `
 Z ) ) )
1312oveq2d 5773 . . . 4  |-  ( ph  ->  ( X  .-  ( Y  .-  Z ) )  =  ( X  .-  ( Y  .+  ( ( inv g `  W
) `  Z )
) ) )
1413sneqd 3594 . . 3  |-  ( ph  ->  { ( X  .-  ( Y  .-  Z ) ) }  =  {
( X  .-  ( Y  .+  ( ( inv g `  W ) `
 Z ) ) ) } )
1514fveq2d 5427 . 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 15786 . . . . . 6  |-  ( W  e.  LVec  ->  W  e. 
LMod )
2219, 21syl 17 . . . . 5  |-  ( ph  ->  W  e.  LMod )
237, 9lmodvnegcl 15592 . . . . 5  |-  ( ( W  e.  LMod  /\  Z  e.  V )  ->  (
( inv g `  W ) `  Z
)  e.  V )
2422, 6, 23syl2anc 645 . . . 4  |-  ( ph  ->  ( ( inv g `  W ) `  Z
)  e.  V )
25 eqid 2256 . . . . . 6  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
267, 25, 18, 22, 3, 6lspprcl 15662 . . . . . 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 15636 . . . . 5  |-  ( ph  ->  X  e.  ( V 
\  {  .0.  }
) )
297, 18, 19, 20, 3, 6, 27lspindpi 15812 . . . . . 6  |-  ( ph  ->  ( ( N `  { X } )  =/=  ( N `  { Y } )  /\  ( N `  { X } )  =/=  ( N `  { Z } ) ) )
3029simpld 447 . . . . 5  |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y } ) )
317, 16, 18, 19, 28, 3, 30lspsnne1 15797 . . . 4  |-  ( ph  ->  -.  X  e.  ( N `  { Y } ) )
32 baerlem3.d . . . . . . . 8  |-  ( ph  ->  ( N `  { Y } )  =/=  ( N `  { Z } ) )
3332necomd 2502 . . . . . . 7  |-  ( ph  ->  ( N `  { Z } )  =/=  ( N `  { Y } ) )
347, 16, 18, 19, 4, 3, 33lspsnne1 15797 . . . . . 6  |-  ( ph  ->  -.  Z  e.  ( N `  { Y } ) )
357, 18, 19, 20, 6, 3, 34, 27lspexchn2 15811 . . . . 5  |-  ( ph  ->  -.  Z  e.  ( N `  { Y ,  X } ) )
36 lmodgrp 15561 . . . . . . . . 9  |-  ( W  e.  LMod  ->  W  e. 
Grp )
3719, 21, 363syl 20 . . . . . . . 8  |-  ( ph  ->  W  e.  Grp )
3837adantr 453 . . . . . . 7  |-  ( (
ph  /\  ( ( inv g `  W ) `
 Z )  e.  ( N `  { Y ,  X }
) )  ->  W  e.  Grp )
396adantr 453 . . . . . . 7  |-  ( (
ph  /\  ( ( inv g `  W ) `
 Z )  e.  ( N `  { Y ,  X }
) )  ->  Z  e.  V )
407, 9grpinvinv 14462 . . . . . . 7  |-  ( ( W  e.  Grp  /\  Z  e.  V )  ->  ( ( inv g `  W ) `  (
( inv g `  W ) `  Z
) )  =  Z )
4138, 39, 40syl2anc 645 . . . . . 6  |-  ( (
ph  /\  ( ( inv g `  W ) `
 Z )  e.  ( N `  { Y ,  X }
) )  ->  (
( inv g `  W ) `  (
( inv g `  W ) `  Z
) )  =  Z )
4222adantr 453 . . . . . . 7  |-  ( (
ph  /\  ( ( inv g `  W ) `
 Z )  e.  ( N `  { Y ,  X }
) )  ->  W  e.  LMod )
437, 25, 18, 22, 3, 20lspprcl 15662 . . . . . . . 8  |-  ( ph  ->  ( N `  { Y ,  X }
)  e.  ( LSubSp `  W ) )
4443adantr 453 . . . . . . 7  |-  ( (
ph  /\  ( ( inv g `  W ) `
 Z )  e.  ( N `  { Y ,  X }
) )  ->  ( N `  { Y ,  X } )  e.  ( LSubSp `  W )
)
45 simpr 449 . . . . . . 7  |-  ( (
ph  /\  ( ( inv g `  W ) `
 Z )  e.  ( N `  { Y ,  X }
) )  ->  (
( inv g `  W ) `  Z
)  e.  ( N `
 { Y ,  X } ) )
4625, 9lssvnegcl 15640 . . . . . . 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 1187 . . . . . 6  |-  ( (
ph  /\  ( ( inv g `  W ) `
 Z )  e.  ( N `  { Y ,  X }
) )  ->  (
( inv g `  W ) `  (
( inv g `  W ) `  Z
) )  e.  ( N `  { Y ,  X } ) )
4841, 47eqeltrrd 2331 . . . . 5  |-  ( (
ph  /\  ( ( inv g `  W ) `
 Z )  e.  ( N `  { Y ,  X }
) )  ->  Z  e.  ( N `  { Y ,  X }
) )
4935, 48mtand 643 . . . 4  |-  ( ph  ->  -.  ( ( inv g `  W ) `
 Z )  e.  ( N `  { Y ,  X }
) )
507, 18, 19, 24, 20, 3, 31, 49lspexchn2 15811 . . 3  |-  ( ph  ->  -.  X  e.  ( N `  { Y ,  ( ( inv g `  W ) `
 Z ) } ) )
517, 9, 18lspsnneg 15690 . . . . 5  |-  ( ( W  e.  LMod  /\  Z  e.  V )  ->  ( N `  { (
( inv g `  W ) `  Z
) } )  =  ( N `  { Z } ) )
5222, 6, 51syl2anc 645 . . . 4  |-  ( ph  ->  ( N `  {
( ( inv g `  W ) `  Z
) } )  =  ( N `  { Z } ) )
5332, 52neeqtrrd 2443 . . 3  |-  ( ph  ->  ( N `  { Y } )  =/=  ( N `  { (
( inv g `  W ) `  Z
) } ) )
547, 16, 9grpinvnzcl 14467 . . . 4  |-  ( ( W  e.  Grp  /\  Z  e.  ( V  \  {  .0.  } ) )  ->  ( ( inv g `  W ) `
 Z )  e.  ( V  \  {  .0.  } ) )
5537, 4, 54syl2anc 645 . . 3  |-  ( ph  ->  ( ( inv g `  W ) `  Z
)  e.  ( V 
\  {  .0.  }
) )
567, 10, 16, 17, 18, 19, 20, 50, 53, 1, 55, 8baerlem5a 31034 . 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 5773 . . 3  |-  ( ph  ->  ( ( N `  { ( X  .-  Y ) } ) 
.(+)  ( N `  { ( ( inv g `  W ) `
 Z ) } ) )  =  ( ( N `  {
( X  .-  Y
) } )  .(+)  ( N `  { Z } ) ) )
587, 8, 10, 9, 37, 20, 6grpsubinv 14468 . . . . . 6  |-  ( ph  ->  ( X  .-  (
( inv g `  W ) `  Z
) )  =  ( X  .+  Z ) )
5958sneqd 3594 . . . . 5  |-  ( ph  ->  { ( X  .-  ( ( inv g `  W ) `  Z
) ) }  =  { ( X  .+  Z ) } )
6059fveq2d 5427 . . . 4  |-  ( ph  ->  ( N `  {
( X  .-  (
( inv g `  W ) `  Z
) ) } )  =  ( N `  { ( X  .+  Z ) } ) )
6160oveq1d 5772 . . 3  |-  ( ph  ->  ( ( N `  { ( X  .-  ( ( inv g `  W ) `  Z
) ) } ) 
.(+)  ( N `  { Y } ) )  =  ( ( N `
 { ( X 
.+  Z ) } )  .(+)  ( N `  { Y } ) ) )
6257, 61ineq12d 3313 . 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 2292 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 5    -> wi 6    /\ wa 360    = wceq 1619    e. wcel 1621    =/= wne 2419    \ cdif 3091    i^i cin 3093   {csn 3581   {cpr 3582   ` cfv 4638  (class class class)co 5757   Basecbs 13075   +g cplusg 13135   0gc0g 13327   Grpcgrp 14289   inv gcminusg 14290   -gcsg 14292   LSSumclsm 14872   LModclmod 15554   LSubSpclss 15616   LSpanclspn 15655   LVecclvec 15782
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 1927  ax-ext 2237  ax-rep 4071  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449  ax-cnex 8726  ax-resscn 8727  ax-1cn 8728  ax-icn 8729  ax-addcl 8730  ax-addrcl 8731  ax-mulcl 8732  ax-mulrcl 8733  ax-mulcom 8734  ax-addass 8735  ax-mulass 8736  ax-distr 8737  ax-i2m1 8738  ax-1ne0 8739  ax-1rid 8740  ax-rnegex 8741  ax-rrecex 8742  ax-cnre 8743  ax-pre-lttri 8744  ax-pre-lttrn 8745  ax-pre-ltadd 8746  ax-pre-mulgt0 8747
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 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-pss 3110  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-tp 3589  df-op 3590  df-uni 3769  df-int 3804  df-iun 3848  df-br 3964  df-opab 4018  df-mpt 4019  df-tr 4054  df-eprel 4242  df-id 4246  df-po 4251  df-so 4252  df-fr 4289  df-we 4291  df-ord 4332  df-on 4333  df-lim 4334  df-suc 4335  df-om 4594  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-1st 6021  df-2nd 6022  df-tpos 6133  df-iota 6190  df-riota 6237  df-recs 6321  df-rdg 6356  df-er 6593  df-en 6797  df-dom 6798  df-sdom 6799  df-pnf 8802  df-mnf 8803  df-xr 8804  df-ltxr 8805  df-le 8806  df-sub 8972  df-neg 8973  df-n 9680  df-2 9737  df-3 9738  df-ndx 13078  df-slot 13079  df-base 13080  df-sets 13081  df-ress 13082  df-plusg 13148  df-mulr 13149  df-0g 13331  df-mnd 14294  df-submnd 14343  df-grp 14416  df-minusg 14417  df-sbg 14418  df-subg 14545  df-cntz 14720  df-lsm 14874  df-cmn 15018  df-abl 15019  df-mgp 15253  df-ring 15267  df-ur 15269  df-oppr 15332  df-dvdsr 15350  df-unit 15351  df-invr 15381  df-drng 15441  df-lmod 15556  df-lss 15617  df-lsp 15656  df-lvec 15783
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