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Theorem lspexch 17551
Description: Exchange property for span of a pair. TODO: see if a version with Y,Z and X,Z reversed will shorten proofs (analogous to lspexchn1 17552 vs. lspexchn2 17553); look for lspexch 17551 and prcom 4098 in same proof. TODO: would a hypothesis of  -.  X  e.  ( N `  { Z } ) instead of  ( N `  { X } )  =/=  ( N { Z } ) ` be better overall? This would be shorter and also satisfy the 
X  =/=  .0. condition. Here and also lspindp* and all proofs affected by them (all in NM's mathbox); there are 58 hypotheses with the 
=/= pattern as of 24-May-2015. (Contributed by NM, 11-Apr-2015.)
Hypotheses
Ref Expression
lspexch.v  |-  V  =  ( Base `  W
)
lspexch.o  |-  .0.  =  ( 0g `  W )
lspexch.n  |-  N  =  ( LSpan `  W )
lspexch.w  |-  ( ph  ->  W  e.  LVec )
lspexch.x  |-  ( ph  ->  X  e.  ( V 
\  {  .0.  }
) )
lspexch.y  |-  ( ph  ->  Y  e.  V )
lspexch.z  |-  ( ph  ->  Z  e.  V )
lspexch.q  |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Z } ) )
lspexch.e  |-  ( ph  ->  X  e.  ( N `
 { Y ,  Z } ) )
Assertion
Ref Expression
lspexch  |-  ( ph  ->  Y  e.  ( N `
 { X ,  Z } ) )

Proof of Theorem lspexch
Dummy variables  j 
k are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lspexch.e . . 3  |-  ( ph  ->  X  e.  ( N `
 { Y ,  Z } ) )
2 lspexch.v . . . 4  |-  V  =  ( Base `  W
)
3 eqid 2460 . . . 4  |-  ( +g  `  W )  =  ( +g  `  W )
4 eqid 2460 . . . 4  |-  (Scalar `  W )  =  (Scalar `  W )
5 eqid 2460 . . . 4  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
6 eqid 2460 . . . 4  |-  ( .s
`  W )  =  ( .s `  W
)
7 lspexch.n . . . 4  |-  N  =  ( LSpan `  W )
8 lspexch.w . . . . 5  |-  ( ph  ->  W  e.  LVec )
9 lveclmod 17528 . . . . 5  |-  ( W  e.  LVec  ->  W  e. 
LMod )
108, 9syl 16 . . . 4  |-  ( ph  ->  W  e.  LMod )
11 lspexch.y . . . 4  |-  ( ph  ->  Y  e.  V )
12 lspexch.z . . . 4  |-  ( ph  ->  Z  e.  V )
132, 3, 4, 5, 6, 7, 10, 11, 12lspprel 17516 . . 3  |-  ( ph  ->  ( X  e.  ( N `  { Y ,  Z } )  <->  E. j  e.  ( Base `  (Scalar `  W ) ) E. k  e.  ( Base `  (Scalar `  W )
) X  =  ( ( j ( .s
`  W ) Y ) ( +g  `  W
) ( k ( .s `  W ) Z ) ) ) )
141, 13mpbid 210 . 2  |-  ( ph  ->  E. j  e.  (
Base `  (Scalar `  W
) ) E. k  e.  ( Base `  (Scalar `  W ) ) X  =  ( ( j ( .s `  W
) Y ) ( +g  `  W ) ( k ( .s
`  W ) Z ) ) )
15 eqid 2460 . . . . . . . 8  |-  ( -g `  W )  =  (
-g `  W )
16 eqid 2460 . . . . . . . 8  |-  ( invg `  (Scalar `  W ) )  =  ( invg `  (Scalar `  W ) )
1783ad2ant1 1012 . . . . . . . . 9  |-  ( (
ph  /\  ( j  e.  ( Base `  (Scalar `  W ) )  /\  k  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( j ( .s `  W
) Y ) ( +g  `  W ) ( k ( .s
`  W ) Z ) ) )  ->  W  e.  LVec )
1817, 9syl 16 . . . . . . . 8  |-  ( (
ph  /\  ( j  e.  ( Base `  (Scalar `  W ) )  /\  k  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( j ( .s `  W
) Y ) ( +g  `  W ) ( k ( .s
`  W ) Z ) ) )  ->  W  e.  LMod )
19 simp2r 1018 . . . . . . . 8  |-  ( (
ph  /\  ( j  e.  ( Base `  (Scalar `  W ) )  /\  k  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( j ( .s `  W
) Y ) ( +g  `  W ) ( k ( .s
`  W ) Z ) ) )  -> 
k  e.  ( Base `  (Scalar `  W )
) )
20 lspexch.x . . . . . . . . . 10  |-  ( ph  ->  X  e.  ( V 
\  {  .0.  }
) )
21203ad2ant1 1012 . . . . . . . . 9  |-  ( (
ph  /\  ( j  e.  ( Base `  (Scalar `  W ) )  /\  k  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( j ( .s `  W
) Y ) ( +g  `  W ) ( k ( .s
`  W ) Z ) ) )  ->  X  e.  ( V  \  {  .0.  } ) )
2221eldifad 3481 . . . . . . . 8  |-  ( (
ph  /\  ( j  e.  ( Base `  (Scalar `  W ) )  /\  k  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( j ( .s `  W
) Y ) ( +g  `  W ) ( k ( .s
`  W ) Z ) ) )  ->  X  e.  V )
23123ad2ant1 1012 . . . . . . . 8  |-  ( (
ph  /\  ( j  e.  ( Base `  (Scalar `  W ) )  /\  k  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( j ( .s `  W
) Y ) ( +g  `  W ) ( k ( .s
`  W ) Z ) ) )  ->  Z  e.  V )
242, 3, 15, 6, 4, 5, 16, 18, 19, 22, 23lmodsubvs 17342 . . . . . . 7  |-  ( (
ph  /\  ( j  e.  ( Base `  (Scalar `  W ) )  /\  k  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( j ( .s `  W
) Y ) ( +g  `  W ) ( k ( .s
`  W ) Z ) ) )  -> 
( X ( -g `  W ) ( k ( .s `  W
) Z ) )  =  ( X ( +g  `  W ) ( ( ( invg `  (Scalar `  W ) ) `  k ) ( .s
`  W ) Z ) ) )
25 simp3 993 . . . . . . . . 9  |-  ( (
ph  /\  ( j  e.  ( Base `  (Scalar `  W ) )  /\  k  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( j ( .s `  W
) Y ) ( +g  `  W ) ( k ( .s
`  W ) Z ) ) )  ->  X  =  ( (
j ( .s `  W ) Y ) ( +g  `  W
) ( k ( .s `  W ) Z ) ) )
2625eqcomd 2468 . . . . . . . 8  |-  ( (
ph  /\  ( j  e.  ( Base `  (Scalar `  W ) )  /\  k  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( j ( .s `  W
) Y ) ( +g  `  W ) ( k ( .s
`  W ) Z ) ) )  -> 
( ( j ( .s `  W ) Y ) ( +g  `  W ) ( k ( .s `  W
) Z ) )  =  X )
27103ad2ant1 1012 . . . . . . . . . 10  |-  ( (
ph  /\  ( j  e.  ( Base `  (Scalar `  W ) )  /\  k  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( j ( .s `  W
) Y ) ( +g  `  W ) ( k ( .s
`  W ) Z ) ) )  ->  W  e.  LMod )
28 lmodgrp 17295 . . . . . . . . . 10  |-  ( W  e.  LMod  ->  W  e. 
Grp )
2927, 28syl 16 . . . . . . . . 9  |-  ( (
ph  /\  ( j  e.  ( Base `  (Scalar `  W ) )  /\  k  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( j ( .s `  W
) Y ) ( +g  `  W ) ( k ( .s
`  W ) Z ) ) )  ->  W  e.  Grp )
302, 4, 6, 5lmodvscl 17305 . . . . . . . . . 10  |-  ( ( W  e.  LMod  /\  k  e.  ( Base `  (Scalar `  W ) )  /\  Z  e.  V )  ->  ( k ( .s
`  W ) Z )  e.  V )
3118, 19, 23, 30syl3anc 1223 . . . . . . . . 9  |-  ( (
ph  /\  ( j  e.  ( Base `  (Scalar `  W ) )  /\  k  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( j ( .s `  W
) Y ) ( +g  `  W ) ( k ( .s
`  W ) Z ) ) )  -> 
( k ( .s
`  W ) Z )  e.  V )
32 simp2l 1017 . . . . . . . . . 10  |-  ( (
ph  /\  ( j  e.  ( Base `  (Scalar `  W ) )  /\  k  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( j ( .s `  W
) Y ) ( +g  `  W ) ( k ( .s
`  W ) Z ) ) )  -> 
j  e.  ( Base `  (Scalar `  W )
) )
33113ad2ant1 1012 . . . . . . . . . 10  |-  ( (
ph  /\  ( j  e.  ( Base `  (Scalar `  W ) )  /\  k  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( j ( .s `  W
) Y ) ( +g  `  W ) ( k ( .s
`  W ) Z ) ) )  ->  Y  e.  V )
342, 4, 6, 5lmodvscl 17305 . . . . . . . . . 10  |-  ( ( W  e.  LMod  /\  j  e.  ( Base `  (Scalar `  W ) )  /\  Y  e.  V )  ->  ( j ( .s
`  W ) Y )  e.  V )
3518, 32, 33, 34syl3anc 1223 . . . . . . . . 9  |-  ( (
ph  /\  ( j  e.  ( Base `  (Scalar `  W ) )  /\  k  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( j ( .s `  W
) Y ) ( +g  `  W ) ( k ( .s
`  W ) Z ) ) )  -> 
( j ( .s
`  W ) Y )  e.  V )
362, 3, 15grpsubadd 15920 . . . . . . . . 9  |-  ( ( W  e.  Grp  /\  ( X  e.  V  /\  ( k ( .s
`  W ) Z )  e.  V  /\  ( j ( .s
`  W ) Y )  e.  V ) )  ->  ( ( X ( -g `  W
) ( k ( .s `  W ) Z ) )  =  ( j ( .s
`  W ) Y )  <->  ( ( j ( .s `  W
) Y ) ( +g  `  W ) ( k ( .s
`  W ) Z ) )  =  X ) )
3729, 22, 31, 35, 36syl13anc 1225 . . . . . . . 8  |-  ( (
ph  /\  ( j  e.  ( Base `  (Scalar `  W ) )  /\  k  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( j ( .s `  W
) Y ) ( +g  `  W ) ( k ( .s
`  W ) Z ) ) )  -> 
( ( X (
-g `  W )
( k ( .s
`  W ) Z ) )  =  ( j ( .s `  W ) Y )  <-> 
( ( j ( .s `  W ) Y ) ( +g  `  W ) ( k ( .s `  W
) Z ) )  =  X ) )
3826, 37mpbird 232 . . . . . . 7  |-  ( (
ph  /\  ( j  e.  ( Base `  (Scalar `  W ) )  /\  k  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( j ( .s `  W
) Y ) ( +g  `  W ) ( k ( .s
`  W ) Z ) ) )  -> 
( X ( -g `  W ) ( k ( .s `  W
) Z ) )  =  ( j ( .s `  W ) Y ) )
3924, 38eqtr3d 2503 . . . . . 6  |-  ( (
ph  /\  ( j  e.  ( Base `  (Scalar `  W ) )  /\  k  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( j ( .s `  W
) Y ) ( +g  `  W ) ( k ( .s
`  W ) Z ) ) )  -> 
( X ( +g  `  W ) ( ( ( invg `  (Scalar `  W ) ) `
 k ) ( .s `  W ) Z ) )  =  ( j ( .s
`  W ) Y ) )
40 eqid 2460 . . . . . . 7  |-  ( 0g
`  (Scalar `  W )
)  =  ( 0g
`  (Scalar `  W )
)
41 eqid 2460 . . . . . . 7  |-  ( invr `  (Scalar `  W )
)  =  ( invr `  (Scalar `  W )
)
42 lspexch.q . . . . . . . . . 10  |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Z } ) )
43423ad2ant1 1012 . . . . . . . . 9  |-  ( (
ph  /\  ( j  e.  ( Base `  (Scalar `  W ) )  /\  k  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( j ( .s `  W
) Y ) ( +g  `  W ) ( k ( .s
`  W ) Z ) ) )  -> 
( N `  { X } )  =/=  ( N `  { Z } ) )
44 lspexch.o . . . . . . . . . . . 12  |-  .0.  =  ( 0g `  W )
4517adantr 465 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
j  e.  ( Base `  (Scalar `  W )
)  /\  k  e.  ( Base `  (Scalar `  W
) ) )  /\  X  =  ( (
j ( .s `  W ) Y ) ( +g  `  W
) ( k ( .s `  W ) Z ) ) )  /\  j  =  ( 0g `  (Scalar `  W ) ) )  ->  W  e.  LVec )
4623adantr 465 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
j  e.  ( Base `  (Scalar `  W )
)  /\  k  e.  ( Base `  (Scalar `  W
) ) )  /\  X  =  ( (
j ( .s `  W ) Y ) ( +g  `  W
) ( k ( .s `  W ) Z ) ) )  /\  j  =  ( 0g `  (Scalar `  W ) ) )  ->  Z  e.  V
)
4725adantr 465 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
j  e.  ( Base `  (Scalar `  W )
)  /\  k  e.  ( Base `  (Scalar `  W
) ) )  /\  X  =  ( (
j ( .s `  W ) Y ) ( +g  `  W
) ( k ( .s `  W ) Z ) ) )  /\  j  =  ( 0g `  (Scalar `  W ) ) )  ->  X  =  ( ( j ( .s
`  W ) Y ) ( +g  `  W
) ( k ( .s `  W ) Z ) ) )
48 oveq1 6282 . . . . . . . . . . . . . . . 16  |-  ( j  =  ( 0g `  (Scalar `  W ) )  ->  ( j ( .s `  W ) Y )  =  ( ( 0g `  (Scalar `  W ) ) ( .s `  W ) Y ) )
4948oveq1d 6290 . . . . . . . . . . . . . . 15  |-  ( j  =  ( 0g `  (Scalar `  W ) )  ->  ( ( j ( .s `  W
) Y ) ( +g  `  W ) ( k ( .s
`  W ) Z ) )  =  ( ( ( 0g `  (Scalar `  W ) ) ( .s `  W
) Y ) ( +g  `  W ) ( k ( .s
`  W ) Z ) ) )
502, 4, 6, 40, 44lmod0vs 17321 . . . . . . . . . . . . . . . . . 18  |-  ( ( W  e.  LMod  /\  Y  e.  V )  ->  (
( 0g `  (Scalar `  W ) ) ( .s `  W ) Y )  =  .0.  )
5118, 33, 50syl2anc 661 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  ( j  e.  ( Base `  (Scalar `  W ) )  /\  k  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( j ( .s `  W
) Y ) ( +g  `  W ) ( k ( .s
`  W ) Z ) ) )  -> 
( ( 0g `  (Scalar `  W ) ) ( .s `  W
) Y )  =  .0.  )
5251oveq1d 6290 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( j  e.  ( Base `  (Scalar `  W ) )  /\  k  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( j ( .s `  W
) Y ) ( +g  `  W ) ( k ( .s
`  W ) Z ) ) )  -> 
( ( ( 0g
`  (Scalar `  W )
) ( .s `  W ) Y ) ( +g  `  W
) ( k ( .s `  W ) Z ) )  =  (  .0.  ( +g  `  W ) ( k ( .s `  W
) Z ) ) )
532, 3, 44lmod0vlid 17318 . . . . . . . . . . . . . . . . 17  |-  ( ( W  e.  LMod  /\  (
k ( .s `  W ) Z )  e.  V )  -> 
(  .0.  ( +g  `  W ) ( k ( .s `  W
) Z ) )  =  ( k ( .s `  W ) Z ) )
5418, 31, 53syl2anc 661 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( j  e.  ( Base `  (Scalar `  W ) )  /\  k  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( j ( .s `  W
) Y ) ( +g  `  W ) ( k ( .s
`  W ) Z ) ) )  -> 
(  .0.  ( +g  `  W ) ( k ( .s `  W
) Z ) )  =  ( k ( .s `  W ) Z ) )
5552, 54eqtrd 2501 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( j  e.  ( Base `  (Scalar `  W ) )  /\  k  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( j ( .s `  W
) Y ) ( +g  `  W ) ( k ( .s
`  W ) Z ) ) )  -> 
( ( ( 0g
`  (Scalar `  W )
) ( .s `  W ) Y ) ( +g  `  W
) ( k ( .s `  W ) Z ) )  =  ( k ( .s
`  W ) Z ) )
5649, 55sylan9eqr 2523 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
j  e.  ( Base `  (Scalar `  W )
)  /\  k  e.  ( Base `  (Scalar `  W
) ) )  /\  X  =  ( (
j ( .s `  W ) Y ) ( +g  `  W
) ( k ( .s `  W ) Z ) ) )  /\  j  =  ( 0g `  (Scalar `  W ) ) )  ->  ( ( j ( .s `  W
) Y ) ( +g  `  W ) ( k ( .s
`  W ) Z ) )  =  ( k ( .s `  W ) Z ) )
5747, 56eqtrd 2501 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
j  e.  ( Base `  (Scalar `  W )
)  /\  k  e.  ( Base `  (Scalar `  W
) ) )  /\  X  =  ( (
j ( .s `  W ) Y ) ( +g  `  W
) ( k ( .s `  W ) Z ) ) )  /\  j  =  ( 0g `  (Scalar `  W ) ) )  ->  X  =  ( k ( .s `  W ) Z ) )
582, 6, 4, 5, 7, 18, 19, 23lspsneli 17423 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( j  e.  ( Base `  (Scalar `  W ) )  /\  k  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( j ( .s `  W
) Y ) ( +g  `  W ) ( k ( .s
`  W ) Z ) ) )  -> 
( k ( .s
`  W ) Z )  e.  ( N `
 { Z }
) )
5958adantr 465 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
j  e.  ( Base `  (Scalar `  W )
)  /\  k  e.  ( Base `  (Scalar `  W
) ) )  /\  X  =  ( (
j ( .s `  W ) Y ) ( +g  `  W
) ( k ( .s `  W ) Z ) ) )  /\  j  =  ( 0g `  (Scalar `  W ) ) )  ->  ( k ( .s `  W ) Z )  e.  ( N `  { Z } ) )
6057, 59eqeltrd 2548 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
j  e.  ( Base `  (Scalar `  W )
)  /\  k  e.  ( Base `  (Scalar `  W
) ) )  /\  X  =  ( (
j ( .s `  W ) Y ) ( +g  `  W
) ( k ( .s `  W ) Z ) ) )  /\  j  =  ( 0g `  (Scalar `  W ) ) )  ->  X  e.  ( N `  { Z } ) )
61 eldifsni 4146 . . . . . . . . . . . . . 14  |-  ( X  e.  ( V  \  {  .0.  } )  ->  X  =/=  .0.  )
6221, 61syl 16 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( j  e.  ( Base `  (Scalar `  W ) )  /\  k  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( j ( .s `  W
) Y ) ( +g  `  W ) ( k ( .s
`  W ) Z ) ) )  ->  X  =/=  .0.  )
6362adantr 465 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
j  e.  ( Base `  (Scalar `  W )
)  /\  k  e.  ( Base `  (Scalar `  W
) ) )  /\  X  =  ( (
j ( .s `  W ) Y ) ( +g  `  W
) ( k ( .s `  W ) Z ) ) )  /\  j  =  ( 0g `  (Scalar `  W ) ) )  ->  X  =/=  .0.  )
642, 44, 7, 45, 46, 60, 63lspsneleq 17537 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
j  e.  ( Base `  (Scalar `  W )
)  /\  k  e.  ( Base `  (Scalar `  W
) ) )  /\  X  =  ( (
j ( .s `  W ) Y ) ( +g  `  W
) ( k ( .s `  W ) Z ) ) )  /\  j  =  ( 0g `  (Scalar `  W ) ) )  ->  ( N `  { X } )  =  ( N `  { Z } ) )
6564ex 434 . . . . . . . . . 10  |-  ( (
ph  /\  ( j  e.  ( Base `  (Scalar `  W ) )  /\  k  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( j ( .s `  W
) Y ) ( +g  `  W ) ( k ( .s
`  W ) Z ) ) )  -> 
( j  =  ( 0g `  (Scalar `  W ) )  -> 
( N `  { X } )  =  ( N `  { Z } ) ) )
6665necon3d 2684 . . . . . . . . 9  |-  ( (
ph  /\  ( j  e.  ( Base `  (Scalar `  W ) )  /\  k  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( j ( .s `  W
) Y ) ( +g  `  W ) ( k ( .s
`  W ) Z ) ) )  -> 
( ( N `  { X } )  =/=  ( N `  { Z } )  ->  j  =/=  ( 0g `  (Scalar `  W ) ) ) )
6743, 66mpd 15 . . . . . . . 8  |-  ( (
ph  /\  ( j  e.  ( Base `  (Scalar `  W ) )  /\  k  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( j ( .s `  W
) Y ) ( +g  `  W ) ( k ( .s
`  W ) Z ) ) )  -> 
j  =/=  ( 0g
`  (Scalar `  W )
) )
68 eldifsn 4145 . . . . . . . 8  |-  ( j  e.  ( ( Base `  (Scalar `  W )
)  \  { ( 0g `  (Scalar `  W
) ) } )  <-> 
( j  e.  (
Base `  (Scalar `  W
) )  /\  j  =/=  ( 0g `  (Scalar `  W ) ) ) )
6932, 67, 68sylanbrc 664 . . . . . . 7  |-  ( (
ph  /\  ( j  e.  ( Base `  (Scalar `  W ) )  /\  k  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( j ( .s `  W
) Y ) ( +g  `  W ) ( k ( .s
`  W ) Z ) ) )  -> 
j  e.  ( (
Base `  (Scalar `  W
) )  \  {
( 0g `  (Scalar `  W ) ) } ) )
704lmodfgrp 17297 . . . . . . . . . . 11  |-  ( W  e.  LMod  ->  (Scalar `  W )  e.  Grp )
7127, 70syl 16 . . . . . . . . . 10  |-  ( (
ph  /\  ( j  e.  ( Base `  (Scalar `  W ) )  /\  k  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( j ( .s `  W
) Y ) ( +g  `  W ) ( k ( .s
`  W ) Z ) ) )  -> 
(Scalar `  W )  e.  Grp )
725, 16grpinvcl 15889 . . . . . . . . . 10  |-  ( ( (Scalar `  W )  e.  Grp  /\  k  e.  ( Base `  (Scalar `  W ) ) )  ->  ( ( invg `  (Scalar `  W ) ) `  k )  e.  (
Base `  (Scalar `  W
) ) )
7371, 19, 72syl2anc 661 . . . . . . . . 9  |-  ( (
ph  /\  ( j  e.  ( Base `  (Scalar `  W ) )  /\  k  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( j ( .s `  W
) Y ) ( +g  `  W ) ( k ( .s
`  W ) Z ) ) )  -> 
( ( invg `  (Scalar `  W )
) `  k )  e.  ( Base `  (Scalar `  W ) ) )
742, 4, 6, 5lmodvscl 17305 . . . . . . . . 9  |-  ( ( W  e.  LMod  /\  (
( invg `  (Scalar `  W ) ) `
 k )  e.  ( Base `  (Scalar `  W ) )  /\  Z  e.  V )  ->  ( ( ( invg `  (Scalar `  W ) ) `  k ) ( .s
`  W ) Z )  e.  V )
7518, 73, 23, 74syl3anc 1223 . . . . . . . 8  |-  ( (
ph  /\  ( j  e.  ( Base `  (Scalar `  W ) )  /\  k  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( j ( .s `  W
) Y ) ( +g  `  W ) ( k ( .s
`  W ) Z ) ) )  -> 
( ( ( invg `  (Scalar `  W ) ) `  k ) ( .s
`  W ) Z )  e.  V )
762, 3lmodvacl 17302 . . . . . . . 8  |-  ( ( W  e.  LMod  /\  X  e.  V  /\  (
( ( invg `  (Scalar `  W )
) `  k )
( .s `  W
) Z )  e.  V )  ->  ( X ( +g  `  W
) ( ( ( invg `  (Scalar `  W ) ) `  k ) ( .s
`  W ) Z ) )  e.  V
)
7718, 22, 75, 76syl3anc 1223 . . . . . . 7  |-  ( (
ph  /\  ( j  e.  ( Base `  (Scalar `  W ) )  /\  k  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( j ( .s `  W
) Y ) ( +g  `  W ) ( k ( .s
`  W ) Z ) ) )  -> 
( X ( +g  `  W ) ( ( ( invg `  (Scalar `  W ) ) `
 k ) ( .s `  W ) Z ) )  e.  V )
782, 6, 4, 5, 40, 41, 17, 69, 77, 33lvecinv 17535 . . . . . 6  |-  ( (
ph  /\  ( j  e.  ( Base `  (Scalar `  W ) )  /\  k  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( j ( .s `  W
) Y ) ( +g  `  W ) ( k ( .s
`  W ) Z ) ) )  -> 
( ( X ( +g  `  W ) ( ( ( invg `  (Scalar `  W ) ) `  k ) ( .s
`  W ) Z ) )  =  ( j ( .s `  W ) Y )  <-> 
Y  =  ( ( ( invr `  (Scalar `  W ) ) `  j ) ( .s
`  W ) ( X ( +g  `  W
) ( ( ( invg `  (Scalar `  W ) ) `  k ) ( .s
`  W ) Z ) ) ) ) )
7939, 78mpbid 210 . . . . 5  |-  ( (
ph  /\  ( j  e.  ( Base `  (Scalar `  W ) )  /\  k  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( j ( .s `  W
) Y ) ( +g  `  W ) ( k ( .s
`  W ) Z ) ) )  ->  Y  =  ( (
( invr `  (Scalar `  W
) ) `  j
) ( .s `  W ) ( X ( +g  `  W
) ( ( ( invg `  (Scalar `  W ) ) `  k ) ( .s
`  W ) Z ) ) ) )
80 eqid 2460 . . . . . . 7  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
812, 80, 7, 18, 22, 23lspprcl 17400 . . . . . 6  |-  ( (
ph  /\  ( j  e.  ( Base `  (Scalar `  W ) )  /\  k  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( j ( .s `  W
) Y ) ( +g  `  W ) ( k ( .s
`  W ) Z ) ) )  -> 
( N `  { X ,  Z }
)  e.  ( LSubSp `  W ) )
824lvecdrng 17527 . . . . . . . 8  |-  ( W  e.  LVec  ->  (Scalar `  W )  e.  DivRing )
8317, 82syl 16 . . . . . . 7  |-  ( (
ph  /\  ( j  e.  ( Base `  (Scalar `  W ) )  /\  k  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( j ( .s `  W
) Y ) ( +g  `  W ) ( k ( .s
`  W ) Z ) ) )  -> 
(Scalar `  W )  e.  DivRing )
845, 40, 41drnginvrcl 17189 . . . . . . 7  |-  ( ( (Scalar `  W )  e.  DivRing  /\  j  e.  ( Base `  (Scalar `  W
) )  /\  j  =/=  ( 0g `  (Scalar `  W ) ) )  ->  ( ( invr `  (Scalar `  W )
) `  j )  e.  ( Base `  (Scalar `  W ) ) )
8583, 32, 67, 84syl3anc 1223 . . . . . 6  |-  ( (
ph  /\  ( j  e.  ( Base `  (Scalar `  W ) )  /\  k  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( j ( .s `  W
) Y ) ( +g  `  W ) ( k ( .s
`  W ) Z ) ) )  -> 
( ( invr `  (Scalar `  W ) ) `  j )  e.  (
Base `  (Scalar `  W
) ) )
86 eqid 2460 . . . . . . . . . 10  |-  ( 1r
`  (Scalar `  W )
)  =  ( 1r
`  (Scalar `  W )
)
872, 4, 6, 86lmodvs1 17316 . . . . . . . . 9  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  (
( 1r `  (Scalar `  W ) ) ( .s `  W ) X )  =  X )
8818, 22, 87syl2anc 661 . . . . . . . 8  |-  ( (
ph  /\  ( j  e.  ( Base `  (Scalar `  W ) )  /\  k  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( j ( .s `  W
) Y ) ( +g  `  W ) ( k ( .s
`  W ) Z ) ) )  -> 
( ( 1r `  (Scalar `  W ) ) ( .s `  W
) X )  =  X )
8988oveq1d 6290 . . . . . . 7  |-  ( (
ph  /\  ( j  e.  ( Base `  (Scalar `  W ) )  /\  k  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( j ( .s `  W
) Y ) ( +g  `  W ) ( k ( .s
`  W ) Z ) ) )  -> 
( ( ( 1r
`  (Scalar `  W )
) ( .s `  W ) X ) ( +g  `  W
) ( ( ( invg `  (Scalar `  W ) ) `  k ) ( .s
`  W ) Z ) )  =  ( X ( +g  `  W
) ( ( ( invg `  (Scalar `  W ) ) `  k ) ( .s
`  W ) Z ) ) )
904lmodrng 17296 . . . . . . . . 9  |-  ( W  e.  LMod  ->  (Scalar `  W )  e.  Ring )
915, 86rngidcl 16999 . . . . . . . . 9  |-  ( (Scalar `  W )  e.  Ring  -> 
( 1r `  (Scalar `  W ) )  e.  ( Base `  (Scalar `  W ) ) )
9218, 90, 913syl 20 . . . . . . . 8  |-  ( (
ph  /\  ( j  e.  ( Base `  (Scalar `  W ) )  /\  k  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( j ( .s `  W
) Y ) ( +g  `  W ) ( k ( .s
`  W ) Z ) ) )  -> 
( 1r `  (Scalar `  W ) )  e.  ( Base `  (Scalar `  W ) ) )
932, 3, 6, 4, 5, 7, 18, 92, 73, 22, 23lsppreli 17512 . . . . . . 7  |-  ( (
ph  /\  ( j  e.  ( Base `  (Scalar `  W ) )  /\  k  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( j ( .s `  W
) Y ) ( +g  `  W ) ( k ( .s
`  W ) Z ) ) )  -> 
( ( ( 1r
`  (Scalar `  W )
) ( .s `  W ) X ) ( +g  `  W
) ( ( ( invg `  (Scalar `  W ) ) `  k ) ( .s
`  W ) Z ) )  e.  ( N `  { X ,  Z } ) )
9489, 93eqeltrrd 2549 . . . . . 6  |-  ( (
ph  /\  ( j  e.  ( Base `  (Scalar `  W ) )  /\  k  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( j ( .s `  W
) Y ) ( +g  `  W ) ( k ( .s
`  W ) Z ) ) )  -> 
( X ( +g  `  W ) ( ( ( invg `  (Scalar `  W ) ) `
 k ) ( .s `  W ) Z ) )  e.  ( N `  { X ,  Z }
) )
954, 6, 5, 80lssvscl 17377 . . . . . 6  |-  ( ( ( W  e.  LMod  /\  ( N `  { X ,  Z }
)  e.  ( LSubSp `  W ) )  /\  ( ( ( invr `  (Scalar `  W )
) `  j )  e.  ( Base `  (Scalar `  W ) )  /\  ( X ( +g  `  W
) ( ( ( invg `  (Scalar `  W ) ) `  k ) ( .s
`  W ) Z ) )  e.  ( N `  { X ,  Z } ) ) )  ->  ( (
( invr `  (Scalar `  W
) ) `  j
) ( .s `  W ) ( X ( +g  `  W
) ( ( ( invg `  (Scalar `  W ) ) `  k ) ( .s
`  W ) Z ) ) )  e.  ( N `  { X ,  Z }
) )
9618, 81, 85, 94, 95syl22anc 1224 . . . . 5  |-  ( (
ph  /\  ( j  e.  ( Base `  (Scalar `  W ) )  /\  k  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( j ( .s `  W
) Y ) ( +g  `  W ) ( k ( .s
`  W ) Z ) ) )  -> 
( ( ( invr `  (Scalar `  W )
) `  j )
( .s `  W
) ( X ( +g  `  W ) ( ( ( invg `  (Scalar `  W ) ) `  k ) ( .s
`  W ) Z ) ) )  e.  ( N `  { X ,  Z }
) )
9779, 96eqeltrd 2548 . . . 4  |-  ( (
ph  /\  ( j  e.  ( Base `  (Scalar `  W ) )  /\  k  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( j ( .s `  W
) Y ) ( +g  `  W ) ( k ( .s
`  W ) Z ) ) )  ->  Y  e.  ( N `  { X ,  Z } ) )
98973exp 1190 . . 3  |-  ( ph  ->  ( ( j  e.  ( Base `  (Scalar `  W ) )  /\  k  e.  ( Base `  (Scalar `  W )
) )  ->  ( X  =  ( (
j ( .s `  W ) Y ) ( +g  `  W
) ( k ( .s `  W ) Z ) )  ->  Y  e.  ( N `  { X ,  Z } ) ) ) )
9998rexlimdvv 2954 . 2  |-  ( ph  ->  ( E. j  e.  ( Base `  (Scalar `  W ) ) E. k  e.  ( Base `  (Scalar `  W )
) X  =  ( ( j ( .s
`  W ) Y ) ( +g  `  W
) ( k ( .s `  W ) Z ) )  ->  Y  e.  ( N `  { X ,  Z } ) ) )
10014, 99mpd 15 1  |-  ( ph  ->  Y  e.  ( N `
 { X ,  Z } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762    =/= wne 2655   E.wrex 2808    \ cdif 3466   {csn 4020   {cpr 4022   ` cfv 5579  (class class class)co 6275   Basecbs 14479   +g cplusg 14544  Scalarcsca 14547   .scvsca 14548   0gc0g 14684   Grpcgrp 15716   invgcminusg 15717   -gcsg 15719   1rcur 16936   Ringcrg 16979   invrcinvr 17097   DivRingcdr 17172   LModclmod 17288   LSubSpclss 17354   LSpanclspn 17393   LVecclvec 17524
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-tpos 6945  df-recs 7032  df-rdg 7066  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-2 10583  df-3 10584  df-ndx 14482  df-slot 14483  df-base 14484  df-sets 14485  df-ress 14486  df-plusg 14557  df-mulr 14558  df-0g 14686  df-mnd 15721  df-submnd 15771  df-grp 15851  df-minusg 15852  df-sbg 15853  df-subg 15986  df-cntz 16143  df-lsm 16445  df-cmn 16589  df-abl 16590  df-mgp 16925  df-ur 16937  df-rng 16981  df-oppr 17049  df-dvdsr 17067  df-unit 17068  df-invr 17098  df-drng 17174  df-lmod 17290  df-lss 17355  df-lsp 17394  df-lvec 17525
This theorem is referenced by:  lspexchn1  17552  lspindp1  17555  mapdh8ab  36449  mapdh8ad  36451  mapdh8b  36452  mapdh8c  36453  mapdh8e  36456
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