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Theorem lspexch 16203
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 16204 vs. lspexchn2 16205); look for lspexch 16203 and prcom 3884 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 2438 . . . 4  |-  ( +g  `  W )  =  ( +g  `  W )
4 eqid 2438 . . . 4  |-  (Scalar `  W )  =  (Scalar `  W )
5 eqid 2438 . . . 4  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
6 eqid 2438 . . . 4  |-  ( .s
`  W )  =  ( .s `  W
)
7 lspexch.n . . . 4  |-  N  =  ( LSpan `  W )
8 lspexch.w . . . . 5  |-  ( ph  ->  W  e.  LVec )
9 lveclmod 16180 . . . . 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 16168 . . 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 203 . 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 2438 . . . . . . . 8  |-  ( -g `  W )  =  (
-g `  W )
16 eqid 2438 . . . . . . . 8  |-  ( inv g `  (Scalar `  W ) )  =  ( inv g `  (Scalar `  W ) )
1783ad2ant1 979 . . . . . . . . 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 985 . . . . . . . 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 979 . . . . . . . . 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 3334 . . . . . . . 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 979 . . . . . . . 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 16002 . . . . . . 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 ) ( ( ( inv g `  (Scalar `  W ) ) `  k ) ( .s
`  W ) Z ) ) )
25 simp3 960 . . . . . . . . 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 2443 . . . . . . . 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 979 . . . . . . . . . 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 15959 . . . . . . . . . 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 15969 . . . . . . . . . 10  |-  ( ( W  e.  LMod  /\  k  e.  ( Base `  (Scalar `  W ) )  /\  Z  e.  V )  ->  ( k ( .s
`  W ) Z )  e.  V )
3118, 19, 23, 30syl3anc 1185 . . . . . . . . 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 984 . . . . . . . . . 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 979 . . . . . . . . . 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 15969 . . . . . . . . . 10  |-  ( ( W  e.  LMod  /\  j  e.  ( Base `  (Scalar `  W ) )  /\  Y  e.  V )  ->  ( j ( .s
`  W ) Y )  e.  V )
3518, 32, 33, 34syl3anc 1185 . . . . . . . . 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 14878 . . . . . . . . 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 1187 . . . . . . . 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 225 . . . . . . 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 2472 . . . . . 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 ) ( ( ( inv g `  (Scalar `  W ) ) `
 k ) ( .s `  W ) Z ) )  =  ( j ( .s
`  W ) Y ) )
40 eqid 2438 . . . . . . 7  |-  ( 0g
`  (Scalar `  W )
)  =  ( 0g
`  (Scalar `  W )
)
41 eqid 2438 . . . . . . 7  |-  ( invr `  (Scalar `  W )
)  =  ( invr `  (Scalar `  W )
)
42 lspexch.q . . . . . . . . . 10  |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Z } ) )
43423ad2ant1 979 . . . . . . . . 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 453 . . . . . . . . . . . 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 453 . . . . . . . . . . . 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 453 . . . . . . . . . . . . . 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 6090 . . . . . . . . . . . . . . . 16  |-  ( j  =  ( 0g `  (Scalar `  W ) )  ->  ( j ( .s `  W ) Y )  =  ( ( 0g `  (Scalar `  W ) ) ( .s `  W ) Y ) )
4948oveq1d 6098 . . . . . . . . . . . . . . 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 15985 . . . . . . . . . . . . . . . . . 18  |-  ( ( W  e.  LMod  /\  Y  e.  V )  ->  (
( 0g `  (Scalar `  W ) ) ( .s `  W ) Y )  =  .0.  )
5118, 33, 50syl2anc 644 . . . . . . . . . . . . . . . . 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 6098 . . . . . . . . . . . . . . . 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 15982 . . . . . . . . . . . . . . . . 17  |-  ( ( W  e.  LMod  /\  (
k ( .s `  W ) Z )  e.  V )  -> 
(  .0.  ( +g  `  W ) ( k ( .s `  W
) Z ) )  =  ( k ( .s `  W ) Z ) )
5418, 31, 53syl2anc 644 . . . . . . . . . . . . . . . 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 2470 . . . . . . . . . . . . . . 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 2492 . . . . . . . . . . . . . 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 2470 . . . . . . . . . . . . 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 16079 . . . . . . . . . . . . . 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 453 . . . . . . . . . . . . 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 2512 . . . . . . . . . . . 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 3930 . . . . . . . . . . . . . 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 453 . . . . . . . . . . . 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 16189 . . . . . . . . . . 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 425 . . . . . . . . . 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 2641 . . . . . . . . 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 3929 . . . . . . . 8  |-  ( j  e.  ( ( Base `  (Scalar `  W )
)  \  { ( 0g `  (Scalar `  W
) ) } )  <-> 
( j  e.  (
Base `  (Scalar `  W
) )  /\  j  =/=  ( 0g `  (Scalar `  W ) ) ) )
6932, 67, 68sylanbrc 647 . . . . . . 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 15961 . . . . . . . . . . 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 14852 . . . . . . . . . 10  |-  ( ( (Scalar `  W )  e.  Grp  /\  k  e.  ( Base `  (Scalar `  W ) ) )  ->  ( ( inv g `  (Scalar `  W ) ) `  k )  e.  (
Base `  (Scalar `  W
) ) )
7371, 19, 72syl2anc 644 . . . . . . . . 9  |-  ( (
ph  /\  ( j  e.  ( Base `  (Scalar `  W ) )  /\  k  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( j ( .s `  W
) Y ) ( +g  `  W ) ( k ( .s
`  W ) Z ) ) )  -> 
( ( inv g `  (Scalar `  W )
) `  k )  e.  ( Base `  (Scalar `  W ) ) )
742, 4, 6, 5lmodvscl 15969 . . . . . . . . 9  |-  ( ( W  e.  LMod  /\  (
( inv g `  (Scalar `  W ) ) `
 k )  e.  ( Base `  (Scalar `  W ) )  /\  Z  e.  V )  ->  ( ( ( inv g `  (Scalar `  W ) ) `  k ) ( .s
`  W ) Z )  e.  V )
7518, 73, 23, 74syl3anc 1185 . . . . . . . 8  |-  ( (
ph  /\  ( j  e.  ( Base `  (Scalar `  W ) )  /\  k  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( j ( .s `  W
) Y ) ( +g  `  W ) ( k ( .s
`  W ) Z ) ) )  -> 
( ( ( inv g `  (Scalar `  W ) ) `  k ) ( .s
`  W ) Z )  e.  V )
762, 3lmodvacl 15966 . . . . . . . 8  |-  ( ( W  e.  LMod  /\  X  e.  V  /\  (
( ( inv g `  (Scalar `  W )
) `  k )
( .s `  W
) Z )  e.  V )  ->  ( X ( +g  `  W
) ( ( ( inv g `  (Scalar `  W ) ) `  k ) ( .s
`  W ) Z ) )  e.  V
)
7718, 22, 75, 76syl3anc 1185 . . . . . . 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 ) ( ( ( inv g `  (Scalar `  W ) ) `
 k ) ( .s `  W ) Z ) )  e.  V )
782, 6, 4, 5, 40, 41, 17, 69, 77, 33lvecinv 16187 . . . . . 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 ) ( ( ( inv g `  (Scalar `  W ) ) `  k ) ( .s
`  W ) Z ) )  =  ( j ( .s `  W ) Y )  <-> 
Y  =  ( ( ( invr `  (Scalar `  W ) ) `  j ) ( .s
`  W ) ( X ( +g  `  W
) ( ( ( inv g `  (Scalar `  W ) ) `  k ) ( .s
`  W ) Z ) ) ) ) )
7939, 78mpbid 203 . . . . 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
) ( ( ( inv g `  (Scalar `  W ) ) `  k ) ( .s
`  W ) Z ) ) ) )
80 eqid 2438 . . . . . . 7  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
812, 80, 7, 18, 22, 23lspprcl 16056 . . . . . 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 16179 . . . . . . . 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 15854 . . . . . . 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 1185 . . . . . 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 2438 . . . . . . . . . 10  |-  ( 1r
`  (Scalar `  W )
)  =  ( 1r
`  (Scalar `  W )
)
872, 4, 6, 86lmodvs1 15980 . . . . . . . . 9  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  (
( 1r `  (Scalar `  W ) ) ( .s `  W ) X )  =  X )
8818, 22, 87syl2anc 644 . . . . . . . 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 6098 . . . . . . 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
) ( ( ( inv g `  (Scalar `  W ) ) `  k ) ( .s
`  W ) Z ) )  =  ( X ( +g  `  W
) ( ( ( inv g `  (Scalar `  W ) ) `  k ) ( .s
`  W ) Z ) ) )
904lmodrng 15960 . . . . . . . . 9  |-  ( W  e.  LMod  ->  (Scalar `  W )  e.  Ring )
915, 86rngidcl 15686 . . . . . . . . 9  |-  ( (Scalar `  W )  e.  Ring  -> 
( 1r `  (Scalar `  W ) )  e.  ( Base `  (Scalar `  W ) ) )
9218, 90, 913syl 19 . . . . . . . 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 16164 . . . . . . 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
) ( ( ( inv g `  (Scalar `  W ) ) `  k ) ( .s
`  W ) Z ) )  e.  ( N `  { X ,  Z } ) )
9489, 93eqeltrrd 2513 . . . . . 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 ) ( ( ( inv g `  (Scalar `  W ) ) `
 k ) ( .s `  W ) Z ) )  e.  ( N `  { X ,  Z }
) )
954, 6, 5, 80lssvscl 16033 . . . . . 6  |-  ( ( ( W  e.  LMod  /\  ( N `  { X ,  Z }
)  e.  ( LSubSp `  W ) )  /\  ( ( ( invr `  (Scalar `  W )
) `  j )  e.  ( Base `  (Scalar `  W ) )  /\  ( X ( +g  `  W
) ( ( ( inv g `  (Scalar `  W ) ) `  k ) ( .s
`  W ) Z ) )  e.  ( N `  { X ,  Z } ) ) )  ->  ( (
( invr `  (Scalar `  W
) ) `  j
) ( .s `  W ) ( X ( +g  `  W
) ( ( ( inv g `  (Scalar `  W ) ) `  k ) ( .s
`  W ) Z ) ) )  e.  ( N `  { X ,  Z }
) )
9618, 81, 85, 94, 95syl22anc 1186 . . . . 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 ) ( ( ( inv g `  (Scalar `  W ) ) `  k ) ( .s
`  W ) Z ) ) )  e.  ( N `  { X ,  Z }
) )
9779, 96eqeltrd 2512 . . . 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 1153 . . 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 2838 . 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 set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   E.wrex 2708    \ cdif 3319   {csn 3816   {cpr 3817   ` cfv 5456  (class class class)co 6083   Basecbs 13471   +g cplusg 13531  Scalarcsca 13534   .scvsca 13535   0gc0g 13725   Grpcgrp 14687   inv gcminusg 14688   -gcsg 14690   Ringcrg 15662   1rcur 15664   invrcinvr 15778   DivRingcdr 15837   LModclmod 15952   LSubSpclss 16010   LSpanclspn 16049   LVecclvec 16176
This theorem is referenced by:  lspexchn1  16204  lspindp1  16207  mapdh8ab  32637  mapdh8ad  32639  mapdh8b  32640  mapdh8c  32641  mapdh8e  32644
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-tpos 6481  df-riota 6551  df-recs 6635  df-rdg 6670  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-nn 10003  df-2 10060  df-3 10061  df-ndx 13474  df-slot 13475  df-base 13476  df-sets 13477  df-ress 13478  df-plusg 13544  df-mulr 13545  df-0g 13729  df-mnd 14692  df-submnd 14741  df-grp 14814  df-minusg 14815  df-sbg 14816  df-subg 14943  df-cntz 15118  df-lsm 15272  df-cmn 15416  df-abl 15417  df-mgp 15651  df-rng 15665  df-ur 15667  df-oppr 15730  df-dvdsr 15748  df-unit 15749  df-invr 15779  df-drng 15839  df-lmod 15954  df-lss 16011  df-lsp 16050  df-lvec 16177
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