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Theorem lspexch 15882
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 15883 vs. lspexchn2 15884); look for lspexch 15882 and prcom 3705 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 2283 . . . 4  |-  ( +g  `  W )  =  ( +g  `  W )
4 eqid 2283 . . . 4  |-  (Scalar `  W )  =  (Scalar `  W )
5 eqid 2283 . . . 4  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
6 eqid 2283 . . . 4  |-  ( .s
`  W )  =  ( .s `  W
)
7 lspexch.n . . . 4  |-  N  =  ( LSpan `  W )
8 lspexch.w . . . . 5  |-  ( ph  ->  W  e.  LVec )
9 lveclmod 15859 . . . . 5  |-  ( W  e.  LVec  ->  W  e. 
LMod )
108, 9syl 15 . . . 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 15847 . . 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 201 . 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 2283 . . . . . . . 8  |-  ( -g `  W )  =  (
-g `  W )
16 eqid 2283 . . . . . . . 8  |-  ( inv g `  (Scalar `  W ) )  =  ( inv g `  (Scalar `  W ) )
1783ad2ant1 976 . . . . . . . . 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 15 . . . . . . . 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 982 . . . . . . . 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 976 . . . . . . . . 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.  } ) )
22 eldifi 3298 . . . . . . . . 9  |-  ( X  e.  ( V  \  {  .0.  } )  ->  X  e.  V )
2321, 22syl 15 . . . . . . . 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 )
24123ad2ant1 976 . . . . . . . 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 )
252, 3, 15, 6, 4, 5, 16, 18, 19, 23, 24lmodsubvs 15681 . . . . . . 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 ) ) )
26 simp3 957 . . . . . . . . 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 ) ) )
2726eqcomd 2288 . . . . . . . 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 )
28103ad2ant1 976 . . . . . . . . . 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 )
29 lmodgrp 15634 . . . . . . . . . 10  |-  ( W  e.  LMod  ->  W  e. 
Grp )
3028, 29syl 15 . . . . . . . . 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 )
312, 4, 6, 5lmodvscl 15644 . . . . . . . . . 10  |-  ( ( W  e.  LMod  /\  k  e.  ( Base `  (Scalar `  W ) )  /\  Z  e.  V )  ->  ( k ( .s
`  W ) Z )  e.  V )
3218, 19, 24, 31syl3anc 1182 . . . . . . . . 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 )
33 simp2l 981 . . . . . . . . . 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 )
) )
34113ad2ant1 976 . . . . . . . . . 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 )
352, 4, 6, 5lmodvscl 15644 . . . . . . . . . 10  |-  ( ( W  e.  LMod  /\  j  e.  ( Base `  (Scalar `  W ) )  /\  Y  e.  V )  ->  ( j ( .s
`  W ) Y )  e.  V )
3618, 33, 34, 35syl3anc 1182 . . . . . . . . 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 )
372, 3, 15grpsubadd 14553 . . . . . . . . 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 ) )
3830, 23, 32, 36, 37syl13anc 1184 . . . . . . . 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 ) )
3927, 38mpbird 223 . . . . . . 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 ) )
4025, 39eqtr3d 2317 . . . . . 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 ) )
41 eqid 2283 . . . . . . 7  |-  ( 0g
`  (Scalar `  W )
)  =  ( 0g
`  (Scalar `  W )
)
42 eqid 2283 . . . . . . 7  |-  ( invr `  (Scalar `  W )
)  =  ( invr `  (Scalar `  W )
)
43 lspexch.q . . . . . . . . . 10  |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Z } ) )
44433ad2ant1 976 . . . . . . . . 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 } ) )
45 lspexch.o . . . . . . . . . . . 12  |-  .0.  =  ( 0g `  W )
4617adantr 451 . . . . . . . . . . . 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 )
4724adantr 451 . . . . . . . . . . . 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
)
4826adantr 451 . . . . . . . . . . . . . 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 ) ) )
49 oveq1 5865 . . . . . . . . . . . . . . . 16  |-  ( j  =  ( 0g `  (Scalar `  W ) )  ->  ( j ( .s `  W ) Y )  =  ( ( 0g `  (Scalar `  W ) ) ( .s `  W ) Y ) )
5049oveq1d 5873 . . . . . . . . . . . . . . 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 ) ) )
512, 4, 6, 41, 45lmod0vs 15663 . . . . . . . . . . . . . . . . . 18  |-  ( ( W  e.  LMod  /\  Y  e.  V )  ->  (
( 0g `  (Scalar `  W ) ) ( .s `  W ) Y )  =  .0.  )
5218, 34, 51syl2anc 642 . . . . . . . . . . . . . . . . 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.  )
5352oveq1d 5873 . . . . . . . . . . . . . . . 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 ) ) )
542, 3, 45lmod0vlid 15660 . . . . . . . . . . . . . . . . 17  |-  ( ( W  e.  LMod  /\  (
k ( .s `  W ) Z )  e.  V )  -> 
(  .0.  ( +g  `  W ) ( k ( .s `  W
) Z ) )  =  ( k ( .s `  W ) Z ) )
5518, 32, 54syl2anc 642 . . . . . . . . . . . . . . . 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 ) )
5653, 55eqtrd 2315 . . . . . . . . . . . . . . 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 ) )
5750, 56sylan9eqr 2337 . . . . . . . . . . . . . 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 ) )
5848, 57eqtrd 2315 . . . . . . . . . . . . 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 ) )
592, 6, 4, 5, 7, 18, 19, 24lspsneli 15758 . . . . . . . . . . . . . 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 }
) )
6059adantr 451 . . . . . . . . . . . . 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 } ) )
6158, 60eqeltrd 2357 . . . . . . . . . . . 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 } ) )
62 eldifsni 3750 . . . . . . . . . . . . . 14  |-  ( X  e.  ( V  \  {  .0.  } )  ->  X  =/=  .0.  )
6321, 62syl 15 . . . . . . . . . . . . 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.  )
6463adantr 451 . . . . . . . . . . . 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.  )
652, 45, 7, 46, 47, 61, 64lspsneleq 15868 . . . . . . . . . . 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 } ) )
6665ex 423 . . . . . . . . . 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 } ) ) )
6766necon3d 2484 . . . . . . . . 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 ) ) ) )
6844, 67mpd 14 . . . . . . . 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 )
) )
69 eldifsn 3749 . . . . . . . 8  |-  ( j  e.  ( ( Base `  (Scalar `  W )
)  \  { ( 0g `  (Scalar `  W
) ) } )  <-> 
( j  e.  (
Base `  (Scalar `  W
) )  /\  j  =/=  ( 0g `  (Scalar `  W ) ) ) )
7033, 68, 69sylanbrc 645 . . . . . . 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 ) ) } ) )
714lmodfgrp 15636 . . . . . . . . . . 11  |-  ( W  e.  LMod  ->  (Scalar `  W )  e.  Grp )
7228, 71syl 15 . . . . . . . . . 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 )
735, 16grpinvcl 14527 . . . . . . . . . 10  |-  ( ( (Scalar `  W )  e.  Grp  /\  k  e.  ( Base `  (Scalar `  W ) ) )  ->  ( ( inv g `  (Scalar `  W ) ) `  k )  e.  (
Base `  (Scalar `  W
) ) )
7472, 19, 73syl2anc 642 . . . . . . . . 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 ) ) )
752, 4, 6, 5lmodvscl 15644 . . . . . . . . 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 )
7618, 74, 24, 75syl3anc 1182 . . . . . . . 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 )
772, 3lmodvacl 15641 . . . . . . . 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
)
7818, 23, 76, 77syl3anc 1182 . . . . . . 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 )
792, 6, 4, 5, 41, 42, 17, 70, 78, 34lvecinv 15866 . . . . . 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 ) ) ) ) )
8040, 79mpbid 201 . . . . 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 ) ) ) )
81 eqid 2283 . . . . . . 7  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
822, 81, 7, 18, 23, 24lspprcl 15735 . . . . . 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 ) )
834lvecdrng 15858 . . . . . . . 8  |-  ( W  e.  LVec  ->  (Scalar `  W )  e.  DivRing )
8417, 83syl 15 . . . . . . 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 )
855, 41, 42drnginvrcl 15529 . . . . . . 7  |-  ( ( (Scalar `  W )  e.  DivRing  /\  j  e.  ( Base `  (Scalar `  W
) )  /\  j  =/=  ( 0g `  (Scalar `  W ) ) )  ->  ( ( invr `  (Scalar `  W )
) `  j )  e.  ( Base `  (Scalar `  W ) ) )
8684, 33, 68, 85syl3anc 1182 . . . . . 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
) ) )
87 eqid 2283 . . . . . . . . . 10  |-  ( 1r
`  (Scalar `  W )
)  =  ( 1r
`  (Scalar `  W )
)
882, 4, 6, 87lmodvs1 15658 . . . . . . . . 9  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  (
( 1r `  (Scalar `  W ) ) ( .s `  W ) X )  =  X )
8918, 23, 88syl2anc 642 . . . . . . . 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 )
9089oveq1d 5873 . . . . . . 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 ) ) )
914lmodrng 15635 . . . . . . . . 9  |-  ( W  e.  LMod  ->  (Scalar `  W )  e.  Ring )
925, 87rngidcl 15361 . . . . . . . . 9  |-  ( (Scalar `  W )  e.  Ring  -> 
( 1r `  (Scalar `  W ) )  e.  ( Base `  (Scalar `  W ) ) )
9318, 91, 923syl 18 . . . . . . . 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 ) ) )
942, 3, 6, 4, 5, 7, 18, 93, 74, 23, 24lsppreli 15843 . . . . . . 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 } ) )
9590, 94eqeltrrd 2358 . . . . . 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 }
) )
964, 6, 5, 81lssvscl 15712 . . . . . 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 }
) )
9718, 82, 86, 95, 96syl22anc 1183 . . . . 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 }
) )
9880, 97eqeltrd 2357 . . . 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 } ) )
99983exp 1150 . . 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 } ) ) ) )
10099rexlimdvv 2673 . 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 } ) ) )
10114, 100mpd 14 1  |-  ( ph  ->  Y  e.  ( N `
 { X ,  Z } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544    \ cdif 3149   {csn 3640   {cpr 3641   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208  Scalarcsca 13211   .scvsca 13212   0gc0g 13400   Grpcgrp 14362   inv gcminusg 14363   -gcsg 14365   Ringcrg 15337   1rcur 15339   invrcinvr 15453   DivRingcdr 15512   LModclmod 15627   LSubSpclss 15689   LSpanclspn 15728   LVecclvec 15855
This theorem is referenced by:  lspexchn1  15883  lspindp1  15886  mapdh8ab  31967  mapdh8ad  31969  mapdh8b  31970  mapdh8c  31971  mapdh8e  31974
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-tpos 6234  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-0g 13404  df-mnd 14367  df-submnd 14416  df-grp 14489  df-minusg 14490  df-sbg 14491  df-subg 14618  df-cntz 14793  df-lsm 14947  df-cmn 15091  df-abl 15092  df-mgp 15326  df-rng 15340  df-ur 15342  df-oppr 15405  df-dvdsr 15423  df-unit 15424  df-invr 15454  df-drng 15514  df-lmod 15629  df-lss 15690  df-lsp 15729  df-lvec 15856
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