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Theorem lspfixed 15897
Description: Show membership in the span of the sum of two vectors, one of which ( Y) is fixed in advance. (Contributed by NM, 27-May-2015.)
Hypotheses
Ref Expression
lspfixed.v  |-  V  =  ( Base `  W
)
lspfixed.p  |-  .+  =  ( +g  `  W )
lspfixed.o  |-  .0.  =  ( 0g `  W )
lspfixed.n  |-  N  =  ( LSpan `  W )
lspfixed.w  |-  ( ph  ->  W  e.  LVec )
lspfixed.x  |-  ( ph  ->  X  e.  V )
lspfixed.y  |-  ( ph  ->  Y  e.  V )
lspfixed.z  |-  ( ph  ->  Z  e.  V )
lspfixed.e  |-  ( ph  ->  -.  X  e.  ( N `  { Y } ) )
lspfixed.f  |-  ( ph  ->  -.  X  e.  ( N `  { Z } ) )
lspfixed.g  |-  ( ph  ->  X  e.  ( N `
 { Y ,  Z } ) )
Assertion
Ref Expression
lspfixed  |-  ( ph  ->  E. z  e.  ( ( N `  { Z } )  \  {  .0.  } ) X  e.  ( N `  {
( Y  .+  z
) } ) )
Distinct variable groups:    z, N    z,  .0.    z,  .+    z, W   
z, X    z, Y    z, Z
Allowed substitution hints:    ph( z)    V( z)

Proof of Theorem lspfixed
Dummy variables  k 
l are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lspfixed.g . . 3  |-  ( ph  ->  X  e.  ( N `
 { Y ,  Z } ) )
2 lspfixed.v . . . 4  |-  V  =  ( Base `  W
)
3 lspfixed.p . . . 4  |-  .+  =  ( +g  `  W )
4 eqid 2296 . . . 4  |-  (Scalar `  W )  =  (Scalar `  W )
5 eqid 2296 . . . 4  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
6 eqid 2296 . . . 4  |-  ( .s
`  W )  =  ( .s `  W
)
7 lspfixed.n . . . 4  |-  N  =  ( LSpan `  W )
8 lspfixed.w . . . . 5  |-  ( ph  ->  W  e.  LVec )
9 lveclmod 15875 . . . . 5  |-  ( W  e.  LVec  ->  W  e. 
LMod )
108, 9syl 15 . . . 4  |-  ( ph  ->  W  e.  LMod )
11 lspfixed.y . . . 4  |-  ( ph  ->  Y  e.  V )
12 lspfixed.z . . . 4  |-  ( ph  ->  Z  e.  V )
132, 3, 4, 5, 6, 7, 10, 11, 12lspprel 15863 . . 3  |-  ( ph  ->  ( X  e.  ( N `  { Y ,  Z } )  <->  E. k  e.  ( Base `  (Scalar `  W ) ) E. l  e.  ( Base `  (Scalar `  W )
) X  =  ( ( k ( .s
`  W ) Y )  .+  ( l ( .s `  W
) Z ) ) ) )
141, 13mpbid 201 . 2  |-  ( ph  ->  E. k  e.  (
Base `  (Scalar `  W
) ) E. l  e.  ( Base `  (Scalar `  W ) ) X  =  ( ( k ( .s `  W
) Y )  .+  ( l ( .s
`  W ) Z ) ) )
15103ad2ant1 976 . . . . . . 7  |-  ( (
ph  /\  ( k  e.  ( Base `  (Scalar `  W ) )  /\  l  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( k ( .s `  W
) Y )  .+  ( l ( .s
`  W ) Z ) ) )  ->  W  e.  LMod )
16 eqid 2296 . . . . . . . . . 10  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
172, 16, 7lspsncl 15750 . . . . . . . . 9  |-  ( ( W  e.  LMod  /\  Z  e.  V )  ->  ( N `  { Z } )  e.  (
LSubSp `  W ) )
1810, 12, 17syl2anc 642 . . . . . . . 8  |-  ( ph  ->  ( N `  { Z } )  e.  (
LSubSp `  W ) )
19183ad2ant1 976 . . . . . . 7  |-  ( (
ph  /\  ( k  e.  ( Base `  (Scalar `  W ) )  /\  l  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( k ( .s `  W
) Y )  .+  ( l ( .s
`  W ) Z ) ) )  -> 
( N `  { Z } )  e.  (
LSubSp `  W ) )
2083ad2ant1 976 . . . . . . . . 9  |-  ( (
ph  /\  ( k  e.  ( Base `  (Scalar `  W ) )  /\  l  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( k ( .s `  W
) Y )  .+  ( l ( .s
`  W ) Z ) ) )  ->  W  e.  LVec )
214lvecdrng 15874 . . . . . . . . 9  |-  ( W  e.  LVec  ->  (Scalar `  W )  e.  DivRing )
2220, 21syl 15 . . . . . . . 8  |-  ( (
ph  /\  ( k  e.  ( Base `  (Scalar `  W ) )  /\  l  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( k ( .s `  W
) Y )  .+  ( l ( .s
`  W ) Z ) ) )  -> 
(Scalar `  W )  e.  DivRing )
23 simp2l 981 . . . . . . . 8  |-  ( (
ph  /\  ( k  e.  ( Base `  (Scalar `  W ) )  /\  l  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( k ( .s `  W
) Y )  .+  ( l ( .s
`  W ) Z ) ) )  -> 
k  e.  ( Base `  (Scalar `  W )
) )
24 lspfixed.f . . . . . . . . . 10  |-  ( ph  ->  -.  X  e.  ( N `  { Z } ) )
25243ad2ant1 976 . . . . . . . . 9  |-  ( (
ph  /\  ( k  e.  ( Base `  (Scalar `  W ) )  /\  l  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( k ( .s `  W
) Y )  .+  ( l ( .s
`  W ) Z ) ) )  ->  -.  X  e.  ( N `  { Z } ) )
26 simpl3 960 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
k  e.  ( Base `  (Scalar `  W )
)  /\  l  e.  ( Base `  (Scalar `  W
) ) )  /\  X  =  ( (
k ( .s `  W ) Y ) 
.+  ( l ( .s `  W ) Z ) ) )  /\  k  =  ( 0g `  (Scalar `  W ) ) )  ->  X  =  ( ( k ( .s
`  W ) Y )  .+  ( l ( .s `  W
) Z ) ) )
27 simpr 447 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
k  e.  ( Base `  (Scalar `  W )
)  /\  l  e.  ( Base `  (Scalar `  W
) ) )  /\  X  =  ( (
k ( .s `  W ) Y ) 
.+  ( l ( .s `  W ) Z ) ) )  /\  k  =  ( 0g `  (Scalar `  W ) ) )  ->  k  =  ( 0g `  (Scalar `  W ) ) )
2827oveq1d 5889 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
k  e.  ( Base `  (Scalar `  W )
)  /\  l  e.  ( Base `  (Scalar `  W
) ) )  /\  X  =  ( (
k ( .s `  W ) Y ) 
.+  ( l ( .s `  W ) Z ) ) )  /\  k  =  ( 0g `  (Scalar `  W ) ) )  ->  ( k ( .s `  W ) Y )  =  ( ( 0g `  (Scalar `  W ) ) ( .s `  W ) Y ) )
29 simpl1 958 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  (
k  e.  ( Base `  (Scalar `  W )
)  /\  l  e.  ( Base `  (Scalar `  W
) ) )  /\  X  =  ( (
k ( .s `  W ) Y ) 
.+  ( l ( .s `  W ) Z ) ) )  /\  k  =  ( 0g `  (Scalar `  W ) ) )  ->  ph )
3029, 10syl 15 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
k  e.  ( Base `  (Scalar `  W )
)  /\  l  e.  ( Base `  (Scalar `  W
) ) )  /\  X  =  ( (
k ( .s `  W ) Y ) 
.+  ( l ( .s `  W ) Z ) ) )  /\  k  =  ( 0g `  (Scalar `  W ) ) )  ->  W  e.  LMod )
3129, 11syl 15 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
k  e.  ( Base `  (Scalar `  W )
)  /\  l  e.  ( Base `  (Scalar `  W
) ) )  /\  X  =  ( (
k ( .s `  W ) Y ) 
.+  ( l ( .s `  W ) Z ) ) )  /\  k  =  ( 0g `  (Scalar `  W ) ) )  ->  Y  e.  V
)
32 eqid 2296 . . . . . . . . . . . . . . . . 17  |-  ( 0g
`  (Scalar `  W )
)  =  ( 0g
`  (Scalar `  W )
)
33 lspfixed.o . . . . . . . . . . . . . . . . 17  |-  .0.  =  ( 0g `  W )
342, 4, 6, 32, 33lmod0vs 15679 . . . . . . . . . . . . . . . 16  |-  ( ( W  e.  LMod  /\  Y  e.  V )  ->  (
( 0g `  (Scalar `  W ) ) ( .s `  W ) Y )  =  .0.  )
3530, 31, 34syl2anc 642 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
k  e.  ( Base `  (Scalar `  W )
)  /\  l  e.  ( Base `  (Scalar `  W
) ) )  /\  X  =  ( (
k ( .s `  W ) Y ) 
.+  ( l ( .s `  W ) Z ) ) )  /\  k  =  ( 0g `  (Scalar `  W ) ) )  ->  ( ( 0g
`  (Scalar `  W )
) ( .s `  W ) Y )  =  .0.  )
3628, 35eqtrd 2328 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
k  e.  ( Base `  (Scalar `  W )
)  /\  l  e.  ( Base `  (Scalar `  W
) ) )  /\  X  =  ( (
k ( .s `  W ) Y ) 
.+  ( l ( .s `  W ) Z ) ) )  /\  k  =  ( 0g `  (Scalar `  W ) ) )  ->  ( k ( .s `  W ) Y )  =  .0.  )
3736oveq1d 5889 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
k  e.  ( Base `  (Scalar `  W )
)  /\  l  e.  ( Base `  (Scalar `  W
) ) )  /\  X  =  ( (
k ( .s `  W ) Y ) 
.+  ( l ( .s `  W ) Z ) ) )  /\  k  =  ( 0g `  (Scalar `  W ) ) )  ->  ( ( k ( .s `  W
) Y )  .+  ( l ( .s
`  W ) Z ) )  =  (  .0.  .+  ( l
( .s `  W
) Z ) ) )
38 simp2r 982 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( k  e.  ( Base `  (Scalar `  W ) )  /\  l  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( k ( .s `  W
) Y )  .+  ( l ( .s
`  W ) Z ) ) )  -> 
l  e.  ( Base `  (Scalar `  W )
) )
39123ad2ant1 976 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( k  e.  ( Base `  (Scalar `  W ) )  /\  l  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( k ( .s `  W
) Y )  .+  ( l ( .s
`  W ) Z ) ) )  ->  Z  e.  V )
402, 4, 6, 5lmodvscl 15660 . . . . . . . . . . . . . . . 16  |-  ( ( W  e.  LMod  /\  l  e.  ( Base `  (Scalar `  W ) )  /\  Z  e.  V )  ->  ( l ( .s
`  W ) Z )  e.  V )
4115, 38, 39, 40syl3anc 1182 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( k  e.  ( Base `  (Scalar `  W ) )  /\  l  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( k ( .s `  W
) Y )  .+  ( l ( .s
`  W ) Z ) ) )  -> 
( l ( .s
`  W ) Z )  e.  V )
4241adantr 451 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
k  e.  ( Base `  (Scalar `  W )
)  /\  l  e.  ( Base `  (Scalar `  W
) ) )  /\  X  =  ( (
k ( .s `  W ) Y ) 
.+  ( l ( .s `  W ) Z ) ) )  /\  k  =  ( 0g `  (Scalar `  W ) ) )  ->  ( l ( .s `  W ) Z )  e.  V
)
432, 3, 33lmod0vlid 15676 . . . . . . . . . . . . . 14  |-  ( ( W  e.  LMod  /\  (
l ( .s `  W ) Z )  e.  V )  -> 
(  .0.  .+  (
l ( .s `  W ) Z ) )  =  ( l ( .s `  W
) Z ) )
4430, 42, 43syl2anc 642 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
k  e.  ( Base `  (Scalar `  W )
)  /\  l  e.  ( Base `  (Scalar `  W
) ) )  /\  X  =  ( (
k ( .s `  W ) Y ) 
.+  ( l ( .s `  W ) Z ) ) )  /\  k  =  ( 0g `  (Scalar `  W ) ) )  ->  (  .0.  .+  ( l ( .s
`  W ) Z ) )  =  ( l ( .s `  W ) Z ) )
4526, 37, 443eqtrd 2332 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
k  e.  ( Base `  (Scalar `  W )
)  /\  l  e.  ( Base `  (Scalar `  W
) ) )  /\  X  =  ( (
k ( .s `  W ) Y ) 
.+  ( l ( .s `  W ) Z ) ) )  /\  k  =  ( 0g `  (Scalar `  W ) ) )  ->  X  =  ( l ( .s `  W ) Z ) )
4629, 18syl 15 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
k  e.  ( Base `  (Scalar `  W )
)  /\  l  e.  ( Base `  (Scalar `  W
) ) )  /\  X  =  ( (
k ( .s `  W ) Y ) 
.+  ( l ( .s `  W ) Z ) ) )  /\  k  =  ( 0g `  (Scalar `  W ) ) )  ->  ( N `  { Z } )  e.  ( LSubSp `  W )
)
47 simpl2r 1009 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
k  e.  ( Base `  (Scalar `  W )
)  /\  l  e.  ( Base `  (Scalar `  W
) ) )  /\  X  =  ( (
k ( .s `  W ) Y ) 
.+  ( l ( .s `  W ) Z ) ) )  /\  k  =  ( 0g `  (Scalar `  W ) ) )  ->  l  e.  (
Base `  (Scalar `  W
) ) )
482, 7lspsnid 15766 . . . . . . . . . . . . . . 15  |-  ( ( W  e.  LMod  /\  Z  e.  V )  ->  Z  e.  ( N `  { Z } ) )
4910, 12, 48syl2anc 642 . . . . . . . . . . . . . 14  |-  ( ph  ->  Z  e.  ( N `
 { Z }
) )
5029, 49syl 15 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
k  e.  ( Base `  (Scalar `  W )
)  /\  l  e.  ( Base `  (Scalar `  W
) ) )  /\  X  =  ( (
k ( .s `  W ) Y ) 
.+  ( l ( .s `  W ) Z ) ) )  /\  k  =  ( 0g `  (Scalar `  W ) ) )  ->  Z  e.  ( N `  { Z } ) )
514, 6, 5, 16lssvscl 15728 . . . . . . . . . . . . 13  |-  ( ( ( W  e.  LMod  /\  ( N `  { Z } )  e.  (
LSubSp `  W ) )  /\  ( l  e.  ( Base `  (Scalar `  W ) )  /\  Z  e.  ( N `  { Z } ) ) )  ->  (
l ( .s `  W ) Z )  e.  ( N `  { Z } ) )
5230, 46, 47, 50, 51syl22anc 1183 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
k  e.  ( Base `  (Scalar `  W )
)  /\  l  e.  ( Base `  (Scalar `  W
) ) )  /\  X  =  ( (
k ( .s `  W ) Y ) 
.+  ( l ( .s `  W ) Z ) ) )  /\  k  =  ( 0g `  (Scalar `  W ) ) )  ->  ( l ( .s `  W ) Z )  e.  ( N `  { Z } ) )
5345, 52eqeltrd 2370 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
k  e.  ( Base `  (Scalar `  W )
)  /\  l  e.  ( Base `  (Scalar `  W
) ) )  /\  X  =  ( (
k ( .s `  W ) Y ) 
.+  ( l ( .s `  W ) Z ) ) )  /\  k  =  ( 0g `  (Scalar `  W ) ) )  ->  X  e.  ( N `  { Z } ) )
5453ex 423 . . . . . . . . . 10  |-  ( (
ph  /\  ( k  e.  ( Base `  (Scalar `  W ) )  /\  l  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( k ( .s `  W
) Y )  .+  ( l ( .s
`  W ) Z ) ) )  -> 
( k  =  ( 0g `  (Scalar `  W ) )  ->  X  e.  ( N `  { Z } ) ) )
5554necon3bd 2496 . . . . . . . . 9  |-  ( (
ph  /\  ( k  e.  ( Base `  (Scalar `  W ) )  /\  l  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( k ( .s `  W
) Y )  .+  ( l ( .s
`  W ) Z ) ) )  -> 
( -.  X  e.  ( N `  { Z } )  ->  k  =/=  ( 0g `  (Scalar `  W ) ) ) )
5625, 55mpd 14 . . . . . . . 8  |-  ( (
ph  /\  ( k  e.  ( Base `  (Scalar `  W ) )  /\  l  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( k ( .s `  W
) Y )  .+  ( l ( .s
`  W ) Z ) ) )  -> 
k  =/=  ( 0g
`  (Scalar `  W )
) )
57 eqid 2296 . . . . . . . . 9  |-  ( invr `  (Scalar `  W )
)  =  ( invr `  (Scalar `  W )
)
585, 32, 57drnginvrcl 15545 . . . . . . . 8  |-  ( ( (Scalar `  W )  e.  DivRing  /\  k  e.  ( Base `  (Scalar `  W
) )  /\  k  =/=  ( 0g `  (Scalar `  W ) ) )  ->  ( ( invr `  (Scalar `  W )
) `  k )  e.  ( Base `  (Scalar `  W ) ) )
5922, 23, 56, 58syl3anc 1182 . . . . . . 7  |-  ( (
ph  /\  ( k  e.  ( Base `  (Scalar `  W ) )  /\  l  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( k ( .s `  W
) Y )  .+  ( l ( .s
`  W ) Z ) ) )  -> 
( ( invr `  (Scalar `  W ) ) `  k )  e.  (
Base `  (Scalar `  W
) ) )
60493ad2ant1 976 . . . . . . . 8  |-  ( (
ph  /\  ( k  e.  ( Base `  (Scalar `  W ) )  /\  l  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( k ( .s `  W
) Y )  .+  ( l ( .s
`  W ) Z ) ) )  ->  Z  e.  ( N `  { Z } ) )
6115, 19, 38, 60, 51syl22anc 1183 . . . . . . 7  |-  ( (
ph  /\  ( k  e.  ( Base `  (Scalar `  W ) )  /\  l  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( k ( .s `  W
) Y )  .+  ( l ( .s
`  W ) Z ) ) )  -> 
( l ( .s
`  W ) Z )  e.  ( N `
 { Z }
) )
624, 6, 5, 16lssvscl 15728 . . . . . . 7  |-  ( ( ( W  e.  LMod  /\  ( N `  { Z } )  e.  (
LSubSp `  W ) )  /\  ( ( (
invr `  (Scalar `  W
) ) `  k
)  e.  ( Base `  (Scalar `  W )
)  /\  ( l
( .s `  W
) Z )  e.  ( N `  { Z } ) ) )  ->  ( ( (
invr `  (Scalar `  W
) ) `  k
) ( .s `  W ) ( l ( .s `  W
) Z ) )  e.  ( N `  { Z } ) )
6315, 19, 59, 61, 62syl22anc 1183 . . . . . 6  |-  ( (
ph  /\  ( k  e.  ( Base `  (Scalar `  W ) )  /\  l  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( k ( .s `  W
) Y )  .+  ( l ( .s
`  W ) Z ) ) )  -> 
( ( ( invr `  (Scalar `  W )
) `  k )
( .s `  W
) ( l ( .s `  W ) Z ) )  e.  ( N `  { Z } ) )
645, 32, 57drnginvrn0 15546 . . . . . . . 8  |-  ( ( (Scalar `  W )  e.  DivRing  /\  k  e.  ( Base `  (Scalar `  W
) )  /\  k  =/=  ( 0g `  (Scalar `  W ) ) )  ->  ( ( invr `  (Scalar `  W )
) `  k )  =/=  ( 0g `  (Scalar `  W ) ) )
6522, 23, 56, 64syl3anc 1182 . . . . . . 7  |-  ( (
ph  /\  ( k  e.  ( Base `  (Scalar `  W ) )  /\  l  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( k ( .s `  W
) Y )  .+  ( l ( .s
`  W ) Z ) ) )  -> 
( ( invr `  (Scalar `  W ) ) `  k )  =/=  ( 0g `  (Scalar `  W
) ) )
66 lspfixed.e . . . . . . . . . 10  |-  ( ph  ->  -.  X  e.  ( N `  { Y } ) )
67663ad2ant1 976 . . . . . . . . 9  |-  ( (
ph  /\  ( k  e.  ( Base `  (Scalar `  W ) )  /\  l  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( k ( .s `  W
) Y )  .+  ( l ( .s
`  W ) Z ) ) )  ->  -.  X  e.  ( N `  { Y } ) )
68 simpl3 960 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
k  e.  ( Base `  (Scalar `  W )
)  /\  l  e.  ( Base `  (Scalar `  W
) ) )  /\  X  =  ( (
k ( .s `  W ) Y ) 
.+  ( l ( .s `  W ) Z ) ) )  /\  l  =  ( 0g `  (Scalar `  W ) ) )  ->  X  =  ( ( k ( .s
`  W ) Y )  .+  ( l ( .s `  W
) Z ) ) )
69 oveq1 5881 . . . . . . . . . . . . . . 15  |-  ( l  =  ( 0g `  (Scalar `  W ) )  ->  ( l ( .s `  W ) Z )  =  ( ( 0g `  (Scalar `  W ) ) ( .s `  W ) Z ) )
702, 4, 6, 32, 33lmod0vs 15679 . . . . . . . . . . . . . . . 16  |-  ( ( W  e.  LMod  /\  Z  e.  V )  ->  (
( 0g `  (Scalar `  W ) ) ( .s `  W ) Z )  =  .0.  )
7115, 39, 70syl2anc 642 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( k  e.  ( Base `  (Scalar `  W ) )  /\  l  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( k ( .s `  W
) Y )  .+  ( l ( .s
`  W ) Z ) ) )  -> 
( ( 0g `  (Scalar `  W ) ) ( .s `  W
) Z )  =  .0.  )
7269, 71sylan9eqr 2350 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
k  e.  ( Base `  (Scalar `  W )
)  /\  l  e.  ( Base `  (Scalar `  W
) ) )  /\  X  =  ( (
k ( .s `  W ) Y ) 
.+  ( l ( .s `  W ) Z ) ) )  /\  l  =  ( 0g `  (Scalar `  W ) ) )  ->  ( l ( .s `  W ) Z )  =  .0.  )
7372oveq2d 5890 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
k  e.  ( Base `  (Scalar `  W )
)  /\  l  e.  ( Base `  (Scalar `  W
) ) )  /\  X  =  ( (
k ( .s `  W ) Y ) 
.+  ( l ( .s `  W ) Z ) ) )  /\  l  =  ( 0g `  (Scalar `  W ) ) )  ->  ( ( k ( .s `  W
) Y )  .+  ( l ( .s
`  W ) Z ) )  =  ( ( k ( .s
`  W ) Y )  .+  .0.  )
)
74113ad2ant1 976 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( k  e.  ( Base `  (Scalar `  W ) )  /\  l  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( k ( .s `  W
) Y )  .+  ( l ( .s
`  W ) Z ) ) )  ->  Y  e.  V )
752, 4, 6, 5lmodvscl 15660 . . . . . . . . . . . . . . . 16  |-  ( ( W  e.  LMod  /\  k  e.  ( Base `  (Scalar `  W ) )  /\  Y  e.  V )  ->  ( k ( .s
`  W ) Y )  e.  V )
7615, 23, 74, 75syl3anc 1182 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( k  e.  ( Base `  (Scalar `  W ) )  /\  l  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( k ( .s `  W
) Y )  .+  ( l ( .s
`  W ) Z ) ) )  -> 
( k ( .s
`  W ) Y )  e.  V )
772, 3, 33lmod0vrid 15677 . . . . . . . . . . . . . . 15  |-  ( ( W  e.  LMod  /\  (
k ( .s `  W ) Y )  e.  V )  -> 
( ( k ( .s `  W ) Y )  .+  .0.  )  =  ( k
( .s `  W
) Y ) )
7815, 76, 77syl2anc 642 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( k  e.  ( Base `  (Scalar `  W ) )  /\  l  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( k ( .s `  W
) Y )  .+  ( l ( .s
`  W ) Z ) ) )  -> 
( ( k ( .s `  W ) Y )  .+  .0.  )  =  ( k
( .s `  W
) Y ) )
7978adantr 451 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
k  e.  ( Base `  (Scalar `  W )
)  /\  l  e.  ( Base `  (Scalar `  W
) ) )  /\  X  =  ( (
k ( .s `  W ) Y ) 
.+  ( l ( .s `  W ) Z ) ) )  /\  l  =  ( 0g `  (Scalar `  W ) ) )  ->  ( ( k ( .s `  W
) Y )  .+  .0.  )  =  (
k ( .s `  W ) Y ) )
8068, 73, 793eqtrd 2332 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
k  e.  ( Base `  (Scalar `  W )
)  /\  l  e.  ( Base `  (Scalar `  W
) ) )  /\  X  =  ( (
k ( .s `  W ) Y ) 
.+  ( l ( .s `  W ) Z ) ) )  /\  l  =  ( 0g `  (Scalar `  W ) ) )  ->  X  =  ( k ( .s `  W ) Y ) )
812, 16, 7lspsncl 15750 . . . . . . . . . . . . . . . 16  |-  ( ( W  e.  LMod  /\  Y  e.  V )  ->  ( N `  { Y } )  e.  (
LSubSp `  W ) )
8210, 11, 81syl2anc 642 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( N `  { Y } )  e.  (
LSubSp `  W ) )
83823ad2ant1 976 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( k  e.  ( Base `  (Scalar `  W ) )  /\  l  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( k ( .s `  W
) Y )  .+  ( l ( .s
`  W ) Z ) ) )  -> 
( N `  { Y } )  e.  (
LSubSp `  W ) )
842, 7lspsnid 15766 . . . . . . . . . . . . . . . 16  |-  ( ( W  e.  LMod  /\  Y  e.  V )  ->  Y  e.  ( N `  { Y } ) )
8510, 11, 84syl2anc 642 . . . . . . . . . . . . . . 15  |-  ( ph  ->  Y  e.  ( N `
 { Y }
) )
86853ad2ant1 976 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( k  e.  ( Base `  (Scalar `  W ) )  /\  l  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( k ( .s `  W
) Y )  .+  ( l ( .s
`  W ) Z ) ) )  ->  Y  e.  ( N `  { Y } ) )
874, 6, 5, 16lssvscl 15728 . . . . . . . . . . . . . 14  |-  ( ( ( W  e.  LMod  /\  ( N `  { Y } )  e.  (
LSubSp `  W ) )  /\  ( k  e.  ( Base `  (Scalar `  W ) )  /\  Y  e.  ( N `  { Y } ) ) )  ->  (
k ( .s `  W ) Y )  e.  ( N `  { Y } ) )
8815, 83, 23, 86, 87syl22anc 1183 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( k  e.  ( Base `  (Scalar `  W ) )  /\  l  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( k ( .s `  W
) Y )  .+  ( l ( .s
`  W ) Z ) ) )  -> 
( k ( .s
`  W ) Y )  e.  ( N `
 { Y }
) )
8988adantr 451 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
k  e.  ( Base `  (Scalar `  W )
)  /\  l  e.  ( Base `  (Scalar `  W
) ) )  /\  X  =  ( (
k ( .s `  W ) Y ) 
.+  ( l ( .s `  W ) Z ) ) )  /\  l  =  ( 0g `  (Scalar `  W ) ) )  ->  ( k ( .s `  W ) Y )  e.  ( N `  { Y } ) )
9080, 89eqeltrd 2370 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
k  e.  ( Base `  (Scalar `  W )
)  /\  l  e.  ( Base `  (Scalar `  W
) ) )  /\  X  =  ( (
k ( .s `  W ) Y ) 
.+  ( l ( .s `  W ) Z ) ) )  /\  l  =  ( 0g `  (Scalar `  W ) ) )  ->  X  e.  ( N `  { Y } ) )
9190ex 423 . . . . . . . . . 10  |-  ( (
ph  /\  ( k  e.  ( Base `  (Scalar `  W ) )  /\  l  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( k ( .s `  W
) Y )  .+  ( l ( .s
`  W ) Z ) ) )  -> 
( l  =  ( 0g `  (Scalar `  W ) )  ->  X  e.  ( N `  { Y } ) ) )
9291necon3bd 2496 . . . . . . . . 9  |-  ( (
ph  /\  ( k  e.  ( Base `  (Scalar `  W ) )  /\  l  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( k ( .s `  W
) Y )  .+  ( l ( .s
`  W ) Z ) ) )  -> 
( -.  X  e.  ( N `  { Y } )  ->  l  =/=  ( 0g `  (Scalar `  W ) ) ) )
9367, 92mpd 14 . . . . . . . 8  |-  ( (
ph  /\  ( k  e.  ( Base `  (Scalar `  W ) )  /\  l  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( k ( .s `  W
) Y )  .+  ( l ( .s
`  W ) Z ) ) )  -> 
l  =/=  ( 0g
`  (Scalar `  W )
) )
94 simpl1 958 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
k  e.  ( Base `  (Scalar `  W )
)  /\  l  e.  ( Base `  (Scalar `  W
) ) )  /\  X  =  ( (
k ( .s `  W ) Y ) 
.+  ( l ( .s `  W ) Z ) ) )  /\  Z  =  .0.  )  ->  ph )
9594, 1syl 15 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
k  e.  ( Base `  (Scalar `  W )
)  /\  l  e.  ( Base `  (Scalar `  W
) ) )  /\  X  =  ( (
k ( .s `  W ) Y ) 
.+  ( l ( .s `  W ) Z ) ) )  /\  Z  =  .0.  )  ->  X  e.  ( N `  { Y ,  Z } ) )
96 preq2 3720 . . . . . . . . . . . . . 14  |-  ( Z  =  .0.  ->  { Y ,  Z }  =  { Y ,  .0.  } )
9796fveq2d 5545 . . . . . . . . . . . . 13  |-  ( Z  =  .0.  ->  ( N `  { Y ,  Z } )  =  ( N `  { Y ,  .0.  } ) )
982, 33, 7, 15, 74lsppr0 15861 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( k  e.  ( Base `  (Scalar `  W ) )  /\  l  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( k ( .s `  W
) Y )  .+  ( l ( .s
`  W ) Z ) ) )  -> 
( N `  { Y ,  .0.  } )  =  ( N `  { Y } ) )
9997, 98sylan9eqr 2350 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
k  e.  ( Base `  (Scalar `  W )
)  /\  l  e.  ( Base `  (Scalar `  W
) ) )  /\  X  =  ( (
k ( .s `  W ) Y ) 
.+  ( l ( .s `  W ) Z ) ) )  /\  Z  =  .0.  )  ->  ( N `  { Y ,  Z } )  =  ( N `  { Y } ) )
10095, 99eleqtrd 2372 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
k  e.  ( Base `  (Scalar `  W )
)  /\  l  e.  ( Base `  (Scalar `  W
) ) )  /\  X  =  ( (
k ( .s `  W ) Y ) 
.+  ( l ( .s `  W ) Z ) ) )  /\  Z  =  .0.  )  ->  X  e.  ( N `  { Y } ) )
101100ex 423 . . . . . . . . . 10  |-  ( (
ph  /\  ( k  e.  ( Base `  (Scalar `  W ) )  /\  l  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( k ( .s `  W
) Y )  .+  ( l ( .s
`  W ) Z ) ) )  -> 
( Z  =  .0. 
->  X  e.  ( N `  { Y } ) ) )
102101necon3bd 2496 . . . . . . . . 9  |-  ( (
ph  /\  ( k  e.  ( Base `  (Scalar `  W ) )  /\  l  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( k ( .s `  W
) Y )  .+  ( l ( .s
`  W ) Z ) ) )  -> 
( -.  X  e.  ( N `  { Y } )  ->  Z  =/=  .0.  ) )
10367, 102mpd 14 . . . . . . . 8  |-  ( (
ph  /\  ( k  e.  ( Base `  (Scalar `  W ) )  /\  l  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( k ( .s `  W
) Y )  .+  ( l ( .s
`  W ) Z ) ) )  ->  Z  =/=  .0.  )
1042, 6, 4, 5, 32, 33, 20, 38, 39lvecvsn0 15878 . . . . . . . 8  |-  ( (
ph  /\  ( k  e.  ( Base `  (Scalar `  W ) )  /\  l  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( k ( .s `  W
) Y )  .+  ( l ( .s
`  W ) Z ) ) )  -> 
( ( l ( .s `  W ) Z )  =/=  .0.  <->  (
l  =/=  ( 0g
`  (Scalar `  W )
)  /\  Z  =/=  .0.  ) ) )
10593, 103, 104mpbir2and 888 . . . . . . 7  |-  ( (
ph  /\  ( k  e.  ( Base `  (Scalar `  W ) )  /\  l  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( k ( .s `  W
) Y )  .+  ( l ( .s
`  W ) Z ) ) )  -> 
( l ( .s
`  W ) Z )  =/=  .0.  )
1062, 6, 4, 5, 32, 33, 20, 59, 41lvecvsn0 15878 . . . . . . 7  |-  ( (
ph  /\  ( k  e.  ( Base `  (Scalar `  W ) )  /\  l  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( k ( .s `  W
) Y )  .+  ( l ( .s
`  W ) Z ) ) )  -> 
( ( ( (
invr `  (Scalar `  W
) ) `  k
) ( .s `  W ) ( l ( .s `  W
) Z ) )  =/=  .0.  <->  ( (
( invr `  (Scalar `  W
) ) `  k
)  =/=  ( 0g
`  (Scalar `  W )
)  /\  ( l
( .s `  W
) Z )  =/= 
.0.  ) ) )
10765, 105, 106mpbir2and 888 . . . . . 6  |-  ( (
ph  /\  ( k  e.  ( Base `  (Scalar `  W ) )  /\  l  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( k ( .s `  W
) Y )  .+  ( l ( .s
`  W ) Z ) ) )  -> 
( ( ( invr `  (Scalar `  W )
) `  k )
( .s `  W
) ( l ( .s `  W ) Z ) )  =/= 
.0.  )
108 eldifsn 3762 . . . . . 6  |-  ( ( ( ( invr `  (Scalar `  W ) ) `  k ) ( .s
`  W ) ( l ( .s `  W ) Z ) )  e.  ( ( N `  { Z } )  \  {  .0.  } )  <->  ( (
( ( invr `  (Scalar `  W ) ) `  k ) ( .s
`  W ) ( l ( .s `  W ) Z ) )  e.  ( N `
 { Z }
)  /\  ( (
( invr `  (Scalar `  W
) ) `  k
) ( .s `  W ) ( l ( .s `  W
) Z ) )  =/=  .0.  ) )
10963, 107, 108sylanbrc 645 . . . . 5  |-  ( (
ph  /\  ( k  e.  ( Base `  (Scalar `  W ) )  /\  l  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( k ( .s `  W
) Y )  .+  ( l ( .s
`  W ) Z ) ) )  -> 
( ( ( invr `  (Scalar `  W )
) `  k )
( .s `  W
) ( l ( .s `  W ) Z ) )  e.  ( ( N `  { Z } )  \  {  .0.  } ) )
110 simp3 957 . . . . . . 7  |-  ( (
ph  /\  ( k  e.  ( Base `  (Scalar `  W ) )  /\  l  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( k ( .s `  W
) Y )  .+  ( l ( .s
`  W ) Z ) ) )  ->  X  =  ( (
k ( .s `  W ) Y ) 
.+  ( l ( .s `  W ) Z ) ) )
1112, 3lmodvacl 15657 . . . . . . . . 9  |-  ( ( W  e.  LMod  /\  (
k ( .s `  W ) Y )  e.  V  /\  (
l ( .s `  W ) Z )  e.  V )  -> 
( ( k ( .s `  W ) Y )  .+  (
l ( .s `  W ) Z ) )  e.  V )
11215, 76, 41, 111syl3anc 1182 . . . . . . . 8  |-  ( (
ph  /\  ( k  e.  ( Base `  (Scalar `  W ) )  /\  l  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( k ( .s `  W
) Y )  .+  ( l ( .s
`  W ) Z ) ) )  -> 
( ( k ( .s `  W ) Y )  .+  (
l ( .s `  W ) Z ) )  e.  V )
1132, 7lspsnid 15766 . . . . . . . 8  |-  ( ( W  e.  LMod  /\  (
( k ( .s
`  W ) Y )  .+  ( l ( .s `  W
) Z ) )  e.  V )  -> 
( ( k ( .s `  W ) Y )  .+  (
l ( .s `  W ) Z ) )  e.  ( N `
 { ( ( k ( .s `  W ) Y ) 
.+  ( l ( .s `  W ) Z ) ) } ) )
11415, 112, 113syl2anc 642 . . . . . . 7  |-  ( (
ph  /\  ( k  e.  ( Base `  (Scalar `  W ) )  /\  l  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( k ( .s `  W
) Y )  .+  ( l ( .s
`  W ) Z ) ) )  -> 
( ( k ( .s `  W ) Y )  .+  (
l ( .s `  W ) Z ) )  e.  ( N `
 { ( ( k ( .s `  W ) Y ) 
.+  ( l ( .s `  W ) Z ) ) } ) )
115110, 114eqeltrd 2370 . . . . . 6  |-  ( (
ph  /\  ( k  e.  ( Base `  (Scalar `  W ) )  /\  l  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( k ( .s `  W
) Y )  .+  ( l ( .s
`  W ) Z ) ) )  ->  X  e.  ( N `  { ( ( k ( .s `  W
) Y )  .+  ( l ( .s
`  W ) Z ) ) } ) )
1162, 4, 6, 5, 32, 7lspsnvs 15883 . . . . . . . 8  |-  ( ( W  e.  LVec  /\  (
( ( invr `  (Scalar `  W ) ) `  k )  e.  (
Base `  (Scalar `  W
) )  /\  (
( invr `  (Scalar `  W
) ) `  k
)  =/=  ( 0g
`  (Scalar `  W )
) )  /\  (
( k ( .s
`  W ) Y )  .+  ( l ( .s `  W
) Z ) )  e.  V )  -> 
( N `  {
( ( ( invr `  (Scalar `  W )
) `  k )
( .s `  W
) ( ( k ( .s `  W
) Y )  .+  ( l ( .s
`  W ) Z ) ) ) } )  =  ( N `
 { ( ( k ( .s `  W ) Y ) 
.+  ( l ( .s `  W ) Z ) ) } ) )
11720, 59, 65, 112, 116syl121anc 1187 . . . . . . 7  |-  ( (
ph  /\  ( k  e.  ( Base `  (Scalar `  W ) )  /\  l  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( k ( .s `  W
) Y )  .+  ( l ( .s
`  W ) Z ) ) )  -> 
( N `  {
( ( ( invr `  (Scalar `  W )
) `  k )
( .s `  W
) ( ( k ( .s `  W
) Y )  .+  ( l ( .s
`  W ) Z ) ) ) } )  =  ( N `
 { ( ( k ( .s `  W ) Y ) 
.+  ( l ( .s `  W ) Z ) ) } ) )
1182, 3, 4, 6, 5lmodvsdi 15666 . . . . . . . . . . 11  |-  ( ( W  e.  LMod  /\  (
( ( invr `  (Scalar `  W ) ) `  k )  e.  (
Base `  (Scalar `  W
) )  /\  (
k ( .s `  W ) Y )  e.  V  /\  (
l ( .s `  W ) Z )  e.  V ) )  ->  ( ( (
invr `  (Scalar `  W
) ) `  k
) ( .s `  W ) ( ( k ( .s `  W ) Y ) 
.+  ( l ( .s `  W ) Z ) ) )  =  ( ( ( ( invr `  (Scalar `  W ) ) `  k ) ( .s
`  W ) ( k ( .s `  W ) Y ) )  .+  ( ( ( invr `  (Scalar `  W ) ) `  k ) ( .s
`  W ) ( l ( .s `  W ) Z ) ) ) )
11915, 59, 76, 41, 118syl13anc 1184 . . . . . . . . . 10  |-  ( (
ph  /\  ( k  e.  ( Base `  (Scalar `  W ) )  /\  l  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( k ( .s `  W
) Y )  .+  ( l ( .s
`  W ) Z ) ) )  -> 
( ( ( invr `  (Scalar `  W )
) `  k )
( .s `  W
) ( ( k ( .s `  W
) Y )  .+  ( l ( .s
`  W ) Z ) ) )  =  ( ( ( (
invr `  (Scalar `  W
) ) `  k
) ( .s `  W ) ( k ( .s `  W
) Y ) ) 
.+  ( ( (
invr `  (Scalar `  W
) ) `  k
) ( .s `  W ) ( l ( .s `  W
) Z ) ) ) )
120 eqid 2296 . . . . . . . . . . . . . . 15  |-  ( .r
`  (Scalar `  W )
)  =  ( .r
`  (Scalar `  W )
)
121 eqid 2296 . . . . . . . . . . . . . . 15  |-  ( 1r
`  (Scalar `  W )
)  =  ( 1r
`  (Scalar `  W )
)
1225, 32, 120, 121, 57drnginvrl 15547 . . . . . . . . . . . . . 14  |-  ( ( (Scalar `  W )  e.  DivRing  /\  k  e.  ( Base `  (Scalar `  W
) )  /\  k  =/=  ( 0g `  (Scalar `  W ) ) )  ->  ( ( (
invr `  (Scalar `  W
) ) `  k
) ( .r `  (Scalar `  W ) ) k )  =  ( 1r `  (Scalar `  W ) ) )
12322, 23, 56, 122syl3anc 1182 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( k  e.  ( Base `  (Scalar `  W ) )  /\  l  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( k ( .s `  W
) Y )  .+  ( l ( .s
`  W ) Z ) ) )  -> 
( ( ( invr `  (Scalar `  W )
) `  k )
( .r `  (Scalar `  W ) ) k )  =  ( 1r
`  (Scalar `  W )
) )
124123oveq1d 5889 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( k  e.  ( Base `  (Scalar `  W ) )  /\  l  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( k ( .s `  W
) Y )  .+  ( l ( .s
`  W ) Z ) ) )  -> 
( ( ( (
invr `  (Scalar `  W
) ) `  k
) ( .r `  (Scalar `  W ) ) k ) ( .s
`  W ) Y )  =  ( ( 1r `  (Scalar `  W ) ) ( .s `  W ) Y ) )
1252, 4, 6, 5, 120lmodvsass 15670 . . . . . . . . . . . . 13  |-  ( ( W  e.  LMod  /\  (
( ( invr `  (Scalar `  W ) ) `  k )  e.  (
Base `  (Scalar `  W
) )  /\  k  e.  ( Base `  (Scalar `  W ) )  /\  Y  e.  V )
)  ->  ( (
( ( invr `  (Scalar `  W ) ) `  k ) ( .r
`  (Scalar `  W )
) k ) ( .s `  W ) Y )  =  ( ( ( invr `  (Scalar `  W ) ) `  k ) ( .s
`  W ) ( k ( .s `  W ) Y ) ) )
12615, 59, 23, 74, 125syl13anc 1184 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( k  e.  ( Base `  (Scalar `  W ) )  /\  l  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( k ( .s `  W
) Y )  .+  ( l ( .s
`  W ) Z ) ) )  -> 
( ( ( (
invr `  (Scalar `  W
) ) `  k
) ( .r `  (Scalar `  W ) ) k ) ( .s
`  W ) Y )  =  ( ( ( invr `  (Scalar `  W ) ) `  k ) ( .s
`  W ) ( k ( .s `  W ) Y ) ) )
1272, 4, 6, 121lmodvs1 15674 . . . . . . . . . . . . 13  |-  ( ( W  e.  LMod  /\  Y  e.  V )  ->  (
( 1r `  (Scalar `  W ) ) ( .s `  W ) Y )  =  Y )
12815, 74, 127syl2anc 642 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( k  e.  ( Base `  (Scalar `  W ) )  /\  l  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( k ( .s `  W
) Y )  .+  ( l ( .s
`  W ) Z ) ) )  -> 
( ( 1r `  (Scalar `  W ) ) ( .s `  W
) Y )  =  Y )
129124, 126, 1283eqtr3d 2336 . . . . . . . . . . 11  |-  ( (
ph  /\  ( k  e.  ( Base `  (Scalar `  W ) )  /\  l  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( k ( .s `  W
) Y )  .+  ( l ( .s
`  W ) Z ) ) )  -> 
( ( ( invr `  (Scalar `  W )
) `  k )
( .s `  W
) ( k ( .s `  W ) Y ) )  =  Y )
130129oveq1d 5889 . . . . . . . . . 10  |-  ( (
ph  /\  ( k  e.  ( Base `  (Scalar `  W ) )  /\  l  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( k ( .s `  W
) Y )  .+  ( l ( .s
`  W ) Z ) ) )  -> 
( ( ( (
invr `  (Scalar `  W
) ) `  k
) ( .s `  W ) ( k ( .s `  W
) Y ) ) 
.+  ( ( (
invr `  (Scalar `  W
) ) `  k
) ( .s `  W ) ( l ( .s `  W
) Z ) ) )  =  ( Y 
.+  ( ( (
invr `  (Scalar `  W
) ) `  k
) ( .s `  W ) ( l ( .s `  W
) Z ) ) ) )
131119, 130eqtrd 2328 . . . . . . . . 9  |-  ( (
ph  /\  ( k  e.  ( Base `  (Scalar `  W ) )  /\  l  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( k ( .s `  W
) Y )  .+  ( l ( .s
`  W ) Z ) ) )  -> 
( ( ( invr `  (Scalar `  W )
) `  k )
( .s `  W
) ( ( k ( .s `  W
) Y )  .+  ( l ( .s
`  W ) Z ) ) )  =  ( Y  .+  (
( ( invr `  (Scalar `  W ) ) `  k ) ( .s
`  W ) ( l ( .s `  W ) Z ) ) ) )
132131sneqd 3666 . . . . . . . 8  |-  ( (
ph  /\  ( k  e.  ( Base `  (Scalar `  W ) )  /\  l  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( k ( .s `  W
) Y )  .+  ( l ( .s
`  W ) Z ) ) )  ->  { ( ( (
invr `  (Scalar `  W
) ) `  k
) ( .s `  W ) ( ( k ( .s `  W ) Y ) 
.+  ( l ( .s `  W ) Z ) ) ) }  =  { ( Y  .+  ( ( ( invr `  (Scalar `  W ) ) `  k ) ( .s
`  W ) ( l ( .s `  W ) Z ) ) ) } )
133132fveq2d 5545 . . . . . . 7  |-  ( (
ph  /\  ( k  e.  ( Base `  (Scalar `  W ) )  /\  l  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( k ( .s `  W
) Y )  .+  ( l ( .s
`  W ) Z ) ) )  -> 
( N `  {
( ( ( invr `  (Scalar `  W )
) `  k )
( .s `  W
) ( ( k ( .s `  W
) Y )  .+  ( l ( .s
`  W ) Z ) ) ) } )  =  ( N `
 { ( Y 
.+  ( ( (
invr `  (Scalar `  W
) ) `  k
) ( .s `  W ) ( l ( .s `  W
) Z ) ) ) } ) )
134117, 133eqtr3d 2330 . . . . . 6  |-  ( (
ph  /\  ( k  e.  ( Base `  (Scalar `  W ) )  /\  l  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( k ( .s `  W
) Y )  .+  ( l ( .s
`  W ) Z ) ) )  -> 
( N `  {
( ( k ( .s `  W ) Y )  .+  (
l ( .s `  W ) Z ) ) } )  =  ( N `  {
( Y  .+  (
( ( invr `  (Scalar `  W ) ) `  k ) ( .s
`  W ) ( l ( .s `  W ) Z ) ) ) } ) )
135115, 134eleqtrd 2372 . . . . 5  |-  ( (
ph  /\  ( k  e.  ( Base `  (Scalar `  W ) )  /\  l  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( k ( .s `  W
) Y )  .+  ( l ( .s
`  W ) Z ) ) )  ->  X  e.  ( N `  { ( Y  .+  ( ( ( invr `  (Scalar `  W )
) `  k )
( .s `  W
) ( l ( .s `  W ) Z ) ) ) } ) )
136 oveq2 5882 . . . . . . . . 9  |-  ( z  =  ( ( (
invr `  (Scalar `  W
) ) `  k
) ( .s `  W ) ( l ( .s `  W
) Z ) )  ->  ( Y  .+  z )  =  ( Y  .+  ( ( ( invr `  (Scalar `  W ) ) `  k ) ( .s
`  W ) ( l ( .s `  W ) Z ) ) ) )
137136sneqd 3666 . . . . . . . 8  |-  ( z  =  ( ( (
invr `  (Scalar `  W
) ) `  k
) ( .s `  W ) ( l ( .s `  W
) Z ) )  ->  { ( Y 
.+  z ) }  =  { ( Y 
.+  ( ( (
invr `  (Scalar `  W
) ) `  k
) ( .s `  W ) ( l ( .s `  W
) Z ) ) ) } )
138137fveq2d 5545 . . . . . . 7  |-  ( z  =  ( ( (
invr `  (Scalar `  W
) ) `  k
) ( .s `  W ) ( l ( .s `  W
) Z ) )  ->  ( N `  { ( Y  .+  z ) } )  =  ( N `  { ( Y  .+  ( ( ( invr `  (Scalar `  W )
) `  k )
( .s `  W
) ( l ( .s `  W ) Z ) ) ) } ) )
139138eleq2d 2363 . . . . . 6  |-  ( z  =  ( ( (
invr `  (Scalar `  W
) ) `  k
) ( .s `  W ) ( l ( .s `  W
) Z ) )  ->  ( X  e.  ( N `  {
( Y  .+  z
) } )  <->  X  e.  ( N `  { ( Y  .+  ( ( ( invr `  (Scalar `  W ) ) `  k ) ( .s
`  W ) ( l ( .s `  W ) Z ) ) ) } ) ) )
140139rspcev 2897 . . . . 5  |-  ( ( ( ( ( invr `  (Scalar `  W )
) `  k )
( .s `  W
) ( l ( .s `  W ) Z ) )  e.  ( ( N `  { Z } )  \  {  .0.  } )  /\  X  e.  ( N `  { ( Y  .+  ( ( ( invr `  (Scalar `  W )
) `  k )
( .s `  W
) ( l ( .s `  W ) Z ) ) ) } ) )  ->  E. z  e.  (
( N `  { Z } )  \  {  .0.  } ) X  e.  ( N `  {
( Y  .+  z
) } ) )
141109, 135, 140syl2anc 642 . . . 4  |-  ( (
ph  /\  ( k  e.  ( Base `  (Scalar `  W ) )  /\  l  e.  ( Base `  (Scalar `  W )
) )  /\  X  =  ( ( k ( .s `  W
) Y )  .+  ( l ( .s
`  W ) Z ) ) )  ->  E. z  e.  (
( N `  { Z } )  \  {  .0.  } ) X  e.  ( N `  {
( Y  .+  z
) } ) )
1421413exp 1150 . . 3  |-  ( ph  ->  ( ( k  e.  ( Base `  (Scalar `  W ) )  /\  l  e.  ( Base `  (Scalar `  W )
) )  ->  ( X  =  ( (
k ( .s `  W ) Y ) 
.+  ( l ( .s `  W ) Z ) )  ->  E. z  e.  (
( N `  { Z } )  \  {  .0.  } ) X  e.  ( N `  {
( Y  .+  z
) } ) ) ) )
143142rexlimdvv 2686 . 2  |-  ( ph  ->  ( E. k  e.  ( Base `  (Scalar `  W ) ) E. l  e.  ( Base `  (Scalar `  W )
) X  =  ( ( k ( .s
`  W ) Y )  .+  ( l ( .s `  W
) Z ) )  ->  E. z  e.  ( ( N `  { Z } )  \  {  .0.  } ) X  e.  ( N `  {
( Y  .+  z
) } ) ) )
14414, 143mpd 14 1  |-  ( ph  ->  E. z  e.  ( ( N `  { Z } )  \  {  .0.  } ) X  e.  ( N `  {
( Y  .+  z
) } ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   E.wrex 2557    \ cdif 3162   {csn 3653   {cpr 3654   ` cfv 5271  (class class class)co 5874   Basecbs 13164   +g cplusg 13224   .rcmulr 13225  Scalarcsca 13227   .scvsca 13228   0gc0g 13416   1rcur 15355   invrcinvr 15469   DivRingcdr 15528   LModclmod 15643   LSubSpclss 15705   LSpanclspn 15744   LVecclvec 15871
This theorem is referenced by:  lsatfixedN  29821
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-tpos 6250  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-0g 13420  df-mnd 14383  df-submnd 14432  df-grp 14505  df-minusg 14506  df-sbg 14507  df-subg 14634  df-cntz 14809  df-lsm 14963  df-cmn 15107  df-abl 15108  df-mgp 15342  df-rng 15356  df-ur 15358  df-oppr 15421  df-dvdsr 15439  df-unit 15440  df-invr 15470  df-drng 15530  df-lmod 15645  df-lss 15706  df-lsp 15745  df-lvec 15872
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