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Theorem lspfixed 16200
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 2436 . . . 4  |-  (Scalar `  W )  =  (Scalar `  W )
5 eqid 2436 . . . 4  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
6 eqid 2436 . . . 4  |-  ( .s
`  W )  =  ( .s `  W
)
7 lspfixed.n . . . 4  |-  N  =  ( LSpan `  W )
8 lspfixed.w . . . . 5  |-  ( ph  ->  W  e.  LVec )
9 lveclmod 16178 . . . . 5  |-  ( W  e.  LVec  ->  W  e. 
LMod )
108, 9syl 16 . . . 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 16166 . . 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 202 . 2  |-  ( ph  ->  E. k  e.  (
Base `  (Scalar `  W
) ) E. l  e.  ( Base `  (Scalar `  W ) ) X  =  ( ( k ( .s `  W
) Y )  .+  ( l ( .s
`  W ) Z ) ) )
15103ad2ant1 978 . . . . . . 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 2436 . . . . . . . . . 10  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
172, 16, 7lspsncl 16053 . . . . . . . . 9  |-  ( ( W  e.  LMod  /\  Z  e.  V )  ->  ( N `  { Z } )  e.  (
LSubSp `  W ) )
1810, 12, 17syl2anc 643 . . . . . . . 8  |-  ( ph  ->  ( N `  { Z } )  e.  (
LSubSp `  W ) )
19183ad2ant1 978 . . . . . . 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 978 . . . . . . . . 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 16177 . . . . . . . . 9  |-  ( W  e.  LVec  ->  (Scalar `  W )  e.  DivRing )
2220, 21syl 16 . . . . . . . 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 983 . . . . . . . 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 978 . . . . . . . . 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 962 . . . . . . . . . . . . 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 448 . . . . . . . . . . . . . . . 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 6096 . . . . . . . . . . . . . . 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 960 . . . . . . . . . . . . . . . . 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 16 . . . . . . . . . . . . . . . 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 16 . . . . . . . . . . . . . . . 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 2436 . . . . . . . . . . . . . . . . 17  |-  ( 0g
`  (Scalar `  W )
)  =  ( 0g
`  (Scalar `  W )
)
33 lspfixed.o . . . . . . . . . . . . . . . . 17  |-  .0.  =  ( 0g `  W )
342, 4, 6, 32, 33lmod0vs 15983 . . . . . . . . . . . . . . . 16  |-  ( ( W  e.  LMod  /\  Y  e.  V )  ->  (
( 0g `  (Scalar `  W ) ) ( .s `  W ) Y )  =  .0.  )
3530, 31, 34syl2anc 643 . . . . . . . . . . . . . . 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 2468 . . . . . . . . . . . . . 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 6096 . . . . . . . . . . . . 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 984 . . . . . . . . . . . . . . . 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 978 . . . . . . . . . . . . . . . 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 15967 . . . . . . . . . . . . . . . 16  |-  ( ( W  e.  LMod  /\  l  e.  ( Base `  (Scalar `  W ) )  /\  Z  e.  V )  ->  ( l ( .s
`  W ) Z )  e.  V )
4115, 38, 39, 40syl3anc 1184 . . . . . . . . . . . . . . 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 452 . . . . . . . . . . . . . 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 15980 . . . . . . . . . . . . . 14  |-  ( ( W  e.  LMod  /\  (
l ( .s `  W ) Z )  e.  V )  -> 
(  .0.  .+  (
l ( .s `  W ) Z ) )  =  ( l ( .s `  W
) Z ) )
4430, 42, 43syl2anc 643 . . . . . . . . . . . . 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 2472 . . . . . . . . . . . 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 16 . . . . . . . . . . . . 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 1011 . . . . . . . . . . . . 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 16069 . . . . . . . . . . . . . . 15  |-  ( ( W  e.  LMod  /\  Z  e.  V )  ->  Z  e.  ( N `  { Z } ) )
4910, 12, 48syl2anc 643 . . . . . . . . . . . . . 14  |-  ( ph  ->  Z  e.  ( N `
 { Z }
) )
5029, 49syl 16 . . . . . . . . . . . . 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 16031 . . . . . . . . . . . . 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 1185 . . . . . . . . . . . 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 2510 . . . . . . . . . . 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 424 . . . . . . . . . 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 2638 . . . . . . . . 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 15 . . . . . . . 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 2436 . . . . . . . . 9  |-  ( invr `  (Scalar `  W )
)  =  ( invr `  (Scalar `  W )
)
585, 32, 57drnginvrcl 15852 . . . . . . . 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 1184 . . . . . . 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 978 . . . . . . . 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 1185 . . . . . . 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 16031 . . . . . . 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 1185 . . . . . 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 15853 . . . . . . . 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 1184 . . . . . . 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 978 . . . . . . . . 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 962 . . . . . . . . . . . . 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 6088 . . . . . . . . . . . . . . 15  |-  ( l  =  ( 0g `  (Scalar `  W ) )  ->  ( l ( .s `  W ) Z )  =  ( ( 0g `  (Scalar `  W ) ) ( .s `  W ) Z ) )
702, 4, 6, 32, 33lmod0vs 15983 . . . . . . . . . . . . . . . 16  |-  ( ( W  e.  LMod  /\  Z  e.  V )  ->  (
( 0g `  (Scalar `  W ) ) ( .s `  W ) Z )  =  .0.  )
7115, 39, 70syl2anc 643 . . . . . . . . . . . . . . 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 2490 . . . . . . . . . . . . . 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 6097 . . . . . . . . . . . . 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 978 . . . . . . . . . . . . . . . 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 15967 . . . . . . . . . . . . . . . 16  |-  ( ( W  e.  LMod  /\  k  e.  ( Base `  (Scalar `  W ) )  /\  Y  e.  V )  ->  ( k ( .s
`  W ) Y )  e.  V )
7615, 23, 74, 75syl3anc 1184 . . . . . . . . . . . . . . 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 15981 . . . . . . . . . . . . . . 15  |-  ( ( W  e.  LMod  /\  (
k ( .s `  W ) Y )  e.  V )  -> 
( ( k ( .s `  W ) Y )  .+  .0.  )  =  ( k
( .s `  W
) Y ) )
7815, 76, 77syl2anc 643 . . . . . . . . . . . . . 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 452 . . . . . . . . . . . . 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 2472 . . . . . . . . . . . 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 16053 . . . . . . . . . . . . . . . 16  |-  ( ( W  e.  LMod  /\  Y  e.  V )  ->  ( N `  { Y } )  e.  (
LSubSp `  W ) )
8210, 11, 81syl2anc 643 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( N `  { Y } )  e.  (
LSubSp `  W ) )
83823ad2ant1 978 . . . . . . . . . . . . . 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 16069 . . . . . . . . . . . . . . . 16  |-  ( ( W  e.  LMod  /\  Y  e.  V )  ->  Y  e.  ( N `  { Y } ) )
8510, 11, 84syl2anc 643 . . . . . . . . . . . . . . 15  |-  ( ph  ->  Y  e.  ( N `
 { Y }
) )
86853ad2ant1 978 . . . . . . . . . . . . . 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 16031 . . . . . . . . . . . . . 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 1185 . . . . . . . . . . . . 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 452 . . . . . . . . . . . 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 2510 . . . . . . . . . . 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 424 . . . . . . . . . 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 2638 . . . . . . . . 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 15 . . . . . . . 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 960 . . . . . . . . . . . . 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 16 . . . . . . . . . . . 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 3884 . . . . . . . . . . . . . 14  |-  ( Z  =  .0.  ->  { Y ,  Z }  =  { Y ,  .0.  } )
9796fveq2d 5732 . . . . . . . . . . . . 13  |-  ( Z  =  .0.  ->  ( N `  { Y ,  Z } )  =  ( N `  { Y ,  .0.  } ) )
982, 33, 7, 15, 74lsppr0 16164 . . . . . . . . . . . . 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 2490 . . . . . . . . . . . 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 2512 . . . . . . . . . . 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 424 . . . . . . . . . 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 2638 . . . . . . . . 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 15 . . . . . . . 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 16181 . . . . . . . 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 889 . . . . . . 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 16181 . . . . . . 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 889 . . . . . 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 3927 . . . . . 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 646 . . . . 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 959 . . . . . . 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 15964 . . . . . . . . 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 1184 . . . . . . . 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 16069 . . . . . . . 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 643 . . . . . . 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 2510 . . . . . 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 16186 . . . . . . . 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 1189 . . . . . . 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 15973 . . . . . . . . . . 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 1186 . . . . . . . . . 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 2436 . . . . . . . . . . . . . . 15  |-  ( .r
`  (Scalar `  W )
)  =  ( .r
`  (Scalar `  W )
)
121 eqid 2436 . . . . . . . . . . . . . . 15  |-  ( 1r
`  (Scalar `  W )
)  =  ( 1r
`  (Scalar `  W )
)
1225, 32, 120, 121, 57drnginvrl 15854 . . . . . . . . . . . . . 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 1184 . . . . . . . . . . . . 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 6096 . . . . . . . . . . . 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 15975 . . . . . . . . . . . . 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 1186 . . . . . . . . . . . 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 15978 . . . . . . . . . . . . 13  |-  ( ( W  e.  LMod  /\  Y  e.  V )  ->  (
( 1r `  (Scalar `  W ) ) ( .s `  W ) Y )  =  Y )
12815, 74, 127syl2anc 643 . . . . . . . . . . . 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 2476 . . . . . . . . . . 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 6096 . . . . . . . . . 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 2468 . . . . . . . . 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 3827 . . . . . . . 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 5732 . . . . . . 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 2470 . . . . . 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 2512 . . . . 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 6089 . . . . . . . . 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 3827 . . . . . . . 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 5732 . . . . . . 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 2503 . . . . . 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 3052 . . . . 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 643 . . . 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 1152 . . 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 2836 . 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 15 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 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2599   E.wrex 2706    \ cdif 3317   {csn 3814   {cpr 3815   ` cfv 5454  (class class class)co 6081   Basecbs 13469   +g cplusg 13529   .rcmulr 13530  Scalarcsca 13532   .scvsca 13533   0gc0g 13723   1rcur 15662   invrcinvr 15776   DivRingcdr 15835   LModclmod 15950   LSubSpclss 16008   LSpanclspn 16047   LVecclvec 16174
This theorem is referenced by:  lsatfixedN  29807
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-tpos 6479  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-3 10059  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-plusg 13542  df-mulr 13543  df-0g 13727  df-mnd 14690  df-submnd 14739  df-grp 14812  df-minusg 14813  df-sbg 14814  df-subg 14941  df-cntz 15116  df-lsm 15270  df-cmn 15414  df-abl 15415  df-mgp 15649  df-rng 15663  df-ur 15665  df-oppr 15728  df-dvdsr 15746  df-unit 15747  df-invr 15777  df-drng 15837  df-lmod 15952  df-lss 16009  df-lsp 16048  df-lvec 16175
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