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Theorem lspfixed 17209
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 2443 . . . 4  |-  (Scalar `  W )  =  (Scalar `  W )
5 eqid 2443 . . . 4  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
6 eqid 2443 . . . 4  |-  ( .s
`  W )  =  ( .s `  W
)
7 lspfixed.n . . . 4  |-  N  =  ( LSpan `  W )
8 lspfixed.w . . . . 5  |-  ( ph  ->  W  e.  LVec )
9 lveclmod 17187 . . . . 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 17175 . . 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 210 . 2  |-  ( ph  ->  E. k  e.  (
Base `  (Scalar `  W
) ) E. l  e.  ( Base `  (Scalar `  W ) ) X  =  ( ( k ( .s `  W
) Y )  .+  ( l ( .s
`  W ) Z ) ) )
15103ad2ant1 1009 . . . . . . 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 2443 . . . . . . . . . 10  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
172, 16, 7lspsncl 17058 . . . . . . . . 9  |-  ( ( W  e.  LMod  /\  Z  e.  V )  ->  ( N `  { Z } )  e.  (
LSubSp `  W ) )
1810, 12, 17syl2anc 661 . . . . . . . 8  |-  ( ph  ->  ( N `  { Z } )  e.  (
LSubSp `  W ) )
19183ad2ant1 1009 . . . . . . 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 1009 . . . . . . . . 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 17186 . . . . . . . . 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 1014 . . . . . . . 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 1009 . . . . . . . . 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 993 . . . . . . . . . . . . 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 461 . . . . . . . . . . . . . . . 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 6106 . . . . . . . . . . . . . . 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 991 . . . . . . . . . . . . . . . . 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 2443 . . . . . . . . . . . . . . . . 17  |-  ( 0g
`  (Scalar `  W )
)  =  ( 0g
`  (Scalar `  W )
)
33 lspfixed.o . . . . . . . . . . . . . . . . 17  |-  .0.  =  ( 0g `  W )
342, 4, 6, 32, 33lmod0vs 16981 . . . . . . . . . . . . . . . 16  |-  ( ( W  e.  LMod  /\  Y  e.  V )  ->  (
( 0g `  (Scalar `  W ) ) ( .s `  W ) Y )  =  .0.  )
3530, 31, 34syl2anc 661 . . . . . . . . . . . . . . 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 2475 . . . . . . . . . . . . . 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 6106 . . . . . . . . . . . . 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 1015 . . . . . . . . . . . . . . . 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 1009 . . . . . . . . . . . . . . . 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 16965 . . . . . . . . . . . . . . . 16  |-  ( ( W  e.  LMod  /\  l  e.  ( Base `  (Scalar `  W ) )  /\  Z  e.  V )  ->  ( l ( .s
`  W ) Z )  e.  V )
4115, 38, 39, 40syl3anc 1218 . . . . . . . . . . . . . . 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 465 . . . . . . . . . . . . . 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 16978 . . . . . . . . . . . . . 14  |-  ( ( W  e.  LMod  /\  (
l ( .s `  W ) Z )  e.  V )  -> 
(  .0.  .+  (
l ( .s `  W ) Z ) )  =  ( l ( .s `  W
) Z ) )
4430, 42, 43syl2anc 661 . . . . . . . . . . . . 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 2479 . . . . . . . . . . . 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 1042 . . . . . . . . . . . . 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 17074 . . . . . . . . . . . . . . 15  |-  ( ( W  e.  LMod  /\  Z  e.  V )  ->  Z  e.  ( N `  { Z } ) )
4910, 12, 48syl2anc 661 . . . . . . . . . . . . . 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 17036 . . . . . . . . . . . . 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 1219 . . . . . . . . . . . 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 2517 . . . . . . . . . . 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 434 . . . . . . . . . 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 2645 . . . . . . . . 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 2443 . . . . . . . . 9  |-  ( invr `  (Scalar `  W )
)  =  ( invr `  (Scalar `  W )
)
585, 32, 57drnginvrcl 16849 . . . . . . . 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 1218 . . . . . . 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 1009 . . . . . . . 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 1219 . . . . . . 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 17036 . . . . . . 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 1219 . . . . . 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 16850 . . . . . . . 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 1218 . . . . . . 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 1009 . . . . . . . . 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 993 . . . . . . . . . . . . 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 6098 . . . . . . . . . . . . . . 15  |-  ( l  =  ( 0g `  (Scalar `  W ) )  ->  ( l ( .s `  W ) Z )  =  ( ( 0g `  (Scalar `  W ) ) ( .s `  W ) Z ) )
702, 4, 6, 32, 33lmod0vs 16981 . . . . . . . . . . . . . . . 16  |-  ( ( W  e.  LMod  /\  Z  e.  V )  ->  (
( 0g `  (Scalar `  W ) ) ( .s `  W ) Z )  =  .0.  )
7115, 39, 70syl2anc 661 . . . . . . . . . . . . . . 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 2497 . . . . . . . . . . . . . 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 6107 . . . . . . . . . . . . 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 1009 . . . . . . . . . . . . . . . 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 16965 . . . . . . . . . . . . . . . 16  |-  ( ( W  e.  LMod  /\  k  e.  ( Base `  (Scalar `  W ) )  /\  Y  e.  V )  ->  ( k ( .s
`  W ) Y )  e.  V )
7615, 23, 74, 75syl3anc 1218 . . . . . . . . . . . . . . 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 16979 . . . . . . . . . . . . . . 15  |-  ( ( W  e.  LMod  /\  (
k ( .s `  W ) Y )  e.  V )  -> 
( ( k ( .s `  W ) Y )  .+  .0.  )  =  ( k
( .s `  W
) Y ) )
7815, 76, 77syl2anc 661 . . . . . . . . . . . . . 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 465 . . . . . . . . . . . . 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 2479 . . . . . . . . . . . 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 17058 . . . . . . . . . . . . . . . 16  |-  ( ( W  e.  LMod  /\  Y  e.  V )  ->  ( N `  { Y } )  e.  (
LSubSp `  W ) )
8210, 11, 81syl2anc 661 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( N `  { Y } )  e.  (
LSubSp `  W ) )
83823ad2ant1 1009 . . . . . . . . . . . . . 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 17074 . . . . . . . . . . . . . . . 16  |-  ( ( W  e.  LMod  /\  Y  e.  V )  ->  Y  e.  ( N `  { Y } ) )
8510, 11, 84syl2anc 661 . . . . . . . . . . . . . . 15  |-  ( ph  ->  Y  e.  ( N `
 { Y }
) )
86853ad2ant1 1009 . . . . . . . . . . . . . 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 17036 . . . . . . . . . . . . . 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 1219 . . . . . . . . . . . . 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 465 . . . . . . . . . . . 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 2517 . . . . . . . . . . 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 434 . . . . . . . . . 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 2645 . . . . . . . . 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 991 . . . . . . . . . . . . 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 3955 . . . . . . . . . . . . . 14  |-  ( Z  =  .0.  ->  { Y ,  Z }  =  { Y ,  .0.  } )
9796fveq2d 5695 . . . . . . . . . . . . 13  |-  ( Z  =  .0.  ->  ( N `  { Y ,  Z } )  =  ( N `  { Y ,  .0.  } ) )
982, 33, 7, 15, 74lsppr0 17173 . . . . . . . . . . . . 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 2497 . . . . . . . . . . . 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 2519 . . . . . . . . . . 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 434 . . . . . . . . . 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 2645 . . . . . . . . 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 17190 . . . . . . . 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 913 . . . . . . 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 17190 . . . . . . 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 913 . . . . . 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 4000 . . . . . 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 664 . . . . 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 990 . . . . . . 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 16962 . . . . . . . . 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 1218 . . . . . . . 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 17074 . . . . . . . 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 661 . . . . . . 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 2517 . . . . . 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 17195 . . . . . . . 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 1223 . . . . . . 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 16971 . . . . . . . . . . 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 1220 . . . . . . . . . 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 2443 . . . . . . . . . . . . . . 15  |-  ( .r
`  (Scalar `  W )
)  =  ( .r
`  (Scalar `  W )
)
121 eqid 2443 . . . . . . . . . . . . . . 15  |-  ( 1r
`  (Scalar `  W )
)  =  ( 1r
`  (Scalar `  W )
)
1225, 32, 120, 121, 57drnginvrl 16851 . . . . . . . . . . . . . 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 1218 . . . . . . . . . . . . 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 6106 . . . . . . . . . . . 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 16973 . . . . . . . . . . . . 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 1220 . . . . . . . . . . . 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 16976 . . . . . . . . . . . . 13  |-  ( ( W  e.  LMod  /\  Y  e.  V )  ->  (
( 1r `  (Scalar `  W ) ) ( .s `  W ) Y )  =  Y )
12815, 74, 127syl2anc 661 . . . . . . . . . . . 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 2483 . . . . . . . . . . 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 6106 . . . . . . . . . 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 2475 . . . . . . . . 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 3889 . . . . . . . 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 5695 . . . . . . 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 2477 . . . . . 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 2519 . . . . 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 6099 . . . . . . . . 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 3889 . . . . . . . 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 5695 . . . . . . 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 2510 . . . . . 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 3073 . . . . 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 661 . . . 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 1186 . . 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 2847 . 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2606   E.wrex 2716    \ cdif 3325   {csn 3877   {cpr 3879   ` cfv 5418  (class class class)co 6091   Basecbs 14174   +g cplusg 14238   .rcmulr 14239  Scalarcsca 14241   .scvsca 14242   0gc0g 14378   1rcur 16603   invrcinvr 16763   DivRingcdr 16832   LModclmod 16948   LSubSpclss 17013   LSpanclspn 17052   LVecclvec 17183
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-tpos 6745  df-recs 6832  df-rdg 6866  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-nn 10323  df-2 10380  df-3 10381  df-ndx 14177  df-slot 14178  df-base 14179  df-sets 14180  df-ress 14181  df-plusg 14251  df-mulr 14252  df-0g 14380  df-mnd 15415  df-submnd 15465  df-grp 15545  df-minusg 15546  df-sbg 15547  df-subg 15678  df-cntz 15835  df-lsm 16135  df-cmn 16279  df-abl 16280  df-mgp 16592  df-ur 16604  df-rng 16647  df-oppr 16715  df-dvdsr 16733  df-unit 16734  df-invr 16764  df-drng 16834  df-lmod 16950  df-lss 17014  df-lsp 17053  df-lvec 17184
This theorem is referenced by:  lsatfixedN  32654
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