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Theorem lspfixed 18094
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 2402 . . . 4  |-  (Scalar `  W )  =  (Scalar `  W )
5 eqid 2402 . . . 4  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
6 eqid 2402 . . . 4  |-  ( .s
`  W )  =  ( .s `  W
)
7 lspfixed.n . . . 4  |-  N  =  ( LSpan `  W )
8 lspfixed.w . . . . 5  |-  ( ph  ->  W  e.  LVec )
9 lveclmod 18072 . . . . 5  |-  ( W  e.  LVec  ->  W  e. 
LMod )
108, 9syl 17 . . . 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 18060 . . 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 1018 . . . . . . 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 2402 . . . . . . . . . 10  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
172, 16, 7lspsncl 17943 . . . . . . . . 9  |-  ( ( W  e.  LMod  /\  Z  e.  V )  ->  ( N `  { Z } )  e.  (
LSubSp `  W ) )
1810, 12, 17syl2anc 659 . . . . . . . 8  |-  ( ph  ->  ( N `  { Z } )  e.  (
LSubSp `  W ) )
19183ad2ant1 1018 . . . . . . 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 1018 . . . . . . . . 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 18071 . . . . . . . . 9  |-  ( W  e.  LVec  ->  (Scalar `  W )  e.  DivRing )
2220, 21syl 17 . . . . . . . 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 1023 . . . . . . . 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 1018 . . . . . . . . 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 1002 . . . . . . . . . . . . 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 459 . . . . . . . . . . . . . . . 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 6293 . . . . . . . . . . . . . . 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 1000 . . . . . . . . . . . . . . . . 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 17 . . . . . . . . . . . . . . . 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 17 . . . . . . . . . . . . . . . 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 2402 . . . . . . . . . . . . . . . . 17  |-  ( 0g
`  (Scalar `  W )
)  =  ( 0g
`  (Scalar `  W )
)
33 lspfixed.o . . . . . . . . . . . . . . . . 17  |-  .0.  =  ( 0g `  W )
342, 4, 6, 32, 33lmod0vs 17865 . . . . . . . . . . . . . . . 16  |-  ( ( W  e.  LMod  /\  Y  e.  V )  ->  (
( 0g `  (Scalar `  W ) ) ( .s `  W ) Y )  =  .0.  )
3530, 31, 34syl2anc 659 . . . . . . . . . . . . . . 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 2443 . . . . . . . . . . . . . 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 6293 . . . . . . . . . . . . 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 1024 . . . . . . . . . . . . . . . 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 1018 . . . . . . . . . . . . . . . 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 17849 . . . . . . . . . . . . . . . 16  |-  ( ( W  e.  LMod  /\  l  e.  ( Base `  (Scalar `  W ) )  /\  Z  e.  V )  ->  ( l ( .s
`  W ) Z )  e.  V )
4115, 38, 39, 40syl3anc 1230 . . . . . . . . . . . . . . 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 463 . . . . . . . . . . . . . 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 17862 . . . . . . . . . . . . . 14  |-  ( ( W  e.  LMod  /\  (
l ( .s `  W ) Z )  e.  V )  -> 
(  .0.  .+  (
l ( .s `  W ) Z ) )  =  ( l ( .s `  W
) Z ) )
4430, 42, 43syl2anc 659 . . . . . . . . . . . . 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 2447 . . . . . . . . . . . 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 17 . . . . . . . . . . . . 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 1051 . . . . . . . . . . . . 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 17959 . . . . . . . . . . . . . . 15  |-  ( ( W  e.  LMod  /\  Z  e.  V )  ->  Z  e.  ( N `  { Z } ) )
4910, 12, 48syl2anc 659 . . . . . . . . . . . . . 14  |-  ( ph  ->  Z  e.  ( N `
 { Z }
) )
5029, 49syl 17 . . . . . . . . . . . . 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 17921 . . . . . . . . . . . . 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 1231 . . . . . . . . . . . 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 2490 . . . . . . . . . . 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 432 . . . . . . . . . 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 2615 . . . . . . . . 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 2402 . . . . . . . . 9  |-  ( invr `  (Scalar `  W )
)  =  ( invr `  (Scalar `  W )
)
585, 32, 57drnginvrcl 17733 . . . . . . . 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 1230 . . . . . . 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 1018 . . . . . . . 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 1231 . . . . . . 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 17921 . . . . . . 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 1231 . . . . . 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 17734 . . . . . . . 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 1230 . . . . . . 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 1018 . . . . . . . . 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 1002 . . . . . . . . . . . . 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 6285 . . . . . . . . . . . . . . 15  |-  ( l  =  ( 0g `  (Scalar `  W ) )  ->  ( l ( .s `  W ) Z )  =  ( ( 0g `  (Scalar `  W ) ) ( .s `  W ) Z ) )
702, 4, 6, 32, 33lmod0vs 17865 . . . . . . . . . . . . . . . 16  |-  ( ( W  e.  LMod  /\  Z  e.  V )  ->  (
( 0g `  (Scalar `  W ) ) ( .s `  W ) Z )  =  .0.  )
7115, 39, 70syl2anc 659 . . . . . . . . . . . . . . 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 2465 . . . . . . . . . . . . . 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 6294 . . . . . . . . . . . . 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 1018 . . . . . . . . . . . . . . . 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 17849 . . . . . . . . . . . . . . . 16  |-  ( ( W  e.  LMod  /\  k  e.  ( Base `  (Scalar `  W ) )  /\  Y  e.  V )  ->  ( k ( .s
`  W ) Y )  e.  V )
7615, 23, 74, 75syl3anc 1230 . . . . . . . . . . . . . . 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 17863 . . . . . . . . . . . . . . 15  |-  ( ( W  e.  LMod  /\  (
k ( .s `  W ) Y )  e.  V )  -> 
( ( k ( .s `  W ) Y )  .+  .0.  )  =  ( k
( .s `  W
) Y ) )
7815, 76, 77syl2anc 659 . . . . . . . . . . . . . 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 463 . . . . . . . . . . . . 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 2447 . . . . . . . . . . . 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 17943 . . . . . . . . . . . . . . . 16  |-  ( ( W  e.  LMod  /\  Y  e.  V )  ->  ( N `  { Y } )  e.  (
LSubSp `  W ) )
8210, 11, 81syl2anc 659 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( N `  { Y } )  e.  (
LSubSp `  W ) )
83823ad2ant1 1018 . . . . . . . . . . . . . 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 17959 . . . . . . . . . . . . . . . 16  |-  ( ( W  e.  LMod  /\  Y  e.  V )  ->  Y  e.  ( N `  { Y } ) )
8510, 11, 84syl2anc 659 . . . . . . . . . . . . . . 15  |-  ( ph  ->  Y  e.  ( N `
 { Y }
) )
86853ad2ant1 1018 . . . . . . . . . . . . . 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 17921 . . . . . . . . . . . . . 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 1231 . . . . . . . . . . . . 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 463 . . . . . . . . . . . 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 2490 . . . . . . . . . . 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 432 . . . . . . . . . 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 2615 . . . . . . . . 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 1000 . . . . . . . . . . . . 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 17 . . . . . . . . . . . 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 4052 . . . . . . . . . . . . . 14  |-  ( Z  =  .0.  ->  { Y ,  Z }  =  { Y ,  .0.  } )
9796fveq2d 5853 . . . . . . . . . . . . 13  |-  ( Z  =  .0.  ->  ( N `  { Y ,  Z } )  =  ( N `  { Y ,  .0.  } ) )
982, 33, 7, 15, 74lsppr0 18058 . . . . . . . . . . . . 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 2465 . . . . . . . . . . . 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 2492 . . . . . . . . . . 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 432 . . . . . . . . . 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 2615 . . . . . . . . 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 18075 . . . . . . . 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 923 . . . . . . 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 18075 . . . . . . 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 923 . . . . . 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 4097 . . . . . 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 662 . . . . 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 999 . . . . . . 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 17846 . . . . . . . . 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 1230 . . . . . . . 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 17959 . . . . . . . 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 659 . . . . . . 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 2490 . . . . . 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 18080 . . . . . . . 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 1235 . . . . . . 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 17855 . . . . . . . . . . 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 1232 . . . . . . . . . 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 2402 . . . . . . . . . . . . . . 15  |-  ( .r
`  (Scalar `  W )
)  =  ( .r
`  (Scalar `  W )
)
121 eqid 2402 . . . . . . . . . . . . . . 15  |-  ( 1r
`  (Scalar `  W )
)  =  ( 1r
`  (Scalar `  W )
)
1225, 32, 120, 121, 57drnginvrl 17735 . . . . . . . . . . . . . 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 1230 . . . . . . . . . . . . 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 6293 . . . . . . . . . . . 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 17857 . . . . . . . . . . . . 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 1232 . . . . . . . . . . . 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 17860 . . . . . . . . . . . . 13  |-  ( ( W  e.  LMod  /\  Y  e.  V )  ->  (
( 1r `  (Scalar `  W ) ) ( .s `  W ) Y )  =  Y )
12815, 74, 127syl2anc 659 . . . . . . . . . . . 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 2451 . . . . . . . . . . 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 6293 . . . . . . . . . 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 2443 . . . . . . . . 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 3984 . . . . . . . 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 5853 . . . . . . 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 2445 . . . . . 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 2492 . . . . 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 6286 . . . . . . . . 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 3984 . . . . . . . 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 5853 . . . . . . 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 2472 . . . . . 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 3160 . . . . 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 659 . . . 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 1196 . . 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 2902 . 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 367    /\ w3a 974    = wceq 1405    e. wcel 1842    =/= wne 2598   E.wrex 2755    \ cdif 3411   {csn 3972   {cpr 3974   ` cfv 5569  (class class class)co 6278   Basecbs 14841   +g cplusg 14909   .rcmulr 14910  Scalarcsca 14912   .scvsca 14913   0gc0g 15054   1rcur 17473   invrcinvr 17640   DivRingcdr 17716   LModclmod 17832   LSubSpclss 17898   LSpanclspn 17937   LVecclvec 18068
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-1st 6784  df-2nd 6785  df-tpos 6958  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-er 7348  df-en 7555  df-dom 7556  df-sdom 7557  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-nn 10577  df-2 10635  df-3 10636  df-ndx 14844  df-slot 14845  df-base 14846  df-sets 14847  df-ress 14848  df-plusg 14922  df-mulr 14923  df-0g 15056  df-mgm 16196  df-sgrp 16235  df-mnd 16245  df-submnd 16291  df-grp 16381  df-minusg 16382  df-sbg 16383  df-subg 16522  df-cntz 16679  df-lsm 16980  df-cmn 17124  df-abl 17125  df-mgp 17462  df-ur 17474  df-ring 17520  df-oppr 17592  df-dvdsr 17610  df-unit 17611  df-invr 17641  df-drng 17718  df-lmod 17834  df-lss 17899  df-lsp 17938  df-lvec 18069
This theorem is referenced by:  lsatfixedN  32027
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