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Theorem lspfixed 17554
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 2467 . . . 4  |-  (Scalar `  W )  =  (Scalar `  W )
5 eqid 2467 . . . 4  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
6 eqid 2467 . . . 4  |-  ( .s
`  W )  =  ( .s `  W
)
7 lspfixed.n . . . 4  |-  N  =  ( LSpan `  W )
8 lspfixed.w . . . . 5  |-  ( ph  ->  W  e.  LVec )
9 lveclmod 17532 . . . . 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 17520 . . 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 1017 . . . . . . 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 2467 . . . . . . . . . 10  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
172, 16, 7lspsncl 17403 . . . . . . . . 9  |-  ( ( W  e.  LMod  /\  Z  e.  V )  ->  ( N `  { Z } )  e.  (
LSubSp `  W ) )
1810, 12, 17syl2anc 661 . . . . . . . 8  |-  ( ph  ->  ( N `  { Z } )  e.  (
LSubSp `  W ) )
19183ad2ant1 1017 . . . . . . 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 1017 . . . . . . . . 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 17531 . . . . . . . . 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 1022 . . . . . . . 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 1017 . . . . . . . . 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 1001 . . . . . . . . . . . . 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 6297 . . . . . . . . . . . . . . 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 999 . . . . . . . . . . . . . . . . 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 2467 . . . . . . . . . . . . . . . . 17  |-  ( 0g
`  (Scalar `  W )
)  =  ( 0g
`  (Scalar `  W )
)
33 lspfixed.o . . . . . . . . . . . . . . . . 17  |-  .0.  =  ( 0g `  W )
342, 4, 6, 32, 33lmod0vs 17325 . . . . . . . . . . . . . . . 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 2508 . . . . . . . . . . . . . 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 6297 . . . . . . . . . . . . 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 1023 . . . . . . . . . . . . . . . 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 1017 . . . . . . . . . . . . . . . 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 17309 . . . . . . . . . . . . . . . 16  |-  ( ( W  e.  LMod  /\  l  e.  ( Base `  (Scalar `  W ) )  /\  Z  e.  V )  ->  ( l ( .s
`  W ) Z )  e.  V )
4115, 38, 39, 40syl3anc 1228 . . . . . . . . . . . . . . 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 17322 . . . . . . . . . . . . . 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 2512 . . . . . . . . . . . 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 1050 . . . . . . . . . . . . 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 17419 . . . . . . . . . . . . . . 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 17381 . . . . . . . . . . . . 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 1229 . . . . . . . . . . . 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 2555 . . . . . . . . . . 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 2679 . . . . . . . . 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 2467 . . . . . . . . 9  |-  ( invr `  (Scalar `  W )
)  =  ( invr `  (Scalar `  W )
)
585, 32, 57drnginvrcl 17193 . . . . . . . 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 1228 . . . . . . 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 1017 . . . . . . . 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 1229 . . . . . . 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 17381 . . . . . . 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 1229 . . . . . 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 17194 . . . . . . . 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 1228 . . . . . . 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 1017 . . . . . . . . 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 1001 . . . . . . . . . . . . 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 6289 . . . . . . . . . . . . . . 15  |-  ( l  =  ( 0g `  (Scalar `  W ) )  ->  ( l ( .s `  W ) Z )  =  ( ( 0g `  (Scalar `  W ) ) ( .s `  W ) Z ) )
702, 4, 6, 32, 33lmod0vs 17325 . . . . . . . . . . . . . . . 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 2530 . . . . . . . . . . . . . 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 6298 . . . . . . . . . . . . 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 1017 . . . . . . . . . . . . . . . 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 17309 . . . . . . . . . . . . . . . 16  |-  ( ( W  e.  LMod  /\  k  e.  ( Base `  (Scalar `  W ) )  /\  Y  e.  V )  ->  ( k ( .s
`  W ) Y )  e.  V )
7615, 23, 74, 75syl3anc 1228 . . . . . . . . . . . . . . 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 17323 . . . . . . . . . . . . . . 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 2512 . . . . . . . . . . . 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 17403 . . . . . . . . . . . . . . . 16  |-  ( ( W  e.  LMod  /\  Y  e.  V )  ->  ( N `  { Y } )  e.  (
LSubSp `  W ) )
8210, 11, 81syl2anc 661 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( N `  { Y } )  e.  (
LSubSp `  W ) )
83823ad2ant1 1017 . . . . . . . . . . . . . 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 17419 . . . . . . . . . . . . . . . 16  |-  ( ( W  e.  LMod  /\  Y  e.  V )  ->  Y  e.  ( N `  { Y } ) )
8510, 11, 84syl2anc 661 . . . . . . . . . . . . . . 15  |-  ( ph  ->  Y  e.  ( N `
 { Y }
) )
86853ad2ant1 1017 . . . . . . . . . . . . . 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 17381 . . . . . . . . . . . . . 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 1229 . . . . . . . . . . . . 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 2555 . . . . . . . . . . 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 2679 . . . . . . . . 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 999 . . . . . . . . . . . . 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 4107 . . . . . . . . . . . . . 14  |-  ( Z  =  .0.  ->  { Y ,  Z }  =  { Y ,  .0.  } )
9796fveq2d 5868 . . . . . . . . . . . . 13  |-  ( Z  =  .0.  ->  ( N `  { Y ,  Z } )  =  ( N `  { Y ,  .0.  } ) )
982, 33, 7, 15, 74lsppr0 17518 . . . . . . . . . . . . 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 2530 . . . . . . . . . . . 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 2557 . . . . . . . . . . 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 2679 . . . . . . . . 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 17535 . . . . . . . 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 920 . . . . . . 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 17535 . . . . . . 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 920 . . . . . 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 4152 . . . . . 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 998 . . . . . . 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 17306 . . . . . . . . 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 1228 . . . . . . . 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 17419 . . . . . . . 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 2555 . . . . . 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 17540 . . . . . . . 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 1233 . . . . . . 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 17315 . . . . . . . . . . 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 1230 . . . . . . . . . 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 2467 . . . . . . . . . . . . . . 15  |-  ( .r
`  (Scalar `  W )
)  =  ( .r
`  (Scalar `  W )
)
121 eqid 2467 . . . . . . . . . . . . . . 15  |-  ( 1r
`  (Scalar `  W )
)  =  ( 1r
`  (Scalar `  W )
)
1225, 32, 120, 121, 57drnginvrl 17195 . . . . . . . . . . . . . 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 1228 . . . . . . . . . . . . 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 6297 . . . . . . . . . . . 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 17317 . . . . . . . . . . . . 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 1230 . . . . . . . . . . . 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 17320 . . . . . . . . . . . . 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 2516 . . . . . . . . . . 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 6297 . . . . . . . . . 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 2508 . . . . . . . . 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 4039 . . . . . . . 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 5868 . . . . . . 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 2510 . . . . . 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 2557 . . . . 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 6290 . . . . . . . . 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 4039 . . . . . . . 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 5868 . . . . . . 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 2537 . . . . . 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 3214 . . . . 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 1195 . . 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 2961 . 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 973    = wceq 1379    e. wcel 1767    =/= wne 2662   E.wrex 2815    \ cdif 3473   {csn 4027   {cpr 4029   ` cfv 5586  (class class class)co 6282   Basecbs 14483   +g cplusg 14548   .rcmulr 14549  Scalarcsca 14551   .scvsca 14552   0gc0g 14688   1rcur 16940   invrcinvr 17101   DivRingcdr 17176   LModclmod 17292   LSubSpclss 17358   LSpanclspn 17397   LVecclvec 17528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-tpos 6952  df-recs 7039  df-rdg 7073  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-3 10591  df-ndx 14486  df-slot 14487  df-base 14488  df-sets 14489  df-ress 14490  df-plusg 14561  df-mulr 14562  df-0g 14690  df-mnd 15725  df-submnd 15775  df-grp 15855  df-minusg 15856  df-sbg 15857  df-subg 15990  df-cntz 16147  df-lsm 16449  df-cmn 16593  df-abl 16594  df-mgp 16929  df-ur 16941  df-rng 16985  df-oppr 17053  df-dvdsr 17071  df-unit 17072  df-invr 17102  df-drng 17178  df-lmod 17294  df-lss 17359  df-lsp 17398  df-lvec 17529
This theorem is referenced by:  lsatfixedN  33806
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