Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  lshpsmreu Unicode version

Theorem lshpsmreu 29746
Description: Lemma for lshpkrex 29755. Show uniqueness of ring multiplier  k when a vector  X is broken down into components, one in a hyperplane and the other outside of it . TODO: do we need the cbvrexv 2925 for 
a to  c? (Contributed by NM, 4-Jan-2015.)
Hypotheses
Ref Expression
lshpsmreu.v  |-  V  =  ( Base `  W
)
lshpsmreu.a  |-  .+  =  ( +g  `  W )
lshpsmreu.n  |-  N  =  ( LSpan `  W )
lshpsmreu.p  |-  .(+)  =  (
LSSum `  W )
lshpsmreu.h  |-  H  =  (LSHyp `  W )
lshpsmreu.w  |-  ( ph  ->  W  e.  LVec )
lshpsmreu.u  |-  ( ph  ->  U  e.  H )
lshpsmreu.z  |-  ( ph  ->  Z  e.  V )
lshpsmreu.x  |-  ( ph  ->  X  e.  V )
lshpsmreu.e  |-  ( ph  ->  ( U  .(+)  ( N `
 { Z }
) )  =  V )
lshpsmreu.d  |-  D  =  (Scalar `  W )
lshpsmreu.k  |-  K  =  ( Base `  D
)
lshpsmreu.t  |-  .x.  =  ( .s `  W )
Assertion
Ref Expression
lshpsmreu  |-  ( ph  ->  E! k  e.  K  E. y  e.  U  X  =  ( y  .+  ( k  .x.  Z
) ) )
Distinct variable groups:    y, k,  .+    k, K    .x. , k, y    U, k, y    k, X, y    k, Z, y
Allowed substitution hints:    ph( y, k)    D( y, k)    .(+) ( y, k)    H( y, k)    K( y)    N( y, k)    V( y, k)    W( y, k)

Proof of Theorem lshpsmreu
Dummy variables  a 
b  c  l  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lshpsmreu.x . . . . . . 7  |-  ( ph  ->  X  e.  V )
2 lshpsmreu.e . . . . . . 7  |-  ( ph  ->  ( U  .(+)  ( N `
 { Z }
) )  =  V )
31, 2eleqtrrd 2512 . . . . . 6  |-  ( ph  ->  X  e.  ( U 
.(+)  ( N `  { Z } ) ) )
4 lshpsmreu.w . . . . . . . . . 10  |-  ( ph  ->  W  e.  LVec )
5 lveclmod 16166 . . . . . . . . . 10  |-  ( W  e.  LVec  ->  W  e. 
LMod )
64, 5syl 16 . . . . . . . . 9  |-  ( ph  ->  W  e.  LMod )
7 eqid 2435 . . . . . . . . . 10  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
87lsssssubg 16022 . . . . . . . . 9  |-  ( W  e.  LMod  ->  ( LSubSp `  W )  C_  (SubGrp `  W ) )
96, 8syl 16 . . . . . . . 8  |-  ( ph  ->  ( LSubSp `  W )  C_  (SubGrp `  W )
)
10 lshpsmreu.h . . . . . . . . 9  |-  H  =  (LSHyp `  W )
11 lshpsmreu.u . . . . . . . . 9  |-  ( ph  ->  U  e.  H )
127, 10, 6, 11lshplss 29618 . . . . . . . 8  |-  ( ph  ->  U  e.  ( LSubSp `  W ) )
139, 12sseldd 3341 . . . . . . 7  |-  ( ph  ->  U  e.  (SubGrp `  W ) )
14 lshpsmreu.z . . . . . . . . 9  |-  ( ph  ->  Z  e.  V )
15 lshpsmreu.v . . . . . . . . . 10  |-  V  =  ( Base `  W
)
16 lshpsmreu.n . . . . . . . . . 10  |-  N  =  ( LSpan `  W )
1715, 7, 16lspsncl 16041 . . . . . . . . 9  |-  ( ( W  e.  LMod  /\  Z  e.  V )  ->  ( N `  { Z } )  e.  (
LSubSp `  W ) )
186, 14, 17syl2anc 643 . . . . . . . 8  |-  ( ph  ->  ( N `  { Z } )  e.  (
LSubSp `  W ) )
199, 18sseldd 3341 . . . . . . 7  |-  ( ph  ->  ( N `  { Z } )  e.  (SubGrp `  W ) )
20 lshpsmreu.a . . . . . . . 8  |-  .+  =  ( +g  `  W )
21 lshpsmreu.p . . . . . . . 8  |-  .(+)  =  (
LSSum `  W )
2220, 21lsmelval 15271 . . . . . . 7  |-  ( ( U  e.  (SubGrp `  W )  /\  ( N `  { Z } )  e.  (SubGrp `  W ) )  -> 
( X  e.  ( U  .(+)  ( N `  { Z } ) )  <->  E. c  e.  U  E. z  e.  ( N `  { Z } ) X  =  ( c  .+  z
) ) )
2313, 19, 22syl2anc 643 . . . . . 6  |-  ( ph  ->  ( X  e.  ( U  .(+)  ( N `  { Z } ) )  <->  E. c  e.  U  E. z  e.  ( N `  { Z } ) X  =  ( c  .+  z
) ) )
243, 23mpbid 202 . . . . 5  |-  ( ph  ->  E. c  e.  U  E. z  e.  ( N `  { Z } ) X  =  ( c  .+  z
) )
25 df-rex 2703 . . . . . . 7  |-  ( E. z  e.  ( N `
 { Z }
) X  =  ( c  .+  z )  <->  E. z ( z  e.  ( N `  { Z } )  /\  X  =  ( c  .+  z ) ) )
26 lshpsmreu.d . . . . . . . . . . . . 13  |-  D  =  (Scalar `  W )
27 lshpsmreu.k . . . . . . . . . . . . 13  |-  K  =  ( Base `  D
)
28 lshpsmreu.t . . . . . . . . . . . . 13  |-  .x.  =  ( .s `  W )
2926, 27, 15, 28, 16lspsnel 16067 . . . . . . . . . . . 12  |-  ( ( W  e.  LMod  /\  Z  e.  V )  ->  (
z  e.  ( N `
 { Z }
)  <->  E. b  e.  K  z  =  ( b  .x.  Z ) ) )
306, 14, 29syl2anc 643 . . . . . . . . . . 11  |-  ( ph  ->  ( z  e.  ( N `  { Z } )  <->  E. b  e.  K  z  =  ( b  .x.  Z
) ) )
3130anbi1d 686 . . . . . . . . . 10  |-  ( ph  ->  ( ( z  e.  ( N `  { Z } )  /\  X  =  ( c  .+  z ) )  <->  ( E. b  e.  K  z  =  ( b  .x.  Z )  /\  X  =  ( c  .+  z ) ) ) )
32 r19.41v 2853 . . . . . . . . . 10  |-  ( E. b  e.  K  ( z  =  ( b 
.x.  Z )  /\  X  =  ( c  .+  z ) )  <->  ( E. b  e.  K  z  =  ( b  .x.  Z )  /\  X  =  ( c  .+  z ) ) )
3331, 32syl6bbr 255 . . . . . . . . 9  |-  ( ph  ->  ( ( z  e.  ( N `  { Z } )  /\  X  =  ( c  .+  z ) )  <->  E. b  e.  K  ( z  =  ( b  .x.  Z )  /\  X  =  ( c  .+  z ) ) ) )
3433exbidv 1636 . . . . . . . 8  |-  ( ph  ->  ( E. z ( z  e.  ( N `
 { Z }
)  /\  X  =  ( c  .+  z
) )  <->  E. z E. b  e.  K  ( z  =  ( b  .x.  Z )  /\  X  =  ( c  .+  z ) ) ) )
35 rexcom4 2967 . . . . . . . . 9  |-  ( E. b  e.  K  E. z ( z  =  ( b  .x.  Z
)  /\  X  =  ( c  .+  z
) )  <->  E. z E. b  e.  K  ( z  =  ( b  .x.  Z )  /\  X  =  ( c  .+  z ) ) )
36 ovex 6097 . . . . . . . . . . 11  |-  ( b 
.x.  Z )  e. 
_V
37 oveq2 6080 . . . . . . . . . . . 12  |-  ( z  =  ( b  .x.  Z )  ->  (
c  .+  z )  =  ( c  .+  ( b  .x.  Z
) ) )
3837eqeq2d 2446 . . . . . . . . . . 11  |-  ( z  =  ( b  .x.  Z )  ->  ( X  =  ( c  .+  z )  <->  X  =  ( c  .+  (
b  .x.  Z )
) ) )
3936, 38ceqsexv 2983 . . . . . . . . . 10  |-  ( E. z ( z  =  ( b  .x.  Z
)  /\  X  =  ( c  .+  z
) )  <->  X  =  ( c  .+  (
b  .x.  Z )
) )
4039rexbii 2722 . . . . . . . . 9  |-  ( E. b  e.  K  E. z ( z  =  ( b  .x.  Z
)  /\  X  =  ( c  .+  z
) )  <->  E. b  e.  K  X  =  ( c  .+  (
b  .x.  Z )
) )
4135, 40bitr3i 243 . . . . . . . 8  |-  ( E. z E. b  e.  K  ( z  =  ( b  .x.  Z
)  /\  X  =  ( c  .+  z
) )  <->  E. b  e.  K  X  =  ( c  .+  (
b  .x.  Z )
) )
4234, 41syl6bb 253 . . . . . . 7  |-  ( ph  ->  ( E. z ( z  e.  ( N `
 { Z }
)  /\  X  =  ( c  .+  z
) )  <->  E. b  e.  K  X  =  ( c  .+  (
b  .x.  Z )
) ) )
4325, 42syl5bb 249 . . . . . 6  |-  ( ph  ->  ( E. z  e.  ( N `  { Z } ) X  =  ( c  .+  z
)  <->  E. b  e.  K  X  =  ( c  .+  ( b  .x.  Z
) ) ) )
4443rexbidv 2718 . . . . 5  |-  ( ph  ->  ( E. c  e.  U  E. z  e.  ( N `  { Z } ) X  =  ( c  .+  z
)  <->  E. c  e.  U  E. b  e.  K  X  =  ( c  .+  ( b  .x.  Z
) ) ) )
4524, 44mpbid 202 . . . 4  |-  ( ph  ->  E. c  e.  U  E. b  e.  K  X  =  ( c  .+  ( b  .x.  Z
) ) )
46 rexcom 2861 . . . 4  |-  ( E. c  e.  U  E. b  e.  K  X  =  ( c  .+  ( b  .x.  Z
) )  <->  E. b  e.  K  E. c  e.  U  X  =  ( c  .+  (
b  .x.  Z )
) )
4745, 46sylib 189 . . 3  |-  ( ph  ->  E. b  e.  K  E. c  e.  U  X  =  ( c  .+  ( b  .x.  Z
) ) )
48 oveq1 6079 . . . . . . . 8  |-  ( c  =  a  ->  (
c  .+  ( b  .x.  Z ) )  =  ( a  .+  (
b  .x.  Z )
) )
4948eqeq2d 2446 . . . . . . 7  |-  ( c  =  a  ->  ( X  =  ( c  .+  ( b  .x.  Z
) )  <->  X  =  ( a  .+  (
b  .x.  Z )
) ) )
5049cbvrexv 2925 . . . . . 6  |-  ( E. c  e.  U  X  =  ( c  .+  ( b  .x.  Z
) )  <->  E. a  e.  U  X  =  ( a  .+  (
b  .x.  Z )
) )
51 eqid 2435 . . . . . . . . . 10  |-  ( 0g
`  W )  =  ( 0g `  W
)
52 eqid 2435 . . . . . . . . . 10  |-  (Cntz `  W )  =  (Cntz `  W )
53 simp11l 1068 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( b  e.  K  /\  l  e.  K
) )  /\  a  e.  U  /\  X  =  ( a  .+  (
b  .x.  Z )
) )  /\  c  e.  U  /\  X  =  ( c  .+  (
l  .x.  Z )
) )  ->  ph )
5453, 13syl 16 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( b  e.  K  /\  l  e.  K
) )  /\  a  e.  U  /\  X  =  ( a  .+  (
b  .x.  Z )
) )  /\  c  e.  U  /\  X  =  ( c  .+  (
l  .x.  Z )
) )  ->  U  e.  (SubGrp `  W )
)
5553, 19syl 16 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( b  e.  K  /\  l  e.  K
) )  /\  a  e.  U  /\  X  =  ( a  .+  (
b  .x.  Z )
) )  /\  c  e.  U  /\  X  =  ( c  .+  (
l  .x.  Z )
) )  ->  ( N `  { Z } )  e.  (SubGrp `  W ) )
5615, 51, 16, 21, 10, 4, 11, 14, 2lshpdisj 29624 . . . . . . . . . . 11  |-  ( ph  ->  ( U  i^i  ( N `  { Z } ) )  =  { ( 0g `  W ) } )
5753, 56syl 16 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( b  e.  K  /\  l  e.  K
) )  /\  a  e.  U  /\  X  =  ( a  .+  (
b  .x.  Z )
) )  /\  c  e.  U  /\  X  =  ( c  .+  (
l  .x.  Z )
) )  ->  ( U  i^i  ( N `  { Z } ) )  =  { ( 0g
`  W ) } )
5853, 4syl 16 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( b  e.  K  /\  l  e.  K
) )  /\  a  e.  U  /\  X  =  ( a  .+  (
b  .x.  Z )
) )  /\  c  e.  U  /\  X  =  ( c  .+  (
l  .x.  Z )
) )  ->  W  e.  LVec )
5958, 5syl 16 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( b  e.  K  /\  l  e.  K
) )  /\  a  e.  U  /\  X  =  ( a  .+  (
b  .x.  Z )
) )  /\  c  e.  U  /\  X  =  ( c  .+  (
l  .x.  Z )
) )  ->  W  e.  LMod )
60 lmodabl 15979 . . . . . . . . . . . 12  |-  ( W  e.  LMod  ->  W  e. 
Abel )
6159, 60syl 16 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( b  e.  K  /\  l  e.  K
) )  /\  a  e.  U  /\  X  =  ( a  .+  (
b  .x.  Z )
) )  /\  c  e.  U  /\  X  =  ( c  .+  (
l  .x.  Z )
) )  ->  W  e.  Abel )
6252, 61, 54, 55ablcntzd 15460 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( b  e.  K  /\  l  e.  K
) )  /\  a  e.  U  /\  X  =  ( a  .+  (
b  .x.  Z )
) )  /\  c  e.  U  /\  X  =  ( c  .+  (
l  .x.  Z )
) )  ->  U  C_  ( (Cntz `  W
) `  ( N `  { Z } ) ) )
63 simp12 988 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( b  e.  K  /\  l  e.  K
) )  /\  a  e.  U  /\  X  =  ( a  .+  (
b  .x.  Z )
) )  /\  c  e.  U  /\  X  =  ( c  .+  (
l  .x.  Z )
) )  ->  a  e.  U )
64 simp2 958 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( b  e.  K  /\  l  e.  K
) )  /\  a  e.  U  /\  X  =  ( a  .+  (
b  .x.  Z )
) )  /\  c  e.  U  /\  X  =  ( c  .+  (
l  .x.  Z )
) )  ->  c  e.  U )
65 simp1rl 1022 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
b  e.  K  /\  l  e.  K )
)  /\  a  e.  U  /\  X  =  ( a  .+  ( b 
.x.  Z ) ) )  ->  b  e.  K )
66653ad2ant1 978 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( b  e.  K  /\  l  e.  K
) )  /\  a  e.  U  /\  X  =  ( a  .+  (
b  .x.  Z )
) )  /\  c  e.  U  /\  X  =  ( c  .+  (
l  .x.  Z )
) )  ->  b  e.  K )
6753, 14syl 16 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( b  e.  K  /\  l  e.  K
) )  /\  a  e.  U  /\  X  =  ( a  .+  (
b  .x.  Z )
) )  /\  c  e.  U  /\  X  =  ( c  .+  (
l  .x.  Z )
) )  ->  Z  e.  V )
6815, 28, 26, 27, 16, 59, 66, 67lspsneli 16065 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( b  e.  K  /\  l  e.  K
) )  /\  a  e.  U  /\  X  =  ( a  .+  (
b  .x.  Z )
) )  /\  c  e.  U  /\  X  =  ( c  .+  (
l  .x.  Z )
) )  ->  (
b  .x.  Z )  e.  ( N `  { Z } ) )
69 simp1rr 1023 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
b  e.  K  /\  l  e.  K )
)  /\  a  e.  U  /\  X  =  ( a  .+  ( b 
.x.  Z ) ) )  ->  l  e.  K )
70693ad2ant1 978 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( b  e.  K  /\  l  e.  K
) )  /\  a  e.  U  /\  X  =  ( a  .+  (
b  .x.  Z )
) )  /\  c  e.  U  /\  X  =  ( c  .+  (
l  .x.  Z )
) )  ->  l  e.  K )
7115, 28, 26, 27, 16, 59, 70, 67lspsneli 16065 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( b  e.  K  /\  l  e.  K
) )  /\  a  e.  U  /\  X  =  ( a  .+  (
b  .x.  Z )
) )  /\  c  e.  U  /\  X  =  ( c  .+  (
l  .x.  Z )
) )  ->  (
l  .x.  Z )  e.  ( N `  { Z } ) )
72 simp13 989 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( b  e.  K  /\  l  e.  K
) )  /\  a  e.  U  /\  X  =  ( a  .+  (
b  .x.  Z )
) )  /\  c  e.  U  /\  X  =  ( c  .+  (
l  .x.  Z )
) )  ->  X  =  ( a  .+  ( b  .x.  Z
) ) )
73 simp3 959 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( b  e.  K  /\  l  e.  K
) )  /\  a  e.  U  /\  X  =  ( a  .+  (
b  .x.  Z )
) )  /\  c  e.  U  /\  X  =  ( c  .+  (
l  .x.  Z )
) )  ->  X  =  ( c  .+  ( l  .x.  Z
) ) )
7472, 73eqtr3d 2469 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( b  e.  K  /\  l  e.  K
) )  /\  a  e.  U  /\  X  =  ( a  .+  (
b  .x.  Z )
) )  /\  c  e.  U  /\  X  =  ( c  .+  (
l  .x.  Z )
) )  ->  (
a  .+  ( b  .x.  Z ) )  =  ( c  .+  (
l  .x.  Z )
) )
7520, 51, 52, 54, 55, 57, 62, 63, 64, 68, 71, 74subgdisj2 15312 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( b  e.  K  /\  l  e.  K
) )  /\  a  e.  U  /\  X  =  ( a  .+  (
b  .x.  Z )
) )  /\  c  e.  U  /\  X  =  ( c  .+  (
l  .x.  Z )
) )  ->  (
b  .x.  Z )  =  ( l  .x.  Z ) )
7653, 11syl 16 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( b  e.  K  /\  l  e.  K
) )  /\  a  e.  U  /\  X  =  ( a  .+  (
b  .x.  Z )
) )  /\  c  e.  U  /\  X  =  ( c  .+  (
l  .x.  Z )
) )  ->  U  e.  H )
7753, 2syl 16 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( b  e.  K  /\  l  e.  K
) )  /\  a  e.  U  /\  X  =  ( a  .+  (
b  .x.  Z )
) )  /\  c  e.  U  /\  X  =  ( c  .+  (
l  .x.  Z )
) )  ->  ( U  .(+)  ( N `  { Z } ) )  =  V )
7815, 16, 21, 10, 51, 59, 76, 67, 77lshpne0 29623 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( b  e.  K  /\  l  e.  K
) )  /\  a  e.  U  /\  X  =  ( a  .+  (
b  .x.  Z )
) )  /\  c  e.  U  /\  X  =  ( c  .+  (
l  .x.  Z )
) )  ->  Z  =/=  ( 0g `  W
) )
7915, 28, 26, 27, 51, 58, 66, 70, 67, 78lvecvscan2 16172 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( b  e.  K  /\  l  e.  K
) )  /\  a  e.  U  /\  X  =  ( a  .+  (
b  .x.  Z )
) )  /\  c  e.  U  /\  X  =  ( c  .+  (
l  .x.  Z )
) )  ->  (
( b  .x.  Z
)  =  ( l 
.x.  Z )  <->  b  =  l ) )
8075, 79mpbid 202 . . . . . . . 8  |-  ( ( ( ( ph  /\  ( b  e.  K  /\  l  e.  K
) )  /\  a  e.  U  /\  X  =  ( a  .+  (
b  .x.  Z )
) )  /\  c  e.  U  /\  X  =  ( c  .+  (
l  .x.  Z )
) )  ->  b  =  l )
8180rexlimdv3a 2824 . . . . . . 7  |-  ( ( ( ph  /\  (
b  e.  K  /\  l  e.  K )
)  /\  a  e.  U  /\  X  =  ( a  .+  ( b 
.x.  Z ) ) )  ->  ( E. c  e.  U  X  =  ( c  .+  ( l  .x.  Z
) )  ->  b  =  l ) )
8281rexlimdv3a 2824 . . . . . 6  |-  ( (
ph  /\  ( b  e.  K  /\  l  e.  K ) )  -> 
( E. a  e.  U  X  =  ( a  .+  ( b 
.x.  Z ) )  ->  ( E. c  e.  U  X  =  ( c  .+  (
l  .x.  Z )
)  ->  b  =  l ) ) )
8350, 82syl5bi 209 . . . . 5  |-  ( (
ph  /\  ( b  e.  K  /\  l  e.  K ) )  -> 
( E. c  e.  U  X  =  ( c  .+  ( b 
.x.  Z ) )  ->  ( E. c  e.  U  X  =  ( c  .+  (
l  .x.  Z )
)  ->  b  =  l ) ) )
8483imp3a 421 . . . 4  |-  ( (
ph  /\  ( b  e.  K  /\  l  e.  K ) )  -> 
( ( E. c  e.  U  X  =  ( c  .+  (
b  .x.  Z )
)  /\  E. c  e.  U  X  =  ( c  .+  (
l  .x.  Z )
) )  ->  b  =  l ) )
8584ralrimivva 2790 . . 3  |-  ( ph  ->  A. b  e.  K  A. l  e.  K  ( ( E. c  e.  U  X  =  ( c  .+  (
b  .x.  Z )
)  /\  E. c  e.  U  X  =  ( c  .+  (
l  .x.  Z )
) )  ->  b  =  l ) )
86 oveq1 6079 . . . . . . 7  |-  ( b  =  l  ->  (
b  .x.  Z )  =  ( l  .x.  Z ) )
8786oveq2d 6088 . . . . . 6  |-  ( b  =  l  ->  (
c  .+  ( b  .x.  Z ) )  =  ( c  .+  (
l  .x.  Z )
) )
8887eqeq2d 2446 . . . . 5  |-  ( b  =  l  ->  ( X  =  ( c  .+  ( b  .x.  Z
) )  <->  X  =  ( c  .+  (
l  .x.  Z )
) ) )
8988rexbidv 2718 . . . 4  |-  ( b  =  l  ->  ( E. c  e.  U  X  =  ( c  .+  ( b  .x.  Z
) )  <->  E. c  e.  U  X  =  ( c  .+  (
l  .x.  Z )
) ) )
9089reu4 3120 . . 3  |-  ( E! b  e.  K  E. c  e.  U  X  =  ( c  .+  ( b  .x.  Z
) )  <->  ( E. b  e.  K  E. c  e.  U  X  =  ( c  .+  ( b  .x.  Z
) )  /\  A. b  e.  K  A. l  e.  K  (
( E. c  e.  U  X  =  ( c  .+  ( b 
.x.  Z ) )  /\  E. c  e.  U  X  =  ( c  .+  ( l 
.x.  Z ) ) )  ->  b  =  l ) ) )
9147, 85, 90sylanbrc 646 . 2  |-  ( ph  ->  E! b  e.  K  E. c  e.  U  X  =  ( c  .+  ( b  .x.  Z
) ) )
92 oveq1 6079 . . . . . . 7  |-  ( b  =  k  ->  (
b  .x.  Z )  =  ( k  .x.  Z ) )
9392oveq2d 6088 . . . . . 6  |-  ( b  =  k  ->  (
c  .+  ( b  .x.  Z ) )  =  ( c  .+  (
k  .x.  Z )
) )
9493eqeq2d 2446 . . . . 5  |-  ( b  =  k  ->  ( X  =  ( c  .+  ( b  .x.  Z
) )  <->  X  =  ( c  .+  (
k  .x.  Z )
) ) )
9594rexbidv 2718 . . . 4  |-  ( b  =  k  ->  ( E. c  e.  U  X  =  ( c  .+  ( b  .x.  Z
) )  <->  E. c  e.  U  X  =  ( c  .+  (
k  .x.  Z )
) ) )
9695cbvreuv 2926 . . 3  |-  ( E! b  e.  K  E. c  e.  U  X  =  ( c  .+  ( b  .x.  Z
) )  <->  E! k  e.  K  E. c  e.  U  X  =  ( c  .+  (
k  .x.  Z )
) )
97 oveq1 6079 . . . . . 6  |-  ( c  =  y  ->  (
c  .+  ( k  .x.  Z ) )  =  ( y  .+  (
k  .x.  Z )
) )
9897eqeq2d 2446 . . . . 5  |-  ( c  =  y  ->  ( X  =  ( c  .+  ( k  .x.  Z
) )  <->  X  =  ( y  .+  (
k  .x.  Z )
) ) )
9998cbvrexv 2925 . . . 4  |-  ( E. c  e.  U  X  =  ( c  .+  ( k  .x.  Z
) )  <->  E. y  e.  U  X  =  ( y  .+  (
k  .x.  Z )
) )
10099reubii 2886 . . 3  |-  ( E! k  e.  K  E. c  e.  U  X  =  ( c  .+  ( k  .x.  Z
) )  <->  E! k  e.  K  E. y  e.  U  X  =  ( y  .+  (
k  .x.  Z )
) )
10196, 100bitri 241 . 2  |-  ( E! b  e.  K  E. c  e.  U  X  =  ( c  .+  ( b  .x.  Z
) )  <->  E! k  e.  K  E. y  e.  U  X  =  ( y  .+  (
k  .x.  Z )
) )
10291, 101sylib 189 1  |-  ( ph  ->  E! k  e.  K  E. y  e.  U  X  =  ( y  .+  ( k  .x.  Z
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936   E.wex 1550    = wceq 1652    e. wcel 1725   A.wral 2697   E.wrex 2698   E!wreu 2699    i^i cin 3311    C_ wss 3312   {csn 3806   ` cfv 5445  (class class class)co 6072   Basecbs 13457   +g cplusg 13517  Scalarcsca 13520   .scvsca 13521   0gc0g 13711  SubGrpcsubg 14926  Cntzccntz 15102   LSSumclsm 15256   Abelcabel 15401   LModclmod 15938   LSubSpclss 15996   LSpanclspn 16035   LVecclvec 16162  LSHypclsh 29612
This theorem is referenced by:  lshpkrlem1  29747  lshpkrlem2  29748  lshpkrlem3  29749  lshpkrcl  29753  dochfl1  32113
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692  ax-cnex 9035  ax-resscn 9036  ax-1cn 9037  ax-icn 9038  ax-addcl 9039  ax-addrcl 9040  ax-mulcl 9041  ax-mulrcl 9042  ax-mulcom 9043  ax-addass 9044  ax-mulass 9045  ax-distr 9046  ax-i2m1 9047  ax-1ne0 9048  ax-1rid 9049  ax-rnegex 9050  ax-rrecex 9051  ax-cnre 9052  ax-pre-lttri 9053  ax-pre-lttrn 9054  ax-pre-ltadd 9055  ax-pre-mulgt0 9056
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4837  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-1st 6340  df-2nd 6341  df-tpos 6470  df-riota 6540  df-recs 6624  df-rdg 6659  df-er 6896  df-en 7101  df-dom 7102  df-sdom 7103  df-pnf 9111  df-mnf 9112  df-xr 9113  df-ltxr 9114  df-le 9115  df-sub 9282  df-neg 9283  df-nn 9990  df-2 10047  df-3 10048  df-ndx 13460  df-slot 13461  df-base 13462  df-sets 13463  df-ress 13464  df-plusg 13530  df-mulr 13531  df-0g 13715  df-mnd 14678  df-submnd 14727  df-grp 14800  df-minusg 14801  df-sbg 14802  df-subg 14929  df-cntz 15104  df-lsm 15258  df-cmn 15402  df-abl 15403  df-mgp 15637  df-rng 15651  df-ur 15653  df-oppr 15716  df-dvdsr 15734  df-unit 15735  df-invr 15765  df-drng 15825  df-lmod 15940  df-lss 15997  df-lsp 16036  df-lvec 16163  df-lshyp 29614
  Copyright terms: Public domain W3C validator