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Theorem lshpkrlem6 34268
Description: Lemma for lshpkrex 34271. Show linearlity of  G. (Contributed by NM, 17-Jul-2014.)
Hypotheses
Ref Expression
lshpkrlem.v  |-  V  =  ( Base `  W
)
lshpkrlem.a  |-  .+  =  ( +g  `  W )
lshpkrlem.n  |-  N  =  ( LSpan `  W )
lshpkrlem.p  |-  .(+)  =  (
LSSum `  W )
lshpkrlem.h  |-  H  =  (LSHyp `  W )
lshpkrlem.w  |-  ( ph  ->  W  e.  LVec )
lshpkrlem.u  |-  ( ph  ->  U  e.  H )
lshpkrlem.z  |-  ( ph  ->  Z  e.  V )
lshpkrlem.x  |-  ( ph  ->  X  e.  V )
lshpkrlem.e  |-  ( ph  ->  ( U  .(+)  ( N `
 { Z }
) )  =  V )
lshpkrlem.d  |-  D  =  (Scalar `  W )
lshpkrlem.k  |-  K  =  ( Base `  D
)
lshpkrlem.t  |-  .x.  =  ( .s `  W )
lshpkrlem.o  |-  .0.  =  ( 0g `  D )
lshpkrlem.g  |-  G  =  ( x  e.  V  |->  ( iota_ k  e.  K  E. y  e.  U  x  =  ( y  .+  ( k  .x.  Z
) ) ) )
Assertion
Ref Expression
lshpkrlem6  |-  ( (
ph  /\  ( l  e.  K  /\  u  e.  V  /\  v  e.  V ) )  -> 
( G `  (
( l  .x.  u
)  .+  v )
)  =  ( ( l ( .r `  D ) ( G `
 u ) ) ( +g  `  D
) ( G `  v ) ) )
Distinct variable groups:    x, k,
y,  .+    k, K, x    .0. , k    .x. , k, x, y    U, k, x, y    x, V    k, X, x, y   
k, Z, x, y    .+ , l    G, l    K, l    U, l    X, l    Z, l, k, x, y    .x. , l    u, k, v, x, y, l
Allowed substitution hints:    ph( x, y, v, u, k, l)    D( x, y, v, u, k, l)    .+ ( v, u)    .(+) (
x, y, v, u, k, l)    .x. ( v, u)    U( v, u)    G( x, y, v, u, k)    H( x, y, v, u, k, l)    K( y, v, u)    N( x, y, v, u, k, l)    V( y, v, u, k, l)    W( x, y, v, u, k, l)    X( v, u)    .0. ( x, y, v, u, l)    Z( v, u)

Proof of Theorem lshpkrlem6
Dummy variables  z 
s  r are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lshpkrlem.v . . 3  |-  V  =  ( Base `  W
)
2 lshpkrlem.a . . 3  |-  .+  =  ( +g  `  W )
3 lshpkrlem.n . . 3  |-  N  =  ( LSpan `  W )
4 lshpkrlem.p . . 3  |-  .(+)  =  (
LSSum `  W )
5 lshpkrlem.h . . 3  |-  H  =  (LSHyp `  W )
6 lshpkrlem.w . . . 4  |-  ( ph  ->  W  e.  LVec )
76adantr 465 . . 3  |-  ( (
ph  /\  ( l  e.  K  /\  u  e.  V  /\  v  e.  V ) )  ->  W  e.  LVec )
8 lshpkrlem.u . . . 4  |-  ( ph  ->  U  e.  H )
98adantr 465 . . 3  |-  ( (
ph  /\  ( l  e.  K  /\  u  e.  V  /\  v  e.  V ) )  ->  U  e.  H )
10 lshpkrlem.z . . . 4  |-  ( ph  ->  Z  e.  V )
1110adantr 465 . . 3  |-  ( (
ph  /\  ( l  e.  K  /\  u  e.  V  /\  v  e.  V ) )  ->  Z  e.  V )
12 simpr2 1003 . . 3  |-  ( (
ph  /\  ( l  e.  K  /\  u  e.  V  /\  v  e.  V ) )  ->  u  e.  V )
13 lshpkrlem.e . . . 4  |-  ( ph  ->  ( U  .(+)  ( N `
 { Z }
) )  =  V )
1413adantr 465 . . 3  |-  ( (
ph  /\  ( l  e.  K  /\  u  e.  V  /\  v  e.  V ) )  -> 
( U  .(+)  ( N `
 { Z }
) )  =  V )
15 lshpkrlem.d . . 3  |-  D  =  (Scalar `  W )
16 lshpkrlem.k . . 3  |-  K  =  ( Base `  D
)
17 lshpkrlem.t . . 3  |-  .x.  =  ( .s `  W )
18 lshpkrlem.o . . 3  |-  .0.  =  ( 0g `  D )
19 lshpkrlem.g . . 3  |-  G  =  ( x  e.  V  |->  ( iota_ k  e.  K  E. y  e.  U  x  =  ( y  .+  ( k  .x.  Z
) ) ) )
201, 2, 3, 4, 5, 7, 9, 11, 12, 14, 15, 16, 17, 18, 19lshpkrlem3 34265 . 2  |-  ( (
ph  /\  ( l  e.  K  /\  u  e.  V  /\  v  e.  V ) )  ->  E. r  e.  U  u  =  ( r  .+  ( ( G `  u )  .x.  Z
) ) )
21 simpr3 1004 . . 3  |-  ( (
ph  /\  ( l  e.  K  /\  u  e.  V  /\  v  e.  V ) )  -> 
v  e.  V )
221, 2, 3, 4, 5, 7, 9, 11, 21, 14, 15, 16, 17, 18, 19lshpkrlem3 34265 . 2  |-  ( (
ph  /\  ( l  e.  K  /\  u  e.  V  /\  v  e.  V ) )  ->  E. s  e.  U  v  =  ( s  .+  ( ( G `  v )  .x.  Z
) ) )
23 lveclmod 17623 . . . . 5  |-  ( W  e.  LVec  ->  W  e. 
LMod )
247, 23syl 16 . . . 4  |-  ( (
ph  /\  ( l  e.  K  /\  u  e.  V  /\  v  e.  V ) )  ->  W  e.  LMod )
25 simpr1 1002 . . . . 5  |-  ( (
ph  /\  ( l  e.  K  /\  u  e.  V  /\  v  e.  V ) )  -> 
l  e.  K )
261, 15, 17, 16lmodvscl 17400 . . . . 5  |-  ( ( W  e.  LMod  /\  l  e.  K  /\  u  e.  V )  ->  (
l  .x.  u )  e.  V )
2724, 25, 12, 26syl3anc 1228 . . . 4  |-  ( (
ph  /\  ( l  e.  K  /\  u  e.  V  /\  v  e.  V ) )  -> 
( l  .x.  u
)  e.  V )
281, 2lmodvacl 17397 . . . 4  |-  ( ( W  e.  LMod  /\  (
l  .x.  u )  e.  V  /\  v  e.  V )  ->  (
( l  .x.  u
)  .+  v )  e.  V )
2924, 27, 21, 28syl3anc 1228 . . 3  |-  ( (
ph  /\  ( l  e.  K  /\  u  e.  V  /\  v  e.  V ) )  -> 
( ( l  .x.  u )  .+  v
)  e.  V )
301, 2, 3, 4, 5, 7, 9, 11, 29, 14, 15, 16, 17, 18, 19lshpkrlem3 34265 . 2  |-  ( (
ph  /\  ( l  e.  K  /\  u  e.  V  /\  v  e.  V ) )  ->  E. z  e.  U  ( ( l  .x.  u )  .+  v
)  =  ( z 
.+  ( ( G `
 ( ( l 
.x.  u )  .+  v ) )  .x.  Z ) ) )
31 3reeanv 3035 . . 3  |-  ( E. r  e.  U  E. s  e.  U  E. z  e.  U  (
u  =  ( r 
.+  ( ( G `
 u )  .x.  Z ) )  /\  v  =  ( s  .+  ( ( G `  v )  .x.  Z
) )  /\  (
( l  .x.  u
)  .+  v )  =  ( z  .+  ( ( G `  ( ( l  .x.  u )  .+  v
) )  .x.  Z
) ) )  <->  ( E. r  e.  U  u  =  ( r  .+  ( ( G `  u )  .x.  Z
) )  /\  E. s  e.  U  v  =  ( s  .+  ( ( G `  v )  .x.  Z
) )  /\  E. z  e.  U  (
( l  .x.  u
)  .+  v )  =  ( z  .+  ( ( G `  ( ( l  .x.  u )  .+  v
) )  .x.  Z
) ) ) )
32 simp1l 1020 . . . . . . . 8  |-  ( ( ( ph  /\  (
l  e.  K  /\  u  e.  V  /\  v  e.  V )
)  /\  ( (
r  e.  U  /\  s  e.  U )  /\  z  e.  U
)  /\  ( u  =  ( r  .+  ( ( G `  u )  .x.  Z
) )  /\  v  =  ( s  .+  ( ( G `  v )  .x.  Z
) )  /\  (
( l  .x.  u
)  .+  v )  =  ( z  .+  ( ( G `  ( ( l  .x.  u )  .+  v
) )  .x.  Z
) ) ) )  ->  ph )
33 simp1r1 1092 . . . . . . . 8  |-  ( ( ( ph  /\  (
l  e.  K  /\  u  e.  V  /\  v  e.  V )
)  /\  ( (
r  e.  U  /\  s  e.  U )  /\  z  e.  U
)  /\  ( u  =  ( r  .+  ( ( G `  u )  .x.  Z
) )  /\  v  =  ( s  .+  ( ( G `  v )  .x.  Z
) )  /\  (
( l  .x.  u
)  .+  v )  =  ( z  .+  ( ( G `  ( ( l  .x.  u )  .+  v
) )  .x.  Z
) ) ) )  ->  l  e.  K
)
34 simp1r2 1093 . . . . . . . 8  |-  ( ( ( ph  /\  (
l  e.  K  /\  u  e.  V  /\  v  e.  V )
)  /\  ( (
r  e.  U  /\  s  e.  U )  /\  z  e.  U
)  /\  ( u  =  ( r  .+  ( ( G `  u )  .x.  Z
) )  /\  v  =  ( s  .+  ( ( G `  v )  .x.  Z
) )  /\  (
( l  .x.  u
)  .+  v )  =  ( z  .+  ( ( G `  ( ( l  .x.  u )  .+  v
) )  .x.  Z
) ) ) )  ->  u  e.  V
)
35 simp1r3 1094 . . . . . . . 8  |-  ( ( ( ph  /\  (
l  e.  K  /\  u  e.  V  /\  v  e.  V )
)  /\  ( (
r  e.  U  /\  s  e.  U )  /\  z  e.  U
)  /\  ( u  =  ( r  .+  ( ( G `  u )  .x.  Z
) )  /\  v  =  ( s  .+  ( ( G `  v )  .x.  Z
) )  /\  (
( l  .x.  u
)  .+  v )  =  ( z  .+  ( ( G `  ( ( l  .x.  u )  .+  v
) )  .x.  Z
) ) ) )  ->  v  e.  V
)
36 simp2ll 1063 . . . . . . . 8  |-  ( ( ( ph  /\  (
l  e.  K  /\  u  e.  V  /\  v  e.  V )
)  /\  ( (
r  e.  U  /\  s  e.  U )  /\  z  e.  U
)  /\  ( u  =  ( r  .+  ( ( G `  u )  .x.  Z
) )  /\  v  =  ( s  .+  ( ( G `  v )  .x.  Z
) )  /\  (
( l  .x.  u
)  .+  v )  =  ( z  .+  ( ( G `  ( ( l  .x.  u )  .+  v
) )  .x.  Z
) ) ) )  ->  r  e.  U
)
37 simp2lr 1064 . . . . . . . . 9  |-  ( ( ( ph  /\  (
l  e.  K  /\  u  e.  V  /\  v  e.  V )
)  /\  ( (
r  e.  U  /\  s  e.  U )  /\  z  e.  U
)  /\  ( u  =  ( r  .+  ( ( G `  u )  .x.  Z
) )  /\  v  =  ( s  .+  ( ( G `  v )  .x.  Z
) )  /\  (
( l  .x.  u
)  .+  v )  =  ( z  .+  ( ( G `  ( ( l  .x.  u )  .+  v
) )  .x.  Z
) ) ) )  ->  s  e.  U
)
38 simp2r 1023 . . . . . . . . 9  |-  ( ( ( ph  /\  (
l  e.  K  /\  u  e.  V  /\  v  e.  V )
)  /\  ( (
r  e.  U  /\  s  e.  U )  /\  z  e.  U
)  /\  ( u  =  ( r  .+  ( ( G `  u )  .x.  Z
) )  /\  v  =  ( s  .+  ( ( G `  v )  .x.  Z
) )  /\  (
( l  .x.  u
)  .+  v )  =  ( z  .+  ( ( G `  ( ( l  .x.  u )  .+  v
) )  .x.  Z
) ) ) )  ->  z  e.  U
)
3937, 38jca 532 . . . . . . . 8  |-  ( ( ( ph  /\  (
l  e.  K  /\  u  e.  V  /\  v  e.  V )
)  /\  ( (
r  e.  U  /\  s  e.  U )  /\  z  e.  U
)  /\  ( u  =  ( r  .+  ( ( G `  u )  .x.  Z
) )  /\  v  =  ( s  .+  ( ( G `  v )  .x.  Z
) )  /\  (
( l  .x.  u
)  .+  v )  =  ( z  .+  ( ( G `  ( ( l  .x.  u )  .+  v
) )  .x.  Z
) ) ) )  ->  ( s  e.  U  /\  z  e.  U ) )
40 simp31 1032 . . . . . . . 8  |-  ( ( ( ph  /\  (
l  e.  K  /\  u  e.  V  /\  v  e.  V )
)  /\  ( (
r  e.  U  /\  s  e.  U )  /\  z  e.  U
)  /\  ( u  =  ( r  .+  ( ( G `  u )  .x.  Z
) )  /\  v  =  ( s  .+  ( ( G `  v )  .x.  Z
) )  /\  (
( l  .x.  u
)  .+  v )  =  ( z  .+  ( ( G `  ( ( l  .x.  u )  .+  v
) )  .x.  Z
) ) ) )  ->  u  =  ( r  .+  ( ( G `  u ) 
.x.  Z ) ) )
41 simp32 1033 . . . . . . . 8  |-  ( ( ( ph  /\  (
l  e.  K  /\  u  e.  V  /\  v  e.  V )
)  /\  ( (
r  e.  U  /\  s  e.  U )  /\  z  e.  U
)  /\  ( u  =  ( r  .+  ( ( G `  u )  .x.  Z
) )  /\  v  =  ( s  .+  ( ( G `  v )  .x.  Z
) )  /\  (
( l  .x.  u
)  .+  v )  =  ( z  .+  ( ( G `  ( ( l  .x.  u )  .+  v
) )  .x.  Z
) ) ) )  ->  v  =  ( s  .+  ( ( G `  v ) 
.x.  Z ) ) )
42 simp33 1034 . . . . . . . 8  |-  ( ( ( ph  /\  (
l  e.  K  /\  u  e.  V  /\  v  e.  V )
)  /\  ( (
r  e.  U  /\  s  e.  U )  /\  z  e.  U
)  /\  ( u  =  ( r  .+  ( ( G `  u )  .x.  Z
) )  /\  v  =  ( s  .+  ( ( G `  v )  .x.  Z
) )  /\  (
( l  .x.  u
)  .+  v )  =  ( z  .+  ( ( G `  ( ( l  .x.  u )  .+  v
) )  .x.  Z
) ) ) )  ->  ( ( l 
.x.  u )  .+  v )  =  ( z  .+  ( ( G `  ( ( l  .x.  u ) 
.+  v ) ) 
.x.  Z ) ) )
431, 2, 3, 4, 5, 6, 8, 10, 10, 13, 15, 16, 17, 18, 19lshpkrlem5 34267 . . . . . . . 8  |-  ( ( ( ph  /\  l  e.  K  /\  u  e.  V )  /\  (
v  e.  V  /\  r  e.  U  /\  ( s  e.  U  /\  z  e.  U
) )  /\  (
u  =  ( r 
.+  ( ( G `
 u )  .x.  Z ) )  /\  v  =  ( s  .+  ( ( G `  v )  .x.  Z
) )  /\  (
( l  .x.  u
)  .+  v )  =  ( z  .+  ( ( G `  ( ( l  .x.  u )  .+  v
) )  .x.  Z
) ) ) )  ->  ( G `  ( ( l  .x.  u )  .+  v
) )  =  ( ( l ( .r
`  D ) ( G `  u ) ) ( +g  `  D
) ( G `  v ) ) )
4432, 33, 34, 35, 36, 39, 40, 41, 42, 43syl333anc 1260 . . . . . . 7  |-  ( ( ( ph  /\  (
l  e.  K  /\  u  e.  V  /\  v  e.  V )
)  /\  ( (
r  e.  U  /\  s  e.  U )  /\  z  e.  U
)  /\  ( u  =  ( r  .+  ( ( G `  u )  .x.  Z
) )  /\  v  =  ( s  .+  ( ( G `  v )  .x.  Z
) )  /\  (
( l  .x.  u
)  .+  v )  =  ( z  .+  ( ( G `  ( ( l  .x.  u )  .+  v
) )  .x.  Z
) ) ) )  ->  ( G `  ( ( l  .x.  u )  .+  v
) )  =  ( ( l ( .r
`  D ) ( G `  u ) ) ( +g  `  D
) ( G `  v ) ) )
45443exp 1195 . . . . . 6  |-  ( (
ph  /\  ( l  e.  K  /\  u  e.  V  /\  v  e.  V ) )  -> 
( ( ( r  e.  U  /\  s  e.  U )  /\  z  e.  U )  ->  (
( u  =  ( r  .+  ( ( G `  u ) 
.x.  Z ) )  /\  v  =  ( s  .+  ( ( G `  v ) 
.x.  Z ) )  /\  ( ( l 
.x.  u )  .+  v )  =  ( z  .+  ( ( G `  ( ( l  .x.  u ) 
.+  v ) ) 
.x.  Z ) ) )  ->  ( G `  ( ( l  .x.  u )  .+  v
) )  =  ( ( l ( .r
`  D ) ( G `  u ) ) ( +g  `  D
) ( G `  v ) ) ) ) )
4645expdimp 437 . . . . 5  |-  ( ( ( ph  /\  (
l  e.  K  /\  u  e.  V  /\  v  e.  V )
)  /\  ( r  e.  U  /\  s  e.  U ) )  -> 
( z  e.  U  ->  ( ( u  =  ( r  .+  (
( G `  u
)  .x.  Z )
)  /\  v  =  ( s  .+  (
( G `  v
)  .x.  Z )
)  /\  ( (
l  .x.  u )  .+  v )  =  ( z  .+  ( ( G `  ( ( l  .x.  u ) 
.+  v ) ) 
.x.  Z ) ) )  ->  ( G `  ( ( l  .x.  u )  .+  v
) )  =  ( ( l ( .r
`  D ) ( G `  u ) ) ( +g  `  D
) ( G `  v ) ) ) ) )
4746rexlimdv 2957 . . . 4  |-  ( ( ( ph  /\  (
l  e.  K  /\  u  e.  V  /\  v  e.  V )
)  /\  ( r  e.  U  /\  s  e.  U ) )  -> 
( E. z  e.  U  ( u  =  ( r  .+  (
( G `  u
)  .x.  Z )
)  /\  v  =  ( s  .+  (
( G `  v
)  .x.  Z )
)  /\  ( (
l  .x.  u )  .+  v )  =  ( z  .+  ( ( G `  ( ( l  .x.  u ) 
.+  v ) ) 
.x.  Z ) ) )  ->  ( G `  ( ( l  .x.  u )  .+  v
) )  =  ( ( l ( .r
`  D ) ( G `  u ) ) ( +g  `  D
) ( G `  v ) ) ) )
4847rexlimdvva 2966 . . 3  |-  ( (
ph  /\  ( l  e.  K  /\  u  e.  V  /\  v  e.  V ) )  -> 
( E. r  e.  U  E. s  e.  U  E. z  e.  U  ( u  =  ( r  .+  (
( G `  u
)  .x.  Z )
)  /\  v  =  ( s  .+  (
( G `  v
)  .x.  Z )
)  /\  ( (
l  .x.  u )  .+  v )  =  ( z  .+  ( ( G `  ( ( l  .x.  u ) 
.+  v ) ) 
.x.  Z ) ) )  ->  ( G `  ( ( l  .x.  u )  .+  v
) )  =  ( ( l ( .r
`  D ) ( G `  u ) ) ( +g  `  D
) ( G `  v ) ) ) )
4931, 48syl5bir 218 . 2  |-  ( (
ph  /\  ( l  e.  K  /\  u  e.  V  /\  v  e.  V ) )  -> 
( ( E. r  e.  U  u  =  ( r  .+  (
( G `  u
)  .x.  Z )
)  /\  E. s  e.  U  v  =  ( s  .+  (
( G `  v
)  .x.  Z )
)  /\  E. z  e.  U  ( (
l  .x.  u )  .+  v )  =  ( z  .+  ( ( G `  ( ( l  .x.  u ) 
.+  v ) ) 
.x.  Z ) ) )  ->  ( G `  ( ( l  .x.  u )  .+  v
) )  =  ( ( l ( .r
`  D ) ( G `  u ) ) ( +g  `  D
) ( G `  v ) ) ) )
5020, 22, 30, 49mp3and 1327 1  |-  ( (
ph  /\  ( l  e.  K  /\  u  e.  V  /\  v  e.  V ) )  -> 
( G `  (
( l  .x.  u
)  .+  v )
)  =  ( ( l ( .r `  D ) ( G `
 u ) ) ( +g  `  D
) ( G `  v ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   E.wrex 2818   {csn 4033    |-> cmpt 4511   ` cfv 5594   iota_crio 6255  (class class class)co 6295   Basecbs 14507   +g cplusg 14572   .rcmulr 14573  Scalarcsca 14575   .scvsca 14576   0gc0g 14712   LSSumclsm 16527   LModclmod 17383   LSpanclspn 17488   LVecclvec 17619  LSHypclsh 34128
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 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
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 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-tpos 6967  df-recs 7054  df-rdg 7088  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-3 10607  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-ress 14514  df-plusg 14585  df-mulr 14586  df-0g 14714  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-submnd 15840  df-grp 15929  df-minusg 15930  df-sbg 15931  df-subg 16070  df-cntz 16227  df-lsm 16529  df-cmn 16673  df-abl 16674  df-mgp 17014  df-ur 17026  df-ring 17072  df-oppr 17144  df-dvdsr 17162  df-unit 17163  df-invr 17193  df-drng 17269  df-lmod 17385  df-lss 17450  df-lsp 17489  df-lvec 17620  df-lshyp 34130
This theorem is referenced by:  lshpkrcl  34269
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