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Theorem lshpkrlem6 33099
Description: Lemma for lshpkrex 33102. 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 995 . . 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 33096 . 2  |-  ( (
ph  /\  ( l  e.  K  /\  u  e.  V  /\  v  e.  V ) )  ->  E. r  e.  U  u  =  ( r  .+  ( ( G `  u )  .x.  Z
) ) )
21 simpr3 996 . . 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 33096 . 2  |-  ( (
ph  /\  ( l  e.  K  /\  u  e.  V  /\  v  e.  V ) )  ->  E. s  e.  U  v  =  ( s  .+  ( ( G `  v )  .x.  Z
) ) )
23 lveclmod 17311 . . . . 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 994 . . . . 5  |-  ( (
ph  /\  ( l  e.  K  /\  u  e.  V  /\  v  e.  V ) )  -> 
l  e.  K )
261, 15, 17, 16lmodvscl 17089 . . . . 5  |-  ( ( W  e.  LMod  /\  l  e.  K  /\  u  e.  V )  ->  (
l  .x.  u )  e.  V )
2724, 25, 12, 26syl3anc 1219 . . . 4  |-  ( (
ph  /\  ( l  e.  K  /\  u  e.  V  /\  v  e.  V ) )  -> 
( l  .x.  u
)  e.  V )
281, 2lmodvacl 17086 . . . 4  |-  ( ( W  e.  LMod  /\  (
l  .x.  u )  e.  V  /\  v  e.  V )  ->  (
( l  .x.  u
)  .+  v )  e.  V )
2924, 27, 21, 28syl3anc 1219 . . 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 33096 . 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 2995 . . 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 1012 . . . . . . . 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 1084 . . . . . . . 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 1085 . . . . . . . 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 1086 . . . . . . . 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 1055 . . . . . . . 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 1056 . . . . . . . . 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 1015 . . . . . . . . 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 1024 . . . . . . . 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 1025 . . . . . . . 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 1026 . . . . . . . 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 33098 . . . . . . . 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 1251 . . . . . . 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 1187 . . . . . 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 2946 . . . 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 2954 . . 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 1318 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 965    = wceq 1370    e. wcel 1758   E.wrex 2800   {csn 3986    |-> cmpt 4459   ` cfv 5527   iota_crio 6161  (class class class)co 6201   Basecbs 14293   +g cplusg 14358   .rcmulr 14359  Scalarcsca 14361   .scvsca 14362   0gc0g 14498   LSSumclsm 16255   LModclmod 17072   LSpanclspn 17176   LVecclvec 17307  LSHypclsh 32959
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-cnex 9450  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470  ax-pre-mulgt0 9471
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-int 4238  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-om 6588  df-1st 6688  df-2nd 6689  df-tpos 6856  df-recs 6943  df-rdg 6977  df-er 7212  df-en 7422  df-dom 7423  df-sdom 7424  df-pnf 9532  df-mnf 9533  df-xr 9534  df-ltxr 9535  df-le 9536  df-sub 9709  df-neg 9710  df-nn 10435  df-2 10492  df-3 10493  df-ndx 14296  df-slot 14297  df-base 14298  df-sets 14299  df-ress 14300  df-plusg 14371  df-mulr 14372  df-0g 14500  df-mnd 15535  df-submnd 15585  df-grp 15665  df-minusg 15666  df-sbg 15667  df-subg 15798  df-cntz 15955  df-lsm 16257  df-cmn 16401  df-abl 16402  df-mgp 16715  df-ur 16727  df-rng 16771  df-oppr 16839  df-dvdsr 16857  df-unit 16858  df-invr 16888  df-drng 16958  df-lmod 17074  df-lss 17138  df-lsp 17177  df-lvec 17308  df-lshyp 32961
This theorem is referenced by:  lshpkrcl  33100
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