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Theorem lshpkrlem6 29305
Description: Lemma for lshpkrex 29308. 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 451 . . 3  |-  ( (
ph  /\  ( l  e.  K  /\  u  e.  V  /\  v  e.  V ) )  ->  W  e.  LVec )
8 lshpkrlem.u . . . 4  |-  ( ph  ->  U  e.  H )
98adantr 451 . . 3  |-  ( (
ph  /\  ( l  e.  K  /\  u  e.  V  /\  v  e.  V ) )  ->  U  e.  H )
10 lshpkrlem.z . . . 4  |-  ( ph  ->  Z  e.  V )
1110adantr 451 . . 3  |-  ( (
ph  /\  ( l  e.  K  /\  u  e.  V  /\  v  e.  V ) )  ->  Z  e.  V )
12 simpr2 962 . . 3  |-  ( (
ph  /\  ( l  e.  K  /\  u  e.  V  /\  v  e.  V ) )  ->  u  e.  V )
13 lshpkrlem.e . . . 4  |-  ( ph  ->  ( U  .(+)  ( N `
 { Z }
) )  =  V )
1413adantr 451 . . 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 29302 . 2  |-  ( (
ph  /\  ( l  e.  K  /\  u  e.  V  /\  v  e.  V ) )  ->  E. r  e.  U  u  =  ( r  .+  ( ( G `  u )  .x.  Z
) ) )
21 simpr3 963 . . 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 29302 . 2  |-  ( (
ph  /\  ( l  e.  K  /\  u  e.  V  /\  v  e.  V ) )  ->  E. s  e.  U  v  =  ( s  .+  ( ( G `  v )  .x.  Z
) ) )
23 lveclmod 15859 . . . . 5  |-  ( W  e.  LVec  ->  W  e. 
LMod )
247, 23syl 15 . . . 4  |-  ( (
ph  /\  ( l  e.  K  /\  u  e.  V  /\  v  e.  V ) )  ->  W  e.  LMod )
25 simpr1 961 . . . . 5  |-  ( (
ph  /\  ( l  e.  K  /\  u  e.  V  /\  v  e.  V ) )  -> 
l  e.  K )
261, 15, 17, 16lmodvscl 15644 . . . . 5  |-  ( ( W  e.  LMod  /\  l  e.  K  /\  u  e.  V )  ->  (
l  .x.  u )  e.  V )
2724, 25, 12, 26syl3anc 1182 . . . 4  |-  ( (
ph  /\  ( l  e.  K  /\  u  e.  V  /\  v  e.  V ) )  -> 
( l  .x.  u
)  e.  V )
281, 2lmodvacl 15641 . . . 4  |-  ( ( W  e.  LMod  /\  (
l  .x.  u )  e.  V  /\  v  e.  V )  ->  (
( l  .x.  u
)  .+  v )  e.  V )
2924, 27, 21, 28syl3anc 1182 . . 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 29302 . 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 2708 . . 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 979 . . . . . . . 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 1051 . . . . . . . 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 1052 . . . . . . . 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 1053 . . . . . . . 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 1022 . . . . . . . 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 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
) ) ) )  ->  s  e.  U
)
38 simp2r 982 . . . . . . . . 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 518 . . . . . . . 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 991 . . . . . . . 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 992 . . . . . . . 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 993 . . . . . . . 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 29304 . . . . . . . 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 1214 . . . . . . 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 1150 . . . . . 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 426 . . . . 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 2666 . . . 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 2674 . . 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 209 . 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 1280 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 set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   E.wrex 2544   {csn 3640    e. cmpt 4077   ` cfv 5255  (class class class)co 5858   iota_crio 6297   Basecbs 13148   +g cplusg 13208   .rcmulr 13209  Scalarcsca 13211   .scvsca 13212   0gc0g 13400   LSSumclsm 14945   LModclmod 15627   LSpanclspn 15728   LVecclvec 15855  LSHypclsh 29165
This theorem is referenced by:  lshpkrcl  29306
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-tpos 6234  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-0g 13404  df-mnd 14367  df-submnd 14416  df-grp 14489  df-minusg 14490  df-sbg 14491  df-subg 14618  df-cntz 14793  df-lsm 14947  df-cmn 15091  df-abl 15092  df-mgp 15326  df-rng 15340  df-ur 15342  df-oppr 15405  df-dvdsr 15423  df-unit 15424  df-invr 15454  df-drng 15514  df-lmod 15629  df-lss 15690  df-lsp 15729  df-lvec 15856  df-lshyp 29167
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