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Theorem lshpkrlem6 29610
Description: Lemma for lshpkrex 29613. 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 452 . . 3  |-  ( (
ph  /\  ( l  e.  K  /\  u  e.  V  /\  v  e.  V ) )  ->  W  e.  LVec )
8 lshpkrlem.u . . . 4  |-  ( ph  ->  U  e.  H )
98adantr 452 . . 3  |-  ( (
ph  /\  ( l  e.  K  /\  u  e.  V  /\  v  e.  V ) )  ->  U  e.  H )
10 lshpkrlem.z . . . 4  |-  ( ph  ->  Z  e.  V )
1110adantr 452 . . 3  |-  ( (
ph  /\  ( l  e.  K  /\  u  e.  V  /\  v  e.  V ) )  ->  Z  e.  V )
12 simpr2 964 . . 3  |-  ( (
ph  /\  ( l  e.  K  /\  u  e.  V  /\  v  e.  V ) )  ->  u  e.  V )
13 lshpkrlem.e . . . 4  |-  ( ph  ->  ( U  .(+)  ( N `
 { Z }
) )  =  V )
1413adantr 452 . . 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 29607 . 2  |-  ( (
ph  /\  ( l  e.  K  /\  u  e.  V  /\  v  e.  V ) )  ->  E. r  e.  U  u  =  ( r  .+  ( ( G `  u )  .x.  Z
) ) )
21 simpr3 965 . . 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 29607 . 2  |-  ( (
ph  /\  ( l  e.  K  /\  u  e.  V  /\  v  e.  V ) )  ->  E. s  e.  U  v  =  ( s  .+  ( ( G `  v )  .x.  Z
) ) )
23 lveclmod 16141 . . . . 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 963 . . . . 5  |-  ( (
ph  /\  ( l  e.  K  /\  u  e.  V  /\  v  e.  V ) )  -> 
l  e.  K )
261, 15, 17, 16lmodvscl 15930 . . . . 5  |-  ( ( W  e.  LMod  /\  l  e.  K  /\  u  e.  V )  ->  (
l  .x.  u )  e.  V )
2724, 25, 12, 26syl3anc 1184 . . . 4  |-  ( (
ph  /\  ( l  e.  K  /\  u  e.  V  /\  v  e.  V ) )  -> 
( l  .x.  u
)  e.  V )
281, 2lmodvacl 15927 . . . 4  |-  ( ( W  e.  LMod  /\  (
l  .x.  u )  e.  V  /\  v  e.  V )  ->  (
( l  .x.  u
)  .+  v )  e.  V )
2924, 27, 21, 28syl3anc 1184 . . 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 29607 . 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 2844 . . 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 981 . . . . . . . 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 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
) ) ) )  ->  l  e.  K
)
34 simp1r2 1054 . . . . . . . 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 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
) ) ) )  ->  v  e.  V
)
36 simp2ll 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
) ) ) )  ->  r  e.  U
)
37 simp2lr 1025 . . . . . . . . 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 984 . . . . . . . . 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 519 . . . . . . . 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 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
) ) ) )  ->  u  =  ( r  .+  ( ( G `  u ) 
.x.  Z ) ) )
41 simp32 994 . . . . . . . 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 995 . . . . . . . 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 29609 . . . . . . . 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 1216 . . . . . . 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 1152 . . . . . 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 427 . . . . 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 2797 . . . 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 2805 . . 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 210 . 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 1282 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 359    /\ w3a 936    = wceq 1649    e. wcel 1721   E.wrex 2675   {csn 3782    e. cmpt 4234   ` cfv 5421  (class class class)co 6048   iota_crio 6509   Basecbs 13432   +g cplusg 13492   .rcmulr 13493  Scalarcsca 13495   .scvsca 13496   0gc0g 13686   LSSumclsm 15231   LModclmod 15913   LSpanclspn 16010   LVecclvec 16137  LSHypclsh 29470
This theorem is referenced by:  lshpkrcl  29611
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-tpos 6446  df-riota 6516  df-recs 6600  df-rdg 6635  df-er 6872  df-en 7077  df-dom 7078  df-sdom 7079  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-nn 9965  df-2 10022  df-3 10023  df-ndx 13435  df-slot 13436  df-base 13437  df-sets 13438  df-ress 13439  df-plusg 13505  df-mulr 13506  df-0g 13690  df-mnd 14653  df-submnd 14702  df-grp 14775  df-minusg 14776  df-sbg 14777  df-subg 14904  df-cntz 15079  df-lsm 15233  df-cmn 15377  df-abl 15378  df-mgp 15612  df-rng 15626  df-ur 15628  df-oppr 15691  df-dvdsr 15709  df-unit 15710  df-invr 15740  df-drng 15800  df-lmod 15915  df-lss 15972  df-lsp 16011  df-lvec 16138  df-lshyp 29472
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