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Theorem lshpkrlem6 30087
Description: Lemma for lshpkrex 30090. 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 453 . . 3  |-  ( (
ph  /\  ( l  e.  K  /\  u  e.  V  /\  v  e.  V ) )  ->  W  e.  LVec )
8 lshpkrlem.u . . . 4  |-  ( ph  ->  U  e.  H )
98adantr 453 . . 3  |-  ( (
ph  /\  ( l  e.  K  /\  u  e.  V  /\  v  e.  V ) )  ->  U  e.  H )
10 lshpkrlem.z . . . 4  |-  ( ph  ->  Z  e.  V )
1110adantr 453 . . 3  |-  ( (
ph  /\  ( l  e.  K  /\  u  e.  V  /\  v  e.  V ) )  ->  Z  e.  V )
12 simpr2 965 . . 3  |-  ( (
ph  /\  ( l  e.  K  /\  u  e.  V  /\  v  e.  V ) )  ->  u  e.  V )
13 lshpkrlem.e . . . 4  |-  ( ph  ->  ( U  .(+)  ( N `
 { Z }
) )  =  V )
1413adantr 453 . . 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 30084 . 2  |-  ( (
ph  /\  ( l  e.  K  /\  u  e.  V  /\  v  e.  V ) )  ->  E. r  e.  U  u  =  ( r  .+  ( ( G `  u )  .x.  Z
) ) )
21 simpr3 966 . . 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 30084 . 2  |-  ( (
ph  /\  ( l  e.  K  /\  u  e.  V  /\  v  e.  V ) )  ->  E. s  e.  U  v  =  ( s  .+  ( ( G `  v )  .x.  Z
) ) )
23 lveclmod 16216 . . . . 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 964 . . . . 5  |-  ( (
ph  /\  ( l  e.  K  /\  u  e.  V  /\  v  e.  V ) )  -> 
l  e.  K )
261, 15, 17, 16lmodvscl 16005 . . . . 5  |-  ( ( W  e.  LMod  /\  l  e.  K  /\  u  e.  V )  ->  (
l  .x.  u )  e.  V )
2724, 25, 12, 26syl3anc 1185 . . . 4  |-  ( (
ph  /\  ( l  e.  K  /\  u  e.  V  /\  v  e.  V ) )  -> 
( l  .x.  u
)  e.  V )
281, 2lmodvacl 16002 . . . 4  |-  ( ( W  e.  LMod  /\  (
l  .x.  u )  e.  V  /\  v  e.  V )  ->  (
( l  .x.  u
)  .+  v )  e.  V )
2924, 27, 21, 28syl3anc 1185 . . 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 30084 . 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 2883 . . 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 982 . . . . . . . 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 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
) ) ) )  ->  l  e.  K
)
34 simp1r2 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
) ) ) )  ->  u  e.  V
)
35 simp1r3 1056 . . . . . . . 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 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
) ) ) )  ->  r  e.  U
)
37 simp2lr 1026 . . . . . . . . 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 985 . . . . . . . . 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 520 . . . . . . . 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 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
) ) ) )  ->  u  =  ( r  .+  ( ( G `  u ) 
.x.  Z ) ) )
41 simp32 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
) ) ) )  ->  v  =  ( s  .+  ( ( G `  v ) 
.x.  Z ) ) )
42 simp33 996 . . . . . . . 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 30086 . . . . . . . 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 1217 . . . . . . 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 1153 . . . . . 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 428 . . . . 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 2836 . . . 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 2844 . . 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 211 . 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 1283 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 360    /\ w3a 937    = wceq 1654    e. wcel 1728   E.wrex 2713   {csn 3843    e. cmpt 4297   ` cfv 5489  (class class class)co 6117   iota_crio 6578   Basecbs 13507   +g cplusg 13567   .rcmulr 13568  Scalarcsca 13570   .scvsca 13571   0gc0g 13761   LSSumclsm 15306   LModclmod 15988   LSpanclspn 16085   LVecclvec 16212  LSHypclsh 29947
This theorem is referenced by:  lshpkrcl  30088
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-13 1730  ax-14 1732  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424  ax-rep 4351  ax-sep 4361  ax-nul 4369  ax-pow 4412  ax-pr 4438  ax-un 4736  ax-cnex 9084  ax-resscn 9085  ax-1cn 9086  ax-icn 9087  ax-addcl 9088  ax-addrcl 9089  ax-mulcl 9090  ax-mulrcl 9091  ax-mulcom 9092  ax-addass 9093  ax-mulass 9094  ax-distr 9095  ax-i2m1 9096  ax-1ne0 9097  ax-1rid 9098  ax-rnegex 9099  ax-rrecex 9100  ax-cnre 9101  ax-pre-lttri 9102  ax-pre-lttrn 9103  ax-pre-ltadd 9104  ax-pre-mulgt0 9105
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1661  df-eu 2292  df-mo 2293  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2717  df-rex 2718  df-reu 2719  df-rmo 2720  df-rab 2721  df-v 2967  df-sbc 3171  df-csb 3271  df-dif 3312  df-un 3314  df-in 3316  df-ss 3323  df-pss 3325  df-nul 3617  df-if 3768  df-pw 3830  df-sn 3849  df-pr 3850  df-tp 3851  df-op 3852  df-uni 4045  df-int 4080  df-iun 4124  df-br 4244  df-opab 4298  df-mpt 4299  df-tr 4334  df-eprel 4529  df-id 4533  df-po 4538  df-so 4539  df-fr 4576  df-we 4578  df-ord 4619  df-on 4620  df-lim 4621  df-suc 4622  df-om 4881  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5453  df-fun 5491  df-fn 5492  df-f 5493  df-f1 5494  df-fo 5495  df-f1o 5496  df-fv 5497  df-ov 6120  df-oprab 6121  df-mpt2 6122  df-1st 6385  df-2nd 6386  df-tpos 6515  df-riota 6585  df-recs 6669  df-rdg 6704  df-er 6941  df-en 7146  df-dom 7147  df-sdom 7148  df-pnf 9160  df-mnf 9161  df-xr 9162  df-ltxr 9163  df-le 9164  df-sub 9331  df-neg 9332  df-nn 10039  df-2 10096  df-3 10097  df-ndx 13510  df-slot 13511  df-base 13512  df-sets 13513  df-ress 13514  df-plusg 13580  df-mulr 13581  df-0g 13765  df-mnd 14728  df-submnd 14777  df-grp 14850  df-minusg 14851  df-sbg 14852  df-subg 14979  df-cntz 15154  df-lsm 15308  df-cmn 15452  df-abl 15453  df-mgp 15687  df-rng 15701  df-ur 15703  df-oppr 15766  df-dvdsr 15784  df-unit 15785  df-invr 15815  df-drng 15875  df-lmod 15990  df-lss 16047  df-lsp 16086  df-lvec 16213  df-lshyp 29949
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