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Theorem coe1tm 16349
Description: Coefficient vector of a polynomial term. (Contributed by Stefan O'Rear, 27-Mar-2015.)
Hypotheses
Ref Expression
coe1tm.z  |-  .0.  =  ( 0g `  R )
coe1tm.k  |-  K  =  ( Base `  R
)
coe1tm.p  |-  P  =  (Poly1 `  R )
coe1tm.x  |-  X  =  (var1 `  R )
coe1tm.m  |-  .x.  =  ( .s `  P )
coe1tm.n  |-  N  =  (mulGrp `  P )
coe1tm.e  |-  .^  =  (.g
`  N )
Assertion
Ref Expression
coe1tm  |-  ( ( R  e.  Ring  /\  C  e.  K  /\  D  e. 
NN0 )  ->  (coe1 `  ( C  .x.  ( D 
.^  X ) ) )  =  ( x  e.  NN0  |->  if ( x  =  D ,  C ,  .0.  )
) )
Distinct variable groups:    x,  .0.    x, C    x, D    x, K    x,  .^    x, N    x, P    x, X    x, R    x, 
.x.

Proof of Theorem coe1tm
Dummy variables  a 
b  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 coe1tm.k . . . 4  |-  K  =  ( Base `  R
)
2 coe1tm.p . . . 4  |-  P  =  (Poly1 `  R )
3 coe1tm.x . . . 4  |-  X  =  (var1 `  R )
4 coe1tm.m . . . 4  |-  .x.  =  ( .s `  P )
5 coe1tm.n . . . 4  |-  N  =  (mulGrp `  P )
6 coe1tm.e . . . 4  |-  .^  =  (.g
`  N )
7 eqid 2283 . . . 4  |-  ( Base `  P )  =  (
Base `  P )
81, 2, 3, 4, 5, 6, 7ply1tmcl 16348 . . 3  |-  ( ( R  e.  Ring  /\  C  e.  K  /\  D  e. 
NN0 )  ->  ( C  .x.  ( D  .^  X ) )  e.  ( Base `  P
) )
9 eqid 2283 . . . 4  |-  (coe1 `  ( C  .x.  ( D  .^  X ) ) )  =  (coe1 `  ( C  .x.  ( D  .^  X ) ) )
10 eqid 2283 . . . 4  |-  ( x  e.  NN0  |->  ( 1o 
X.  { x }
) )  =  ( x  e.  NN0  |->  ( 1o 
X.  { x }
) )
119, 7, 2, 10coe1fval2 16291 . . 3  |-  ( ( C  .x.  ( D 
.^  X ) )  e.  ( Base `  P
)  ->  (coe1 `  ( C  .x.  ( D  .^  X ) ) )  =  ( ( C 
.x.  ( D  .^  X ) )  o.  ( x  e.  NN0  |->  ( 1o  X.  { x } ) ) ) )
128, 11syl 15 . 2  |-  ( ( R  e.  Ring  /\  C  e.  K  /\  D  e. 
NN0 )  ->  (coe1 `  ( C  .x.  ( D 
.^  X ) ) )  =  ( ( C  .x.  ( D 
.^  X ) )  o.  ( x  e. 
NN0  |->  ( 1o  X.  { x } ) ) ) )
13 fconst6g 5430 . . . . 5  |-  ( x  e.  NN0  ->  ( 1o 
X.  { x }
) : 1o --> NN0 )
14 nn0ex 9971 . . . . . 6  |-  NN0  e.  _V
15 1on 6486 . . . . . . 7  |-  1o  e.  On
1615elexi 2797 . . . . . 6  |-  1o  e.  _V
1714, 16elmap 6796 . . . . 5  |-  ( ( 1o  X.  { x } )  e.  ( NN0  ^m  1o )  <-> 
( 1o  X.  {
x } ) : 1o --> NN0 )
1813, 17sylibr 203 . . . 4  |-  ( x  e.  NN0  ->  ( 1o 
X.  { x }
)  e.  ( NN0 
^m  1o ) )
1918adantl 452 . . 3  |-  ( ( ( R  e.  Ring  /\  C  e.  K  /\  D  e.  NN0 )  /\  x  e.  NN0 )  -> 
( 1o  X.  {
x } )  e.  ( NN0  ^m  1o ) )
20 eqidd 2284 . . 3  |-  ( ( R  e.  Ring  /\  C  e.  K  /\  D  e. 
NN0 )  ->  (
x  e.  NN0  |->  ( 1o 
X.  { x }
) )  =  ( x  e.  NN0  |->  ( 1o 
X.  { x }
) ) )
21 eqid 2283 . . . . . . . 8  |-  (.g `  (mulGrp `  ( 1o mPoly  R )
) )  =  (.g `  (mulGrp `  ( 1o mPoly  R ) ) )
225, 7mgpbas 15331 . . . . . . . . 9  |-  ( Base `  P )  =  (
Base `  N )
2322a1i 10 . . . . . . . 8  |-  ( R  e.  Ring  ->  ( Base `  P )  =  (
Base `  N )
)
24 eqid 2283 . . . . . . . . . 10  |-  (mulGrp `  ( 1o mPoly  R ) )  =  (mulGrp `  ( 1o mPoly  R ) )
25 eqid 2283 . . . . . . . . . . 11  |-  (PwSer1 `  R
)  =  (PwSer1 `  R
)
262, 25, 7ply1bas 16274 . . . . . . . . . 10  |-  ( Base `  P )  =  (
Base `  ( 1o mPoly  R ) )
2724, 26mgpbas 15331 . . . . . . . . 9  |-  ( Base `  P )  =  (
Base `  (mulGrp `  ( 1o mPoly  R ) ) )
2827a1i 10 . . . . . . . 8  |-  ( R  e.  Ring  ->  ( Base `  P )  =  (
Base `  (mulGrp `  ( 1o mPoly  R ) ) ) )
29 ssv 3198 . . . . . . . . 9  |-  ( Base `  P )  C_  _V
3029a1i 10 . . . . . . . 8  |-  ( R  e.  Ring  ->  ( Base `  P )  C_  _V )
31 ovex 5883 . . . . . . . . 9  |-  ( x ( +g  `  N
) y )  e. 
_V
3231a1i 10 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  (
x  e.  _V  /\  y  e.  _V )
)  ->  ( x
( +g  `  N ) y )  e.  _V )
33 eqid 2283 . . . . . . . . . . . 12  |-  ( .r
`  P )  =  ( .r `  P
)
345, 33mgpplusg 15329 . . . . . . . . . . 11  |-  ( .r
`  P )  =  ( +g  `  N
)
35 eqid 2283 . . . . . . . . . . . . 13  |-  ( 1o mPoly  R )  =  ( 1o mPoly  R )
362, 35, 33ply1mulr 16305 . . . . . . . . . . . 12  |-  ( .r
`  P )  =  ( .r `  ( 1o mPoly  R ) )
3724, 36mgpplusg 15329 . . . . . . . . . . 11  |-  ( .r
`  P )  =  ( +g  `  (mulGrp `  ( 1o mPoly  R )
) )
3834, 37eqtr3i 2305 . . . . . . . . . 10  |-  ( +g  `  N )  =  ( +g  `  (mulGrp `  ( 1o mPoly  R ) ) )
3938a1i 10 . . . . . . . . 9  |-  ( R  e.  Ring  ->  ( +g  `  N )  =  ( +g  `  (mulGrp `  ( 1o mPoly  R ) ) ) )
4039proplem3 13593 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  (
x  e.  _V  /\  y  e.  _V )
)  ->  ( x
( +g  `  N ) y )  =  ( x ( +g  `  (mulGrp `  ( 1o mPoly  R )
) ) y ) )
416, 21, 23, 28, 30, 32, 40mulgpropd 14600 . . . . . . 7  |-  ( R  e.  Ring  ->  .^  =  (.g
`  (mulGrp `  ( 1o mPoly  R ) ) ) )
42413ad2ant1 976 . . . . . 6  |-  ( ( R  e.  Ring  /\  C  e.  K  /\  D  e. 
NN0 )  ->  .^  =  (.g
`  (mulGrp `  ( 1o mPoly  R ) ) ) )
43 eqidd 2284 . . . . . 6  |-  ( ( R  e.  Ring  /\  C  e.  K  /\  D  e. 
NN0 )  ->  D  =  D )
443vr1val 16271 . . . . . . 7  |-  X  =  ( ( 1o mVar  R
) `  (/) )
4544a1i 10 . . . . . 6  |-  ( ( R  e.  Ring  /\  C  e.  K  /\  D  e. 
NN0 )  ->  X  =  ( ( 1o mVar  R ) `  (/) ) )
4642, 43, 45oveq123d 5879 . . . . 5  |-  ( ( R  e.  Ring  /\  C  e.  K  /\  D  e. 
NN0 )  ->  ( D  .^  X )  =  ( D (.g `  (mulGrp `  ( 1o mPoly  R )
) ) ( ( 1o mVar  R ) `  (/) ) ) )
4746oveq2d 5874 . . . 4  |-  ( ( R  e.  Ring  /\  C  e.  K  /\  D  e. 
NN0 )  ->  ( C  .x.  ( D  .^  X ) )  =  ( C  .x.  ( D (.g `  (mulGrp `  ( 1o mPoly  R ) ) ) ( ( 1o mVar  R
) `  (/) ) ) ) )
48 psr1baslem 16264 . . . . . 6  |-  ( NN0 
^m  1o )  =  { a  e.  ( NN0  ^m  1o )  |  ( `' a
" NN )  e. 
Fin }
49 coe1tm.z . . . . . 6  |-  .0.  =  ( 0g `  R )
50 eqid 2283 . . . . . 6  |-  ( 1r
`  R )  =  ( 1r `  R
)
5115a1i 10 . . . . . 6  |-  ( ( R  e.  Ring  /\  C  e.  K  /\  D  e. 
NN0 )  ->  1o  e.  On )
52 eqid 2283 . . . . . 6  |-  ( 1o mVar  R )  =  ( 1o mVar  R )
53 simp1 955 . . . . . 6  |-  ( ( R  e.  Ring  /\  C  e.  K  /\  D  e. 
NN0 )  ->  R  e.  Ring )
54 0lt1o 6503 . . . . . . 7  |-  (/)  e.  1o
5554a1i 10 . . . . . 6  |-  ( ( R  e.  Ring  /\  C  e.  K  /\  D  e. 
NN0 )  ->  (/)  e.  1o )
56 simp3 957 . . . . . 6  |-  ( ( R  e.  Ring  /\  C  e.  K  /\  D  e. 
NN0 )  ->  D  e.  NN0 )
5735, 48, 49, 50, 51, 24, 21, 52, 53, 55, 56mplcoe3 16210 . . . . 5  |-  ( ( R  e.  Ring  /\  C  e.  K  /\  D  e. 
NN0 )  ->  (
y  e.  ( NN0 
^m  1o )  |->  if ( y  =  ( b  e.  1o  |->  if ( b  =  (/) ,  D ,  0 ) ) ,  ( 1r
`  R ) ,  .0.  ) )  =  ( D (.g `  (mulGrp `  ( 1o mPoly  R )
) ) ( ( 1o mVar  R ) `  (/) ) ) )
5857oveq2d 5874 . . . 4  |-  ( ( R  e.  Ring  /\  C  e.  K  /\  D  e. 
NN0 )  ->  ( C  .x.  ( y  e.  ( NN0  ^m  1o )  |->  if ( y  =  ( b  e.  1o  |->  if ( b  =  (/) ,  D , 
0 ) ) ,  ( 1r `  R
) ,  .0.  )
) )  =  ( C  .x.  ( D (.g `  (mulGrp `  ( 1o mPoly  R ) ) ) ( ( 1o mVar  R
) `  (/) ) ) ) )
592, 35, 4ply1vsca 16304 . . . . 5  |-  .x.  =  ( .s `  ( 1o mPoly  R ) )
60 elsni 3664 . . . . . . . . . . 11  |-  ( b  e.  { (/) }  ->  b  =  (/) )
61 df1o2 6491 . . . . . . . . . . 11  |-  1o  =  { (/) }
6260, 61eleq2s 2375 . . . . . . . . . 10  |-  ( b  e.  1o  ->  b  =  (/) )
63 iftrue 3571 . . . . . . . . . 10  |-  ( b  =  (/)  ->  if ( b  =  (/) ,  D ,  0 )  =  D )
6462, 63syl 15 . . . . . . . . 9  |-  ( b  e.  1o  ->  if ( b  =  (/) ,  D ,  0 )  =  D )
6564adantl 452 . . . . . . . 8  |-  ( ( ( R  e.  Ring  /\  C  e.  K  /\  D  e.  NN0 )  /\  b  e.  1o )  ->  if ( b  =  (/) ,  D ,  0 )  =  D )
6665mpteq2dva 4106 . . . . . . 7  |-  ( ( R  e.  Ring  /\  C  e.  K  /\  D  e. 
NN0 )  ->  (
b  e.  1o  |->  if ( b  =  (/) ,  D ,  0 ) )  =  ( b  e.  1o  |->  D ) )
67 fconstmpt 4732 . . . . . . 7  |-  ( 1o 
X.  { D }
)  =  ( b  e.  1o  |->  D )
6866, 67syl6eqr 2333 . . . . . 6  |-  ( ( R  e.  Ring  /\  C  e.  K  /\  D  e. 
NN0 )  ->  (
b  e.  1o  |->  if ( b  =  (/) ,  D ,  0 ) )  =  ( 1o 
X.  { D }
) )
69 fconst6g 5430 . . . . . . . 8  |-  ( D  e.  NN0  ->  ( 1o 
X.  { D }
) : 1o --> NN0 )
7014, 16elmap 6796 . . . . . . . 8  |-  ( ( 1o  X.  { D } )  e.  ( NN0  ^m  1o )  <-> 
( 1o  X.  { D } ) : 1o --> NN0 )
7169, 70sylibr 203 . . . . . . 7  |-  ( D  e.  NN0  ->  ( 1o 
X.  { D }
)  e.  ( NN0 
^m  1o ) )
72713ad2ant3 978 . . . . . 6  |-  ( ( R  e.  Ring  /\  C  e.  K  /\  D  e. 
NN0 )  ->  ( 1o  X.  { D }
)  e.  ( NN0 
^m  1o ) )
7368, 72eqeltrd 2357 . . . . 5  |-  ( ( R  e.  Ring  /\  C  e.  K  /\  D  e. 
NN0 )  ->  (
b  e.  1o  |->  if ( b  =  (/) ,  D ,  0 ) )  e.  ( NN0 
^m  1o ) )
74 simp2 956 . . . . 5  |-  ( ( R  e.  Ring  /\  C  e.  K  /\  D  e. 
NN0 )  ->  C  e.  K )
7535, 59, 48, 50, 49, 1, 51, 53, 73, 74mplmon2 16234 . . . 4  |-  ( ( R  e.  Ring  /\  C  e.  K  /\  D  e. 
NN0 )  ->  ( C  .x.  ( y  e.  ( NN0  ^m  1o )  |->  if ( y  =  ( b  e.  1o  |->  if ( b  =  (/) ,  D , 
0 ) ) ,  ( 1r `  R
) ,  .0.  )
) )  =  ( y  e.  ( NN0 
^m  1o )  |->  if ( y  =  ( b  e.  1o  |->  if ( b  =  (/) ,  D ,  0 ) ) ,  C ,  .0.  ) ) )
7647, 58, 753eqtr2d 2321 . . 3  |-  ( ( R  e.  Ring  /\  C  e.  K  /\  D  e. 
NN0 )  ->  ( C  .x.  ( D  .^  X ) )  =  ( y  e.  ( NN0  ^m  1o ) 
|->  if ( y  =  ( b  e.  1o  |->  if ( b  =  (/) ,  D ,  0 ) ) ,  C ,  .0.  ) ) )
77 eqeq1 2289 . . . 4  |-  ( y  =  ( 1o  X.  { x } )  ->  ( y  =  ( b  e.  1o  |->  if ( b  =  (/) ,  D ,  0 ) )  <->  ( 1o  X.  { x } )  =  ( b  e.  1o  |->  if ( b  =  (/) ,  D , 
0 ) ) ) )
7877ifbid 3583 . . 3  |-  ( y  =  ( 1o  X.  { x } )  ->  if ( y  =  ( b  e.  1o  |->  if ( b  =  (/) ,  D , 
0 ) ) ,  C ,  .0.  )  =  if ( ( 1o 
X.  { x }
)  =  ( b  e.  1o  |->  if ( b  =  (/) ,  D ,  0 ) ) ,  C ,  .0.  ) )
7919, 20, 76, 78fmptco 5691 . 2  |-  ( ( R  e.  Ring  /\  C  e.  K  /\  D  e. 
NN0 )  ->  (
( C  .x.  ( D  .^  X ) )  o.  ( x  e. 
NN0  |->  ( 1o  X.  { x } ) ) )  =  ( x  e.  NN0  |->  if ( ( 1o  X.  {
x } )  =  ( b  e.  1o  |->  if ( b  =  (/) ,  D ,  0 ) ) ,  C ,  .0.  ) ) )
8068adantr 451 . . . . . 6  |-  ( ( ( R  e.  Ring  /\  C  e.  K  /\  D  e.  NN0 )  /\  x  e.  NN0 )  -> 
( b  e.  1o  |->  if ( b  =  (/) ,  D ,  0 ) )  =  ( 1o 
X.  { D }
) )
8180eqeq2d 2294 . . . . 5  |-  ( ( ( R  e.  Ring  /\  C  e.  K  /\  D  e.  NN0 )  /\  x  e.  NN0 )  -> 
( ( 1o  X.  { x } )  =  ( b  e.  1o  |->  if ( b  =  (/) ,  D , 
0 ) )  <->  ( 1o  X.  { x } )  =  ( 1o  X.  { D } ) ) )
82 fveq1 5524 . . . . . . 7  |-  ( ( 1o  X.  { x } )  =  ( 1o  X.  { D } )  ->  (
( 1o  X.  {
x } ) `  (/) )  =  ( ( 1o  X.  { D } ) `  (/) ) )
83 vex 2791 . . . . . . . . . 10  |-  x  e. 
_V
8483fvconst2 5729 . . . . . . . . 9  |-  ( (/)  e.  1o  ->  ( ( 1o  X.  { x }
) `  (/) )  =  x )
8554, 84mp1i 11 . . . . . . . 8  |-  ( ( ( R  e.  Ring  /\  C  e.  K  /\  D  e.  NN0 )  /\  x  e.  NN0 )  -> 
( ( 1o  X.  { x } ) `
 (/) )  =  x )
86 simpl3 960 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  C  e.  K  /\  D  e.  NN0 )  /\  x  e.  NN0 )  ->  D  e.  NN0 )
87 fvconst2g 5727 . . . . . . . . 9  |-  ( ( D  e.  NN0  /\  (/) 
e.  1o )  -> 
( ( 1o  X.  { D } ) `  (/) )  =  D )
8886, 54, 87sylancl 643 . . . . . . . 8  |-  ( ( ( R  e.  Ring  /\  C  e.  K  /\  D  e.  NN0 )  /\  x  e.  NN0 )  -> 
( ( 1o  X.  { D } ) `  (/) )  =  D )
8985, 88eqeq12d 2297 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  C  e.  K  /\  D  e.  NN0 )  /\  x  e.  NN0 )  -> 
( ( ( 1o 
X.  { x }
) `  (/) )  =  ( ( 1o  X.  { D } ) `  (/) )  <->  x  =  D
) )
9082, 89syl5ib 210 . . . . . 6  |-  ( ( ( R  e.  Ring  /\  C  e.  K  /\  D  e.  NN0 )  /\  x  e.  NN0 )  -> 
( ( 1o  X.  { x } )  =  ( 1o  X.  { D } )  ->  x  =  D )
)
91 sneq 3651 . . . . . . 7  |-  ( x  =  D  ->  { x }  =  { D } )
9291xpeq2d 4713 . . . . . 6  |-  ( x  =  D  ->  ( 1o  X.  { x }
)  =  ( 1o 
X.  { D }
) )
9390, 92impbid1 194 . . . . 5  |-  ( ( ( R  e.  Ring  /\  C  e.  K  /\  D  e.  NN0 )  /\  x  e.  NN0 )  -> 
( ( 1o  X.  { x } )  =  ( 1o  X.  { D } )  <->  x  =  D ) )
9481, 93bitrd 244 . . . 4  |-  ( ( ( R  e.  Ring  /\  C  e.  K  /\  D  e.  NN0 )  /\  x  e.  NN0 )  -> 
( ( 1o  X.  { x } )  =  ( b  e.  1o  |->  if ( b  =  (/) ,  D , 
0 ) )  <->  x  =  D ) )
9594ifbid 3583 . . 3  |-  ( ( ( R  e.  Ring  /\  C  e.  K  /\  D  e.  NN0 )  /\  x  e.  NN0 )  ->  if ( ( 1o  X.  { x } )  =  ( b  e.  1o  |->  if ( b  =  (/) ,  D , 
0 ) ) ,  C ,  .0.  )  =  if ( x  =  D ,  C ,  .0.  ) )
9695mpteq2dva 4106 . 2  |-  ( ( R  e.  Ring  /\  C  e.  K  /\  D  e. 
NN0 )  ->  (
x  e.  NN0  |->  if ( ( 1o  X.  {
x } )  =  ( b  e.  1o  |->  if ( b  =  (/) ,  D ,  0 ) ) ,  C ,  .0.  ) )  =  ( x  e.  NN0  |->  if ( x  =  D ,  C ,  .0.  )
) )
9712, 79, 963eqtrd 2319 1  |-  ( ( R  e.  Ring  /\  C  e.  K  /\  D  e. 
NN0 )  ->  (coe1 `  ( C  .x.  ( D 
.^  X ) ) )  =  ( x  e.  NN0  |->  if ( x  =  D ,  C ,  .0.  )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   _Vcvv 2788    C_ wss 3152   (/)c0 3455   ifcif 3565   {csn 3640    e. cmpt 4077   Oncon0 4392    X. cxp 4687    o. ccom 4693   -->wf 5251   ` cfv 5255  (class class class)co 5858   1oc1o 6472    ^m cmap 6772   0cc0 8737   NN0cn0 9965   Basecbs 13148   +g cplusg 13208   .rcmulr 13209   .scvsca 13212   0gc0g 13400  .gcmg 14366  mulGrpcmgp 15325   Ringcrg 15337   1rcur 15339   mVar cmvr 16088   mPoly cmpl 16089  PwSer1cps1 16250  var1cv1 16251  Poly1cpl1 16252  coe1cco1 16255
This theorem is referenced by:  coe1tmfv1  16350  coe1tmfv2  16351  coe1scl  16362
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-inf2 7342  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-iin 3908  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-se 4353  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-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-ofr 6079  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-oi 7225  df-card 7572  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-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-fzo 10871  df-seq 11047  df-hash 11338  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-0g 13404  df-gsum 13405  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-mhm 14415  df-submnd 14416  df-grp 14489  df-minusg 14490  df-sbg 14491  df-mulg 14492  df-subg 14618  df-ghm 14681  df-cntz 14793  df-cmn 15091  df-abl 15092  df-mgp 15326  df-rng 15340  df-ur 15342  df-subrg 15543  df-lmod 15629  df-lss 15690  df-psr 16098  df-mvr 16099  df-mpl 16100  df-opsr 16106  df-psr1 16257  df-vr1 16258  df-ply1 16259  df-coe1 16262
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