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Theorem coe1tm 16657
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 2435 . . . 4  |-  ( Base `  P )  =  (
Base `  P )
81, 2, 3, 4, 5, 6, 7ply1tmcl 16656 . . 3  |-  ( ( R  e.  Ring  /\  C  e.  K  /\  D  e. 
NN0 )  ->  ( C  .x.  ( D  .^  X ) )  e.  ( Base `  P
) )
9 eqid 2435 . . . 4  |-  (coe1 `  ( C  .x.  ( D  .^  X ) ) )  =  (coe1 `  ( C  .x.  ( D  .^  X ) ) )
10 eqid 2435 . . . 4  |-  ( x  e.  NN0  |->  ( 1o 
X.  { x }
) )  =  ( x  e.  NN0  |->  ( 1o 
X.  { x }
) )
119, 7, 2, 10coe1fval2 16600 . . 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 16 . 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 5624 . . . . 5  |-  ( x  e.  NN0  ->  ( 1o 
X.  { x }
) : 1o --> NN0 )
14 nn0ex 10219 . . . . . 6  |-  NN0  e.  _V
15 1on 6723 . . . . . . 7  |-  1o  e.  On
1615elexi 2957 . . . . . 6  |-  1o  e.  _V
1714, 16elmap 7034 . . . . 5  |-  ( ( 1o  X.  { x } )  e.  ( NN0  ^m  1o )  <-> 
( 1o  X.  {
x } ) : 1o --> NN0 )
1813, 17sylibr 204 . . . 4  |-  ( x  e.  NN0  ->  ( 1o 
X.  { x }
)  e.  ( NN0 
^m  1o ) )
1918adantl 453 . . 3  |-  ( ( ( R  e.  Ring  /\  C  e.  K  /\  D  e.  NN0 )  /\  x  e.  NN0 )  -> 
( 1o  X.  {
x } )  e.  ( NN0  ^m  1o ) )
20 eqidd 2436 . . 3  |-  ( ( R  e.  Ring  /\  C  e.  K  /\  D  e. 
NN0 )  ->  (
x  e.  NN0  |->  ( 1o 
X.  { x }
) )  =  ( x  e.  NN0  |->  ( 1o 
X.  { x }
) ) )
21 eqid 2435 . . . . . . . 8  |-  (.g `  (mulGrp `  ( 1o mPoly  R )
) )  =  (.g `  (mulGrp `  ( 1o mPoly  R ) ) )
225, 7mgpbas 15646 . . . . . . . . 9  |-  ( Base `  P )  =  (
Base `  N )
2322a1i 11 . . . . . . . 8  |-  ( R  e.  Ring  ->  ( Base `  P )  =  (
Base `  N )
)
24 eqid 2435 . . . . . . . . . 10  |-  (mulGrp `  ( 1o mPoly  R ) )  =  (mulGrp `  ( 1o mPoly  R ) )
25 eqid 2435 . . . . . . . . . . 11  |-  (PwSer1 `  R
)  =  (PwSer1 `  R
)
262, 25, 7ply1bas 16585 . . . . . . . . . 10  |-  ( Base `  P )  =  (
Base `  ( 1o mPoly  R ) )
2724, 26mgpbas 15646 . . . . . . . . 9  |-  ( Base `  P )  =  (
Base `  (mulGrp `  ( 1o mPoly  R ) ) )
2827a1i 11 . . . . . . . 8  |-  ( R  e.  Ring  ->  ( Base `  P )  =  (
Base `  (mulGrp `  ( 1o mPoly  R ) ) ) )
29 ssv 3360 . . . . . . . . 9  |-  ( Base `  P )  C_  _V
3029a1i 11 . . . . . . . 8  |-  ( R  e.  Ring  ->  ( Base `  P )  C_  _V )
31 ovex 6098 . . . . . . . . 9  |-  ( x ( +g  `  N
) y )  e. 
_V
3231a1i 11 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  (
x  e.  _V  /\  y  e.  _V )
)  ->  ( x
( +g  `  N ) y )  e.  _V )
33 eqid 2435 . . . . . . . . . . . 12  |-  ( .r
`  P )  =  ( .r `  P
)
345, 33mgpplusg 15644 . . . . . . . . . . 11  |-  ( .r
`  P )  =  ( +g  `  N
)
35 eqid 2435 . . . . . . . . . . . . 13  |-  ( 1o mPoly  R )  =  ( 1o mPoly  R )
362, 35, 33ply1mulr 16613 . . . . . . . . . . . 12  |-  ( .r
`  P )  =  ( .r `  ( 1o mPoly  R ) )
3724, 36mgpplusg 15644 . . . . . . . . . . 11  |-  ( .r
`  P )  =  ( +g  `  (mulGrp `  ( 1o mPoly  R )
) )
3834, 37eqtr3i 2457 . . . . . . . . . 10  |-  ( +g  `  N )  =  ( +g  `  (mulGrp `  ( 1o mPoly  R ) ) )
3938a1i 11 . . . . . . . . 9  |-  ( R  e.  Ring  ->  ( +g  `  N )  =  ( +g  `  (mulGrp `  ( 1o mPoly  R ) ) ) )
4039proplem3 13908 . . . . . . . 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 14915 . . . . . . 7  |-  ( R  e.  Ring  ->  .^  =  (.g
`  (mulGrp `  ( 1o mPoly  R ) ) ) )
42413ad2ant1 978 . . . . . 6  |-  ( ( R  e.  Ring  /\  C  e.  K  /\  D  e. 
NN0 )  ->  .^  =  (.g
`  (mulGrp `  ( 1o mPoly  R ) ) ) )
43 eqidd 2436 . . . . . 6  |-  ( ( R  e.  Ring  /\  C  e.  K  /\  D  e. 
NN0 )  ->  D  =  D )
443vr1val 16582 . . . . . . 7  |-  X  =  ( ( 1o mVar  R
) `  (/) )
4544a1i 11 . . . . . 6  |-  ( ( R  e.  Ring  /\  C  e.  K  /\  D  e. 
NN0 )  ->  X  =  ( ( 1o mVar  R ) `  (/) ) )
4642, 43, 45oveq123d 6094 . . . . 5  |-  ( ( R  e.  Ring  /\  C  e.  K  /\  D  e. 
NN0 )  ->  ( D  .^  X )  =  ( D (.g `  (mulGrp `  ( 1o mPoly  R )
) ) ( ( 1o mVar  R ) `  (/) ) ) )
4746oveq2d 6089 . . . 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 16575 . . . . . 6  |-  ( NN0 
^m  1o )  =  { a  e.  ( NN0  ^m  1o )  |  ( `' a
" NN )  e. 
Fin }
49 coe1tm.z . . . . . 6  |-  .0.  =  ( 0g `  R )
50 eqid 2435 . . . . . 6  |-  ( 1r
`  R )  =  ( 1r `  R
)
5115a1i 11 . . . . . 6  |-  ( ( R  e.  Ring  /\  C  e.  K  /\  D  e. 
NN0 )  ->  1o  e.  On )
52 eqid 2435 . . . . . 6  |-  ( 1o mVar  R )  =  ( 1o mVar  R )
53 simp1 957 . . . . . 6  |-  ( ( R  e.  Ring  /\  C  e.  K  /\  D  e. 
NN0 )  ->  R  e.  Ring )
54 0lt1o 6740 . . . . . . 7  |-  (/)  e.  1o
5554a1i 11 . . . . . 6  |-  ( ( R  e.  Ring  /\  C  e.  K  /\  D  e. 
NN0 )  ->  (/)  e.  1o )
56 simp3 959 . . . . . 6  |-  ( ( R  e.  Ring  /\  C  e.  K  /\  D  e. 
NN0 )  ->  D  e.  NN0 )
5735, 48, 49, 50, 51, 24, 21, 52, 53, 55, 56mplcoe3 16521 . . . . 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 6089 . . . 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 16612 . . . . 5  |-  .x.  =  ( .s `  ( 1o mPoly  R ) )
60 elsni 3830 . . . . . . . . . . 11  |-  ( b  e.  { (/) }  ->  b  =  (/) )
61 df1o2 6728 . . . . . . . . . . 11  |-  1o  =  { (/) }
6260, 61eleq2s 2527 . . . . . . . . . 10  |-  ( b  e.  1o  ->  b  =  (/) )
63 iftrue 3737 . . . . . . . . . 10  |-  ( b  =  (/)  ->  if ( b  =  (/) ,  D ,  0 )  =  D )
6462, 63syl 16 . . . . . . . . 9  |-  ( b  e.  1o  ->  if ( b  =  (/) ,  D ,  0 )  =  D )
6564adantl 453 . . . . . . . 8  |-  ( ( ( R  e.  Ring  /\  C  e.  K  /\  D  e.  NN0 )  /\  b  e.  1o )  ->  if ( b  =  (/) ,  D ,  0 )  =  D )
6665mpteq2dva 4287 . . . . . . 7  |-  ( ( R  e.  Ring  /\  C  e.  K  /\  D  e. 
NN0 )  ->  (
b  e.  1o  |->  if ( b  =  (/) ,  D ,  0 ) )  =  ( b  e.  1o  |->  D ) )
67 fconstmpt 4913 . . . . . . 7  |-  ( 1o 
X.  { D }
)  =  ( b  e.  1o  |->  D )
6866, 67syl6eqr 2485 . . . . . 6  |-  ( ( R  e.  Ring  /\  C  e.  K  /\  D  e. 
NN0 )  ->  (
b  e.  1o  |->  if ( b  =  (/) ,  D ,  0 ) )  =  ( 1o 
X.  { D }
) )
69 fconst6g 5624 . . . . . . . 8  |-  ( D  e.  NN0  ->  ( 1o 
X.  { D }
) : 1o --> NN0 )
7014, 16elmap 7034 . . . . . . . 8  |-  ( ( 1o  X.  { D } )  e.  ( NN0  ^m  1o )  <-> 
( 1o  X.  { D } ) : 1o --> NN0 )
7169, 70sylibr 204 . . . . . . 7  |-  ( D  e.  NN0  ->  ( 1o 
X.  { D }
)  e.  ( NN0 
^m  1o ) )
72713ad2ant3 980 . . . . . 6  |-  ( ( R  e.  Ring  /\  C  e.  K  /\  D  e. 
NN0 )  ->  ( 1o  X.  { D }
)  e.  ( NN0 
^m  1o ) )
7368, 72eqeltrd 2509 . . . . 5  |-  ( ( R  e.  Ring  /\  C  e.  K  /\  D  e. 
NN0 )  ->  (
b  e.  1o  |->  if ( b  =  (/) ,  D ,  0 ) )  e.  ( NN0 
^m  1o ) )
74 simp2 958 . . . . 5  |-  ( ( R  e.  Ring  /\  C  e.  K  /\  D  e. 
NN0 )  ->  C  e.  K )
7535, 59, 48, 50, 49, 1, 51, 53, 73, 74mplmon2 16545 . . . 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 2473 . . 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 2441 . . . 4  |-  ( y  =  ( 1o  X.  { x } )  ->  ( y  =  ( b  e.  1o  |->  if ( b  =  (/) ,  D ,  0 ) )  <->  ( 1o  X.  { x } )  =  ( b  e.  1o  |->  if ( b  =  (/) ,  D , 
0 ) ) ) )
7877ifbid 3749 . . 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 5893 . 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 452 . . . . . 6  |-  ( ( ( R  e.  Ring  /\  C  e.  K  /\  D  e.  NN0 )  /\  x  e.  NN0 )  -> 
( b  e.  1o  |->  if ( b  =  (/) ,  D ,  0 ) )  =  ( 1o 
X.  { D }
) )
8180eqeq2d 2446 . . . . 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 5719 . . . . . . 7  |-  ( ( 1o  X.  { x } )  =  ( 1o  X.  { D } )  ->  (
( 1o  X.  {
x } ) `  (/) )  =  ( ( 1o  X.  { D } ) `  (/) ) )
83 vex 2951 . . . . . . . . . 10  |-  x  e. 
_V
8483fvconst2 5939 . . . . . . . . 9  |-  ( (/)  e.  1o  ->  ( ( 1o  X.  { x }
) `  (/) )  =  x )
8554, 84mp1i 12 . . . . . . . 8  |-  ( ( ( R  e.  Ring  /\  C  e.  K  /\  D  e.  NN0 )  /\  x  e.  NN0 )  -> 
( ( 1o  X.  { x } ) `
 (/) )  =  x )
86 simpl3 962 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  C  e.  K  /\  D  e.  NN0 )  /\  x  e.  NN0 )  ->  D  e.  NN0 )
87 fvconst2g 5937 . . . . . . . . 9  |-  ( ( D  e.  NN0  /\  (/) 
e.  1o )  -> 
( ( 1o  X.  { D } ) `  (/) )  =  D )
8886, 54, 87sylancl 644 . . . . . . . 8  |-  ( ( ( R  e.  Ring  /\  C  e.  K  /\  D  e.  NN0 )  /\  x  e.  NN0 )  -> 
( ( 1o  X.  { D } ) `  (/) )  =  D )
8985, 88eqeq12d 2449 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  C  e.  K  /\  D  e.  NN0 )  /\  x  e.  NN0 )  -> 
( ( ( 1o 
X.  { x }
) `  (/) )  =  ( ( 1o  X.  { D } ) `  (/) )  <->  x  =  D
) )
9082, 89syl5ib 211 . . . . . 6  |-  ( ( ( R  e.  Ring  /\  C  e.  K  /\  D  e.  NN0 )  /\  x  e.  NN0 )  -> 
( ( 1o  X.  { x } )  =  ( 1o  X.  { D } )  ->  x  =  D )
)
91 sneq 3817 . . . . . . 7  |-  ( x  =  D  ->  { x }  =  { D } )
9291xpeq2d 4894 . . . . . 6  |-  ( x  =  D  ->  ( 1o  X.  { x }
)  =  ( 1o 
X.  { D }
) )
9390, 92impbid1 195 . . . . 5  |-  ( ( ( R  e.  Ring  /\  C  e.  K  /\  D  e.  NN0 )  /\  x  e.  NN0 )  -> 
( ( 1o  X.  { x } )  =  ( 1o  X.  { D } )  <->  x  =  D ) )
9481, 93bitrd 245 . . . 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 3749 . . 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 4287 . 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 2471 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 359    /\ w3a 936    = wceq 1652    e. wcel 1725   _Vcvv 2948    C_ wss 3312   (/)c0 3620   ifcif 3731   {csn 3806    e. cmpt 4258   Oncon0 4573    X. cxp 4868    o. ccom 4874   -->wf 5442   ` cfv 5446  (class class class)co 6073   1oc1o 6709    ^m cmap 7010   0cc0 8982   NN0cn0 10213   Basecbs 13461   +g cplusg 13521   .rcmulr 13522   .scvsca 13525   0gc0g 13715  .gcmg 14681  mulGrpcmgp 15640   Ringcrg 15652   1rcur 15654   mVar cmvr 16399   mPoly cmpl 16400  PwSer1cps1 16561  var1cv1 16562  Poly1cpl1 16563  coe1cco1 16566
This theorem is referenced by:  coe1tmfv1  16658  coe1tmfv2  16659  coe1scl  16670
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-ofr 6298  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-2o 6717  df-oadd 6720  df-er 6897  df-map 7012  df-pm 7013  df-ixp 7056  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-oi 7471  df-card 7818  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-7 10055  df-8 10056  df-9 10057  df-10 10058  df-n0 10214  df-z 10275  df-uz 10481  df-fz 11036  df-fzo 11128  df-seq 11316  df-hash 11611  df-struct 13463  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-mulr 13535  df-sca 13537  df-vsca 13538  df-tset 13540  df-ple 13541  df-0g 13719  df-gsum 13720  df-mre 13803  df-mrc 13804  df-acs 13806  df-mnd 14682  df-mhm 14730  df-submnd 14731  df-grp 14804  df-minusg 14805  df-sbg 14806  df-mulg 14807  df-subg 14933  df-ghm 14996  df-cntz 15108  df-cmn 15406  df-abl 15407  df-mgp 15641  df-rng 15655  df-ur 15657  df-subrg 15858  df-lmod 15944  df-lss 16001  df-psr 16409  df-mvr 16410  df-mpl 16411  df-opsr 16417  df-psr1 16568  df-vr1 16569  df-ply1 16570  df-coe1 16573
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