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Theorem coe1tm 16365
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 2296 . . . 4  |-  ( Base `  P )  =  (
Base `  P )
81, 2, 3, 4, 5, 6, 7ply1tmcl 16364 . . 3  |-  ( ( R  e.  Ring  /\  C  e.  K  /\  D  e. 
NN0 )  ->  ( C  .x.  ( D  .^  X ) )  e.  ( Base `  P
) )
9 eqid 2296 . . . 4  |-  (coe1 `  ( C  .x.  ( D  .^  X ) ) )  =  (coe1 `  ( C  .x.  ( D  .^  X ) ) )
10 eqid 2296 . . . 4  |-  ( x  e.  NN0  |->  ( 1o 
X.  { x }
) )  =  ( x  e.  NN0  |->  ( 1o 
X.  { x }
) )
119, 7, 2, 10coe1fval2 16307 . . 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 5446 . . . . 5  |-  ( x  e.  NN0  ->  ( 1o 
X.  { x }
) : 1o --> NN0 )
14 nn0ex 9987 . . . . . 6  |-  NN0  e.  _V
15 1on 6502 . . . . . . 7  |-  1o  e.  On
1615elexi 2810 . . . . . 6  |-  1o  e.  _V
1714, 16elmap 6812 . . . . 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 2297 . . 3  |-  ( ( R  e.  Ring  /\  C  e.  K  /\  D  e. 
NN0 )  ->  (
x  e.  NN0  |->  ( 1o 
X.  { x }
) )  =  ( x  e.  NN0  |->  ( 1o 
X.  { x }
) ) )
21 eqid 2296 . . . . . . . 8  |-  (.g `  (mulGrp `  ( 1o mPoly  R )
) )  =  (.g `  (mulGrp `  ( 1o mPoly  R ) ) )
225, 7mgpbas 15347 . . . . . . . . 9  |-  ( Base `  P )  =  (
Base `  N )
2322a1i 10 . . . . . . . 8  |-  ( R  e.  Ring  ->  ( Base `  P )  =  (
Base `  N )
)
24 eqid 2296 . . . . . . . . . 10  |-  (mulGrp `  ( 1o mPoly  R ) )  =  (mulGrp `  ( 1o mPoly  R ) )
25 eqid 2296 . . . . . . . . . . 11  |-  (PwSer1 `  R
)  =  (PwSer1 `  R
)
262, 25, 7ply1bas 16290 . . . . . . . . . 10  |-  ( Base `  P )  =  (
Base `  ( 1o mPoly  R ) )
2724, 26mgpbas 15347 . . . . . . . . 9  |-  ( Base `  P )  =  (
Base `  (mulGrp `  ( 1o mPoly  R ) ) )
2827a1i 10 . . . . . . . 8  |-  ( R  e.  Ring  ->  ( Base `  P )  =  (
Base `  (mulGrp `  ( 1o mPoly  R ) ) ) )
29 ssv 3211 . . . . . . . . 9  |-  ( Base `  P )  C_  _V
3029a1i 10 . . . . . . . 8  |-  ( R  e.  Ring  ->  ( Base `  P )  C_  _V )
31 ovex 5899 . . . . . . . . 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 2296 . . . . . . . . . . . 12  |-  ( .r
`  P )  =  ( .r `  P
)
345, 33mgpplusg 15345 . . . . . . . . . . 11  |-  ( .r
`  P )  =  ( +g  `  N
)
35 eqid 2296 . . . . . . . . . . . . 13  |-  ( 1o mPoly  R )  =  ( 1o mPoly  R )
362, 35, 33ply1mulr 16321 . . . . . . . . . . . 12  |-  ( .r
`  P )  =  ( .r `  ( 1o mPoly  R ) )
3724, 36mgpplusg 15345 . . . . . . . . . . 11  |-  ( .r
`  P )  =  ( +g  `  (mulGrp `  ( 1o mPoly  R )
) )
3834, 37eqtr3i 2318 . . . . . . . . . 10  |-  ( +g  `  N )  =  ( +g  `  (mulGrp `  ( 1o mPoly  R ) ) )
3938a1i 10 . . . . . . . . 9  |-  ( R  e.  Ring  ->  ( +g  `  N )  =  ( +g  `  (mulGrp `  ( 1o mPoly  R ) ) ) )
4039proplem3 13609 . . . . . . . 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 14616 . . . . . . 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 2297 . . . . . 6  |-  ( ( R  e.  Ring  /\  C  e.  K  /\  D  e. 
NN0 )  ->  D  =  D )
443vr1val 16287 . . . . . . 7  |-  X  =  ( ( 1o mVar  R
) `  (/) )
4544a1i 10 . . . . . 6  |-  ( ( R  e.  Ring  /\  C  e.  K  /\  D  e. 
NN0 )  ->  X  =  ( ( 1o mVar  R ) `  (/) ) )
4642, 43, 45oveq123d 5895 . . . . 5  |-  ( ( R  e.  Ring  /\  C  e.  K  /\  D  e. 
NN0 )  ->  ( D  .^  X )  =  ( D (.g `  (mulGrp `  ( 1o mPoly  R )
) ) ( ( 1o mVar  R ) `  (/) ) ) )
4746oveq2d 5890 . . . 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 16280 . . . . . 6  |-  ( NN0 
^m  1o )  =  { a  e.  ( NN0  ^m  1o )  |  ( `' a
" NN )  e. 
Fin }
49 coe1tm.z . . . . . 6  |-  .0.  =  ( 0g `  R )
50 eqid 2296 . . . . . 6  |-  ( 1r
`  R )  =  ( 1r `  R
)
5115a1i 10 . . . . . 6  |-  ( ( R  e.  Ring  /\  C  e.  K  /\  D  e. 
NN0 )  ->  1o  e.  On )
52 eqid 2296 . . . . . 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 6519 . . . . . . 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 16226 . . . . 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 5890 . . . 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 16320 . . . . 5  |-  .x.  =  ( .s `  ( 1o mPoly  R ) )
60 elsni 3677 . . . . . . . . . . 11  |-  ( b  e.  { (/) }  ->  b  =  (/) )
61 df1o2 6507 . . . . . . . . . . 11  |-  1o  =  { (/) }
6260, 61eleq2s 2388 . . . . . . . . . 10  |-  ( b  e.  1o  ->  b  =  (/) )
63 iftrue 3584 . . . . . . . . . 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 4122 . . . . . . 7  |-  ( ( R  e.  Ring  /\  C  e.  K  /\  D  e. 
NN0 )  ->  (
b  e.  1o  |->  if ( b  =  (/) ,  D ,  0 ) )  =  ( b  e.  1o  |->  D ) )
67 fconstmpt 4748 . . . . . . 7  |-  ( 1o 
X.  { D }
)  =  ( b  e.  1o  |->  D )
6866, 67syl6eqr 2346 . . . . . 6  |-  ( ( R  e.  Ring  /\  C  e.  K  /\  D  e. 
NN0 )  ->  (
b  e.  1o  |->  if ( b  =  (/) ,  D ,  0 ) )  =  ( 1o 
X.  { D }
) )
69 fconst6g 5446 . . . . . . . 8  |-  ( D  e.  NN0  ->  ( 1o 
X.  { D }
) : 1o --> NN0 )
7014, 16elmap 6812 . . . . . . . 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 2370 . . . . 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 16250 . . . 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 2334 . . 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 2302 . . . 4  |-  ( y  =  ( 1o  X.  { x } )  ->  ( y  =  ( b  e.  1o  |->  if ( b  =  (/) ,  D ,  0 ) )  <->  ( 1o  X.  { x } )  =  ( b  e.  1o  |->  if ( b  =  (/) ,  D , 
0 ) ) ) )
7877ifbid 3596 . . 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 5707 . 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 2307 . . . . 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 5540 . . . . . . 7  |-  ( ( 1o  X.  { x } )  =  ( 1o  X.  { D } )  ->  (
( 1o  X.  {
x } ) `  (/) )  =  ( ( 1o  X.  { D } ) `  (/) ) )
83 vex 2804 . . . . . . . . . 10  |-  x  e. 
_V
8483fvconst2 5745 . . . . . . . . 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 5743 . . . . . . . . 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 2310 . . . . . . 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 3664 . . . . . . 7  |-  ( x  =  D  ->  { x }  =  { D } )
9291xpeq2d 4729 . . . . . 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 3596 . . 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 4122 . 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 2332 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 1632    e. wcel 1696   _Vcvv 2801    C_ wss 3165   (/)c0 3468   ifcif 3578   {csn 3653    e. cmpt 4093   Oncon0 4408    X. cxp 4703    o. ccom 4709   -->wf 5267   ` cfv 5271  (class class class)co 5874   1oc1o 6488    ^m cmap 6788   0cc0 8753   NN0cn0 9981   Basecbs 13164   +g cplusg 13224   .rcmulr 13225   .scvsca 13228   0gc0g 13416  .gcmg 14382  mulGrpcmgp 15341   Ringcrg 15353   1rcur 15355   mVar cmvr 16104   mPoly cmpl 16105  PwSer1cps1 16266  var1cv1 16267  Poly1cpl1 16268  coe1cco1 16271
This theorem is referenced by:  coe1tmfv1  16366  coe1tmfv2  16367  coe1scl  16378
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-ofr 6095  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-oi 7241  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-fzo 10887  df-seq 11063  df-hash 11354  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-0g 13420  df-gsum 13421  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-mhm 14431  df-submnd 14432  df-grp 14505  df-minusg 14506  df-sbg 14507  df-mulg 14508  df-subg 14634  df-ghm 14697  df-cntz 14809  df-cmn 15107  df-abl 15108  df-mgp 15342  df-rng 15356  df-ur 15358  df-subrg 15559  df-lmod 15645  df-lss 15706  df-psr 16114  df-mvr 16115  df-mpl 16116  df-opsr 16122  df-psr1 16273  df-vr1 16274  df-ply1 16275  df-coe1 16278
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