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Theorem mdegle0 20005
Description: A polynomial has nonpositive degree iff it is a constant. (Contributed by Stefan O'Rear, 29-Mar-2015.)
Hypotheses
Ref Expression
mdegaddle.y  |-  Y  =  ( I mPoly  R )
mdegaddle.d  |-  D  =  ( I mDeg  R )
mdegaddle.i  |-  ( ph  ->  I  e.  V )
mdegaddle.r  |-  ( ph  ->  R  e.  Ring )
mdegle0.b  |-  B  =  ( Base `  Y
)
mdegle0.a  |-  A  =  (algSc `  Y )
mdegle0.f  |-  ( ph  ->  F  e.  B )
Assertion
Ref Expression
mdegle0  |-  ( ph  ->  ( ( D `  F )  <_  0  <->  F  =  ( A `  ( F `  ( I  X.  { 0 } ) ) ) ) )

Proof of Theorem mdegle0
Dummy variables  x  a  b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mdegle0.f . . 3  |-  ( ph  ->  F  e.  B )
2 0xr 9136 . . 3  |-  0  e.  RR*
3 mdegaddle.d . . . 4  |-  D  =  ( I mDeg  R )
4 mdegaddle.y . . . 4  |-  Y  =  ( I mPoly  R )
5 mdegle0.b . . . 4  |-  B  =  ( Base `  Y
)
6 eqid 2438 . . . 4  |-  ( 0g
`  R )  =  ( 0g `  R
)
7 eqid 2438 . . . 4  |-  { a  e.  ( NN0  ^m  I )  |  ( `' a " NN )  e.  Fin }  =  { a  e.  ( NN0  ^m  I )  |  ( `' a
" NN )  e. 
Fin }
8 eqid 2438 . . . 4  |-  ( b  e.  { a  e.  ( NN0  ^m  I
)  |  ( `' a " NN )  e.  Fin }  |->  (fld  gsumg  b ) )  =  ( b  e.  { a  e.  ( NN0  ^m  I
)  |  ( `' a " NN )  e.  Fin }  |->  (fld  gsumg  b ) )
93, 4, 5, 6, 7, 8mdegleb 19992 . . 3  |-  ( ( F  e.  B  /\  0  e.  RR* )  -> 
( ( D `  F )  <_  0  <->  A. x  e.  { a  e.  ( NN0  ^m  I )  |  ( `' a " NN )  e.  Fin }  (
0  <  ( (
b  e.  { a  e.  ( NN0  ^m  I )  |  ( `' a " NN )  e.  Fin }  |->  (fld  gsumg  b ) ) `  x )  ->  ( F `  x )  =  ( 0g `  R ) ) ) )
101, 2, 9sylancl 645 . 2  |-  ( ph  ->  ( ( D `  F )  <_  0  <->  A. x  e.  { a  e.  ( NN0  ^m  I )  |  ( `' a " NN )  e.  Fin }  (
0  <  ( (
b  e.  { a  e.  ( NN0  ^m  I )  |  ( `' a " NN )  e.  Fin }  |->  (fld  gsumg  b ) ) `  x )  ->  ( F `  x )  =  ( 0g `  R ) ) ) )
11 mdegaddle.i . . . . . . . . . 10  |-  ( ph  ->  I  e.  V )
127, 8tdeglem1 19986 . . . . . . . . . 10  |-  ( I  e.  V  ->  (
b  e.  { a  e.  ( NN0  ^m  I )  |  ( `' a " NN )  e.  Fin }  |->  (fld  gsumg  b ) ) : { a  e.  ( NN0  ^m  I )  |  ( `' a " NN )  e.  Fin } --> NN0 )
1311, 12syl 16 . . . . . . . . 9  |-  ( ph  ->  ( b  e.  {
a  e.  ( NN0 
^m  I )  |  ( `' a " NN )  e.  Fin } 
|->  (fld 
gsumg  b ) ) : { a  e.  ( NN0  ^m  I )  |  ( `' a
" NN )  e. 
Fin } --> NN0 )
1413ffvelrnda 5873 . . . . . . . 8  |-  ( (
ph  /\  x  e.  { a  e.  ( NN0 
^m  I )  |  ( `' a " NN )  e.  Fin } )  ->  ( (
b  e.  { a  e.  ( NN0  ^m  I )  |  ( `' a " NN )  e.  Fin }  |->  (fld  gsumg  b ) ) `  x )  e.  NN0 )
15 nn0re 10235 . . . . . . . . 9  |-  ( ( ( b  e.  {
a  e.  ( NN0 
^m  I )  |  ( `' a " NN )  e.  Fin } 
|->  (fld 
gsumg  b ) ) `  x )  e.  NN0  ->  ( ( b  e. 
{ a  e.  ( NN0  ^m  I )  |  ( `' a
" NN )  e. 
Fin }  |->  (fld  gsumg  b ) ) `  x )  e.  RR )
16 nn0ge0 10252 . . . . . . . . 9  |-  ( ( ( b  e.  {
a  e.  ( NN0 
^m  I )  |  ( `' a " NN )  e.  Fin } 
|->  (fld 
gsumg  b ) ) `  x )  e.  NN0  ->  0  <_  ( (
b  e.  { a  e.  ( NN0  ^m  I )  |  ( `' a " NN )  e.  Fin }  |->  (fld  gsumg  b ) ) `  x ) )
1715, 16jca 520 . . . . . . . 8  |-  ( ( ( b  e.  {
a  e.  ( NN0 
^m  I )  |  ( `' a " NN )  e.  Fin } 
|->  (fld 
gsumg  b ) ) `  x )  e.  NN0  ->  ( ( ( b  e.  { a  e.  ( NN0  ^m  I
)  |  ( `' a " NN )  e.  Fin }  |->  (fld  gsumg  b ) ) `  x )  e.  RR  /\  0  <_  ( ( b  e. 
{ a  e.  ( NN0  ^m  I )  |  ( `' a
" NN )  e. 
Fin }  |->  (fld  gsumg  b ) ) `  x ) ) )
18 ne0gt0 9183 . . . . . . . 8  |-  ( ( ( ( b  e. 
{ a  e.  ( NN0  ^m  I )  |  ( `' a
" NN )  e. 
Fin }  |->  (fld  gsumg  b ) ) `  x )  e.  RR  /\  0  <_  ( (
b  e.  { a  e.  ( NN0  ^m  I )  |  ( `' a " NN )  e.  Fin }  |->  (fld  gsumg  b ) ) `  x ) )  ->  ( (
( b  e.  {
a  e.  ( NN0 
^m  I )  |  ( `' a " NN )  e.  Fin } 
|->  (fld 
gsumg  b ) ) `  x )  =/=  0  <->  0  <  ( ( b  e.  { a  e.  ( NN0  ^m  I
)  |  ( `' a " NN )  e.  Fin }  |->  (fld  gsumg  b ) ) `  x ) ) )
1914, 17, 183syl 19 . . . . . . 7  |-  ( (
ph  /\  x  e.  { a  e.  ( NN0 
^m  I )  |  ( `' a " NN )  e.  Fin } )  ->  ( (
( b  e.  {
a  e.  ( NN0 
^m  I )  |  ( `' a " NN )  e.  Fin } 
|->  (fld 
gsumg  b ) ) `  x )  =/=  0  <->  0  <  ( ( b  e.  { a  e.  ( NN0  ^m  I
)  |  ( `' a " NN )  e.  Fin }  |->  (fld  gsumg  b ) ) `  x ) ) )
207, 8tdeglem4 19988 . . . . . . . . 9  |-  ( ( I  e.  V  /\  x  e.  { a  e.  ( NN0  ^m  I
)  |  ( `' a " NN )  e.  Fin } )  ->  ( ( ( b  e.  { a  e.  ( NN0  ^m  I )  |  ( `' a " NN )  e.  Fin }  |->  (fld  gsumg  b ) ) `  x )  =  0  <->  x  =  ( I  X.  { 0 } ) ) )
2111, 20sylan 459 . . . . . . . 8  |-  ( (
ph  /\  x  e.  { a  e.  ( NN0 
^m  I )  |  ( `' a " NN )  e.  Fin } )  ->  ( (
( b  e.  {
a  e.  ( NN0 
^m  I )  |  ( `' a " NN )  e.  Fin } 
|->  (fld 
gsumg  b ) ) `  x )  =  0  <-> 
x  =  ( I  X.  { 0 } ) ) )
2221necon3abid 2636 . . . . . . 7  |-  ( (
ph  /\  x  e.  { a  e.  ( NN0 
^m  I )  |  ( `' a " NN )  e.  Fin } )  ->  ( (
( b  e.  {
a  e.  ( NN0 
^m  I )  |  ( `' a " NN )  e.  Fin } 
|->  (fld 
gsumg  b ) ) `  x )  =/=  0  <->  -.  x  =  ( I  X.  { 0 } ) ) )
2319, 22bitr3d 248 . . . . . 6  |-  ( (
ph  /\  x  e.  { a  e.  ( NN0 
^m  I )  |  ( `' a " NN )  e.  Fin } )  ->  ( 0  <  ( ( b  e.  { a  e.  ( NN0  ^m  I
)  |  ( `' a " NN )  e.  Fin }  |->  (fld  gsumg  b ) ) `  x )  <->  -.  x  =  (
I  X.  { 0 } ) ) )
2423imbi1d 310 . . . . 5  |-  ( (
ph  /\  x  e.  { a  e.  ( NN0 
^m  I )  |  ( `' a " NN )  e.  Fin } )  ->  ( (
0  <  ( (
b  e.  { a  e.  ( NN0  ^m  I )  |  ( `' a " NN )  e.  Fin }  |->  (fld  gsumg  b ) ) `  x )  ->  ( F `  x )  =  ( 0g `  R ) )  <->  ( -.  x  =  ( I  X.  { 0 } )  ->  ( F `  x )  =  ( 0g `  R ) ) ) )
25 eqeq2 2447 . . . . . . . 8  |-  ( ( F `  ( I  X.  { 0 } ) )  =  if ( x  =  ( I  X.  { 0 } ) ,  ( F `  ( I  X.  { 0 } ) ) ,  ( 0g `  R ) )  ->  ( ( F `  x )  =  ( F `  ( I  X.  { 0 } ) )  <->  ( F `  x )  =  if ( x  =  ( I  X.  { 0 } ) ,  ( F `  ( I  X.  { 0 } ) ) ,  ( 0g `  R ) ) ) )
2625bibi1d 312 . . . . . . 7  |-  ( ( F `  ( I  X.  { 0 } ) )  =  if ( x  =  ( I  X.  { 0 } ) ,  ( F `  ( I  X.  { 0 } ) ) ,  ( 0g `  R ) )  ->  ( (
( F `  x
)  =  ( F `
 ( I  X.  { 0 } ) )  <->  ( -.  x  =  ( I  X.  { 0 } )  ->  ( F `  x )  =  ( 0g `  R ) ) )  <->  ( ( F `  x )  =  if ( x  =  ( I  X.  {
0 } ) ,  ( F `  (
I  X.  { 0 } ) ) ,  ( 0g `  R
) )  <->  ( -.  x  =  ( I  X.  { 0 } )  ->  ( F `  x )  =  ( 0g `  R ) ) ) ) )
27 eqeq2 2447 . . . . . . . 8  |-  ( ( 0g `  R )  =  if ( x  =  ( I  X.  { 0 } ) ,  ( F `  ( I  X.  { 0 } ) ) ,  ( 0g `  R
) )  ->  (
( F `  x
)  =  ( 0g
`  R )  <->  ( F `  x )  =  if ( x  =  ( I  X.  { 0 } ) ,  ( F `  ( I  X.  { 0 } ) ) ,  ( 0g `  R ) ) ) )
2827bibi1d 312 . . . . . . 7  |-  ( ( 0g `  R )  =  if ( x  =  ( I  X.  { 0 } ) ,  ( F `  ( I  X.  { 0 } ) ) ,  ( 0g `  R
) )  ->  (
( ( F `  x )  =  ( 0g `  R )  <-> 
( -.  x  =  ( I  X.  {
0 } )  -> 
( F `  x
)  =  ( 0g
`  R ) ) )  <->  ( ( F `
 x )  =  if ( x  =  ( I  X.  {
0 } ) ,  ( F `  (
I  X.  { 0 } ) ) ,  ( 0g `  R
) )  <->  ( -.  x  =  ( I  X.  { 0 } )  ->  ( F `  x )  =  ( 0g `  R ) ) ) ) )
29 fveq2 5731 . . . . . . . . 9  |-  ( x  =  ( I  X.  { 0 } )  ->  ( F `  x )  =  ( F `  ( I  X.  { 0 } ) ) )
30 pm2.24 104 . . . . . . . . 9  |-  ( x  =  ( I  X.  { 0 } )  ->  ( -.  x  =  ( I  X.  { 0 } )  ->  ( F `  x )  =  ( 0g `  R ) ) )
3129, 302thd 233 . . . . . . . 8  |-  ( x  =  ( I  X.  { 0 } )  ->  ( ( F `
 x )  =  ( F `  (
I  X.  { 0 } ) )  <->  ( -.  x  =  ( I  X.  { 0 } )  ->  ( F `  x )  =  ( 0g `  R ) ) ) )
3231adantl 454 . . . . . . 7  |-  ( (
ph  /\  x  =  ( I  X.  { 0 } ) )  -> 
( ( F `  x )  =  ( F `  ( I  X.  { 0 } ) )  <->  ( -.  x  =  ( I  X.  { 0 } )  ->  ( F `  x )  =  ( 0g `  R ) ) ) )
33 biimt 327 . . . . . . . 8  |-  ( -.  x  =  ( I  X.  { 0 } )  ->  ( ( F `  x )  =  ( 0g `  R )  <->  ( -.  x  =  ( I  X.  { 0 } )  ->  ( F `  x )  =  ( 0g `  R ) ) ) )
3433adantl 454 . . . . . . 7  |-  ( (
ph  /\  -.  x  =  ( I  X.  { 0 } ) )  ->  ( ( F `  x )  =  ( 0g `  R )  <->  ( -.  x  =  ( I  X.  { 0 } )  ->  ( F `  x )  =  ( 0g `  R ) ) ) )
3526, 28, 32, 34ifbothda 3771 . . . . . 6  |-  ( ph  ->  ( ( F `  x )  =  if ( x  =  ( I  X.  { 0 } ) ,  ( F `  ( I  X.  { 0 } ) ) ,  ( 0g `  R ) )  <->  ( -.  x  =  ( I  X.  { 0 } )  ->  ( F `  x )  =  ( 0g `  R ) ) ) )
3635adantr 453 . . . . 5  |-  ( (
ph  /\  x  e.  { a  e.  ( NN0 
^m  I )  |  ( `' a " NN )  e.  Fin } )  ->  ( ( F `  x )  =  if ( x  =  ( I  X.  {
0 } ) ,  ( F `  (
I  X.  { 0 } ) ) ,  ( 0g `  R
) )  <->  ( -.  x  =  ( I  X.  { 0 } )  ->  ( F `  x )  =  ( 0g `  R ) ) ) )
3724, 36bitr4d 249 . . . 4  |-  ( (
ph  /\  x  e.  { a  e.  ( NN0 
^m  I )  |  ( `' a " NN )  e.  Fin } )  ->  ( (
0  <  ( (
b  e.  { a  e.  ( NN0  ^m  I )  |  ( `' a " NN )  e.  Fin }  |->  (fld  gsumg  b ) ) `  x )  ->  ( F `  x )  =  ( 0g `  R ) )  <->  ( F `  x )  =  if ( x  =  ( I  X.  { 0 } ) ,  ( F `  ( I  X.  { 0 } ) ) ,  ( 0g `  R ) ) ) )
3837ralbidva 2723 . . 3  |-  ( ph  ->  ( A. x  e. 
{ a  e.  ( NN0  ^m  I )  |  ( `' a
" NN )  e. 
Fin }  ( 0  <  ( ( b  e.  { a  e.  ( NN0  ^m  I
)  |  ( `' a " NN )  e.  Fin }  |->  (fld  gsumg  b ) ) `  x )  ->  ( F `  x )  =  ( 0g `  R ) )  <->  A. x  e.  {
a  e.  ( NN0 
^m  I )  |  ( `' a " NN )  e.  Fin }  ( F `  x
)  =  if ( x  =  ( I  X.  { 0 } ) ,  ( F `
 ( I  X.  { 0 } ) ) ,  ( 0g
`  R ) ) ) )
39 eqid 2438 . . . . . . 7  |-  ( Base `  R )  =  (
Base `  R )
404, 39, 5, 7, 1mplelf 16502 . . . . . 6  |-  ( ph  ->  F : { a  e.  ( NN0  ^m  I )  |  ( `' a " NN )  e.  Fin } --> ( Base `  R ) )
4140feqmptd 5782 . . . . 5  |-  ( ph  ->  F  =  ( x  e.  { a  e.  ( NN0  ^m  I
)  |  ( `' a " NN )  e.  Fin }  |->  ( F `  x ) ) )
42 mdegle0.a . . . . . 6  |-  A  =  (algSc `  Y )
43 mdegaddle.r . . . . . 6  |-  ( ph  ->  R  e.  Ring )
447psrbag0 16559 . . . . . . . 8  |-  ( I  e.  V  ->  (
I  X.  { 0 } )  e.  {
a  e.  ( NN0 
^m  I )  |  ( `' a " NN )  e.  Fin } )
4511, 44syl 16 . . . . . . 7  |-  ( ph  ->  ( I  X.  {
0 } )  e. 
{ a  e.  ( NN0  ^m  I )  |  ( `' a
" NN )  e. 
Fin } )
4640, 45ffvelrnd 5874 . . . . . 6  |-  ( ph  ->  ( F `  (
I  X.  { 0 } ) )  e.  ( Base `  R
) )
474, 7, 6, 39, 42, 11, 43, 46mplascl 16561 . . . . 5  |-  ( ph  ->  ( A `  ( F `  ( I  X.  { 0 } ) ) )  =  ( x  e.  { a  e.  ( NN0  ^m  I )  |  ( `' a " NN )  e.  Fin }  |->  if ( x  =  ( I  X.  { 0 } ) ,  ( F `  ( I  X.  { 0 } ) ) ,  ( 0g `  R ) ) ) )
4841, 47eqeq12d 2452 . . . 4  |-  ( ph  ->  ( F  =  ( A `  ( F `
 ( I  X.  { 0 } ) ) )  <->  ( x  e.  { a  e.  ( NN0  ^m  I )  |  ( `' a
" NN )  e. 
Fin }  |->  ( F `
 x ) )  =  ( x  e. 
{ a  e.  ( NN0  ^m  I )  |  ( `' a
" NN )  e. 
Fin }  |->  if ( x  =  ( I  X.  { 0 } ) ,  ( F `
 ( I  X.  { 0 } ) ) ,  ( 0g
`  R ) ) ) ) )
49 fvex 5745 . . . . . 6  |-  ( F `
 x )  e. 
_V
5049rgenw 2775 . . . . 5  |-  A. x  e.  { a  e.  ( NN0  ^m  I )  |  ( `' a
" NN )  e. 
Fin }  ( F `  x )  e.  _V
51 mpteqb 5822 . . . . 5  |-  ( A. x  e.  { a  e.  ( NN0  ^m  I
)  |  ( `' a " NN )  e.  Fin }  ( F `  x )  e.  _V  ->  ( (
x  e.  { a  e.  ( NN0  ^m  I )  |  ( `' a " NN )  e.  Fin }  |->  ( F `  x ) )  =  ( x  e.  { a  e.  ( NN0  ^m  I
)  |  ( `' a " NN )  e.  Fin }  |->  if ( x  =  ( I  X.  { 0 } ) ,  ( F `  ( I  X.  { 0 } ) ) ,  ( 0g `  R ) ) )  <->  A. x  e.  { a  e.  ( NN0  ^m  I )  |  ( `' a
" NN )  e. 
Fin }  ( F `  x )  =  if ( x  =  ( I  X.  { 0 } ) ,  ( F `  ( I  X.  { 0 } ) ) ,  ( 0g `  R ) ) ) )
5250, 51mp1i 12 . . . 4  |-  ( ph  ->  ( ( x  e. 
{ a  e.  ( NN0  ^m  I )  |  ( `' a
" NN )  e. 
Fin }  |->  ( F `
 x ) )  =  ( x  e. 
{ a  e.  ( NN0  ^m  I )  |  ( `' a
" NN )  e. 
Fin }  |->  if ( x  =  ( I  X.  { 0 } ) ,  ( F `
 ( I  X.  { 0 } ) ) ,  ( 0g
`  R ) ) )  <->  A. x  e.  {
a  e.  ( NN0 
^m  I )  |  ( `' a " NN )  e.  Fin }  ( F `  x
)  =  if ( x  =  ( I  X.  { 0 } ) ,  ( F `
 ( I  X.  { 0 } ) ) ,  ( 0g
`  R ) ) ) )
5348, 52bitrd 246 . . 3  |-  ( ph  ->  ( F  =  ( A `  ( F `
 ( I  X.  { 0 } ) ) )  <->  A. x  e.  { a  e.  ( NN0  ^m  I )  |  ( `' a
" NN )  e. 
Fin }  ( F `  x )  =  if ( x  =  ( I  X.  { 0 } ) ,  ( F `  ( I  X.  { 0 } ) ) ,  ( 0g `  R ) ) ) )
5438, 53bitr4d 249 . 2  |-  ( ph  ->  ( A. x  e. 
{ a  e.  ( NN0  ^m  I )  |  ( `' a
" NN )  e. 
Fin }  ( 0  <  ( ( b  e.  { a  e.  ( NN0  ^m  I
)  |  ( `' a " NN )  e.  Fin }  |->  (fld  gsumg  b ) ) `  x )  ->  ( F `  x )  =  ( 0g `  R ) )  <->  F  =  ( A `  ( F `  ( I  X.  {
0 } ) ) ) ) )
5510, 54bitrd 246 1  |-  ( ph  ->  ( ( D `  F )  <_  0  <->  F  =  ( A `  ( F `  ( I  X.  { 0 } ) ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2601   A.wral 2707   {crab 2711   _Vcvv 2958   ifcif 3741   {csn 3816   class class class wbr 4215    e. cmpt 4269    X. cxp 4879   `'ccnv 4880   "cima 4884   -->wf 5453   ` cfv 5457  (class class class)co 6084    ^m cmap 7021   Fincfn 7112   RRcr 8994   0cc0 8995   RR*cxr 9124    < clt 9125    <_ cle 9126   NNcn 10005   NN0cn0 10226   Basecbs 13474   0gc0g 13728    gsumg cgsu 13729   Ringcrg 15665  algSccascl 16376   mPoly cmpl 16413  ℂfldccnfld 16708   mDeg cmdg 19981
This theorem is referenced by:  deg1le0  20039
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 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-inf2 7599  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072  ax-pre-sup 9073  ax-addf 9074  ax-mulf 9075
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 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-iin 4098  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-se 4545  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-isom 5466  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-of 6308  df-ofr 6309  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-2o 6728  df-oadd 6731  df-er 6908  df-map 7023  df-pm 7024  df-ixp 7067  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-sup 7449  df-oi 7482  df-card 7831  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-nn 10006  df-2 10063  df-3 10064  df-4 10065  df-5 10066  df-6 10067  df-7 10068  df-8 10069  df-9 10070  df-10 10071  df-n0 10227  df-z 10288  df-dec 10388  df-uz 10494  df-fz 11049  df-fzo 11141  df-seq 11329  df-hash 11624  df-struct 13476  df-ndx 13477  df-slot 13478  df-base 13479  df-sets 13480  df-ress 13481  df-plusg 13547  df-mulr 13548  df-starv 13549  df-sca 13550  df-vsca 13551  df-tset 13553  df-ple 13554  df-ds 13556  df-unif 13557  df-0g 13732  df-gsum 13733  df-mre 13816  df-mrc 13817  df-acs 13819  df-mnd 14695  df-mhm 14743  df-submnd 14744  df-grp 14817  df-minusg 14818  df-mulg 14820  df-subg 14946  df-ghm 15009  df-cntz 15121  df-cmn 15419  df-abl 15420  df-mgp 15654  df-rng 15668  df-cring 15669  df-ur 15670  df-subrg 15871  df-ascl 16379  df-psr 16422  df-mpl 16424  df-cnfld 16709  df-mdeg 19983
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