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Theorem mdegle0 22228
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 9639 . . 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 2467 . . . 4  |-  ( 0g
`  R )  =  ( 0g `  R
)
7 eqid 2467 . . . 4  |-  { a  e.  ( NN0  ^m  I )  |  ( `' a " NN )  e.  Fin }  =  { a  e.  ( NN0  ^m  I )  |  ( `' a
" NN )  e. 
Fin }
8 eqid 2467 . . . 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 22215 . . 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 662 . 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 22207 . . . . . . . . . 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 6020 . . . . . . . 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 10803 . . . . . . . . 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 10820 . . . . . . . . 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 532 . . . . . . . 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 9688 . . . . . . . 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 20 . . . . . . 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 22209 . . . . . . . . 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 471 . . . . . . . 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 2713 . . . . . . 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 255 . . . . . 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 317 . . . . 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 2482 . . . . . . . 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 319 . . . . . . 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 2482 . . . . . . . 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 319 . . . . . . 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 5865 . . . . . . . . 9  |-  ( x  =  ( I  X.  { 0 } )  ->  ( F `  x )  =  ( F `  ( I  X.  { 0 } ) ) )
30 pm2.24 109 . . . . . . . . 9  |-  ( x  =  ( I  X.  { 0 } )  ->  ( -.  x  =  ( I  X.  { 0 } )  ->  ( F `  x )  =  ( 0g `  R ) ) )
3129, 302thd 240 . . . . . . . 8  |-  ( x  =  ( I  X.  { 0 } )  ->  ( ( F `
 x )  =  ( F `  (
I  X.  { 0 } ) )  <->  ( -.  x  =  ( I  X.  { 0 } )  ->  ( F `  x )  =  ( 0g `  R ) ) ) )
3231adantl 466 . . . . . . 7  |-  ( (
ph  /\  x  =  ( I  X.  { 0 } ) )  -> 
( ( F `  x )  =  ( F `  ( I  X.  { 0 } ) )  <->  ( -.  x  =  ( I  X.  { 0 } )  ->  ( F `  x )  =  ( 0g `  R ) ) ) )
33 biimt 335 . . . . . . . 8  |-  ( -.  x  =  ( I  X.  { 0 } )  ->  ( ( F `  x )  =  ( 0g `  R )  <->  ( -.  x  =  ( I  X.  { 0 } )  ->  ( F `  x )  =  ( 0g `  R ) ) ) )
3433adantl 466 . . . . . . 7  |-  ( (
ph  /\  -.  x  =  ( I  X.  { 0 } ) )  ->  ( ( F `  x )  =  ( 0g `  R )  <->  ( -.  x  =  ( I  X.  { 0 } )  ->  ( F `  x )  =  ( 0g `  R ) ) ) )
3526, 28, 32, 34ifbothda 3974 . . . . . 6  |-  ( ph  ->  ( ( F `  x )  =  if ( x  =  ( I  X.  { 0 } ) ,  ( F `  ( I  X.  { 0 } ) ) ,  ( 0g `  R ) )  <->  ( -.  x  =  ( I  X.  { 0 } )  ->  ( F `  x )  =  ( 0g `  R ) ) ) )
3635adantr 465 . . . . 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 256 . . . 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 2900 . . 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 2467 . . . . . . 7  |-  ( Base `  R )  =  (
Base `  R )
404, 39, 5, 7, 1mplelf 17879 . . . . . 6  |-  ( ph  ->  F : { a  e.  ( NN0  ^m  I )  |  ( `' a " NN )  e.  Fin } --> ( Base `  R ) )
4140feqmptd 5919 . . . . 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 17946 . . . . . . . 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 6021 . . . . . 6  |-  ( ph  ->  ( F `  (
I  X.  { 0 } ) )  e.  ( Base `  R
) )
474, 7, 6, 39, 42, 11, 43, 46mplascl 17948 . . . . 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 2489 . . . 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 5875 . . . . . 6  |-  ( F `
 x )  e. 
_V
5049rgenw 2825 . . . . 5  |-  A. x  e.  { a  e.  ( NN0  ^m  I )  |  ( `' a
" NN )  e. 
Fin }  ( F `  x )  e.  _V
51 mpteqb 5963 . . . . 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 253 . . 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 256 . 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 253 1  |-  ( ph  ->  ( ( D `  F )  <_  0  <->  F  =  ( A `  ( F `  ( I  X.  { 0 } ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   {crab 2818   _Vcvv 3113   ifcif 3939   {csn 4027   class class class wbr 4447    |-> cmpt 4505    X. cxp 4997   `'ccnv 4998   "cima 5002   -->wf 5583   ` cfv 5587  (class class class)co 6283    ^m cmap 7420   Fincfn 7516   RRcr 9490   0cc0 9491   RR*cxr 9626    < clt 9627    <_ cle 9628   NNcn 10535   NN0cn0 10794   Basecbs 14489   0gc0g 14694    gsumg cgsu 14695   Ringcrg 16995  algSccascl 17747   mPoly cmpl 17789  ℂfldccnfld 18207   mDeg cmdg 22202
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575  ax-inf2 8057  ax-cnex 9547  ax-resscn 9548  ax-1cn 9549  ax-icn 9550  ax-addcl 9551  ax-addrcl 9552  ax-mulcl 9553  ax-mulrcl 9554  ax-mulcom 9555  ax-addass 9556  ax-mulass 9557  ax-distr 9558  ax-i2m1 9559  ax-1ne0 9560  ax-1rid 9561  ax-rnegex 9562  ax-rrecex 9563  ax-cnre 9564  ax-pre-lttri 9565  ax-pre-lttrn 9566  ax-pre-ltadd 9567  ax-pre-mulgt0 9568  ax-pre-sup 9569  ax-addf 9570  ax-mulf 9571
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-isom 5596  df-riota 6244  df-ov 6286  df-oprab 6287  df-mpt2 6288  df-of 6523  df-ofr 6524  df-om 6680  df-1st 6784  df-2nd 6785  df-supp 6902  df-recs 7042  df-rdg 7076  df-1o 7130  df-2o 7131  df-oadd 7134  df-er 7311  df-map 7422  df-pm 7423  df-ixp 7470  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-fsupp 7829  df-sup 7900  df-oi 7934  df-card 8319  df-pnf 9629  df-mnf 9630  df-xr 9631  df-ltxr 9632  df-le 9633  df-sub 9806  df-neg 9807  df-nn 10536  df-2 10593  df-3 10594  df-4 10595  df-5 10596  df-6 10597  df-7 10598  df-8 10599  df-9 10600  df-10 10601  df-n0 10795  df-z 10864  df-dec 10976  df-uz 11082  df-fz 11672  df-fzo 11792  df-seq 12075  df-hash 12373  df-struct 14491  df-ndx 14492  df-slot 14493  df-base 14494  df-sets 14495  df-ress 14496  df-plusg 14567  df-mulr 14568  df-starv 14569  df-sca 14570  df-vsca 14571  df-tset 14573  df-ple 14574  df-ds 14576  df-unif 14577  df-0g 14696  df-gsum 14697  df-mre 14840  df-mrc 14841  df-acs 14843  df-mnd 15731  df-mhm 15783  df-submnd 15784  df-grp 15864  df-minusg 15865  df-mulg 15867  df-subg 16000  df-ghm 16067  df-cntz 16157  df-cmn 16603  df-abl 16604  df-mgp 16941  df-ur 16953  df-rng 16997  df-cring 16998  df-subrg 17222  df-ascl 17750  df-psr 17792  df-mpl 17794  df-cnfld 18208  df-mdeg 22204
This theorem is referenced by:  deg1le0  22263
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