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Theorem mdegle0 21548
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 9430 . . 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 2443 . . . 4  |-  ( 0g
`  R )  =  ( 0g `  R
)
7 eqid 2443 . . . 4  |-  { a  e.  ( NN0  ^m  I )  |  ( `' a " NN )  e.  Fin }  =  { a  e.  ( NN0  ^m  I )  |  ( `' a
" NN )  e. 
Fin }
8 eqid 2443 . . . 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 21535 . . 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 21527 . . . . . . . . . 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 5843 . . . . . . . 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 10588 . . . . . . . . 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 10605 . . . . . . . . 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 9479 . . . . . . . 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 21529 . . . . . . . . 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 2641 . . . . . . 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 2452 . . . . . . . 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 2452 . . . . . . . 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 5691 . . . . . . . . 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 3824 . . . . . 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 2731 . . 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 2443 . . . . . . 7  |-  ( Base `  R )  =  (
Base `  R )
404, 39, 5, 7, 1mplelf 17509 . . . . . 6  |-  ( ph  ->  F : { a  e.  ( NN0  ^m  I )  |  ( `' a " NN )  e.  Fin } --> ( Base `  R ) )
4140feqmptd 5744 . . . . 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 17576 . . . . . . . 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 5844 . . . . . 6  |-  ( ph  ->  ( F `  (
I  X.  { 0 } ) )  e.  ( Base `  R
) )
474, 7, 6, 39, 42, 11, 43, 46mplascl 17578 . . . . 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 2457 . . . 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 5701 . . . . . 6  |-  ( F `
 x )  e. 
_V
5049rgenw 2783 . . . . 5  |-  A. x  e.  { a  e.  ( NN0  ^m  I )  |  ( `' a
" NN )  e. 
Fin }  ( F `  x )  e.  _V
51 mpteqb 5788 . . . . 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 1369    e. wcel 1756    =/= wne 2606   A.wral 2715   {crab 2719   _Vcvv 2972   ifcif 3791   {csn 3877   class class class wbr 4292    e. cmpt 4350    X. cxp 4838   `'ccnv 4839   "cima 4843   -->wf 5414   ` cfv 5418  (class class class)co 6091    ^m cmap 7214   Fincfn 7310   RRcr 9281   0cc0 9282   RR*cxr 9417    < clt 9418    <_ cle 9419   NNcn 10322   NN0cn0 10579   Basecbs 14174   0gc0g 14378    gsumg cgsu 14379   Ringcrg 16645  algSccascl 17383   mPoly cmpl 17420  ℂfldccnfld 17818   mDeg cmdg 21522
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-inf2 7847  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-pre-sup 9360  ax-addf 9361  ax-mulf 9362
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-iin 4174  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-se 4680  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-isom 5427  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-of 6320  df-ofr 6321  df-om 6477  df-1st 6577  df-2nd 6578  df-supp 6691  df-recs 6832  df-rdg 6866  df-1o 6920  df-2o 6921  df-oadd 6924  df-er 7101  df-map 7216  df-pm 7217  df-ixp 7264  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-fsupp 7621  df-sup 7691  df-oi 7724  df-card 8109  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-nn 10323  df-2 10380  df-3 10381  df-4 10382  df-5 10383  df-6 10384  df-7 10385  df-8 10386  df-9 10387  df-10 10388  df-n0 10580  df-z 10647  df-dec 10756  df-uz 10862  df-fz 11438  df-fzo 11549  df-seq 11807  df-hash 12104  df-struct 14176  df-ndx 14177  df-slot 14178  df-base 14179  df-sets 14180  df-ress 14181  df-plusg 14251  df-mulr 14252  df-starv 14253  df-sca 14254  df-vsca 14255  df-tset 14257  df-ple 14258  df-ds 14260  df-unif 14261  df-0g 14380  df-gsum 14381  df-mre 14524  df-mrc 14525  df-acs 14527  df-mnd 15415  df-mhm 15464  df-submnd 15465  df-grp 15545  df-minusg 15546  df-mulg 15548  df-subg 15678  df-ghm 15745  df-cntz 15835  df-cmn 16279  df-abl 16280  df-mgp 16592  df-ur 16604  df-rng 16647  df-cring 16648  df-subrg 16863  df-ascl 17386  df-psr 17423  df-mpl 17425  df-cnfld 17819  df-mdeg 21524
This theorem is referenced by:  deg1le0  21583
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