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Theorem mdegle0 19463
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 8878 . . 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 2283 . . . 4  |-  ( 0g
`  R )  =  ( 0g `  R
)
7 eqid 2283 . . . 4  |-  { a  e.  ( NN0  ^m  I )  |  ( `' a " NN )  e.  Fin }  =  { a  e.  ( NN0  ^m  I )  |  ( `' a
" NN )  e. 
Fin }
8 eqid 2283 . . . 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 19450 . . 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 643 . 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 19444 . . . . . . . . . 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 15 . . . . . . . . 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 )
14 ffvelrn 5663 . . . . . . . . 9  |-  ( ( ( b  e.  {
a  e.  ( NN0 
^m  I )  |  ( `' a " NN )  e.  Fin } 
|->  (fld 
gsumg  b ) ) : { a  e.  ( NN0  ^m  I )  |  ( `' a
" NN )  e. 
Fin } --> NN0  /\  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 )
1513, 14sylan 457 . . . . . . . 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 )
16 nn0re 9974 . . . . . . . . 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 )
17 nn0ge0 9991 . . . . . . . . 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 ) )
1816, 17jca 518 . . . . . . . 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 ) ) )
19 ne0gt0 8925 . . . . . . . 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 ) ) )
2015, 18, 193syl 18 . . . . . . 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 ) ) )
217, 8tdeglem4 19446 . . . . . . . . 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 } ) ) )
2211, 21sylan 457 . . . . . . . 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 } ) ) )
2322necon3abid 2479 . . . . . . 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 } ) ) )
2420, 23bitr3d 246 . . . . . 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 } ) ) )
2524imbi1d 308 . . . . 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 ) ) ) )
26 eqeq2 2292 . . . . . . . 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 ) ) ) )
2726bibi1d 310 . . . . . . 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 ) ) ) ) )
28 eqeq2 2292 . . . . . . . 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 ) ) ) )
2928bibi1d 310 . . . . . . 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 ) ) ) ) )
30 fveq2 5525 . . . . . . . . 9  |-  ( x  =  ( I  X.  { 0 } )  ->  ( F `  x )  =  ( F `  ( I  X.  { 0 } ) ) )
31 pm2.24 101 . . . . . . . . 9  |-  ( x  =  ( I  X.  { 0 } )  ->  ( -.  x  =  ( I  X.  { 0 } )  ->  ( F `  x )  =  ( 0g `  R ) ) )
3230, 312thd 231 . . . . . . . 8  |-  ( x  =  ( I  X.  { 0 } )  ->  ( ( F `
 x )  =  ( F `  (
I  X.  { 0 } ) )  <->  ( -.  x  =  ( I  X.  { 0 } )  ->  ( F `  x )  =  ( 0g `  R ) ) ) )
3332adantl 452 . . . . . . 7  |-  ( (
ph  /\  x  =  ( I  X.  { 0 } ) )  -> 
( ( F `  x )  =  ( F `  ( I  X.  { 0 } ) )  <->  ( -.  x  =  ( I  X.  { 0 } )  ->  ( F `  x )  =  ( 0g `  R ) ) ) )
34 biimt 325 . . . . . . . 8  |-  ( -.  x  =  ( I  X.  { 0 } )  ->  ( ( F `  x )  =  ( 0g `  R )  <->  ( -.  x  =  ( I  X.  { 0 } )  ->  ( F `  x )  =  ( 0g `  R ) ) ) )
3534adantl 452 . . . . . . 7  |-  ( (
ph  /\  -.  x  =  ( I  X.  { 0 } ) )  ->  ( ( F `  x )  =  ( 0g `  R )  <->  ( -.  x  =  ( I  X.  { 0 } )  ->  ( F `  x )  =  ( 0g `  R ) ) ) )
3627, 29, 33, 35ifbothda 3595 . . . . . 6  |-  ( ph  ->  ( ( F `  x )  =  if ( x  =  ( I  X.  { 0 } ) ,  ( F `  ( I  X.  { 0 } ) ) ,  ( 0g `  R ) )  <->  ( -.  x  =  ( I  X.  { 0 } )  ->  ( F `  x )  =  ( 0g `  R ) ) ) )
3736adantr 451 . . . . 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 ) ) ) )
3825, 37bitr4d 247 . . . 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 ) ) ) )
3938ralbidva 2559 . . 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 ) ) ) )
40 eqid 2283 . . . . . . 7  |-  ( Base `  R )  =  (
Base `  R )
414, 40, 5, 7, 1mplelf 16178 . . . . . 6  |-  ( ph  ->  F : { a  e.  ( NN0  ^m  I )  |  ( `' a " NN )  e.  Fin } --> ( Base `  R ) )
4241feqmptd 5575 . . . . 5  |-  ( ph  ->  F  =  ( x  e.  { a  e.  ( NN0  ^m  I
)  |  ( `' a " NN )  e.  Fin }  |->  ( F `  x ) ) )
43 mdegle0.a . . . . . 6  |-  A  =  (algSc `  Y )
44 mdegaddle.r . . . . . 6  |-  ( ph  ->  R  e.  Ring )
457psrbag0 16235 . . . . . . . 8  |-  ( I  e.  V  ->  (
I  X.  { 0 } )  e.  {
a  e.  ( NN0 
^m  I )  |  ( `' a " NN )  e.  Fin } )
4611, 45syl 15 . . . . . . 7  |-  ( ph  ->  ( I  X.  {
0 } )  e. 
{ a  e.  ( NN0  ^m  I )  |  ( `' a
" NN )  e. 
Fin } )
47 ffvelrn 5663 . . . . . . 7  |-  ( ( F : { a  e.  ( NN0  ^m  I )  |  ( `' a " NN )  e.  Fin } --> ( Base `  R )  /\  (
I  X.  { 0 } )  e.  {
a  e.  ( NN0 
^m  I )  |  ( `' a " NN )  e.  Fin } )  ->  ( F `  ( I  X.  {
0 } ) )  e.  ( Base `  R
) )
4841, 46, 47syl2anc 642 . . . . . 6  |-  ( ph  ->  ( F `  (
I  X.  { 0 } ) )  e.  ( Base `  R
) )
494, 7, 6, 40, 43, 11, 44, 48mplascl 16237 . . . . 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 ) ) ) )
5042, 49eqeq12d 2297 . . . 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 ) ) ) ) )
51 fvex 5539 . . . . . 6  |-  ( F `
 x )  e. 
_V
5251rgenw 2610 . . . . 5  |-  A. x  e.  { a  e.  ( NN0  ^m  I )  |  ( `' a
" NN )  e. 
Fin }  ( F `  x )  e.  _V
53 mpteqb 5614 . . . . 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 ) ) ) )
5452, 53mp1i 11 . . . 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 ) ) ) )
5550, 54bitrd 244 . . 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 ) ) ) )
5639, 55bitr4d 247 . 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 } ) ) ) ) )
5710, 56bitrd 244 1  |-  ( ph  ->  ( ( D `  F )  <_  0  <->  F  =  ( A `  ( F `  ( I  X.  { 0 } ) ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   {crab 2547   _Vcvv 2788   ifcif 3565   {csn 3640   class class class wbr 4023    e. cmpt 4077    X. cxp 4687   `'ccnv 4688   "cima 4692   -->wf 5251   ` cfv 5255  (class class class)co 5858    ^m cmap 6772   Fincfn 6863   RRcr 8736   0cc0 8737   RR*cxr 8866    < clt 8867    <_ cle 8868   NNcn 9746   NN0cn0 9965   Basecbs 13148   0gc0g 13400    gsumg cgsu 13401   Ringcrg 15337  algSccascl 16052   mPoly cmpl 16089  ℂfldccnfld 16377   mDeg cmdg 19439
This theorem is referenced by:  deg1le0  19497
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-ofr 6079  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-fz 10783  df-fzo 10871  df-seq 11047  df-hash 11338  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-0g 13404  df-gsum 13405  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-mhm 14415  df-submnd 14416  df-grp 14489  df-minusg 14490  df-mulg 14492  df-subg 14618  df-ghm 14681  df-cntz 14793  df-cmn 15091  df-abl 15092  df-mgp 15326  df-rng 15340  df-cring 15341  df-ur 15342  df-subrg 15543  df-ascl 16055  df-psr 16098  df-mpl 16100  df-cnfld 16378  df-mdeg 19441
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