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Theorem mdegle0 19567
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 8968 . . 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 2358 . . . 4  |-  ( 0g
`  R )  =  ( 0g `  R
)
7 eqid 2358 . . . 4  |-  { a  e.  ( NN0  ^m  I )  |  ( `' a " NN )  e.  Fin }  =  { a  e.  ( NN0  ^m  I )  |  ( `' a
" NN )  e. 
Fin }
8 eqid 2358 . . . 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 19554 . . 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 19548 . . . . . . . . . 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 5746 . . . . . . . . 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 10066 . . . . . . . . 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 10083 . . . . . . . . 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 9015 . . . . . . . 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 19550 . . . . . . . . 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 2554 . . . . . . 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 2367 . . . . . . . 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 2367 . . . . . . . 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 5608 . . . . . . . . 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 3671 . . . . . 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 2635 . . 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 2358 . . . . . . 7  |-  ( Base `  R )  =  (
Base `  R )
414, 40, 5, 7, 1mplelf 16277 . . . . . 6  |-  ( ph  ->  F : { a  e.  ( NN0  ^m  I )  |  ( `' a " NN )  e.  Fin } --> ( Base `  R ) )
4241feqmptd 5658 . . . . 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 16334 . . . . . . . 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 5746 . . . . . . 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 16336 . . . . 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 2372 . . . 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 5622 . . . . . 6  |-  ( F `
 x )  e. 
_V
5251rgenw 2686 . . . . 5  |-  A. x  e.  { a  e.  ( NN0  ^m  I )  |  ( `' a
" NN )  e. 
Fin }  ( F `  x )  e.  _V
53 mpteqb 5697 . . . . 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 1642    e. wcel 1710    =/= wne 2521   A.wral 2619   {crab 2623   _Vcvv 2864   ifcif 3641   {csn 3716   class class class wbr 4104    e. cmpt 4158    X. cxp 4769   `'ccnv 4770   "cima 4774   -->wf 5333   ` cfv 5337  (class class class)co 5945    ^m cmap 6860   Fincfn 6951   RRcr 8826   0cc0 8827   RR*cxr 8956    < clt 8957    <_ cle 8958   NNcn 9836   NN0cn0 10057   Basecbs 13245   0gc0g 13499    gsumg cgsu 13500   Ringcrg 15436  algSccascl 16151   mPoly cmpl 16188  ℂfldccnfld 16482   mDeg cmdg 19543
This theorem is referenced by:  deg1le0  19601
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-inf2 7432  ax-cnex 8883  ax-resscn 8884  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-addrcl 8888  ax-mulcl 8889  ax-mulrcl 8890  ax-mulcom 8891  ax-addass 8892  ax-mulass 8893  ax-distr 8894  ax-i2m1 8895  ax-1ne0 8896  ax-1rid 8897  ax-rnegex 8898  ax-rrecex 8899  ax-cnre 8900  ax-pre-lttri 8901  ax-pre-lttrn 8902  ax-pre-ltadd 8903  ax-pre-mulgt0 8904  ax-pre-sup 8905  ax-addf 8906  ax-mulf 8907
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-int 3944  df-iun 3988  df-iin 3989  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-se 4435  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-isom 5346  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-of 6165  df-ofr 6166  df-1st 6209  df-2nd 6210  df-riota 6391  df-recs 6475  df-rdg 6510  df-1o 6566  df-2o 6567  df-oadd 6570  df-er 6747  df-map 6862  df-pm 6863  df-ixp 6906  df-en 6952  df-dom 6953  df-sdom 6954  df-fin 6955  df-sup 7284  df-oi 7315  df-card 7662  df-pnf 8959  df-mnf 8960  df-xr 8961  df-ltxr 8962  df-le 8963  df-sub 9129  df-neg 9130  df-nn 9837  df-2 9894  df-3 9895  df-4 9896  df-5 9897  df-6 9898  df-7 9899  df-8 9900  df-9 9901  df-10 9902  df-n0 10058  df-z 10117  df-dec 10217  df-uz 10323  df-fz 10875  df-fzo 10963  df-seq 11139  df-hash 11431  df-struct 13247  df-ndx 13248  df-slot 13249  df-base 13250  df-sets 13251  df-ress 13252  df-plusg 13318  df-mulr 13319  df-starv 13320  df-sca 13321  df-vsca 13322  df-tset 13324  df-ple 13325  df-ds 13327  df-unif 13328  df-0g 13503  df-gsum 13504  df-mre 13587  df-mrc 13588  df-acs 13590  df-mnd 14466  df-mhm 14514  df-submnd 14515  df-grp 14588  df-minusg 14589  df-mulg 14591  df-subg 14717  df-ghm 14780  df-cntz 14892  df-cmn 15190  df-abl 15191  df-mgp 15425  df-rng 15439  df-cring 15440  df-ur 15441  df-subrg 15642  df-ascl 16154  df-psr 16197  df-mpl 16199  df-cnfld 16483  df-mdeg 19545
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