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Theorem quotcan 20231
Description: Exact division with a multiple. (Contributed by Mario Carneiro, 26-Jul-2014.)
Hypothesis
Ref Expression
quotcan.1  |-  H  =  ( F  o F  x.  G )
Assertion
Ref Expression
quotcan  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0 p )  ->  ( H quot  G )  =  F )

Proof of Theorem quotcan
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 plyssc 20124 . . . . . . . . 9  |-  (Poly `  S )  C_  (Poly `  CC )
2 simp2 959 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0 p )  ->  G  e.  (Poly `  S )
)
31, 2sseldi 3348 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0 p )  ->  G  e.  (Poly `  CC )
)
4 simp1 958 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0 p )  ->  F  e.  (Poly `  S )
)
51, 4sseldi 3348 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0 p )  ->  F  e.  (Poly `  CC )
)
6 quotcan.1 . . . . . . . . . . . 12  |-  H  =  ( F  o F  x.  G )
7 plymulcl 20145 . . . . . . . . . . . 12  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( F  o F  x.  G
)  e.  (Poly `  CC ) )
86, 7syl5eqel 2522 . . . . . . . . . . 11  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  H  e.  (Poly `  CC ) )
983adant3 978 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0 p )  ->  H  e.  (Poly `  CC )
)
10 simp3 960 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0 p )  ->  G  =/=  0 p )
11 quotcl2 20224 . . . . . . . . . 10  |-  ( ( H  e.  (Poly `  CC )  /\  G  e.  (Poly `  CC )  /\  G  =/=  0 p )  ->  ( H quot  G )  e.  (Poly `  CC ) )
129, 3, 10, 11syl3anc 1185 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0 p )  ->  ( H quot  G )  e.  (Poly `  CC ) )
13 plysubcl 20146 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  CC )  /\  ( H quot  G )  e.  (Poly `  CC ) )  -> 
( F  o F  -  ( H quot  G
) )  e.  (Poly `  CC ) )
145, 12, 13syl2anc 644 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0 p )  ->  ( F  o F  -  ( H quot  G ) )  e.  (Poly `  CC )
)
15 plymul0or 20203 . . . . . . . 8  |-  ( ( G  e.  (Poly `  CC )  /\  ( F  o F  -  ( H quot  G ) )  e.  (Poly `  CC )
)  ->  ( ( G  o F  x.  ( F  o F  -  ( H quot  G ) ) )  =  0 p  <->  ( G  =  0 p  \/  ( F  o F  -  ( H quot  G
) )  =  0 p ) ) )
163, 14, 15syl2anc 644 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0 p )  ->  (
( G  o F  x.  ( F  o F  -  ( H quot  G ) ) )  =  0 p  <->  ( G  =  0 p  \/  ( F  o F  -  ( H quot  G
) )  =  0 p ) ) )
17 cnex 9076 . . . . . . . . . . . . 13  |-  CC  e.  _V
1817a1i 11 . . . . . . . . . . . 12  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0 p )  ->  CC  e.  _V )
19 plyf 20122 . . . . . . . . . . . . 13  |-  ( F  e.  (Poly `  S
)  ->  F : CC
--> CC )
204, 19syl 16 . . . . . . . . . . . 12  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0 p )  ->  F : CC --> CC )
21 plyf 20122 . . . . . . . . . . . . 13  |-  ( G  e.  (Poly `  S
)  ->  G : CC
--> CC )
222, 21syl 16 . . . . . . . . . . . 12  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0 p )  ->  G : CC --> CC )
23 mulcom 9081 . . . . . . . . . . . . 13  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  x.  y
)  =  ( y  x.  x ) )
2423adantl 454 . . . . . . . . . . . 12  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0 p )  /\  (
x  e.  CC  /\  y  e.  CC )
)  ->  ( x  x.  y )  =  ( y  x.  x ) )
2518, 20, 22, 24caofcom 6339 . . . . . . . . . . 11  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0 p )  ->  ( F  o F  x.  G
)  =  ( G  o F  x.  F
) )
266, 25syl5eq 2482 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0 p )  ->  H  =  ( G  o F  x.  F )
)
2726oveq1d 6099 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0 p )  ->  ( H  o F  -  ( G  o F  x.  ( H quot  G ) ) )  =  ( ( G  o F  x.  F
)  o F  -  ( G  o F  x.  ( H quot  G ) ) ) )
28 plyf 20122 . . . . . . . . . . 11  |-  ( ( H quot  G )  e.  (Poly `  CC )  ->  ( H quot  G ) : CC --> CC )
2912, 28syl 16 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0 p )  ->  ( H quot  G ) : CC --> CC )
30 subdi 9472 . . . . . . . . . . 11  |-  ( ( x  e.  CC  /\  y  e.  CC  /\  z  e.  CC )  ->  (
x  x.  ( y  -  z ) )  =  ( ( x  x.  y )  -  ( x  x.  z
) ) )
3130adantl 454 . . . . . . . . . 10  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0 p )  /\  (
x  e.  CC  /\  y  e.  CC  /\  z  e.  CC ) )  -> 
( x  x.  (
y  -  z ) )  =  ( ( x  x.  y )  -  ( x  x.  z ) ) )
3218, 22, 20, 29, 31caofdi 6343 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0 p )  ->  ( G  o F  x.  ( F  o F  -  ( H quot  G ) ) )  =  ( ( G  o F  x.  F
)  o F  -  ( G  o F  x.  ( H quot  G ) ) ) )
3327, 32eqtr4d 2473 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0 p )  ->  ( H  o F  -  ( G  o F  x.  ( H quot  G ) ) )  =  ( G  o F  x.  ( F  o F  -  ( H quot  G ) ) ) )
3433eqeq1d 2446 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0 p )  ->  (
( H  o F  -  ( G  o F  x.  ( H quot  G ) ) )  =  0 p  <->  ( G  o F  x.  ( F  o F  -  ( H quot  G ) ) )  =  0 p ) )
3510neneqd 2619 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0 p )  ->  -.  G  =  0 p
)
36 biorf 396 . . . . . . . 8  |-  ( -.  G  =  0 p  ->  ( ( F  o F  -  ( H quot  G ) )  =  0 p  <->  ( G  =  0 p  \/  ( F  o F  -  ( H quot  G
) )  =  0 p ) ) )
3735, 36syl 16 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0 p )  ->  (
( F  o F  -  ( H quot  G
) )  =  0 p  <->  ( G  =  0 p  \/  ( F  o F  -  ( H quot  G ) )  =  0 p ) ) )
3816, 34, 373bitr4d 278 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0 p )  ->  (
( H  o F  -  ( G  o F  x.  ( H quot  G ) ) )  =  0 p  <->  ( F  o F  -  ( H quot  G ) )  =  0 p ) )
3938biimpd 200 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0 p )  ->  (
( H  o F  -  ( G  o F  x.  ( H quot  G ) ) )  =  0 p  ->  ( F  o F  -  ( H quot  G ) )  =  0 p ) )
40 eqid 2438 . . . . . . . . . . 11  |-  (deg `  G )  =  (deg
`  G )
41 eqid 2438 . . . . . . . . . . 11  |-  (deg `  ( F  o F  -  ( H quot  G
) ) )  =  (deg `  ( F  o F  -  ( H quot  G ) ) )
4240, 41dgrmul 20193 . . . . . . . . . 10  |-  ( ( ( G  e.  (Poly `  CC )  /\  G  =/=  0 p )  /\  ( ( F  o F  -  ( H quot  G ) )  e.  (Poly `  CC )  /\  ( F  o F  -  ( H quot  G ) )  =/=  0 p ) )  ->  (deg `  ( G  o F  x.  ( F  o F  -  ( H quot  G ) ) ) )  =  ( (deg
`  G )  +  (deg `  ( F  o F  -  ( H quot  G ) ) ) ) )
4342expr 600 . . . . . . . . 9  |-  ( ( ( G  e.  (Poly `  CC )  /\  G  =/=  0 p )  /\  ( F  o F  -  ( H quot  G
) )  e.  (Poly `  CC ) )  -> 
( ( F  o F  -  ( H quot  G ) )  =/=  0 p  ->  (deg `  ( G  o F  x.  ( F  o F  -  ( H quot  G ) ) ) )  =  ( (deg
`  G )  +  (deg `  ( F  o F  -  ( H quot  G ) ) ) ) ) )
443, 10, 14, 43syl21anc 1184 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0 p )  ->  (
( F  o F  -  ( H quot  G
) )  =/=  0 p  ->  (deg `  ( G  o F  x.  ( F  o F  -  ( H quot  G ) ) ) )  =  ( (deg
`  G )  +  (deg `  ( F  o F  -  ( H quot  G ) ) ) ) ) )
45 dgrcl 20157 . . . . . . . . . . . 12  |-  ( G  e.  (Poly `  S
)  ->  (deg `  G
)  e.  NN0 )
462, 45syl 16 . . . . . . . . . . 11  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0 p )  ->  (deg `  G )  e.  NN0 )
4746nn0red 10280 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0 p )  ->  (deg `  G )  e.  RR )
48 dgrcl 20157 . . . . . . . . . . 11  |-  ( ( F  o F  -  ( H quot  G )
)  e.  (Poly `  CC )  ->  (deg `  ( F  o F  -  ( H quot  G
) ) )  e. 
NN0 )
4914, 48syl 16 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0 p )  ->  (deg `  ( F  o F  -  ( H quot  G
) ) )  e. 
NN0 )
50 nn0addge1 10271 . . . . . . . . . 10  |-  ( ( (deg `  G )  e.  RR  /\  (deg `  ( F  o F  -  ( H quot  G
) ) )  e. 
NN0 )  ->  (deg `  G )  <_  (
(deg `  G )  +  (deg `  ( F  o F  -  ( H quot  G ) ) ) ) )
5147, 49, 50syl2anc 644 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0 p )  ->  (deg `  G )  <_  (
(deg `  G )  +  (deg `  ( F  o F  -  ( H quot  G ) ) ) ) )
52 breq2 4219 . . . . . . . . 9  |-  ( (deg
`  ( G  o F  x.  ( F  o F  -  ( H quot  G ) ) ) )  =  ( (deg
`  G )  +  (deg `  ( F  o F  -  ( H quot  G ) ) ) )  ->  ( (deg `  G )  <_  (deg `  ( G  o F  x.  ( F  o F  -  ( H quot  G ) ) ) )  <-> 
(deg `  G )  <_  ( (deg `  G
)  +  (deg `  ( F  o F  -  ( H quot  G
) ) ) ) ) )
5351, 52syl5ibrcom 215 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0 p )  ->  (
(deg `  ( G  o F  x.  ( F  o F  -  ( H quot  G ) ) ) )  =  ( (deg
`  G )  +  (deg `  ( F  o F  -  ( H quot  G ) ) ) )  ->  (deg `  G
)  <_  (deg `  ( G  o F  x.  ( F  o F  -  ( H quot  G ) ) ) ) ) )
5444, 53syld 43 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0 p )  ->  (
( F  o F  -  ( H quot  G
) )  =/=  0 p  ->  (deg `  G
)  <_  (deg `  ( G  o F  x.  ( F  o F  -  ( H quot  G ) ) ) ) ) )
5533fveq2d 5735 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0 p )  ->  (deg `  ( H  o F  -  ( G  o F  x.  ( H quot  G ) ) ) )  =  (deg `  ( G  o F  x.  ( F  o F  -  ( H quot  G ) ) ) ) )
5655breq2d 4227 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0 p )  ->  (
(deg `  G )  <_  (deg `  ( H  o F  -  ( G  o F  x.  ( H quot  G ) ) ) )  <->  (deg `  G )  <_  (deg `  ( G  o F  x.  ( F  o F  -  ( H quot  G ) ) ) ) ) )
57 plymulcl 20145 . . . . . . . . . . . . 13  |-  ( ( G  e.  (Poly `  CC )  /\  ( H quot  G )  e.  (Poly `  CC ) )  -> 
( G  o F  x.  ( H quot  G
) )  e.  (Poly `  CC ) )
583, 12, 57syl2anc 644 . . . . . . . . . . . 12  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0 p )  ->  ( G  o F  x.  ( H quot  G ) )  e.  (Poly `  CC )
)
59 plysubcl 20146 . . . . . . . . . . . 12  |-  ( ( H  e.  (Poly `  CC )  /\  ( G  o F  x.  ( H quot  G ) )  e.  (Poly `  CC )
)  ->  ( H  o F  -  ( G  o F  x.  ( H quot  G ) ) )  e.  (Poly `  CC ) )
609, 58, 59syl2anc 644 . . . . . . . . . . 11  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0 p )  ->  ( H  o F  -  ( G  o F  x.  ( H quot  G ) ) )  e.  (Poly `  CC ) )
61 dgrcl 20157 . . . . . . . . . . 11  |-  ( ( H  o F  -  ( G  o F  x.  ( H quot  G ) ) )  e.  (Poly `  CC )  ->  (deg `  ( H  o F  -  ( G  o F  x.  ( H quot  G ) ) ) )  e.  NN0 )
6260, 61syl 16 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0 p )  ->  (deg `  ( H  o F  -  ( G  o F  x.  ( H quot  G ) ) ) )  e.  NN0 )
6362nn0red 10280 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0 p )  ->  (deg `  ( H  o F  -  ( G  o F  x.  ( H quot  G ) ) ) )  e.  RR )
6447, 63lenltd 9224 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0 p )  ->  (
(deg `  G )  <_  (deg `  ( H  o F  -  ( G  o F  x.  ( H quot  G ) ) ) )  <->  -.  (deg `  ( H  o F  -  ( G  o F  x.  ( H quot  G ) ) ) )  <  (deg `  G ) ) )
6556, 64bitr3d 248 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0 p )  ->  (
(deg `  G )  <_  (deg `  ( G  o F  x.  ( F  o F  -  ( H quot  G ) ) ) )  <->  -.  (deg `  ( H  o F  -  ( G  o F  x.  ( H quot  G ) ) ) )  <  (deg `  G ) ) )
6654, 65sylibd 207 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0 p )  ->  (
( F  o F  -  ( H quot  G
) )  =/=  0 p  ->  -.  (deg `  ( H  o F  -  ( G  o F  x.  ( H quot  G ) ) ) )  <  (deg `  G ) ) )
6766necon4ad 2667 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0 p )  ->  (
(deg `  ( H  o F  -  ( G  o F  x.  ( H quot  G ) ) ) )  <  (deg `  G )  ->  ( F  o F  -  ( H quot  G ) )  =  0 p ) )
68 eqid 2438 . . . . . . 7  |-  ( H  o F  -  ( G  o F  x.  ( H quot  G ) ) )  =  ( H  o F  -  ( G  o F  x.  ( H quot  G ) ) )
6968quotdgr 20225 . . . . . 6  |-  ( ( H  e.  (Poly `  CC )  /\  G  e.  (Poly `  CC )  /\  G  =/=  0 p )  ->  (
( H  o F  -  ( G  o F  x.  ( H quot  G ) ) )  =  0 p  \/  (deg `  ( H  o F  -  ( G  o F  x.  ( H quot  G ) ) ) )  <  (deg `  G
) ) )
709, 3, 10, 69syl3anc 1185 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0 p )  ->  (
( H  o F  -  ( G  o F  x.  ( H quot  G ) ) )  =  0 p  \/  (deg `  ( H  o F  -  ( G  o F  x.  ( H quot  G ) ) ) )  <  (deg `  G
) ) )
7139, 67, 70mpjaod 372 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0 p )  ->  ( F  o F  -  ( H quot  G ) )  =  0 p )
72 df-0p 19565 . . . 4  |-  0 p  =  ( CC  X.  { 0 } )
7371, 72syl6eq 2486 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0 p )  ->  ( F  o F  -  ( H quot  G ) )  =  ( CC  X.  {
0 } ) )
74 ofsubeq0 10002 . . . 4  |-  ( ( CC  e.  _V  /\  F : CC --> CC  /\  ( H quot  G ) : CC --> CC )  -> 
( ( F  o F  -  ( H quot  G ) )  =  ( CC  X.  { 0 } )  <->  F  =  ( H quot  G )
) )
7518, 20, 29, 74syl3anc 1185 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0 p )  ->  (
( F  o F  -  ( H quot  G
) )  =  ( CC  X.  { 0 } )  <->  F  =  ( H quot  G )
) )
7673, 75mpbid 203 . 2  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0 p )  ->  F  =  ( H quot  G
) )
7776eqcomd 2443 1  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0 p )  ->  ( H quot  G )  =  F )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    \/ wo 359    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   _Vcvv 2958   {csn 3816   class class class wbr 4215    X. cxp 4879   -->wf 5453   ` cfv 5457  (class class class)co 6084    o Fcof 6306   CCcc 8993   RRcr 8994   0cc0 8995    + caddc 8998    x. cmul 9000    < clt 9125    <_ cle 9126    - cmin 9296   NN0cn0 10226   0 pc0p 19564  Polycply 20108  degcdgr 20111   quot cquot 20212
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
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-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-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-oadd 6731  df-er 6908  df-map 7023  df-pm 7024  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-div 9683  df-nn 10006  df-2 10063  df-3 10064  df-n0 10227  df-z 10288  df-uz 10494  df-rp 10618  df-fz 11049  df-fzo 11141  df-fl 11207  df-seq 11329  df-exp 11388  df-hash 11624  df-cj 11909  df-re 11910  df-im 11911  df-sqr 12045  df-abs 12046  df-clim 12287  df-rlim 12288  df-sum 12485  df-0p 19565  df-ply 20112  df-coe 20114  df-dgr 20115  df-quot 20213
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