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Theorem quotcan 19689
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 19582 . . . . . . . . 9  |-  (Poly `  S )  C_  (Poly `  CC )
2 simp2 956 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0 p )  ->  G  e.  (Poly `  S )
)
31, 2sseldi 3178 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0 p )  ->  G  e.  (Poly `  CC )
)
4 simp1 955 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0 p )  ->  F  e.  (Poly `  S )
)
51, 4sseldi 3178 . . . . . . . . 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 19603 . . . . . . . . . . . 12  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( F  o F  x.  G
)  e.  (Poly `  CC ) )
86, 7syl5eqel 2367 . . . . . . . . . . 11  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  H  e.  (Poly `  CC ) )
983adant3 975 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0 p )  ->  H  e.  (Poly `  CC )
)
10 simp3 957 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0 p )  ->  G  =/=  0 p )
11 quotcl2 19682 . . . . . . . . . 10  |-  ( ( H  e.  (Poly `  CC )  /\  G  e.  (Poly `  CC )  /\  G  =/=  0 p )  ->  ( H quot  G )  e.  (Poly `  CC ) )
129, 3, 10, 11syl3anc 1182 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0 p )  ->  ( H quot  G )  e.  (Poly `  CC ) )
13 plysubcl 19604 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  CC )  /\  ( H quot  G )  e.  (Poly `  CC ) )  -> 
( F  o F  -  ( H quot  G
) )  e.  (Poly `  CC ) )
145, 12, 13syl2anc 642 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0 p )  ->  ( F  o F  -  ( H quot  G ) )  e.  (Poly `  CC )
)
15 plymul0or 19661 . . . . . . . 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 642 . . . . . . 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 8818 . . . . . . . . . . . . 13  |-  CC  e.  _V
1817a1i 10 . . . . . . . . . . . 12  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0 p )  ->  CC  e.  _V )
19 plyf 19580 . . . . . . . . . . . . 13  |-  ( F  e.  (Poly `  S
)  ->  F : CC
--> CC )
204, 19syl 15 . . . . . . . . . . . 12  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0 p )  ->  F : CC --> CC )
21 plyf 19580 . . . . . . . . . . . . 13  |-  ( G  e.  (Poly `  S
)  ->  G : CC
--> CC )
222, 21syl 15 . . . . . . . . . . . 12  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0 p )  ->  G : CC --> CC )
23 mulcom 8823 . . . . . . . . . . . . 13  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  x.  y
)  =  ( y  x.  x ) )
2423adantl 452 . . . . . . . . . . . 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 6109 . . . . . . . . . . 11  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0 p )  ->  ( F  o F  x.  G
)  =  ( G  o F  x.  F
) )
266, 25syl5eq 2327 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0 p )  ->  H  =  ( G  o F  x.  F )
)
2726oveq1d 5873 . . . . . . . . 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 19580 . . . . . . . . . . 11  |-  ( ( H quot  G )  e.  (Poly `  CC )  ->  ( H quot  G ) : CC --> CC )
2912, 28syl 15 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0 p )  ->  ( H quot  G ) : CC --> CC )
30 subdi 9213 . . . . . . . . . . 11  |-  ( ( x  e.  CC  /\  y  e.  CC  /\  z  e.  CC )  ->  (
x  x.  ( y  -  z ) )  =  ( ( x  x.  y )  -  ( x  x.  z
) ) )
3130adantl 452 . . . . . . . . . 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 6113 . . . . . . . . 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 2318 . . . . . . . 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 2291 . . . . . . 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 2462 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0 p )  ->  -.  G  =  0 p
)
36 biorf 394 . . . . . . . 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 15 . . . . . . 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 276 . . . . . 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 198 . . . . 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 2283 . . . . . . . . . . 11  |-  (deg `  G )  =  (deg
`  G )
41 eqid 2283 . . . . . . . . . . 11  |-  (deg `  ( F  o F  -  ( H quot  G
) ) )  =  (deg `  ( F  o F  -  ( H quot  G ) ) )
4240, 41dgrmul 19651 . . . . . . . . . 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 598 . . . . . . . . 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 1181 . . . . . . . 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 19615 . . . . . . . . . . . 12  |-  ( G  e.  (Poly `  S
)  ->  (deg `  G
)  e.  NN0 )
462, 45syl 15 . . . . . . . . . . 11  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0 p )  ->  (deg `  G )  e.  NN0 )
4746nn0red 10019 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0 p )  ->  (deg `  G )  e.  RR )
48 dgrcl 19615 . . . . . . . . . . 11  |-  ( ( F  o F  -  ( H quot  G )
)  e.  (Poly `  CC )  ->  (deg `  ( F  o F  -  ( H quot  G
) ) )  e. 
NN0 )
4914, 48syl 15 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0 p )  ->  (deg `  ( F  o F  -  ( H quot  G
) ) )  e. 
NN0 )
50 nn0addge1 10010 . . . . . . . . . 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 642 . . . . . . . . 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 4027 . . . . . . . . 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 213 . . . . . . . 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 40 . . . . . . 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 5529 . . . . . . . . 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 4035 . . . . . . . 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 19603 . . . . . . . . . . . . 13  |-  ( ( G  e.  (Poly `  CC )  /\  ( H quot  G )  e.  (Poly `  CC ) )  -> 
( G  o F  x.  ( H quot  G
) )  e.  (Poly `  CC ) )
583, 12, 57syl2anc 642 . . . . . . . . . . . 12  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0 p )  ->  ( G  o F  x.  ( H quot  G ) )  e.  (Poly `  CC )
)
59 plysubcl 19604 . . . . . . . . . . . 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 642 . . . . . . . . . . 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 19615 . . . . . . . . . . 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 15 . . . . . . . . . 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 10019 . . . . . . . . 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 8965 . . . . . . . 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 246 . . . . . . 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 205 . . . . . 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 2507 . . . . 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 2283 . . . . . . 7  |-  ( H  o F  -  ( G  o F  x.  ( H quot  G ) ) )  =  ( H  o F  -  ( G  o F  x.  ( H quot  G ) ) )
6968quotdgr 19683 . . . . . 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 1182 . . . . 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 370 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0 p )  ->  ( F  o F  -  ( H quot  G ) )  =  0 p )
72 df-0p 19025 . . . 4  |-  0 p  =  ( CC  X.  { 0 } )
7371, 72syl6eq 2331 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0 p )  ->  ( F  o F  -  ( H quot  G ) )  =  ( CC  X.  {
0 } ) )
74 ofsubeq0 9743 . . . 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 1182 . . 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 201 . 2  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0 p )  ->  F  =  ( H quot  G
) )
7776eqcomd 2288 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 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   _Vcvv 2788   {csn 3640   class class class wbr 4023    X. cxp 4687   -->wf 5251   ` cfv 5255  (class class class)co 5858    o Fcof 6076   CCcc 8735   RRcr 8736   0cc0 8737    + caddc 8740    x. cmul 8742    < clt 8867    <_ cle 8868    - cmin 9037   NN0cn0 9965   0 pc0p 19024  Polycply 19566  degcdgr 19569   quot cquot 19670
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
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-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-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  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-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-fz 10783  df-fzo 10871  df-fl 10925  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-rlim 11963  df-sum 12159  df-0p 19025  df-ply 19570  df-coe 19572  df-dgr 19573  df-quot 19671
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