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Theorem quotcan 20187
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 20080 . . . . . . . . 9  |-  (Poly `  S )  C_  (Poly `  CC )
2 simp2 958 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0 p )  ->  G  e.  (Poly `  S )
)
31, 2sseldi 3314 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0 p )  ->  G  e.  (Poly `  CC )
)
4 simp1 957 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0 p )  ->  F  e.  (Poly `  S )
)
51, 4sseldi 3314 . . . . . . . . 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 20101 . . . . . . . . . . . 12  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( F  o F  x.  G
)  e.  (Poly `  CC ) )
86, 7syl5eqel 2496 . . . . . . . . . . 11  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  H  e.  (Poly `  CC ) )
983adant3 977 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0 p )  ->  H  e.  (Poly `  CC )
)
10 simp3 959 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0 p )  ->  G  =/=  0 p )
11 quotcl2 20180 . . . . . . . . . 10  |-  ( ( H  e.  (Poly `  CC )  /\  G  e.  (Poly `  CC )  /\  G  =/=  0 p )  ->  ( H quot  G )  e.  (Poly `  CC ) )
129, 3, 10, 11syl3anc 1184 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0 p )  ->  ( H quot  G )  e.  (Poly `  CC ) )
13 plysubcl 20102 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  CC )  /\  ( H quot  G )  e.  (Poly `  CC ) )  -> 
( F  o F  -  ( H quot  G
) )  e.  (Poly `  CC ) )
145, 12, 13syl2anc 643 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0 p )  ->  ( F  o F  -  ( H quot  G ) )  e.  (Poly `  CC )
)
15 plymul0or 20159 . . . . . . . 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 643 . . . . . . 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 9035 . . . . . . . . . . . . 13  |-  CC  e.  _V
1817a1i 11 . . . . . . . . . . . 12  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0 p )  ->  CC  e.  _V )
19 plyf 20078 . . . . . . . . . . . . 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 20078 . . . . . . . . . . . . 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 9040 . . . . . . . . . . . . 13  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  x.  y
)  =  ( y  x.  x ) )
2423adantl 453 . . . . . . . . . . . 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 6303 . . . . . . . . . . 11  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0 p )  ->  ( F  o F  x.  G
)  =  ( G  o F  x.  F
) )
266, 25syl5eq 2456 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0 p )  ->  H  =  ( G  o F  x.  F )
)
2726oveq1d 6063 . . . . . . . . 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 20078 . . . . . . . . . . 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 9431 . . . . . . . . . . 11  |-  ( ( x  e.  CC  /\  y  e.  CC  /\  z  e.  CC )  ->  (
x  x.  ( y  -  z ) )  =  ( ( x  x.  y )  -  ( x  x.  z
) ) )
3130adantl 453 . . . . . . . . . 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 6307 . . . . . . . . 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 2447 . . . . . . . 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 2420 . . . . . . 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 2591 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0 p )  ->  -.  G  =  0 p
)
36 biorf 395 . . . . . . . 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 277 . . . . . 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 199 . . . . 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 2412 . . . . . . . . . . 11  |-  (deg `  G )  =  (deg
`  G )
41 eqid 2412 . . . . . . . . . . 11  |-  (deg `  ( F  o F  -  ( H quot  G
) ) )  =  (deg `  ( F  o F  -  ( H quot  G ) ) )
4240, 41dgrmul 20149 . . . . . . . . . 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 599 . . . . . . . . 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 1183 . . . . . . . 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 20113 . . . . . . . . . . . 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 10239 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0 p )  ->  (deg `  G )  e.  RR )
48 dgrcl 20113 . . . . . . . . . . 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 10230 . . . . . . . . . 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 643 . . . . . . . . 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 4184 . . . . . . . . 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 214 . . . . . . . 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 42 . . . . . . 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 5699 . . . . . . . . 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 4192 . . . . . . . 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 20101 . . . . . . . . . . . . 13  |-  ( ( G  e.  (Poly `  CC )  /\  ( H quot  G )  e.  (Poly `  CC ) )  -> 
( G  o F  x.  ( H quot  G
) )  e.  (Poly `  CC ) )
583, 12, 57syl2anc 643 . . . . . . . . . . . 12  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0 p )  ->  ( G  o F  x.  ( H quot  G ) )  e.  (Poly `  CC )
)
59 plysubcl 20102 . . . . . . . . . . . 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 643 . . . . . . . . . . 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 20113 . . . . . . . . . . 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 10239 . . . . . . . . 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 9183 . . . . . . . 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 247 . . . . . . 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 206 . . . . . 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 2636 . . . . 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 2412 . . . . . . 7  |-  ( H  o F  -  ( G  o F  x.  ( H quot  G ) ) )  =  ( H  o F  -  ( G  o F  x.  ( H quot  G ) ) )
6968quotdgr 20181 . . . . . 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 1184 . . . . 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 371 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0 p )  ->  ( F  o F  -  ( H quot  G ) )  =  0 p )
72 df-0p 19523 . . . 4  |-  0 p  =  ( CC  X.  { 0 } )
7371, 72syl6eq 2460 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0 p )  ->  ( F  o F  -  ( H quot  G ) )  =  ( CC  X.  {
0 } ) )
74 ofsubeq0 9961 . . . 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 1184 . . 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 202 . 2  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0 p )  ->  F  =  ( H quot  G
) )
7776eqcomd 2417 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 177    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2575   _Vcvv 2924   {csn 3782   class class class wbr 4180    X. cxp 4843   -->wf 5417   ` cfv 5421  (class class class)co 6048    o Fcof 6270   CCcc 8952   RRcr 8953   0cc0 8954    + caddc 8957    x. cmul 8959    < clt 9084    <_ cle 9085    - cmin 9255   NN0cn0 10185   0 pc0p 19522  Polycply 20064  degcdgr 20067   quot cquot 20168
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-inf2 7560  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031  ax-pre-sup 9032  ax-addf 9033
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-se 4510  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-isom 5430  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-of 6272  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-1o 6691  df-oadd 6695  df-er 6872  df-map 6987  df-pm 6988  df-en 7077  df-dom 7078  df-sdom 7079  df-fin 7080  df-sup 7412  df-oi 7443  df-card 7790  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-div 9642  df-nn 9965  df-2 10022  df-3 10023  df-n0 10186  df-z 10247  df-uz 10453  df-rp 10577  df-fz 11008  df-fzo 11099  df-fl 11165  df-seq 11287  df-exp 11346  df-hash 11582  df-cj 11867  df-re 11868  df-im 11869  df-sqr 12003  df-abs 12004  df-clim 12245  df-rlim 12246  df-sum 12443  df-0p 19523  df-ply 20068  df-coe 20070  df-dgr 20071  df-quot 20169
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