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Theorem quotcan 23130
Description: Exact division with a multiple. (Contributed by Mario Carneiro, 26-Jul-2014.)
Hypothesis
Ref Expression
quotcan.1  |-  H  =  ( F  oF  x.  G )
Assertion
Ref Expression
quotcan  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  ( 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 23022 . . . . . . . . 9  |-  (Poly `  S )  C_  (Poly `  CC )
2 simp2 1006 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  G  e.  (Poly `  S )
)
31, 2sseldi 3468 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  G  e.  (Poly `  CC )
)
4 simp1 1005 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  F  e.  (Poly `  S )
)
51, 4sseldi 3468 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  F  e.  (Poly `  CC )
)
6 quotcan.1 . . . . . . . . . . . 12  |-  H  =  ( F  oF  x.  G )
7 plymulcl 23043 . . . . . . . . . . . 12  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( F  oF  x.  G
)  e.  (Poly `  CC ) )
86, 7syl5eqel 2521 . . . . . . . . . . 11  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  H  e.  (Poly `  CC ) )
983adant3 1025 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  H  e.  (Poly `  CC )
)
10 simp3 1007 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  G  =/=  0p )
11 quotcl2 23123 . . . . . . . . . 10  |-  ( ( H  e.  (Poly `  CC )  /\  G  e.  (Poly `  CC )  /\  G  =/=  0p )  ->  ( H quot  G )  e.  (Poly `  CC ) )
129, 3, 10, 11syl3anc 1264 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  ( H quot  G )  e.  (Poly `  CC ) )
13 plysubcl 23044 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  CC )  /\  ( H quot  G )  e.  (Poly `  CC ) )  -> 
( F  oF  -  ( H quot  G
) )  e.  (Poly `  CC ) )
145, 12, 13syl2anc 665 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  ( F  oF  -  ( H quot  G ) )  e.  (Poly `  CC )
)
15 plymul0or 23102 . . . . . . . 8  |-  ( ( G  e.  (Poly `  CC )  /\  ( F  oF  -  ( H quot  G ) )  e.  (Poly `  CC )
)  ->  ( ( G  oF  x.  ( F  oF  -  ( H quot  G ) ) )  =  0p  <->  ( G  =  0p  \/  ( F  oF  -  ( H quot  G
) )  =  0p ) ) )
163, 14, 15syl2anc 665 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  (
( G  oF  x.  ( F  oF  -  ( H quot  G ) ) )  =  0p  <->  ( G  =  0p  \/  ( F  oF  -  ( H quot  G
) )  =  0p ) ) )
17 cnex 9619 . . . . . . . . . . . . 13  |-  CC  e.  _V
1817a1i 11 . . . . . . . . . . . 12  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  CC  e.  _V )
19 plyf 23020 . . . . . . . . . . . . 13  |-  ( F  e.  (Poly `  S
)  ->  F : CC
--> CC )
204, 19syl 17 . . . . . . . . . . . 12  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  F : CC --> CC )
21 plyf 23020 . . . . . . . . . . . . 13  |-  ( G  e.  (Poly `  S
)  ->  G : CC
--> CC )
222, 21syl 17 . . . . . . . . . . . 12  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  G : CC --> CC )
23 mulcom 9624 . . . . . . . . . . . . 13  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  x.  y
)  =  ( y  x.  x ) )
2423adantl 467 . . . . . . . . . . . 12  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  /\  (
x  e.  CC  /\  y  e.  CC )
)  ->  ( x  x.  y )  =  ( y  x.  x ) )
2518, 20, 22, 24caofcom 6577 . . . . . . . . . . 11  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  ( F  oF  x.  G
)  =  ( G  oF  x.  F
) )
266, 25syl5eq 2482 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  H  =  ( G  oF  x.  F )
)
2726oveq1d 6320 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  ( H  oF  -  ( G  oF  x.  ( H quot  G ) ) )  =  ( ( G  oF  x.  F
)  oF  -  ( G  oF  x.  ( H quot  G ) ) ) )
28 plyf 23020 . . . . . . . . . . 11  |-  ( ( H quot  G )  e.  (Poly `  CC )  ->  ( H quot  G ) : CC --> CC )
2912, 28syl 17 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  ( H quot  G ) : CC --> CC )
30 subdi 10051 . . . . . . . . . . 11  |-  ( ( x  e.  CC  /\  y  e.  CC  /\  z  e.  CC )  ->  (
x  x.  ( y  -  z ) )  =  ( ( x  x.  y )  -  ( x  x.  z
) ) )
3130adantl 467 . . . . . . . . . 10  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  /\  (
x  e.  CC  /\  y  e.  CC  /\  z  e.  CC ) )  -> 
( x  x.  (
y  -  z ) )  =  ( ( x  x.  y )  -  ( x  x.  z ) ) )
3218, 22, 20, 29, 31caofdi 6581 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  ( G  oF  x.  ( F  oF  -  ( H quot  G ) ) )  =  ( ( G  oF  x.  F
)  oF  -  ( G  oF  x.  ( H quot  G ) ) ) )
3327, 32eqtr4d 2473 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  ( H  oF  -  ( G  oF  x.  ( H quot  G ) ) )  =  ( G  oF  x.  ( F  oF  -  ( H quot  G ) ) ) )
3433eqeq1d 2431 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  (
( H  oF  -  ( G  oF  x.  ( H quot  G ) ) )  =  0p  <->  ( G  oF  x.  ( F  oF  -  ( H quot  G ) ) )  =  0p ) )
3510neneqd 2632 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  -.  G  =  0p
)
36 biorf 406 . . . . . . . 8  |-  ( -.  G  =  0p  ->  ( ( F  oF  -  ( H quot  G ) )  =  0p  <->  ( G  =  0p  \/  ( F  oF  -  ( H quot  G
) )  =  0p ) ) )
3735, 36syl 17 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  (
( F  oF  -  ( H quot  G
) )  =  0p  <->  ( G  =  0p  \/  ( F  oF  -  ( H quot  G ) )  =  0p ) ) )
3816, 34, 373bitr4d 288 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  (
( H  oF  -  ( G  oF  x.  ( H quot  G ) ) )  =  0p  <->  ( F  oF  -  ( H quot  G ) )  =  0p ) )
3938biimpd 210 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  (
( H  oF  -  ( G  oF  x.  ( H quot  G ) ) )  =  0p  ->  ( F  oF  -  ( H quot  G ) )  =  0p ) )
40 eqid 2429 . . . . . . . . . . 11  |-  (deg `  G )  =  (deg
`  G )
41 eqid 2429 . . . . . . . . . . 11  |-  (deg `  ( F  oF  -  ( H quot  G
) ) )  =  (deg `  ( F  oF  -  ( H quot  G ) ) )
4240, 41dgrmul 23092 . . . . . . . . . 10  |-  ( ( ( G  e.  (Poly `  CC )  /\  G  =/=  0p )  /\  ( ( F  oF  -  ( H quot  G ) )  e.  (Poly `  CC )  /\  ( F  oF  -  ( H quot  G ) )  =/=  0p ) )  ->  (deg `  ( G  oF  x.  ( F  oF  -  ( H quot  G ) ) ) )  =  ( (deg
`  G )  +  (deg `  ( F  oF  -  ( H quot  G ) ) ) ) )
4342expr 618 . . . . . . . . 9  |-  ( ( ( G  e.  (Poly `  CC )  /\  G  =/=  0p )  /\  ( F  oF  -  ( H quot  G
) )  e.  (Poly `  CC ) )  -> 
( ( F  oF  -  ( H quot  G ) )  =/=  0p  ->  (deg `  ( G  oF  x.  ( F  oF  -  ( H quot  G ) ) ) )  =  ( (deg
`  G )  +  (deg `  ( F  oF  -  ( H quot  G ) ) ) ) ) )
443, 10, 14, 43syl21anc 1263 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  (
( F  oF  -  ( H quot  G
) )  =/=  0p  ->  (deg `  ( G  oF  x.  ( F  oF  -  ( H quot  G ) ) ) )  =  ( (deg
`  G )  +  (deg `  ( F  oF  -  ( H quot  G ) ) ) ) ) )
45 dgrcl 23055 . . . . . . . . . . . 12  |-  ( G  e.  (Poly `  S
)  ->  (deg `  G
)  e.  NN0 )
462, 45syl 17 . . . . . . . . . . 11  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  (deg `  G )  e.  NN0 )
4746nn0red 10926 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  (deg `  G )  e.  RR )
48 dgrcl 23055 . . . . . . . . . . 11  |-  ( ( F  oF  -  ( H quot  G )
)  e.  (Poly `  CC )  ->  (deg `  ( F  oF  -  ( H quot  G
) ) )  e. 
NN0 )
4914, 48syl 17 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  (deg `  ( F  oF  -  ( H quot  G
) ) )  e. 
NN0 )
50 nn0addge1 10916 . . . . . . . . . 10  |-  ( ( (deg `  G )  e.  RR  /\  (deg `  ( F  oF  -  ( H quot  G
) ) )  e. 
NN0 )  ->  (deg `  G )  <_  (
(deg `  G )  +  (deg `  ( F  oF  -  ( H quot  G ) ) ) ) )
5147, 49, 50syl2anc 665 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  (deg `  G )  <_  (
(deg `  G )  +  (deg `  ( F  oF  -  ( H quot  G ) ) ) ) )
52 breq2 4430 . . . . . . . . 9  |-  ( (deg
`  ( G  oF  x.  ( F  oF  -  ( H quot  G ) ) ) )  =  ( (deg
`  G )  +  (deg `  ( F  oF  -  ( H quot  G ) ) ) )  ->  ( (deg `  G )  <_  (deg `  ( G  oF  x.  ( F  oF  -  ( H quot  G ) ) ) )  <-> 
(deg `  G )  <_  ( (deg `  G
)  +  (deg `  ( F  oF  -  ( H quot  G
) ) ) ) ) )
5351, 52syl5ibrcom 225 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  (
(deg `  ( G  oF  x.  ( F  oF  -  ( H quot  G ) ) ) )  =  ( (deg
`  G )  +  (deg `  ( F  oF  -  ( H quot  G ) ) ) )  ->  (deg `  G
)  <_  (deg `  ( G  oF  x.  ( F  oF  -  ( H quot  G ) ) ) ) ) )
5444, 53syld 45 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  (
( F  oF  -  ( H quot  G
) )  =/=  0p  ->  (deg `  G
)  <_  (deg `  ( G  oF  x.  ( F  oF  -  ( H quot  G ) ) ) ) ) )
5533fveq2d 5885 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  (deg `  ( H  oF  -  ( G  oF  x.  ( H quot  G ) ) ) )  =  (deg `  ( G  oF  x.  ( F  oF  -  ( H quot  G ) ) ) ) )
5655breq2d 4438 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  (
(deg `  G )  <_  (deg `  ( H  oF  -  ( G  oF  x.  ( H quot  G ) ) ) )  <->  (deg `  G )  <_  (deg `  ( G  oF  x.  ( F  oF  -  ( H quot  G ) ) ) ) ) )
57 plymulcl 23043 . . . . . . . . . . . . 13  |-  ( ( G  e.  (Poly `  CC )  /\  ( H quot  G )  e.  (Poly `  CC ) )  -> 
( G  oF  x.  ( H quot  G
) )  e.  (Poly `  CC ) )
583, 12, 57syl2anc 665 . . . . . . . . . . . 12  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  ( G  oF  x.  ( H quot  G ) )  e.  (Poly `  CC )
)
59 plysubcl 23044 . . . . . . . . . . . 12  |-  ( ( H  e.  (Poly `  CC )  /\  ( G  oF  x.  ( H quot  G ) )  e.  (Poly `  CC )
)  ->  ( H  oF  -  ( G  oF  x.  ( H quot  G ) ) )  e.  (Poly `  CC ) )
609, 58, 59syl2anc 665 . . . . . . . . . . 11  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  ( H  oF  -  ( G  oF  x.  ( H quot  G ) ) )  e.  (Poly `  CC ) )
61 dgrcl 23055 . . . . . . . . . . 11  |-  ( ( H  oF  -  ( G  oF  x.  ( H quot  G ) ) )  e.  (Poly `  CC )  ->  (deg `  ( H  oF  -  ( G  oF  x.  ( H quot  G ) ) ) )  e.  NN0 )
6260, 61syl 17 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  (deg `  ( H  oF  -  ( G  oF  x.  ( H quot  G ) ) ) )  e.  NN0 )
6362nn0red 10926 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  (deg `  ( H  oF  -  ( G  oF  x.  ( H quot  G ) ) ) )  e.  RR )
6447, 63lenltd 9780 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  (
(deg `  G )  <_  (deg `  ( H  oF  -  ( G  oF  x.  ( H quot  G ) ) ) )  <->  -.  (deg `  ( H  oF  -  ( G  oF  x.  ( H quot  G ) ) ) )  <  (deg `  G ) ) )
6556, 64bitr3d 258 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  (
(deg `  G )  <_  (deg `  ( G  oF  x.  ( F  oF  -  ( H quot  G ) ) ) )  <->  -.  (deg `  ( H  oF  -  ( G  oF  x.  ( H quot  G ) ) ) )  <  (deg `  G ) ) )
6654, 65sylibd 217 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  (
( F  oF  -  ( H quot  G
) )  =/=  0p  ->  -.  (deg `  ( H  oF  -  ( G  oF  x.  ( H quot  G ) ) ) )  <  (deg `  G ) ) )
6766necon4ad 2651 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  (
(deg `  ( H  oF  -  ( G  oF  x.  ( H quot  G ) ) ) )  <  (deg `  G )  ->  ( F  oF  -  ( H quot  G ) )  =  0p ) )
68 eqid 2429 . . . . . . 7  |-  ( H  oF  -  ( G  oF  x.  ( H quot  G ) ) )  =  ( H  oF  -  ( G  oF  x.  ( H quot  G ) ) )
6968quotdgr 23124 . . . . . 6  |-  ( ( H  e.  (Poly `  CC )  /\  G  e.  (Poly `  CC )  /\  G  =/=  0p )  ->  (
( H  oF  -  ( G  oF  x.  ( H quot  G ) ) )  =  0p  \/  (deg `  ( H  oF  -  ( G  oF  x.  ( H quot  G ) ) ) )  <  (deg `  G
) ) )
709, 3, 10, 69syl3anc 1264 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  (
( H  oF  -  ( G  oF  x.  ( H quot  G ) ) )  =  0p  \/  (deg `  ( H  oF  -  ( G  oF  x.  ( H quot  G ) ) ) )  <  (deg `  G
) ) )
7139, 67, 70mpjaod 382 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  ( F  oF  -  ( H quot  G ) )  =  0p )
72 df-0p 22505 . . . 4  |-  0p  =  ( CC  X.  { 0 } )
7371, 72syl6eq 2486 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  ( F  oF  -  ( H quot  G ) )  =  ( CC  X.  {
0 } ) )
74 ofsubeq0 10606 . . . 4  |-  ( ( CC  e.  _V  /\  F : CC --> CC  /\  ( H quot  G ) : CC --> CC )  -> 
( ( F  oF  -  ( H quot  G ) )  =  ( CC  X.  { 0 } )  <->  F  =  ( H quot  G )
) )
7518, 20, 29, 74syl3anc 1264 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  (
( F  oF  -  ( H quot  G
) )  =  ( CC  X.  { 0 } )  <->  F  =  ( H quot  G )
) )
7673, 75mpbid 213 . 2  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  F  =  ( H quot  G
) )
7776eqcomd 2437 1  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  ( H quot  G )  =  F )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    \/ wo 369    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870    =/= wne 2625   _Vcvv 3087   {csn 4002   class class class wbr 4426    X. cxp 4852   -->wf 5597   ` cfv 5601  (class class class)co 6305    oFcof 6543   CCcc 9536   RRcr 9537   0cc0 9538    + caddc 9541    x. cmul 9543    < clt 9674    <_ cle 9675    - cmin 9859   NN0cn0 10869   0pc0p 22504  Polycply 23006  degcdgr 23009   quot cquot 23111
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-inf2 8146  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-pre-sup 9616  ax-addf 9617
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-se 4814  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-of 6545  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-oadd 7194  df-er 7371  df-map 7482  df-pm 7483  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-sup 7962  df-inf 7963  df-oi 8025  df-card 8372  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  df-nn 10610  df-2 10668  df-3 10669  df-n0 10870  df-z 10938  df-uz 11160  df-rp 11303  df-fz 11783  df-fzo 11914  df-fl 12025  df-seq 12211  df-exp 12270  df-hash 12513  df-cj 13141  df-re 13142  df-im 13143  df-sqrt 13277  df-abs 13278  df-clim 13530  df-rlim 13531  df-sum 13731  df-0p 22505  df-ply 23010  df-coe 23012  df-dgr 23013  df-quot 23112
This theorem is referenced by: (None)
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