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Theorem quotcan 22467
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 22360 . . . . . . . . 9  |-  (Poly `  S )  C_  (Poly `  CC )
2 simp2 997 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  G  e.  (Poly `  S )
)
31, 2sseldi 3502 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  G  e.  (Poly `  CC )
)
4 simp1 996 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  F  e.  (Poly `  S )
)
51, 4sseldi 3502 . . . . . . . . 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 22381 . . . . . . . . . . . 12  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( F  oF  x.  G
)  e.  (Poly `  CC ) )
86, 7syl5eqel 2559 . . . . . . . . . . 11  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  H  e.  (Poly `  CC ) )
983adant3 1016 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  H  e.  (Poly `  CC )
)
10 simp3 998 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  G  =/=  0p )
11 quotcl2 22460 . . . . . . . . . 10  |-  ( ( H  e.  (Poly `  CC )  /\  G  e.  (Poly `  CC )  /\  G  =/=  0p )  ->  ( H quot  G )  e.  (Poly `  CC ) )
129, 3, 10, 11syl3anc 1228 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  ( H quot  G )  e.  (Poly `  CC ) )
13 plysubcl 22382 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  CC )  /\  ( H quot  G )  e.  (Poly `  CC ) )  -> 
( F  oF  -  ( H quot  G
) )  e.  (Poly `  CC ) )
145, 12, 13syl2anc 661 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  ( F  oF  -  ( H quot  G ) )  e.  (Poly `  CC )
)
15 plymul0or 22439 . . . . . . . 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 661 . . . . . . 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 9573 . . . . . . . . . . . . 13  |-  CC  e.  _V
1817a1i 11 . . . . . . . . . . . 12  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  CC  e.  _V )
19 plyf 22358 . . . . . . . . . . . . 13  |-  ( F  e.  (Poly `  S
)  ->  F : CC
--> CC )
204, 19syl 16 . . . . . . . . . . . 12  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  F : CC --> CC )
21 plyf 22358 . . . . . . . . . . . . 13  |-  ( G  e.  (Poly `  S
)  ->  G : CC
--> CC )
222, 21syl 16 . . . . . . . . . . . 12  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  G : CC --> CC )
23 mulcom 9578 . . . . . . . . . . . . 13  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  x.  y
)  =  ( y  x.  x ) )
2423adantl 466 . . . . . . . . . . . 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 6556 . . . . . . . . . . 11  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  ( F  oF  x.  G
)  =  ( G  oF  x.  F
) )
266, 25syl5eq 2520 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  H  =  ( G  oF  x.  F )
)
2726oveq1d 6299 . . . . . . . . 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 22358 . . . . . . . . . . 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  =/=  0p )  ->  ( H quot  G ) : CC --> CC )
30 subdi 9990 . . . . . . . . . . 11  |-  ( ( x  e.  CC  /\  y  e.  CC  /\  z  e.  CC )  ->  (
x  x.  ( y  -  z ) )  =  ( ( x  x.  y )  -  ( x  x.  z
) ) )
3130adantl 466 . . . . . . . . . 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 6560 . . . . . . . . 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 2511 . . . . . . . 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 2469 . . . . . . 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 2669 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  -.  G  =  0p
)
36 biorf 405 . . . . . . . 8  |-  ( -.  G  =  0p  ->  ( ( F  oF  -  ( H quot  G ) )  =  0p  <->  ( G  =  0p  \/  ( F  oF  -  ( H quot  G
) )  =  0p ) ) )
3735, 36syl 16 . . . . . . 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 285 . . . . . 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 207 . . . . 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 2467 . . . . . . . . . . 11  |-  (deg `  G )  =  (deg
`  G )
41 eqid 2467 . . . . . . . . . . 11  |-  (deg `  ( F  oF  -  ( H quot  G
) ) )  =  (deg `  ( F  oF  -  ( H quot  G ) ) )
4240, 41dgrmul 22429 . . . . . . . . . 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 615 . . . . . . . . 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 1227 . . . . . . . 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 22393 . . . . . . . . . . . 12  |-  ( G  e.  (Poly `  S
)  ->  (deg `  G
)  e.  NN0 )
462, 45syl 16 . . . . . . . . . . 11  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  (deg `  G )  e.  NN0 )
4746nn0red 10853 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  (deg `  G )  e.  RR )
48 dgrcl 22393 . . . . . . . . . . 11  |-  ( ( F  oF  -  ( H quot  G )
)  e.  (Poly `  CC )  ->  (deg `  ( F  oF  -  ( H quot  G
) ) )  e. 
NN0 )
4914, 48syl 16 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  (deg `  ( F  oF  -  ( H quot  G
) ) )  e. 
NN0 )
50 nn0addge1 10842 . . . . . . . . . 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 661 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  (deg `  G )  <_  (
(deg `  G )  +  (deg `  ( F  oF  -  ( H quot  G ) ) ) ) )
52 breq2 4451 . . . . . . . . 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 222 . . . . . . . 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 44 . . . . . . 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 5870 . . . . . . . . 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 4459 . . . . . . . 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 22381 . . . . . . . . . . . . 13  |-  ( ( G  e.  (Poly `  CC )  /\  ( H quot  G )  e.  (Poly `  CC ) )  -> 
( G  oF  x.  ( H quot  G
) )  e.  (Poly `  CC ) )
583, 12, 57syl2anc 661 . . . . . . . . . . . 12  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  ( G  oF  x.  ( H quot  G ) )  e.  (Poly `  CC )
)
59 plysubcl 22382 . . . . . . . . . . . 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 661 . . . . . . . . . . 11  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  ( H  oF  -  ( G  oF  x.  ( H quot  G ) ) )  e.  (Poly `  CC ) )
61 dgrcl 22393 . . . . . . . . . . 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 16 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  (deg `  ( H  oF  -  ( G  oF  x.  ( H quot  G ) ) ) )  e.  NN0 )
6362nn0red 10853 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  (deg `  ( H  oF  -  ( G  oF  x.  ( H quot  G ) ) ) )  e.  RR )
6447, 63lenltd 9730 . . . . . . . 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 255 . . . . . . 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 214 . . . . . 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 2687 . . . . 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 2467 . . . . . . 7  |-  ( H  oF  -  ( G  oF  x.  ( H quot  G ) ) )  =  ( H  oF  -  ( G  oF  x.  ( H quot  G ) ) )
6968quotdgr 22461 . . . . . 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 1228 . . . . 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 381 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  ( F  oF  -  ( H quot  G ) )  =  0p )
72 df-0p 21840 . . . 4  |-  0p  =  ( CC  X.  { 0 } )
7371, 72syl6eq 2524 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  ( F  oF  -  ( H quot  G ) )  =  ( CC  X.  {
0 } ) )
74 ofsubeq0 10533 . . . 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 1228 . . 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 210 . 2  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  F  =  ( H quot  G
) )
7776eqcomd 2475 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 184    \/ wo 368    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   _Vcvv 3113   {csn 4027   class class class wbr 4447    X. cxp 4997   -->wf 5584   ` cfv 5588  (class class class)co 6284    oFcof 6522   CCcc 9490   RRcr 9491   0cc0 9492    + caddc 9495    x. cmul 9497    < clt 9628    <_ cle 9629    - cmin 9805   NN0cn0 10795   0pc0p 21839  Polycply 22344  degcdgr 22347   quot cquot 22448
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-inf2 8058  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569  ax-pre-sup 9570  ax-addf 9571
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-of 6524  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-oadd 7134  df-er 7311  df-map 7422  df-pm 7423  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-sup 7901  df-oi 7935  df-card 8320  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-div 10207  df-nn 10537  df-2 10594  df-3 10595  df-n0 10796  df-z 10865  df-uz 11083  df-rp 11221  df-fz 11673  df-fzo 11793  df-fl 11897  df-seq 12076  df-exp 12135  df-hash 12374  df-cj 12895  df-re 12896  df-im 12897  df-sqrt 13031  df-abs 13032  df-clim 13274  df-rlim 13275  df-sum 13472  df-0p 21840  df-ply 22348  df-coe 22350  df-dgr 22351  df-quot 22449
This theorem is referenced by: (None)
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