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Theorem quotcan 21718
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 21611 . . . . . . . . 9  |-  (Poly `  S )  C_  (Poly `  CC )
2 simp2 984 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  G  e.  (Poly `  S )
)
31, 2sseldi 3351 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  G  e.  (Poly `  CC )
)
4 simp1 983 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  F  e.  (Poly `  S )
)
51, 4sseldi 3351 . . . . . . . . 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 21632 . . . . . . . . . . . 12  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( F  oF  x.  G
)  e.  (Poly `  CC ) )
86, 7syl5eqel 2525 . . . . . . . . . . 11  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  H  e.  (Poly `  CC ) )
983adant3 1003 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  H  e.  (Poly `  CC )
)
10 simp3 985 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  G  =/=  0p )
11 quotcl2 21711 . . . . . . . . . 10  |-  ( ( H  e.  (Poly `  CC )  /\  G  e.  (Poly `  CC )  /\  G  =/=  0p )  ->  ( H quot  G )  e.  (Poly `  CC ) )
129, 3, 10, 11syl3anc 1213 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  ( H quot  G )  e.  (Poly `  CC ) )
13 plysubcl 21633 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  CC )  /\  ( H quot  G )  e.  (Poly `  CC ) )  -> 
( F  oF  -  ( H quot  G
) )  e.  (Poly `  CC ) )
145, 12, 13syl2anc 656 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  ( F  oF  -  ( H quot  G ) )  e.  (Poly `  CC )
)
15 plymul0or 21690 . . . . . . . 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 656 . . . . . . 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 9359 . . . . . . . . . . . . 13  |-  CC  e.  _V
1817a1i 11 . . . . . . . . . . . 12  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  CC  e.  _V )
19 plyf 21609 . . . . . . . . . . . . 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 21609 . . . . . . . . . . . . 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 9364 . . . . . . . . . . . . 13  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  x.  y
)  =  ( y  x.  x ) )
2423adantl 463 . . . . . . . . . . . 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 6351 . . . . . . . . . . 11  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  ( F  oF  x.  G
)  =  ( G  oF  x.  F
) )
266, 25syl5eq 2485 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  H  =  ( G  oF  x.  F )
)
2726oveq1d 6105 . . . . . . . . 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 21609 . . . . . . . . . . 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 9774 . . . . . . . . . . 11  |-  ( ( x  e.  CC  /\  y  e.  CC  /\  z  e.  CC )  ->  (
x  x.  ( y  -  z ) )  =  ( ( x  x.  y )  -  ( x  x.  z
) ) )
3130adantl 463 . . . . . . . . . 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 6355 . . . . . . . . 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 2476 . . . . . . . 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 2449 . . . . . . 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 2622 . . . . . . . 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 2441 . . . . . . . . . . 11  |-  (deg `  G )  =  (deg
`  G )
41 eqid 2441 . . . . . . . . . . 11  |-  (deg `  ( F  oF  -  ( H quot  G
) ) )  =  (deg `  ( F  oF  -  ( H quot  G ) ) )
4240, 41dgrmul 21680 . . . . . . . . . 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 612 . . . . . . . . 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 1212 . . . . . . . 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 21644 . . . . . . . . . . . 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 10633 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  (deg `  G )  e.  RR )
48 dgrcl 21644 . . . . . . . . . . 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 10622 . . . . . . . . . 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 656 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  (deg `  G )  <_  (
(deg `  G )  +  (deg `  ( F  oF  -  ( H quot  G ) ) ) ) )
52 breq2 4293 . . . . . . . . 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 5692 . . . . . . . . 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 4301 . . . . . . . 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 21632 . . . . . . . . . . . . 13  |-  ( ( G  e.  (Poly `  CC )  /\  ( H quot  G )  e.  (Poly `  CC ) )  -> 
( G  oF  x.  ( H quot  G
) )  e.  (Poly `  CC ) )
583, 12, 57syl2anc 656 . . . . . . . . . . . 12  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  ( G  oF  x.  ( H quot  G ) )  e.  (Poly `  CC )
)
59 plysubcl 21633 . . . . . . . . . . . 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 656 . . . . . . . . . . 11  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  ( H  oF  -  ( G  oF  x.  ( H quot  G ) ) )  e.  (Poly `  CC ) )
61 dgrcl 21644 . . . . . . . . . . 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 10633 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  (deg `  ( H  oF  -  ( G  oF  x.  ( H quot  G ) ) ) )  e.  RR )
6447, 63lenltd 9516 . . . . . . . 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 2670 . . . . 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 2441 . . . . . . 7  |-  ( H  oF  -  ( G  oF  x.  ( H quot  G ) ) )  =  ( H  oF  -  ( G  oF  x.  ( H quot  G ) ) )
6968quotdgr 21712 . . . . . 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 1213 . . . . 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 21048 . . . 4  |-  0p  =  ( CC  X.  { 0 } )
7371, 72syl6eq 2489 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  ( F  oF  -  ( H quot  G ) )  =  ( CC  X.  {
0 } ) )
74 ofsubeq0 10315 . . . 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 1213 . . 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 2446 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 960    = wceq 1364    e. wcel 1761    =/= wne 2604   _Vcvv 2970   {csn 3874   class class class wbr 4289    X. cxp 4834   -->wf 5411   ` cfv 5415  (class class class)co 6090    oFcof 6317   CCcc 9276   RRcr 9277   0cc0 9278    + caddc 9281    x. cmul 9283    < clt 9414    <_ cle 9415    - cmin 9591   NN0cn0 10575   0pc0p 21047  Polycply 21595  degcdgr 21598   quot cquot 21699
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356  ax-addf 9357
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-map 7212  df-pm 7213  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-sup 7687  df-oi 7720  df-card 8105  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-n0 10576  df-z 10643  df-uz 10858  df-rp 10988  df-fz 11434  df-fzo 11545  df-fl 11638  df-seq 11803  df-exp 11862  df-hash 12100  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-clim 12962  df-rlim 12963  df-sum 13160  df-0p 21048  df-ply 21599  df-coe 21601  df-dgr 21602  df-quot 21700
This theorem is referenced by: (None)
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