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Theorem quotcan 21775
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 21668 . . . . . . . . 9  |-  (Poly `  S )  C_  (Poly `  CC )
2 simp2 989 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  G  e.  (Poly `  S )
)
31, 2sseldi 3354 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  G  e.  (Poly `  CC )
)
4 simp1 988 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  F  e.  (Poly `  S )
)
51, 4sseldi 3354 . . . . . . . . 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 21689 . . . . . . . . . . . 12  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( F  oF  x.  G
)  e.  (Poly `  CC ) )
86, 7syl5eqel 2527 . . . . . . . . . . 11  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  H  e.  (Poly `  CC ) )
983adant3 1008 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  H  e.  (Poly `  CC )
)
10 simp3 990 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  G  =/=  0p )
11 quotcl2 21768 . . . . . . . . . 10  |-  ( ( H  e.  (Poly `  CC )  /\  G  e.  (Poly `  CC )  /\  G  =/=  0p )  ->  ( H quot  G )  e.  (Poly `  CC ) )
129, 3, 10, 11syl3anc 1218 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  ( H quot  G )  e.  (Poly `  CC ) )
13 plysubcl 21690 . . . . . . . . 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 21747 . . . . . . . 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 9363 . . . . . . . . . . . . 13  |-  CC  e.  _V
1817a1i 11 . . . . . . . . . . . 12  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  CC  e.  _V )
19 plyf 21666 . . . . . . . . . . . . 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 21666 . . . . . . . . . . . . 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 9368 . . . . . . . . . . . . 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 6352 . . . . . . . . . . 11  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  ( F  oF  x.  G
)  =  ( G  oF  x.  F
) )
266, 25syl5eq 2487 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  H  =  ( G  oF  x.  F )
)
2726oveq1d 6106 . . . . . . . . 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 21666 . . . . . . . . . . 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 9778 . . . . . . . . . . 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 6356 . . . . . . . . 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 2478 . . . . . . . 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 2451 . . . . . . 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 2624 . . . . . . . 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 2443 . . . . . . . . . . 11  |-  (deg `  G )  =  (deg
`  G )
41 eqid 2443 . . . . . . . . . . 11  |-  (deg `  ( F  oF  -  ( H quot  G
) ) )  =  (deg `  ( F  oF  -  ( H quot  G ) ) )
4240, 41dgrmul 21737 . . . . . . . . . 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 1217 . . . . . . . 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 21701 . . . . . . . . . . . 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 10637 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  (deg `  G )  e.  RR )
48 dgrcl 21701 . . . . . . . . . . 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 10626 . . . . . . . . . 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 4296 . . . . . . . . 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 5695 . . . . . . . . 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 4304 . . . . . . . 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 21689 . . . . . . . . . . . . 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 21690 . . . . . . . . . . . 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 21701 . . . . . . . . . . 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 10637 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  (deg `  ( H  oF  -  ( G  oF  x.  ( H quot  G ) ) ) )  e.  RR )
6447, 63lenltd 9520 . . . . . . . 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 2672 . . . . 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 2443 . . . . . . 7  |-  ( H  oF  -  ( G  oF  x.  ( H quot  G ) ) )  =  ( H  oF  -  ( G  oF  x.  ( H quot  G ) ) )
6968quotdgr 21769 . . . . . 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 1218 . . . . 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 21148 . . . 4  |-  0p  =  ( CC  X.  { 0 } )
7371, 72syl6eq 2491 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  ( F  oF  -  ( H quot  G ) )  =  ( CC  X.  {
0 } ) )
74 ofsubeq0 10319 . . . 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 1218 . . 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 2448 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 965    = wceq 1369    e. wcel 1756    =/= wne 2606   _Vcvv 2972   {csn 3877   class class class wbr 4292    X. cxp 4838   -->wf 5414   ` cfv 5418  (class class class)co 6091    oFcof 6318   CCcc 9280   RRcr 9281   0cc0 9282    + caddc 9285    x. cmul 9287    < clt 9418    <_ cle 9419    - cmin 9595   NN0cn0 10579   0pc0p 21147  Polycply 21652  degcdgr 21655   quot cquot 21756
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-inf2 7847  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-pre-sup 9360  ax-addf 9361
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-se 4680  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-isom 5427  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-of 6320  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-1o 6920  df-oadd 6924  df-er 7101  df-map 7216  df-pm 7217  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-sup 7691  df-oi 7724  df-card 8109  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-2 10380  df-3 10381  df-n0 10580  df-z 10647  df-uz 10862  df-rp 10992  df-fz 11438  df-fzo 11549  df-fl 11642  df-seq 11807  df-exp 11866  df-hash 12104  df-cj 12588  df-re 12589  df-im 12590  df-sqr 12724  df-abs 12725  df-clim 12966  df-rlim 12967  df-sum 13164  df-0p 21148  df-ply 21656  df-coe 21658  df-dgr 21659  df-quot 21757
This theorem is referenced by: (None)
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