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Theorem crre 12897
Description: The real part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132. (Contributed by NM, 12-May-2005.) (Revised by Mario Carneiro, 7-Nov-2013.)
Assertion
Ref Expression
crre  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( Re `  ( A  +  ( _i  x.  B ) ) )  =  A )

Proof of Theorem crre
StepHypRef Expression
1 recn 9571 . . . 4  |-  ( A  e.  RR  ->  A  e.  CC )
2 ax-icn 9540 . . . . 5  |-  _i  e.  CC
3 recn 9571 . . . . 5  |-  ( B  e.  RR  ->  B  e.  CC )
4 mulcl 9565 . . . . 5  |-  ( ( _i  e.  CC  /\  B  e.  CC )  ->  ( _i  x.  B
)  e.  CC )
52, 3, 4sylancr 663 . . . 4  |-  ( B  e.  RR  ->  (
_i  x.  B )  e.  CC )
6 addcl 9563 . . . 4  |-  ( ( A  e.  CC  /\  ( _i  x.  B
)  e.  CC )  ->  ( A  +  ( _i  x.  B
) )  e.  CC )
71, 5, 6syl2an 477 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  +  ( _i  x.  B ) )  e.  CC )
8 reval 12889 . . 3  |-  ( ( A  +  ( _i  x.  B ) )  e.  CC  ->  (
Re `  ( A  +  ( _i  x.  B ) ) )  =  ( ( ( A  +  ( _i  x.  B ) )  +  ( * `  ( A  +  (
_i  x.  B )
) ) )  / 
2 ) )
97, 8syl 16 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( Re `  ( A  +  ( _i  x.  B ) ) )  =  ( ( ( A  +  ( _i  x.  B ) )  +  ( * `  ( A  +  (
_i  x.  B )
) ) )  / 
2 ) )
10 cjcl 12888 . . . . . 6  |-  ( ( A  +  ( _i  x.  B ) )  e.  CC  ->  (
* `  ( A  +  ( _i  x.  B ) ) )  e.  CC )
117, 10syl 16 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( * `  ( A  +  ( _i  x.  B ) ) )  e.  CC )
127, 11addcld 9604 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  +  ( _i  x.  B
) )  +  ( * `  ( A  +  ( _i  x.  B ) ) ) )  e.  CC )
1312halfcld 10772 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( A  +  ( _i  x.  B ) )  +  ( * `  ( A  +  ( _i  x.  B ) ) ) )  /  2 )  e.  CC )
141adantr 465 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  A  e.  CC )
15 recl 12893 . . . . . . 7  |-  ( ( A  +  ( _i  x.  B ) )  e.  CC  ->  (
Re `  ( A  +  ( _i  x.  B ) ) )  e.  RR )
167, 15syl 16 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( Re `  ( A  +  ( _i  x.  B ) ) )  e.  RR )
179, 16eqeltrrd 2549 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( A  +  ( _i  x.  B ) )  +  ( * `  ( A  +  ( _i  x.  B ) ) ) )  /  2 )  e.  RR )
18 simpl 457 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  A  e.  RR )
1917, 18resubcld 9976 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( ( A  +  ( _i  x.  B ) )  +  ( * `  ( A  +  (
_i  x.  B )
) ) )  / 
2 )  -  A
)  e.  RR )
202a1i 11 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  _i  e.  CC )
213adantl 466 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  B  e.  CC )
222, 21, 4sylancr 663 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( _i  x.  B
)  e.  CC )
237, 11subcld 9919 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  +  ( _i  x.  B
) )  -  (
* `  ( A  +  ( _i  x.  B ) ) ) )  e.  CC )
2423halfcld 10772 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( A  +  ( _i  x.  B ) )  -  ( * `  ( A  +  ( _i  x.  B ) ) ) )  /  2 )  e.  CC )
2520, 22, 24subdid 10001 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( _i  x.  (
( _i  x.  B
)  -  ( ( ( A  +  ( _i  x.  B ) )  -  ( * `
 ( A  +  ( _i  x.  B
) ) ) )  /  2 ) ) )  =  ( ( _i  x.  ( _i  x.  B ) )  -  ( _i  x.  ( ( ( A  +  ( _i  x.  B ) )  -  ( * `  ( A  +  ( _i  x.  B ) ) ) )  /  2 ) ) ) )
2614, 22, 14pnpcand 9956 . . . . . . . . . . . . . 14  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  +  ( _i  x.  B
) )  -  ( A  +  A )
)  =  ( ( _i  x.  B )  -  A ) )
2722, 14, 22pnpcan2d 9957 . . . . . . . . . . . . . 14  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( _i  x.  B )  +  ( _i  x.  B
) )  -  ( A  +  ( _i  x.  B ) ) )  =  ( ( _i  x.  B )  -  A ) )
2826, 27eqtr4d 2504 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  +  ( _i  x.  B
) )  -  ( A  +  A )
)  =  ( ( ( _i  x.  B
)  +  ( _i  x.  B ) )  -  ( A  +  ( _i  x.  B
) ) ) )
2928oveq1d 6290 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( A  +  ( _i  x.  B ) )  -  ( A  +  A
) )  +  ( * `  ( A  +  ( _i  x.  B ) ) ) )  =  ( ( ( ( _i  x.  B )  +  ( _i  x.  B ) )  -  ( A  +  ( _i  x.  B ) ) )  +  ( * `  ( A  +  (
_i  x.  B )
) ) ) )
3014, 14addcld 9604 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  +  A
)  e.  CC )
317, 11, 30addsubd 9940 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( A  +  ( _i  x.  B ) )  +  ( * `  ( A  +  ( _i  x.  B ) ) ) )  -  ( A  +  A ) )  =  ( ( ( A  +  ( _i  x.  B ) )  -  ( A  +  A ) )  +  ( * `  ( A  +  ( _i  x.  B ) ) ) ) )
3222, 22addcld 9604 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( _i  x.  B )  +  ( _i  x.  B ) )  e.  CC )
3332, 7, 11subsubd 9947 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( _i  x.  B )  +  ( _i  x.  B
) )  -  (
( A  +  ( _i  x.  B ) )  -  ( * `
 ( A  +  ( _i  x.  B
) ) ) ) )  =  ( ( ( ( _i  x.  B )  +  ( _i  x.  B ) )  -  ( A  +  ( _i  x.  B ) ) )  +  ( * `  ( A  +  (
_i  x.  B )
) ) ) )
3429, 31, 333eqtr4d 2511 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( A  +  ( _i  x.  B ) )  +  ( * `  ( A  +  ( _i  x.  B ) ) ) )  -  ( A  +  A ) )  =  ( ( ( _i  x.  B )  +  ( _i  x.  B ) )  -  ( ( A  +  ( _i  x.  B
) )  -  (
* `  ( A  +  ( _i  x.  B ) ) ) ) ) )
35142timesd 10770 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 2  x.  A
)  =  ( A  +  A ) )
3635oveq2d 6291 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( A  +  ( _i  x.  B ) )  +  ( * `  ( A  +  ( _i  x.  B ) ) ) )  -  ( 2  x.  A ) )  =  ( ( ( A  +  ( _i  x.  B ) )  +  ( * `  ( A  +  (
_i  x.  B )
) ) )  -  ( A  +  A
) ) )
37222timesd 10770 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 2  x.  (
_i  x.  B )
)  =  ( ( _i  x.  B )  +  ( _i  x.  B ) ) )
3837oveq1d 6290 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( 2  x.  ( _i  x.  B
) )  -  (
( A  +  ( _i  x.  B ) )  -  ( * `
 ( A  +  ( _i  x.  B
) ) ) ) )  =  ( ( ( _i  x.  B
)  +  ( _i  x.  B ) )  -  ( ( A  +  ( _i  x.  B ) )  -  ( * `  ( A  +  ( _i  x.  B ) ) ) ) ) )
3934, 36, 383eqtr4d 2511 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( A  +  ( _i  x.  B ) )  +  ( * `  ( A  +  ( _i  x.  B ) ) ) )  -  ( 2  x.  A ) )  =  ( ( 2  x.  ( _i  x.  B ) )  -  ( ( A  +  ( _i  x.  B
) )  -  (
* `  ( A  +  ( _i  x.  B ) ) ) ) ) )
4039oveq1d 6290 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( ( A  +  ( _i  x.  B ) )  +  ( * `  ( A  +  (
_i  x.  B )
) ) )  -  ( 2  x.  A
) )  /  2
)  =  ( ( ( 2  x.  (
_i  x.  B )
)  -  ( ( A  +  ( _i  x.  B ) )  -  ( * `  ( A  +  (
_i  x.  B )
) ) ) )  /  2 ) )
41 2cn 10595 . . . . . . . . . . 11  |-  2  e.  CC
42 mulcl 9565 . . . . . . . . . . 11  |-  ( ( 2  e.  CC  /\  A  e.  CC )  ->  ( 2  x.  A
)  e.  CC )
4341, 14, 42sylancr 663 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 2  x.  A
)  e.  CC )
4441a1i 11 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  2  e.  CC )
45 2ne0 10617 . . . . . . . . . . 11  |-  2  =/=  0
4645a1i 11 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  2  =/=  0 )
4712, 43, 44, 46divsubdird 10348 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( ( A  +  ( _i  x.  B ) )  +  ( * `  ( A  +  (
_i  x.  B )
) ) )  -  ( 2  x.  A
) )  /  2
)  =  ( ( ( ( A  +  ( _i  x.  B
) )  +  ( * `  ( A  +  ( _i  x.  B ) ) ) )  /  2 )  -  ( ( 2  x.  A )  / 
2 ) ) )
48 mulcl 9565 . . . . . . . . . . 11  |-  ( ( 2  e.  CC  /\  ( _i  x.  B
)  e.  CC )  ->  ( 2  x.  ( _i  x.  B
) )  e.  CC )
4941, 22, 48sylancr 663 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 2  x.  (
_i  x.  B )
)  e.  CC )
5049, 23, 44, 46divsubdird 10348 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( 2  x.  ( _i  x.  B ) )  -  ( ( A  +  ( _i  x.  B
) )  -  (
* `  ( A  +  ( _i  x.  B ) ) ) ) )  /  2
)  =  ( ( ( 2  x.  (
_i  x.  B )
)  /  2 )  -  ( ( ( A  +  ( _i  x.  B ) )  -  ( * `  ( A  +  (
_i  x.  B )
) ) )  / 
2 ) ) )
5140, 47, 503eqtr3d 2509 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( ( A  +  ( _i  x.  B ) )  +  ( * `  ( A  +  (
_i  x.  B )
) ) )  / 
2 )  -  (
( 2  x.  A
)  /  2 ) )  =  ( ( ( 2  x.  (
_i  x.  B )
)  /  2 )  -  ( ( ( A  +  ( _i  x.  B ) )  -  ( * `  ( A  +  (
_i  x.  B )
) ) )  / 
2 ) ) )
5214, 44, 46divcan3d 10314 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( 2  x.  A )  /  2
)  =  A )
5352oveq2d 6291 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( ( A  +  ( _i  x.  B ) )  +  ( * `  ( A  +  (
_i  x.  B )
) ) )  / 
2 )  -  (
( 2  x.  A
)  /  2 ) )  =  ( ( ( ( A  +  ( _i  x.  B
) )  +  ( * `  ( A  +  ( _i  x.  B ) ) ) )  /  2 )  -  A ) )
5422, 44, 46divcan3d 10314 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( 2  x.  ( _i  x.  B
) )  /  2
)  =  ( _i  x.  B ) )
5554oveq1d 6290 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( 2  x.  ( _i  x.  B ) )  / 
2 )  -  (
( ( A  +  ( _i  x.  B
) )  -  (
* `  ( A  +  ( _i  x.  B ) ) ) )  /  2 ) )  =  ( ( _i  x.  B )  -  ( ( ( A  +  ( _i  x.  B ) )  -  ( * `  ( A  +  (
_i  x.  B )
) ) )  / 
2 ) ) )
5651, 53, 553eqtr3d 2509 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( ( A  +  ( _i  x.  B ) )  +  ( * `  ( A  +  (
_i  x.  B )
) ) )  / 
2 )  -  A
)  =  ( ( _i  x.  B )  -  ( ( ( A  +  ( _i  x.  B ) )  -  ( * `  ( A  +  (
_i  x.  B )
) ) )  / 
2 ) ) )
5756oveq2d 6291 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( _i  x.  (
( ( ( A  +  ( _i  x.  B ) )  +  ( * `  ( A  +  ( _i  x.  B ) ) ) )  /  2 )  -  A ) )  =  ( _i  x.  ( ( _i  x.  B )  -  (
( ( A  +  ( _i  x.  B
) )  -  (
* `  ( A  +  ( _i  x.  B ) ) ) )  /  2 ) ) ) )
5820, 20, 21mulassd 9608 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( _i  x.  _i )  x.  B
)  =  ( _i  x.  ( _i  x.  B ) ) )
5920, 23, 44, 46divassd 10344 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( _i  x.  ( ( A  +  ( _i  x.  B
) )  -  (
* `  ( A  +  ( _i  x.  B ) ) ) ) )  /  2
)  =  ( _i  x.  ( ( ( A  +  ( _i  x.  B ) )  -  ( * `  ( A  +  (
_i  x.  B )
) ) )  / 
2 ) ) )
6058, 59oveq12d 6293 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( _i  x.  _i )  x.  B )  -  (
( _i  x.  (
( A  +  ( _i  x.  B ) )  -  ( * `
 ( A  +  ( _i  x.  B
) ) ) ) )  /  2 ) )  =  ( ( _i  x.  ( _i  x.  B ) )  -  ( _i  x.  ( ( ( A  +  ( _i  x.  B ) )  -  ( * `  ( A  +  ( _i  x.  B ) ) ) )  /  2 ) ) ) )
6125, 57, 603eqtr4d 2511 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( _i  x.  (
( ( ( A  +  ( _i  x.  B ) )  +  ( * `  ( A  +  ( _i  x.  B ) ) ) )  /  2 )  -  A ) )  =  ( ( ( _i  x.  _i )  x.  B )  -  ( ( _i  x.  ( ( A  +  ( _i  x.  B
) )  -  (
* `  ( A  +  ( _i  x.  B ) ) ) ) )  /  2
) ) )
62 ixi 10167 . . . . . . . 8  |-  ( _i  x.  _i )  = 
-u 1
63 neg1rr 10629 . . . . . . . 8  |-  -u 1  e.  RR
6462, 63eqeltri 2544 . . . . . . 7  |-  ( _i  x.  _i )  e.  RR
65 simpr 461 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  B  e.  RR )
66 remulcl 9566 . . . . . . 7  |-  ( ( ( _i  x.  _i )  e.  RR  /\  B  e.  RR )  ->  (
( _i  x.  _i )  x.  B )  e.  RR )
6764, 65, 66sylancr 663 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( _i  x.  _i )  x.  B
)  e.  RR )
68 cjth 12886 . . . . . . . . 9  |-  ( ( A  +  ( _i  x.  B ) )  e.  CC  ->  (
( ( A  +  ( _i  x.  B
) )  +  ( * `  ( A  +  ( _i  x.  B ) ) ) )  e.  RR  /\  ( _i  x.  (
( A  +  ( _i  x.  B ) )  -  ( * `
 ( A  +  ( _i  x.  B
) ) ) ) )  e.  RR ) )
6968simprd 463 . . . . . . . 8  |-  ( ( A  +  ( _i  x.  B ) )  e.  CC  ->  (
_i  x.  ( ( A  +  ( _i  x.  B ) )  -  ( * `  ( A  +  ( _i  x.  B ) ) ) ) )  e.  RR )
707, 69syl 16 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( _i  x.  (
( A  +  ( _i  x.  B ) )  -  ( * `
 ( A  +  ( _i  x.  B
) ) ) ) )  e.  RR )
7170rehalfcld 10774 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( _i  x.  ( ( A  +  ( _i  x.  B
) )  -  (
* `  ( A  +  ( _i  x.  B ) ) ) ) )  /  2
)  e.  RR )
7267, 71resubcld 9976 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( _i  x.  _i )  x.  B )  -  (
( _i  x.  (
( A  +  ( _i  x.  B ) )  -  ( * `
 ( A  +  ( _i  x.  B
) ) ) ) )  /  2 ) )  e.  RR )
7361, 72eqeltrd 2548 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( _i  x.  (
( ( ( A  +  ( _i  x.  B ) )  +  ( * `  ( A  +  ( _i  x.  B ) ) ) )  /  2 )  -  A ) )  e.  RR )
74 rimul 10516 . . . 4  |-  ( ( ( ( ( ( A  +  ( _i  x.  B ) )  +  ( * `  ( A  +  (
_i  x.  B )
) ) )  / 
2 )  -  A
)  e.  RR  /\  ( _i  x.  (
( ( ( A  +  ( _i  x.  B ) )  +  ( * `  ( A  +  ( _i  x.  B ) ) ) )  /  2 )  -  A ) )  e.  RR )  -> 
( ( ( ( A  +  ( _i  x.  B ) )  +  ( * `  ( A  +  (
_i  x.  B )
) ) )  / 
2 )  -  A
)  =  0 )
7519, 73, 74syl2anc 661 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( ( A  +  ( _i  x.  B ) )  +  ( * `  ( A  +  (
_i  x.  B )
) ) )  / 
2 )  -  A
)  =  0 )
7613, 14, 75subeq0d 9927 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( A  +  ( _i  x.  B ) )  +  ( * `  ( A  +  ( _i  x.  B ) ) ) )  /  2 )  =  A )
779, 76eqtrd 2501 1  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( Re `  ( A  +  ( _i  x.  B ) ) )  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1374    e. wcel 1762    =/= wne 2655   ` cfv 5579  (class class class)co 6275   CCcc 9479   RRcr 9480   0cc0 9481   1c1 9482   _ici 9483    + caddc 9484    x. cmul 9486    - cmin 9794   -ucneg 9795    / cdiv 10195   2c2 10574   *ccj 12879   Recre 12880
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 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-po 4793  df-so 4794  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196  df-2 10583  df-cj 12882  df-re 12883
This theorem is referenced by:  crim  12898  replim  12899  mulre  12904  recj  12907  reneg  12908  readd  12909  remullem  12911  rei  12939  crrei  12975  crred  13014  rennim  13022  absreimsq  13075  4sqlem4  14318  2sqlem2  23360  cnre2csqima  27379
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