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Theorem crre 9945
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 7220 . . . 4  |-  ( A  e.  RR  ->  A  e.  CC )
2 ax-icn 7185 . . . . 5  |-  _i  e.  CC
3 recn 7220 . . . . 5  |-  ( B  e.  RR  ->  B  e.  CC )
4 mulcl 7214 . . . . 5  |-  ( ( _i  e.  CC  /\  B  e.  CC )  ->  ( _i  x.  B
)  e.  CC )
52, 3, 4sylancr 405 . . . 4  |-  ( B  e.  RR  ->  (
_i  x.  B )  e.  CC )
6 addcl 7212 . . . 4  |-  ( ( A  e.  CC  /\  ( _i  x.  B
)  e.  CC )  ->  ( A  +  ( _i  x.  B
) )  e.  CC )
71, 5, 6syl2an 283 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  +  ( _i  x.  B ) )  e.  CC )
8 reval 9937 . . 3  |-  ( ( A  +  ( _i  x.  B ) )  e.  CC  ->  (
Re `  ( A  +  ( _i  x.  B ) ) )  =  ( ( ( A  +  ( _i  x.  B ) )  +  ( * `  ( A  +  (
_i  x.  B )
) ) )  / 
2 ) )
97, 8syl 14 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( Re `  ( A  +  ( _i  x.  B ) ) )  =  ( ( ( A  +  ( _i  x.  B ) )  +  ( * `  ( A  +  (
_i  x.  B )
) ) )  / 
2 ) )
10 cjcl 9936 . . . . . 6  |-  ( ( A  +  ( _i  x.  B ) )  e.  CC  ->  (
* `  ( A  +  ( _i  x.  B ) ) )  e.  CC )
117, 10syl 14 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( * `  ( A  +  ( _i  x.  B ) ) )  e.  CC )
127, 11addcld 7252 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  +  ( _i  x.  B
) )  +  ( * `  ( A  +  ( _i  x.  B ) ) ) )  e.  CC )
1312halfcld 8394 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( A  +  ( _i  x.  B ) )  +  ( * `  ( A  +  ( _i  x.  B ) ) ) )  /  2 )  e.  CC )
141adantr 270 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  A  e.  CC )
15 recl 9941 . . . . . . 7  |-  ( ( A  +  ( _i  x.  B ) )  e.  CC  ->  (
Re `  ( A  +  ( _i  x.  B ) ) )  e.  RR )
167, 15syl 14 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( Re `  ( A  +  ( _i  x.  B ) ) )  e.  RR )
179, 16eqeltrrd 2160 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( A  +  ( _i  x.  B ) )  +  ( * `  ( A  +  ( _i  x.  B ) ) ) )  /  2 )  e.  RR )
18 simpl 107 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  A  e.  RR )
1917, 18resubcld 7604 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( ( A  +  ( _i  x.  B ) )  +  ( * `  ( A  +  (
_i  x.  B )
) ) )  / 
2 )  -  A
)  e.  RR )
202a1i 9 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  _i  e.  CC )
213adantl 271 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  B  e.  CC )
222, 21, 4sylancr 405 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( _i  x.  B
)  e.  CC )
237, 11subcld 7538 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  +  ( _i  x.  B
) )  -  (
* `  ( A  +  ( _i  x.  B ) ) ) )  e.  CC )
2423halfcld 8394 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( A  +  ( _i  x.  B ) )  -  ( * `  ( A  +  ( _i  x.  B ) ) ) )  /  2 )  e.  CC )
2520, 22, 24subdid 7637 . . . . . 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 7575 . . . . . . . . . . . . . 14  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  +  ( _i  x.  B
) )  -  ( A  +  A )
)  =  ( ( _i  x.  B )  -  A ) )
2722, 14, 22pnpcan2d 7576 . . . . . . . . . . . . . 14  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( _i  x.  B )  +  ( _i  x.  B
) )  -  ( A  +  ( _i  x.  B ) ) )  =  ( ( _i  x.  B )  -  A ) )
2826, 27eqtr4d 2118 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  +  ( _i  x.  B
) )  -  ( A  +  A )
)  =  ( ( ( _i  x.  B
)  +  ( _i  x.  B ) )  -  ( A  +  ( _i  x.  B
) ) ) )
2928oveq1d 5578 . . . . . . . . . . . 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 7252 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  +  A
)  e.  CC )
317, 11, 30addsubd 7559 . . . . . . . . . . . 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 7252 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( _i  x.  B )  +  ( _i  x.  B ) )  e.  CC )
3332, 7, 11subsubd 7566 . . . . . . . . . . . 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 2125 . . . . . . . . . . 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 8392 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 2  x.  A
)  =  ( A  +  A ) )
3635oveq2d 5579 . . . . . . . . . . 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 8392 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 2  x.  (
_i  x.  B )
)  =  ( ( _i  x.  B )  +  ( _i  x.  B ) ) )
3837oveq1d 5578 . . . . . . . . . . 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 2125 . . . . . . . . . 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 5578 . . . . . . . . 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 8229 . . . . . . . . . . 11  |-  2  e.  CC
42 mulcl 7214 . . . . . . . . . . 11  |-  ( ( 2  e.  CC  /\  A  e.  CC )  ->  ( 2  x.  A
)  e.  CC )
4341, 14, 42sylancr 405 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 2  x.  A
)  e.  CC )
4441a1i 9 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  2  e.  CC )
45 2ap0 8251 . . . . . . . . . . 11  |-  2 #  0
4645a1i 9 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  2 #  0 )
4712, 43, 44, 46divsubdirapd 8035 . . . . . . . . 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 7214 . . . . . . . . . . 11  |-  ( ( 2  e.  CC  /\  ( _i  x.  B
)  e.  CC )  ->  ( 2  x.  ( _i  x.  B
) )  e.  CC )
4941, 22, 48sylancr 405 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 2  x.  (
_i  x.  B )
)  e.  CC )
5049, 23, 44, 46divsubdirapd 8035 . . . . . . . . 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 2123 . . . . . . . 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, 46divcanap3d 8001 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( 2  x.  A )  /  2
)  =  A )
5352oveq2d 5579 . . . . . . . 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, 46divcanap3d 8001 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( 2  x.  ( _i  x.  B
) )  /  2
)  =  ( _i  x.  B ) )
5554oveq1d 5578 . . . . . . . 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 2123 . . . . . . 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 5579 . . . . . 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 7256 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( _i  x.  _i )  x.  B
)  =  ( _i  x.  ( _i  x.  B ) ) )
5920, 23, 44, 46divassapd 8031 . . . . . . 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 5581 . . . . . 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 2125 . . . . 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 7802 . . . . . . . 8  |-  ( _i  x.  _i )  = 
-u 1
63 neg1rr 8264 . . . . . . . 8  |-  -u 1  e.  RR
6462, 63eqeltri 2155 . . . . . . 7  |-  ( _i  x.  _i )  e.  RR
65 simpr 108 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  B  e.  RR )
66 remulcl 7215 . . . . . . 7  |-  ( ( ( _i  x.  _i )  e.  RR  /\  B  e.  RR )  ->  (
( _i  x.  _i )  x.  B )  e.  RR )
6764, 65, 66sylancr 405 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( _i  x.  _i )  x.  B
)  e.  RR )
68 cjth 9934 . . . . . . . . 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 112 . . . . . . . 8  |-  ( ( A  +  ( _i  x.  B ) )  e.  CC  ->  (
_i  x.  ( ( A  +  ( _i  x.  B ) )  -  ( * `  ( A  +  ( _i  x.  B ) ) ) ) )  e.  RR )
707, 69syl 14 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( _i  x.  (
( A  +  ( _i  x.  B ) )  -  ( * `
 ( A  +  ( _i  x.  B
) ) ) ) )  e.  RR )
7170rehalfcld 8396 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( _i  x.  ( ( A  +  ( _i  x.  B
) )  -  (
* `  ( A  +  ( _i  x.  B ) ) ) ) )  /  2
)  e.  RR )
7267, 71resubcld 7604 . . . . 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 2159 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( _i  x.  (
( ( ( A  +  ( _i  x.  B ) )  +  ( * `  ( A  +  ( _i  x.  B ) ) ) )  /  2 )  -  A ) )  e.  RR )
74 rimul 7804 . . . 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 403 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( ( A  +  ( _i  x.  B ) )  +  ( * `  ( A  +  (
_i  x.  B )
) ) )  / 
2 )  -  A
)  =  0 )
7613, 14, 75subeq0d 7546 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( A  +  ( _i  x.  B ) )  +  ( * `  ( A  +  ( _i  x.  B ) ) ) )  /  2 )  =  A )
779, 76eqtrd 2115 1  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( Re `  ( A  +  ( _i  x.  B ) ) )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1285    e. wcel 1434   class class class wbr 3805   ` cfv 4952  (class class class)co 5563   CCcc 7093   RRcr 7094   0cc0 7095   1c1 7096   _ici 7097    + caddc 7098    x. cmul 7100    - cmin 7398   -ucneg 7399   # cap 7800    / cdiv 7879   2c2 8208   *ccj 9927   Recre 9928
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-cnex 7181  ax-resscn 7182  ax-1cn 7183  ax-1re 7184  ax-icn 7185  ax-addcl 7186  ax-addrcl 7187  ax-mulcl 7188  ax-mulrcl 7189  ax-addcom 7190  ax-mulcom 7191  ax-addass 7192  ax-mulass 7193  ax-distr 7194  ax-i2m1 7195  ax-0lt1 7196  ax-1rid 7197  ax-0id 7198  ax-rnegex 7199  ax-precex 7200  ax-cnre 7201  ax-pre-ltirr 7202  ax-pre-ltwlin 7203  ax-pre-lttrn 7204  ax-pre-apti 7205  ax-pre-ltadd 7206  ax-pre-mulgt0 7207  ax-pre-mulext 7208
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rmo 2361  df-rab 2362  df-v 2612  df-sbc 2825  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-br 3806  df-opab 3860  df-mpt 3861  df-id 4076  df-po 4079  df-iso 4080  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-fv 4960  df-riota 5519  df-ov 5566  df-oprab 5567  df-mpt2 5568  df-pnf 7269  df-mnf 7270  df-xr 7271  df-ltxr 7272  df-le 7273  df-sub 7400  df-neg 7401  df-reap 7794  df-ap 7801  df-div 7880  df-2 8217  df-cj 9930  df-re 9931
This theorem is referenced by:  crim  9946  replim  9947  mulreap  9952  recj  9955  reneg  9956  readd  9957  remullem  9959  rei  9987  crrei  10024  crred  10064  rennim  10089  absreimsq  10154
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