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Theorem crre 10256
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 7454 . . . 4  |-  ( A  e.  RR  ->  A  e.  CC )
2 ax-icn 7419 . . . . 5  |-  _i  e.  CC
3 recn 7454 . . . . 5  |-  ( B  e.  RR  ->  B  e.  CC )
4 mulcl 7448 . . . . 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 7446 . . . 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 10248 . . 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 10247 . . . . . 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 7486 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  +  ( _i  x.  B
) )  +  ( * `  ( A  +  ( _i  x.  B ) ) ) )  e.  CC )
1312halfcld 8630 . . 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 10252 . . . . . . 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 2165 . . . . 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 7838 . . . 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 7772 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  +  ( _i  x.  B
) )  -  (
* `  ( A  +  ( _i  x.  B ) ) ) )  e.  CC )
2423halfcld 8630 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( A  +  ( _i  x.  B ) )  -  ( * `  ( A  +  ( _i  x.  B ) ) ) )  /  2 )  e.  CC )
2520, 22, 24subdid 7871 . . . . . 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 7809 . . . . . . . . . . . . . 14  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  +  ( _i  x.  B
) )  -  ( A  +  A )
)  =  ( ( _i  x.  B )  -  A ) )
2722, 14, 22pnpcan2d 7810 . . . . . . . . . . . . . 14  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( _i  x.  B )  +  ( _i  x.  B
) )  -  ( A  +  ( _i  x.  B ) ) )  =  ( ( _i  x.  B )  -  A ) )
2826, 27eqtr4d 2123 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  +  ( _i  x.  B
) )  -  ( A  +  A )
)  =  ( ( ( _i  x.  B
)  +  ( _i  x.  B ) )  -  ( A  +  ( _i  x.  B
) ) ) )
2928oveq1d 5649 . . . . . . . . . . . 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 7486 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  +  A
)  e.  CC )
317, 11, 30addsubd 7793 . . . . . . . . . . . 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 7486 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( _i  x.  B )  +  ( _i  x.  B ) )  e.  CC )
3332, 7, 11subsubd 7800 . . . . . . . . . . . 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 2130 . . . . . . . . . . 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 8628 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 2  x.  A
)  =  ( A  +  A ) )
3635oveq2d 5650 . . . . . . . . . . 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 8628 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 2  x.  (
_i  x.  B )
)  =  ( ( _i  x.  B )  +  ( _i  x.  B ) ) )
3837oveq1d 5649 . . . . . . . . . . 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 2130 . . . . . . . . . 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 5649 . . . . . . . . 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 8464 . . . . . . . . . . 11  |-  2  e.  CC
42 mulcl 7448 . . . . . . . . . . 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 8486 . . . . . . . . . . 11  |-  2 #  0
4645a1i 9 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  2 #  0 )
4712, 43, 44, 46divsubdirapd 8269 . . . . . . . . 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 7448 . . . . . . . . . . 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 8269 . . . . . . . . 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 2128 . . . . . . . 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 8235 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( 2  x.  A )  /  2
)  =  A )
5352oveq2d 5650 . . . . . . . 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 8235 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( 2  x.  ( _i  x.  B
) )  /  2
)  =  ( _i  x.  B ) )
5554oveq1d 5649 . . . . . . . 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 2128 . . . . . . 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 5650 . . . . . 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 7490 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( _i  x.  _i )  x.  B
)  =  ( _i  x.  ( _i  x.  B ) ) )
5920, 23, 44, 46divassapd 8265 . . . . . . 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 5652 . . . . . 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 2130 . . . . 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 8036 . . . . . . . 8  |-  ( _i  x.  _i )  = 
-u 1
63 neg1rr 8499 . . . . . . . 8  |-  -u 1  e.  RR
6462, 63eqeltri 2160 . . . . . . 7  |-  ( _i  x.  _i )  e.  RR
65 simpr 108 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  B  e.  RR )
66 remulcl 7449 . . . . . . 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 10245 . . . . . . . . 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 8632 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( _i  x.  ( ( A  +  ( _i  x.  B
) )  -  (
* `  ( A  +  ( _i  x.  B ) ) ) ) )  /  2
)  e.  RR )
7267, 71resubcld 7838 . . . . 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 2164 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( _i  x.  (
( ( ( A  +  ( _i  x.  B ) )  +  ( * `  ( A  +  ( _i  x.  B ) ) ) )  /  2 )  -  A ) )  e.  RR )
74 rimul 8038 . . . 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 7780 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( A  +  ( _i  x.  B ) )  +  ( * `  ( A  +  ( _i  x.  B ) ) ) )  /  2 )  =  A )
779, 76eqtrd 2120 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 1289    e. wcel 1438   class class class wbr 3837   ` cfv 5002  (class class class)co 5634   CCcc 7327   RRcr 7328   0cc0 7329   1c1 7330   _ici 7331    + caddc 7332    x. cmul 7334    - cmin 7632   -ucneg 7633   # cap 8034    / cdiv 8113   2c2 8444   *ccj 10238   Recre 10239
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-cnex 7415  ax-resscn 7416  ax-1cn 7417  ax-1re 7418  ax-icn 7419  ax-addcl 7420  ax-addrcl 7421  ax-mulcl 7422  ax-mulrcl 7423  ax-addcom 7424  ax-mulcom 7425  ax-addass 7426  ax-mulass 7427  ax-distr 7428  ax-i2m1 7429  ax-0lt1 7430  ax-1rid 7431  ax-0id 7432  ax-rnegex 7433  ax-precex 7434  ax-cnre 7435  ax-pre-ltirr 7436  ax-pre-ltwlin 7437  ax-pre-lttrn 7438  ax-pre-apti 7439  ax-pre-ltadd 7440  ax-pre-mulgt0 7441  ax-pre-mulext 7442
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367  df-rab 2368  df-v 2621  df-sbc 2839  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-mpt 3893  df-id 4111  df-po 4114  df-iso 4115  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-fv 5010  df-riota 5590  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-pnf 7503  df-mnf 7504  df-xr 7505  df-ltxr 7506  df-le 7507  df-sub 7634  df-neg 7635  df-reap 8028  df-ap 8035  df-div 8114  df-2 8452  df-cj 10241  df-re 10242
This theorem is referenced by:  crim  10257  replim  10258  mulreap  10263  recj  10266  reneg  10267  readd  10268  remullem  10270  rei  10298  crrei  10335  crred  10375  rennim  10400  absreimsq  10465
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