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Theorem crre 11476
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 8707 . . . 4  |-  ( A  e.  RR  ->  A  e.  CC )
2 ax-icn 8676 . . . . 5  |-  _i  e.  CC
3 recn 8707 . . . . 5  |-  ( B  e.  RR  ->  B  e.  CC )
4 mulcl 8701 . . . . 5  |-  ( ( _i  e.  CC  /\  B  e.  CC )  ->  ( _i  x.  B
)  e.  CC )
52, 3, 4sylancr 647 . . . 4  |-  ( B  e.  RR  ->  (
_i  x.  B )  e.  CC )
6 addcl 8699 . . . 4  |-  ( ( A  e.  CC  /\  ( _i  x.  B
)  e.  CC )  ->  ( A  +  ( _i  x.  B
) )  e.  CC )
71, 5, 6syl2an 465 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  +  ( _i  x.  B ) )  e.  CC )
8 reval 11468 . . 3  |-  ( ( A  +  ( _i  x.  B ) )  e.  CC  ->  (
Re `  ( A  +  ( _i  x.  B ) ) )  =  ( ( ( A  +  ( _i  x.  B ) )  +  ( * `  ( A  +  (
_i  x.  B )
) ) )  / 
2 ) )
97, 8syl 17 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( Re `  ( A  +  ( _i  x.  B ) ) )  =  ( ( ( A  +  ( _i  x.  B ) )  +  ( * `  ( A  +  (
_i  x.  B )
) ) )  / 
2 ) )
10 cjcl 11467 . . . . . 6  |-  ( ( A  +  ( _i  x.  B ) )  e.  CC  ->  (
* `  ( A  +  ( _i  x.  B ) ) )  e.  CC )
117, 10syl 17 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( * `  ( A  +  ( _i  x.  B ) ) )  e.  CC )
127, 11addcld 8734 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  +  ( _i  x.  B
) )  +  ( * `  ( A  +  ( _i  x.  B ) ) ) )  e.  CC )
1312halfcld 9835 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( A  +  ( _i  x.  B ) )  +  ( * `  ( A  +  ( _i  x.  B ) ) ) )  /  2 )  e.  CC )
141adantr 453 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  A  e.  CC )
15 recl 11472 . . . . . . 7  |-  ( ( A  +  ( _i  x.  B ) )  e.  CC  ->  (
Re `  ( A  +  ( _i  x.  B ) ) )  e.  RR )
167, 15syl 17 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( Re `  ( A  +  ( _i  x.  B ) ) )  e.  RR )
179, 16eqeltrrd 2328 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( A  +  ( _i  x.  B ) )  +  ( * `  ( A  +  ( _i  x.  B ) ) ) )  /  2 )  e.  RR )
18 simpl 445 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  A  e.  RR )
1917, 18resubcld 9091 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( ( A  +  ( _i  x.  B ) )  +  ( * `  ( A  +  (
_i  x.  B )
) ) )  / 
2 )  -  A
)  e.  RR )
202a1i 12 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  _i  e.  CC )
213adantl 454 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  B  e.  CC )
222, 21, 4sylancr 647 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( _i  x.  B
)  e.  CC )
237, 11subcld 9037 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  +  ( _i  x.  B
) )  -  (
* `  ( A  +  ( _i  x.  B ) ) ) )  e.  CC )
2423halfcld 9835 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( A  +  ( _i  x.  B ) )  -  ( * `  ( A  +  ( _i  x.  B ) ) ) )  /  2 )  e.  CC )
2520, 22, 24subdid 9115 . . . . . 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 ) ) ) )
26 ax-mulcl 8679 . . . . . . . . . . . . . . . 16  |-  ( ( _i  e.  CC  /\  B  e.  CC )  ->  ( _i  x.  B
)  e.  CC )
272, 21, 26sylancr 647 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( _i  x.  B
)  e.  CC )
2814, 27, 14pnpcand 9074 . . . . . . . . . . . . . 14  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  +  ( _i  x.  B
) )  -  ( A  +  A )
)  =  ( ( _i  x.  B )  -  A ) )
2927, 14, 27pnpcan2d 9075 . . . . . . . . . . . . . 14  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( _i  x.  B )  +  ( _i  x.  B
) )  -  ( A  +  ( _i  x.  B ) ) )  =  ( ( _i  x.  B )  -  A ) )
3028, 29eqtr4d 2288 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  +  ( _i  x.  B
) )  -  ( A  +  A )
)  =  ( ( ( _i  x.  B
)  +  ( _i  x.  B ) )  -  ( A  +  ( _i  x.  B
) ) ) )
3130oveq1d 5725 . . . . . . . . . . . 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 )
) ) ) )
3214, 14addcld 8734 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  +  A
)  e.  CC )
337, 11, 32addsubd 9058 . . . . . . . . . . . 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 ) ) ) ) )
3427, 27addcld 8734 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( _i  x.  B )  +  ( _i  x.  B ) )  e.  CC )
3534, 7, 11subsubd 9065 . . . . . . . . . . . 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 )
) ) ) )
3631, 33, 353eqtr4d 2295 . . . . . . . . . . 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 ) ) ) ) ) )
37142timesd 9833 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 2  x.  A
)  =  ( A  +  A ) )
3837oveq2d 5726 . . . . . . . . . . 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
) ) )
39272timesd 9833 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 2  x.  (
_i  x.  B )
)  =  ( ( _i  x.  B )  +  ( _i  x.  B ) ) )
4039oveq1d 5725 . . . . . . . . . . 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 ) ) ) ) ) )
4136, 38, 403eqtr4d 2295 . . . . . . . . . 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 ) ) ) ) ) )
4241oveq1d 5725 . . . . . . . . 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 ) )
43 2cn 9696 . . . . . . . . . . 11  |-  2  e.  CC
44 ax-mulcl 8679 . . . . . . . . . . 11  |-  ( ( 2  e.  CC  /\  A  e.  CC )  ->  ( 2  x.  A
)  e.  CC )
4543, 14, 44sylancr 647 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 2  x.  A
)  e.  CC )
4643a1i 12 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  2  e.  CC )
47 2ne0 9709 . . . . . . . . . . 11  |-  2  =/=  0
4847a1i 12 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  2  =/=  0 )
4912, 45, 46, 48divsubdird 9455 . . . . . . . . 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 ) ) )
50 ax-mulcl 8679 . . . . . . . . . . 11  |-  ( ( 2  e.  CC  /\  ( _i  x.  B
)  e.  CC )  ->  ( 2  x.  ( _i  x.  B
) )  e.  CC )
5143, 27, 50sylancr 647 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 2  x.  (
_i  x.  B )
)  e.  CC )
5251, 23, 46, 48divsubdird 9455 . . . . . . . . 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 ) ) )
5342, 49, 523eqtr3d 2293 . . . . . . . 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 ) ) )
5414, 46, 48divcan3d 9421 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( 2  x.  A )  /  2
)  =  A )
5554oveq2d 5726 . . . . . . . 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 ) )
5627, 46, 48divcan3d 9421 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( 2  x.  ( _i  x.  B
) )  /  2
)  =  ( _i  x.  B ) )
5756oveq1d 5725 . . . . . . . 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 ) ) )
5853, 55, 573eqtr3d 2293 . . . . . . 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 ) ) )
5958oveq2d 5726 . . . . . 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 ) ) ) )
6020, 20, 21mulassd 8738 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( _i  x.  _i )  x.  B
)  =  ( _i  x.  ( _i  x.  B ) ) )
6120, 23, 46, 48divassd 9451 . . . . . . 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 ) ) )
6260, 61oveq12d 5728 . . . . . 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 ) ) ) )
6325, 59, 623eqtr4d 2295 . . . . 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
) ) )
64 ixi 9277 . . . . . . . 8  |-  ( _i  x.  _i )  = 
-u 1
65 1re 8717 . . . . . . . . 9  |-  1  e.  RR
6665renegcli 8988 . . . . . . . 8  |-  -u 1  e.  RR
6764, 66eqeltri 2323 . . . . . . 7  |-  ( _i  x.  _i )  e.  RR
68 simpr 449 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  B  e.  RR )
69 ax-mulrcl 8680 . . . . . . 7  |-  ( ( ( _i  x.  _i )  e.  RR  /\  B  e.  RR )  ->  (
( _i  x.  _i )  x.  B )  e.  RR )
7067, 68, 69sylancr 647 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( _i  x.  _i )  x.  B
)  e.  RR )
71 cjth 11465 . . . . . . . . 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 ) )
7271simprd 451 . . . . . . . 8  |-  ( ( A  +  ( _i  x.  B ) )  e.  CC  ->  (
_i  x.  ( ( A  +  ( _i  x.  B ) )  -  ( * `  ( A  +  ( _i  x.  B ) ) ) ) )  e.  RR )
737, 72syl 17 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( _i  x.  (
( A  +  ( _i  x.  B ) )  -  ( * `
 ( A  +  ( _i  x.  B
) ) ) ) )  e.  RR )
7473rehalfcld 9837 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( _i  x.  ( ( A  +  ( _i  x.  B
) )  -  (
* `  ( A  +  ( _i  x.  B ) ) ) ) )  /  2
)  e.  RR )
7570, 74resubcld 9091 . . . . 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 )
7663, 75eqeltrd 2327 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( _i  x.  (
( ( ( A  +  ( _i  x.  B ) )  +  ( * `  ( A  +  ( _i  x.  B ) ) ) )  /  2 )  -  A ) )  e.  RR )
77 rimul 9617 . . . 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 )
7819, 76, 77syl2anc 645 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( ( A  +  ( _i  x.  B ) )  +  ( * `  ( A  +  (
_i  x.  B )
) ) )  / 
2 )  -  A
)  =  0 )
7913, 14, 78subeq0d 9045 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( A  +  ( _i  x.  B ) )  +  ( * `  ( A  +  ( _i  x.  B ) ) ) )  /  2 )  =  A )
809, 79eqtrd 2285 1  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( Re `  ( A  +  ( _i  x.  B ) ) )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    = wceq 1619    e. wcel 1621    =/= wne 2412   ` cfv 4592  (class class class)co 5710   CCcc 8615   RRcr 8616   0cc0 8617   1c1 8618   _ici 8619    + caddc 8620    x. cmul 8622    - cmin 8917   -ucneg 8918    / cdiv 9303   2c2 9675   *ccj 11458   Recre 11459
This theorem is referenced by:  crim  11477  replim  11478  mulre  11483  recj  11486  reneg  11487  readd  11488  remullem  11490  rei  11518  crrei  11554  crred  11593  rennim  11601  absreimsq  11654  4sqlem4  12873  2sqlem2  20435
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-resscn 8674  ax-1cn 8675  ax-icn 8676  ax-addcl 8677  ax-addrcl 8678  ax-mulcl 8679  ax-mulrcl 8680  ax-mulcom 8681  ax-addass 8682  ax-mulass 8683  ax-distr 8684  ax-i2m1 8685  ax-1ne0 8686  ax-1rid 8687  ax-rnegex 8688  ax-rrecex 8689  ax-cnre 8690  ax-pre-lttri 8691  ax-pre-lttrn 8692  ax-pre-ltadd 8693  ax-pre-mulgt0 8694
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-po 4207  df-so 4208  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-iota 6143  df-riota 6190  df-er 6546  df-en 6750  df-dom 6751  df-sdom 6752  df-pnf 8749  df-mnf 8750  df-xr 8751  df-ltxr 8752  df-le 8753  df-sub 8919  df-neg 8920  df-div 9304  df-2 9684  df-cj 11461  df-re 11462
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