MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  atandm2 Structured version   Unicode version

Theorem atandm2 22284
Description: This form of atandm 22283 is a bit more useful for showing that the logarithms in df-atan 22274 are well-defined. (Contributed by Mario Carneiro, 31-Mar-2015.)
Assertion
Ref Expression
atandm2  |-  ( A  e.  dom arctan  <->  ( A  e.  CC  /\  ( 1  -  ( _i  x.  A ) )  =/=  0  /\  ( 1  +  ( _i  x.  A ) )  =/=  0 ) )

Proof of Theorem atandm2
StepHypRef Expression
1 atandm 22283 . 2  |-  ( A  e.  dom arctan  <->  ( A  e.  CC  /\  A  =/=  -u _i  /\  A  =/= 
_i ) )
2 3anass 969 . . 3  |-  ( ( A  e.  CC  /\  ( 1  -  (
_i  x.  A )
)  =/=  0  /\  ( 1  +  ( _i  x.  A ) )  =/=  0 )  <-> 
( A  e.  CC  /\  ( ( 1  -  ( _i  x.  A
) )  =/=  0  /\  ( 1  +  ( _i  x.  A ) )  =/=  0 ) ) )
3 ax-1cn 9352 . . . . . . . . . 10  |-  1  e.  CC
4 ax-icn 9353 . . . . . . . . . . 11  |-  _i  e.  CC
5 mulcl 9378 . . . . . . . . . . 11  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( _i  x.  A
)  e.  CC )
64, 5mpan 670 . . . . . . . . . 10  |-  ( A  e.  CC  ->  (
_i  x.  A )  e.  CC )
7 subeq0 9647 . . . . . . . . . 10  |-  ( ( 1  e.  CC  /\  ( _i  x.  A
)  e.  CC )  ->  ( ( 1  -  ( _i  x.  A ) )  =  0  <->  1  =  ( _i  x.  A ) ) )
83, 6, 7sylancr 663 . . . . . . . . 9  |-  ( A  e.  CC  ->  (
( 1  -  (
_i  x.  A )
)  =  0  <->  1  =  ( _i  x.  A ) ) )
94, 4mulneg2i 9803 . . . . . . . . . . . 12  |-  ( _i  x.  -u _i )  = 
-u ( _i  x.  _i )
10 ixi 9977 . . . . . . . . . . . . 13  |-  ( _i  x.  _i )  = 
-u 1
1110negeqi 9615 . . . . . . . . . . . 12  |-  -u (
_i  x.  _i )  =  -u -u 1
12 negneg1e1 10441 . . . . . . . . . . . 12  |-  -u -u 1  =  1
139, 11, 123eqtri 2467 . . . . . . . . . . 11  |-  ( _i  x.  -u _i )  =  1
1413eqeq2i 2453 . . . . . . . . . 10  |-  ( ( _i  x.  A )  =  ( _i  x.  -u _i )  <->  ( _i  x.  A )  =  1 )
15 eqcom 2445 . . . . . . . . . 10  |-  ( ( _i  x.  A )  =  1  <->  1  =  ( _i  x.  A
) )
1614, 15bitri 249 . . . . . . . . 9  |-  ( ( _i  x.  A )  =  ( _i  x.  -u _i )  <->  1  =  ( _i  x.  A
) )
178, 16syl6bbr 263 . . . . . . . 8  |-  ( A  e.  CC  ->  (
( 1  -  (
_i  x.  A )
)  =  0  <->  (
_i  x.  A )  =  ( _i  x.  -u _i ) ) )
18 id 22 . . . . . . . . 9  |-  ( A  e.  CC  ->  A  e.  CC )
194negcli 9688 . . . . . . . . . 10  |-  -u _i  e.  CC
2019a1i 11 . . . . . . . . 9  |-  ( A  e.  CC  ->  -u _i  e.  CC )
214a1i 11 . . . . . . . . 9  |-  ( A  e.  CC  ->  _i  e.  CC )
22 ine0 9792 . . . . . . . . . 10  |-  _i  =/=  0
2322a1i 11 . . . . . . . . 9  |-  ( A  e.  CC  ->  _i  =/=  0 )
2418, 20, 21, 23mulcand 9981 . . . . . . . 8  |-  ( A  e.  CC  ->  (
( _i  x.  A
)  =  ( _i  x.  -u _i )  <->  A  =  -u _i ) )
2517, 24bitrd 253 . . . . . . 7  |-  ( A  e.  CC  ->  (
( 1  -  (
_i  x.  A )
)  =  0  <->  A  =  -u _i ) )
2625necon3bid 2655 . . . . . 6  |-  ( A  e.  CC  ->  (
( 1  -  (
_i  x.  A )
)  =/=  0  <->  A  =/=  -u _i ) )
27 addcom 9567 . . . . . . . . . . . . 13  |-  ( ( 1  e.  CC  /\  ( _i  x.  A
)  e.  CC )  ->  ( 1  +  ( _i  x.  A
) )  =  ( ( _i  x.  A
)  +  1 ) )
283, 6, 27sylancr 663 . . . . . . . . . . . 12  |-  ( A  e.  CC  ->  (
1  +  ( _i  x.  A ) )  =  ( ( _i  x.  A )  +  1 ) )
29 subneg 9670 . . . . . . . . . . . . 13  |-  ( ( ( _i  x.  A
)  e.  CC  /\  1  e.  CC )  ->  ( ( _i  x.  A )  -  -u 1
)  =  ( ( _i  x.  A )  +  1 ) )
306, 3, 29sylancl 662 . . . . . . . . . . . 12  |-  ( A  e.  CC  ->  (
( _i  x.  A
)  -  -u 1
)  =  ( ( _i  x.  A )  +  1 ) )
3128, 30eqtr4d 2478 . . . . . . . . . . 11  |-  ( A  e.  CC  ->  (
1  +  ( _i  x.  A ) )  =  ( ( _i  x.  A )  -  -u 1 ) )
3231eqeq1d 2451 . . . . . . . . . 10  |-  ( A  e.  CC  ->  (
( 1  +  ( _i  x.  A ) )  =  0  <->  (
( _i  x.  A
)  -  -u 1
)  =  0 ) )
333negcli 9688 . . . . . . . . . . 11  |-  -u 1  e.  CC
34 subeq0 9647 . . . . . . . . . . 11  |-  ( ( ( _i  x.  A
)  e.  CC  /\  -u 1  e.  CC )  ->  ( ( ( _i  x.  A )  -  -u 1 )  =  0  <->  ( _i  x.  A )  =  -u
1 ) )
356, 33, 34sylancl 662 . . . . . . . . . 10  |-  ( A  e.  CC  ->  (
( ( _i  x.  A )  -  -u 1
)  =  0  <->  (
_i  x.  A )  =  -u 1 ) )
3632, 35bitrd 253 . . . . . . . . 9  |-  ( A  e.  CC  ->  (
( 1  +  ( _i  x.  A ) )  =  0  <->  (
_i  x.  A )  =  -u 1 ) )
3710eqeq2i 2453 . . . . . . . . 9  |-  ( ( _i  x.  A )  =  ( _i  x.  _i )  <->  ( _i  x.  A )  =  -u
1 )
3836, 37syl6bbr 263 . . . . . . . 8  |-  ( A  e.  CC  ->  (
( 1  +  ( _i  x.  A ) )  =  0  <->  (
_i  x.  A )  =  ( _i  x.  _i ) ) )
3918, 21, 21, 23mulcand 9981 . . . . . . . 8  |-  ( A  e.  CC  ->  (
( _i  x.  A
)  =  ( _i  x.  _i )  <->  A  =  _i ) )
4038, 39bitrd 253 . . . . . . 7  |-  ( A  e.  CC  ->  (
( 1  +  ( _i  x.  A ) )  =  0  <->  A  =  _i ) )
4140necon3bid 2655 . . . . . 6  |-  ( A  e.  CC  ->  (
( 1  +  ( _i  x.  A ) )  =/=  0  <->  A  =/=  _i ) )
4226, 41anbi12d 710 . . . . 5  |-  ( A  e.  CC  ->  (
( ( 1  -  ( _i  x.  A
) )  =/=  0  /\  ( 1  +  ( _i  x.  A ) )  =/=  0 )  <-> 
( A  =/=  -u _i  /\  A  =/=  _i ) ) )
4342pm5.32i 637 . . . 4  |-  ( ( A  e.  CC  /\  ( ( 1  -  ( _i  x.  A
) )  =/=  0  /\  ( 1  +  ( _i  x.  A ) )  =/=  0 ) )  <->  ( A  e.  CC  /\  ( A  =/=  -u _i  /\  A  =/=  _i ) ) )
44 3anass 969 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  -u _i  /\  A  =/=  _i )  <->  ( A  e.  CC  /\  ( A  =/=  -u _i  /\  A  =/=  _i ) ) )
4543, 44bitr4i 252 . . 3  |-  ( ( A  e.  CC  /\  ( ( 1  -  ( _i  x.  A
) )  =/=  0  /\  ( 1  +  ( _i  x.  A ) )  =/=  0 ) )  <->  ( A  e.  CC  /\  A  =/=  -u _i  /\  A  =/= 
_i ) )
462, 45bitri 249 . 2  |-  ( ( A  e.  CC  /\  ( 1  -  (
_i  x.  A )
)  =/=  0  /\  ( 1  +  ( _i  x.  A ) )  =/=  0 )  <-> 
( A  e.  CC  /\  A  =/=  -u _i  /\  A  =/=  _i ) )
471, 46bitr4i 252 1  |-  ( A  e.  dom arctan  <->  ( A  e.  CC  /\  ( 1  -  ( _i  x.  A ) )  =/=  0  /\  ( 1  +  ( _i  x.  A ) )  =/=  0 ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2618   dom cdm 4852  (class class class)co 6103   CCcc 9292   0cc0 9294   1c1 9295   _ici 9296    + caddc 9297    x. cmul 9299    - cmin 9607   -ucneg 9608  arctancatan 22271
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-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-mulcom 9358  ax-addass 9359  ax-mulass 9360  ax-distr 9361  ax-i2m1 9362  ax-1ne0 9363  ax-1rid 9364  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367  ax-pre-lttri 9368  ax-pre-lttrn 9369  ax-pre-ltadd 9370  ax-pre-mulgt0 9371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  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 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-br 4305  df-opab 4363  df-mpt 4364  df-id 4648  df-po 4653  df-so 4654  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-er 7113  df-en 7323  df-dom 7324  df-sdom 7325  df-pnf 9432  df-mnf 9433  df-xr 9434  df-ltxr 9435  df-le 9436  df-sub 9609  df-neg 9610  df-atan 22274
This theorem is referenced by:  atanf  22287  atanneg  22314  atancj  22317  efiatan  22319  atanlogaddlem  22320  atanlogadd  22321  atanlogsublem  22322  atanlogsub  22323  efiatan2  22324  2efiatan  22325  atantan  22330  atanbndlem  22332  dvatan  22342  atantayl  22344
  Copyright terms: Public domain W3C validator