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

Theorem atandm2 23077
Description: This form of atandm 23076 is a bit more useful for showing that the logarithms in df-atan 23067 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 23076 . 2  |-  ( A  e.  dom arctan  <->  ( A  e.  CC  /\  A  =/=  -u _i  /\  A  =/= 
_i ) )
2 3anass 976 . . 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 9550 . . . . . . . . . 10  |-  1  e.  CC
4 ax-icn 9551 . . . . . . . . . . 11  |-  _i  e.  CC
5 mulcl 9576 . . . . . . . . . . 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 9847 . . . . . . . . . 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 10006 . . . . . . . . . . . 12  |-  ( _i  x.  -u _i )  = 
-u ( _i  x.  _i )
10 ixi 10181 . . . . . . . . . . . . 13  |-  ( _i  x.  _i )  = 
-u 1
1110negeqi 9815 . . . . . . . . . . . 12  |-  -u (
_i  x.  _i )  =  -u -u 1
12 negneg1e1 10646 . . . . . . . . . . . 12  |-  -u -u 1  =  1
139, 11, 123eqtri 2474 . . . . . . . . . . 11  |-  ( _i  x.  -u _i )  =  1
1413eqeq2i 2459 . . . . . . . . . 10  |-  ( ( _i  x.  A )  =  ( _i  x.  -u _i )  <->  ( _i  x.  A )  =  1 )
15 eqcom 2450 . . . . . . . . . 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 9889 . . . . . . . . . 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 9995 . . . . . . . . . 10  |-  _i  =/=  0
2322a1i 11 . . . . . . . . 9  |-  ( A  e.  CC  ->  _i  =/=  0 )
2418, 20, 21, 23mulcand 10185 . . . . . . . 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 2699 . . . . . 6  |-  ( A  e.  CC  ->  (
( 1  -  (
_i  x.  A )
)  =/=  0  <->  A  =/=  -u _i ) )
27 addcom 9766 . . . . . . . . . . . . 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 9870 . . . . . . . . . . . . 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 2485 . . . . . . . . . . 11  |-  ( A  e.  CC  ->  (
1  +  ( _i  x.  A ) )  =  ( ( _i  x.  A )  -  -u 1 ) )
3231eqeq1d 2443 . . . . . . . . . 10  |-  ( A  e.  CC  ->  (
( 1  +  ( _i  x.  A ) )  =  0  <->  (
( _i  x.  A
)  -  -u 1
)  =  0 ) )
333negcli 9889 . . . . . . . . . . 11  |-  -u 1  e.  CC
34 subeq0 9847 . . . . . . . . . . 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 2459 . . . . . . . . 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 10185 . . . . . . . 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 2699 . . . . . 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 976 . . . 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 972    = wceq 1381    e. wcel 1802    =/= wne 2636   dom cdm 4986  (class class class)co 6278   CCcc 9490   0cc0 9492   1c1 9493   _ici 9494    + caddc 9495    x. cmul 9497    - cmin 9807   -ucneg 9808  arctancatan 23064
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4555  ax-nul 4563  ax-pow 4612  ax-pr 4673  ax-un 6574  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3419  df-dif 3462  df-un 3464  df-in 3466  df-ss 3473  df-nul 3769  df-if 3924  df-pw 3996  df-sn 4012  df-pr 4014  df-op 4018  df-uni 4232  df-br 4435  df-opab 4493  df-mpt 4494  df-id 4782  df-po 4787  df-so 4788  df-xp 4992  df-rel 4993  df-cnv 4994  df-co 4995  df-dm 4996  df-rn 4997  df-res 4998  df-ima 4999  df-iota 5538  df-fun 5577  df-fn 5578  df-f 5579  df-f1 5580  df-fo 5581  df-f1o 5582  df-fv 5583  df-riota 6239  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-er 7310  df-en 7516  df-dom 7517  df-sdom 7518  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9809  df-neg 9810  df-atan 23067
This theorem is referenced by:  atanf  23080  atanneg  23107  atancj  23110  efiatan  23112  atanlogaddlem  23113  atanlogadd  23114  atanlogsublem  23115  atanlogsub  23116  efiatan2  23117  2efiatan  23118  atantan  23123  atanbndlem  23125  dvatan  23135  atantayl  23137
  Copyright terms: Public domain W3C validator