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Theorem ang180lem4 22227
Description: Lemma for ang180 22229. Reduce the statement to one variable. (Contributed by Mario Carneiro, 23-Sep-2014.)
Hypothesis
Ref Expression
ang.1  |-  F  =  ( x  e.  ( CC  \  { 0 } ) ,  y  e.  ( CC  \  { 0 } ) 
|->  ( Im `  ( log `  ( y  /  x ) ) ) )
Assertion
Ref Expression
ang180lem4  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( ( ( 1  -  A ) F 1 )  +  ( A F ( A  -  1 ) ) )  +  ( 1 F A ) )  e.  { -u pi ,  pi } )
Distinct variable group:    x, y, A
Allowed substitution hints:    F( x, y)

Proof of Theorem ang180lem4
StepHypRef Expression
1 ang.1 . . . . . . 7  |-  F  =  ( x  e.  ( CC  \  { 0 } ) ,  y  e.  ( CC  \  { 0 } ) 
|->  ( Im `  ( log `  ( y  /  x ) ) ) )
2 1cnd 9421 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  1  e.  CC )
3 simp1 988 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  A  e.  CC )
42, 3subcld 9738 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
1  -  A )  e.  CC )
5 simp3 990 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  A  =/=  1 )
65necomd 2636 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  1  =/=  A )
72, 3, 6subne0d 9747 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
1  -  A )  =/=  0 )
8 ax-1ne0 9370 . . . . . . . 8  |-  1  =/=  0
98a1i 11 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  1  =/=  0 )
101, 4, 7, 2, 9angvald 22219 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( 1  -  A
) F 1 )  =  ( Im `  ( log `  ( 1  /  ( 1  -  A ) ) ) ) )
11 simp2 989 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  A  =/=  0 )
123, 2subcld 9738 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( A  -  1 )  e.  CC )
133, 2, 5subne0d 9747 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( A  -  1 )  =/=  0 )
141, 3, 11, 12, 13angvald 22219 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( A F ( A  - 
1 ) )  =  ( Im `  ( log `  ( ( A  -  1 )  /  A ) ) ) )
1510, 14oveq12d 6128 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( ( 1  -  A ) F 1 )  +  ( A F ( A  - 
1 ) ) )  =  ( ( Im
`  ( log `  (
1  /  ( 1  -  A ) ) ) )  +  ( Im `  ( log `  ( ( A  - 
1 )  /  A
) ) ) ) )
162, 4, 7divcld 10126 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
1  /  ( 1  -  A ) )  e.  CC )
174, 7recne0d 10120 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
1  /  ( 1  -  A ) )  =/=  0 )
1816, 17logcld 22041 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( log `  ( 1  / 
( 1  -  A
) ) )  e.  CC )
1912, 3, 11divcld 10126 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( A  -  1 )  /  A )  e.  CC )
2012, 3, 13, 11divne0d 10142 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( A  -  1 )  /  A )  =/=  0 )
2119, 20logcld 22041 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( log `  ( ( A  -  1 )  /  A ) )  e.  CC )
2218, 21imaddd 12723 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
Im `  ( ( log `  ( 1  / 
( 1  -  A
) ) )  +  ( log `  (
( A  -  1 )  /  A ) ) ) )  =  ( ( Im `  ( log `  ( 1  /  ( 1  -  A ) ) ) )  +  ( Im
`  ( log `  (
( A  -  1 )  /  A ) ) ) ) )
2315, 22eqtr4d 2478 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( ( 1  -  A ) F 1 )  +  ( A F ( A  - 
1 ) ) )  =  ( Im `  ( ( log `  (
1  /  ( 1  -  A ) ) )  +  ( log `  ( ( A  - 
1 )  /  A
) ) ) ) )
241, 2, 9, 3, 11angvald 22219 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
1 F A )  =  ( Im `  ( log `  ( A  /  1 ) ) ) )
253div1d 10118 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( A  /  1 )  =  A )
2625fveq2d 5714 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( log `  ( A  / 
1 ) )  =  ( log `  A
) )
2726fveq2d 5714 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
Im `  ( log `  ( A  /  1
) ) )  =  ( Im `  ( log `  A ) ) )
2824, 27eqtrd 2475 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
1 F A )  =  ( Im `  ( log `  A ) ) )
2923, 28oveq12d 6128 . . 3  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( ( ( 1  -  A ) F 1 )  +  ( A F ( A  -  1 ) ) )  +  ( 1 F A ) )  =  ( ( Im
`  ( ( log `  ( 1  /  (
1  -  A ) ) )  +  ( log `  ( ( A  -  1 )  /  A ) ) ) )  +  ( Im `  ( log `  A ) ) ) )
3018, 21addcld 9424 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( log `  (
1  /  ( 1  -  A ) ) )  +  ( log `  ( ( A  - 
1 )  /  A
) ) )  e.  CC )
313, 11logcld 22041 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( log `  A )  e.  CC )
3230, 31imaddd 12723 . . 3  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
Im `  ( (
( log `  (
1  /  ( 1  -  A ) ) )  +  ( log `  ( ( A  - 
1 )  /  A
) ) )  +  ( log `  A
) ) )  =  ( ( Im `  ( ( log `  (
1  /  ( 1  -  A ) ) )  +  ( log `  ( ( A  - 
1 )  /  A
) ) ) )  +  ( Im `  ( log `  A ) ) ) )
3329, 32eqtr4d 2478 . 2  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( ( ( 1  -  A ) F 1 )  +  ( A F ( A  -  1 ) ) )  +  ( 1 F A ) )  =  ( Im `  ( ( ( log `  ( 1  /  (
1  -  A ) ) )  +  ( log `  ( ( A  -  1 )  /  A ) ) )  +  ( log `  A ) ) ) )
34 eqid 2443 . . . 4  |-  ( ( ( log `  (
1  /  ( 1  -  A ) ) )  +  ( log `  ( ( A  - 
1 )  /  A
) ) )  +  ( log `  A
) )  =  ( ( ( log `  (
1  /  ( 1  -  A ) ) )  +  ( log `  ( ( A  - 
1 )  /  A
) ) )  +  ( log `  A
) )
35 eqid 2443 . . . 4  |-  ( ( ( ( ( ( log `  ( 1  /  ( 1  -  A ) ) )  +  ( log `  (
( A  -  1 )  /  A ) ) )  +  ( log `  A ) )  /  _i )  /  ( 2  x.  pi ) )  -  ( 1  /  2
) )  =  ( ( ( ( ( ( log `  (
1  /  ( 1  -  A ) ) )  +  ( log `  ( ( A  - 
1 )  /  A
) ) )  +  ( log `  A
) )  /  _i )  /  ( 2  x.  pi ) )  -  ( 1  /  2
) )
361, 34, 35ang180lem3 22226 . . 3  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( ( log `  (
1  /  ( 1  -  A ) ) )  +  ( log `  ( ( A  - 
1 )  /  A
) ) )  +  ( log `  A
) )  e.  { -u ( _i  x.  pi ) ,  ( _i  x.  pi ) } )
37 fveq2 5710 . . . . . 6  |-  ( ( ( ( log `  (
1  /  ( 1  -  A ) ) )  +  ( log `  ( ( A  - 
1 )  /  A
) ) )  +  ( log `  A
) )  =  -u ( _i  x.  pi )  ->  ( Im `  ( ( ( log `  ( 1  /  (
1  -  A ) ) )  +  ( log `  ( ( A  -  1 )  /  A ) ) )  +  ( log `  A ) ) )  =  ( Im `  -u ( _i  x.  pi ) ) )
38 ax-icn 9360 . . . . . . . . 9  |-  _i  e.  CC
39 picn 21941 . . . . . . . . 9  |-  pi  e.  CC
4038, 39mulcli 9410 . . . . . . . 8  |-  ( _i  x.  pi )  e.  CC
4140imnegi 12689 . . . . . . 7  |-  ( Im
`  -u ( _i  x.  pi ) )  =  -u ( Im `  ( _i  x.  pi ) )
4240addid2i 9576 . . . . . . . . . 10  |-  ( 0  +  ( _i  x.  pi ) )  =  ( _i  x.  pi )
4342fveq2i 5713 . . . . . . . . 9  |-  ( Im
`  ( 0  +  ( _i  x.  pi ) ) )  =  ( Im `  (
_i  x.  pi )
)
44 0re 9405 . . . . . . . . . 10  |-  0  e.  RR
45 pire 21940 . . . . . . . . . 10  |-  pi  e.  RR
4644, 45crimi 12701 . . . . . . . . 9  |-  ( Im
`  ( 0  +  ( _i  x.  pi ) ) )  =  pi
4743, 46eqtr3i 2465 . . . . . . . 8  |-  ( Im
`  ( _i  x.  pi ) )  =  pi
4847negeqi 9622 . . . . . . 7  |-  -u (
Im `  ( _i  x.  pi ) )  = 
-u pi
4941, 48eqtri 2463 . . . . . 6  |-  ( Im
`  -u ( _i  x.  pi ) )  =  -u pi
5037, 49syl6eq 2491 . . . . 5  |-  ( ( ( ( log `  (
1  /  ( 1  -  A ) ) )  +  ( log `  ( ( A  - 
1 )  /  A
) ) )  +  ( log `  A
) )  =  -u ( _i  x.  pi )  ->  ( Im `  ( ( ( log `  ( 1  /  (
1  -  A ) ) )  +  ( log `  ( ( A  -  1 )  /  A ) ) )  +  ( log `  A ) ) )  =  -u pi )
51 fveq2 5710 . . . . . 6  |-  ( ( ( ( log `  (
1  /  ( 1  -  A ) ) )  +  ( log `  ( ( A  - 
1 )  /  A
) ) )  +  ( log `  A
) )  =  ( _i  x.  pi )  ->  ( Im `  ( ( ( log `  ( 1  /  (
1  -  A ) ) )  +  ( log `  ( ( A  -  1 )  /  A ) ) )  +  ( log `  A ) ) )  =  ( Im `  ( _i  x.  pi ) ) )
5251, 47syl6eq 2491 . . . . 5  |-  ( ( ( ( log `  (
1  /  ( 1  -  A ) ) )  +  ( log `  ( ( A  - 
1 )  /  A
) ) )  +  ( log `  A
) )  =  ( _i  x.  pi )  ->  ( Im `  ( ( ( log `  ( 1  /  (
1  -  A ) ) )  +  ( log `  ( ( A  -  1 )  /  A ) ) )  +  ( log `  A ) ) )  =  pi )
5350, 52orim12i 516 . . . 4  |-  ( ( ( ( ( log `  ( 1  /  (
1  -  A ) ) )  +  ( log `  ( ( A  -  1 )  /  A ) ) )  +  ( log `  A ) )  = 
-u ( _i  x.  pi )  \/  (
( ( log `  (
1  /  ( 1  -  A ) ) )  +  ( log `  ( ( A  - 
1 )  /  A
) ) )  +  ( log `  A
) )  =  ( _i  x.  pi ) )  ->  ( (
Im `  ( (
( log `  (
1  /  ( 1  -  A ) ) )  +  ( log `  ( ( A  - 
1 )  /  A
) ) )  +  ( log `  A
) ) )  = 
-u pi  \/  (
Im `  ( (
( log `  (
1  /  ( 1  -  A ) ) )  +  ( log `  ( ( A  - 
1 )  /  A
) ) )  +  ( log `  A
) ) )  =  pi ) )
54 ovex 6135 . . . . 5  |-  ( ( ( log `  (
1  /  ( 1  -  A ) ) )  +  ( log `  ( ( A  - 
1 )  /  A
) ) )  +  ( log `  A
) )  e.  _V
5554elpr 3914 . . . 4  |-  ( ( ( ( log `  (
1  /  ( 1  -  A ) ) )  +  ( log `  ( ( A  - 
1 )  /  A
) ) )  +  ( log `  A
) )  e.  { -u ( _i  x.  pi ) ,  ( _i  x.  pi ) }  <->  ( (
( ( log `  (
1  /  ( 1  -  A ) ) )  +  ( log `  ( ( A  - 
1 )  /  A
) ) )  +  ( log `  A
) )  =  -u ( _i  x.  pi )  \/  ( (
( log `  (
1  /  ( 1  -  A ) ) )  +  ( log `  ( ( A  - 
1 )  /  A
) ) )  +  ( log `  A
) )  =  ( _i  x.  pi ) ) )
56 fvex 5720 . . . . 5  |-  ( Im
`  ( ( ( log `  ( 1  /  ( 1  -  A ) ) )  +  ( log `  (
( A  -  1 )  /  A ) ) )  +  ( log `  A ) ) )  e.  _V
5756elpr 3914 . . . 4  |-  ( ( Im `  ( ( ( log `  (
1  /  ( 1  -  A ) ) )  +  ( log `  ( ( A  - 
1 )  /  A
) ) )  +  ( log `  A
) ) )  e. 
{ -u pi ,  pi } 
<->  ( ( Im `  ( ( ( log `  ( 1  /  (
1  -  A ) ) )  +  ( log `  ( ( A  -  1 )  /  A ) ) )  +  ( log `  A ) ) )  =  -u pi  \/  (
Im `  ( (
( log `  (
1  /  ( 1  -  A ) ) )  +  ( log `  ( ( A  - 
1 )  /  A
) ) )  +  ( log `  A
) ) )  =  pi ) )
5853, 55, 573imtr4i 266 . . 3  |-  ( ( ( ( log `  (
1  /  ( 1  -  A ) ) )  +  ( log `  ( ( A  - 
1 )  /  A
) ) )  +  ( log `  A
) )  e.  { -u ( _i  x.  pi ) ,  ( _i  x.  pi ) }  ->  ( Im `  ( ( ( log `  (
1  /  ( 1  -  A ) ) )  +  ( log `  ( ( A  - 
1 )  /  A
) ) )  +  ( log `  A
) ) )  e. 
{ -u pi ,  pi } )
5936, 58syl 16 . 2  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
Im `  ( (
( log `  (
1  /  ( 1  -  A ) ) )  +  ( log `  ( ( A  - 
1 )  /  A
) ) )  +  ( log `  A
) ) )  e. 
{ -u pi ,  pi } )
6033, 59eqeltrd 2517 1  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( ( ( 1  -  A ) F 1 )  +  ( A F ( A  -  1 ) ) )  +  ( 1 F A ) )  e.  { -u pi ,  pi } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2620    \ cdif 3344   {csn 3896   {cpr 3898   ` cfv 5437  (class class class)co 6110    e. cmpt2 6112   CCcc 9299   0cc0 9301   1c1 9302   _ici 9303    + caddc 9304    x. cmul 9306    - cmin 9614   -ucneg 9615    / cdiv 10012   2c2 10390   Imcim 12606   picpi 13371   logclog 22025
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-rep 4422  ax-sep 4432  ax-nul 4440  ax-pow 4489  ax-pr 4550  ax-un 6391  ax-inf2 7866  ax-cnex 9357  ax-resscn 9358  ax-1cn 9359  ax-icn 9360  ax-addcl 9361  ax-addrcl 9362  ax-mulcl 9363  ax-mulrcl 9364  ax-mulcom 9365  ax-addass 9366  ax-mulass 9367  ax-distr 9368  ax-i2m1 9369  ax-1ne0 9370  ax-1rid 9371  ax-rnegex 9372  ax-rrecex 9373  ax-cnre 9374  ax-pre-lttri 9375  ax-pre-lttrn 9376  ax-pre-ltadd 9377  ax-pre-mulgt0 9378  ax-pre-sup 9379  ax-addf 9380  ax-mulf 9381
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  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 2622  df-nel 2623  df-ral 2739  df-rex 2740  df-reu 2741  df-rmo 2742  df-rab 2743  df-v 2993  df-sbc 3206  df-csb 3308  df-dif 3350  df-un 3352  df-in 3354  df-ss 3361  df-pss 3363  df-nul 3657  df-if 3811  df-pw 3881  df-sn 3897  df-pr 3899  df-tp 3901  df-op 3903  df-uni 4111  df-int 4148  df-iun 4192  df-iin 4193  df-br 4312  df-opab 4370  df-mpt 4371  df-tr 4405  df-eprel 4651  df-id 4655  df-po 4660  df-so 4661  df-fr 4698  df-se 4699  df-we 4700  df-ord 4741  df-on 4742  df-lim 4743  df-suc 4744  df-xp 4865  df-rel 4866  df-cnv 4867  df-co 4868  df-dm 4869  df-rn 4870  df-res 4871  df-ima 4872  df-iota 5400  df-fun 5439  df-fn 5440  df-f 5441  df-f1 5442  df-fo 5443  df-f1o 5444  df-fv 5445  df-isom 5446  df-riota 6071  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-of 6339  df-om 6496  df-1st 6596  df-2nd 6597  df-supp 6710  df-recs 6851  df-rdg 6885  df-1o 6939  df-2o 6940  df-oadd 6943  df-er 7120  df-map 7235  df-pm 7236  df-ixp 7283  df-en 7330  df-dom 7331  df-sdom 7332  df-fin 7333  df-fsupp 7640  df-fi 7680  df-sup 7710  df-oi 7743  df-card 8128  df-cda 8356  df-pnf 9439  df-mnf 9440  df-xr 9441  df-ltxr 9442  df-le 9443  df-sub 9616  df-neg 9617  df-div 10013  df-nn 10342  df-2 10399  df-3 10400  df-4 10401  df-5 10402  df-6 10403  df-7 10404  df-8 10405  df-9 10406  df-10 10407  df-n0 10599  df-z 10666  df-dec 10775  df-uz 10881  df-q 10973  df-rp 11011  df-xneg 11108  df-xadd 11109  df-xmul 11110  df-ioo 11323  df-ioc 11324  df-ico 11325  df-icc 11326  df-fz 11457  df-fzo 11568  df-fl 11661  df-mod 11728  df-seq 11826  df-exp 11885  df-fac 12071  df-bc 12098  df-hash 12123  df-shft 12575  df-cj 12607  df-re 12608  df-im 12609  df-sqr 12743  df-abs 12744  df-limsup 12968  df-clim 12985  df-rlim 12986  df-sum 13183  df-ef 13372  df-sin 13374  df-cos 13375  df-pi 13377  df-struct 14195  df-ndx 14196  df-slot 14197  df-base 14198  df-sets 14199  df-ress 14200  df-plusg 14270  df-mulr 14271  df-starv 14272  df-sca 14273  df-vsca 14274  df-ip 14275  df-tset 14276  df-ple 14277  df-ds 14279  df-unif 14280  df-hom 14281  df-cco 14282  df-rest 14380  df-topn 14381  df-0g 14399  df-gsum 14400  df-topgen 14401  df-pt 14402  df-prds 14405  df-xrs 14459  df-qtop 14464  df-imas 14465  df-xps 14467  df-mre 14543  df-mrc 14544  df-acs 14546  df-mnd 15434  df-submnd 15484  df-mulg 15567  df-cntz 15854  df-cmn 16298  df-psmet 17828  df-xmet 17829  df-met 17830  df-bl 17831  df-mopn 17832  df-fbas 17833  df-fg 17834  df-cnfld 17838  df-top 18522  df-bases 18524  df-topon 18525  df-topsp 18526  df-cld 18642  df-ntr 18643  df-cls 18644  df-nei 18721  df-lp 18759  df-perf 18760  df-cn 18850  df-cnp 18851  df-haus 18938  df-tx 19154  df-hmeo 19347  df-fil 19438  df-fm 19530  df-flim 19531  df-flf 19532  df-xms 19914  df-ms 19915  df-tms 19916  df-cncf 20473  df-limc 21360  df-dv 21361  df-log 22027
This theorem is referenced by:  ang180lem5  22228
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