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Theorem ang180lem4 23471
Description: Lemma for ang180 23473. 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 9642 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  1  e.  CC )
3 simp1 997 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  A  e.  CC )
42, 3subcld 9967 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
1  -  A )  e.  CC )
5 simp3 999 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  A  =/=  1 )
65necomd 2674 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  1  =/=  A )
72, 3, 6subne0d 9976 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
1  -  A )  =/=  0 )
8 ax-1ne0 9591 . . . . . . . 8  |-  1  =/=  0
98a1i 11 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  1  =/=  0 )
101, 4, 7, 2, 9angvald 23463 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( 1  -  A
) F 1 )  =  ( Im `  ( log `  ( 1  /  ( 1  -  A ) ) ) ) )
11 simp2 998 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  A  =/=  0 )
123, 2subcld 9967 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( A  -  1 )  e.  CC )
133, 2, 5subne0d 9976 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( A  -  1 )  =/=  0 )
141, 3, 11, 12, 13angvald 23463 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( A F ( A  - 
1 ) )  =  ( Im `  ( log `  ( ( A  -  1 )  /  A ) ) ) )
1510, 14oveq12d 6296 . . . . 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 10361 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
1  /  ( 1  -  A ) )  e.  CC )
174, 7recne0d 10355 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
1  /  ( 1  -  A ) )  =/=  0 )
1816, 17logcld 23250 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( log `  ( 1  / 
( 1  -  A
) ) )  e.  CC )
1912, 3, 11divcld 10361 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( A  -  1 )  /  A )  e.  CC )
2012, 3, 13, 11divne0d 10377 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( A  -  1 )  /  A )  =/=  0 )
2119, 20logcld 23250 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( log `  ( ( A  -  1 )  /  A ) )  e.  CC )
2218, 21imaddd 13197 . . . . 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 2446 . . . 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 23463 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
1 F A )  =  ( Im `  ( log `  ( A  /  1 ) ) ) )
253div1d 10353 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( A  /  1 )  =  A )
2625fveq2d 5853 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( log `  ( A  / 
1 ) )  =  ( log `  A
) )
2726fveq2d 5853 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
Im `  ( log `  ( A  /  1
) ) )  =  ( Im `  ( log `  A ) ) )
2824, 27eqtrd 2443 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
1 F A )  =  ( Im `  ( log `  A ) ) )
2923, 28oveq12d 6296 . . 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 9645 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( log `  (
1  /  ( 1  -  A ) ) )  +  ( log `  ( ( A  - 
1 )  /  A
) ) )  e.  CC )
313, 11logcld 23250 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( log `  A )  e.  CC )
3230, 31imaddd 13197 . . 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 2446 . 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 2402 . . . 4  |-  ( ( ( log `  (
1  /  ( 1  -  A ) ) )  +  ( log `  ( ( A  - 
1 )  /  A
) ) )  +  ( log `  A
) )  =  ( ( ( log `  (
1  /  ( 1  -  A ) ) )  +  ( log `  ( ( A  - 
1 )  /  A
) ) )  +  ( log `  A
) )
35 eqid 2402 . . . 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 23470 . . 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 5849 . . . . . 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 9581 . . . . . . . . 9  |-  _i  e.  CC
39 picn 23144 . . . . . . . . 9  |-  pi  e.  CC
4038, 39mulcli 9631 . . . . . . . 8  |-  ( _i  x.  pi )  e.  CC
4140imnegi 13163 . . . . . . 7  |-  ( Im
`  -u ( _i  x.  pi ) )  =  -u ( Im `  ( _i  x.  pi ) )
4240addid2i 9802 . . . . . . . . . 10  |-  ( 0  +  ( _i  x.  pi ) )  =  ( _i  x.  pi )
4342fveq2i 5852 . . . . . . . . 9  |-  ( Im
`  ( 0  +  ( _i  x.  pi ) ) )  =  ( Im `  (
_i  x.  pi )
)
44 0re 9626 . . . . . . . . . 10  |-  0  e.  RR
45 pire 23143 . . . . . . . . . 10  |-  pi  e.  RR
4644, 45crimi 13175 . . . . . . . . 9  |-  ( Im
`  ( 0  +  ( _i  x.  pi ) ) )  =  pi
4743, 46eqtr3i 2433 . . . . . . . 8  |-  ( Im
`  ( _i  x.  pi ) )  =  pi
4847negeqi 9849 . . . . . . 7  |-  -u (
Im `  ( _i  x.  pi ) )  = 
-u pi
4941, 48eqtri 2431 . . . . . 6  |-  ( Im
`  -u ( _i  x.  pi ) )  =  -u pi
5037, 49syl6eq 2459 . . . . 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 5849 . . . . . 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 2459 . . . . 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 514 . . . 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 6306 . . . . 5  |-  ( ( ( log `  (
1  /  ( 1  -  A ) ) )  +  ( log `  ( ( A  - 
1 )  /  A
) ) )  +  ( log `  A
) )  e.  _V
5554elpr 3990 . . . 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 5859 . . . . 5  |-  ( Im
`  ( ( ( log `  ( 1  /  ( 1  -  A ) ) )  +  ( log `  (
( A  -  1 )  /  A ) ) )  +  ( log `  A ) ) )  e.  _V
5756elpr 3990 . . . 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 17 . 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 2490 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 366    /\ w3a 974    = wceq 1405    e. wcel 1842    =/= wne 2598    \ cdif 3411   {csn 3972   {cpr 3974   ` cfv 5569  (class class class)co 6278    |-> cmpt2 6280   CCcc 9520   0cc0 9522   1c1 9523   _ici 9524    + caddc 9525    x. cmul 9527    - cmin 9841   -ucneg 9842    / cdiv 10247   2c2 10626   Imcim 13080   picpi 14011   logclog 23234
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-inf2 8091  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599  ax-pre-sup 9600  ax-addf 9601  ax-mulf 9602
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-iin 4274  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-se 4783  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-isom 5578  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-of 6521  df-om 6684  df-1st 6784  df-2nd 6785  df-supp 6903  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-2o 7168  df-oadd 7171  df-er 7348  df-map 7459  df-pm 7460  df-ixp 7508  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-fsupp 7864  df-fi 7905  df-sup 7935  df-oi 7969  df-card 8352  df-cda 8580  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-div 10248  df-nn 10577  df-2 10635  df-3 10636  df-4 10637  df-5 10638  df-6 10639  df-7 10640  df-8 10641  df-9 10642  df-10 10643  df-n0 10837  df-z 10906  df-dec 11020  df-uz 11128  df-q 11228  df-rp 11266  df-xneg 11371  df-xadd 11372  df-xmul 11373  df-ioo 11586  df-ioc 11587  df-ico 11588  df-icc 11589  df-fz 11727  df-fzo 11855  df-fl 11966  df-mod 12035  df-seq 12152  df-exp 12211  df-fac 12398  df-bc 12425  df-hash 12453  df-shft 13049  df-cj 13081  df-re 13082  df-im 13083  df-sqrt 13217  df-abs 13218  df-limsup 13443  df-clim 13460  df-rlim 13461  df-sum 13658  df-ef 14012  df-sin 14014  df-cos 14015  df-pi 14017  df-struct 14843  df-ndx 14844  df-slot 14845  df-base 14846  df-sets 14847  df-ress 14848  df-plusg 14922  df-mulr 14923  df-starv 14924  df-sca 14925  df-vsca 14926  df-ip 14927  df-tset 14928  df-ple 14929  df-ds 14931  df-unif 14932  df-hom 14933  df-cco 14934  df-rest 15037  df-topn 15038  df-0g 15056  df-gsum 15057  df-topgen 15058  df-pt 15059  df-prds 15062  df-xrs 15116  df-qtop 15121  df-imas 15122  df-xps 15124  df-mre 15200  df-mrc 15201  df-acs 15203  df-mgm 16196  df-sgrp 16235  df-mnd 16245  df-submnd 16291  df-mulg 16384  df-cntz 16679  df-cmn 17124  df-psmet 18731  df-xmet 18732  df-met 18733  df-bl 18734  df-mopn 18735  df-fbas 18736  df-fg 18737  df-cnfld 18741  df-top 19691  df-bases 19693  df-topon 19694  df-topsp 19695  df-cld 19812  df-ntr 19813  df-cls 19814  df-nei 19892  df-lp 19930  df-perf 19931  df-cn 20021  df-cnp 20022  df-haus 20109  df-tx 20355  df-hmeo 20548  df-fil 20639  df-fm 20731  df-flim 20732  df-flf 20733  df-xms 21115  df-ms 21116  df-tms 21117  df-cncf 21674  df-limc 22562  df-dv 22563  df-log 23236
This theorem is referenced by:  ang180lem5  23472
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