MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ang180lem4 Structured version   Unicode version

Theorem ang180lem4 20654
Description: Lemma for ang180 20656. 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 ax-1cn 9048 . . . . . . . . 9  |-  1  e.  CC
32a1i 11 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  1  e.  CC )
4 simp1 957 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  A  e.  CC )
53, 4subcld 9411 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
1  -  A )  e.  CC )
6 simp3 959 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  A  =/=  1 )
76necomd 2687 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  1  =/=  A )
83, 4, 7subne0d 9420 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
1  -  A )  =/=  0 )
9 ax-1ne0 9059 . . . . . . . 8  |-  1  =/=  0
109a1i 11 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  1  =/=  0 )
111, 5, 8, 3, 10angvald 20646 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( 1  -  A
) F 1 )  =  ( Im `  ( log `  ( 1  /  ( 1  -  A ) ) ) ) )
12 simp2 958 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  A  =/=  0 )
134, 3subcld 9411 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( A  -  1 )  e.  CC )
144, 3, 6subne0d 9420 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( A  -  1 )  =/=  0 )
151, 4, 12, 13, 14angvald 20646 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( A F ( A  - 
1 ) )  =  ( Im `  ( log `  ( ( A  -  1 )  /  A ) ) ) )
1611, 15oveq12d 6099 . . . . 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
) ) ) ) )
173, 5, 8divcld 9790 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
1  /  ( 1  -  A ) )  e.  CC )
185, 8recne0d 9784 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
1  /  ( 1  -  A ) )  =/=  0 )
1917, 18logcld 20468 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( log `  ( 1  / 
( 1  -  A
) ) )  e.  CC )
2013, 4, 12divcld 9790 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( A  -  1 )  /  A )  e.  CC )
2113, 4, 14, 12divne0d 9806 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( A  -  1 )  /  A )  =/=  0 )
2220, 21logcld 20468 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( log `  ( ( A  -  1 )  /  A ) )  e.  CC )
2319, 22imaddd 12020 . . . . 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 ) ) ) ) )
2416, 23eqtr4d 2471 . . . 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
) ) ) ) )
251, 3, 10, 4, 12angvald 20646 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
1 F A )  =  ( Im `  ( log `  ( A  /  1 ) ) ) )
264div1d 9782 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( A  /  1 )  =  A )
2726fveq2d 5732 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( log `  ( A  / 
1 ) )  =  ( log `  A
) )
2827fveq2d 5732 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
Im `  ( log `  ( A  /  1
) ) )  =  ( Im `  ( log `  A ) ) )
2925, 28eqtrd 2468 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
1 F A )  =  ( Im `  ( log `  A ) ) )
3024, 29oveq12d 6099 . . 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 ) ) ) )
3119, 22addcld 9107 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( log `  (
1  /  ( 1  -  A ) ) )  +  ( log `  ( ( A  - 
1 )  /  A
) ) )  e.  CC )
324, 12logcld 20468 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( log `  A )  e.  CC )
3331, 32imaddd 12020 . . 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 ) ) ) )
3430, 33eqtr4d 2471 . 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 ) ) ) )
35 eqid 2436 . . . 4  |-  ( ( ( log `  (
1  /  ( 1  -  A ) ) )  +  ( log `  ( ( A  - 
1 )  /  A
) ) )  +  ( log `  A
) )  =  ( ( ( log `  (
1  /  ( 1  -  A ) ) )  +  ( log `  ( ( A  - 
1 )  /  A
) ) )  +  ( log `  A
) )
36 eqid 2436 . . . 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
) )
371, 35, 36ang180lem3 20653 . . 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 ) } )
38 fveq2 5728 . . . . . 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 ) ) )
39 ax-icn 9049 . . . . . . . . 9  |-  _i  e.  CC
40 pire 20372 . . . . . . . . . 10  |-  pi  e.  RR
4140recni 9102 . . . . . . . . 9  |-  pi  e.  CC
4239, 41mulcli 9095 . . . . . . . 8  |-  ( _i  x.  pi )  e.  CC
4342imnegi 11986 . . . . . . 7  |-  ( Im
`  -u ( _i  x.  pi ) )  =  -u ( Im `  ( _i  x.  pi ) )
4442addid2i 9254 . . . . . . . . . 10  |-  ( 0  +  ( _i  x.  pi ) )  =  ( _i  x.  pi )
4544fveq2i 5731 . . . . . . . . 9  |-  ( Im
`  ( 0  +  ( _i  x.  pi ) ) )  =  ( Im `  (
_i  x.  pi )
)
46 0re 9091 . . . . . . . . . 10  |-  0  e.  RR
4746, 40crimi 11998 . . . . . . . . 9  |-  ( Im
`  ( 0  +  ( _i  x.  pi ) ) )  =  pi
4845, 47eqtr3i 2458 . . . . . . . 8  |-  ( Im
`  ( _i  x.  pi ) )  =  pi
4948negeqi 9299 . . . . . . 7  |-  -u (
Im `  ( _i  x.  pi ) )  = 
-u pi
5043, 49eqtri 2456 . . . . . 6  |-  ( Im
`  -u ( _i  x.  pi ) )  =  -u pi
5138, 50syl6eq 2484 . . . . 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 )
52 fveq2 5728 . . . . . 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 ) ) )
5352, 48syl6eq 2484 . . . . 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 )
5451, 53orim12i 503 . . . 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 ) )
55 ovex 6106 . . . . 5  |-  ( ( ( log `  (
1  /  ( 1  -  A ) ) )  +  ( log `  ( ( A  - 
1 )  /  A
) ) )  +  ( log `  A
) )  e.  _V
5655elpr 3832 . . . 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 ) ) )
57 fvex 5742 . . . . 5  |-  ( Im
`  ( ( ( log `  ( 1  /  ( 1  -  A ) ) )  +  ( log `  (
( A  -  1 )  /  A ) ) )  +  ( log `  A ) ) )  e.  _V
5857elpr 3832 . . . 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 ) )
5954, 56, 583imtr4i 258 . . 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 } )
6037, 59syl 16 . 2  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
Im `  ( (
( log `  (
1  /  ( 1  -  A ) ) )  +  ( log `  ( ( A  - 
1 )  /  A
) ) )  +  ( log `  A
) ) )  e. 
{ -u pi ,  pi } )
6134, 60eqeltrd 2510 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 set class
Syntax hints:    -> wi 4    \/ wo 358    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2599    \ cdif 3317   {csn 3814   {cpr 3815   ` cfv 5454  (class class class)co 6081    e. cmpt2 6083   CCcc 8988   0cc0 8990   1c1 8991   _ici 8992    + caddc 8993    x. cmul 8995    - cmin 9291   -ucneg 9292    / cdiv 9677   2c2 10049   Imcim 11903   picpi 12669   logclog 20452
This theorem is referenced by:  ang180lem5  20655
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068  ax-addf 9069  ax-mulf 9070
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-iin 4096  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-of 6305  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-2o 6725  df-oadd 6728  df-er 6905  df-map 7020  df-pm 7021  df-ixp 7064  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-fi 7416  df-sup 7446  df-oi 7479  df-card 7826  df-cda 8048  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-4 10060  df-5 10061  df-6 10062  df-7 10063  df-8 10064  df-9 10065  df-10 10066  df-n0 10222  df-z 10283  df-dec 10383  df-uz 10489  df-q 10575  df-rp 10613  df-xneg 10710  df-xadd 10711  df-xmul 10712  df-ioo 10920  df-ioc 10921  df-ico 10922  df-icc 10923  df-fz 11044  df-fzo 11136  df-fl 11202  df-mod 11251  df-seq 11324  df-exp 11383  df-fac 11567  df-bc 11594  df-hash 11619  df-shft 11882  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-limsup 12265  df-clim 12282  df-rlim 12283  df-sum 12480  df-ef 12670  df-sin 12672  df-cos 12673  df-pi 12675  df-struct 13471  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-plusg 13542  df-mulr 13543  df-starv 13544  df-sca 13545  df-vsca 13546  df-tset 13548  df-ple 13549  df-ds 13551  df-unif 13552  df-hom 13553  df-cco 13554  df-rest 13650  df-topn 13651  df-topgen 13667  df-pt 13668  df-prds 13671  df-xrs 13726  df-0g 13727  df-gsum 13728  df-qtop 13733  df-imas 13734  df-xps 13736  df-mre 13811  df-mrc 13812  df-acs 13814  df-mnd 14690  df-submnd 14739  df-mulg 14815  df-cntz 15116  df-cmn 15414  df-psmet 16694  df-xmet 16695  df-met 16696  df-bl 16697  df-mopn 16698  df-fbas 16699  df-fg 16700  df-cnfld 16704  df-top 16963  df-bases 16965  df-topon 16966  df-topsp 16967  df-cld 17083  df-ntr 17084  df-cls 17085  df-nei 17162  df-lp 17200  df-perf 17201  df-cn 17291  df-cnp 17292  df-haus 17379  df-tx 17594  df-hmeo 17787  df-fil 17878  df-fm 17970  df-flim 17971  df-flf 17972  df-xms 18350  df-ms 18351  df-tms 18352  df-cncf 18908  df-limc 19753  df-dv 19754  df-log 20454
  Copyright terms: Public domain W3C validator