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Theorem isosctrlem1ALT 32689
Description: Lemma for isosctr 22876. This proof was automatically derived by completeusersproof from its Virtual Deduction proof counterpart http://us.metamath.org/other/completeusersproof/isosctrlem1altvd.html. As it is verified by the Metamath program, isosctrlem1ALT 32689 verifies http://us.metamath.org/other/completeusersproof/isosctrlem1altvd.html. (Contributed by Alan Sare, 22-Apr-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
isosctrlem1ALT  |-  ( ( A  e.  CC  /\  ( abs `  A )  =  1  /\  -.  1  =  A )  ->  ( Im `  ( log `  ( 1  -  A ) ) )  =/=  pi )

Proof of Theorem isosctrlem1ALT
StepHypRef Expression
1 ax-1cn 9539 . . . . . . . 8  |-  1  e.  CC
21a1i 11 . . . . . . 7  |-  ( A  e.  CC  ->  1  e.  CC )
3 id 22 . . . . . . 7  |-  ( A  e.  CC  ->  A  e.  CC )
42, 3subcld 9919 . . . . . 6  |-  ( A  e.  CC  ->  (
1  -  A )  e.  CC )
54adantr 465 . . . . 5  |-  ( ( A  e.  CC  /\  -.  1  =  A
)  ->  ( 1  -  A )  e.  CC )
6 subeq0 9834 . . . . . . . . . . 11  |-  ( ( 1  e.  CC  /\  A  e.  CC )  ->  ( ( 1  -  A )  =  0  <->  1  =  A ) )
76biimpd 207 . . . . . . . . . 10  |-  ( ( 1  e.  CC  /\  A  e.  CC )  ->  ( ( 1  -  A )  =  0  ->  1  =  A ) )
87idiALT 32172 . . . . . . . . 9  |-  ( ( 1  e.  CC  /\  A  e.  CC )  ->  ( ( 1  -  A )  =  0  ->  1  =  A ) )
91, 3, 8sylancr 663 . . . . . . . 8  |-  ( A  e.  CC  ->  (
( 1  -  A
)  =  0  -> 
1  =  A ) )
109con3d 133 . . . . . . 7  |-  ( A  e.  CC  ->  ( -.  1  =  A  ->  -.  ( 1  -  A )  =  0 ) )
11 df-ne 2657 . . . . . . . 8  |-  ( ( 1  -  A )  =/=  0  <->  -.  (
1  -  A )  =  0 )
1211biimpri 206 . . . . . . 7  |-  ( -.  ( 1  -  A
)  =  0  -> 
( 1  -  A
)  =/=  0 )
1310, 12syl6 33 . . . . . 6  |-  ( A  e.  CC  ->  ( -.  1  =  A  ->  ( 1  -  A
)  =/=  0 ) )
1413imp 429 . . . . 5  |-  ( ( A  e.  CC  /\  -.  1  =  A
)  ->  ( 1  -  A )  =/=  0 )
155, 14logcld 22679 . . . 4  |-  ( ( A  e.  CC  /\  -.  1  =  A
)  ->  ( log `  ( 1  -  A
) )  e.  CC )
1615imcld 12978 . . 3  |-  ( ( A  e.  CC  /\  -.  1  =  A
)  ->  ( Im `  ( log `  (
1  -  A ) ) )  e.  RR )
17163adant2 1010 . 2  |-  ( ( A  e.  CC  /\  ( abs `  A )  =  1  /\  -.  1  =  A )  ->  ( Im `  ( log `  ( 1  -  A ) ) )  e.  RR )
18 pire 22578 . . . . 5  |-  pi  e.  RR
19 2re 10594 . . . . 5  |-  2  e.  RR
20 2ne0 10617 . . . . 5  |-  2  =/=  0
2118, 19, 20redivcli 10300 . . . 4  |-  ( pi 
/  2 )  e.  RR
2221a1i 11 . . 3  |-  ( ( A  e.  CC  /\  ( abs `  A )  =  1  /\  -.  1  =  A )  ->  ( pi  /  2
)  e.  RR )
2318a1i 11 . . 3  |-  ( ( A  e.  CC  /\  ( abs `  A )  =  1  /\  -.  1  =  A )  ->  pi  e.  RR )
24 neghalfpirx 22585 . . . 4  |-  -u (
pi  /  2 )  e.  RR*
2521rexri 9635 . . . 4  |-  ( pi 
/  2 )  e. 
RR*
263recld 12977 . . . . . . . . . 10  |-  ( A  e.  CC  ->  (
Re `  A )  e.  RR )
2726recnd 9611 . . . . . . . . 9  |-  ( A  e.  CC  ->  (
Re `  A )  e.  CC )
2827subidd 9907 . . . . . . . 8  |-  ( A  e.  CC  ->  (
( Re `  A
)  -  ( Re
`  A ) )  =  0 )
2928adantr 465 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( abs `  A )  =  1 )  -> 
( ( Re `  A )  -  (
Re `  A )
)  =  0 )
30 1re 9584 . . . . . . . . . 10  |-  1  e.  RR
3130a1i 11 . . . . . . . . 9  |-  ( 1  e.  CC  ->  1  e.  RR )
321, 31ax-mp 5 . . . . . . . 8  |-  1  e.  RR
333releabsd 13231 . . . . . . . . . 10  |-  ( A  e.  CC  ->  (
Re `  A )  <_  ( abs `  A
) )
3433adantr 465 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  ( abs `  A )  =  1 )  -> 
( Re `  A
)  <_  ( abs `  A ) )
35 id 22 . . . . . . . . . 10  |-  ( ( abs `  A )  =  1  ->  ( abs `  A )  =  1 )
3635adantl 466 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  ( abs `  A )  =  1 )  -> 
( abs `  A
)  =  1 )
3734, 36breqtrd 4464 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ( abs `  A )  =  1 )  -> 
( Re `  A
)  <_  1 )
38 lesub1 10035 . . . . . . . . . 10  |-  ( ( ( Re `  A
)  e.  RR  /\  1  e.  RR  /\  (
Re `  A )  e.  RR )  ->  (
( Re `  A
)  <_  1  <->  ( (
Re `  A )  -  ( Re `  A ) )  <_ 
( 1  -  (
Re `  A )
) ) )
39383impcombi 32569 . . . . . . . . 9  |-  ( ( 1  e.  RR  /\  ( Re `  A )  e.  RR  /\  (
Re `  A )  <_  1 )  ->  (
( Re `  A
)  -  ( Re
`  A ) )  <_  ( 1  -  ( Re `  A
) ) )
4039idiALT 32172 . . . . . . . 8  |-  ( ( 1  e.  RR  /\  ( Re `  A )  e.  RR  /\  (
Re `  A )  <_  1 )  ->  (
( Re `  A
)  -  ( Re
`  A ) )  <_  ( 1  -  ( Re `  A
) ) )
4132, 26, 37, 40eel0121 32453 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( abs `  A )  =  1 )  -> 
( ( Re `  A )  -  (
Re `  A )
)  <_  ( 1  -  ( Re `  A ) ) )
4229, 41eqbrtrrd 4462 . . . . . 6  |-  ( ( A  e.  CC  /\  ( abs `  A )  =  1 )  -> 
0  <_  ( 1  -  ( Re `  A ) ) )
4332a1i 11 . . . . . . . . . . 11  |-  ( T. 
->  1  e.  RR )
4443rered 13007 . . . . . . . . . 10  |-  ( T. 
->  ( Re `  1
)  =  1 )
4544trud 1383 . . . . . . . . 9  |-  ( Re
`  1 )  =  1
46 oveq1 6282 . . . . . . . . . 10  |-  ( ( Re `  1 )  =  1  ->  (
( Re `  1
)  -  ( Re
`  A ) )  =  ( 1  -  ( Re `  A
) ) )
4746eqcomd 2468 . . . . . . . . 9  |-  ( ( Re `  1 )  =  1  ->  (
1  -  ( Re
`  A ) )  =  ( ( Re
`  1 )  -  ( Re `  A ) ) )
4845, 47ax-mp 5 . . . . . . . 8  |-  ( 1  -  ( Re `  A ) )  =  ( ( Re ` 
1 )  -  (
Re `  A )
)
49 resub 12910 . . . . . . . . . . 11  |-  ( ( 1  e.  CC  /\  A  e.  CC )  ->  ( Re `  (
1  -  A ) )  =  ( ( Re `  1 )  -  ( Re `  A ) ) )
5049eqcomd 2468 . . . . . . . . . 10  |-  ( ( 1  e.  CC  /\  A  e.  CC )  ->  ( ( Re ` 
1 )  -  (
Re `  A )
)  =  ( Re
`  ( 1  -  A ) ) )
5150idiALT 32172 . . . . . . . . 9  |-  ( ( 1  e.  CC  /\  A  e.  CC )  ->  ( ( Re ` 
1 )  -  (
Re `  A )
)  =  ( Re
`  ( 1  -  A ) ) )
521, 3, 51sylancr 663 . . . . . . . 8  |-  ( A  e.  CC  ->  (
( Re `  1
)  -  ( Re
`  A ) )  =  ( Re `  ( 1  -  A
) ) )
5348, 52syl5eq 2513 . . . . . . 7  |-  ( A  e.  CC  ->  (
1  -  ( Re
`  A ) )  =  ( Re `  ( 1  -  A
) ) )
5453adantr 465 . . . . . 6  |-  ( ( A  e.  CC  /\  ( abs `  A )  =  1 )  -> 
( 1  -  (
Re `  A )
)  =  ( Re
`  ( 1  -  A ) ) )
5542, 54breqtrd 4464 . . . . 5  |-  ( ( A  e.  CC  /\  ( abs `  A )  =  1 )  -> 
0  <_  ( Re `  ( 1  -  A
) ) )
56 argrege0 22717 . . . . . . 7  |-  ( ( ( 1  -  A
)  e.  CC  /\  ( 1  -  A
)  =/=  0  /\  0  <_  ( Re `  ( 1  -  A
) ) )  -> 
( Im `  ( log `  ( 1  -  A ) ) )  e.  ( -u (
pi  /  2 ) [,] ( pi  / 
2 ) ) )
57563coml 1198 . . . . . 6  |-  ( ( ( 1  -  A
)  =/=  0  /\  0  <_  ( Re `  ( 1  -  A
) )  /\  (
1  -  A )  e.  CC )  -> 
( Im `  ( log `  ( 1  -  A ) ) )  e.  ( -u (
pi  /  2 ) [,] ( pi  / 
2 ) ) )
58573com13 1196 . . . . 5  |-  ( ( ( 1  -  A
)  e.  CC  /\  0  <_  ( Re `  ( 1  -  A
) )  /\  (
1  -  A )  =/=  0 )  -> 
( Im `  ( log `  ( 1  -  A ) ) )  e.  ( -u (
pi  /  2 ) [,] ( pi  / 
2 ) ) )
594, 55, 14, 58eel12131 32461 . . . 4  |-  ( ( A  e.  CC  /\  ( abs `  A )  =  1  /\  -.  1  =  A )  ->  ( Im `  ( log `  ( 1  -  A ) ) )  e.  ( -u (
pi  /  2 ) [,] ( pi  / 
2 ) ) )
60 iccleub 11569 . . . 4  |-  ( (
-u ( pi  / 
2 )  e.  RR*  /\  ( pi  /  2
)  e.  RR*  /\  (
Im `  ( log `  ( 1  -  A
) ) )  e.  ( -u ( pi 
/  2 ) [,] ( pi  /  2
) ) )  -> 
( Im `  ( log `  ( 1  -  A ) ) )  <_  ( pi  / 
2 ) )
6124, 25, 59, 60eel001 32456 . . 3  |-  ( ( A  e.  CC  /\  ( abs `  A )  =  1  /\  -.  1  =  A )  ->  ( Im `  ( log `  ( 1  -  A ) ) )  <_  ( pi  / 
2 ) )
62 pipos 22580 . . . . . 6  |-  0  <  pi
6318, 62elrpii 11212 . . . . 5  |-  pi  e.  RR+
64 rphalflt 11235 . . . . 5  |-  ( pi  e.  RR+  ->  ( pi 
/  2 )  < 
pi )
6563, 64ax-mp 5 . . . 4  |-  ( pi 
/  2 )  < 
pi
6665a1i 11 . . 3  |-  ( ( A  e.  CC  /\  ( abs `  A )  =  1  /\  -.  1  =  A )  ->  ( pi  /  2
)  <  pi )
6717, 22, 23, 61, 66lelttrd 9728 . 2  |-  ( ( A  e.  CC  /\  ( abs `  A )  =  1  /\  -.  1  =  A )  ->  ( Im `  ( log `  ( 1  -  A ) ) )  <  pi )
6817, 67ltned 9709 1  |-  ( ( A  e.  CC  /\  ( abs `  A )  =  1  /\  -.  1  =  A )  ->  ( Im `  ( log `  ( 1  -  A ) ) )  =/=  pi )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 968    = wceq 1374   T. wtru 1375    e. wcel 1762    =/= wne 2655   class class class wbr 4440   ` cfv 5579  (class class class)co 6275   CCcc 9479   RRcr 9480   0cc0 9481   1c1 9482   RR*cxr 9616    < clt 9617    <_ cle 9618    - cmin 9794   -ucneg 9795    / cdiv 10195   2c2 10574   RR+crp 11209   [,]cicc 11521   Recre 12880   Imcim 12881   abscabs 13017   picpi 13653   logclog 22663
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-inf2 8047  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559  ax-addf 9560  ax-mulf 9561
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-fal 1380  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-iin 4321  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-se 4832  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-isom 5588  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-of 6515  df-om 6672  df-1st 6774  df-2nd 6775  df-supp 6892  df-recs 7032  df-rdg 7066  df-1o 7120  df-2o 7121  df-oadd 7124  df-er 7301  df-map 7412  df-pm 7413  df-ixp 7460  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-fsupp 7819  df-fi 7860  df-sup 7890  df-oi 7924  df-card 8309  df-cda 8537  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196  df-nn 10526  df-2 10583  df-3 10584  df-4 10585  df-5 10586  df-6 10587  df-7 10588  df-8 10589  df-9 10590  df-10 10591  df-n0 10785  df-z 10854  df-dec 10966  df-uz 11072  df-q 11172  df-rp 11210  df-xneg 11307  df-xadd 11308  df-xmul 11309  df-ioo 11522  df-ioc 11523  df-ico 11524  df-icc 11525  df-fz 11662  df-fzo 11782  df-fl 11886  df-mod 11953  df-seq 12064  df-exp 12123  df-fac 12309  df-bc 12336  df-hash 12361  df-shft 12850  df-cj 12882  df-re 12883  df-im 12884  df-sqr 13018  df-abs 13019  df-limsup 13243  df-clim 13260  df-rlim 13261  df-sum 13458  df-ef 13654  df-sin 13656  df-cos 13657  df-pi 13659  df-struct 14481  df-ndx 14482  df-slot 14483  df-base 14484  df-sets 14485  df-ress 14486  df-plusg 14557  df-mulr 14558  df-starv 14559  df-sca 14560  df-vsca 14561  df-ip 14562  df-tset 14563  df-ple 14564  df-ds 14566  df-unif 14567  df-hom 14568  df-cco 14569  df-rest 14667  df-topn 14668  df-0g 14686  df-gsum 14687  df-topgen 14688  df-pt 14689  df-prds 14692  df-xrs 14746  df-qtop 14751  df-imas 14752  df-xps 14754  df-mre 14830  df-mrc 14831  df-acs 14833  df-mnd 15721  df-submnd 15771  df-mulg 15854  df-cntz 16143  df-cmn 16589  df-psmet 18175  df-xmet 18176  df-met 18177  df-bl 18178  df-mopn 18179  df-fbas 18180  df-fg 18181  df-cnfld 18185  df-top 19159  df-bases 19161  df-topon 19162  df-topsp 19163  df-cld 19279  df-ntr 19280  df-cls 19281  df-nei 19358  df-lp 19396  df-perf 19397  df-cn 19487  df-cnp 19488  df-haus 19575  df-tx 19791  df-hmeo 19984  df-fil 20075  df-fm 20167  df-flim 20168  df-flf 20169  df-xms 20551  df-ms 20552  df-tms 20553  df-cncf 21110  df-limc 21998  df-dv 21999  df-log 22665
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator