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Theorem ang180lem5 23011
Description: Lemma for ang180 23012: Reduce the statement to two variables. (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
ang180lem5  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  A  =/=  B )  ->  (
( ( ( A  -  B ) F A )  +  ( B F ( B  -  A ) ) )  +  ( A F B ) )  e.  { -u pi ,  pi } )
Distinct variable groups:    x, y, A    x, B, y
Allowed substitution hints:    F( x, y)

Proof of Theorem ang180lem5
StepHypRef Expression
1 simp1l 1020 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  A  =/=  B )  ->  A  e.  CC )
2 1cnd 9624 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  A  =/=  B )  ->  1  e.  CC )
3 simp2l 1022 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  A  =/=  B )  ->  B  e.  CC )
4 simp1r 1021 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  A  =/=  B )  ->  A  =/=  0 )
53, 1, 4divcld 10332 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  A  =/=  B )  ->  ( B  /  A )  e.  CC )
61, 2, 5subdid 10024 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  A  =/=  B )  ->  ( A  x.  ( 1  -  ( B  /  A ) ) )  =  ( ( A  x.  1 )  -  ( A  x.  ( B  /  A ) ) ) )
71mulid1d 9625 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  A  =/=  B )  ->  ( A  x.  1 )  =  A )
83, 1, 4divcan2d 10334 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  A  =/=  B )  ->  ( A  x.  ( B  /  A ) )  =  B )
97, 8oveq12d 6313 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  A  =/=  B )  ->  (
( A  x.  1 )  -  ( A  x.  ( B  /  A ) ) )  =  ( A  -  B ) )
106, 9eqtrd 2508 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  A  =/=  B )  ->  ( A  x.  ( 1  -  ( B  /  A ) ) )  =  ( A  -  B ) )
1110, 7oveq12d 6313 . . . . 5  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  A  =/=  B )  ->  (
( A  x.  (
1  -  ( B  /  A ) ) ) F ( A  x.  1 ) )  =  ( ( A  -  B ) F A ) )
122, 5subcld 9942 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  A  =/=  B )  ->  (
1  -  ( B  /  A ) )  e.  CC )
13 simp3 998 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  A  =/=  B )  ->  A  =/=  B )
1413necomd 2738 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  A  =/=  B )  ->  B  =/=  A )
153, 1, 4, 14divne1d 10343 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  A  =/=  B )  ->  ( B  /  A )  =/=  1 )
1615necomd 2738 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  A  =/=  B )  ->  1  =/=  ( B  /  A
) )
172, 5, 16subne0d 9951 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  A  =/=  B )  ->  (
1  -  ( B  /  A ) )  =/=  0 )
18 ax-1ne0 9573 . . . . . . 7  |-  1  =/=  0
1918a1i 11 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  A  =/=  B )  ->  1  =/=  0 )
20 ang.1 . . . . . . 7  |-  F  =  ( x  e.  ( CC  \  { 0 } ) ,  y  e.  ( CC  \  { 0 } ) 
|->  ( Im `  ( log `  ( y  /  x ) ) ) )
2120angcan 23000 . . . . . 6  |-  ( ( ( ( 1  -  ( B  /  A
) )  e.  CC  /\  ( 1  -  ( B  /  A ) )  =/=  0 )  /\  ( 1  e.  CC  /\  1  =/=  0 )  /\  ( A  e.  CC  /\  A  =/=  0 ) )  -> 
( ( A  x.  ( 1  -  ( B  /  A ) ) ) F ( A  x.  1 ) )  =  ( ( 1  -  ( B  /  A ) ) F 1 ) )
2212, 17, 2, 19, 1, 4, 21syl222anc 1244 . . . . 5  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  A  =/=  B )  ->  (
( A  x.  (
1  -  ( B  /  A ) ) ) F ( A  x.  1 ) )  =  ( ( 1  -  ( B  /  A ) ) F 1 ) )
2311, 22eqtr3d 2510 . . . 4  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  A  =/=  B )  ->  (
( A  -  B
) F A )  =  ( ( 1  -  ( B  /  A ) ) F 1 ) )
241, 5, 2subdid 10024 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  A  =/=  B )  ->  ( A  x.  ( ( B  /  A )  - 
1 ) )  =  ( ( A  x.  ( B  /  A
) )  -  ( A  x.  1 ) ) )
258, 7oveq12d 6313 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  A  =/=  B )  ->  (
( A  x.  ( B  /  A ) )  -  ( A  x.  1 ) )  =  ( B  -  A
) )
2624, 25eqtrd 2508 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  A  =/=  B )  ->  ( A  x.  ( ( B  /  A )  - 
1 ) )  =  ( B  -  A
) )
278, 26oveq12d 6313 . . . . 5  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  A  =/=  B )  ->  (
( A  x.  ( B  /  A ) ) F ( A  x.  ( ( B  /  A )  -  1 ) ) )  =  ( B F ( B  -  A ) ) )
28 simp2r 1023 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  A  =/=  B )  ->  B  =/=  0 )
293, 1, 28, 4divne0d 10348 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  A  =/=  B )  ->  ( B  /  A )  =/=  0 )
305, 2subcld 9942 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  A  =/=  B )  ->  (
( B  /  A
)  -  1 )  e.  CC )
315, 2, 15subne0d 9951 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  A  =/=  B )  ->  (
( B  /  A
)  -  1 )  =/=  0 )
3220angcan 23000 . . . . . 6  |-  ( ( ( ( B  /  A )  e.  CC  /\  ( B  /  A
)  =/=  0 )  /\  ( ( ( B  /  A )  -  1 )  e.  CC  /\  ( ( B  /  A )  -  1 )  =/=  0 )  /\  ( A  e.  CC  /\  A  =/=  0 ) )  -> 
( ( A  x.  ( B  /  A
) ) F ( A  x.  ( ( B  /  A )  -  1 ) ) )  =  ( ( B  /  A ) F ( ( B  /  A )  - 
1 ) ) )
335, 29, 30, 31, 1, 4, 32syl222anc 1244 . . . . 5  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  A  =/=  B )  ->  (
( A  x.  ( B  /  A ) ) F ( A  x.  ( ( B  /  A )  -  1 ) ) )  =  ( ( B  /  A ) F ( ( B  /  A
)  -  1 ) ) )
3427, 33eqtr3d 2510 . . . 4  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  A  =/=  B )  ->  ( B F ( B  -  A ) )  =  ( ( B  /  A ) F ( ( B  /  A
)  -  1 ) ) )
3523, 34oveq12d 6313 . . 3  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  A  =/=  B )  ->  (
( ( A  -  B ) F A )  +  ( B F ( B  -  A ) ) )  =  ( ( ( 1  -  ( B  /  A ) ) F 1 )  +  ( ( B  /  A ) F ( ( B  /  A
)  -  1 ) ) ) )
367, 8oveq12d 6313 . . . 4  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  A  =/=  B )  ->  (
( A  x.  1 ) F ( A  x.  ( B  /  A ) ) )  =  ( A F B ) )
3720angcan 23000 . . . . 5  |-  ( ( ( 1  e.  CC  /\  1  =/=  0 )  /\  ( ( B  /  A )  e.  CC  /\  ( B  /  A )  =/=  0 )  /\  ( A  e.  CC  /\  A  =/=  0 ) )  -> 
( ( A  x.  1 ) F ( A  x.  ( B  /  A ) ) )  =  ( 1 F ( B  /  A ) ) )
382, 19, 5, 29, 1, 4, 37syl222anc 1244 . . . 4  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  A  =/=  B )  ->  (
( A  x.  1 ) F ( A  x.  ( B  /  A ) ) )  =  ( 1 F ( B  /  A
) ) )
3936, 38eqtr3d 2510 . . 3  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  A  =/=  B )  ->  ( A F B )  =  ( 1 F ( B  /  A ) ) )
4035, 39oveq12d 6313 . 2  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  A  =/=  B )  ->  (
( ( ( A  -  B ) F A )  +  ( B F ( B  -  A ) ) )  +  ( A F B ) )  =  ( ( ( ( 1  -  ( B  /  A ) ) F 1 )  +  ( ( B  /  A ) F ( ( B  /  A
)  -  1 ) ) )  +  ( 1 F ( B  /  A ) ) ) )
4120ang180lem4 23010 . . 3  |-  ( ( ( B  /  A
)  e.  CC  /\  ( B  /  A
)  =/=  0  /\  ( B  /  A
)  =/=  1 )  ->  ( ( ( ( 1  -  ( B  /  A ) ) F 1 )  +  ( ( B  /  A ) F ( ( B  /  A
)  -  1 ) ) )  +  ( 1 F ( B  /  A ) ) )  e.  { -u pi ,  pi }
)
425, 29, 15, 41syl3anc 1228 . 2  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  A  =/=  B )  ->  (
( ( ( 1  -  ( B  /  A ) ) F 1 )  +  ( ( B  /  A
) F ( ( B  /  A )  -  1 ) ) )  +  ( 1 F ( B  /  A ) ) )  e.  { -u pi ,  pi } )
4340, 42eqeltrd 2555 1  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  A  =/=  B )  ->  (
( ( ( A  -  B ) F A )  +  ( B F ( B  -  A ) ) )  +  ( A F B ) )  e.  { -u pi ,  pi } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662    \ cdif 3478   {csn 4033   {cpr 4035   ` cfv 5594  (class class class)co 6295    |-> cmpt2 6297   CCcc 9502   0cc0 9504   1c1 9505    + caddc 9507    x. cmul 9509    - cmin 9817   -ucneg 9818    / cdiv 10218   Imcim 12911   picpi 13681   logclog 22808
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582  ax-addf 9583  ax-mulf 9584
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-iin 4334  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-of 6535  df-om 6696  df-1st 6795  df-2nd 6796  df-supp 6914  df-recs 7054  df-rdg 7088  df-1o 7142  df-2o 7143  df-oadd 7146  df-er 7323  df-map 7434  df-pm 7435  df-ixp 7482  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-fsupp 7842  df-fi 7883  df-sup 7913  df-oi 7947  df-card 8332  df-cda 8560  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-9 10613  df-10 10614  df-n0 10808  df-z 10877  df-dec 10989  df-uz 11095  df-q 11195  df-rp 11233  df-xneg 11330  df-xadd 11331  df-xmul 11332  df-ioo 11545  df-ioc 11546  df-ico 11547  df-icc 11548  df-fz 11685  df-fzo 11805  df-fl 11909  df-mod 11977  df-seq 12088  df-exp 12147  df-fac 12334  df-bc 12361  df-hash 12386  df-shft 12880  df-cj 12912  df-re 12913  df-im 12914  df-sqrt 13048  df-abs 13049  df-limsup 13274  df-clim 13291  df-rlim 13292  df-sum 13489  df-ef 13682  df-sin 13684  df-cos 13685  df-pi 13687  df-struct 14509  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-ress 14514  df-plusg 14585  df-mulr 14586  df-starv 14587  df-sca 14588  df-vsca 14589  df-ip 14590  df-tset 14591  df-ple 14592  df-ds 14594  df-unif 14595  df-hom 14596  df-cco 14597  df-rest 14695  df-topn 14696  df-0g 14714  df-gsum 14715  df-topgen 14716  df-pt 14717  df-prds 14720  df-xrs 14774  df-qtop 14779  df-imas 14780  df-xps 14782  df-mre 14858  df-mrc 14859  df-acs 14861  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-submnd 15840  df-mulg 15932  df-cntz 16227  df-cmn 16673  df-psmet 18281  df-xmet 18282  df-met 18283  df-bl 18284  df-mopn 18285  df-fbas 18286  df-fg 18287  df-cnfld 18291  df-top 19268  df-bases 19270  df-topon 19271  df-topsp 19272  df-cld 19388  df-ntr 19389  df-cls 19390  df-nei 19467  df-lp 19505  df-perf 19506  df-cn 19596  df-cnp 19597  df-haus 19684  df-tx 19931  df-hmeo 20124  df-fil 20215  df-fm 20307  df-flim 20308  df-flf 20309  df-xms 20691  df-ms 20692  df-tms 20693  df-cncf 21250  df-limc 22138  df-dv 22139  df-log 22810
This theorem is referenced by:  ang180  23012
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