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Theorem ang180lem5 20515
Description: Lemma for ang180 20516: 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 981 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  A  =/=  B )  ->  A  e.  CC )
2 ax-1cn 8974 . . . . . . . . 9  |-  1  e.  CC
32a1i 11 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  A  =/=  B )  ->  1  e.  CC )
4 simp2l 983 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  A  =/=  B )  ->  B  e.  CC )
5 simp1r 982 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  A  =/=  B )  ->  A  =/=  0 )
64, 1, 5divcld 9715 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  A  =/=  B )  ->  ( B  /  A )  e.  CC )
71, 3, 6subdid 9414 . . . . . . 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 ) ) ) )
81mulid1d 9031 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  A  =/=  B )  ->  ( A  x.  1 )  =  A )
94, 1, 5divcan2d 9717 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  A  =/=  B )  ->  ( A  x.  ( B  /  A ) )  =  B )
108, 9oveq12d 6031 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  A  =/=  B )  ->  (
( A  x.  1 )  -  ( A  x.  ( B  /  A ) ) )  =  ( A  -  B ) )
117, 10eqtrd 2412 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  A  =/=  B )  ->  ( A  x.  ( 1  -  ( B  /  A ) ) )  =  ( A  -  B ) )
1211, 8oveq12d 6031 . . . . 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 ) )
133, 6subcld 9336 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  A  =/=  B )  ->  (
1  -  ( B  /  A ) )  e.  CC )
14 simp3 959 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  A  =/=  B )  ->  A  =/=  B )
1514necomd 2626 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  A  =/=  B )  ->  B  =/=  A )
164, 1, 5, 15divne1d 9726 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  A  =/=  B )  ->  ( B  /  A )  =/=  1 )
1716necomd 2626 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  A  =/=  B )  ->  1  =/=  ( B  /  A
) )
183, 6, 17subne0d 9345 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  A  =/=  B )  ->  (
1  -  ( B  /  A ) )  =/=  0 )
19 ax-1ne0 8985 . . . . . . 7  |-  1  =/=  0
2019a1i 11 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  A  =/=  B )  ->  1  =/=  0 )
21 ang.1 . . . . . . 7  |-  F  =  ( x  e.  ( CC  \  { 0 } ) ,  y  e.  ( CC  \  { 0 } ) 
|->  ( Im `  ( log `  ( y  /  x ) ) ) )
2221angcan 20504 . . . . . 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 ) )
2313, 18, 3, 20, 1, 5, 22syl222anc 1200 . . . . 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 ) )
2412, 23eqtr3d 2414 . . . 4  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  A  =/=  B )  ->  (
( A  -  B
) F A )  =  ( ( 1  -  ( B  /  A ) ) F 1 ) )
251, 6, 3subdid 9414 . . . . . . 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 ) ) )
269, 8oveq12d 6031 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  A  =/=  B )  ->  (
( A  x.  ( B  /  A ) )  -  ( A  x.  1 ) )  =  ( B  -  A
) )
2725, 26eqtrd 2412 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  A  =/=  B )  ->  ( A  x.  ( ( B  /  A )  - 
1 ) )  =  ( B  -  A
) )
289, 27oveq12d 6031 . . . . 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 ) ) )
29 simp2r 984 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  A  =/=  B )  ->  B  =/=  0 )
304, 1, 29, 5divne0d 9731 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  A  =/=  B )  ->  ( B  /  A )  =/=  0 )
316, 3subcld 9336 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  A  =/=  B )  ->  (
( B  /  A
)  -  1 )  e.  CC )
326, 3, 16subne0d 9345 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  A  =/=  B )  ->  (
( B  /  A
)  -  1 )  =/=  0 )
3321angcan 20504 . . . . . 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 ) ) )
346, 30, 31, 32, 1, 5, 33syl222anc 1200 . . . . 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 ) ) )
3528, 34eqtr3d 2414 . . . 4  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  A  =/=  B )  ->  ( B F ( B  -  A ) )  =  ( ( B  /  A ) F ( ( B  /  A
)  -  1 ) ) )
3624, 35oveq12d 6031 . . 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 ) ) ) )
378, 9oveq12d 6031 . . . 4  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  A  =/=  B )  ->  (
( A  x.  1 ) F ( A  x.  ( B  /  A ) ) )  =  ( A F B ) )
3821angcan 20504 . . . . 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 ) ) )
393, 20, 6, 30, 1, 5, 38syl222anc 1200 . . . 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
) ) )
4037, 39eqtr3d 2414 . . 3  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  A  =/=  B )  ->  ( A F B )  =  ( 1 F ( B  /  A ) ) )
4136, 40oveq12d 6031 . 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 ) ) ) )
4221ang180lem4 20514 . . 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 }
)
436, 30, 16, 42syl3anc 1184 . 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 } )
4441, 43eqeltrd 2454 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 set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2543    \ cdif 3253   {csn 3750   {cpr 3751   ` cfv 5387  (class class class)co 6013    e. cmpt2 6015   CCcc 8914   0cc0 8916   1c1 8917    + caddc 8919    x. cmul 8921    - cmin 9216   -ucneg 9217    / cdiv 9602   Imcim 11823   picpi 12589   logclog 20312
This theorem is referenced by:  ang180  20516
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-inf2 7522  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993  ax-pre-sup 8994  ax-addf 8995  ax-mulf 8996
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-int 3986  df-iun 4030  df-iin 4031  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-se 4476  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-isom 5396  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-of 6237  df-1st 6281  df-2nd 6282  df-riota 6478  df-recs 6562  df-rdg 6597  df-1o 6653  df-2o 6654  df-oadd 6657  df-er 6834  df-map 6949  df-pm 6950  df-ixp 6993  df-en 7039  df-dom 7040  df-sdom 7041  df-fin 7042  df-fi 7344  df-sup 7374  df-oi 7405  df-card 7752  df-cda 7974  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-div 9603  df-nn 9926  df-2 9983  df-3 9984  df-4 9985  df-5 9986  df-6 9987  df-7 9988  df-8 9989  df-9 9990  df-10 9991  df-n0 10147  df-z 10208  df-dec 10308  df-uz 10414  df-q 10500  df-rp 10538  df-xneg 10635  df-xadd 10636  df-xmul 10637  df-ioo 10845  df-ioc 10846  df-ico 10847  df-icc 10848  df-fz 10969  df-fzo 11059  df-fl 11122  df-mod 11171  df-seq 11244  df-exp 11303  df-fac 11487  df-bc 11514  df-hash 11539  df-shft 11802  df-cj 11824  df-re 11825  df-im 11826  df-sqr 11960  df-abs 11961  df-limsup 12185  df-clim 12202  df-rlim 12203  df-sum 12400  df-ef 12590  df-sin 12592  df-cos 12593  df-pi 12595  df-struct 13391  df-ndx 13392  df-slot 13393  df-base 13394  df-sets 13395  df-ress 13396  df-plusg 13462  df-mulr 13463  df-starv 13464  df-sca 13465  df-vsca 13466  df-tset 13468  df-ple 13469  df-ds 13471  df-unif 13472  df-hom 13473  df-cco 13474  df-rest 13570  df-topn 13571  df-topgen 13587  df-pt 13588  df-prds 13591  df-xrs 13646  df-0g 13647  df-gsum 13648  df-qtop 13653  df-imas 13654  df-xps 13656  df-mre 13731  df-mrc 13732  df-acs 13734  df-mnd 14610  df-submnd 14659  df-mulg 14735  df-cntz 15036  df-cmn 15334  df-xmet 16612  df-met 16613  df-bl 16614  df-mopn 16615  df-fbas 16616  df-fg 16617  df-cnfld 16620  df-top 16879  df-bases 16881  df-topon 16882  df-topsp 16883  df-cld 16999  df-ntr 17000  df-cls 17001  df-nei 17078  df-lp 17116  df-perf 17117  df-cn 17206  df-cnp 17207  df-haus 17294  df-tx 17508  df-hmeo 17701  df-fil 17792  df-fm 17884  df-flim 17885  df-flf 17886  df-xms 18252  df-ms 18253  df-tms 18254  df-cncf 18772  df-limc 19613  df-dv 19614  df-log 20314
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