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Theorem ang180lem5 20648
Description: Lemma for ang180 20649: 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 9041 . . . . . . . . 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 9783 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  A  =/=  B )  ->  ( B  /  A )  e.  CC )
71, 3, 6subdid 9482 . . . . . . 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 9098 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  A  =/=  B )  ->  ( A  x.  1 )  =  A )
94, 1, 5divcan2d 9785 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  A  =/=  B )  ->  ( A  x.  ( B  /  A ) )  =  B )
108, 9oveq12d 6092 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  A  =/=  B )  ->  (
( A  x.  1 )  -  ( A  x.  ( B  /  A ) ) )  =  ( A  -  B ) )
117, 10eqtrd 2468 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  A  =/=  B )  ->  ( A  x.  ( 1  -  ( B  /  A ) ) )  =  ( A  -  B ) )
1211, 8oveq12d 6092 . . . . 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 9404 . . . . . 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 2682 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  A  =/=  B )  ->  B  =/=  A )
164, 1, 5, 15divne1d 9794 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  A  =/=  B )  ->  ( B  /  A )  =/=  1 )
1716necomd 2682 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  A  =/=  B )  ->  1  =/=  ( B  /  A
) )
183, 6, 17subne0d 9413 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  A  =/=  B )  ->  (
1  -  ( B  /  A ) )  =/=  0 )
19 ax-1ne0 9052 . . . . . . 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 20637 . . . . . 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 2470 . . . 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 9482 . . . . . . 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 6092 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  A  =/=  B )  ->  (
( A  x.  ( B  /  A ) )  -  ( A  x.  1 ) )  =  ( B  -  A
) )
2725, 26eqtrd 2468 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  A  =/=  B )  ->  ( A  x.  ( ( B  /  A )  - 
1 ) )  =  ( B  -  A
) )
289, 27oveq12d 6092 . . . . 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 9799 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  A  =/=  B )  ->  ( B  /  A )  =/=  0 )
316, 3subcld 9404 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  A  =/=  B )  ->  (
( B  /  A
)  -  1 )  e.  CC )
326, 3, 16subne0d 9413 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  A  =/=  B )  ->  (
( B  /  A
)  -  1 )  =/=  0 )
3321angcan 20637 . . . . . 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 2470 . . . 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 6092 . . 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 6092 . . . 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 20637 . . . . 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 2470 . . 3  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  A  =/=  B )  ->  ( A F B )  =  ( 1 F ( B  /  A ) ) )
4136, 40oveq12d 6092 . 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 20647 . . 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 2510 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 1652    e. wcel 1725    =/= wne 2599    \ cdif 3310   {csn 3807   {cpr 3808   ` cfv 5447  (class class class)co 6074    e. cmpt2 6076   CCcc 8981   0cc0 8983   1c1 8984    + caddc 8986    x. cmul 8988    - cmin 9284   -ucneg 9285    / cdiv 9670   Imcim 11896   picpi 12662   logclog 20445
This theorem is referenced by:  ang180  20649
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 4313  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396  ax-un 4694  ax-inf2 7589  ax-cnex 9039  ax-resscn 9040  ax-1cn 9041  ax-icn 9042  ax-addcl 9043  ax-addrcl 9044  ax-mulcl 9045  ax-mulrcl 9046  ax-mulcom 9047  ax-addass 9048  ax-mulass 9049  ax-distr 9050  ax-i2m1 9051  ax-1ne0 9052  ax-1rid 9053  ax-rnegex 9054  ax-rrecex 9055  ax-cnre 9056  ax-pre-lttri 9057  ax-pre-lttrn 9058  ax-pre-ltadd 9059  ax-pre-mulgt0 9060  ax-pre-sup 9061  ax-addf 9062  ax-mulf 9063
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 2703  df-rex 2704  df-reu 2705  df-rmo 2706  df-rab 2707  df-v 2951  df-sbc 3155  df-csb 3245  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-pss 3329  df-nul 3622  df-if 3733  df-pw 3794  df-sn 3813  df-pr 3814  df-tp 3815  df-op 3816  df-uni 4009  df-int 4044  df-iun 4088  df-iin 4089  df-br 4206  df-opab 4260  df-mpt 4261  df-tr 4296  df-eprel 4487  df-id 4491  df-po 4496  df-so 4497  df-fr 4534  df-se 4535  df-we 4536  df-ord 4577  df-on 4578  df-lim 4579  df-suc 4580  df-om 4839  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-res 4883  df-ima 4884  df-iota 5411  df-fun 5449  df-fn 5450  df-f 5451  df-f1 5452  df-fo 5453  df-f1o 5454  df-fv 5455  df-isom 5456  df-ov 6077  df-oprab 6078  df-mpt2 6079  df-of 6298  df-1st 6342  df-2nd 6343  df-riota 6542  df-recs 6626  df-rdg 6661  df-1o 6717  df-2o 6718  df-oadd 6721  df-er 6898  df-map 7013  df-pm 7014  df-ixp 7057  df-en 7103  df-dom 7104  df-sdom 7105  df-fin 7106  df-fi 7409  df-sup 7439  df-oi 7472  df-card 7819  df-cda 8041  df-pnf 9115  df-mnf 9116  df-xr 9117  df-ltxr 9118  df-le 9119  df-sub 9286  df-neg 9287  df-div 9671  df-nn 9994  df-2 10051  df-3 10052  df-4 10053  df-5 10054  df-6 10055  df-7 10056  df-8 10057  df-9 10058  df-10 10059  df-n0 10215  df-z 10276  df-dec 10376  df-uz 10482  df-q 10568  df-rp 10606  df-xneg 10703  df-xadd 10704  df-xmul 10705  df-ioo 10913  df-ioc 10914  df-ico 10915  df-icc 10916  df-fz 11037  df-fzo 11129  df-fl 11195  df-mod 11244  df-seq 11317  df-exp 11376  df-fac 11560  df-bc 11587  df-hash 11612  df-shft 11875  df-cj 11897  df-re 11898  df-im 11899  df-sqr 12033  df-abs 12034  df-limsup 12258  df-clim 12275  df-rlim 12276  df-sum 12473  df-ef 12663  df-sin 12665  df-cos 12666  df-pi 12668  df-struct 13464  df-ndx 13465  df-slot 13466  df-base 13467  df-sets 13468  df-ress 13469  df-plusg 13535  df-mulr 13536  df-starv 13537  df-sca 13538  df-vsca 13539  df-tset 13541  df-ple 13542  df-ds 13544  df-unif 13545  df-hom 13546  df-cco 13547  df-rest 13643  df-topn 13644  df-topgen 13660  df-pt 13661  df-prds 13664  df-xrs 13719  df-0g 13720  df-gsum 13721  df-qtop 13726  df-imas 13727  df-xps 13729  df-mre 13804  df-mrc 13805  df-acs 13807  df-mnd 14683  df-submnd 14732  df-mulg 14808  df-cntz 15109  df-cmn 15407  df-psmet 16687  df-xmet 16688  df-met 16689  df-bl 16690  df-mopn 16691  df-fbas 16692  df-fg 16693  df-cnfld 16697  df-top 16956  df-bases 16958  df-topon 16959  df-topsp 16960  df-cld 17076  df-ntr 17077  df-cls 17078  df-nei 17155  df-lp 17193  df-perf 17194  df-cn 17284  df-cnp 17285  df-haus 17372  df-tx 17587  df-hmeo 17780  df-fil 17871  df-fm 17963  df-flim 17964  df-flf 17965  df-xms 18343  df-ms 18344  df-tms 18345  df-cncf 18901  df-limc 19746  df-dv 19747  df-log 20447
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