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Theorem chordthmlem 20541
Description: If M is the midpoint of AB and AQ = BQ, then QMB is a right angle. The proof uses ssscongptld 20534 to observe that, since AMQ and BMQ have equal sides, the angles QMB and QMA must be equal. Since they are supplementary, both must be right. (Contributed by David Moews, 28-Feb-2017.)
Hypotheses
Ref Expression
chordthmlem.angdef  |-  F  =  ( x  e.  ( CC  \  { 0 } ) ,  y  e.  ( CC  \  { 0 } ) 
|->  ( Im `  ( log `  ( y  /  x ) ) ) )
chordthmlem.A  |-  ( ph  ->  A  e.  CC )
chordthmlem.B  |-  ( ph  ->  B  e.  CC )
chordthmlem.Q  |-  ( ph  ->  Q  e.  CC )
chordthmlem.M  |-  ( ph  ->  M  =  ( ( A  +  B )  /  2 ) )
chordthmlem.ABequidistQ  |-  ( ph  ->  ( abs `  ( A  -  Q )
)  =  ( abs `  ( B  -  Q
) ) )
chordthmlem.AneB  |-  ( ph  ->  A  =/=  B )
chordthmlem.QneM  |-  ( ph  ->  Q  =/=  M )
Assertion
Ref Expression
chordthmlem  |-  ( ph  ->  ( ( Q  -  M ) F ( B  -  M ) )  e.  { ( pi  /  2 ) ,  -u ( pi  / 
2 ) } )
Distinct variable groups:    x, y, A    x, B, y    x, M, y    x, Q, y
Allowed substitution hints:    ph( x, y)    F( x, y)

Proof of Theorem chordthmlem
StepHypRef Expression
1 negpitopissre 20310 . . . . . 6  |-  ( -u pi (,] pi )  C_  RR
2 chordthmlem.angdef . . . . . . 7  |-  F  =  ( x  e.  ( CC  \  { 0 } ) ,  y  e.  ( CC  \  { 0 } ) 
|->  ( Im `  ( log `  ( y  /  x ) ) ) )
3 chordthmlem.Q . . . . . . . 8  |-  ( ph  ->  Q  e.  CC )
4 chordthmlem.M . . . . . . . . 9  |-  ( ph  ->  M  =  ( ( A  +  B )  /  2 ) )
5 chordthmlem.A . . . . . . . . . . 11  |-  ( ph  ->  A  e.  CC )
6 chordthmlem.B . . . . . . . . . . 11  |-  ( ph  ->  B  e.  CC )
75, 6addcld 9041 . . . . . . . . . 10  |-  ( ph  ->  ( A  +  B
)  e.  CC )
87halfcld 10145 . . . . . . . . 9  |-  ( ph  ->  ( ( A  +  B )  /  2
)  e.  CC )
94, 8eqeltrd 2462 . . . . . . . 8  |-  ( ph  ->  M  e.  CC )
103, 9subcld 9344 . . . . . . 7  |-  ( ph  ->  ( Q  -  M
)  e.  CC )
11 chordthmlem.QneM . . . . . . . 8  |-  ( ph  ->  Q  =/=  M )
123, 9, 11subne0d 9353 . . . . . . 7  |-  ( ph  ->  ( Q  -  M
)  =/=  0 )
136, 9subcld 9344 . . . . . . 7  |-  ( ph  ->  ( B  -  M
)  e.  CC )
144oveq1d 6036 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( M  x.  2 )  =  ( ( ( A  +  B
)  /  2 )  x.  2 ) )
159times2d 10144 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( M  x.  2 )  =  ( M  +  M ) )
16 2cn 10003 . . . . . . . . . . . . . . . 16  |-  2  e.  CC
1716a1i 11 . . . . . . . . . . . . . . 15  |-  ( ph  ->  2  e.  CC )
18 2ne0 10016 . . . . . . . . . . . . . . . 16  |-  2  =/=  0
1918a1i 11 . . . . . . . . . . . . . . 15  |-  ( ph  ->  2  =/=  0 )
207, 17, 19divcan1d 9724 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( ( A  +  B )  / 
2 )  x.  2 )  =  ( A  +  B ) )
2114, 15, 203eqtr3d 2428 . . . . . . . . . . . . 13  |-  ( ph  ->  ( M  +  M
)  =  ( A  +  B ) )
22 chordthmlem.AneB . . . . . . . . . . . . . 14  |-  ( ph  ->  A  =/=  B )
235, 6, 6, 22addneintr2d 9207 . . . . . . . . . . . . 13  |-  ( ph  ->  ( A  +  B
)  =/=  ( B  +  B ) )
2421, 23eqnetrd 2569 . . . . . . . . . . . 12  |-  ( ph  ->  ( M  +  M
)  =/=  ( B  +  B ) )
2524neneqd 2567 . . . . . . . . . . 11  |-  ( ph  ->  -.  ( M  +  M )  =  ( B  +  B ) )
26 oveq12 6030 . . . . . . . . . . . 12  |-  ( ( M  =  B  /\  M  =  B )  ->  ( M  +  M
)  =  ( B  +  B ) )
2726anidms 627 . . . . . . . . . . 11  |-  ( M  =  B  ->  ( M  +  M )  =  ( B  +  B ) )
2825, 27nsyl 115 . . . . . . . . . 10  |-  ( ph  ->  -.  M  =  B )
2928neneqad 2621 . . . . . . . . 9  |-  ( ph  ->  M  =/=  B )
3029necomd 2634 . . . . . . . 8  |-  ( ph  ->  B  =/=  M )
316, 9, 30subne0d 9353 . . . . . . 7  |-  ( ph  ->  ( B  -  M
)  =/=  0 )
322, 10, 12, 13, 31angcld 20515 . . . . . 6  |-  ( ph  ->  ( ( Q  -  M ) F ( B  -  M ) )  e.  ( -u pi (,] pi ) )
331, 32sseldi 3290 . . . . 5  |-  ( ph  ->  ( ( Q  -  M ) F ( B  -  M ) )  e.  RR )
3433recnd 9048 . . . 4  |-  ( ph  ->  ( ( Q  -  M ) F ( B  -  M ) )  e.  CC )
3534coscld 12660 . . 3  |-  ( ph  ->  ( cos `  (
( Q  -  M
) F ( B  -  M ) ) )  e.  CC )
366, 9negsubdi2d 9360 . . . . . . 7  |-  ( ph  -> 
-u ( B  -  M )  =  ( M  -  B ) )
379, 9, 5, 6addsubeq4d 9395 . . . . . . . 8  |-  ( ph  ->  ( ( M  +  M )  =  ( A  +  B )  <-> 
( A  -  M
)  =  ( M  -  B ) ) )
3821, 37mpbid 202 . . . . . . 7  |-  ( ph  ->  ( A  -  M
)  =  ( M  -  B ) )
3936, 38eqtr4d 2423 . . . . . 6  |-  ( ph  -> 
-u ( B  -  M )  =  ( A  -  M ) )
4039oveq2d 6037 . . . . 5  |-  ( ph  ->  ( ( Q  -  M ) F -u ( B  -  M
) )  =  ( ( Q  -  M
) F ( A  -  M ) ) )
4140fveq2d 5673 . . . 4  |-  ( ph  ->  ( cos `  (
( Q  -  M
) F -u ( B  -  M )
) )  =  ( cos `  ( ( Q  -  M ) F ( A  -  M ) ) ) )
422, 10, 12, 13, 31cosangneg2d 20517 . . . 4  |-  ( ph  ->  ( cos `  (
( Q  -  M
) F -u ( B  -  M )
) )  =  -u ( cos `  ( ( Q  -  M ) F ( B  -  M ) ) ) )
435, 5, 6, 22addneintrd 9206 . . . . . . . . . 10  |-  ( ph  ->  ( A  +  A
)  =/=  ( A  +  B ) )
4443, 21neeqtrrd 2575 . . . . . . . . 9  |-  ( ph  ->  ( A  +  A
)  =/=  ( M  +  M ) )
4544necomd 2634 . . . . . . . 8  |-  ( ph  ->  ( M  +  M
)  =/=  ( A  +  A ) )
4645neneqd 2567 . . . . . . 7  |-  ( ph  ->  -.  ( M  +  M )  =  ( A  +  A ) )
47 oveq12 6030 . . . . . . . 8  |-  ( ( M  =  A  /\  M  =  A )  ->  ( M  +  M
)  =  ( A  +  A ) )
4847anidms 627 . . . . . . 7  |-  ( M  =  A  ->  ( M  +  M )  =  ( A  +  A ) )
4946, 48nsyl 115 . . . . . 6  |-  ( ph  ->  -.  M  =  A )
5049neneqad 2621 . . . . 5  |-  ( ph  ->  M  =/=  A )
51 eqidd 2389 . . . . 5  |-  ( ph  ->  ( abs `  ( Q  -  M )
)  =  ( abs `  ( Q  -  M
) ) )
525, 9subcld 9344 . . . . . . 7  |-  ( ph  ->  ( A  -  M
)  e.  CC )
5352absnegd 12179 . . . . . 6  |-  ( ph  ->  ( abs `  -u ( A  -  M )
)  =  ( abs `  ( A  -  M
) ) )
545, 9negsubdi2d 9360 . . . . . . 7  |-  ( ph  -> 
-u ( A  -  M )  =  ( M  -  A ) )
5554fveq2d 5673 . . . . . 6  |-  ( ph  ->  ( abs `  -u ( A  -  M )
)  =  ( abs `  ( M  -  A
) ) )
5638fveq2d 5673 . . . . . 6  |-  ( ph  ->  ( abs `  ( A  -  M )
)  =  ( abs `  ( M  -  B
) ) )
5753, 55, 563eqtr3d 2428 . . . . 5  |-  ( ph  ->  ( abs `  ( M  -  A )
)  =  ( abs `  ( M  -  B
) ) )
58 chordthmlem.ABequidistQ . . . . 5  |-  ( ph  ->  ( abs `  ( A  -  Q )
)  =  ( abs `  ( B  -  Q
) ) )
592, 3, 9, 5, 3, 9, 6, 11, 50, 11, 29, 51, 57, 58ssscongptld 20534 . . . 4  |-  ( ph  ->  ( cos `  (
( Q  -  M
) F ( A  -  M ) ) )  =  ( cos `  ( ( Q  -  M ) F ( B  -  M ) ) ) )
6041, 42, 593eqtr3rd 2429 . . 3  |-  ( ph  ->  ( cos `  (
( Q  -  M
) F ( B  -  M ) ) )  =  -u ( cos `  ( ( Q  -  M ) F ( B  -  M
) ) ) )
6135, 60eqnegad 9669 . 2  |-  ( ph  ->  ( cos `  (
( Q  -  M
) F ( B  -  M ) ) )  =  0 )
62 coseq0negpitopi 20279 . . 3  |-  ( ( ( Q  -  M
) F ( B  -  M ) )  e.  ( -u pi (,] pi )  ->  (
( cos `  (
( Q  -  M
) F ( B  -  M ) ) )  =  0  <->  (
( Q  -  M
) F ( B  -  M ) )  e.  { ( pi 
/  2 ) , 
-u ( pi  / 
2 ) } ) )
6332, 62syl 16 . 2  |-  ( ph  ->  ( ( cos `  (
( Q  -  M
) F ( B  -  M ) ) )  =  0  <->  (
( Q  -  M
) F ( B  -  M ) )  e.  { ( pi 
/  2 ) , 
-u ( pi  / 
2 ) } ) )
6461, 63mpbid 202 1  |-  ( ph  ->  ( ( Q  -  M ) F ( B  -  M ) )  e.  { ( pi  /  2 ) ,  -u ( pi  / 
2 ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1649    e. wcel 1717    =/= wne 2551    \ cdif 3261   {csn 3758   {cpr 3759   ` cfv 5395  (class class class)co 6021    e. cmpt2 6023   CCcc 8922   RRcr 8923   0cc0 8924    + caddc 8927    x. cmul 8929    - cmin 9224   -ucneg 9225    / cdiv 9610   2c2 9982   (,]cioc 10850   Imcim 11831   abscabs 11967   cosccos 12595   picpi 12597   logclog 20320
This theorem is referenced by:  chordthmlem2  20542
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 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-inf2 7530  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001  ax-pre-sup 9002  ax-addf 9003  ax-mulf 9004
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-int 3994  df-iun 4038  df-iin 4039  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-se 4484  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-isom 5404  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-of 6245  df-1st 6289  df-2nd 6290  df-riota 6486  df-recs 6570  df-rdg 6605  df-1o 6661  df-2o 6662  df-oadd 6665  df-er 6842  df-map 6957  df-pm 6958  df-ixp 7001  df-en 7047  df-dom 7048  df-sdom 7049  df-fin 7050  df-fi 7352  df-sup 7382  df-oi 7413  df-card 7760  df-cda 7982  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-div 9611  df-nn 9934  df-2 9991  df-3 9992  df-4 9993  df-5 9994  df-6 9995  df-7 9996  df-8 9997  df-9 9998  df-10 9999  df-n0 10155  df-z 10216  df-dec 10316  df-uz 10422  df-q 10508  df-rp 10546  df-xneg 10643  df-xadd 10644  df-xmul 10645  df-ioo 10853  df-ioc 10854  df-ico 10855  df-icc 10856  df-fz 10977  df-fzo 11067  df-fl 11130  df-mod 11179  df-seq 11252  df-exp 11311  df-fac 11495  df-bc 11522  df-hash 11547  df-shft 11810  df-cj 11832  df-re 11833  df-im 11834  df-sqr 11968  df-abs 11969  df-limsup 12193  df-clim 12210  df-rlim 12211  df-sum 12408  df-ef 12598  df-sin 12600  df-cos 12601  df-pi 12603  df-struct 13399  df-ndx 13400  df-slot 13401  df-base 13402  df-sets 13403  df-ress 13404  df-plusg 13470  df-mulr 13471  df-starv 13472  df-sca 13473  df-vsca 13474  df-tset 13476  df-ple 13477  df-ds 13479  df-unif 13480  df-hom 13481  df-cco 13482  df-rest 13578  df-topn 13579  df-topgen 13595  df-pt 13596  df-prds 13599  df-xrs 13654  df-0g 13655  df-gsum 13656  df-qtop 13661  df-imas 13662  df-xps 13664  df-mre 13739  df-mrc 13740  df-acs 13742  df-mnd 14618  df-submnd 14667  df-mulg 14743  df-cntz 15044  df-cmn 15342  df-xmet 16620  df-met 16621  df-bl 16622  df-mopn 16623  df-fbas 16624  df-fg 16625  df-cnfld 16628  df-top 16887  df-bases 16889  df-topon 16890  df-topsp 16891  df-cld 17007  df-ntr 17008  df-cls 17009  df-nei 17086  df-lp 17124  df-perf 17125  df-cn 17214  df-cnp 17215  df-haus 17302  df-tx 17516  df-hmeo 17709  df-fil 17800  df-fm 17892  df-flim 17893  df-flf 17894  df-xms 18260  df-ms 18261  df-tms 18262  df-cncf 18780  df-limc 19621  df-dv 19622  df-log 20322
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