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Theorem chordthmlem 20665
Description: If M is the midpoint of AB and AQ = BQ, then QMB is a right angle. The proof uses ssscongptld 20658 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 20434 . . . . . 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 9099 . . . . . . . . . 10  |-  ( ph  ->  ( A  +  B
)  e.  CC )
87halfcld 10204 . . . . . . . . 9  |-  ( ph  ->  ( ( A  +  B )  /  2
)  e.  CC )
94, 8eqeltrd 2509 . . . . . . . 8  |-  ( ph  ->  M  e.  CC )
103, 9subcld 9403 . . . . . . 7  |-  ( ph  ->  ( Q  -  M
)  e.  CC )
11 chordthmlem.QneM . . . . . . . 8  |-  ( ph  ->  Q  =/=  M )
123, 9, 11subne0d 9412 . . . . . . 7  |-  ( ph  ->  ( Q  -  M
)  =/=  0 )
136, 9subcld 9403 . . . . . . 7  |-  ( ph  ->  ( B  -  M
)  e.  CC )
144oveq1d 6088 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( M  x.  2 )  =  ( ( ( A  +  B
)  /  2 )  x.  2 ) )
159times2d 10203 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( M  x.  2 )  =  ( M  +  M ) )
16 2cn 10062 . . . . . . . . . . . . . . . 16  |-  2  e.  CC
1716a1i 11 . . . . . . . . . . . . . . 15  |-  ( ph  ->  2  e.  CC )
18 2ne0 10075 . . . . . . . . . . . . . . . 16  |-  2  =/=  0
1918a1i 11 . . . . . . . . . . . . . . 15  |-  ( ph  ->  2  =/=  0 )
207, 17, 19divcan1d 9783 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( ( A  +  B )  / 
2 )  x.  2 )  =  ( A  +  B ) )
2114, 15, 203eqtr3d 2475 . . . . . . . . . . . . 13  |-  ( ph  ->  ( M  +  M
)  =  ( A  +  B ) )
22 chordthmlem.AneB . . . . . . . . . . . . . 14  |-  ( ph  ->  A  =/=  B )
235, 6, 6, 22addneintr2d 9266 . . . . . . . . . . . . 13  |-  ( ph  ->  ( A  +  B
)  =/=  ( B  +  B ) )
2421, 23eqnetrd 2616 . . . . . . . . . . . 12  |-  ( ph  ->  ( M  +  M
)  =/=  ( B  +  B ) )
2524neneqd 2614 . . . . . . . . . . 11  |-  ( ph  ->  -.  ( M  +  M )  =  ( B  +  B ) )
26 oveq12 6082 . . . . . . . . . . . 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 2668 . . . . . . . . 9  |-  ( ph  ->  M  =/=  B )
3029necomd 2681 . . . . . . . 8  |-  ( ph  ->  B  =/=  M )
316, 9, 30subne0d 9412 . . . . . . 7  |-  ( ph  ->  ( B  -  M
)  =/=  0 )
322, 10, 12, 13, 31angcld 20639 . . . . . 6  |-  ( ph  ->  ( ( Q  -  M ) F ( B  -  M ) )  e.  ( -u pi (,] pi ) )
331, 32sseldi 3338 . . . . 5  |-  ( ph  ->  ( ( Q  -  M ) F ( B  -  M ) )  e.  RR )
3433recnd 9106 . . . 4  |-  ( ph  ->  ( ( Q  -  M ) F ( B  -  M ) )  e.  CC )
3534coscld 12724 . . 3  |-  ( ph  ->  ( cos `  (
( Q  -  M
) F ( B  -  M ) ) )  e.  CC )
366, 9negsubdi2d 9419 . . . . . . 7  |-  ( ph  -> 
-u ( B  -  M )  =  ( M  -  B ) )
379, 9, 5, 6addsubeq4d 9454 . . . . . . . 8  |-  ( ph  ->  ( ( M  +  M )  =  ( A  +  B )  <-> 
( A  -  M
)  =  ( M  -  B ) ) )
3821, 37mpbid 202 . . . . . . 7  |-  ( ph  ->  ( A  -  M
)  =  ( M  -  B ) )
3936, 38eqtr4d 2470 . . . . . 6  |-  ( ph  -> 
-u ( B  -  M )  =  ( A  -  M ) )
4039oveq2d 6089 . . . . 5  |-  ( ph  ->  ( ( Q  -  M ) F -u ( B  -  M
) )  =  ( ( Q  -  M
) F ( A  -  M ) ) )
4140fveq2d 5724 . . . 4  |-  ( ph  ->  ( cos `  (
( Q  -  M
) F -u ( B  -  M )
) )  =  ( cos `  ( ( Q  -  M ) F ( A  -  M ) ) ) )
422, 10, 12, 13, 31cosangneg2d 20641 . . . 4  |-  ( ph  ->  ( cos `  (
( Q  -  M
) F -u ( B  -  M )
) )  =  -u ( cos `  ( ( Q  -  M ) F ( B  -  M ) ) ) )
435, 5, 6, 22addneintrd 9265 . . . . . . . . . 10  |-  ( ph  ->  ( A  +  A
)  =/=  ( A  +  B ) )
4443, 21neeqtrrd 2622 . . . . . . . . 9  |-  ( ph  ->  ( A  +  A
)  =/=  ( M  +  M ) )
4544necomd 2681 . . . . . . . 8  |-  ( ph  ->  ( M  +  M
)  =/=  ( A  +  A ) )
4645neneqd 2614 . . . . . . 7  |-  ( ph  ->  -.  ( M  +  M )  =  ( A  +  A ) )
47 oveq12 6082 . . . . . . . 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 2668 . . . . 5  |-  ( ph  ->  M  =/=  A )
51 eqidd 2436 . . . . 5  |-  ( ph  ->  ( abs `  ( Q  -  M )
)  =  ( abs `  ( Q  -  M
) ) )
525, 9subcld 9403 . . . . . . 7  |-  ( ph  ->  ( A  -  M
)  e.  CC )
5352absnegd 12243 . . . . . 6  |-  ( ph  ->  ( abs `  -u ( A  -  M )
)  =  ( abs `  ( A  -  M
) ) )
545, 9negsubdi2d 9419 . . . . . . 7  |-  ( ph  -> 
-u ( A  -  M )  =  ( M  -  A ) )
5554fveq2d 5724 . . . . . 6  |-  ( ph  ->  ( abs `  -u ( A  -  M )
)  =  ( abs `  ( M  -  A
) ) )
5638fveq2d 5724 . . . . . 6  |-  ( ph  ->  ( abs `  ( A  -  M )
)  =  ( abs `  ( M  -  B
) ) )
5753, 55, 563eqtr3d 2475 . . . . 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 20658 . . . 4  |-  ( ph  ->  ( cos `  (
( Q  -  M
) F ( A  -  M ) ) )  =  ( cos `  ( ( Q  -  M ) F ( B  -  M ) ) ) )
6041, 42, 593eqtr3rd 2476 . . 3  |-  ( ph  ->  ( cos `  (
( Q  -  M
) F ( B  -  M ) ) )  =  -u ( cos `  ( ( Q  -  M ) F ( B  -  M
) ) ) )
6135, 60eqnegad 9728 . 2  |-  ( ph  ->  ( cos `  (
( Q  -  M
) F ( B  -  M ) ) )  =  0 )
62 coseq0negpitopi 20403 . . 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 1652    e. wcel 1725    =/= wne 2598    \ cdif 3309   {csn 3806   {cpr 3807   ` cfv 5446  (class class class)co 6073    e. cmpt2 6075   CCcc 8980   RRcr 8981   0cc0 8982    + caddc 8985    x. cmul 8987    - cmin 9283   -ucneg 9284    / cdiv 9669   2c2 10041   (,]cioc 10909   Imcim 11895   abscabs 12031   cosccos 12659   picpi 12661   logclog 20444
This theorem is referenced by:  chordthmlem2  20666
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 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060  ax-addf 9061  ax-mulf 9062
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-2o 6717  df-oadd 6720  df-er 6897  df-map 7012  df-pm 7013  df-ixp 7056  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-fi 7408  df-sup 7438  df-oi 7471  df-card 7818  df-cda 8040  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-7 10055  df-8 10056  df-9 10057  df-10 10058  df-n0 10214  df-z 10275  df-dec 10375  df-uz 10481  df-q 10567  df-rp 10605  df-xneg 10702  df-xadd 10703  df-xmul 10704  df-ioo 10912  df-ioc 10913  df-ico 10914  df-icc 10915  df-fz 11036  df-fzo 11128  df-fl 11194  df-mod 11243  df-seq 11316  df-exp 11375  df-fac 11559  df-bc 11586  df-hash 11611  df-shft 11874  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-limsup 12257  df-clim 12274  df-rlim 12275  df-sum 12472  df-ef 12662  df-sin 12664  df-cos 12665  df-pi 12667  df-struct 13463  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-mulr 13535  df-starv 13536  df-sca 13537  df-vsca 13538  df-tset 13540  df-ple 13541  df-ds 13543  df-unif 13544  df-hom 13545  df-cco 13546  df-rest 13642  df-topn 13643  df-topgen 13659  df-pt 13660  df-prds 13663  df-xrs 13718  df-0g 13719  df-gsum 13720  df-qtop 13725  df-imas 13726  df-xps 13728  df-mre 13803  df-mrc 13804  df-acs 13806  df-mnd 14682  df-submnd 14731  df-mulg 14807  df-cntz 15108  df-cmn 15406  df-psmet 16686  df-xmet 16687  df-met 16688  df-bl 16689  df-mopn 16690  df-fbas 16691  df-fg 16692  df-cnfld 16696  df-top 16955  df-bases 16957  df-topon 16958  df-topsp 16959  df-cld 17075  df-ntr 17076  df-cls 17077  df-nei 17154  df-lp 17192  df-perf 17193  df-cn 17283  df-cnp 17284  df-haus 17371  df-tx 17586  df-hmeo 17779  df-fil 17870  df-fm 17962  df-flim 17963  df-flf 17964  df-xms 18342  df-ms 18343  df-tms 18344  df-cncf 18900  df-limc 19745  df-dv 19746  df-log 20446
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