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Theorem chordthmlem 20634
Description: If M is the midpoint of AB and AQ = BQ, then QMB is a right angle. The proof uses ssscongptld 20627 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 20403 . . . . . 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 9071 . . . . . . . . . 10  |-  ( ph  ->  ( A  +  B
)  e.  CC )
87halfcld 10176 . . . . . . . . 9  |-  ( ph  ->  ( ( A  +  B )  /  2
)  e.  CC )
94, 8eqeltrd 2486 . . . . . . . 8  |-  ( ph  ->  M  e.  CC )
103, 9subcld 9375 . . . . . . 7  |-  ( ph  ->  ( Q  -  M
)  e.  CC )
11 chordthmlem.QneM . . . . . . . 8  |-  ( ph  ->  Q  =/=  M )
123, 9, 11subne0d 9384 . . . . . . 7  |-  ( ph  ->  ( Q  -  M
)  =/=  0 )
136, 9subcld 9375 . . . . . . 7  |-  ( ph  ->  ( B  -  M
)  e.  CC )
144oveq1d 6063 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( M  x.  2 )  =  ( ( ( A  +  B
)  /  2 )  x.  2 ) )
159times2d 10175 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( M  x.  2 )  =  ( M  +  M ) )
16 2cn 10034 . . . . . . . . . . . . . . . 16  |-  2  e.  CC
1716a1i 11 . . . . . . . . . . . . . . 15  |-  ( ph  ->  2  e.  CC )
18 2ne0 10047 . . . . . . . . . . . . . . . 16  |-  2  =/=  0
1918a1i 11 . . . . . . . . . . . . . . 15  |-  ( ph  ->  2  =/=  0 )
207, 17, 19divcan1d 9755 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( ( A  +  B )  / 
2 )  x.  2 )  =  ( A  +  B ) )
2114, 15, 203eqtr3d 2452 . . . . . . . . . . . . 13  |-  ( ph  ->  ( M  +  M
)  =  ( A  +  B ) )
22 chordthmlem.AneB . . . . . . . . . . . . . 14  |-  ( ph  ->  A  =/=  B )
235, 6, 6, 22addneintr2d 9238 . . . . . . . . . . . . 13  |-  ( ph  ->  ( A  +  B
)  =/=  ( B  +  B ) )
2421, 23eqnetrd 2593 . . . . . . . . . . . 12  |-  ( ph  ->  ( M  +  M
)  =/=  ( B  +  B ) )
2524neneqd 2591 . . . . . . . . . . 11  |-  ( ph  ->  -.  ( M  +  M )  =  ( B  +  B ) )
26 oveq12 6057 . . . . . . . . . . . 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 2645 . . . . . . . . 9  |-  ( ph  ->  M  =/=  B )
3029necomd 2658 . . . . . . . 8  |-  ( ph  ->  B  =/=  M )
316, 9, 30subne0d 9384 . . . . . . 7  |-  ( ph  ->  ( B  -  M
)  =/=  0 )
322, 10, 12, 13, 31angcld 20608 . . . . . 6  |-  ( ph  ->  ( ( Q  -  M ) F ( B  -  M ) )  e.  ( -u pi (,] pi ) )
331, 32sseldi 3314 . . . . 5  |-  ( ph  ->  ( ( Q  -  M ) F ( B  -  M ) )  e.  RR )
3433recnd 9078 . . . 4  |-  ( ph  ->  ( ( Q  -  M ) F ( B  -  M ) )  e.  CC )
3534coscld 12695 . . 3  |-  ( ph  ->  ( cos `  (
( Q  -  M
) F ( B  -  M ) ) )  e.  CC )
366, 9negsubdi2d 9391 . . . . . . 7  |-  ( ph  -> 
-u ( B  -  M )  =  ( M  -  B ) )
379, 9, 5, 6addsubeq4d 9426 . . . . . . . 8  |-  ( ph  ->  ( ( M  +  M )  =  ( A  +  B )  <-> 
( A  -  M
)  =  ( M  -  B ) ) )
3821, 37mpbid 202 . . . . . . 7  |-  ( ph  ->  ( A  -  M
)  =  ( M  -  B ) )
3936, 38eqtr4d 2447 . . . . . 6  |-  ( ph  -> 
-u ( B  -  M )  =  ( A  -  M ) )
4039oveq2d 6064 . . . . 5  |-  ( ph  ->  ( ( Q  -  M ) F -u ( B  -  M
) )  =  ( ( Q  -  M
) F ( A  -  M ) ) )
4140fveq2d 5699 . . . 4  |-  ( ph  ->  ( cos `  (
( Q  -  M
) F -u ( B  -  M )
) )  =  ( cos `  ( ( Q  -  M ) F ( A  -  M ) ) ) )
422, 10, 12, 13, 31cosangneg2d 20610 . . . 4  |-  ( ph  ->  ( cos `  (
( Q  -  M
) F -u ( B  -  M )
) )  =  -u ( cos `  ( ( Q  -  M ) F ( B  -  M ) ) ) )
435, 5, 6, 22addneintrd 9237 . . . . . . . . . 10  |-  ( ph  ->  ( A  +  A
)  =/=  ( A  +  B ) )
4443, 21neeqtrrd 2599 . . . . . . . . 9  |-  ( ph  ->  ( A  +  A
)  =/=  ( M  +  M ) )
4544necomd 2658 . . . . . . . 8  |-  ( ph  ->  ( M  +  M
)  =/=  ( A  +  A ) )
4645neneqd 2591 . . . . . . 7  |-  ( ph  ->  -.  ( M  +  M )  =  ( A  +  A ) )
47 oveq12 6057 . . . . . . . 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 2645 . . . . 5  |-  ( ph  ->  M  =/=  A )
51 eqidd 2413 . . . . 5  |-  ( ph  ->  ( abs `  ( Q  -  M )
)  =  ( abs `  ( Q  -  M
) ) )
525, 9subcld 9375 . . . . . . 7  |-  ( ph  ->  ( A  -  M
)  e.  CC )
5352absnegd 12214 . . . . . 6  |-  ( ph  ->  ( abs `  -u ( A  -  M )
)  =  ( abs `  ( A  -  M
) ) )
545, 9negsubdi2d 9391 . . . . . . 7  |-  ( ph  -> 
-u ( A  -  M )  =  ( M  -  A ) )
5554fveq2d 5699 . . . . . 6  |-  ( ph  ->  ( abs `  -u ( A  -  M )
)  =  ( abs `  ( M  -  A
) ) )
5638fveq2d 5699 . . . . . 6  |-  ( ph  ->  ( abs `  ( A  -  M )
)  =  ( abs `  ( M  -  B
) ) )
5753, 55, 563eqtr3d 2452 . . . . 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 20627 . . . 4  |-  ( ph  ->  ( cos `  (
( Q  -  M
) F ( A  -  M ) ) )  =  ( cos `  ( ( Q  -  M ) F ( B  -  M ) ) ) )
6041, 42, 593eqtr3rd 2453 . . 3  |-  ( ph  ->  ( cos `  (
( Q  -  M
) F ( B  -  M ) ) )  =  -u ( cos `  ( ( Q  -  M ) F ( B  -  M
) ) ) )
6135, 60eqnegad 9700 . 2  |-  ( ph  ->  ( cos `  (
( Q  -  M
) F ( B  -  M ) ) )  =  0 )
62 coseq0negpitopi 20372 . . 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 1721    =/= wne 2575    \ cdif 3285   {csn 3782   {cpr 3783   ` cfv 5421  (class class class)co 6048    e. cmpt2 6050   CCcc 8952   RRcr 8953   0cc0 8954    + caddc 8957    x. cmul 8959    - cmin 9255   -ucneg 9256    / cdiv 9641   2c2 10013   (,]cioc 10881   Imcim 11866   abscabs 12002   cosccos 12630   picpi 12632   logclog 20413
This theorem is referenced by:  chordthmlem2  20635
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-inf2 7560  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031  ax-pre-sup 9032  ax-addf 9033  ax-mulf 9034
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 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-iin 4064  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-se 4510  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-isom 5430  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-of 6272  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-1o 6691  df-2o 6692  df-oadd 6695  df-er 6872  df-map 6987  df-pm 6988  df-ixp 7031  df-en 7077  df-dom 7078  df-sdom 7079  df-fin 7080  df-fi 7382  df-sup 7412  df-oi 7443  df-card 7790  df-cda 8012  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-div 9642  df-nn 9965  df-2 10022  df-3 10023  df-4 10024  df-5 10025  df-6 10026  df-7 10027  df-8 10028  df-9 10029  df-10 10030  df-n0 10186  df-z 10247  df-dec 10347  df-uz 10453  df-q 10539  df-rp 10577  df-xneg 10674  df-xadd 10675  df-xmul 10676  df-ioo 10884  df-ioc 10885  df-ico 10886  df-icc 10887  df-fz 11008  df-fzo 11099  df-fl 11165  df-mod 11214  df-seq 11287  df-exp 11346  df-fac 11530  df-bc 11557  df-hash 11582  df-shft 11845  df-cj 11867  df-re 11868  df-im 11869  df-sqr 12003  df-abs 12004  df-limsup 12228  df-clim 12245  df-rlim 12246  df-sum 12443  df-ef 12633  df-sin 12635  df-cos 12636  df-pi 12638  df-struct 13434  df-ndx 13435  df-slot 13436  df-base 13437  df-sets 13438  df-ress 13439  df-plusg 13505  df-mulr 13506  df-starv 13507  df-sca 13508  df-vsca 13509  df-tset 13511  df-ple 13512  df-ds 13514  df-unif 13515  df-hom 13516  df-cco 13517  df-rest 13613  df-topn 13614  df-topgen 13630  df-pt 13631  df-prds 13634  df-xrs 13689  df-0g 13690  df-gsum 13691  df-qtop 13696  df-imas 13697  df-xps 13699  df-mre 13774  df-mrc 13775  df-acs 13777  df-mnd 14653  df-submnd 14702  df-mulg 14778  df-cntz 15079  df-cmn 15377  df-psmet 16657  df-xmet 16658  df-met 16659  df-bl 16660  df-mopn 16661  df-fbas 16662  df-fg 16663  df-cnfld 16667  df-top 16926  df-bases 16928  df-topon 16929  df-topsp 16930  df-cld 17046  df-ntr 17047  df-cls 17048  df-nei 17125  df-lp 17163  df-perf 17164  df-cn 17253  df-cnp 17254  df-haus 17341  df-tx 17555  df-hmeo 17748  df-fil 17839  df-fm 17931  df-flim 17932  df-flf 17933  df-xms 18311  df-ms 18312  df-tms 18313  df-cncf 18869  df-limc 19714  df-dv 19715  df-log 20415
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