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Theorem chordthmlem2 20542
Description: If M is the midpoint of AB, AQ = BQ, and P is on the line AB, then QMP is a right angle. This is proven by reduction to the special case chordthmlem 20541, where P = B, and using angrtmuld 20518 to observe that QMP is right iff QMB is. (Contributed by David Moews, 28-Feb-2017.)
Hypotheses
Ref Expression
chordthmlem2.angdef  |-  F  =  ( x  e.  ( CC  \  { 0 } ) ,  y  e.  ( CC  \  { 0 } ) 
|->  ( Im `  ( log `  ( y  /  x ) ) ) )
chordthmlem2.A  |-  ( ph  ->  A  e.  CC )
chordthmlem2.B  |-  ( ph  ->  B  e.  CC )
chordthmlem2.Q  |-  ( ph  ->  Q  e.  CC )
chordthmlem2.X  |-  ( ph  ->  X  e.  RR )
chordthmlem2.M  |-  ( ph  ->  M  =  ( ( A  +  B )  /  2 ) )
chordthmlem2.P  |-  ( ph  ->  P  =  ( ( X  x.  A )  +  ( ( 1  -  X )  x.  B ) ) )
chordthmlem2.ABequidistQ  |-  ( ph  ->  ( abs `  ( A  -  Q )
)  =  ( abs `  ( B  -  Q
) ) )
chordthmlem2.PneM  |-  ( ph  ->  P  =/=  M )
chordthmlem2.QneM  |-  ( ph  ->  Q  =/=  M )
Assertion
Ref Expression
chordthmlem2  |-  ( ph  ->  ( ( Q  -  M ) F ( P  -  M ) )  e.  { ( pi  /  2 ) ,  -u ( pi  / 
2 ) } )
Distinct variable groups:    x, y, Q    x, P, y    x, M, y    x, B, y   
x, A, y
Allowed substitution hints:    ph( x, y)    F( x, y)    X( x, y)

Proof of Theorem chordthmlem2
StepHypRef Expression
1 chordthmlem2.angdef . . 3  |-  F  =  ( x  e.  ( CC  \  { 0 } ) ,  y  e.  ( CC  \  { 0 } ) 
|->  ( Im `  ( log `  ( y  /  x ) ) ) )
2 chordthmlem2.A . . 3  |-  ( ph  ->  A  e.  CC )
3 chordthmlem2.B . . 3  |-  ( ph  ->  B  e.  CC )
4 chordthmlem2.Q . . 3  |-  ( ph  ->  Q  e.  CC )
5 chordthmlem2.M . . 3  |-  ( ph  ->  M  =  ( ( A  +  B )  /  2 ) )
6 chordthmlem2.ABequidistQ . . 3  |-  ( ph  ->  ( abs `  ( A  -  Q )
)  =  ( abs `  ( B  -  Q
) ) )
7 2re 10002 . . . . . . . . . 10  |-  2  e.  RR
87a1i 11 . . . . . . . . 9  |-  ( ph  ->  2  e.  RR )
9 2ne0 10016 . . . . . . . . . 10  |-  2  =/=  0
109a1i 11 . . . . . . . . 9  |-  ( ph  ->  2  =/=  0 )
118, 10rereccld 9774 . . . . . . . 8  |-  ( ph  ->  ( 1  /  2
)  e.  RR )
12 chordthmlem2.X . . . . . . . 8  |-  ( ph  ->  X  e.  RR )
1311, 12resubcld 9398 . . . . . . 7  |-  ( ph  ->  ( ( 1  / 
2 )  -  X
)  e.  RR )
1413recnd 9048 . . . . . 6  |-  ( ph  ->  ( ( 1  / 
2 )  -  X
)  e.  CC )
153, 2subcld 9344 . . . . . 6  |-  ( ph  ->  ( B  -  A
)  e.  CC )
1611recnd 9048 . . . . . . . . 9  |-  ( ph  ->  ( 1  /  2
)  e.  CC )
1712recnd 9048 . . . . . . . . 9  |-  ( ph  ->  X  e.  CC )
1816, 17, 15subdird 9423 . . . . . . . 8  |-  ( ph  ->  ( ( ( 1  /  2 )  -  X )  x.  ( B  -  A )
)  =  ( ( ( 1  /  2
)  x.  ( B  -  A ) )  -  ( X  x.  ( B  -  A
) ) ) )
19 2cn 10003 . . . . . . . . . . . . . . 15  |-  2  e.  CC
2019a1i 11 . . . . . . . . . . . . . 14  |-  ( ph  ->  2  e.  CC )
213, 20, 10divcan4d 9729 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( B  x.  2 )  /  2
)  =  B )
223times2d 10144 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( B  x.  2 )  =  ( B  +  B ) )
2322oveq1d 6036 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( B  x.  2 )  /  2
)  =  ( ( B  +  B )  /  2 ) )
2421, 23eqtr3d 2422 . . . . . . . . . . . 12  |-  ( ph  ->  B  =  ( ( B  +  B )  /  2 ) )
2524, 5oveq12d 6039 . . . . . . . . . . 11  |-  ( ph  ->  ( B  -  M
)  =  ( ( ( B  +  B
)  /  2 )  -  ( ( A  +  B )  / 
2 ) ) )
263, 3addcld 9041 . . . . . . . . . . . 12  |-  ( ph  ->  ( B  +  B
)  e.  CC )
272, 3addcld 9041 . . . . . . . . . . . 12  |-  ( ph  ->  ( A  +  B
)  e.  CC )
2826, 27, 20, 10divsubdird 9762 . . . . . . . . . . 11  |-  ( ph  ->  ( ( ( B  +  B )  -  ( A  +  B
) )  /  2
)  =  ( ( ( B  +  B
)  /  2 )  -  ( ( A  +  B )  / 
2 ) ) )
293, 2, 3pnpcan2d 9382 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( B  +  B )  -  ( A  +  B )
)  =  ( B  -  A ) )
3029oveq1d 6036 . . . . . . . . . . 11  |-  ( ph  ->  ( ( ( B  +  B )  -  ( A  +  B
) )  /  2
)  =  ( ( B  -  A )  /  2 ) )
3125, 28, 303eqtr2d 2426 . . . . . . . . . 10  |-  ( ph  ->  ( B  -  M
)  =  ( ( B  -  A )  /  2 ) )
3215, 20, 10divrec2d 9727 . . . . . . . . . 10  |-  ( ph  ->  ( ( B  -  A )  /  2
)  =  ( ( 1  /  2 )  x.  ( B  -  A ) ) )
3331, 32eqtrd 2420 . . . . . . . . 9  |-  ( ph  ->  ( B  -  M
)  =  ( ( 1  /  2 )  x.  ( B  -  A ) ) )
34 chordthmlem2.P . . . . . . . . . 10  |-  ( ph  ->  P  =  ( ( X  x.  A )  +  ( ( 1  -  X )  x.  B ) ) )
3517, 2mulcld 9042 . . . . . . . . . . . . 13  |-  ( ph  ->  ( X  x.  A
)  e.  CC )
36 ax-1cn 8982 . . . . . . . . . . . . . . . 16  |-  1  e.  CC
3736a1i 11 . . . . . . . . . . . . . . 15  |-  ( ph  ->  1  e.  CC )
3837, 17subcld 9344 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( 1  -  X
)  e.  CC )
3938, 3mulcld 9042 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( 1  -  X )  x.  B
)  e.  CC )
4035, 39addcld 9041 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( X  x.  A )  +  ( ( 1  -  X
)  x.  B ) )  e.  CC )
4134, 40eqeltrd 2462 . . . . . . . . . . 11  |-  ( ph  ->  P  e.  CC )
422, 41, 3, 17affineequiv 20535 . . . . . . . . . 10  |-  ( ph  ->  ( P  =  ( ( X  x.  A
)  +  ( ( 1  -  X )  x.  B ) )  <-> 
( B  -  P
)  =  ( X  x.  ( B  -  A ) ) ) )
4334, 42mpbid 202 . . . . . . . . 9  |-  ( ph  ->  ( B  -  P
)  =  ( X  x.  ( B  -  A ) ) )
4433, 43oveq12d 6039 . . . . . . . 8  |-  ( ph  ->  ( ( B  -  M )  -  ( B  -  P )
)  =  ( ( ( 1  /  2
)  x.  ( B  -  A ) )  -  ( X  x.  ( B  -  A
) ) ) )
4527halfcld 10145 . . . . . . . . . 10  |-  ( ph  ->  ( ( A  +  B )  /  2
)  e.  CC )
465, 45eqeltrd 2462 . . . . . . . . 9  |-  ( ph  ->  M  e.  CC )
473, 46, 41nnncan1d 9378 . . . . . . . 8  |-  ( ph  ->  ( ( B  -  M )  -  ( B  -  P )
)  =  ( P  -  M ) )
4818, 44, 473eqtr2rd 2427 . . . . . . 7  |-  ( ph  ->  ( P  -  M
)  =  ( ( ( 1  /  2
)  -  X )  x.  ( B  -  A ) ) )
49 chordthmlem2.PneM . . . . . . . 8  |-  ( ph  ->  P  =/=  M )
5041, 46, 49subne0d 9353 . . . . . . 7  |-  ( ph  ->  ( P  -  M
)  =/=  0 )
5148, 50eqnetrrd 2571 . . . . . 6  |-  ( ph  ->  ( ( ( 1  /  2 )  -  X )  x.  ( B  -  A )
)  =/=  0 )
5214, 15, 51mulne0bbd 9609 . . . . 5  |-  ( ph  ->  ( B  -  A
)  =/=  0 )
533, 2, 52subne0ad 9355 . . . 4  |-  ( ph  ->  B  =/=  A )
5453necomd 2634 . . 3  |-  ( ph  ->  A  =/=  B )
55 chordthmlem2.QneM . . 3  |-  ( ph  ->  Q  =/=  M )
561, 2, 3, 4, 5, 6, 54, 55chordthmlem 20541 . 2  |-  ( ph  ->  ( ( Q  -  M ) F ( B  -  M ) )  e.  { ( pi  /  2 ) ,  -u ( pi  / 
2 ) } )
574, 46subcld 9344 . . 3  |-  ( ph  ->  ( Q  -  M
)  e.  CC )
5841, 46subcld 9344 . . 3  |-  ( ph  ->  ( P  -  M
)  e.  CC )
593, 46subcld 9344 . . 3  |-  ( ph  ->  ( B  -  M
)  e.  CC )
604, 46, 55subne0d 9353 . . 3  |-  ( ph  ->  ( Q  -  M
)  =/=  0 )
6120, 10recne0d 9717 . . . . 5  |-  ( ph  ->  ( 1  /  2
)  =/=  0 )
6216, 15, 61, 52mulne0d 9607 . . . 4  |-  ( ph  ->  ( ( 1  / 
2 )  x.  ( B  -  A )
)  =/=  0 )
6333, 62eqnetrd 2569 . . 3  |-  ( ph  ->  ( B  -  M
)  =/=  0 )
6433, 48oveq12d 6039 . . . . 5  |-  ( ph  ->  ( ( B  -  M )  /  ( P  -  M )
)  =  ( ( ( 1  /  2
)  x.  ( B  -  A ) )  /  ( ( ( 1  /  2 )  -  X )  x.  ( B  -  A
) ) ) )
6514, 15, 51mulne0bad 9608 . . . . . 6  |-  ( ph  ->  ( ( 1  / 
2 )  -  X
)  =/=  0 )
6616, 14, 15, 65, 52divcan5rd 9750 . . . . 5  |-  ( ph  ->  ( ( ( 1  /  2 )  x.  ( B  -  A
) )  /  (
( ( 1  / 
2 )  -  X
)  x.  ( B  -  A ) ) )  =  ( ( 1  /  2 )  /  ( ( 1  /  2 )  -  X ) ) )
6764, 66eqtrd 2420 . . . 4  |-  ( ph  ->  ( ( B  -  M )  /  ( P  -  M )
)  =  ( ( 1  /  2 )  /  ( ( 1  /  2 )  -  X ) ) )
6811, 13, 65redivcld 9775 . . . 4  |-  ( ph  ->  ( ( 1  / 
2 )  /  (
( 1  /  2
)  -  X ) )  e.  RR )
6967, 68eqeltrd 2462 . . 3  |-  ( ph  ->  ( ( B  -  M )  /  ( P  -  M )
)  e.  RR )
701, 57, 58, 59, 60, 50, 63, 69angrtmuld 20518 . 2  |-  ( ph  ->  ( ( ( Q  -  M ) F ( P  -  M
) )  e.  {
( pi  /  2
) ,  -u (
pi  /  2 ) }  <->  ( ( Q  -  M ) F ( B  -  M
) )  e.  {
( pi  /  2
) ,  -u (
pi  /  2 ) } ) )
7156, 70mpbird 224 1  |-  ( ph  ->  ( ( Q  -  M ) F ( P  -  M ) )  e.  { ( pi  /  2 ) ,  -u ( pi  / 
2 ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = 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   1c1 8925    + caddc 8927    x. cmul 8929    - cmin 9224   -ucneg 9225    / cdiv 9610   2c2 9982   Imcim 11831   abscabs 11967   picpi 12597   logclog 20320
This theorem is referenced by:  chordthmlem3  20543
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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