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Theorem chordthmlem2 20130
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 20129, where P = B, and using angrtmuld 20106 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 9815 . . . . . . . . . 10  |-  2  e.  RR
87a1i 10 . . . . . . . . 9  |-  ( ph  ->  2  e.  RR )
9 2ne0 9829 . . . . . . . . . 10  |-  2  =/=  0
109a1i 10 . . . . . . . . 9  |-  ( ph  ->  2  =/=  0 )
118, 10rereccld 9587 . . . . . . . 8  |-  ( ph  ->  ( 1  /  2
)  e.  RR )
12 chordthmlem2.X . . . . . . . 8  |-  ( ph  ->  X  e.  RR )
1311, 12resubcld 9211 . . . . . . 7  |-  ( ph  ->  ( ( 1  / 
2 )  -  X
)  e.  RR )
1413recnd 8861 . . . . . 6  |-  ( ph  ->  ( ( 1  / 
2 )  -  X
)  e.  CC )
153, 2subcld 9157 . . . . . 6  |-  ( ph  ->  ( B  -  A
)  e.  CC )
1611recnd 8861 . . . . . . . . 9  |-  ( ph  ->  ( 1  /  2
)  e.  CC )
1712recnd 8861 . . . . . . . . 9  |-  ( ph  ->  X  e.  CC )
1816, 17, 15subdird 9236 . . . . . . . 8  |-  ( ph  ->  ( ( ( 1  /  2 )  -  X )  x.  ( B  -  A )
)  =  ( ( ( 1  /  2
)  x.  ( B  -  A ) )  -  ( X  x.  ( B  -  A
) ) ) )
19 2cn 9816 . . . . . . . . . . . . . . 15  |-  2  e.  CC
2019a1i 10 . . . . . . . . . . . . . 14  |-  ( ph  ->  2  e.  CC )
213, 20, 10divcan4d 9542 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( B  x.  2 )  /  2
)  =  B )
223times2d 9955 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( B  x.  2 )  =  ( B  +  B ) )
2322oveq1d 5873 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( B  x.  2 )  /  2
)  =  ( ( B  +  B )  /  2 ) )
2421, 23eqtr3d 2317 . . . . . . . . . . . 12  |-  ( ph  ->  B  =  ( ( B  +  B )  /  2 ) )
2524, 5oveq12d 5876 . . . . . . . . . . 11  |-  ( ph  ->  ( B  -  M
)  =  ( ( ( B  +  B
)  /  2 )  -  ( ( A  +  B )  / 
2 ) ) )
263, 3addcld 8854 . . . . . . . . . . . 12  |-  ( ph  ->  ( B  +  B
)  e.  CC )
272, 3addcld 8854 . . . . . . . . . . . 12  |-  ( ph  ->  ( A  +  B
)  e.  CC )
2826, 27, 20, 10divsubdird 9575 . . . . . . . . . . 11  |-  ( ph  ->  ( ( ( B  +  B )  -  ( A  +  B
) )  /  2
)  =  ( ( ( B  +  B
)  /  2 )  -  ( ( A  +  B )  / 
2 ) ) )
293, 2, 3pnpcan2d 9195 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( B  +  B )  -  ( A  +  B )
)  =  ( B  -  A ) )
3029oveq1d 5873 . . . . . . . . . . 11  |-  ( ph  ->  ( ( ( B  +  B )  -  ( A  +  B
) )  /  2
)  =  ( ( B  -  A )  /  2 ) )
3125, 28, 303eqtr2d 2321 . . . . . . . . . 10  |-  ( ph  ->  ( B  -  M
)  =  ( ( B  -  A )  /  2 ) )
3215, 20, 10divrec2d 9540 . . . . . . . . . 10  |-  ( ph  ->  ( ( B  -  A )  /  2
)  =  ( ( 1  /  2 )  x.  ( B  -  A ) ) )
3331, 32eqtrd 2315 . . . . . . . . 9  |-  ( ph  ->  ( B  -  M
)  =  ( ( 1  /  2 )  x.  ( B  -  A ) ) )
34 chordthmlem2.P . . . . . . . . . 10  |-  ( ph  ->  P  =  ( ( X  x.  A )  +  ( ( 1  -  X )  x.  B ) ) )
3517, 2mulcld 8855 . . . . . . . . . . . . 13  |-  ( ph  ->  ( X  x.  A
)  e.  CC )
36 ax-1cn 8795 . . . . . . . . . . . . . . . 16  |-  1  e.  CC
3736a1i 10 . . . . . . . . . . . . . . 15  |-  ( ph  ->  1  e.  CC )
3837, 17subcld 9157 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( 1  -  X
)  e.  CC )
3938, 3mulcld 8855 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( 1  -  X )  x.  B
)  e.  CC )
4035, 39addcld 8854 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( X  x.  A )  +  ( ( 1  -  X
)  x.  B ) )  e.  CC )
4134, 40eqeltrd 2357 . . . . . . . . . . 11  |-  ( ph  ->  P  e.  CC )
422, 41, 3, 17affineequiv 20123 . . . . . . . . . 10  |-  ( ph  ->  ( P  =  ( ( X  x.  A
)  +  ( ( 1  -  X )  x.  B ) )  <-> 
( B  -  P
)  =  ( X  x.  ( B  -  A ) ) ) )
4334, 42mpbid 201 . . . . . . . . 9  |-  ( ph  ->  ( B  -  P
)  =  ( X  x.  ( B  -  A ) ) )
4433, 43oveq12d 5876 . . . . . . . 8  |-  ( ph  ->  ( ( B  -  M )  -  ( B  -  P )
)  =  ( ( ( 1  /  2
)  x.  ( B  -  A ) )  -  ( X  x.  ( B  -  A
) ) ) )
4527halfcld 9956 . . . . . . . . . 10  |-  ( ph  ->  ( ( A  +  B )  /  2
)  e.  CC )
465, 45eqeltrd 2357 . . . . . . . . 9  |-  ( ph  ->  M  e.  CC )
473, 46, 41nnncan1d 9191 . . . . . . . 8  |-  ( ph  ->  ( ( B  -  M )  -  ( B  -  P )
)  =  ( P  -  M ) )
4818, 44, 473eqtr2rd 2322 . . . . . . 7  |-  ( ph  ->  ( P  -  M
)  =  ( ( ( 1  /  2
)  -  X )  x.  ( B  -  A ) ) )
49 chordthmlem2.PneM . . . . . . . 8  |-  ( ph  ->  P  =/=  M )
5041, 46, 49subne0d 9166 . . . . . . 7  |-  ( ph  ->  ( P  -  M
)  =/=  0 )
5148, 50eqnetrrd 2466 . . . . . 6  |-  ( ph  ->  ( ( ( 1  /  2 )  -  X )  x.  ( B  -  A )
)  =/=  0 )
5214, 15, 51mulne0bbd 9422 . . . . 5  |-  ( ph  ->  ( B  -  A
)  =/=  0 )
533, 2, 52subne0ad 9168 . . . 4  |-  ( ph  ->  B  =/=  A )
5453necomd 2529 . . 3  |-  ( ph  ->  A  =/=  B )
55 chordthmlem2.QneM . . 3  |-  ( ph  ->  Q  =/=  M )
561, 2, 3, 4, 5, 6, 54, 55chordthmlem 20129 . 2  |-  ( ph  ->  ( ( Q  -  M ) F ( B  -  M ) )  e.  { ( pi  /  2 ) ,  -u ( pi  / 
2 ) } )
574, 46subcld 9157 . . 3  |-  ( ph  ->  ( Q  -  M
)  e.  CC )
5841, 46subcld 9157 . . 3  |-  ( ph  ->  ( P  -  M
)  e.  CC )
593, 46subcld 9157 . . 3  |-  ( ph  ->  ( B  -  M
)  e.  CC )
604, 46, 55subne0d 9166 . . 3  |-  ( ph  ->  ( Q  -  M
)  =/=  0 )
6120, 10recne0d 9530 . . . . 5  |-  ( ph  ->  ( 1  /  2
)  =/=  0 )
6216, 15, 61, 52mulne0d 9420 . . . 4  |-  ( ph  ->  ( ( 1  / 
2 )  x.  ( B  -  A )
)  =/=  0 )
6333, 62eqnetrd 2464 . . 3  |-  ( ph  ->  ( B  -  M
)  =/=  0 )
6433, 48oveq12d 5876 . . . . 5  |-  ( ph  ->  ( ( B  -  M )  /  ( P  -  M )
)  =  ( ( ( 1  /  2
)  x.  ( B  -  A ) )  /  ( ( ( 1  /  2 )  -  X )  x.  ( B  -  A
) ) ) )
6514, 15, 51mulne0bad 9421 . . . . . 6  |-  ( ph  ->  ( ( 1  / 
2 )  -  X
)  =/=  0 )
6616, 14, 15, 65, 52divcan5rd 9563 . . . . 5  |-  ( ph  ->  ( ( ( 1  /  2 )  x.  ( B  -  A
) )  /  (
( ( 1  / 
2 )  -  X
)  x.  ( B  -  A ) ) )  =  ( ( 1  /  2 )  /  ( ( 1  /  2 )  -  X ) ) )
6764, 66eqtrd 2315 . . . 4  |-  ( ph  ->  ( ( B  -  M )  /  ( P  -  M )
)  =  ( ( 1  /  2 )  /  ( ( 1  /  2 )  -  X ) ) )
6811, 13, 65redivcld 9588 . . . 4  |-  ( ph  ->  ( ( 1  / 
2 )  /  (
( 1  /  2
)  -  X ) )  e.  RR )
6967, 68eqeltrd 2357 . . 3  |-  ( ph  ->  ( ( B  -  M )  /  ( P  -  M )
)  e.  RR )
701, 57, 58, 59, 60, 50, 63, 69angrtmuld 20106 . 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 223 1  |-  ( ph  ->  ( ( Q  -  M ) F ( P  -  M ) )  e.  { ( pi  /  2 ) ,  -u ( pi  / 
2 ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684    =/= wne 2446    \ cdif 3149   {csn 3640   {cpr 3641   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742    - cmin 9037   -ucneg 9038    / cdiv 9423   2c2 9795   Imcim 11583   abscabs 11719   picpi 12348   logclog 19912
This theorem is referenced by:  chordthmlem3  20131
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ioo 10660  df-ioc 10661  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-fac 11289  df-bc 11316  df-hash 11338  df-shft 11562  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-limsup 11945  df-clim 11962  df-rlim 11963  df-sum 12159  df-ef 12349  df-sin 12351  df-cos 12352  df-pi 12354  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-rest 13327  df-topn 13328  df-topgen 13344  df-pt 13345  df-prds 13348  df-xrs 13403  df-0g 13404  df-gsum 13405  df-qtop 13410  df-imas 13411  df-xps 13413  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-mulg 14492  df-cntz 14793  df-cmn 15091  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cld 16756  df-ntr 16757  df-cls 16758  df-nei 16835  df-lp 16868  df-perf 16869  df-cn 16957  df-cnp 16958  df-haus 17043  df-tx 17257  df-hmeo 17446  df-fbas 17520  df-fg 17521  df-fil 17541  df-fm 17633  df-flim 17634  df-flf 17635  df-xms 17885  df-ms 17886  df-tms 17887  df-cncf 18382  df-limc 19216  df-dv 19217  df-log 19914
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