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Theorem chordthmlem 20124
Description: If M is the midpoint of AB and AQ = BQ, then QMB is a right angle. The proof uses ssscongptld 20117 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 19897 . . . . . 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 8850 . . . . . . . . . 10  |-  ( ph  ->  ( A  +  B
)  e.  CC )
87halfcld 9952 . . . . . . . . 9  |-  ( ph  ->  ( ( A  +  B )  /  2
)  e.  CC )
94, 8eqeltrd 2359 . . . . . . . 8  |-  ( ph  ->  M  e.  CC )
103, 9subcld 9153 . . . . . . 7  |-  ( ph  ->  ( Q  -  M
)  e.  CC )
11 chordthmlem.QneM . . . . . . . 8  |-  ( ph  ->  Q  =/=  M )
123, 9, 11subne0d 9162 . . . . . . 7  |-  ( ph  ->  ( Q  -  M
)  =/=  0 )
136, 9subcld 9153 . . . . . . 7  |-  ( ph  ->  ( B  -  M
)  e.  CC )
144oveq1d 5835 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( M  x.  2 )  =  ( ( ( A  +  B
)  /  2 )  x.  2 ) )
159times2d 9951 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( M  x.  2 )  =  ( M  +  M ) )
16 2cn 9812 . . . . . . . . . . . . . . . 16  |-  2  e.  CC
1716a1i 12 . . . . . . . . . . . . . . 15  |-  ( ph  ->  2  e.  CC )
18 2ne0 9825 . . . . . . . . . . . . . . . 16  |-  2  =/=  0
1918a1i 12 . . . . . . . . . . . . . . 15  |-  ( ph  ->  2  =/=  0 )
207, 17, 19divcan1d 9533 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( ( A  +  B )  / 
2 )  x.  2 )  =  ( A  +  B ) )
2114, 15, 203eqtr3d 2325 . . . . . . . . . . . . 13  |-  ( ph  ->  ( M  +  M
)  =  ( A  +  B ) )
22 chordthmlem.AneB . . . . . . . . . . . . . 14  |-  ( ph  ->  A  =/=  B )
235, 6, 6, 22addneintr2d 9016 . . . . . . . . . . . . 13  |-  ( ph  ->  ( A  +  B
)  =/=  ( B  +  B ) )
2421, 23eqnetrd 2466 . . . . . . . . . . . 12  |-  ( ph  ->  ( M  +  M
)  =/=  ( B  +  B ) )
2524neneqd 2464 . . . . . . . . . . 11  |-  ( ph  ->  -.  ( M  +  M )  =  ( B  +  B ) )
26 oveq12 5829 . . . . . . . . . . . 12  |-  ( ( M  =  B  /\  M  =  B )  ->  ( M  +  M
)  =  ( B  +  B ) )
2726anidms 628 . . . . . . . . . . 11  |-  ( M  =  B  ->  ( M  +  M )  =  ( B  +  B ) )
2825, 27nsyl 115 . . . . . . . . . 10  |-  ( ph  ->  -.  M  =  B )
2928neneqad 2518 . . . . . . . . 9  |-  ( ph  ->  M  =/=  B )
3029necomd 2531 . . . . . . . 8  |-  ( ph  ->  B  =/=  M )
316, 9, 30subne0d 9162 . . . . . . 7  |-  ( ph  ->  ( B  -  M
)  =/=  0 )
322, 10, 12, 13, 31angcld 20098 . . . . . 6  |-  ( ph  ->  ( ( Q  -  M ) F ( B  -  M ) )  e.  ( -u pi (,] pi ) )
331, 32sseldi 3180 . . . . 5  |-  ( ph  ->  ( ( Q  -  M ) F ( B  -  M ) )  e.  RR )
3433recnd 8857 . . . 4  |-  ( ph  ->  ( ( Q  -  M ) F ( B  -  M ) )  e.  CC )
3534coscld 12406 . . 3  |-  ( ph  ->  ( cos `  (
( Q  -  M
) F ( B  -  M ) ) )  e.  CC )
366, 9negsubdi2d 9169 . . . . . . 7  |-  ( ph  -> 
-u ( B  -  M )  =  ( M  -  B ) )
379, 9, 5, 6addsubeq4d 9204 . . . . . . . 8  |-  ( ph  ->  ( ( M  +  M )  =  ( A  +  B )  <-> 
( A  -  M
)  =  ( M  -  B ) ) )
3821, 37mpbid 203 . . . . . . 7  |-  ( ph  ->  ( A  -  M
)  =  ( M  -  B ) )
3936, 38eqtr4d 2320 . . . . . 6  |-  ( ph  -> 
-u ( B  -  M )  =  ( A  -  M ) )
4039oveq2d 5836 . . . . 5  |-  ( ph  ->  ( ( Q  -  M ) F -u ( B  -  M
) )  =  ( ( Q  -  M
) F ( A  -  M ) ) )
4140fveq2d 5490 . . . 4  |-  ( ph  ->  ( cos `  (
( Q  -  M
) F -u ( B  -  M )
) )  =  ( cos `  ( ( Q  -  M ) F ( A  -  M ) ) ) )
422, 10, 12, 13, 31cosangneg2d 20100 . . . 4  |-  ( ph  ->  ( cos `  (
( Q  -  M
) F -u ( B  -  M )
) )  =  -u ( cos `  ( ( Q  -  M ) F ( B  -  M ) ) ) )
435, 5, 6, 22addneintrd 9015 . . . . . . . . . 10  |-  ( ph  ->  ( A  +  A
)  =/=  ( A  +  B ) )
4443, 21neeqtrrd 2472 . . . . . . . . 9  |-  ( ph  ->  ( A  +  A
)  =/=  ( M  +  M ) )
4544necomd 2531 . . . . . . . 8  |-  ( ph  ->  ( M  +  M
)  =/=  ( A  +  A ) )
4645neneqd 2464 . . . . . . 7  |-  ( ph  ->  -.  ( M  +  M )  =  ( A  +  A ) )
47 oveq12 5829 . . . . . . . 8  |-  ( ( M  =  A  /\  M  =  A )  ->  ( M  +  M
)  =  ( A  +  A ) )
4847anidms 628 . . . . . . 7  |-  ( M  =  A  ->  ( M  +  M )  =  ( A  +  A ) )
4946, 48nsyl 115 . . . . . 6  |-  ( ph  ->  -.  M  =  A )
5049neneqad 2518 . . . . 5  |-  ( ph  ->  M  =/=  A )
51 eqidd 2286 . . . . 5  |-  ( ph  ->  ( abs `  ( Q  -  M )
)  =  ( abs `  ( Q  -  M
) ) )
525, 9subcld 9153 . . . . . . 7  |-  ( ph  ->  ( A  -  M
)  e.  CC )
5352absnegd 11926 . . . . . 6  |-  ( ph  ->  ( abs `  -u ( A  -  M )
)  =  ( abs `  ( A  -  M
) ) )
545, 9negsubdi2d 9169 . . . . . . 7  |-  ( ph  -> 
-u ( A  -  M )  =  ( M  -  A ) )
5554fveq2d 5490 . . . . . 6  |-  ( ph  ->  ( abs `  -u ( A  -  M )
)  =  ( abs `  ( M  -  A
) ) )
5638fveq2d 5490 . . . . . 6  |-  ( ph  ->  ( abs `  ( A  -  M )
)  =  ( abs `  ( M  -  B
) ) )
5753, 55, 563eqtr3d 2325 . . . . 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 20117 . . . 4  |-  ( ph  ->  ( cos `  (
( Q  -  M
) F ( A  -  M ) ) )  =  ( cos `  ( ( Q  -  M ) F ( B  -  M ) ) ) )
6041, 42, 593eqtr3rd 2326 . . 3  |-  ( ph  ->  ( cos `  (
( Q  -  M
) F ( B  -  M ) ) )  =  -u ( cos `  ( ( Q  -  M ) F ( B  -  M
) ) ) )
6135, 60eqnegad 9478 . 2  |-  ( ph  ->  ( cos `  (
( Q  -  M
) F ( B  -  M ) ) )  =  0 )
62 coseq0negpitopi 19866 . . 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 17 . 2  |-  ( ph  ->  ( ( cos `  (
( Q  -  M
) F ( B  -  M ) ) )  =  0  <->  (
( Q  -  M
) F ( B  -  M ) )  e.  { ( pi 
/  2 ) , 
-u ( pi  / 
2 ) } ) )
6461, 63mpbid 203 1  |-  ( ph  ->  ( ( Q  -  M ) F ( B  -  M ) )  e.  { ( pi  /  2 ) ,  -u ( pi  / 
2 ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    = wceq 1624    e. wcel 1685    =/= wne 2448    \ cdif 3151   {csn 3642   {cpr 3643   ` cfv 5222  (class class class)co 5820    e. cmpt2 5822   CCcc 8731   RRcr 8732   0cc0 8733    + caddc 8736    x. cmul 8738    - cmin 9033   -ucneg 9034    / cdiv 9419   2c2 9791   (,]cioc 10652   Imcim 11578   abscabs 11714   cosccos 12341   picpi 12343   logclog 19907
This theorem is referenced by:  chordthmlem2  20125
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7338  ax-cnex 8789  ax-resscn 8790  ax-1cn 8791  ax-icn 8792  ax-addcl 8793  ax-addrcl 8794  ax-mulcl 8795  ax-mulrcl 8796  ax-mulcom 8797  ax-addass 8798  ax-mulass 8799  ax-distr 8800  ax-i2m1 8801  ax-1ne0 8802  ax-1rid 8803  ax-rnegex 8804  ax-rrecex 8805  ax-cnre 8806  ax-pre-lttri 8807  ax-pre-lttrn 8808  ax-pre-ltadd 8809  ax-pre-mulgt0 8810  ax-pre-sup 8811  ax-addf 8812  ax-mulf 8813
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-int 3865  df-iun 3909  df-iin 3910  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  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-fun 5224  df-fn 5225  df-f 5226  df-f1 5227  df-fo 5228  df-f1o 5229  df-fv 5230  df-isom 5231  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-of 6040  df-1st 6084  df-2nd 6085  df-iota 6253  df-riota 6300  df-recs 6384  df-rdg 6419  df-1o 6475  df-2o 6476  df-oadd 6479  df-er 6656  df-map 6770  df-pm 6771  df-ixp 6814  df-en 6860  df-dom 6861  df-sdom 6862  df-fin 6863  df-fi 7161  df-sup 7190  df-oi 7221  df-card 7568  df-cda 7790  df-pnf 8865  df-mnf 8866  df-xr 8867  df-ltxr 8868  df-le 8869  df-sub 9035  df-neg 9036  df-div 9420  df-nn 9743  df-2 9800  df-3 9801  df-4 9802  df-5 9803  df-6 9804  df-7 9805  df-8 9806  df-9 9807  df-10 9808  df-n0 9962  df-z 10021  df-dec 10121  df-uz 10227  df-q 10313  df-rp 10351  df-xneg 10448  df-xadd 10449  df-xmul 10450  df-ioo 10655  df-ioc 10656  df-ico 10657  df-icc 10658  df-fz 10778  df-fzo 10866  df-fl 10920  df-mod 10969  df-seq 11042  df-exp 11100  df-fac 11284  df-bc 11311  df-hash 11333  df-shft 11557  df-cj 11579  df-re 11580  df-im 11581  df-sqr 11715  df-abs 11716  df-limsup 11940  df-clim 11957  df-rlim 11958  df-sum 12154  df-ef 12344  df-sin 12346  df-cos 12347  df-pi 12349  df-struct 13145  df-ndx 13146  df-slot 13147  df-base 13148  df-sets 13149  df-ress 13150  df-plusg 13216  df-mulr 13217  df-starv 13218  df-sca 13219  df-vsca 13220  df-tset 13222  df-ple 13223  df-ds 13225  df-hom 13227  df-cco 13228  df-rest 13322  df-topn 13323  df-topgen 13339  df-pt 13340  df-prds 13343  df-xrs 13398  df-0g 13399  df-gsum 13400  df-qtop 13405  df-imas 13406  df-xps 13408  df-mre 13483  df-mrc 13484  df-acs 13486  df-mnd 14362  df-submnd 14411  df-mulg 14487  df-cntz 14788  df-cmn 15086  df-xmet 16368  df-met 16369  df-bl 16370  df-mopn 16371  df-cnfld 16373  df-top 16631  df-bases 16633  df-topon 16634  df-topsp 16635  df-cld 16751  df-ntr 16752  df-cls 16753  df-nei 16830  df-lp 16863  df-perf 16864  df-cn 16952  df-cnp 16953  df-haus 17038  df-tx 17252  df-hmeo 17441  df-fbas 17515  df-fg 17516  df-fil 17536  df-fm 17628  df-flim 17629  df-flf 17630  df-xms 17880  df-ms 17881  df-tms 17882  df-cncf 18377  df-limc 19211  df-dv 19212  df-log 19909
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