MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  chordthmlem Structured version   Unicode version

Theorem chordthmlem 22907
Description: If M is the midpoint of AB and AQ = BQ, then QMB is a right angle. The proof uses ssscongptld 22900 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 22676 . . . . . 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 9614 . . . . . . . . . 10  |-  ( ph  ->  ( A  +  B
)  e.  CC )
87halfcld 10782 . . . . . . . . 9  |-  ( ph  ->  ( ( A  +  B )  /  2
)  e.  CC )
94, 8eqeltrd 2555 . . . . . . . 8  |-  ( ph  ->  M  e.  CC )
103, 9subcld 9929 . . . . . . 7  |-  ( ph  ->  ( Q  -  M
)  e.  CC )
11 chordthmlem.QneM . . . . . . . 8  |-  ( ph  ->  Q  =/=  M )
123, 9, 11subne0d 9938 . . . . . . 7  |-  ( ph  ->  ( Q  -  M
)  =/=  0 )
136, 9subcld 9929 . . . . . . 7  |-  ( ph  ->  ( B  -  M
)  e.  CC )
144oveq1d 6298 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( M  x.  2 )  =  ( ( ( A  +  B
)  /  2 )  x.  2 ) )
159times2d 10781 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( M  x.  2 )  =  ( M  +  M ) )
16 2cnd 10607 . . . . . . . . . . . . . . 15  |-  ( ph  ->  2  e.  CC )
17 2ne0 10627 . . . . . . . . . . . . . . . 16  |-  2  =/=  0
1817a1i 11 . . . . . . . . . . . . . . 15  |-  ( ph  ->  2  =/=  0 )
197, 16, 18divcan1d 10320 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( ( A  +  B )  / 
2 )  x.  2 )  =  ( A  +  B ) )
2014, 15, 193eqtr3d 2516 . . . . . . . . . . . . 13  |-  ( ph  ->  ( M  +  M
)  =  ( A  +  B ) )
21 chordthmlem.AneB . . . . . . . . . . . . . 14  |-  ( ph  ->  A  =/=  B )
225, 6, 6, 21addneintr2d 9786 . . . . . . . . . . . . 13  |-  ( ph  ->  ( A  +  B
)  =/=  ( B  +  B ) )
2320, 22eqnetrd 2760 . . . . . . . . . . . 12  |-  ( ph  ->  ( M  +  M
)  =/=  ( B  +  B ) )
2423neneqd 2669 . . . . . . . . . . 11  |-  ( ph  ->  -.  ( M  +  M )  =  ( B  +  B ) )
25 oveq12 6292 . . . . . . . . . . . 12  |-  ( ( M  =  B  /\  M  =  B )  ->  ( M  +  M
)  =  ( B  +  B ) )
2625anidms 645 . . . . . . . . . . 11  |-  ( M  =  B  ->  ( M  +  M )  =  ( B  +  B ) )
2724, 26nsyl 121 . . . . . . . . . 10  |-  ( ph  ->  -.  M  =  B )
2827neqned 2670 . . . . . . . . 9  |-  ( ph  ->  M  =/=  B )
2928necomd 2738 . . . . . . . 8  |-  ( ph  ->  B  =/=  M )
306, 9, 29subne0d 9938 . . . . . . 7  |-  ( ph  ->  ( B  -  M
)  =/=  0 )
312, 10, 12, 13, 30angcld 22881 . . . . . 6  |-  ( ph  ->  ( ( Q  -  M ) F ( B  -  M ) )  e.  ( -u pi (,] pi ) )
321, 31sseldi 3502 . . . . 5  |-  ( ph  ->  ( ( Q  -  M ) F ( B  -  M ) )  e.  RR )
3332recnd 9621 . . . 4  |-  ( ph  ->  ( ( Q  -  M ) F ( B  -  M ) )  e.  CC )
3433coscld 13726 . . 3  |-  ( ph  ->  ( cos `  (
( Q  -  M
) F ( B  -  M ) ) )  e.  CC )
356, 9negsubdi2d 9945 . . . . . . 7  |-  ( ph  -> 
-u ( B  -  M )  =  ( M  -  B ) )
369, 9, 5, 6addsubeq4d 9980 . . . . . . . 8  |-  ( ph  ->  ( ( M  +  M )  =  ( A  +  B )  <-> 
( A  -  M
)  =  ( M  -  B ) ) )
3720, 36mpbid 210 . . . . . . 7  |-  ( ph  ->  ( A  -  M
)  =  ( M  -  B ) )
3835, 37eqtr4d 2511 . . . . . 6  |-  ( ph  -> 
-u ( B  -  M )  =  ( A  -  M ) )
3938oveq2d 6299 . . . . 5  |-  ( ph  ->  ( ( Q  -  M ) F -u ( B  -  M
) )  =  ( ( Q  -  M
) F ( A  -  M ) ) )
4039fveq2d 5869 . . . 4  |-  ( ph  ->  ( cos `  (
( Q  -  M
) F -u ( B  -  M )
) )  =  ( cos `  ( ( Q  -  M ) F ( A  -  M ) ) ) )
412, 10, 12, 13, 30cosangneg2d 22883 . . . 4  |-  ( ph  ->  ( cos `  (
( Q  -  M
) F -u ( B  -  M )
) )  =  -u ( cos `  ( ( Q  -  M ) F ( B  -  M ) ) ) )
425, 5, 6, 21addneintrd 9785 . . . . . . . . . 10  |-  ( ph  ->  ( A  +  A
)  =/=  ( A  +  B ) )
4342, 20neeqtrrd 2767 . . . . . . . . 9  |-  ( ph  ->  ( A  +  A
)  =/=  ( M  +  M ) )
4443necomd 2738 . . . . . . . 8  |-  ( ph  ->  ( M  +  M
)  =/=  ( A  +  A ) )
4544neneqd 2669 . . . . . . 7  |-  ( ph  ->  -.  ( M  +  M )  =  ( A  +  A ) )
46 oveq12 6292 . . . . . . . 8  |-  ( ( M  =  A  /\  M  =  A )  ->  ( M  +  M
)  =  ( A  +  A ) )
4746anidms 645 . . . . . . 7  |-  ( M  =  A  ->  ( M  +  M )  =  ( A  +  A ) )
4845, 47nsyl 121 . . . . . 6  |-  ( ph  ->  -.  M  =  A )
4948neqned 2670 . . . . 5  |-  ( ph  ->  M  =/=  A )
50 eqidd 2468 . . . . 5  |-  ( ph  ->  ( abs `  ( Q  -  M )
)  =  ( abs `  ( Q  -  M
) ) )
515, 9subcld 9929 . . . . . . 7  |-  ( ph  ->  ( A  -  M
)  e.  CC )
5251absnegd 13242 . . . . . 6  |-  ( ph  ->  ( abs `  -u ( A  -  M )
)  =  ( abs `  ( A  -  M
) ) )
535, 9negsubdi2d 9945 . . . . . . 7  |-  ( ph  -> 
-u ( A  -  M )  =  ( M  -  A ) )
5453fveq2d 5869 . . . . . 6  |-  ( ph  ->  ( abs `  -u ( A  -  M )
)  =  ( abs `  ( M  -  A
) ) )
5537fveq2d 5869 . . . . . 6  |-  ( ph  ->  ( abs `  ( A  -  M )
)  =  ( abs `  ( M  -  B
) ) )
5652, 54, 553eqtr3d 2516 . . . . 5  |-  ( ph  ->  ( abs `  ( M  -  A )
)  =  ( abs `  ( M  -  B
) ) )
57 chordthmlem.ABequidistQ . . . . 5  |-  ( ph  ->  ( abs `  ( A  -  Q )
)  =  ( abs `  ( B  -  Q
) ) )
582, 3, 9, 5, 3, 9, 6, 11, 49, 11, 28, 50, 56, 57ssscongptld 22900 . . . 4  |-  ( ph  ->  ( cos `  (
( Q  -  M
) F ( A  -  M ) ) )  =  ( cos `  ( ( Q  -  M ) F ( B  -  M ) ) ) )
5940, 41, 583eqtr3rd 2517 . . 3  |-  ( ph  ->  ( cos `  (
( Q  -  M
) F ( B  -  M ) ) )  =  -u ( cos `  ( ( Q  -  M ) F ( B  -  M
) ) ) )
6034, 59eqnegad 10265 . 2  |-  ( ph  ->  ( cos `  (
( Q  -  M
) F ( B  -  M ) ) )  =  0 )
61 coseq0negpitopi 22645 . . 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 ) } ) )
6231, 61syl 16 . 2  |-  ( ph  ->  ( ( cos `  (
( Q  -  M
) F ( B  -  M ) ) )  =  0  <->  (
( Q  -  M
) F ( B  -  M ) )  e.  { ( pi 
/  2 ) , 
-u ( pi  / 
2 ) } ) )
6360, 62mpbid 210 1  |-  ( ph  ->  ( ( Q  -  M ) F ( B  -  M ) )  e.  { ( pi  /  2 ) ,  -u ( pi  / 
2 ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1379    e. wcel 1767    =/= wne 2662    \ cdif 3473   {csn 4027   {cpr 4029   ` cfv 5587  (class class class)co 6283    |-> cmpt2 6285   CCcc 9489   RRcr 9490   0cc0 9491    + caddc 9494    x. cmul 9496    - cmin 9804   -ucneg 9805    / cdiv 10205   2c2 10584   (,]cioc 11529   Imcim 12893   abscabs 13029   cosccos 13661   picpi 13663   logclog 22686
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575  ax-inf2 8057  ax-cnex 9547  ax-resscn 9548  ax-1cn 9549  ax-icn 9550  ax-addcl 9551  ax-addrcl 9552  ax-mulcl 9553  ax-mulrcl 9554  ax-mulcom 9555  ax-addass 9556  ax-mulass 9557  ax-distr 9558  ax-i2m1 9559  ax-1ne0 9560  ax-1rid 9561  ax-rnegex 9562  ax-rrecex 9563  ax-cnre 9564  ax-pre-lttri 9565  ax-pre-lttrn 9566  ax-pre-ltadd 9567  ax-pre-mulgt0 9568  ax-pre-sup 9569  ax-addf 9570  ax-mulf 9571
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-isom 5596  df-riota 6244  df-ov 6286  df-oprab 6287  df-mpt2 6288  df-of 6523  df-om 6680  df-1st 6784  df-2nd 6785  df-supp 6902  df-recs 7042  df-rdg 7076  df-1o 7130  df-2o 7131  df-oadd 7134  df-er 7311  df-map 7422  df-pm 7423  df-ixp 7470  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-fsupp 7829  df-fi 7870  df-sup 7900  df-oi 7934  df-card 8319  df-cda 8547  df-pnf 9629  df-mnf 9630  df-xr 9631  df-ltxr 9632  df-le 9633  df-sub 9806  df-neg 9807  df-div 10206  df-nn 10536  df-2 10593  df-3 10594  df-4 10595  df-5 10596  df-6 10597  df-7 10598  df-8 10599  df-9 10600  df-10 10601  df-n0 10795  df-z 10864  df-dec 10976  df-uz 11082  df-q 11182  df-rp 11220  df-xneg 11317  df-xadd 11318  df-xmul 11319  df-ioo 11532  df-ioc 11533  df-ico 11534  df-icc 11535  df-fz 11672  df-fzo 11792  df-fl 11896  df-mod 11964  df-seq 12075  df-exp 12134  df-fac 12321  df-bc 12348  df-hash 12373  df-shft 12862  df-cj 12894  df-re 12895  df-im 12896  df-sqrt 13030  df-abs 13031  df-limsup 13256  df-clim 13273  df-rlim 13274  df-sum 13471  df-ef 13664  df-sin 13666  df-cos 13667  df-pi 13669  df-struct 14491  df-ndx 14492  df-slot 14493  df-base 14494  df-sets 14495  df-ress 14496  df-plusg 14567  df-mulr 14568  df-starv 14569  df-sca 14570  df-vsca 14571  df-ip 14572  df-tset 14573  df-ple 14574  df-ds 14576  df-unif 14577  df-hom 14578  df-cco 14579  df-rest 14677  df-topn 14678  df-0g 14696  df-gsum 14697  df-topgen 14698  df-pt 14699  df-prds 14702  df-xrs 14756  df-qtop 14761  df-imas 14762  df-xps 14764  df-mre 14840  df-mrc 14841  df-acs 14843  df-mnd 15731  df-submnd 15784  df-mulg 15867  df-cntz 16157  df-cmn 16603  df-psmet 18198  df-xmet 18199  df-met 18200  df-bl 18201  df-mopn 18202  df-fbas 18203  df-fg 18204  df-cnfld 18208  df-top 19182  df-bases 19184  df-topon 19185  df-topsp 19186  df-cld 19302  df-ntr 19303  df-cls 19304  df-nei 19381  df-lp 19419  df-perf 19420  df-cn 19510  df-cnp 19511  df-haus 19598  df-tx 19814  df-hmeo 20007  df-fil 20098  df-fm 20190  df-flim 20191  df-flf 20192  df-xms 20574  df-ms 20575  df-tms 20576  df-cncf 21133  df-limc 22021  df-dv 22022  df-log 22688
This theorem is referenced by:  chordthmlem2  22908
  Copyright terms: Public domain W3C validator