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Theorem chordthmlem3 20543
Description: If M is the midpoint of AB, AQ = BQ, and P is on the line AB, then PQ 2 = QM 2  + PM 2 . This follows from chordthmlem2 20542 and the Pythagorean theorem (pythag 20527) in the case where P and Q are unequal to M. If either P or Q equals M, the result is trivial. (Contributed by David Moews, 28-Feb-2017.)
Hypotheses
Ref Expression
chordthmlem3.A  |-  ( ph  ->  A  e.  CC )
chordthmlem3.B  |-  ( ph  ->  B  e.  CC )
chordthmlem3.Q  |-  ( ph  ->  Q  e.  CC )
chordthmlem3.X  |-  ( ph  ->  X  e.  RR )
chordthmlem3.M  |-  ( ph  ->  M  =  ( ( A  +  B )  /  2 ) )
chordthmlem3.P  |-  ( ph  ->  P  =  ( ( X  x.  A )  +  ( ( 1  -  X )  x.  B ) ) )
chordthmlem3.ABequidistQ  |-  ( ph  ->  ( abs `  ( A  -  Q )
)  =  ( abs `  ( B  -  Q
) ) )
Assertion
Ref Expression
chordthmlem3  |-  ( ph  ->  ( ( abs `  ( P  -  Q )
) ^ 2 )  =  ( ( ( abs `  ( Q  -  M ) ) ^ 2 )  +  ( ( abs `  ( P  -  M )
) ^ 2 ) ) )

Proof of Theorem chordthmlem3
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 chordthmlem3.Q . . . . . . . . 9  |-  ( ph  ->  Q  e.  CC )
2 chordthmlem3.M . . . . . . . . . 10  |-  ( ph  ->  M  =  ( ( A  +  B )  /  2 ) )
3 chordthmlem3.A . . . . . . . . . . . 12  |-  ( ph  ->  A  e.  CC )
4 chordthmlem3.B . . . . . . . . . . . 12  |-  ( ph  ->  B  e.  CC )
53, 4addcld 9041 . . . . . . . . . . 11  |-  ( ph  ->  ( A  +  B
)  e.  CC )
65halfcld 10145 . . . . . . . . . 10  |-  ( ph  ->  ( ( A  +  B )  /  2
)  e.  CC )
72, 6eqeltrd 2462 . . . . . . . . 9  |-  ( ph  ->  M  e.  CC )
81, 7subcld 9344 . . . . . . . 8  |-  ( ph  ->  ( Q  -  M
)  e.  CC )
98abscld 12166 . . . . . . 7  |-  ( ph  ->  ( abs `  ( Q  -  M )
)  e.  RR )
109recnd 9048 . . . . . 6  |-  ( ph  ->  ( abs `  ( Q  -  M )
)  e.  CC )
1110sqcld 11449 . . . . 5  |-  ( ph  ->  ( ( abs `  ( Q  -  M )
) ^ 2 )  e.  CC )
1211adantr 452 . . . 4  |-  ( (
ph  /\  P  =  M )  ->  (
( abs `  ( Q  -  M )
) ^ 2 )  e.  CC )
1312addid1d 9199 . . 3  |-  ( (
ph  /\  P  =  M )  ->  (
( ( abs `  ( Q  -  M )
) ^ 2 )  +  0 )  =  ( ( abs `  ( Q  -  M )
) ^ 2 ) )
14 chordthmlem3.P . . . . . . . . 9  |-  ( ph  ->  P  =  ( ( X  x.  A )  +  ( ( 1  -  X )  x.  B ) ) )
15 chordthmlem3.X . . . . . . . . . . . 12  |-  ( ph  ->  X  e.  RR )
1615recnd 9048 . . . . . . . . . . 11  |-  ( ph  ->  X  e.  CC )
1716, 3mulcld 9042 . . . . . . . . . 10  |-  ( ph  ->  ( X  x.  A
)  e.  CC )
18 ax-1cn 8982 . . . . . . . . . . . . 13  |-  1  e.  CC
1918a1i 11 . . . . . . . . . . . 12  |-  ( ph  ->  1  e.  CC )
2019, 16subcld 9344 . . . . . . . . . . 11  |-  ( ph  ->  ( 1  -  X
)  e.  CC )
2120, 4mulcld 9042 . . . . . . . . . 10  |-  ( ph  ->  ( ( 1  -  X )  x.  B
)  e.  CC )
2217, 21addcld 9041 . . . . . . . . 9  |-  ( ph  ->  ( ( X  x.  A )  +  ( ( 1  -  X
)  x.  B ) )  e.  CC )
2314, 22eqeltrd 2462 . . . . . . . 8  |-  ( ph  ->  P  e.  CC )
2423adantr 452 . . . . . . 7  |-  ( (
ph  /\  P  =  M )  ->  P  e.  CC )
25 simpr 448 . . . . . . 7  |-  ( (
ph  /\  P  =  M )  ->  P  =  M )
2624, 25subeq0bd 9396 . . . . . 6  |-  ( (
ph  /\  P  =  M )  ->  ( P  -  M )  =  0 )
2726abs00bd 12024 . . . . 5  |-  ( (
ph  /\  P  =  M )  ->  ( abs `  ( P  -  M ) )  =  0 )
2827sq0id 11403 . . . 4  |-  ( (
ph  /\  P  =  M )  ->  (
( abs `  ( P  -  M )
) ^ 2 )  =  0 )
2928oveq2d 6037 . . 3  |-  ( (
ph  /\  P  =  M )  ->  (
( ( abs `  ( Q  -  M )
) ^ 2 )  +  ( ( abs `  ( P  -  M
) ) ^ 2 ) )  =  ( ( ( abs `  ( Q  -  M )
) ^ 2 )  +  0 ) )
301adantr 452 . . . . . 6  |-  ( (
ph  /\  P  =  M )  ->  Q  e.  CC )
3130, 24abssubd 12183 . . . . 5  |-  ( (
ph  /\  P  =  M )  ->  ( abs `  ( Q  -  P ) )  =  ( abs `  ( P  -  Q )
) )
3225oveq2d 6037 . . . . . 6  |-  ( (
ph  /\  P  =  M )  ->  ( Q  -  P )  =  ( Q  -  M ) )
3332fveq2d 5673 . . . . 5  |-  ( (
ph  /\  P  =  M )  ->  ( abs `  ( Q  -  P ) )  =  ( abs `  ( Q  -  M )
) )
3431, 33eqtr3d 2422 . . . 4  |-  ( (
ph  /\  P  =  M )  ->  ( abs `  ( P  -  Q ) )  =  ( abs `  ( Q  -  M )
) )
3534oveq1d 6036 . . 3  |-  ( (
ph  /\  P  =  M )  ->  (
( abs `  ( P  -  Q )
) ^ 2 )  =  ( ( abs `  ( Q  -  M
) ) ^ 2 ) )
3613, 29, 353eqtr4rd 2431 . 2  |-  ( (
ph  /\  P  =  M )  ->  (
( abs `  ( P  -  Q )
) ^ 2 )  =  ( ( ( abs `  ( Q  -  M ) ) ^ 2 )  +  ( ( abs `  ( P  -  M )
) ^ 2 ) ) )
3723, 7subcld 9344 . . . . . . . 8  |-  ( ph  ->  ( P  -  M
)  e.  CC )
3837abscld 12166 . . . . . . 7  |-  ( ph  ->  ( abs `  ( P  -  M )
)  e.  RR )
3938recnd 9048 . . . . . 6  |-  ( ph  ->  ( abs `  ( P  -  M )
)  e.  CC )
4039sqcld 11449 . . . . 5  |-  ( ph  ->  ( ( abs `  ( P  -  M )
) ^ 2 )  e.  CC )
4140adantr 452 . . . 4  |-  ( (
ph  /\  Q  =  M )  ->  (
( abs `  ( P  -  M )
) ^ 2 )  e.  CC )
4241addid2d 9200 . . 3  |-  ( (
ph  /\  Q  =  M )  ->  (
0  +  ( ( abs `  ( P  -  M ) ) ^ 2 ) )  =  ( ( abs `  ( P  -  M
) ) ^ 2 ) )
431adantr 452 . . . . . . 7  |-  ( (
ph  /\  Q  =  M )  ->  Q  e.  CC )
44 simpr 448 . . . . . . 7  |-  ( (
ph  /\  Q  =  M )  ->  Q  =  M )
4543, 44subeq0bd 9396 . . . . . 6  |-  ( (
ph  /\  Q  =  M )  ->  ( Q  -  M )  =  0 )
4645abs00bd 12024 . . . . 5  |-  ( (
ph  /\  Q  =  M )  ->  ( abs `  ( Q  -  M ) )  =  0 )
4746sq0id 11403 . . . 4  |-  ( (
ph  /\  Q  =  M )  ->  (
( abs `  ( Q  -  M )
) ^ 2 )  =  0 )
4847oveq1d 6036 . . 3  |-  ( (
ph  /\  Q  =  M )  ->  (
( ( abs `  ( Q  -  M )
) ^ 2 )  +  ( ( abs `  ( P  -  M
) ) ^ 2 ) )  =  ( 0  +  ( ( abs `  ( P  -  M ) ) ^ 2 ) ) )
4944oveq2d 6037 . . . . 5  |-  ( (
ph  /\  Q  =  M )  ->  ( P  -  Q )  =  ( P  -  M ) )
5049fveq2d 5673 . . . 4  |-  ( (
ph  /\  Q  =  M )  ->  ( abs `  ( P  -  Q ) )  =  ( abs `  ( P  -  M )
) )
5150oveq1d 6036 . . 3  |-  ( (
ph  /\  Q  =  M )  ->  (
( abs `  ( P  -  Q )
) ^ 2 )  =  ( ( abs `  ( P  -  M
) ) ^ 2 ) )
5242, 48, 513eqtr4rd 2431 . 2  |-  ( (
ph  /\  Q  =  M )  ->  (
( abs `  ( P  -  Q )
) ^ 2 )  =  ( ( ( abs `  ( Q  -  M ) ) ^ 2 )  +  ( ( abs `  ( P  -  M )
) ^ 2 ) ) )
5323adantr 452 . . 3  |-  ( (
ph  /\  ( P  =/=  M  /\  Q  =/= 
M ) )  ->  P  e.  CC )
541adantr 452 . . 3  |-  ( (
ph  /\  ( P  =/=  M  /\  Q  =/= 
M ) )  ->  Q  e.  CC )
557adantr 452 . . 3  |-  ( (
ph  /\  ( P  =/=  M  /\  Q  =/= 
M ) )  ->  M  e.  CC )
56 simprl 733 . . 3  |-  ( (
ph  /\  ( P  =/=  M  /\  Q  =/= 
M ) )  ->  P  =/=  M )
57 simprr 734 . . 3  |-  ( (
ph  /\  ( P  =/=  M  /\  Q  =/= 
M ) )  ->  Q  =/=  M )
58 eqid 2388 . . . 4  |-  ( x  e.  ( CC  \  { 0 } ) ,  y  e.  ( CC  \  { 0 } )  |->  ( Im
`  ( log `  (
y  /  x ) ) ) )  =  ( x  e.  ( CC  \  { 0 } ) ,  y  e.  ( CC  \  { 0 } ) 
|->  ( Im `  ( log `  ( y  /  x ) ) ) )
593adantr 452 . . . 4  |-  ( (
ph  /\  ( P  =/=  M  /\  Q  =/= 
M ) )  ->  A  e.  CC )
604adantr 452 . . . 4  |-  ( (
ph  /\  ( P  =/=  M  /\  Q  =/= 
M ) )  ->  B  e.  CC )
6115adantr 452 . . . 4  |-  ( (
ph  /\  ( P  =/=  M  /\  Q  =/= 
M ) )  ->  X  e.  RR )
622adantr 452 . . . 4  |-  ( (
ph  /\  ( P  =/=  M  /\  Q  =/= 
M ) )  ->  M  =  ( ( A  +  B )  /  2 ) )
6314adantr 452 . . . 4  |-  ( (
ph  /\  ( P  =/=  M  /\  Q  =/= 
M ) )  ->  P  =  ( ( X  x.  A )  +  ( ( 1  -  X )  x.  B ) ) )
64 chordthmlem3.ABequidistQ . . . . 5  |-  ( ph  ->  ( abs `  ( A  -  Q )
)  =  ( abs `  ( B  -  Q
) ) )
6564adantr 452 . . . 4  |-  ( (
ph  /\  ( P  =/=  M  /\  Q  =/= 
M ) )  -> 
( abs `  ( A  -  Q )
)  =  ( abs `  ( B  -  Q
) ) )
6658, 59, 60, 54, 61, 62, 63, 65, 56, 57chordthmlem2 20542 . . 3  |-  ( (
ph  /\  ( P  =/=  M  /\  Q  =/= 
M ) )  -> 
( ( Q  -  M ) ( x  e.  ( CC  \  { 0 } ) ,  y  e.  ( CC  \  { 0 } )  |->  ( Im
`  ( log `  (
y  /  x ) ) ) ) ( P  -  M ) )  e.  { ( pi  /  2 ) ,  -u ( pi  / 
2 ) } )
67 eqid 2388 . . . 4  |-  ( abs `  ( Q  -  M
) )  =  ( abs `  ( Q  -  M ) )
68 eqid 2388 . . . 4  |-  ( abs `  ( P  -  M
) )  =  ( abs `  ( P  -  M ) )
69 eqid 2388 . . . 4  |-  ( abs `  ( P  -  Q
) )  =  ( abs `  ( P  -  Q ) )
70 eqid 2388 . . . 4  |-  ( ( Q  -  M ) ( x  e.  ( CC  \  { 0 } ) ,  y  e.  ( CC  \  { 0 } ) 
|->  ( Im `  ( log `  ( y  /  x ) ) ) ) ( P  -  M ) )  =  ( ( Q  -  M ) ( x  e.  ( CC  \  { 0 } ) ,  y  e.  ( CC  \  { 0 } )  |->  ( Im
`  ( log `  (
y  /  x ) ) ) ) ( P  -  M ) )
7158, 67, 68, 69, 70pythag 20527 . . 3  |-  ( ( ( P  e.  CC  /\  Q  e.  CC  /\  M  e.  CC )  /\  ( P  =/=  M  /\  Q  =/=  M
)  /\  ( ( Q  -  M )
( x  e.  ( CC  \  { 0 } ) ,  y  e.  ( CC  \  { 0 } ) 
|->  ( Im `  ( log `  ( y  /  x ) ) ) ) ( P  -  M ) )  e. 
{ ( pi  / 
2 ) ,  -u ( pi  /  2
) } )  -> 
( ( abs `  ( P  -  Q )
) ^ 2 )  =  ( ( ( abs `  ( Q  -  M ) ) ^ 2 )  +  ( ( abs `  ( P  -  M )
) ^ 2 ) ) )
7253, 54, 55, 56, 57, 66, 71syl321anc 1206 . 2  |-  ( (
ph  /\  ( P  =/=  M  /\  Q  =/= 
M ) )  -> 
( ( abs `  ( P  -  Q )
) ^ 2 )  =  ( ( ( abs `  ( Q  -  M ) ) ^ 2 )  +  ( ( abs `  ( P  -  M )
) ^ 2 ) ) )
7336, 52, 72pm2.61da2ne 2630 1  |-  ( ph  ->  ( ( abs `  ( P  -  Q )
) ^ 2 )  =  ( ( ( abs `  ( Q  -  M ) ) ^ 2 )  +  ( ( abs `  ( P  -  M )
) ^ 2 ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = 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   ^cexp 11310   Imcim 11831   abscabs 11967   picpi 12597   logclog 20320
This theorem is referenced by:  chordthmlem5  20545
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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