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Theorem chordthmlem3 20131
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 20130 and the Pythagorean theorem (pythag 20115) 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 8854 . . . . . . . . . . 11  |-  ( ph  ->  ( A  +  B
)  e.  CC )
65halfcld 9956 . . . . . . . . . 10  |-  ( ph  ->  ( ( A  +  B )  /  2
)  e.  CC )
72, 6eqeltrd 2357 . . . . . . . . 9  |-  ( ph  ->  M  e.  CC )
81, 7subcld 9157 . . . . . . . 8  |-  ( ph  ->  ( Q  -  M
)  e.  CC )
98abscld 11918 . . . . . . 7  |-  ( ph  ->  ( abs `  ( Q  -  M )
)  e.  RR )
109recnd 8861 . . . . . 6  |-  ( ph  ->  ( abs `  ( Q  -  M )
)  e.  CC )
1110sqcld 11243 . . . . 5  |-  ( ph  ->  ( ( abs `  ( Q  -  M )
) ^ 2 )  e.  CC )
1211adantr 451 . . . 4  |-  ( (
ph  /\  P  =  M )  ->  (
( abs `  ( Q  -  M )
) ^ 2 )  e.  CC )
1312addid1d 9012 . . 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 8861 . . . . . . . . . . 11  |-  ( ph  ->  X  e.  CC )
1716, 3mulcld 8855 . . . . . . . . . 10  |-  ( ph  ->  ( X  x.  A
)  e.  CC )
18 ax-1cn 8795 . . . . . . . . . . . . 13  |-  1  e.  CC
1918a1i 10 . . . . . . . . . . . 12  |-  ( ph  ->  1  e.  CC )
2019, 16subcld 9157 . . . . . . . . . . 11  |-  ( ph  ->  ( 1  -  X
)  e.  CC )
2120, 4mulcld 8855 . . . . . . . . . 10  |-  ( ph  ->  ( ( 1  -  X )  x.  B
)  e.  CC )
2217, 21addcld 8854 . . . . . . . . 9  |-  ( ph  ->  ( ( X  x.  A )  +  ( ( 1  -  X
)  x.  B ) )  e.  CC )
2314, 22eqeltrd 2357 . . . . . . . 8  |-  ( ph  ->  P  e.  CC )
2423adantr 451 . . . . . . 7  |-  ( (
ph  /\  P  =  M )  ->  P  e.  CC )
25 simpr 447 . . . . . . 7  |-  ( (
ph  /\  P  =  M )  ->  P  =  M )
2624, 25subeq0bd 9209 . . . . . 6  |-  ( (
ph  /\  P  =  M )  ->  ( P  -  M )  =  0 )
2726abs00bd 11776 . . . . 5  |-  ( (
ph  /\  P  =  M )  ->  ( abs `  ( P  -  M ) )  =  0 )
2827sq0id 11197 . . . 4  |-  ( (
ph  /\  P  =  M )  ->  (
( abs `  ( P  -  M )
) ^ 2 )  =  0 )
2928oveq2d 5874 . . 3  |-  ( (
ph  /\  P  =  M )  ->  (
( ( abs `  ( Q  -  M )
) ^ 2 )  +  ( ( abs `  ( P  -  M
) ) ^ 2 ) )  =  ( ( ( abs `  ( Q  -  M )
) ^ 2 )  +  0 ) )
301adantr 451 . . . . . 6  |-  ( (
ph  /\  P  =  M )  ->  Q  e.  CC )
3130, 24abssubd 11935 . . . . 5  |-  ( (
ph  /\  P  =  M )  ->  ( abs `  ( Q  -  P ) )  =  ( abs `  ( P  -  Q )
) )
3225oveq2d 5874 . . . . . 6  |-  ( (
ph  /\  P  =  M )  ->  ( Q  -  P )  =  ( Q  -  M ) )
3332fveq2d 5529 . . . . 5  |-  ( (
ph  /\  P  =  M )  ->  ( abs `  ( Q  -  P ) )  =  ( abs `  ( Q  -  M )
) )
3431, 33eqtr3d 2317 . . . 4  |-  ( (
ph  /\  P  =  M )  ->  ( abs `  ( P  -  Q ) )  =  ( abs `  ( Q  -  M )
) )
3534oveq1d 5873 . . 3  |-  ( (
ph  /\  P  =  M )  ->  (
( abs `  ( P  -  Q )
) ^ 2 )  =  ( ( abs `  ( Q  -  M
) ) ^ 2 ) )
3613, 29, 353eqtr4rd 2326 . 2  |-  ( (
ph  /\  P  =  M )  ->  (
( abs `  ( P  -  Q )
) ^ 2 )  =  ( ( ( abs `  ( Q  -  M ) ) ^ 2 )  +  ( ( abs `  ( P  -  M )
) ^ 2 ) ) )
3723, 7subcld 9157 . . . . . . . 8  |-  ( ph  ->  ( P  -  M
)  e.  CC )
3837abscld 11918 . . . . . . 7  |-  ( ph  ->  ( abs `  ( P  -  M )
)  e.  RR )
3938recnd 8861 . . . . . 6  |-  ( ph  ->  ( abs `  ( P  -  M )
)  e.  CC )
4039sqcld 11243 . . . . 5  |-  ( ph  ->  ( ( abs `  ( P  -  M )
) ^ 2 )  e.  CC )
4140adantr 451 . . . 4  |-  ( (
ph  /\  Q  =  M )  ->  (
( abs `  ( P  -  M )
) ^ 2 )  e.  CC )
4241addid2d 9013 . . 3  |-  ( (
ph  /\  Q  =  M )  ->  (
0  +  ( ( abs `  ( P  -  M ) ) ^ 2 ) )  =  ( ( abs `  ( P  -  M
) ) ^ 2 ) )
431adantr 451 . . . . . . 7  |-  ( (
ph  /\  Q  =  M )  ->  Q  e.  CC )
44 simpr 447 . . . . . . 7  |-  ( (
ph  /\  Q  =  M )  ->  Q  =  M )
4543, 44subeq0bd 9209 . . . . . 6  |-  ( (
ph  /\  Q  =  M )  ->  ( Q  -  M )  =  0 )
4645abs00bd 11776 . . . . 5  |-  ( (
ph  /\  Q  =  M )  ->  ( abs `  ( Q  -  M ) )  =  0 )
4746sq0id 11197 . . . 4  |-  ( (
ph  /\  Q  =  M )  ->  (
( abs `  ( Q  -  M )
) ^ 2 )  =  0 )
4847oveq1d 5873 . . 3  |-  ( (
ph  /\  Q  =  M )  ->  (
( ( abs `  ( Q  -  M )
) ^ 2 )  +  ( ( abs `  ( P  -  M
) ) ^ 2 ) )  =  ( 0  +  ( ( abs `  ( P  -  M ) ) ^ 2 ) ) )
4944oveq2d 5874 . . . . 5  |-  ( (
ph  /\  Q  =  M )  ->  ( P  -  Q )  =  ( P  -  M ) )
5049fveq2d 5529 . . . 4  |-  ( (
ph  /\  Q  =  M )  ->  ( abs `  ( P  -  Q ) )  =  ( abs `  ( P  -  M )
) )
5150oveq1d 5873 . . 3  |-  ( (
ph  /\  Q  =  M )  ->  (
( abs `  ( P  -  Q )
) ^ 2 )  =  ( ( abs `  ( P  -  M
) ) ^ 2 ) )
5242, 48, 513eqtr4rd 2326 . 2  |-  ( (
ph  /\  Q  =  M )  ->  (
( abs `  ( P  -  Q )
) ^ 2 )  =  ( ( ( abs `  ( Q  -  M ) ) ^ 2 )  +  ( ( abs `  ( P  -  M )
) ^ 2 ) ) )
5323adantr 451 . . 3  |-  ( (
ph  /\  ( P  =/=  M  /\  Q  =/= 
M ) )  ->  P  e.  CC )
541adantr 451 . . 3  |-  ( (
ph  /\  ( P  =/=  M  /\  Q  =/= 
M ) )  ->  Q  e.  CC )
557adantr 451 . . 3  |-  ( (
ph  /\  ( P  =/=  M  /\  Q  =/= 
M ) )  ->  M  e.  CC )
56 simprl 732 . . 3  |-  ( (
ph  /\  ( P  =/=  M  /\  Q  =/= 
M ) )  ->  P  =/=  M )
57 simprr 733 . . 3  |-  ( (
ph  /\  ( P  =/=  M  /\  Q  =/= 
M ) )  ->  Q  =/=  M )
58 eqid 2283 . . . 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 451 . . . 4  |-  ( (
ph  /\  ( P  =/=  M  /\  Q  =/= 
M ) )  ->  A  e.  CC )
604adantr 451 . . . 4  |-  ( (
ph  /\  ( P  =/=  M  /\  Q  =/= 
M ) )  ->  B  e.  CC )
6115adantr 451 . . . 4  |-  ( (
ph  /\  ( P  =/=  M  /\  Q  =/= 
M ) )  ->  X  e.  RR )
622adantr 451 . . . 4  |-  ( (
ph  /\  ( P  =/=  M  /\  Q  =/= 
M ) )  ->  M  =  ( ( A  +  B )  /  2 ) )
6314adantr 451 . . . 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 451 . . . 4  |-  ( (
ph  /\  ( P  =/=  M  /\  Q  =/= 
M ) )  -> 
( abs `  ( A  -  Q )
)  =  ( abs `  ( B  -  Q
) ) )
6658, 59, 60, 54, 61, 62, 63, 65, 56, 57chordthmlem2 20130 . . 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 2283 . . . 4  |-  ( abs `  ( Q  -  M
) )  =  ( abs `  ( Q  -  M ) )
68 eqid 2283 . . . 4  |-  ( abs `  ( P  -  M
) )  =  ( abs `  ( P  -  M ) )
69 eqid 2283 . . . 4  |-  ( abs `  ( P  -  Q
) )  =  ( abs `  ( P  -  Q ) )
70 eqid 2283 . . . 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 20115 . . 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 1204 . 2  |-  ( (
ph  /\  ( P  =/=  M  /\  Q  =/= 
M ) )  -> 
( ( abs `  ( P  -  Q )
) ^ 2 )  =  ( ( ( abs `  ( Q  -  M ) ) ^ 2 )  +  ( ( abs `  ( P  -  M )
) ^ 2 ) ) )
7336, 52, 72pm2.61da2ne 2525 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 358    = 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   ^cexp 11104   Imcim 11583   abscabs 11719   picpi 12348   logclog 19912
This theorem is referenced by:  chordthmlem5  20133
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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