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Theorem chordthmlem3 20126
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 20125 and the Pythagorean theorem (pythag 20110) 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 ) ) )
Dummy variables  x  y are mutually distinct and distinct from all other variables.

Proof of Theorem chordthmlem3
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 8850 . . . . . . . . . . 11  |-  ( ph  ->  ( A  +  B
)  e.  CC )
65halfcld 9952 . . . . . . . . . 10  |-  ( ph  ->  ( ( A  +  B )  /  2
)  e.  CC )
72, 6eqeltrd 2359 . . . . . . . . 9  |-  ( ph  ->  M  e.  CC )
81, 7subcld 9153 . . . . . . . 8  |-  ( ph  ->  ( Q  -  M
)  e.  CC )
98abscld 11913 . . . . . . 7  |-  ( ph  ->  ( abs `  ( Q  -  M )
)  e.  RR )
109recnd 8857 . . . . . 6  |-  ( ph  ->  ( abs `  ( Q  -  M )
)  e.  CC )
1110sqcld 11238 . . . . 5  |-  ( ph  ->  ( ( abs `  ( Q  -  M )
) ^ 2 )  e.  CC )
1211adantr 453 . . . 4  |-  ( (
ph  /\  P  =  M )  ->  (
( abs `  ( Q  -  M )
) ^ 2 )  e.  CC )
1312addid1d 9008 . . 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 8857 . . . . . . . . . . 11  |-  ( ph  ->  X  e.  CC )
1716, 3mulcld 8851 . . . . . . . . . 10  |-  ( ph  ->  ( X  x.  A
)  e.  CC )
18 ax-1cn 8791 . . . . . . . . . . . . 13  |-  1  e.  CC
1918a1i 12 . . . . . . . . . . . 12  |-  ( ph  ->  1  e.  CC )
2019, 16subcld 9153 . . . . . . . . . . 11  |-  ( ph  ->  ( 1  -  X
)  e.  CC )
2120, 4mulcld 8851 . . . . . . . . . 10  |-  ( ph  ->  ( ( 1  -  X )  x.  B
)  e.  CC )
2217, 21addcld 8850 . . . . . . . . 9  |-  ( ph  ->  ( ( X  x.  A )  +  ( ( 1  -  X
)  x.  B ) )  e.  CC )
2314, 22eqeltrd 2359 . . . . . . . 8  |-  ( ph  ->  P  e.  CC )
2423adantr 453 . . . . . . 7  |-  ( (
ph  /\  P  =  M )  ->  P  e.  CC )
25 simpr 449 . . . . . . 7  |-  ( (
ph  /\  P  =  M )  ->  P  =  M )
2624, 25subeq0bd 9205 . . . . . 6  |-  ( (
ph  /\  P  =  M )  ->  ( P  -  M )  =  0 )
2726abs00bd 11771 . . . . 5  |-  ( (
ph  /\  P  =  M )  ->  ( abs `  ( P  -  M ) )  =  0 )
2827sq0id 11192 . . . 4  |-  ( (
ph  /\  P  =  M )  ->  (
( abs `  ( P  -  M )
) ^ 2 )  =  0 )
2928oveq2d 5836 . . 3  |-  ( (
ph  /\  P  =  M )  ->  (
( ( abs `  ( Q  -  M )
) ^ 2 )  +  ( ( abs `  ( P  -  M
) ) ^ 2 ) )  =  ( ( ( abs `  ( Q  -  M )
) ^ 2 )  +  0 ) )
301adantr 453 . . . . . 6  |-  ( (
ph  /\  P  =  M )  ->  Q  e.  CC )
3130, 24abssubd 11930 . . . . 5  |-  ( (
ph  /\  P  =  M )  ->  ( abs `  ( Q  -  P ) )  =  ( abs `  ( P  -  Q )
) )
3225oveq2d 5836 . . . . . 6  |-  ( (
ph  /\  P  =  M )  ->  ( Q  -  P )  =  ( Q  -  M ) )
3332fveq2d 5490 . . . . 5  |-  ( (
ph  /\  P  =  M )  ->  ( abs `  ( Q  -  P ) )  =  ( abs `  ( Q  -  M )
) )
3431, 33eqtr3d 2319 . . . 4  |-  ( (
ph  /\  P  =  M )  ->  ( abs `  ( P  -  Q ) )  =  ( abs `  ( Q  -  M )
) )
3534oveq1d 5835 . . 3  |-  ( (
ph  /\  P  =  M )  ->  (
( abs `  ( P  -  Q )
) ^ 2 )  =  ( ( abs `  ( Q  -  M
) ) ^ 2 ) )
3613, 29, 353eqtr4rd 2328 . 2  |-  ( (
ph  /\  P  =  M )  ->  (
( abs `  ( P  -  Q )
) ^ 2 )  =  ( ( ( abs `  ( Q  -  M ) ) ^ 2 )  +  ( ( abs `  ( P  -  M )
) ^ 2 ) ) )
3723, 7subcld 9153 . . . . . . . 8  |-  ( ph  ->  ( P  -  M
)  e.  CC )
3837abscld 11913 . . . . . . 7  |-  ( ph  ->  ( abs `  ( P  -  M )
)  e.  RR )
3938recnd 8857 . . . . . 6  |-  ( ph  ->  ( abs `  ( P  -  M )
)  e.  CC )
4039sqcld 11238 . . . . 5  |-  ( ph  ->  ( ( abs `  ( P  -  M )
) ^ 2 )  e.  CC )
4140adantr 453 . . . 4  |-  ( (
ph  /\  Q  =  M )  ->  (
( abs `  ( P  -  M )
) ^ 2 )  e.  CC )
4241addid2d 9009 . . 3  |-  ( (
ph  /\  Q  =  M )  ->  (
0  +  ( ( abs `  ( P  -  M ) ) ^ 2 ) )  =  ( ( abs `  ( P  -  M
) ) ^ 2 ) )
431adantr 453 . . . . . . 7  |-  ( (
ph  /\  Q  =  M )  ->  Q  e.  CC )
44 simpr 449 . . . . . . 7  |-  ( (
ph  /\  Q  =  M )  ->  Q  =  M )
4543, 44subeq0bd 9205 . . . . . 6  |-  ( (
ph  /\  Q  =  M )  ->  ( Q  -  M )  =  0 )
4645abs00bd 11771 . . . . 5  |-  ( (
ph  /\  Q  =  M )  ->  ( abs `  ( Q  -  M ) )  =  0 )
4746sq0id 11192 . . . 4  |-  ( (
ph  /\  Q  =  M )  ->  (
( abs `  ( Q  -  M )
) ^ 2 )  =  0 )
4847oveq1d 5835 . . 3  |-  ( (
ph  /\  Q  =  M )  ->  (
( ( abs `  ( Q  -  M )
) ^ 2 )  +  ( ( abs `  ( P  -  M
) ) ^ 2 ) )  =  ( 0  +  ( ( abs `  ( P  -  M ) ) ^ 2 ) ) )
4944oveq2d 5836 . . . . 5  |-  ( (
ph  /\  Q  =  M )  ->  ( P  -  Q )  =  ( P  -  M ) )
5049fveq2d 5490 . . . 4  |-  ( (
ph  /\  Q  =  M )  ->  ( abs `  ( P  -  Q ) )  =  ( abs `  ( P  -  M )
) )
5150oveq1d 5835 . . 3  |-  ( (
ph  /\  Q  =  M )  ->  (
( abs `  ( P  -  Q )
) ^ 2 )  =  ( ( abs `  ( P  -  M
) ) ^ 2 ) )
5242, 48, 513eqtr4rd 2328 . 2  |-  ( (
ph  /\  Q  =  M )  ->  (
( abs `  ( P  -  Q )
) ^ 2 )  =  ( ( ( abs `  ( Q  -  M ) ) ^ 2 )  +  ( ( abs `  ( P  -  M )
) ^ 2 ) ) )
5323adantr 453 . . 3  |-  ( (
ph  /\  ( P  =/=  M  /\  Q  =/= 
M ) )  ->  P  e.  CC )
541adantr 453 . . 3  |-  ( (
ph  /\  ( P  =/=  M  /\  Q  =/= 
M ) )  ->  Q  e.  CC )
557adantr 453 . . 3  |-  ( (
ph  /\  ( P  =/=  M  /\  Q  =/= 
M ) )  ->  M  e.  CC )
56 simprl 734 . . 3  |-  ( (
ph  /\  ( P  =/=  M  /\  Q  =/= 
M ) )  ->  P  =/=  M )
57 simprr 735 . . 3  |-  ( (
ph  /\  ( P  =/=  M  /\  Q  =/= 
M ) )  ->  Q  =/=  M )
58 eqid 2285 . . . 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 453 . . . 4  |-  ( (
ph  /\  ( P  =/=  M  /\  Q  =/= 
M ) )  ->  A  e.  CC )
604adantr 453 . . . 4  |-  ( (
ph  /\  ( P  =/=  M  /\  Q  =/= 
M ) )  ->  B  e.  CC )
6115adantr 453 . . . 4  |-  ( (
ph  /\  ( P  =/=  M  /\  Q  =/= 
M ) )  ->  X  e.  RR )
622adantr 453 . . . 4  |-  ( (
ph  /\  ( P  =/=  M  /\  Q  =/= 
M ) )  ->  M  =  ( ( A  +  B )  /  2 ) )
6314adantr 453 . . . 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 453 . . . 4  |-  ( (
ph  /\  ( P  =/=  M  /\  Q  =/= 
M ) )  -> 
( abs `  ( A  -  Q )
)  =  ( abs `  ( B  -  Q
) ) )
6658, 59, 60, 54, 61, 62, 63, 65, 56, 57chordthmlem2 20125 . . 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 2285 . . . 4  |-  ( abs `  ( Q  -  M
) )  =  ( abs `  ( Q  -  M ) )
68 eqid 2285 . . . 4  |-  ( abs `  ( P  -  M
) )  =  ( abs `  ( P  -  M ) )
69 eqid 2285 . . . 4  |-  ( abs `  ( P  -  Q
) )  =  ( abs `  ( P  -  Q ) )
70 eqid 2285 . . . 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 20110 . . 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 2527 1  |-  ( ph  ->  ( ( abs `  ( P  -  Q )
) ^ 2 )  =  ( ( ( abs `  ( Q  -  M ) ) ^ 2 )  +  ( ( abs `  ( P  -  M )
) ^ 2 ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    = 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   1c1 8734    + caddc 8736    x. cmul 8738    - cmin 9033   -ucneg 9034    / cdiv 9419   2c2 9791   ^cexp 11099   Imcim 11578   abscabs 11714   picpi 12343   logclog 19907
This theorem is referenced by:  chordthmlem5  20128
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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