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Theorem chordthmlem3 20079
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 20078 and the Pythagorean theorem (pythag 20063) 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
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 8808 . . . . . . . . . . 11  |-  ( ph  ->  ( A  +  B
)  e.  CC )
65halfcld 9909 . . . . . . . . . 10  |-  ( ph  ->  ( ( A  +  B )  /  2
)  e.  CC )
72, 6eqeltrd 2330 . . . . . . . . 9  |-  ( ph  ->  M  e.  CC )
81, 7subcld 9111 . . . . . . . 8  |-  ( ph  ->  ( Q  -  M
)  e.  CC )
98abscld 11869 . . . . . . 7  |-  ( ph  ->  ( abs `  ( Q  -  M )
)  e.  RR )
109recnd 8815 . . . . . 6  |-  ( ph  ->  ( abs `  ( Q  -  M )
)  e.  CC )
1110sqcld 11195 . . . . 5  |-  ( ph  ->  ( ( abs `  ( Q  -  M )
) ^ 2 )  e.  CC )
1211adantr 453 . . . 4  |-  ( (
ph  /\  P  =  M )  ->  (
( abs `  ( Q  -  M )
) ^ 2 )  e.  CC )
1312addid1d 8966 . . 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 8815 . . . . . . . . . . 11  |-  ( ph  ->  X  e.  CC )
1716, 3mulcld 8809 . . . . . . . . . 10  |-  ( ph  ->  ( X  x.  A
)  e.  CC )
18 ax-1cn 8749 . . . . . . . . . . . . 13  |-  1  e.  CC
1918a1i 12 . . . . . . . . . . . 12  |-  ( ph  ->  1  e.  CC )
2019, 16subcld 9111 . . . . . . . . . . 11  |-  ( ph  ->  ( 1  -  X
)  e.  CC )
2120, 4mulcld 8809 . . . . . . . . . 10  |-  ( ph  ->  ( ( 1  -  X )  x.  B
)  e.  CC )
2217, 21addcld 8808 . . . . . . . . 9  |-  ( ph  ->  ( ( X  x.  A )  +  ( ( 1  -  X
)  x.  B ) )  e.  CC )
2314, 22eqeltrd 2330 . . . . . . . 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 9163 . . . . . 6  |-  ( (
ph  /\  P  =  M )  ->  ( P  -  M )  =  0 )
2726abs00bd 11727 . . . . 5  |-  ( (
ph  /\  P  =  M )  ->  ( abs `  ( P  -  M ) )  =  0 )
2827sq0id 11149 . . . 4  |-  ( (
ph  /\  P  =  M )  ->  (
( abs `  ( P  -  M )
) ^ 2 )  =  0 )
2928oveq2d 5794 . . 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 11886 . . . . 5  |-  ( (
ph  /\  P  =  M )  ->  ( abs `  ( Q  -  P ) )  =  ( abs `  ( P  -  Q )
) )
3225oveq2d 5794 . . . . . 6  |-  ( (
ph  /\  P  =  M )  ->  ( Q  -  P )  =  ( Q  -  M ) )
3332fveq2d 5448 . . . . 5  |-  ( (
ph  /\  P  =  M )  ->  ( abs `  ( Q  -  P ) )  =  ( abs `  ( Q  -  M )
) )
3431, 33eqtr3d 2290 . . . 4  |-  ( (
ph  /\  P  =  M )  ->  ( abs `  ( P  -  Q ) )  =  ( abs `  ( Q  -  M )
) )
3534oveq1d 5793 . . 3  |-  ( (
ph  /\  P  =  M )  ->  (
( abs `  ( P  -  Q )
) ^ 2 )  =  ( ( abs `  ( Q  -  M
) ) ^ 2 ) )
3613, 29, 353eqtr4rd 2299 . 2  |-  ( (
ph  /\  P  =  M )  ->  (
( abs `  ( P  -  Q )
) ^ 2 )  =  ( ( ( abs `  ( Q  -  M ) ) ^ 2 )  +  ( ( abs `  ( P  -  M )
) ^ 2 ) ) )
3723, 7subcld 9111 . . . . . . . 8  |-  ( ph  ->  ( P  -  M
)  e.  CC )
3837abscld 11869 . . . . . . 7  |-  ( ph  ->  ( abs `  ( P  -  M )
)  e.  RR )
3938recnd 8815 . . . . . 6  |-  ( ph  ->  ( abs `  ( P  -  M )
)  e.  CC )
4039sqcld 11195 . . . . 5  |-  ( ph  ->  ( ( abs `  ( P  -  M )
) ^ 2 )  e.  CC )
4140adantr 453 . . . 4  |-  ( (
ph  /\  Q  =  M )  ->  (
( abs `  ( P  -  M )
) ^ 2 )  e.  CC )
4241addid2d 8967 . . 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 9163 . . . . . 6  |-  ( (
ph  /\  Q  =  M )  ->  ( Q  -  M )  =  0 )
4645abs00bd 11727 . . . . 5  |-  ( (
ph  /\  Q  =  M )  ->  ( abs `  ( Q  -  M ) )  =  0 )
4746sq0id 11149 . . . 4  |-  ( (
ph  /\  Q  =  M )  ->  (
( abs `  ( Q  -  M )
) ^ 2 )  =  0 )
4847oveq1d 5793 . . 3  |-  ( (
ph  /\  Q  =  M )  ->  (
( ( abs `  ( Q  -  M )
) ^ 2 )  +  ( ( abs `  ( P  -  M
) ) ^ 2 ) )  =  ( 0  +  ( ( abs `  ( P  -  M ) ) ^ 2 ) ) )
4944oveq2d 5794 . . . . 5  |-  ( (
ph  /\  Q  =  M )  ->  ( P  -  Q )  =  ( P  -  M ) )
5049fveq2d 5448 . . . 4  |-  ( (
ph  /\  Q  =  M )  ->  ( abs `  ( P  -  Q ) )  =  ( abs `  ( P  -  M )
) )
5150oveq1d 5793 . . 3  |-  ( (
ph  /\  Q  =  M )  ->  (
( abs `  ( P  -  Q )
) ^ 2 )  =  ( ( abs `  ( P  -  M
) ) ^ 2 ) )
5242, 48, 513eqtr4rd 2299 . 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 735 . . 3  |-  ( (
ph  /\  ( P  =/=  M  /\  Q  =/= 
M ) )  ->  P  =/=  M )
57 simprr 736 . . 3  |-  ( (
ph  /\  ( P  =/=  M  /\  Q  =/= 
M ) )  ->  Q  =/=  M )
58 eqid 2256 . . . 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 20078 . . 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 2256 . . . 4  |-  ( abs `  ( Q  -  M
) )  =  ( abs `  ( Q  -  M ) )
68 eqid 2256 . . . 4  |-  ( abs `  ( P  -  M
) )  =  ( abs `  ( P  -  M ) )
69 eqid 2256 . . . 4  |-  ( abs `  ( P  -  Q
) )  =  ( abs `  ( P  -  Q ) )
70 eqid 2256 . . . 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 20063 . . 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 1209 . 2  |-  ( (
ph  /\  ( P  =/=  M  /\  Q  =/= 
M ) )  -> 
( ( abs `  ( P  -  Q )
) ^ 2 )  =  ( ( ( abs `  ( Q  -  M ) ) ^ 2 )  +  ( ( abs `  ( P  -  M )
) ^ 2 ) ) )
7336, 52, 72pm2.61da2ne 2498 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 1619    e. wcel 1621    =/= wne 2419    \ cdif 3110   {csn 3600   {cpr 3601   ` cfv 4659  (class class class)co 5778    e. cmpt2 5780   CCcc 8689   RRcr 8690   0cc0 8691   1c1 8692    + caddc 8694    x. cmul 8696    - cmin 8991   -ucneg 8992    / cdiv 9377   2c2 9749   ^cexp 11056   Imcim 11534   abscabs 11670   picpi 12296   logclog 19860
This theorem is referenced by:  chordthmlem5  20081
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4091  ax-sep 4101  ax-nul 4109  ax-pow 4146  ax-pr 4172  ax-un 4470  ax-inf2 7296  ax-cnex 8747  ax-resscn 8748  ax-1cn 8749  ax-icn 8750  ax-addcl 8751  ax-addrcl 8752  ax-mulcl 8753  ax-mulrcl 8754  ax-mulcom 8755  ax-addass 8756  ax-mulass 8757  ax-distr 8758  ax-i2m1 8759  ax-1ne0 8760  ax-1rid 8761  ax-rnegex 8762  ax-rrecex 8763  ax-cnre 8764  ax-pre-lttri 8765  ax-pre-lttrn 8766  ax-pre-ltadd 8767  ax-pre-mulgt0 8768  ax-pre-sup 8769  ax-addf 8770  ax-mulf 8771
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2521  df-rex 2522  df-reu 2523  df-rmo 2524  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-pss 3129  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-tp 3608  df-op 3609  df-uni 3788  df-int 3823  df-iun 3867  df-iin 3868  df-br 3984  df-opab 4038  df-mpt 4039  df-tr 4074  df-eprel 4263  df-id 4267  df-po 4272  df-so 4273  df-fr 4310  df-se 4311  df-we 4312  df-ord 4353  df-on 4354  df-lim 4355  df-suc 4356  df-om 4615  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-isom 4676  df-ov 5781  df-oprab 5782  df-mpt2 5783  df-of 5998  df-1st 6042  df-2nd 6043  df-iota 6211  df-riota 6258  df-recs 6342  df-rdg 6377  df-1o 6433  df-2o 6434  df-oadd 6437  df-er 6614  df-map 6728  df-pm 6729  df-ixp 6772  df-en 6818  df-dom 6819  df-sdom 6820  df-fin 6821  df-fi 7119  df-sup 7148  df-oi 7179  df-card 7526  df-cda 7748  df-pnf 8823  df-mnf 8824  df-xr 8825  df-ltxr 8826  df-le 8827  df-sub 8993  df-neg 8994  df-div 9378  df-n 9701  df-2 9758  df-3 9759  df-4 9760  df-5 9761  df-6 9762  df-7 9763  df-8 9764  df-9 9765  df-10 9766  df-n0 9919  df-z 9978  df-dec 10078  df-uz 10184  df-q 10270  df-rp 10308  df-xneg 10405  df-xadd 10406  df-xmul 10407  df-ioo 10612  df-ioc 10613  df-ico 10614  df-icc 10615  df-fz 10735  df-fzo 10823  df-fl 10877  df-mod 10926  df-seq 10999  df-exp 11057  df-fac 11241  df-bc 11268  df-hash 11290  df-shft 11513  df-cj 11535  df-re 11536  df-im 11537  df-sqr 11671  df-abs 11672  df-limsup 11896  df-clim 11913  df-rlim 11914  df-sum 12110  df-ef 12297  df-sin 12299  df-cos 12300  df-pi 12302  df-struct 13098  df-ndx 13099  df-slot 13100  df-base 13101  df-sets 13102  df-ress 13103  df-plusg 13169  df-mulr 13170  df-starv 13171  df-sca 13172  df-vsca 13173  df-tset 13175  df-ple 13176  df-ds 13178  df-hom 13180  df-cco 13181  df-rest 13275  df-topn 13276  df-topgen 13292  df-pt 13293  df-prds 13296  df-xrs 13351  df-0g 13352  df-gsum 13353  df-qtop 13358  df-imas 13359  df-xps 13361  df-mre 13436  df-mrc 13437  df-acs 13439  df-mnd 14315  df-submnd 14364  df-mulg 14440  df-cntz 14741  df-cmn 15039  df-xmet 16321  df-met 16322  df-bl 16323  df-mopn 16324  df-cnfld 16326  df-top 16584  df-bases 16586  df-topon 16587  df-topsp 16588  df-cld 16704  df-ntr 16705  df-cls 16706  df-nei 16783  df-lp 16816  df-perf 16817  df-cn 16905  df-cnp 16906  df-haus 16991  df-tx 17205  df-hmeo 17394  df-fbas 17468  df-fg 17469  df-fil 17489  df-fm 17581  df-flim 17582  df-flf 17583  df-xms 17833  df-ms 17834  df-tms 17835  df-cncf 18330  df-limc 19164  df-dv 19165  df-log 19862
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