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Theorem chordthmlem3 22909
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 22908 and the Pythagorean theorem (pythag 22893) 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 9614 . . . . . . . . . . 11  |-  ( ph  ->  ( A  +  B
)  e.  CC )
65halfcld 10782 . . . . . . . . . 10  |-  ( ph  ->  ( ( A  +  B )  /  2
)  e.  CC )
72, 6eqeltrd 2555 . . . . . . . . 9  |-  ( ph  ->  M  e.  CC )
81, 7subcld 9929 . . . . . . . 8  |-  ( ph  ->  ( Q  -  M
)  e.  CC )
98abscld 13229 . . . . . . 7  |-  ( ph  ->  ( abs `  ( Q  -  M )
)  e.  RR )
109recnd 9621 . . . . . 6  |-  ( ph  ->  ( abs `  ( Q  -  M )
)  e.  CC )
1110sqcld 12275 . . . . 5  |-  ( ph  ->  ( ( abs `  ( Q  -  M )
) ^ 2 )  e.  CC )
1211adantr 465 . . . 4  |-  ( (
ph  /\  P  =  M )  ->  (
( abs `  ( Q  -  M )
) ^ 2 )  e.  CC )
1312addid1d 9778 . . 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 9621 . . . . . . . . . . 11  |-  ( ph  ->  X  e.  CC )
1716, 3mulcld 9615 . . . . . . . . . 10  |-  ( ph  ->  ( X  x.  A
)  e.  CC )
18 1cnd 9611 . . . . . . . . . . . 12  |-  ( ph  ->  1  e.  CC )
1918, 16subcld 9929 . . . . . . . . . . 11  |-  ( ph  ->  ( 1  -  X
)  e.  CC )
2019, 4mulcld 9615 . . . . . . . . . 10  |-  ( ph  ->  ( ( 1  -  X )  x.  B
)  e.  CC )
2117, 20addcld 9614 . . . . . . . . 9  |-  ( ph  ->  ( ( X  x.  A )  +  ( ( 1  -  X
)  x.  B ) )  e.  CC )
2214, 21eqeltrd 2555 . . . . . . . 8  |-  ( ph  ->  P  e.  CC )
2322adantr 465 . . . . . . 7  |-  ( (
ph  /\  P  =  M )  ->  P  e.  CC )
24 simpr 461 . . . . . . 7  |-  ( (
ph  /\  P  =  M )  ->  P  =  M )
2523, 24subeq0bd 9984 . . . . . 6  |-  ( (
ph  /\  P  =  M )  ->  ( P  -  M )  =  0 )
2625abs00bd 13086 . . . . 5  |-  ( (
ph  /\  P  =  M )  ->  ( abs `  ( P  -  M ) )  =  0 )
2726sq0id 12228 . . . 4  |-  ( (
ph  /\  P  =  M )  ->  (
( abs `  ( P  -  M )
) ^ 2 )  =  0 )
2827oveq2d 6299 . . 3  |-  ( (
ph  /\  P  =  M )  ->  (
( ( abs `  ( Q  -  M )
) ^ 2 )  +  ( ( abs `  ( P  -  M
) ) ^ 2 ) )  =  ( ( ( abs `  ( Q  -  M )
) ^ 2 )  +  0 ) )
291adantr 465 . . . . . 6  |-  ( (
ph  /\  P  =  M )  ->  Q  e.  CC )
3029, 23abssubd 13246 . . . . 5  |-  ( (
ph  /\  P  =  M )  ->  ( abs `  ( Q  -  P ) )  =  ( abs `  ( P  -  Q )
) )
3124oveq2d 6299 . . . . . 6  |-  ( (
ph  /\  P  =  M )  ->  ( Q  -  P )  =  ( Q  -  M ) )
3231fveq2d 5869 . . . . 5  |-  ( (
ph  /\  P  =  M )  ->  ( abs `  ( Q  -  P ) )  =  ( abs `  ( Q  -  M )
) )
3330, 32eqtr3d 2510 . . . 4  |-  ( (
ph  /\  P  =  M )  ->  ( abs `  ( P  -  Q ) )  =  ( abs `  ( Q  -  M )
) )
3433oveq1d 6298 . . 3  |-  ( (
ph  /\  P  =  M )  ->  (
( abs `  ( P  -  Q )
) ^ 2 )  =  ( ( abs `  ( Q  -  M
) ) ^ 2 ) )
3513, 28, 343eqtr4rd 2519 . 2  |-  ( (
ph  /\  P  =  M )  ->  (
( abs `  ( P  -  Q )
) ^ 2 )  =  ( ( ( abs `  ( Q  -  M ) ) ^ 2 )  +  ( ( abs `  ( P  -  M )
) ^ 2 ) ) )
3622, 7subcld 9929 . . . . . . . 8  |-  ( ph  ->  ( P  -  M
)  e.  CC )
3736abscld 13229 . . . . . . 7  |-  ( ph  ->  ( abs `  ( P  -  M )
)  e.  RR )
3837recnd 9621 . . . . . 6  |-  ( ph  ->  ( abs `  ( P  -  M )
)  e.  CC )
3938sqcld 12275 . . . . 5  |-  ( ph  ->  ( ( abs `  ( P  -  M )
) ^ 2 )  e.  CC )
4039adantr 465 . . . 4  |-  ( (
ph  /\  Q  =  M )  ->  (
( abs `  ( P  -  M )
) ^ 2 )  e.  CC )
4140addid2d 9779 . . 3  |-  ( (
ph  /\  Q  =  M )  ->  (
0  +  ( ( abs `  ( P  -  M ) ) ^ 2 ) )  =  ( ( abs `  ( P  -  M
) ) ^ 2 ) )
421adantr 465 . . . . . . 7  |-  ( (
ph  /\  Q  =  M )  ->  Q  e.  CC )
43 simpr 461 . . . . . . 7  |-  ( (
ph  /\  Q  =  M )  ->  Q  =  M )
4442, 43subeq0bd 9984 . . . . . 6  |-  ( (
ph  /\  Q  =  M )  ->  ( Q  -  M )  =  0 )
4544abs00bd 13086 . . . . 5  |-  ( (
ph  /\  Q  =  M )  ->  ( abs `  ( Q  -  M ) )  =  0 )
4645sq0id 12228 . . . 4  |-  ( (
ph  /\  Q  =  M )  ->  (
( abs `  ( Q  -  M )
) ^ 2 )  =  0 )
4746oveq1d 6298 . . 3  |-  ( (
ph  /\  Q  =  M )  ->  (
( ( abs `  ( Q  -  M )
) ^ 2 )  +  ( ( abs `  ( P  -  M
) ) ^ 2 ) )  =  ( 0  +  ( ( abs `  ( P  -  M ) ) ^ 2 ) ) )
4843oveq2d 6299 . . . . 5  |-  ( (
ph  /\  Q  =  M )  ->  ( P  -  Q )  =  ( P  -  M ) )
4948fveq2d 5869 . . . 4  |-  ( (
ph  /\  Q  =  M )  ->  ( abs `  ( P  -  Q ) )  =  ( abs `  ( P  -  M )
) )
5049oveq1d 6298 . . 3  |-  ( (
ph  /\  Q  =  M )  ->  (
( abs `  ( P  -  Q )
) ^ 2 )  =  ( ( abs `  ( P  -  M
) ) ^ 2 ) )
5141, 47, 503eqtr4rd 2519 . 2  |-  ( (
ph  /\  Q  =  M )  ->  (
( abs `  ( P  -  Q )
) ^ 2 )  =  ( ( ( abs `  ( Q  -  M ) ) ^ 2 )  +  ( ( abs `  ( P  -  M )
) ^ 2 ) ) )
5222adantr 465 . . 3  |-  ( (
ph  /\  ( P  =/=  M  /\  Q  =/= 
M ) )  ->  P  e.  CC )
531adantr 465 . . 3  |-  ( (
ph  /\  ( P  =/=  M  /\  Q  =/= 
M ) )  ->  Q  e.  CC )
547adantr 465 . . 3  |-  ( (
ph  /\  ( P  =/=  M  /\  Q  =/= 
M ) )  ->  M  e.  CC )
55 simprl 755 . . 3  |-  ( (
ph  /\  ( P  =/=  M  /\  Q  =/= 
M ) )  ->  P  =/=  M )
56 simprr 756 . . 3  |-  ( (
ph  /\  ( P  =/=  M  /\  Q  =/= 
M ) )  ->  Q  =/=  M )
57 eqid 2467 . . . 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 ) ) ) )
583adantr 465 . . . 4  |-  ( (
ph  /\  ( P  =/=  M  /\  Q  =/= 
M ) )  ->  A  e.  CC )
594adantr 465 . . . 4  |-  ( (
ph  /\  ( P  =/=  M  /\  Q  =/= 
M ) )  ->  B  e.  CC )
6015adantr 465 . . . 4  |-  ( (
ph  /\  ( P  =/=  M  /\  Q  =/= 
M ) )  ->  X  e.  RR )
612adantr 465 . . . 4  |-  ( (
ph  /\  ( P  =/=  M  /\  Q  =/= 
M ) )  ->  M  =  ( ( A  +  B )  /  2 ) )
6214adantr 465 . . . 4  |-  ( (
ph  /\  ( P  =/=  M  /\  Q  =/= 
M ) )  ->  P  =  ( ( X  x.  A )  +  ( ( 1  -  X )  x.  B ) ) )
63 chordthmlem3.ABequidistQ . . . . 5  |-  ( ph  ->  ( abs `  ( A  -  Q )
)  =  ( abs `  ( B  -  Q
) ) )
6463adantr 465 . . . 4  |-  ( (
ph  /\  ( P  =/=  M  /\  Q  =/= 
M ) )  -> 
( abs `  ( A  -  Q )
)  =  ( abs `  ( B  -  Q
) ) )
6557, 58, 59, 53, 60, 61, 62, 64, 55, 56chordthmlem2 22908 . . 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 ) } )
66 eqid 2467 . . . 4  |-  ( abs `  ( Q  -  M
) )  =  ( abs `  ( Q  -  M ) )
67 eqid 2467 . . . 4  |-  ( abs `  ( P  -  M
) )  =  ( abs `  ( P  -  M ) )
68 eqid 2467 . . . 4  |-  ( abs `  ( P  -  Q
) )  =  ( abs `  ( P  -  Q ) )
69 eqid 2467 . . . 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 ) )
7057, 66, 67, 68, 69pythag 22893 . . 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 ) ) )
7152, 53, 54, 55, 56, 65, 70syl321anc 1250 . 2  |-  ( (
ph  /\  ( P  =/=  M  /\  Q  =/= 
M ) )  -> 
( ( abs `  ( P  -  Q )
) ^ 2 )  =  ( ( ( abs `  ( Q  -  M ) ) ^ 2 )  +  ( ( abs `  ( P  -  M )
) ^ 2 ) ) )
7235, 51, 71pm2.61da2ne 2786 1  |-  ( ph  ->  ( ( abs `  ( P  -  Q )
) ^ 2 )  =  ( ( ( abs `  ( Q  -  M ) ) ^ 2 )  +  ( ( abs `  ( P  -  M )
) ^ 2 ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = 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   1c1 9492    + caddc 9494    x. cmul 9496    - cmin 9804   -ucneg 9805    / cdiv 10205   2c2 10584   ^cexp 12133   Imcim 12893   abscabs 13029   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:  chordthmlem5  22911
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