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Theorem chordthmlem4 19876
Description: If P is on the segment AB and M is the midpoint of AB, then PA  x. PB = BM2  - PM2. If all lengths are reexpressed as fractions of AB, this reduces to the identity  X  x.  (
1  -  X )  =  ( 1  / 
2 )2  -  ( ( 1  /  2 )  -  X )2. (Contributed by David Moews, 28-Feb-2017.)
Hypotheses
Ref Expression
chordthmlem4.A  |-  ( ph  ->  A  e.  CC )
chordthmlem4.B  |-  ( ph  ->  B  e.  CC )
chordthmlem4.X  |-  ( ph  ->  X  e.  ( 0 [,] 1 ) )
chordthmlem4.M  |-  ( ph  ->  M  =  ( ( A  +  B )  /  2 ) )
chordthmlem4.P  |-  ( ph  ->  P  =  ( ( X  x.  A )  +  ( ( 1  -  X )  x.  B ) ) )
Assertion
Ref Expression
chordthmlem4  |-  ( ph  ->  ( ( abs `  ( P  -  A )
)  x.  ( abs `  ( P  -  B
) ) )  =  ( ( ( abs `  ( B  -  M
) ) ^ 2 )  -  ( ( abs `  ( P  -  M ) ) ^ 2 ) ) )

Proof of Theorem chordthmlem4
StepHypRef Expression
1 1re 8717 . . . . . . . . 9  |-  1  e.  RR
21a1i 12 . . . . . . . 8  |-  ( ph  ->  1  e.  RR )
3 unitssre 10659 . . . . . . . . 9  |-  ( 0 [,] 1 )  C_  RR
4 chordthmlem4.X . . . . . . . . 9  |-  ( ph  ->  X  e.  ( 0 [,] 1 ) )
53, 4sseldi 3101 . . . . . . . 8  |-  ( ph  ->  X  e.  RR )
62, 5resubcld 9091 . . . . . . 7  |-  ( ph  ->  ( 1  -  X
)  e.  RR )
76recnd 8741 . . . . . 6  |-  ( ph  ->  ( 1  -  X
)  e.  CC )
87abscld 11795 . . . . 5  |-  ( ph  ->  ( abs `  (
1  -  X ) )  e.  RR )
98recnd 8741 . . . 4  |-  ( ph  ->  ( abs `  (
1  -  X ) )  e.  CC )
10 chordthmlem4.B . . . . . . 7  |-  ( ph  ->  B  e.  CC )
11 chordthmlem4.A . . . . . . 7  |-  ( ph  ->  A  e.  CC )
1210, 11subcld 9037 . . . . . 6  |-  ( ph  ->  ( B  -  A
)  e.  CC )
1312abscld 11795 . . . . 5  |-  ( ph  ->  ( abs `  ( B  -  A )
)  e.  RR )
1413recnd 8741 . . . 4  |-  ( ph  ->  ( abs `  ( B  -  A )
)  e.  CC )
155recnd 8741 . . . . . 6  |-  ( ph  ->  X  e.  CC )
1615abscld 11795 . . . . 5  |-  ( ph  ->  ( abs `  X
)  e.  RR )
1716recnd 8741 . . . 4  |-  ( ph  ->  ( abs `  X
)  e.  CC )
189, 14, 17, 14mul4d 8904 . . 3  |-  ( ph  ->  ( ( ( abs `  ( 1  -  X
) )  x.  ( abs `  ( B  -  A ) ) )  x.  ( ( abs `  X )  x.  ( abs `  ( B  -  A ) ) ) )  =  ( ( ( abs `  (
1  -  X ) )  x.  ( abs `  X ) )  x.  ( ( abs `  ( B  -  A )
)  x.  ( abs `  ( B  -  A
) ) ) ) )
19 chordthmlem4.P . . . . . . 7  |-  ( ph  ->  P  =  ( ( X  x.  A )  +  ( ( 1  -  X )  x.  B ) ) )
2015, 11mulcld 8735 . . . . . . . . . 10  |-  ( ph  ->  ( X  x.  A
)  e.  CC )
217, 10mulcld 8735 . . . . . . . . . 10  |-  ( ph  ->  ( ( 1  -  X )  x.  B
)  e.  CC )
2220, 21addcld 8734 . . . . . . . . 9  |-  ( ph  ->  ( ( X  x.  A )  +  ( ( 1  -  X
)  x.  B ) )  e.  CC )
2319, 22eqeltrd 2327 . . . . . . . 8  |-  ( ph  ->  P  e.  CC )
2411, 23, 10, 15affineequiv2 19868 . . . . . . 7  |-  ( ph  ->  ( P  =  ( ( X  x.  A
)  +  ( ( 1  -  X )  x.  B ) )  <-> 
( P  -  A
)  =  ( ( 1  -  X )  x.  ( B  -  A ) ) ) )
2519, 24mpbid 203 . . . . . 6  |-  ( ph  ->  ( P  -  A
)  =  ( ( 1  -  X )  x.  ( B  -  A ) ) )
2625fveq2d 5381 . . . . 5  |-  ( ph  ->  ( abs `  ( P  -  A )
)  =  ( abs `  ( ( 1  -  X )  x.  ( B  -  A )
) ) )
277, 12absmuld 11813 . . . . 5  |-  ( ph  ->  ( abs `  (
( 1  -  X
)  x.  ( B  -  A ) ) )  =  ( ( abs `  ( 1  -  X ) )  x.  ( abs `  ( B  -  A )
) ) )
2826, 27eqtrd 2285 . . . 4  |-  ( ph  ->  ( abs `  ( P  -  A )
)  =  ( ( abs `  ( 1  -  X ) )  x.  ( abs `  ( B  -  A )
) ) )
2923, 10abssubd 11812 . . . . 5  |-  ( ph  ->  ( abs `  ( P  -  B )
)  =  ( abs `  ( B  -  P
) ) )
3011, 23, 10, 15affineequiv 19867 . . . . . . 7  |-  ( ph  ->  ( P  =  ( ( X  x.  A
)  +  ( ( 1  -  X )  x.  B ) )  <-> 
( B  -  P
)  =  ( X  x.  ( B  -  A ) ) ) )
3119, 30mpbid 203 . . . . . 6  |-  ( ph  ->  ( B  -  P
)  =  ( X  x.  ( B  -  A ) ) )
3231fveq2d 5381 . . . . 5  |-  ( ph  ->  ( abs `  ( B  -  P )
)  =  ( abs `  ( X  x.  ( B  -  A )
) ) )
3315, 12absmuld 11813 . . . . 5  |-  ( ph  ->  ( abs `  ( X  x.  ( B  -  A ) ) )  =  ( ( abs `  X )  x.  ( abs `  ( B  -  A ) ) ) )
3429, 32, 333eqtrd 2289 . . . 4  |-  ( ph  ->  ( abs `  ( P  -  B )
)  =  ( ( abs `  X )  x.  ( abs `  ( B  -  A )
) ) )
3528, 34oveq12d 5728 . . 3  |-  ( ph  ->  ( ( abs `  ( P  -  A )
)  x.  ( abs `  ( P  -  B
) ) )  =  ( ( ( abs `  ( 1  -  X
) )  x.  ( abs `  ( B  -  A ) ) )  x.  ( ( abs `  X )  x.  ( abs `  ( B  -  A ) ) ) ) )
3614sqvald 11120 . . . 4  |-  ( ph  ->  ( ( abs `  ( B  -  A )
) ^ 2 )  =  ( ( abs `  ( B  -  A
) )  x.  ( abs `  ( B  -  A ) ) ) )
3736oveq2d 5726 . . 3  |-  ( ph  ->  ( ( ( abs `  ( 1  -  X
) )  x.  ( abs `  X ) )  x.  ( ( abs `  ( B  -  A
) ) ^ 2 ) )  =  ( ( ( abs `  (
1  -  X ) )  x.  ( abs `  X ) )  x.  ( ( abs `  ( B  -  A )
)  x.  ( abs `  ( B  -  A
) ) ) ) )
3818, 35, 373eqtr4d 2295 . 2  |-  ( ph  ->  ( ( abs `  ( P  -  A )
)  x.  ( abs `  ( P  -  B
) ) )  =  ( ( ( abs `  ( 1  -  X
) )  x.  ( abs `  X ) )  x.  ( ( abs `  ( B  -  A
) ) ^ 2 ) ) )
392recnd 8741 . . . . . 6  |-  ( ph  ->  1  e.  CC )
4039halfcld 9835 . . . . 5  |-  ( ph  ->  ( 1  /  2
)  e.  CC )
4140sqcld 11121 . . . 4  |-  ( ph  ->  ( ( 1  / 
2 ) ^ 2 )  e.  CC )
422rehalfcld 9837 . . . . . . . . 9  |-  ( ph  ->  ( 1  /  2
)  e.  RR )
4342, 5resubcld 9091 . . . . . . . 8  |-  ( ph  ->  ( ( 1  / 
2 )  -  X
)  e.  RR )
4443recnd 8741 . . . . . . 7  |-  ( ph  ->  ( ( 1  / 
2 )  -  X
)  e.  CC )
4544abscld 11795 . . . . . 6  |-  ( ph  ->  ( abs `  (
( 1  /  2
)  -  X ) )  e.  RR )
4645recnd 8741 . . . . 5  |-  ( ph  ->  ( abs `  (
( 1  /  2
)  -  X ) )  e.  CC )
4746sqcld 11121 . . . 4  |-  ( ph  ->  ( ( abs `  (
( 1  /  2
)  -  X ) ) ^ 2 )  e.  CC )
4814sqcld 11121 . . . 4  |-  ( ph  ->  ( ( abs `  ( B  -  A )
) ^ 2 )  e.  CC )
4941, 47, 48subdird 9116 . . 3  |-  ( ph  ->  ( ( ( ( 1  /  2 ) ^ 2 )  -  ( ( abs `  (
( 1  /  2
)  -  X ) ) ^ 2 ) )  x.  ( ( abs `  ( B  -  A ) ) ^ 2 ) )  =  ( ( ( ( 1  /  2
) ^ 2 )  x.  ( ( abs `  ( B  -  A
) ) ^ 2 ) )  -  (
( ( abs `  (
( 1  /  2
)  -  X ) ) ^ 2 )  x.  ( ( abs `  ( B  -  A
) ) ^ 2 ) ) ) )
50 subsq 11088 . . . . . . 7  |-  ( ( ( 1  /  2
)  e.  CC  /\  ( ( 1  / 
2 )  -  X
)  e.  CC )  ->  ( ( ( 1  /  2 ) ^ 2 )  -  ( ( ( 1  /  2 )  -  X ) ^ 2 ) )  =  ( ( ( 1  / 
2 )  +  ( ( 1  /  2
)  -  X ) )  x.  ( ( 1  /  2 )  -  ( ( 1  /  2 )  -  X ) ) ) )
5140, 44, 50syl2anc 645 . . . . . 6  |-  ( ph  ->  ( ( ( 1  /  2 ) ^
2 )  -  (
( ( 1  / 
2 )  -  X
) ^ 2 ) )  =  ( ( ( 1  /  2
)  +  ( ( 1  /  2 )  -  X ) )  x.  ( ( 1  /  2 )  -  ( ( 1  / 
2 )  -  X
) ) ) )
5240, 40, 15addsubassd 9057 . . . . . . . 8  |-  ( ph  ->  ( ( ( 1  /  2 )  +  ( 1  /  2
) )  -  X
)  =  ( ( 1  /  2 )  +  ( ( 1  /  2 )  -  X ) ) )
53392halvesd 9836 . . . . . . . . 9  |-  ( ph  ->  ( ( 1  / 
2 )  +  ( 1  /  2 ) )  =  1 )
5453oveq1d 5725 . . . . . . . 8  |-  ( ph  ->  ( ( ( 1  /  2 )  +  ( 1  /  2
) )  -  X
)  =  ( 1  -  X ) )
5552, 54eqtr3d 2287 . . . . . . 7  |-  ( ph  ->  ( ( 1  / 
2 )  +  ( ( 1  /  2
)  -  X ) )  =  ( 1  -  X ) )
5640, 15nncand 9042 . . . . . . 7  |-  ( ph  ->  ( ( 1  / 
2 )  -  (
( 1  /  2
)  -  X ) )  =  X )
5755, 56oveq12d 5728 . . . . . 6  |-  ( ph  ->  ( ( ( 1  /  2 )  +  ( ( 1  / 
2 )  -  X
) )  x.  (
( 1  /  2
)  -  ( ( 1  /  2 )  -  X ) ) )  =  ( ( 1  -  X )  x.  X ) )
5851, 57eqtr2d 2286 . . . . 5  |-  ( ph  ->  ( ( 1  -  X )  x.  X
)  =  ( ( ( 1  /  2
) ^ 2 )  -  ( ( ( 1  /  2 )  -  X ) ^
2 ) ) )
59 0re 8718 . . . . . . . . . 10  |-  0  e.  RR
6059, 1elicc2i 10594 . . . . . . . . 9  |-  ( X  e.  ( 0 [,] 1 )  <->  ( X  e.  RR  /\  0  <_  X  /\  X  <_  1
) )
614, 60sylib 190 . . . . . . . 8  |-  ( ph  ->  ( X  e.  RR  /\  0  <_  X  /\  X  <_  1 ) )
6261simp3d 974 . . . . . . 7  |-  ( ph  ->  X  <_  1 )
635, 2, 62abssubge0d 11791 . . . . . 6  |-  ( ph  ->  ( abs `  (
1  -  X ) )  =  ( 1  -  X ) )
6461simp2d 973 . . . . . . 7  |-  ( ph  ->  0  <_  X )
655, 64absidd 11782 . . . . . 6  |-  ( ph  ->  ( abs `  X
)  =  X )
6663, 65oveq12d 5728 . . . . 5  |-  ( ph  ->  ( ( abs `  (
1  -  X ) )  x.  ( abs `  X ) )  =  ( ( 1  -  X )  x.  X
) )
67 absresq 11664 . . . . . . 7  |-  ( ( ( 1  /  2
)  -  X )  e.  RR  ->  (
( abs `  (
( 1  /  2
)  -  X ) ) ^ 2 )  =  ( ( ( 1  /  2 )  -  X ) ^
2 ) )
6843, 67syl 17 . . . . . 6  |-  ( ph  ->  ( ( abs `  (
( 1  /  2
)  -  X ) ) ^ 2 )  =  ( ( ( 1  /  2 )  -  X ) ^
2 ) )
6968oveq2d 5726 . . . . 5  |-  ( ph  ->  ( ( ( 1  /  2 ) ^
2 )  -  (
( abs `  (
( 1  /  2
)  -  X ) ) ^ 2 ) )  =  ( ( ( 1  /  2
) ^ 2 )  -  ( ( ( 1  /  2 )  -  X ) ^
2 ) ) )
7058, 66, 693eqtr4d 2295 . . . 4  |-  ( ph  ->  ( ( abs `  (
1  -  X ) )  x.  ( abs `  X ) )  =  ( ( ( 1  /  2 ) ^
2 )  -  (
( abs `  (
( 1  /  2
)  -  X ) ) ^ 2 ) ) )
7170oveq1d 5725 . . 3  |-  ( ph  ->  ( ( ( abs `  ( 1  -  X
) )  x.  ( abs `  X ) )  x.  ( ( abs `  ( B  -  A
) ) ^ 2 ) )  =  ( ( ( ( 1  /  2 ) ^
2 )  -  (
( abs `  (
( 1  /  2
)  -  X ) ) ^ 2 ) )  x.  ( ( abs `  ( B  -  A ) ) ^ 2 ) ) )
72 2cn 9696 . . . . . . . . . . . . . 14  |-  2  e.  CC
7372a1i 12 . . . . . . . . . . . . 13  |-  ( ph  ->  2  e.  CC )
74 2ne0 9709 . . . . . . . . . . . . . 14  |-  2  =/=  0
7574a1i 12 . . . . . . . . . . . . 13  |-  ( ph  ->  2  =/=  0 )
7610, 73, 75divcan4d 9422 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( B  x.  2 )  /  2
)  =  B )
7710times2d 9834 . . . . . . . . . . . . 13  |-  ( ph  ->  ( B  x.  2 )  =  ( B  +  B ) )
7877oveq1d 5725 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( B  x.  2 )  /  2
)  =  ( ( B  +  B )  /  2 ) )
7976, 78eqtr3d 2287 . . . . . . . . . . 11  |-  ( ph  ->  B  =  ( ( B  +  B )  /  2 ) )
80 chordthmlem4.M . . . . . . . . . . 11  |-  ( ph  ->  M  =  ( ( A  +  B )  /  2 ) )
8179, 80oveq12d 5728 . . . . . . . . . 10  |-  ( ph  ->  ( B  -  M
)  =  ( ( ( B  +  B
)  /  2 )  -  ( ( A  +  B )  / 
2 ) ) )
8210, 10addcld 8734 . . . . . . . . . . 11  |-  ( ph  ->  ( B  +  B
)  e.  CC )
8311, 10addcld 8734 . . . . . . . . . . 11  |-  ( ph  ->  ( A  +  B
)  e.  CC )
8482, 83, 73, 75divsubdird 9455 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( B  +  B )  -  ( A  +  B
) )  /  2
)  =  ( ( ( B  +  B
)  /  2 )  -  ( ( A  +  B )  / 
2 ) ) )
8510, 11, 10pnpcan2d 9075 . . . . . . . . . . 11  |-  ( ph  ->  ( ( B  +  B )  -  ( A  +  B )
)  =  ( B  -  A ) )
8685oveq1d 5725 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( B  +  B )  -  ( A  +  B
) )  /  2
)  =  ( ( B  -  A )  /  2 ) )
8781, 84, 863eqtr2d 2291 . . . . . . . . 9  |-  ( ph  ->  ( B  -  M
)  =  ( ( B  -  A )  /  2 ) )
8812, 73, 75divrec2d 9420 . . . . . . . . 9  |-  ( ph  ->  ( ( B  -  A )  /  2
)  =  ( ( 1  /  2 )  x.  ( B  -  A ) ) )
8987, 88eqtrd 2285 . . . . . . . 8  |-  ( ph  ->  ( B  -  M
)  =  ( ( 1  /  2 )  x.  ( B  -  A ) ) )
9089fveq2d 5381 . . . . . . 7  |-  ( ph  ->  ( abs `  ( B  -  M )
)  =  ( abs `  ( ( 1  / 
2 )  x.  ( B  -  A )
) ) )
9140, 12absmuld 11813 . . . . . . 7  |-  ( ph  ->  ( abs `  (
( 1  /  2
)  x.  ( B  -  A ) ) )  =  ( ( abs `  ( 1  /  2 ) )  x.  ( abs `  ( B  -  A )
) ) )
9259a1i 12 . . . . . . . . . 10  |-  ( ph  ->  0  e.  RR )
93 halfgt0 9811 . . . . . . . . . . 11  |-  0  <  ( 1  /  2
)
9493a1i 12 . . . . . . . . . 10  |-  ( ph  ->  0  <  ( 1  /  2 ) )
9592, 42, 94ltled 8847 . . . . . . . . 9  |-  ( ph  ->  0  <_  ( 1  /  2 ) )
9642, 95absidd 11782 . . . . . . . 8  |-  ( ph  ->  ( abs `  (
1  /  2 ) )  =  ( 1  /  2 ) )
9796oveq1d 5725 . . . . . . 7  |-  ( ph  ->  ( ( abs `  (
1  /  2 ) )  x.  ( abs `  ( B  -  A
) ) )  =  ( ( 1  / 
2 )  x.  ( abs `  ( B  -  A ) ) ) )
9890, 91, 973eqtrd 2289 . . . . . 6  |-  ( ph  ->  ( abs `  ( B  -  M )
)  =  ( ( 1  /  2 )  x.  ( abs `  ( B  -  A )
) ) )
9998oveq1d 5725 . . . . 5  |-  ( ph  ->  ( ( abs `  ( B  -  M )
) ^ 2 )  =  ( ( ( 1  /  2 )  x.  ( abs `  ( B  -  A )
) ) ^ 2 ) )
10040, 14sqmuld 11135 . . . . 5  |-  ( ph  ->  ( ( ( 1  /  2 )  x.  ( abs `  ( B  -  A )
) ) ^ 2 )  =  ( ( ( 1  /  2
) ^ 2 )  x.  ( ( abs `  ( B  -  A
) ) ^ 2 ) ) )
10199, 100eqtrd 2285 . . . 4  |-  ( ph  ->  ( ( abs `  ( B  -  M )
) ^ 2 )  =  ( ( ( 1  /  2 ) ^ 2 )  x.  ( ( abs `  ( B  -  A )
) ^ 2 ) ) )
10240, 15, 12subdird 9116 . . . . . . . . 9  |-  ( ph  ->  ( ( ( 1  /  2 )  -  X )  x.  ( B  -  A )
)  =  ( ( ( 1  /  2
)  x.  ( B  -  A ) )  -  ( X  x.  ( B  -  A
) ) ) )
10389, 31oveq12d 5728 . . . . . . . . 9  |-  ( ph  ->  ( ( B  -  M )  -  ( B  -  P )
)  =  ( ( ( 1  /  2
)  x.  ( B  -  A ) )  -  ( X  x.  ( B  -  A
) ) ) )
10483halfcld 9835 . . . . . . . . . . 11  |-  ( ph  ->  ( ( A  +  B )  /  2
)  e.  CC )
10580, 104eqeltrd 2327 . . . . . . . . . 10  |-  ( ph  ->  M  e.  CC )
10610, 105, 23nnncan1d 9071 . . . . . . . . 9  |-  ( ph  ->  ( ( B  -  M )  -  ( B  -  P )
)  =  ( P  -  M ) )
107102, 103, 1063eqtr2rd 2292 . . . . . . . 8  |-  ( ph  ->  ( P  -  M
)  =  ( ( ( 1  /  2
)  -  X )  x.  ( B  -  A ) ) )
108107fveq2d 5381 . . . . . . 7  |-  ( ph  ->  ( abs `  ( P  -  M )
)  =  ( abs `  ( ( ( 1  /  2 )  -  X )  x.  ( B  -  A )
) ) )
10944, 12absmuld 11813 . . . . . . 7  |-  ( ph  ->  ( abs `  (
( ( 1  / 
2 )  -  X
)  x.  ( B  -  A ) ) )  =  ( ( abs `  ( ( 1  /  2 )  -  X ) )  x.  ( abs `  ( B  -  A )
) ) )
110108, 109eqtrd 2285 . . . . . 6  |-  ( ph  ->  ( abs `  ( P  -  M )
)  =  ( ( abs `  ( ( 1  /  2 )  -  X ) )  x.  ( abs `  ( B  -  A )
) ) )
111110oveq1d 5725 . . . . 5  |-  ( ph  ->  ( ( abs `  ( P  -  M )
) ^ 2 )  =  ( ( ( abs `  ( ( 1  /  2 )  -  X ) )  x.  ( abs `  ( B  -  A )
) ) ^ 2 ) )
11246, 14sqmuld 11135 . . . . 5  |-  ( ph  ->  ( ( ( abs `  ( ( 1  / 
2 )  -  X
) )  x.  ( abs `  ( B  -  A ) ) ) ^ 2 )  =  ( ( ( abs `  ( ( 1  / 
2 )  -  X
) ) ^ 2 )  x.  ( ( abs `  ( B  -  A ) ) ^ 2 ) ) )
113111, 112eqtrd 2285 . . . 4  |-  ( ph  ->  ( ( abs `  ( P  -  M )
) ^ 2 )  =  ( ( ( abs `  ( ( 1  /  2 )  -  X ) ) ^ 2 )  x.  ( ( abs `  ( B  -  A )
) ^ 2 ) ) )
114101, 113oveq12d 5728 . . 3  |-  ( ph  ->  ( ( ( abs `  ( B  -  M
) ) ^ 2 )  -  ( ( abs `  ( P  -  M ) ) ^ 2 ) )  =  ( ( ( ( 1  /  2
) ^ 2 )  x.  ( ( abs `  ( B  -  A
) ) ^ 2 ) )  -  (
( ( abs `  (
( 1  /  2
)  -  X ) ) ^ 2 )  x.  ( ( abs `  ( B  -  A
) ) ^ 2 ) ) ) )
11549, 71, 1143eqtr4rd 2296 . 2  |-  ( ph  ->  ( ( ( abs `  ( B  -  M
) ) ^ 2 )  -  ( ( abs `  ( P  -  M ) ) ^ 2 ) )  =  ( ( ( abs `  ( 1  -  X ) )  x.  ( abs `  X
) )  x.  (
( abs `  ( B  -  A )
) ^ 2 ) ) )
11638, 115eqtr4d 2288 1  |-  ( ph  ->  ( ( abs `  ( P  -  A )
)  x.  ( abs `  ( P  -  B
) ) )  =  ( ( ( abs `  ( B  -  M
) ) ^ 2 )  -  ( ( abs `  ( P  -  M ) ) ^ 2 ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ w3a 939    = wceq 1619    e. wcel 1621    =/= wne 2412   class class class wbr 3920   ` cfv 4592  (class class class)co 5710   CCcc 8615   RRcr 8616   0cc0 8617   1c1 8618    + caddc 8620    x. cmul 8622    < clt 8747    <_ cle 8748    - cmin 8917    / cdiv 9303   2c2 9675   [,]cicc 10537   ^cexp 10982   abscabs 11596
This theorem is referenced by:  chordthmlem5  19877
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 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-cnex 8673  ax-resscn 8674  ax-1cn 8675  ax-icn 8676  ax-addcl 8677  ax-addrcl 8678  ax-mulcl 8679  ax-mulrcl 8680  ax-mulcom 8681  ax-addass 8682  ax-mulass 8683  ax-distr 8684  ax-i2m1 8685  ax-1ne0 8686  ax-1rid 8687  ax-rnegex 8688  ax-rrecex 8689  ax-cnre 8690  ax-pre-lttri 8691  ax-pre-lttrn 8692  ax-pre-ltadd 8693  ax-pre-mulgt0 8694  ax-pre-sup 8695
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-2nd 5975  df-iota 6143  df-riota 6190  df-recs 6274  df-rdg 6309  df-er 6546  df-en 6750  df-dom 6751  df-sdom 6752  df-sup 7078  df-pnf 8749  df-mnf 8750  df-xr 8751  df-ltxr 8752  df-le 8753  df-sub 8919  df-neg 8920  df-div 9304  df-n 9627  df-2 9684  df-3 9685  df-n0 9845  df-z 9904  df-uz 10110  df-rp 10234  df-icc 10541  df-seq 10925  df-exp 10983  df-cj 11461  df-re 11462  df-im 11463  df-sqr 11597  df-abs 11598
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