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Theorem chordthmALT 37369
Description: The intersecting chords theorem. If points A, B, C, and D lie on a circle (with center Q, say), and the point P is on the interior of the segments AB and CD, then the two products of lengths PA  x. PB and PC  x. PD are equal. The Euclidean plane is identified with the complex plane, and the fact that P is on AB and on CD is expressed by the hypothesis that the angles APB and CPD are equal to  pi. The result is proven by using chordthmlem5 23810 twice to show that PA  x. PB and PC  x. PD both equal BQ 2  - PQ 2 . This is similar to the proof of the theorem given in Euclid's Elements, where it is Proposition III.35. Proven by David Moews on 28-Feb-2017 as chordthm 23811. http://us.metamath.org/other/completeusersproof/chordthmaltvd.html is a Virtual Deduction User's Proof transcription of chordthm 23811. That VD User's Proof was input into completeusersproof, automatically generating this chordthmALT 37369 Metamath proof. (Contributed by Alan Sare, 19-Sep-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
chordthmALT.angdef  |-  F  =  ( x  e.  ( CC  \  { 0 } ) ,  y  e.  ( CC  \  { 0 } ) 
|->  ( Im `  ( log `  ( y  /  x ) ) ) )
chordthmALT.A  |-  ( ph  ->  A  e.  CC )
chordthmALT.B  |-  ( ph  ->  B  e.  CC )
chordthmALT.C  |-  ( ph  ->  C  e.  CC )
chordthmALT.D  |-  ( ph  ->  D  e.  CC )
chordthmALT.P  |-  ( ph  ->  P  e.  CC )
chordthmALT.AneP  |-  ( ph  ->  A  =/=  P )
chordthmALT.BneP  |-  ( ph  ->  B  =/=  P )
chordthmALT.CneP  |-  ( ph  ->  C  =/=  P )
chordthmALT.DneP  |-  ( ph  ->  D  =/=  P )
chordthmALT.APB  |-  ( ph  ->  ( ( A  -  P ) F ( B  -  P ) )  =  pi )
chordthmALT.CPD  |-  ( ph  ->  ( ( C  -  P ) F ( D  -  P ) )  =  pi )
chordthmALT.Q  |-  ( ph  ->  Q  e.  CC )
chordthmALT.ABcirc  |-  ( ph  ->  ( abs `  ( A  -  Q )
)  =  ( abs `  ( B  -  Q
) ) )
chordthmALT.ACcirc  |-  ( ph  ->  ( abs `  ( A  -  Q )
)  =  ( abs `  ( C  -  Q
) ) )
chordthmALT.ADcirc  |-  ( ph  ->  ( abs `  ( A  -  Q )
)  =  ( abs `  ( D  -  Q
) ) )
Assertion
Ref Expression
chordthmALT  |-  ( ph  ->  ( ( abs `  ( P  -  A )
)  x.  ( abs `  ( P  -  B
) ) )  =  ( ( abs `  ( P  -  C )
)  x.  ( abs `  ( P  -  D
) ) ) )
Distinct variable groups:    x, A, y    x, B, y    x, C, y    x, D, y   
x, P, y
Allowed substitution hints:    ph( x, y)    Q( x, y)    F( x, y)

Proof of Theorem chordthmALT
Dummy variables  v  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 chordthmALT.CPD . . . 4  |-  ( ph  ->  ( ( C  -  P ) F ( D  -  P ) )  =  pi )
2 chordthmALT.angdef . . . . 5  |-  F  =  ( x  e.  ( CC  \  { 0 } ) ,  y  e.  ( CC  \  { 0 } ) 
|->  ( Im `  ( log `  ( y  /  x ) ) ) )
3 chordthmALT.C . . . . 5  |-  ( ph  ->  C  e.  CC )
4 chordthmALT.P . . . . 5  |-  ( ph  ->  P  e.  CC )
5 chordthmALT.D . . . . 5  |-  ( ph  ->  D  e.  CC )
6 chordthmALT.CneP . . . . 5  |-  ( ph  ->  C  =/=  P )
7 chordthmALT.DneP . . . . . 6  |-  ( ph  ->  D  =/=  P )
87necomd 2690 . . . . 5  |-  ( ph  ->  P  =/=  D )
92, 3, 4, 5, 6, 8angpieqvd 23805 . . . 4  |-  ( ph  ->  ( ( ( C  -  P ) F ( D  -  P
) )  =  pi  <->  E. v  e.  ( 0 (,) 1 ) P  =  ( ( v  x.  C )  +  ( ( 1  -  v )  x.  D
) ) ) )
101, 9mpbid 215 . . 3  |-  ( ph  ->  E. v  e.  ( 0 (,) 1 ) P  =  ( ( v  x.  C )  +  ( ( 1  -  v )  x.  D ) ) )
11 df-rex 2754 . . . 4  |-  ( E. v  e.  ( 0 (,) 1 ) P  =  ( ( v  x.  C )  +  ( ( 1  -  v )  x.  D
) )  <->  E. v
( v  e.  ( 0 (,) 1 )  /\  P  =  ( ( v  x.  C
)  +  ( ( 1  -  v )  x.  D ) ) ) )
1211biimpi 199 . . 3  |-  ( E. v  e.  ( 0 (,) 1 ) P  =  ( ( v  x.  C )  +  ( ( 1  -  v )  x.  D
) )  ->  E. v
( v  e.  ( 0 (,) 1 )  /\  P  =  ( ( v  x.  C
)  +  ( ( 1  -  v )  x.  D ) ) ) )
1310, 12syl 17 . 2  |-  ( ph  ->  E. v ( v  e.  ( 0 (,) 1 )  /\  P  =  ( ( v  x.  C )  +  ( ( 1  -  v )  x.  D
) ) ) )
14 chordthmALT.APB . . . . . . . 8  |-  ( ph  ->  ( ( A  -  P ) F ( B  -  P ) )  =  pi )
15 chordthmALT.A . . . . . . . . 9  |-  ( ph  ->  A  e.  CC )
16 chordthmALT.B . . . . . . . . 9  |-  ( ph  ->  B  e.  CC )
17 chordthmALT.AneP . . . . . . . . 9  |-  ( ph  ->  A  =/=  P )
18 chordthmALT.BneP . . . . . . . . . 10  |-  ( ph  ->  B  =/=  P )
1918necomd 2690 . . . . . . . . 9  |-  ( ph  ->  P  =/=  B )
202, 15, 4, 16, 17, 19angpieqvd 23805 . . . . . . . 8  |-  ( ph  ->  ( ( ( A  -  P ) F ( B  -  P
) )  =  pi  <->  E. w  e.  ( 0 (,) 1 ) P  =  ( ( w  x.  A )  +  ( ( 1  -  w )  x.  B
) ) ) )
2114, 20mpbid 215 . . . . . . 7  |-  ( ph  ->  E. w  e.  ( 0 (,) 1 ) P  =  ( ( w  x.  A )  +  ( ( 1  -  w )  x.  B ) ) )
22 df-rex 2754 . . . . . . . 8  |-  ( E. w  e.  ( 0 (,) 1 ) P  =  ( ( w  x.  A )  +  ( ( 1  -  w )  x.  B
) )  <->  E. w
( w  e.  ( 0 (,) 1 )  /\  P  =  ( ( w  x.  A
)  +  ( ( 1  -  w )  x.  B ) ) ) )
2322biimpi 199 . . . . . . 7  |-  ( E. w  e.  ( 0 (,) 1 ) P  =  ( ( w  x.  A )  +  ( ( 1  -  w )  x.  B
) )  ->  E. w
( w  e.  ( 0 (,) 1 )  /\  P  =  ( ( w  x.  A
)  +  ( ( 1  -  w )  x.  B ) ) ) )
2421, 23syl 17 . . . . . 6  |-  ( ph  ->  E. w ( w  e.  ( 0 (,) 1 )  /\  P  =  ( ( w  x.  A )  +  ( ( 1  -  w )  x.  B
) ) ) )
2524adantr 471 . . . . 5  |-  ( (
ph  /\  ( v  e.  ( 0 (,) 1
)  /\  P  =  ( ( v  x.  C )  +  ( ( 1  -  v
)  x.  D ) ) ) )  ->  E. w ( w  e.  ( 0 (,) 1
)  /\  P  =  ( ( w  x.  A )  +  ( ( 1  -  w
)  x.  B ) ) ) )
26 chordthmALT.ABcirc . . . . . . . . . . . 12  |-  ( ph  ->  ( abs `  ( A  -  Q )
)  =  ( abs `  ( B  -  Q
) ) )
27 chordthmALT.ADcirc . . . . . . . . . . . 12  |-  ( ph  ->  ( abs `  ( A  -  Q )
)  =  ( abs `  ( D  -  Q
) ) )
2826, 27eqtr3d 2497 . . . . . . . . . . 11  |-  ( ph  ->  ( abs `  ( B  -  Q )
)  =  ( abs `  ( D  -  Q
) ) )
2928oveq1d 6329 . . . . . . . . . 10  |-  ( ph  ->  ( ( abs `  ( B  -  Q )
) ^ 2 )  =  ( ( abs `  ( D  -  Q
) ) ^ 2 ) )
3029oveq1d 6329 . . . . . . . . 9  |-  ( ph  ->  ( ( ( abs `  ( B  -  Q
) ) ^ 2 )  -  ( ( abs `  ( P  -  Q ) ) ^ 2 ) )  =  ( ( ( abs `  ( D  -  Q ) ) ^ 2 )  -  ( ( abs `  ( P  -  Q )
) ^ 2 ) ) )
31303ad2ant1 1035 . . . . . . . 8  |-  ( (
ph  /\  ( v  e.  ( 0 (,) 1
)  /\  P  =  ( ( v  x.  C )  +  ( ( 1  -  v
)  x.  D ) ) )  /\  (
w  e.  ( 0 (,) 1 )  /\  P  =  ( (
w  x.  A )  +  ( ( 1  -  w )  x.  B ) ) ) )  ->  ( (
( abs `  ( B  -  Q )
) ^ 2 )  -  ( ( abs `  ( P  -  Q
) ) ^ 2 ) )  =  ( ( ( abs `  ( D  -  Q )
) ^ 2 )  -  ( ( abs `  ( P  -  Q
) ) ^ 2 ) ) )
32153ad2ant1 1035 . . . . . . . . . . 11  |-  ( (
ph  /\  w  e.  ( 0 (,) 1
)  /\  P  =  ( ( w  x.  A )  +  ( ( 1  -  w
)  x.  B ) ) )  ->  A  e.  CC )
33163ad2ant1 1035 . . . . . . . . . . 11  |-  ( (
ph  /\  w  e.  ( 0 (,) 1
)  /\  P  =  ( ( w  x.  A )  +  ( ( 1  -  w
)  x.  B ) ) )  ->  B  e.  CC )
34 chordthmALT.Q . . . . . . . . . . . 12  |-  ( ph  ->  Q  e.  CC )
35343ad2ant1 1035 . . . . . . . . . . 11  |-  ( (
ph  /\  w  e.  ( 0 (,) 1
)  /\  P  =  ( ( w  x.  A )  +  ( ( 1  -  w
)  x.  B ) ) )  ->  Q  e.  CC )
36 ioossicc 11748 . . . . . . . . . . . . 13  |-  ( 0 (,) 1 )  C_  ( 0 [,] 1
)
37 id 22 . . . . . . . . . . . . 13  |-  ( w  e.  ( 0 (,) 1 )  ->  w  e.  ( 0 (,) 1
) )
3836, 37sseldi 3441 . . . . . . . . . . . 12  |-  ( w  e.  ( 0 (,) 1 )  ->  w  e.  ( 0 [,] 1
) )
39383ad2ant2 1036 . . . . . . . . . . 11  |-  ( (
ph  /\  w  e.  ( 0 (,) 1
)  /\  P  =  ( ( w  x.  A )  +  ( ( 1  -  w
)  x.  B ) ) )  ->  w  e.  ( 0 [,] 1
) )
40 id 22 . . . . . . . . . . . 12  |-  ( P  =  ( ( w  x.  A )  +  ( ( 1  -  w )  x.  B
) )  ->  P  =  ( ( w  x.  A )  +  ( ( 1  -  w )  x.  B
) ) )
41403ad2ant3 1037 . . . . . . . . . . 11  |-  ( (
ph  /\  w  e.  ( 0 (,) 1
)  /\  P  =  ( ( w  x.  A )  +  ( ( 1  -  w
)  x.  B ) ) )  ->  P  =  ( ( w  x.  A )  +  ( ( 1  -  w )  x.  B
) ) )
42263ad2ant1 1035 . . . . . . . . . . 11  |-  ( (
ph  /\  w  e.  ( 0 (,) 1
)  /\  P  =  ( ( w  x.  A )  +  ( ( 1  -  w
)  x.  B ) ) )  ->  ( abs `  ( A  -  Q ) )  =  ( abs `  ( B  -  Q )
) )
4332, 33, 35, 39, 41, 42chordthmlem5 23810 . . . . . . . . . 10  |-  ( (
ph  /\  w  e.  ( 0 (,) 1
)  /\  P  =  ( ( w  x.  A )  +  ( ( 1  -  w
)  x.  B ) ) )  ->  (
( abs `  ( P  -  A )
)  x.  ( abs `  ( P  -  B
) ) )  =  ( ( ( abs `  ( B  -  Q
) ) ^ 2 )  -  ( ( abs `  ( P  -  Q ) ) ^ 2 ) ) )
44433expb 1216 . . . . . . . . 9  |-  ( (
ph  /\  ( w  e.  ( 0 (,) 1
)  /\  P  =  ( ( w  x.  A )  +  ( ( 1  -  w
)  x.  B ) ) ) )  -> 
( ( abs `  ( P  -  A )
)  x.  ( abs `  ( P  -  B
) ) )  =  ( ( ( abs `  ( B  -  Q
) ) ^ 2 )  -  ( ( abs `  ( P  -  Q ) ) ^ 2 ) ) )
45443adant2 1033 . . . . . . . 8  |-  ( (
ph  /\  ( v  e.  ( 0 (,) 1
)  /\  P  =  ( ( v  x.  C )  +  ( ( 1  -  v
)  x.  D ) ) )  /\  (
w  e.  ( 0 (,) 1 )  /\  P  =  ( (
w  x.  A )  +  ( ( 1  -  w )  x.  B ) ) ) )  ->  ( ( abs `  ( P  -  A ) )  x.  ( abs `  ( P  -  B )
) )  =  ( ( ( abs `  ( B  -  Q )
) ^ 2 )  -  ( ( abs `  ( P  -  Q
) ) ^ 2 ) ) )
4633ad2ant1 1035 . . . . . . . . . . 11  |-  ( (
ph  /\  v  e.  ( 0 (,) 1
)  /\  P  =  ( ( v  x.  C )  +  ( ( 1  -  v
)  x.  D ) ) )  ->  C  e.  CC )
4753ad2ant1 1035 . . . . . . . . . . 11  |-  ( (
ph  /\  v  e.  ( 0 (,) 1
)  /\  P  =  ( ( v  x.  C )  +  ( ( 1  -  v
)  x.  D ) ) )  ->  D  e.  CC )
48343ad2ant1 1035 . . . . . . . . . . 11  |-  ( (
ph  /\  v  e.  ( 0 (,) 1
)  /\  P  =  ( ( v  x.  C )  +  ( ( 1  -  v
)  x.  D ) ) )  ->  Q  e.  CC )
49 id 22 . . . . . . . . . . . . 13  |-  ( v  e.  ( 0 (,) 1 )  ->  v  e.  ( 0 (,) 1
) )
5036, 49sseldi 3441 . . . . . . . . . . . 12  |-  ( v  e.  ( 0 (,) 1 )  ->  v  e.  ( 0 [,] 1
) )
51503ad2ant2 1036 . . . . . . . . . . 11  |-  ( (
ph  /\  v  e.  ( 0 (,) 1
)  /\  P  =  ( ( v  x.  C )  +  ( ( 1  -  v
)  x.  D ) ) )  ->  v  e.  ( 0 [,] 1
) )
52 id 22 . . . . . . . . . . . 12  |-  ( P  =  ( ( v  x.  C )  +  ( ( 1  -  v )  x.  D
) )  ->  P  =  ( ( v  x.  C )  +  ( ( 1  -  v )  x.  D
) ) )
53523ad2ant3 1037 . . . . . . . . . . 11  |-  ( (
ph  /\  v  e.  ( 0 (,) 1
)  /\  P  =  ( ( v  x.  C )  +  ( ( 1  -  v
)  x.  D ) ) )  ->  P  =  ( ( v  x.  C )  +  ( ( 1  -  v )  x.  D
) ) )
54 chordthmALT.ACcirc . . . . . . . . . . . . 13  |-  ( ph  ->  ( abs `  ( A  -  Q )
)  =  ( abs `  ( C  -  Q
) ) )
5554, 27eqtr3d 2497 . . . . . . . . . . . 12  |-  ( ph  ->  ( abs `  ( C  -  Q )
)  =  ( abs `  ( D  -  Q
) ) )
56553ad2ant1 1035 . . . . . . . . . . 11  |-  ( (
ph  /\  v  e.  ( 0 (,) 1
)  /\  P  =  ( ( v  x.  C )  +  ( ( 1  -  v
)  x.  D ) ) )  ->  ( abs `  ( C  -  Q ) )  =  ( abs `  ( D  -  Q )
) )
5746, 47, 48, 51, 53, 56chordthmlem5 23810 . . . . . . . . . 10  |-  ( (
ph  /\  v  e.  ( 0 (,) 1
)  /\  P  =  ( ( v  x.  C )  +  ( ( 1  -  v
)  x.  D ) ) )  ->  (
( abs `  ( P  -  C )
)  x.  ( abs `  ( P  -  D
) ) )  =  ( ( ( abs `  ( D  -  Q
) ) ^ 2 )  -  ( ( abs `  ( P  -  Q ) ) ^ 2 ) ) )
58573expb 1216 . . . . . . . . 9  |-  ( (
ph  /\  ( v  e.  ( 0 (,) 1
)  /\  P  =  ( ( v  x.  C )  +  ( ( 1  -  v
)  x.  D ) ) ) )  -> 
( ( abs `  ( P  -  C )
)  x.  ( abs `  ( P  -  D
) ) )  =  ( ( ( abs `  ( D  -  Q
) ) ^ 2 )  -  ( ( abs `  ( P  -  Q ) ) ^ 2 ) ) )
59583adant3 1034 . . . . . . . 8  |-  ( (
ph  /\  ( v  e.  ( 0 (,) 1
)  /\  P  =  ( ( v  x.  C )  +  ( ( 1  -  v
)  x.  D ) ) )  /\  (
w  e.  ( 0 (,) 1 )  /\  P  =  ( (
w  x.  A )  +  ( ( 1  -  w )  x.  B ) ) ) )  ->  ( ( abs `  ( P  -  C ) )  x.  ( abs `  ( P  -  D )
) )  =  ( ( ( abs `  ( D  -  Q )
) ^ 2 )  -  ( ( abs `  ( P  -  Q
) ) ^ 2 ) ) )
6031, 45, 593eqtr4d 2505 . . . . . . 7  |-  ( (
ph  /\  ( v  e.  ( 0 (,) 1
)  /\  P  =  ( ( v  x.  C )  +  ( ( 1  -  v
)  x.  D ) ) )  /\  (
w  e.  ( 0 (,) 1 )  /\  P  =  ( (
w  x.  A )  +  ( ( 1  -  w )  x.  B ) ) ) )  ->  ( ( abs `  ( P  -  A ) )  x.  ( abs `  ( P  -  B )
) )  =  ( ( abs `  ( P  -  C )
)  x.  ( abs `  ( P  -  D
) ) ) )
61603expia 1217 . . . . . 6  |-  ( (
ph  /\  ( v  e.  ( 0 (,) 1
)  /\  P  =  ( ( v  x.  C )  +  ( ( 1  -  v
)  x.  D ) ) ) )  -> 
( ( w  e.  ( 0 (,) 1
)  /\  P  =  ( ( w  x.  A )  +  ( ( 1  -  w
)  x.  B ) ) )  ->  (
( abs `  ( P  -  A )
)  x.  ( abs `  ( P  -  B
) ) )  =  ( ( abs `  ( P  -  C )
)  x.  ( abs `  ( P  -  D
) ) ) ) )
6261exlimdv 1789 . . . . 5  |-  ( (
ph  /\  ( v  e.  ( 0 (,) 1
)  /\  P  =  ( ( v  x.  C )  +  ( ( 1  -  v
)  x.  D ) ) ) )  -> 
( E. w ( w  e.  ( 0 (,) 1 )  /\  P  =  ( (
w  x.  A )  +  ( ( 1  -  w )  x.  B ) ) )  ->  ( ( abs `  ( P  -  A
) )  x.  ( abs `  ( P  -  B ) ) )  =  ( ( abs `  ( P  -  C
) )  x.  ( abs `  ( P  -  D ) ) ) ) )
6325, 62mpd 15 . . . 4  |-  ( (
ph  /\  ( v  e.  ( 0 (,) 1
)  /\  P  =  ( ( v  x.  C )  +  ( ( 1  -  v
)  x.  D ) ) ) )  -> 
( ( abs `  ( P  -  A )
)  x.  ( abs `  ( P  -  B
) ) )  =  ( ( abs `  ( P  -  C )
)  x.  ( abs `  ( P  -  D
) ) ) )
6463ex 440 . . 3  |-  ( ph  ->  ( ( v  e.  ( 0 (,) 1
)  /\  P  =  ( ( v  x.  C )  +  ( ( 1  -  v
)  x.  D ) ) )  ->  (
( abs `  ( P  -  A )
)  x.  ( abs `  ( P  -  B
) ) )  =  ( ( abs `  ( P  -  C )
)  x.  ( abs `  ( P  -  D
) ) ) ) )
6564exlimdv 1789 . 2  |-  ( ph  ->  ( E. v ( v  e.  ( 0 (,) 1 )  /\  P  =  ( (
v  x.  C )  +  ( ( 1  -  v )  x.  D ) ) )  ->  ( ( abs `  ( P  -  A
) )  x.  ( abs `  ( P  -  B ) ) )  =  ( ( abs `  ( P  -  C
) )  x.  ( abs `  ( P  -  D ) ) ) ) )
6613, 65mpd 15 1  |-  ( ph  ->  ( ( abs `  ( P  -  A )
)  x.  ( abs `  ( P  -  B
) ) )  =  ( ( abs `  ( P  -  C )
)  x.  ( abs `  ( P  -  D
) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 375    /\ w3a 991    = wceq 1454   E.wex 1673    e. wcel 1897    =/= wne 2632   E.wrex 2749    \ cdif 3412   {csn 3979   ` cfv 5600  (class class class)co 6314    |-> cmpt2 6316   CCcc 9562   0cc0 9564   1c1 9565    + caddc 9567    x. cmul 9569    - cmin 9885    / cdiv 10296   2c2 10686   (,)cioo 11663   [,]cicc 11666   ^cexp 12303   Imcim 13209   abscabs 13345   picpi 14167   logclog 23552
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-rep 4528  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6609  ax-inf2 8171  ax-cnex 9620  ax-resscn 9621  ax-1cn 9622  ax-icn 9623  ax-addcl 9624  ax-addrcl 9625  ax-mulcl 9626  ax-mulrcl 9627  ax-mulcom 9628  ax-addass 9629  ax-mulass 9630  ax-distr 9631  ax-i2m1 9632  ax-1ne0 9633  ax-1rid 9634  ax-rnegex 9635  ax-rrecex 9636  ax-cnre 9637  ax-pre-lttri 9638  ax-pre-lttrn 9639  ax-pre-ltadd 9640  ax-pre-mulgt0 9641  ax-pre-sup 9642  ax-addf 9643  ax-mulf 9644
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1457  df-fal 1460  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-nel 2635  df-ral 2753  df-rex 2754  df-reu 2755  df-rmo 2756  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-pss 3431  df-nul 3743  df-if 3893  df-pw 3964  df-sn 3980  df-pr 3982  df-tp 3984  df-op 3986  df-uni 4212  df-int 4248  df-iun 4293  df-iin 4294  df-br 4416  df-opab 4475  df-mpt 4476  df-tr 4511  df-eprel 4763  df-id 4767  df-po 4773  df-so 4774  df-fr 4811  df-se 4812  df-we 4813  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-pred 5398  df-ord 5444  df-on 5445  df-lim 5446  df-suc 5447  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-fv 5608  df-isom 5609  df-riota 6276  df-ov 6317  df-oprab 6318  df-mpt2 6319  df-of 6557  df-om 6719  df-1st 6819  df-2nd 6820  df-supp 6941  df-wrecs 7053  df-recs 7115  df-rdg 7153  df-1o 7207  df-2o 7208  df-oadd 7211  df-er 7388  df-map 7499  df-pm 7500  df-ixp 7548  df-en 7595  df-dom 7596  df-sdom 7597  df-fin 7598  df-fsupp 7909  df-fi 7950  df-sup 7981  df-inf 7982  df-oi 8050  df-card 8398  df-cda 8623  df-pnf 9702  df-mnf 9703  df-xr 9704  df-ltxr 9705  df-le 9706  df-sub 9887  df-neg 9888  df-div 10297  df-nn 10637  df-2 10695  df-3 10696  df-4 10697  df-5 10698  df-6 10699  df-7 10700  df-8 10701  df-9 10702  df-10 10703  df-n0 10898  df-z 10966  df-dec 11080  df-uz 11188  df-q 11293  df-rp 11331  df-xneg 11437  df-xadd 11438  df-xmul 11439  df-ioo 11667  df-ioc 11668  df-ico 11669  df-icc 11670  df-fz 11813  df-fzo 11946  df-fl 12059  df-mod 12128  df-seq 12245  df-exp 12304  df-fac 12491  df-bc 12519  df-hash 12547  df-shft 13178  df-cj 13210  df-re 13211  df-im 13212  df-sqrt 13346  df-abs 13347  df-limsup 13574  df-clim 13600  df-rlim 13601  df-sum 13801  df-ef 14169  df-sin 14171  df-cos 14172  df-pi 14174  df-struct 15171  df-ndx 15172  df-slot 15173  df-base 15174  df-sets 15175  df-ress 15176  df-plusg 15251  df-mulr 15252  df-starv 15253  df-sca 15254  df-vsca 15255  df-ip 15256  df-tset 15257  df-ple 15258  df-ds 15260  df-unif 15261  df-hom 15262  df-cco 15263  df-rest 15369  df-topn 15370  df-0g 15388  df-gsum 15389  df-topgen 15390  df-pt 15391  df-prds 15394  df-xrs 15448  df-qtop 15454  df-imas 15455  df-xps 15458  df-mre 15540  df-mrc 15541  df-acs 15543  df-mgm 16536  df-sgrp 16575  df-mnd 16585  df-submnd 16631  df-mulg 16724  df-cntz 17019  df-cmn 17480  df-psmet 19010  df-xmet 19011  df-met 19012  df-bl 19013  df-mopn 19014  df-fbas 19015  df-fg 19016  df-cnfld 19019  df-top 19969  df-bases 19970  df-topon 19971  df-topsp 19972  df-cld 20082  df-ntr 20083  df-cls 20084  df-nei 20162  df-lp 20200  df-perf 20201  df-cn 20291  df-cnp 20292  df-haus 20379  df-tx 20625  df-hmeo 20818  df-fil 20909  df-fm 21001  df-flim 21002  df-flf 21003  df-xms 21383  df-ms 21384  df-tms 21385  df-cncf 21958  df-limc 22869  df-dv 22870  df-log 23554
This theorem is referenced by: (None)
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