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Theorem lterpq 8782
Description: Compatibility of ordering on equivalent fractions. (Contributed by Mario Carneiro, 9-May-2013.) (New usage is discouraged.)
Assertion
Ref Expression
lterpq  |-  ( A 
<pQ  B  <->  ( /Q `  A )  <Q  ( /Q `  B ) )

Proof of Theorem lterpq
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ltpq 8722 . . . 4  |-  <pQ  =  { <. x ,  y >.  |  ( ( x  e.  ( N.  X.  N. )  /\  y  e.  ( N.  X.  N. ) )  /\  (
( 1st `  x
)  .N  ( 2nd `  y ) )  <N 
( ( 1st `  y
)  .N  ( 2nd `  x ) ) ) }
2 opabssxp 4892 . . . 4  |-  { <. x ,  y >.  |  ( ( x  e.  ( N.  X.  N. )  /\  y  e.  ( N.  X.  N. ) )  /\  ( ( 1st `  x )  .N  ( 2nd `  y ) ) 
<N  ( ( 1st `  y
)  .N  ( 2nd `  x ) ) ) }  C_  ( ( N.  X.  N. )  X.  ( N.  X.  N. ) )
31, 2eqsstri 3323 . . 3  |-  <pQ  C_  (
( N.  X.  N. )  X.  ( N.  X.  N. ) )
43brel 4868 . 2  |-  ( A 
<pQ  B  ->  ( A  e.  ( N.  X.  N. )  /\  B  e.  ( N.  X.  N. )
) )
5 ltrelnq 8738 . . . 4  |-  <Q  C_  ( Q.  X.  Q. )
65brel 4868 . . 3  |-  ( ( /Q `  A ) 
<Q  ( /Q `  B
)  ->  ( ( /Q `  A )  e. 
Q.  /\  ( /Q `  B )  e.  Q. ) )
7 elpqn 8737 . . . 4  |-  ( ( /Q `  A )  e.  Q.  ->  ( /Q `  A )  e.  ( N.  X.  N. ) )
8 elpqn 8737 . . . 4  |-  ( ( /Q `  B )  e.  Q.  ->  ( /Q `  B )  e.  ( N.  X.  N. ) )
9 nqerf 8742 . . . . . . 7  |-  /Q :
( N.  X.  N. )
--> Q.
109fdmi 5538 . . . . . 6  |-  dom  /Q  =  ( N.  X.  N. )
11 0nelxp 4848 . . . . . 6  |-  -.  (/)  e.  ( N.  X.  N. )
1210, 11ndmfvrcl 5698 . . . . 5  |-  ( ( /Q `  A )  e.  ( N.  X.  N. )  ->  A  e.  ( N.  X.  N. ) )
1310, 11ndmfvrcl 5698 . . . . 5  |-  ( ( /Q `  B )  e.  ( N.  X.  N. )  ->  B  e.  ( N.  X.  N. ) )
1412, 13anim12i 550 . . . 4  |-  ( ( ( /Q `  A
)  e.  ( N. 
X.  N. )  /\  ( /Q `  B )  e.  ( N.  X.  N. ) )  ->  ( A  e.  ( N.  X.  N. )  /\  B  e.  ( N.  X.  N. ) ) )
157, 8, 14syl2an 464 . . 3  |-  ( ( ( /Q `  A
)  e.  Q.  /\  ( /Q `  B )  e.  Q. )  -> 
( A  e.  ( N.  X.  N. )  /\  B  e.  ( N.  X.  N. ) ) )
166, 15syl 16 . 2  |-  ( ( /Q `  A ) 
<Q  ( /Q `  B
)  ->  ( A  e.  ( N.  X.  N. )  /\  B  e.  ( N.  X.  N. )
) )
17 xp1st 6317 . . . . 5  |-  ( A  e.  ( N.  X.  N. )  ->  ( 1st `  A )  e.  N. )
18 xp2nd 6318 . . . . 5  |-  ( B  e.  ( N.  X.  N. )  ->  ( 2nd `  B )  e.  N. )
19 mulclpi 8705 . . . . 5  |-  ( ( ( 1st `  A
)  e.  N.  /\  ( 2nd `  B )  e.  N. )  -> 
( ( 1st `  A
)  .N  ( 2nd `  B ) )  e. 
N. )
2017, 18, 19syl2an 464 . . . 4  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. ) )  ->  (
( 1st `  A
)  .N  ( 2nd `  B ) )  e. 
N. )
21 ltmpi 8716 . . . 4  |-  ( ( ( 1st `  A
)  .N  ( 2nd `  B ) )  e. 
N.  ->  ( ( ( 1st `  ( /Q
`  A ) )  .N  ( 2nd `  ( /Q `  B ) ) )  <N  ( ( 1st `  ( /Q `  B ) )  .N  ( 2nd `  ( /Q `  A ) ) )  <->  ( ( ( 1st `  A )  .N  ( 2nd `  B
) )  .N  (
( 1st `  ( /Q `  A ) )  .N  ( 2nd `  ( /Q `  B ) ) ) )  <N  (
( ( 1st `  A
)  .N  ( 2nd `  B ) )  .N  ( ( 1st `  ( /Q `  B ) )  .N  ( 2nd `  ( /Q `  A ) ) ) ) ) )
2220, 21syl 16 . . 3  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. ) )  ->  (
( ( 1st `  ( /Q `  A ) )  .N  ( 2nd `  ( /Q `  B ) ) )  <N  ( ( 1st `  ( /Q `  B ) )  .N  ( 2nd `  ( /Q `  A ) ) )  <->  ( ( ( 1st `  A )  .N  ( 2nd `  B
) )  .N  (
( 1st `  ( /Q `  A ) )  .N  ( 2nd `  ( /Q `  B ) ) ) )  <N  (
( ( 1st `  A
)  .N  ( 2nd `  B ) )  .N  ( ( 1st `  ( /Q `  B ) )  .N  ( 2nd `  ( /Q `  A ) ) ) ) ) )
23 nqercl 8743 . . . 4  |-  ( A  e.  ( N.  X.  N. )  ->  ( /Q
`  A )  e. 
Q. )
24 nqercl 8743 . . . 4  |-  ( B  e.  ( N.  X.  N. )  ->  ( /Q
`  B )  e. 
Q. )
25 ordpinq 8755 . . . 4  |-  ( ( ( /Q `  A
)  e.  Q.  /\  ( /Q `  B )  e.  Q. )  -> 
( ( /Q `  A )  <Q  ( /Q `  B )  <->  ( ( 1st `  ( /Q `  A ) )  .N  ( 2nd `  ( /Q `  B ) ) )  <N  ( ( 1st `  ( /Q `  B ) )  .N  ( 2nd `  ( /Q `  A ) ) ) ) )
2623, 24, 25syl2an 464 . . 3  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. ) )  ->  (
( /Q `  A
)  <Q  ( /Q `  B )  <->  ( ( 1st `  ( /Q `  A ) )  .N  ( 2nd `  ( /Q `  B ) ) )  <N  ( ( 1st `  ( /Q `  B ) )  .N  ( 2nd `  ( /Q `  A ) ) ) ) )
27 1st2nd2 6327 . . . . . 6  |-  ( A  e.  ( N.  X.  N. )  ->  A  = 
<. ( 1st `  A
) ,  ( 2nd `  A ) >. )
28 1st2nd2 6327 . . . . . 6  |-  ( B  e.  ( N.  X.  N. )  ->  B  = 
<. ( 1st `  B
) ,  ( 2nd `  B ) >. )
2927, 28breqan12d 4170 . . . . 5  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. ) )  ->  ( A  <pQ  B  <->  <. ( 1st `  A ) ,  ( 2nd `  A )
>.  <pQ  <. ( 1st `  B
) ,  ( 2nd `  B ) >. )
)
30 ordpipq 8754 . . . . 5  |-  ( <.
( 1st `  A
) ,  ( 2nd `  A ) >.  <pQ  <. ( 1st `  B ) ,  ( 2nd `  B
) >. 
<->  ( ( 1st `  A
)  .N  ( 2nd `  B ) )  <N 
( ( 1st `  B
)  .N  ( 2nd `  A ) ) )
3129, 30syl6bb 253 . . . 4  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. ) )  ->  ( A  <pQ  B  <->  ( ( 1st `  A )  .N  ( 2nd `  B
) )  <N  (
( 1st `  B
)  .N  ( 2nd `  A ) ) ) )
32 xp1st 6317 . . . . . . 7  |-  ( ( /Q `  A )  e.  ( N.  X.  N. )  ->  ( 1st `  ( /Q `  A
) )  e.  N. )
3323, 7, 323syl 19 . . . . . 6  |-  ( A  e.  ( N.  X.  N. )  ->  ( 1st `  ( /Q `  A
) )  e.  N. )
34 xp2nd 6318 . . . . . . 7  |-  ( ( /Q `  B )  e.  ( N.  X.  N. )  ->  ( 2nd `  ( /Q `  B
) )  e.  N. )
3524, 8, 343syl 19 . . . . . 6  |-  ( B  e.  ( N.  X.  N. )  ->  ( 2nd `  ( /Q `  B
) )  e.  N. )
36 mulclpi 8705 . . . . . 6  |-  ( ( ( 1st `  ( /Q `  A ) )  e.  N.  /\  ( 2nd `  ( /Q `  B ) )  e. 
N. )  ->  (
( 1st `  ( /Q `  A ) )  .N  ( 2nd `  ( /Q `  B ) ) )  e.  N. )
3733, 35, 36syl2an 464 . . . . 5  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. ) )  ->  (
( 1st `  ( /Q `  A ) )  .N  ( 2nd `  ( /Q `  B ) ) )  e.  N. )
38 ltmpi 8716 . . . . 5  |-  ( ( ( 1st `  ( /Q `  A ) )  .N  ( 2nd `  ( /Q `  B ) ) )  e.  N.  ->  ( ( ( 1st `  A
)  .N  ( 2nd `  B ) )  <N 
( ( 1st `  B
)  .N  ( 2nd `  A ) )  <->  ( (
( 1st `  ( /Q `  A ) )  .N  ( 2nd `  ( /Q `  B ) ) )  .N  ( ( 1st `  A )  .N  ( 2nd `  B
) ) )  <N 
( ( ( 1st `  ( /Q `  A
) )  .N  ( 2nd `  ( /Q `  B ) ) )  .N  ( ( 1st `  B )  .N  ( 2nd `  A ) ) ) ) )
3937, 38syl 16 . . . 4  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. ) )  ->  (
( ( 1st `  A
)  .N  ( 2nd `  B ) )  <N 
( ( 1st `  B
)  .N  ( 2nd `  A ) )  <->  ( (
( 1st `  ( /Q `  A ) )  .N  ( 2nd `  ( /Q `  B ) ) )  .N  ( ( 1st `  A )  .N  ( 2nd `  B
) ) )  <N 
( ( ( 1st `  ( /Q `  A
) )  .N  ( 2nd `  ( /Q `  B ) ) )  .N  ( ( 1st `  B )  .N  ( 2nd `  A ) ) ) ) )
40 mulcompi 8708 . . . . . 6  |-  ( ( ( 1st `  ( /Q `  A ) )  .N  ( 2nd `  ( /Q `  B ) ) )  .N  ( ( 1st `  A )  .N  ( 2nd `  B
) ) )  =  ( ( ( 1st `  A )  .N  ( 2nd `  B ) )  .N  ( ( 1st `  ( /Q `  A
) )  .N  ( 2nd `  ( /Q `  B ) ) ) )
4140a1i 11 . . . . 5  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. ) )  ->  (
( ( 1st `  ( /Q `  A ) )  .N  ( 2nd `  ( /Q `  B ) ) )  .N  ( ( 1st `  A )  .N  ( 2nd `  B
) ) )  =  ( ( ( 1st `  A )  .N  ( 2nd `  B ) )  .N  ( ( 1st `  ( /Q `  A
) )  .N  ( 2nd `  ( /Q `  B ) ) ) ) )
42 nqerrel 8744 . . . . . . . . 9  |-  ( A  e.  ( N.  X.  N. )  ->  A  ~Q  ( /Q `  A ) )
4323, 7syl 16 . . . . . . . . . 10  |-  ( A  e.  ( N.  X.  N. )  ->  ( /Q
`  A )  e.  ( N.  X.  N. ) )
44 enqbreq2 8732 . . . . . . . . . 10  |-  ( ( A  e.  ( N. 
X.  N. )  /\  ( /Q `  A )  e.  ( N.  X.  N. ) )  ->  ( A  ~Q  ( /Q `  A )  <->  ( ( 1st `  A )  .N  ( 2nd `  ( /Q `  A ) ) )  =  ( ( 1st `  ( /Q
`  A ) )  .N  ( 2nd `  A
) ) ) )
4543, 44mpdan 650 . . . . . . . . 9  |-  ( A  e.  ( N.  X.  N. )  ->  ( A  ~Q  ( /Q `  A )  <->  ( ( 1st `  A )  .N  ( 2nd `  ( /Q `  A ) ) )  =  ( ( 1st `  ( /Q
`  A ) )  .N  ( 2nd `  A
) ) ) )
4642, 45mpbid 202 . . . . . . . 8  |-  ( A  e.  ( N.  X.  N. )  ->  ( ( 1st `  A )  .N  ( 2nd `  ( /Q `  A ) ) )  =  ( ( 1st `  ( /Q
`  A ) )  .N  ( 2nd `  A
) ) )
4746eqcomd 2394 . . . . . . 7  |-  ( A  e.  ( N.  X.  N. )  ->  ( ( 1st `  ( /Q
`  A ) )  .N  ( 2nd `  A
) )  =  ( ( 1st `  A
)  .N  ( 2nd `  ( /Q `  A
) ) ) )
48 nqerrel 8744 . . . . . . . 8  |-  ( B  e.  ( N.  X.  N. )  ->  B  ~Q  ( /Q `  B ) )
4924, 8syl 16 . . . . . . . . 9  |-  ( B  e.  ( N.  X.  N. )  ->  ( /Q
`  B )  e.  ( N.  X.  N. ) )
50 enqbreq2 8732 . . . . . . . . 9  |-  ( ( B  e.  ( N. 
X.  N. )  /\  ( /Q `  B )  e.  ( N.  X.  N. ) )  ->  ( B  ~Q  ( /Q `  B )  <->  ( ( 1st `  B )  .N  ( 2nd `  ( /Q `  B ) ) )  =  ( ( 1st `  ( /Q
`  B ) )  .N  ( 2nd `  B
) ) ) )
5149, 50mpdan 650 . . . . . . . 8  |-  ( B  e.  ( N.  X.  N. )  ->  ( B  ~Q  ( /Q `  B )  <->  ( ( 1st `  B )  .N  ( 2nd `  ( /Q `  B ) ) )  =  ( ( 1st `  ( /Q
`  B ) )  .N  ( 2nd `  B
) ) ) )
5248, 51mpbid 202 . . . . . . 7  |-  ( B  e.  ( N.  X.  N. )  ->  ( ( 1st `  B )  .N  ( 2nd `  ( /Q `  B ) ) )  =  ( ( 1st `  ( /Q
`  B ) )  .N  ( 2nd `  B
) ) )
5347, 52oveqan12d 6041 . . . . . 6  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. ) )  ->  (
( ( 1st `  ( /Q `  A ) )  .N  ( 2nd `  A
) )  .N  (
( 1st `  B
)  .N  ( 2nd `  ( /Q `  B
) ) ) )  =  ( ( ( 1st `  A )  .N  ( 2nd `  ( /Q `  A ) ) )  .N  ( ( 1st `  ( /Q
`  B ) )  .N  ( 2nd `  B
) ) ) )
54 mulcompi 8708 . . . . . . 7  |-  ( ( ( 1st `  ( /Q `  A ) )  .N  ( 2nd `  ( /Q `  B ) ) )  .N  ( ( 1st `  B )  .N  ( 2nd `  A
) ) )  =  ( ( ( 1st `  B )  .N  ( 2nd `  A ) )  .N  ( ( 1st `  ( /Q `  A
) )  .N  ( 2nd `  ( /Q `  B ) ) ) )
55 fvex 5684 . . . . . . . 8  |-  ( 1st `  B )  e.  _V
56 fvex 5684 . . . . . . . 8  |-  ( 2nd `  A )  e.  _V
57 fvex 5684 . . . . . . . 8  |-  ( 1st `  ( /Q `  A
) )  e.  _V
58 mulcompi 8708 . . . . . . . 8  |-  ( x  .N  y )  =  ( y  .N  x
)
59 mulasspi 8709 . . . . . . . 8  |-  ( ( x  .N  y )  .N  z )  =  ( x  .N  (
y  .N  z ) )
60 fvex 5684 . . . . . . . 8  |-  ( 2nd `  ( /Q `  B
) )  e.  _V
6155, 56, 57, 58, 59, 60caov411 6220 . . . . . . 7  |-  ( ( ( 1st `  B
)  .N  ( 2nd `  A ) )  .N  ( ( 1st `  ( /Q `  A ) )  .N  ( 2nd `  ( /Q `  B ) ) ) )  =  ( ( ( 1st `  ( /Q `  A ) )  .N  ( 2nd `  A
) )  .N  (
( 1st `  B
)  .N  ( 2nd `  ( /Q `  B
) ) ) )
6254, 61eqtri 2409 . . . . . 6  |-  ( ( ( 1st `  ( /Q `  A ) )  .N  ( 2nd `  ( /Q `  B ) ) )  .N  ( ( 1st `  B )  .N  ( 2nd `  A
) ) )  =  ( ( ( 1st `  ( /Q `  A
) )  .N  ( 2nd `  A ) )  .N  ( ( 1st `  B )  .N  ( 2nd `  ( /Q `  B ) ) ) )
63 mulcompi 8708 . . . . . . 7  |-  ( ( ( 1st `  A
)  .N  ( 2nd `  B ) )  .N  ( ( 1st `  ( /Q `  B ) )  .N  ( 2nd `  ( /Q `  A ) ) ) )  =  ( ( ( 1st `  ( /Q `  B ) )  .N  ( 2nd `  ( /Q `  A ) ) )  .N  ( ( 1st `  A )  .N  ( 2nd `  B
) ) )
64 fvex 5684 . . . . . . . 8  |-  ( 1st `  ( /Q `  B
) )  e.  _V
65 fvex 5684 . . . . . . . 8  |-  ( 2nd `  ( /Q `  A
) )  e.  _V
66 fvex 5684 . . . . . . . 8  |-  ( 1st `  A )  e.  _V
67 fvex 5684 . . . . . . . 8  |-  ( 2nd `  B )  e.  _V
6864, 65, 66, 58, 59, 67caov411 6220 . . . . . . 7  |-  ( ( ( 1st `  ( /Q `  B ) )  .N  ( 2nd `  ( /Q `  A ) ) )  .N  ( ( 1st `  A )  .N  ( 2nd `  B
) ) )  =  ( ( ( 1st `  A )  .N  ( 2nd `  ( /Q `  A ) ) )  .N  ( ( 1st `  ( /Q `  B
) )  .N  ( 2nd `  B ) ) )
6963, 68eqtri 2409 . . . . . 6  |-  ( ( ( 1st `  A
)  .N  ( 2nd `  B ) )  .N  ( ( 1st `  ( /Q `  B ) )  .N  ( 2nd `  ( /Q `  A ) ) ) )  =  ( ( ( 1st `  A
)  .N  ( 2nd `  ( /Q `  A
) ) )  .N  ( ( 1st `  ( /Q `  B ) )  .N  ( 2nd `  B
) ) )
7053, 62, 693eqtr4g 2446 . . . . 5  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. ) )  ->  (
( ( 1st `  ( /Q `  A ) )  .N  ( 2nd `  ( /Q `  B ) ) )  .N  ( ( 1st `  B )  .N  ( 2nd `  A
) ) )  =  ( ( ( 1st `  A )  .N  ( 2nd `  B ) )  .N  ( ( 1st `  ( /Q `  B
) )  .N  ( 2nd `  ( /Q `  A ) ) ) ) )
7141, 70breq12d 4168 . . . 4  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. ) )  ->  (
( ( ( 1st `  ( /Q `  A
) )  .N  ( 2nd `  ( /Q `  B ) ) )  .N  ( ( 1st `  A )  .N  ( 2nd `  B ) ) )  <N  ( (
( 1st `  ( /Q `  A ) )  .N  ( 2nd `  ( /Q `  B ) ) )  .N  ( ( 1st `  B )  .N  ( 2nd `  A
) ) )  <->  ( (
( 1st `  A
)  .N  ( 2nd `  B ) )  .N  ( ( 1st `  ( /Q `  A ) )  .N  ( 2nd `  ( /Q `  B ) ) ) )  <N  (
( ( 1st `  A
)  .N  ( 2nd `  B ) )  .N  ( ( 1st `  ( /Q `  B ) )  .N  ( 2nd `  ( /Q `  A ) ) ) ) ) )
7231, 39, 713bitrd 271 . . 3  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. ) )  ->  ( A  <pQ  B  <->  ( (
( 1st `  A
)  .N  ( 2nd `  B ) )  .N  ( ( 1st `  ( /Q `  A ) )  .N  ( 2nd `  ( /Q `  B ) ) ) )  <N  (
( ( 1st `  A
)  .N  ( 2nd `  B ) )  .N  ( ( 1st `  ( /Q `  B ) )  .N  ( 2nd `  ( /Q `  A ) ) ) ) ) )
7322, 26, 723bitr4rd 278 . 2  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. ) )  ->  ( A  <pQ  B  <->  ( /Q `  A )  <Q  ( /Q `  B ) ) )
744, 16, 73pm5.21nii 343 1  |-  ( A 
<pQ  B  <->  ( /Q `  A )  <Q  ( /Q `  B ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   <.cop 3762   class class class wbr 4155   {copab 4208    X. cxp 4818   ` cfv 5396  (class class class)co 6022   1stc1st 6288   2ndc2nd 6289   N.cnpi 8654    .N cmi 8656    <N clti 8657    <pQ cltpq 8660    ~Q ceq 8661   Q.cnq 8662   /Qcerq 8664    <Q cltq 8668
This theorem is referenced by:  ltanq  8783  ltmnq  8784  1lt2nq  8785
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-recs 6571  df-rdg 6606  df-1o 6662  df-oadd 6666  df-omul 6667  df-er 6843  df-ni 8684  df-mi 8686  df-lti 8687  df-ltpq 8722  df-enq 8723  df-nq 8724  df-erq 8725  df-1nq 8728  df-ltnq 8730
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