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Theorem lterpq 8839
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 8779 . . . 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 4942 . . . 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 3370 . . 3  |-  <pQ  C_  (
( N.  X.  N. )  X.  ( N.  X.  N. ) )
43brel 4918 . 2  |-  ( A 
<pQ  B  ->  ( A  e.  ( N.  X.  N. )  /\  B  e.  ( N.  X.  N. )
) )
5 ltrelnq 8795 . . . 4  |-  <Q  C_  ( Q.  X.  Q. )
65brel 4918 . . 3  |-  ( ( /Q `  A ) 
<Q  ( /Q `  B
)  ->  ( ( /Q `  A )  e. 
Q.  /\  ( /Q `  B )  e.  Q. ) )
7 elpqn 8794 . . . 4  |-  ( ( /Q `  A )  e.  Q.  ->  ( /Q `  A )  e.  ( N.  X.  N. ) )
8 elpqn 8794 . . . 4  |-  ( ( /Q `  B )  e.  Q.  ->  ( /Q `  B )  e.  ( N.  X.  N. ) )
9 nqerf 8799 . . . . . . 7  |-  /Q :
( N.  X.  N. )
--> Q.
109fdmi 5588 . . . . . 6  |-  dom  /Q  =  ( N.  X.  N. )
11 0nelxp 4898 . . . . . 6  |-  -.  (/)  e.  ( N.  X.  N. )
1210, 11ndmfvrcl 5748 . . . . 5  |-  ( ( /Q `  A )  e.  ( N.  X.  N. )  ->  A  e.  ( N.  X.  N. ) )
1310, 11ndmfvrcl 5748 . . . . 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 6368 . . . . 5  |-  ( A  e.  ( N.  X.  N. )  ->  ( 1st `  A )  e.  N. )
18 xp2nd 6369 . . . . 5  |-  ( B  e.  ( N.  X.  N. )  ->  ( 2nd `  B )  e.  N. )
19 mulclpi 8762 . . . . 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 8773 . . . 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 8800 . . . 4  |-  ( A  e.  ( N.  X.  N. )  ->  ( /Q
`  A )  e. 
Q. )
24 nqercl 8800 . . . 4  |-  ( B  e.  ( N.  X.  N. )  ->  ( /Q
`  B )  e. 
Q. )
25 ordpinq 8812 . . . 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 6378 . . . . . 6  |-  ( A  e.  ( N.  X.  N. )  ->  A  = 
<. ( 1st `  A
) ,  ( 2nd `  A ) >. )
28 1st2nd2 6378 . . . . . 6  |-  ( B  e.  ( N.  X.  N. )  ->  B  = 
<. ( 1st `  B
) ,  ( 2nd `  B ) >. )
2927, 28breqan12d 4219 . . . . 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 8811 . . . . 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 6368 . . . . . . 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 6369 . . . . . . 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 8762 . . . . . 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 8773 . . . . 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 8765 . . . . . 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 8801 . . . . . . . . 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 8789 . . . . . . . . . 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 2440 . . . . . . 7  |-  ( A  e.  ( N.  X.  N. )  ->  ( ( 1st `  ( /Q
`  A ) )  .N  ( 2nd `  A
) )  =  ( ( 1st `  A
)  .N  ( 2nd `  ( /Q `  A
) ) ) )
48 nqerrel 8801 . . . . . . . 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 8789 . . . . . . . . 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 6092 . . . . . 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 8765 . . . . . . 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 5734 . . . . . . . 8  |-  ( 1st `  B )  e.  _V
56 fvex 5734 . . . . . . . 8  |-  ( 2nd `  A )  e.  _V
57 fvex 5734 . . . . . . . 8  |-  ( 1st `  ( /Q `  A
) )  e.  _V
58 mulcompi 8765 . . . . . . . 8  |-  ( x  .N  y )  =  ( y  .N  x
)
59 mulasspi 8766 . . . . . . . 8  |-  ( ( x  .N  y )  .N  z )  =  ( x  .N  (
y  .N  z ) )
60 fvex 5734 . . . . . . . 8  |-  ( 2nd `  ( /Q `  B
) )  e.  _V
6155, 56, 57, 58, 59, 60caov411 6271 . . . . . . 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 2455 . . . . . 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 8765 . . . . . . 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 5734 . . . . . . . 8  |-  ( 1st `  ( /Q `  B
) )  e.  _V
65 fvex 5734 . . . . . . . 8  |-  ( 2nd `  ( /Q `  A
) )  e.  _V
66 fvex 5734 . . . . . . . 8  |-  ( 1st `  A )  e.  _V
67 fvex 5734 . . . . . . . 8  |-  ( 2nd `  B )  e.  _V
6864, 65, 66, 58, 59, 67caov411 6271 . . . . . . 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 2455 . . . . . 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 2492 . . . . 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 4217 . . . 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 1652    e. wcel 1725   <.cop 3809   class class class wbr 4204   {copab 4257    X. cxp 4868   ` cfv 5446  (class class class)co 6073   1stc1st 6339   2ndc2nd 6340   N.cnpi 8711    .N cmi 8713    <N clti 8714    <pQ cltpq 8717    ~Q ceq 8718   Q.cnq 8719   /Qcerq 8721    <Q cltq 8725
This theorem is referenced by:  ltanq  8840  ltmnq  8841  1lt2nq  8842
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-omul 6721  df-er 6897  df-ni 8741  df-mi 8743  df-lti 8744  df-ltpq 8779  df-enq 8780  df-nq 8781  df-erq 8782  df-1nq 8785  df-ltnq 8787
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