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Theorem lterpq 8594
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 8534 . . . 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 4762 . . . 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 3208 . . 3  |-  <pQ  C_  (
( N.  X.  N. )  X.  ( N.  X.  N. ) )
43brel 4737 . 2  |-  ( A 
<pQ  B  ->  ( A  e.  ( N.  X.  N. )  /\  B  e.  ( N.  X.  N. )
) )
5 ltrelnq 8550 . . . 4  |-  <Q  C_  ( Q.  X.  Q. )
65brel 4737 . . 3  |-  ( ( /Q `  A ) 
<Q  ( /Q `  B
)  ->  ( ( /Q `  A )  e. 
Q.  /\  ( /Q `  B )  e.  Q. ) )
7 elpqn 8549 . . . 4  |-  ( ( /Q `  A )  e.  Q.  ->  ( /Q `  A )  e.  ( N.  X.  N. ) )
8 elpqn 8549 . . . 4  |-  ( ( /Q `  B )  e.  Q.  ->  ( /Q `  B )  e.  ( N.  X.  N. ) )
9 nqerf 8554 . . . . . . 7  |-  /Q :
( N.  X.  N. )
--> Q.
109fdmi 5394 . . . . . 6  |-  dom  /Q  =  ( N.  X.  N. )
11 0nelxp 4717 . . . . . 6  |-  -.  (/)  e.  ( N.  X.  N. )
1210, 11ndmfvrcl 5553 . . . . 5  |-  ( ( /Q `  A )  e.  ( N.  X.  N. )  ->  A  e.  ( N.  X.  N. ) )
1310, 11ndmfvrcl 5553 . . . . 5  |-  ( ( /Q `  B )  e.  ( N.  X.  N. )  ->  B  e.  ( N.  X.  N. ) )
1412, 13anim12i 549 . . . 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 463 . . 3  |-  ( ( ( /Q `  A
)  e.  Q.  /\  ( /Q `  B )  e.  Q. )  -> 
( A  e.  ( N.  X.  N. )  /\  B  e.  ( N.  X.  N. ) ) )
166, 15syl 15 . 2  |-  ( ( /Q `  A ) 
<Q  ( /Q `  B
)  ->  ( A  e.  ( N.  X.  N. )  /\  B  e.  ( N.  X.  N. )
) )
17 xp1st 6149 . . . . 5  |-  ( A  e.  ( N.  X.  N. )  ->  ( 1st `  A )  e.  N. )
18 xp2nd 6150 . . . . 5  |-  ( B  e.  ( N.  X.  N. )  ->  ( 2nd `  B )  e.  N. )
19 mulclpi 8517 . . . . 5  |-  ( ( ( 1st `  A
)  e.  N.  /\  ( 2nd `  B )  e.  N. )  -> 
( ( 1st `  A
)  .N  ( 2nd `  B ) )  e. 
N. )
2017, 18, 19syl2an 463 . . . 4  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. ) )  ->  (
( 1st `  A
)  .N  ( 2nd `  B ) )  e. 
N. )
21 ltmpi 8528 . . . 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 15 . . 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 8555 . . . 4  |-  ( A  e.  ( N.  X.  N. )  ->  ( /Q
`  A )  e. 
Q. )
24 nqercl 8555 . . . 4  |-  ( B  e.  ( N.  X.  N. )  ->  ( /Q
`  B )  e. 
Q. )
25 ordpinq 8567 . . . 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 463 . . 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 6159 . . . . . 6  |-  ( A  e.  ( N.  X.  N. )  ->  A  = 
<. ( 1st `  A
) ,  ( 2nd `  A ) >. )
28 1st2nd2 6159 . . . . . 6  |-  ( B  e.  ( N.  X.  N. )  ->  B  = 
<. ( 1st `  B
) ,  ( 2nd `  B ) >. )
2927, 28breqan12d 4038 . . . . 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 8566 . . . . 5  |-  ( <.
( 1st `  A
) ,  ( 2nd `  A ) >.  <pQ  <. ( 1st `  B ) ,  ( 2nd `  B
) >. 
<->  ( ( 1st `  A
)  .N  ( 2nd `  B ) )  <N 
( ( 1st `  B
)  .N  ( 2nd `  A ) ) )
3129, 30syl6bb 252 . . . 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 6149 . . . . . . 7  |-  ( ( /Q `  A )  e.  ( N.  X.  N. )  ->  ( 1st `  ( /Q `  A
) )  e.  N. )
3323, 7, 323syl 18 . . . . . 6  |-  ( A  e.  ( N.  X.  N. )  ->  ( 1st `  ( /Q `  A
) )  e.  N. )
34 xp2nd 6150 . . . . . . 7  |-  ( ( /Q `  B )  e.  ( N.  X.  N. )  ->  ( 2nd `  ( /Q `  B
) )  e.  N. )
3524, 8, 343syl 18 . . . . . 6  |-  ( B  e.  ( N.  X.  N. )  ->  ( 2nd `  ( /Q `  B
) )  e.  N. )
36 mulclpi 8517 . . . . . 6  |-  ( ( ( 1st `  ( /Q `  A ) )  e.  N.  /\  ( 2nd `  ( /Q `  B ) )  e. 
N. )  ->  (
( 1st `  ( /Q `  A ) )  .N  ( 2nd `  ( /Q `  B ) ) )  e.  N. )
3733, 35, 36syl2an 463 . . . . 5  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. ) )  ->  (
( 1st `  ( /Q `  A ) )  .N  ( 2nd `  ( /Q `  B ) ) )  e.  N. )
38 ltmpi 8528 . . . . 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 15 . . . 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 8520 . . . . . 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 10 . . . . 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 8556 . . . . . . . . 9  |-  ( A  e.  ( N.  X.  N. )  ->  A  ~Q  ( /Q `  A ) )
4323, 7syl 15 . . . . . . . . . 10  |-  ( A  e.  ( N.  X.  N. )  ->  ( /Q
`  A )  e.  ( N.  X.  N. ) )
44 enqbreq2 8544 . . . . . . . . . 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 649 . . . . . . . . 9  |-  ( A  e.  ( N.  X.  N. )  ->  ( A  ~Q  ( /Q `  A )  <->  ( ( 1st `  A )  .N  ( 2nd `  ( /Q `  A ) ) )  =  ( ( 1st `  ( /Q
`  A ) )  .N  ( 2nd `  A
) ) ) )
4642, 45mpbid 201 . . . . . . . 8  |-  ( A  e.  ( N.  X.  N. )  ->  ( ( 1st `  A )  .N  ( 2nd `  ( /Q `  A ) ) )  =  ( ( 1st `  ( /Q
`  A ) )  .N  ( 2nd `  A
) ) )
4746eqcomd 2288 . . . . . . 7  |-  ( A  e.  ( N.  X.  N. )  ->  ( ( 1st `  ( /Q
`  A ) )  .N  ( 2nd `  A
) )  =  ( ( 1st `  A
)  .N  ( 2nd `  ( /Q `  A
) ) ) )
48 nqerrel 8556 . . . . . . . 8  |-  ( B  e.  ( N.  X.  N. )  ->  B  ~Q  ( /Q `  B ) )
4924, 8syl 15 . . . . . . . . 9  |-  ( B  e.  ( N.  X.  N. )  ->  ( /Q
`  B )  e.  ( N.  X.  N. ) )
50 enqbreq2 8544 . . . . . . . . 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 649 . . . . . . . 8  |-  ( B  e.  ( N.  X.  N. )  ->  ( B  ~Q  ( /Q `  B )  <->  ( ( 1st `  B )  .N  ( 2nd `  ( /Q `  B ) ) )  =  ( ( 1st `  ( /Q
`  B ) )  .N  ( 2nd `  B
) ) ) )
5248, 51mpbid 201 . . . . . . 7  |-  ( B  e.  ( N.  X.  N. )  ->  ( ( 1st `  B )  .N  ( 2nd `  ( /Q `  B ) ) )  =  ( ( 1st `  ( /Q
`  B ) )  .N  ( 2nd `  B
) ) )
5347, 52oveqan12d 5877 . . . . . 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 8520 . . . . . . 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 5539 . . . . . . . 8  |-  ( 1st `  B )  e.  _V
56 fvex 5539 . . . . . . . 8  |-  ( 2nd `  A )  e.  _V
57 fvex 5539 . . . . . . . 8  |-  ( 1st `  ( /Q `  A
) )  e.  _V
58 mulcompi 8520 . . . . . . . 8  |-  ( x  .N  y )  =  ( y  .N  x
)
59 mulasspi 8521 . . . . . . . 8  |-  ( ( x  .N  y )  .N  z )  =  ( x  .N  (
y  .N  z ) )
60 fvex 5539 . . . . . . . 8  |-  ( 2nd `  ( /Q `  B
) )  e.  _V
6155, 56, 57, 58, 59, 60caov411 6052 . . . . . . 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 2303 . . . . . 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 8520 . . . . . . 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 5539 . . . . . . . 8  |-  ( 1st `  ( /Q `  B
) )  e.  _V
65 fvex 5539 . . . . . . . 8  |-  ( 2nd `  ( /Q `  A
) )  e.  _V
66 fvex 5539 . . . . . . . 8  |-  ( 1st `  A )  e.  _V
67 fvex 5539 . . . . . . . 8  |-  ( 2nd `  B )  e.  _V
6864, 65, 66, 58, 59, 67caov411 6052 . . . . . . 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 2303 . . . . . 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 2340 . . . . 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 4036 . . . 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 270 . . 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 277 . 2  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. ) )  ->  ( A  <pQ  B  <->  ( /Q `  A )  <Q  ( /Q `  B ) ) )
744, 16, 73pm5.21nii 342 1  |-  ( A 
<pQ  B  <->  ( /Q `  A )  <Q  ( /Q `  B ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   <.cop 3643   class class class wbr 4023   {copab 4076    X. cxp 4687   ` cfv 5255  (class class class)co 5858   1stc1st 6120   2ndc2nd 6121   N.cnpi 8466    .N cmi 8468    <N clti 8469    <pQ cltpq 8472    ~Q ceq 8473   Q.cnq 8474   /Qcerq 8476    <Q cltq 8480
This theorem is referenced by:  ltanq  8595  ltmnq  8596  1lt2nq  8597
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-omul 6484  df-er 6660  df-ni 8496  df-mi 8498  df-lti 8499  df-ltpq 8534  df-enq 8535  df-nq 8536  df-erq 8537  df-1nq 8540  df-ltnq 8542
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