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Theorem lterpq 9344
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 9284 . . . 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 5072 . . . 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 3534 . . 3  |-  <pQ  C_  (
( N.  X.  N. )  X.  ( N.  X.  N. ) )
43brel 5047 . 2  |-  ( A 
<pQ  B  ->  ( A  e.  ( N.  X.  N. )  /\  B  e.  ( N.  X.  N. )
) )
5 ltrelnq 9300 . . . 4  |-  <Q  C_  ( Q.  X.  Q. )
65brel 5047 . . 3  |-  ( ( /Q `  A ) 
<Q  ( /Q `  B
)  ->  ( ( /Q `  A )  e. 
Q.  /\  ( /Q `  B )  e.  Q. ) )
7 elpqn 9299 . . . 4  |-  ( ( /Q `  A )  e.  Q.  ->  ( /Q `  A )  e.  ( N.  X.  N. ) )
8 elpqn 9299 . . . 4  |-  ( ( /Q `  B )  e.  Q.  ->  ( /Q `  B )  e.  ( N.  X.  N. ) )
9 nqerf 9304 . . . . . . 7  |-  /Q :
( N.  X.  N. )
--> Q.
109fdmi 5734 . . . . . 6  |-  dom  /Q  =  ( N.  X.  N. )
11 0nelxp 5026 . . . . . 6  |-  -.  (/)  e.  ( N.  X.  N. )
1210, 11ndmfvrcl 5889 . . . . 5  |-  ( ( /Q `  A )  e.  ( N.  X.  N. )  ->  A  e.  ( N.  X.  N. ) )
1310, 11ndmfvrcl 5889 . . . . 5  |-  ( ( /Q `  B )  e.  ( N.  X.  N. )  ->  B  e.  ( N.  X.  N. ) )
1412, 13anim12i 566 . . . 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 477 . . 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 6811 . . . . 5  |-  ( A  e.  ( N.  X.  N. )  ->  ( 1st `  A )  e.  N. )
18 xp2nd 6812 . . . . 5  |-  ( B  e.  ( N.  X.  N. )  ->  ( 2nd `  B )  e.  N. )
19 mulclpi 9267 . . . . 5  |-  ( ( ( 1st `  A
)  e.  N.  /\  ( 2nd `  B )  e.  N. )  -> 
( ( 1st `  A
)  .N  ( 2nd `  B ) )  e. 
N. )
2017, 18, 19syl2an 477 . . . 4  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. ) )  ->  (
( 1st `  A
)  .N  ( 2nd `  B ) )  e. 
N. )
21 ltmpi 9278 . . . 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 9305 . . . 4  |-  ( A  e.  ( N.  X.  N. )  ->  ( /Q
`  A )  e. 
Q. )
24 nqercl 9305 . . . 4  |-  ( B  e.  ( N.  X.  N. )  ->  ( /Q
`  B )  e. 
Q. )
25 ordpinq 9317 . . . 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 477 . . 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 6818 . . . . . 6  |-  ( A  e.  ( N.  X.  N. )  ->  A  = 
<. ( 1st `  A
) ,  ( 2nd `  A ) >. )
28 1st2nd2 6818 . . . . . 6  |-  ( B  e.  ( N.  X.  N. )  ->  B  = 
<. ( 1st `  B
) ,  ( 2nd `  B ) >. )
2927, 28breqan12d 4462 . . . . 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 9316 . . . . 5  |-  ( <.
( 1st `  A
) ,  ( 2nd `  A ) >.  <pQ  <. ( 1st `  B ) ,  ( 2nd `  B
) >. 
<->  ( ( 1st `  A
)  .N  ( 2nd `  B ) )  <N 
( ( 1st `  B
)  .N  ( 2nd `  A ) ) )
3129, 30syl6bb 261 . . . 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 6811 . . . . . . 7  |-  ( ( /Q `  A )  e.  ( N.  X.  N. )  ->  ( 1st `  ( /Q `  A
) )  e.  N. )
3323, 7, 323syl 20 . . . . . 6  |-  ( A  e.  ( N.  X.  N. )  ->  ( 1st `  ( /Q `  A
) )  e.  N. )
34 xp2nd 6812 . . . . . . 7  |-  ( ( /Q `  B )  e.  ( N.  X.  N. )  ->  ( 2nd `  ( /Q `  B
) )  e.  N. )
3524, 8, 343syl 20 . . . . . 6  |-  ( B  e.  ( N.  X.  N. )  ->  ( 2nd `  ( /Q `  B
) )  e.  N. )
36 mulclpi 9267 . . . . . 6  |-  ( ( ( 1st `  ( /Q `  A ) )  e.  N.  /\  ( 2nd `  ( /Q `  B ) )  e. 
N. )  ->  (
( 1st `  ( /Q `  A ) )  .N  ( 2nd `  ( /Q `  B ) ) )  e.  N. )
3733, 35, 36syl2an 477 . . . . 5  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. ) )  ->  (
( 1st `  ( /Q `  A ) )  .N  ( 2nd `  ( /Q `  B ) ) )  e.  N. )
38 ltmpi 9278 . . . . 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 9270 . . . . . 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 9306 . . . . . . . . 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 9294 . . . . . . . . . 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 668 . . . . . . . . 9  |-  ( A  e.  ( N.  X.  N. )  ->  ( A  ~Q  ( /Q `  A )  <->  ( ( 1st `  A )  .N  ( 2nd `  ( /Q `  A ) ) )  =  ( ( 1st `  ( /Q
`  A ) )  .N  ( 2nd `  A
) ) ) )
4642, 45mpbid 210 . . . . . . . 8  |-  ( A  e.  ( N.  X.  N. )  ->  ( ( 1st `  A )  .N  ( 2nd `  ( /Q `  A ) ) )  =  ( ( 1st `  ( /Q
`  A ) )  .N  ( 2nd `  A
) ) )
4746eqcomd 2475 . . . . . . 7  |-  ( A  e.  ( N.  X.  N. )  ->  ( ( 1st `  ( /Q
`  A ) )  .N  ( 2nd `  A
) )  =  ( ( 1st `  A
)  .N  ( 2nd `  ( /Q `  A
) ) ) )
48 nqerrel 9306 . . . . . . . 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 9294 . . . . . . . . 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 668 . . . . . . . 8  |-  ( B  e.  ( N.  X.  N. )  ->  ( B  ~Q  ( /Q `  B )  <->  ( ( 1st `  B )  .N  ( 2nd `  ( /Q `  B ) ) )  =  ( ( 1st `  ( /Q
`  B ) )  .N  ( 2nd `  B
) ) ) )
5248, 51mpbid 210 . . . . . . 7  |-  ( B  e.  ( N.  X.  N. )  ->  ( ( 1st `  B )  .N  ( 2nd `  ( /Q `  B ) ) )  =  ( ( 1st `  ( /Q
`  B ) )  .N  ( 2nd `  B
) ) )
5347, 52oveqan12d 6301 . . . . . 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 9270 . . . . . . 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 5874 . . . . . . . 8  |-  ( 1st `  B )  e.  _V
56 fvex 5874 . . . . . . . 8  |-  ( 2nd `  A )  e.  _V
57 fvex 5874 . . . . . . . 8  |-  ( 1st `  ( /Q `  A
) )  e.  _V
58 mulcompi 9270 . . . . . . . 8  |-  ( x  .N  y )  =  ( y  .N  x
)
59 mulasspi 9271 . . . . . . . 8  |-  ( ( x  .N  y )  .N  z )  =  ( x  .N  (
y  .N  z ) )
60 fvex 5874 . . . . . . . 8  |-  ( 2nd `  ( /Q `  B
) )  e.  _V
6155, 56, 57, 58, 59, 60caov411 6489 . . . . . . 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 2496 . . . . . 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 9270 . . . . . . 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 5874 . . . . . . . 8  |-  ( 1st `  ( /Q `  B
) )  e.  _V
65 fvex 5874 . . . . . . . 8  |-  ( 2nd `  ( /Q `  A
) )  e.  _V
66 fvex 5874 . . . . . . . 8  |-  ( 1st `  A )  e.  _V
67 fvex 5874 . . . . . . . 8  |-  ( 2nd `  B )  e.  _V
6864, 65, 66, 58, 59, 67caov411 6489 . . . . . . 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 2496 . . . . . 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 2533 . . . . 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 4460 . . . 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 279 . . 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 286 . 2  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. ) )  ->  ( A  <pQ  B  <->  ( /Q `  A )  <Q  ( /Q `  B ) ) )
744, 16, 73pm5.21nii 353 1  |-  ( A 
<pQ  B  <->  ( /Q `  A )  <Q  ( /Q `  B ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   <.cop 4033   class class class wbr 4447   {copab 4504    X. cxp 4997   ` cfv 5586  (class class class)co 6282   1stc1st 6779   2ndc2nd 6780   N.cnpi 9218    .N cmi 9220    <N clti 9221    <pQ cltpq 9224    ~Q ceq 9225   Q.cnq 9226   /Qcerq 9228    <Q cltq 9232
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-omul 7132  df-er 7308  df-ni 9246  df-mi 9248  df-lti 9249  df-ltpq 9284  df-enq 9285  df-nq 9286  df-erq 9287  df-1nq 9290  df-ltnq 9292
This theorem is referenced by:  ltanq  9345  ltmnq  9346  1lt2nq  9347
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