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Theorem lterpq 9365
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 9305 . . . 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 5083 . . . 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 3529 . . 3  |-  <pQ  C_  (
( N.  X.  N. )  X.  ( N.  X.  N. ) )
43brel 5057 . 2  |-  ( A 
<pQ  B  ->  ( A  e.  ( N.  X.  N. )  /\  B  e.  ( N.  X.  N. )
) )
5 ltrelnq 9321 . . . 4  |-  <Q  C_  ( Q.  X.  Q. )
65brel 5057 . . 3  |-  ( ( /Q `  A ) 
<Q  ( /Q `  B
)  ->  ( ( /Q `  A )  e. 
Q.  /\  ( /Q `  B )  e.  Q. ) )
7 elpqn 9320 . . . 4  |-  ( ( /Q `  A )  e.  Q.  ->  ( /Q `  A )  e.  ( N.  X.  N. ) )
8 elpqn 9320 . . . 4  |-  ( ( /Q `  B )  e.  Q.  ->  ( /Q `  B )  e.  ( N.  X.  N. ) )
9 nqerf 9325 . . . . . . 7  |-  /Q :
( N.  X.  N. )
--> Q.
109fdmi 5742 . . . . . 6  |-  dom  /Q  =  ( N.  X.  N. )
11 0nelxp 5036 . . . . . 6  |-  -.  (/)  e.  ( N.  X.  N. )
1210, 11ndmfvrcl 5897 . . . . 5  |-  ( ( /Q `  A )  e.  ( N.  X.  N. )  ->  A  e.  ( N.  X.  N. ) )
1310, 11ndmfvrcl 5897 . . . . 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 6829 . . . . 5  |-  ( A  e.  ( N.  X.  N. )  ->  ( 1st `  A )  e.  N. )
18 xp2nd 6830 . . . . 5  |-  ( B  e.  ( N.  X.  N. )  ->  ( 2nd `  B )  e.  N. )
19 mulclpi 9288 . . . . 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 9299 . . . 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 9326 . . . 4  |-  ( A  e.  ( N.  X.  N. )  ->  ( /Q
`  A )  e. 
Q. )
24 nqercl 9326 . . . 4  |-  ( B  e.  ( N.  X.  N. )  ->  ( /Q
`  B )  e. 
Q. )
25 ordpinq 9338 . . . 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 6836 . . . . . 6  |-  ( A  e.  ( N.  X.  N. )  ->  A  = 
<. ( 1st `  A
) ,  ( 2nd `  A ) >. )
28 1st2nd2 6836 . . . . . 6  |-  ( B  e.  ( N.  X.  N. )  ->  B  = 
<. ( 1st `  B
) ,  ( 2nd `  B ) >. )
2927, 28breqan12d 4471 . . . . 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 9337 . . . . 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 6829 . . . . . . 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 6830 . . . . . . 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 9288 . . . . . 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 9299 . . . . 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 9291 . . . . . 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 9327 . . . . . . . . 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 9315 . . . . . . . . . 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 2465 . . . . . . 7  |-  ( A  e.  ( N.  X.  N. )  ->  ( ( 1st `  ( /Q
`  A ) )  .N  ( 2nd `  A
) )  =  ( ( 1st `  A
)  .N  ( 2nd `  ( /Q `  A
) ) ) )
48 nqerrel 9327 . . . . . . . 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 9315 . . . . . . . . 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 6315 . . . . . 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 9291 . . . . . . 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 5882 . . . . . . . 8  |-  ( 1st `  B )  e.  _V
56 fvex 5882 . . . . . . . 8  |-  ( 2nd `  A )  e.  _V
57 fvex 5882 . . . . . . . 8  |-  ( 1st `  ( /Q `  A
) )  e.  _V
58 mulcompi 9291 . . . . . . . 8  |-  ( x  .N  y )  =  ( y  .N  x
)
59 mulasspi 9292 . . . . . . . 8  |-  ( ( x  .N  y )  .N  z )  =  ( x  .N  (
y  .N  z ) )
60 fvex 5882 . . . . . . . 8  |-  ( 2nd `  ( /Q `  B
) )  e.  _V
6155, 56, 57, 58, 59, 60caov411 6506 . . . . . . 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 2486 . . . . . 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 9291 . . . . . . 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 5882 . . . . . . . 8  |-  ( 1st `  ( /Q `  B
) )  e.  _V
65 fvex 5882 . . . . . . . 8  |-  ( 2nd `  ( /Q `  A
) )  e.  _V
66 fvex 5882 . . . . . . . 8  |-  ( 1st `  A )  e.  _V
67 fvex 5882 . . . . . . . 8  |-  ( 2nd `  B )  e.  _V
6864, 65, 66, 58, 59, 67caov411 6506 . . . . . . 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 2486 . . . . . 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 2523 . . . . 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 4469 . . . 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 1395    e. wcel 1819   <.cop 4038   class class class wbr 4456   {copab 4514    X. cxp 5006   ` cfv 5594  (class class class)co 6296   1stc1st 6797   2ndc2nd 6798   N.cnpi 9239    .N cmi 9241    <N clti 9242    <pQ cltpq 9245    ~Q ceq 9246   Q.cnq 9247   /Qcerq 9249    <Q cltq 9253
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-omul 7153  df-er 7329  df-ni 9267  df-mi 9269  df-lti 9270  df-ltpq 9305  df-enq 9306  df-nq 9307  df-erq 9308  df-1nq 9311  df-ltnq 9313
This theorem is referenced by:  ltanq  9366  ltmnq  9367  1lt2nq  9368
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