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Theorem recmulnq 8468
Description: Relationship between reciprocal and multiplication on positive fractions. (Contributed by NM, 6-Mar-1996.) (Revised by Mario Carneiro, 28-Apr-2015.) (New usage is discouraged.)
Assertion
Ref Expression
recmulnq  |-  ( A  e.  Q.  ->  (
( *Q `  A
)  =  B  <->  ( A  .Q  B )  =  1Q ) )

Proof of Theorem recmulnq
StepHypRef Expression
1 fvex 5391 . . . 4  |-  ( *Q
`  A )  e. 
_V
21a1i 12 . . 3  |-  ( A  e.  Q.  ->  ( *Q `  A )  e. 
_V )
3 eleq1 2313 . . 3  |-  ( ( *Q `  A )  =  B  ->  (
( *Q `  A
)  e.  _V  <->  B  e.  _V ) )
42, 3syl5ibcom 213 . 2  |-  ( A  e.  Q.  ->  (
( *Q `  A
)  =  B  ->  B  e.  _V )
)
5 id 21 . . . . . . 7  |-  ( ( A  .Q  B )  =  1Q  ->  ( A  .Q  B )  =  1Q )
6 1nq 8432 . . . . . . 7  |-  1Q  e.  Q.
75, 6syl6eqel 2341 . . . . . 6  |-  ( ( A  .Q  B )  =  1Q  ->  ( A  .Q  B )  e. 
Q. )
8 mulnqf 8453 . . . . . . . 8  |-  .Q  :
( Q.  X.  Q. )
--> Q.
98fdmi 5251 . . . . . . 7  |-  dom  .Q  =  ( Q.  X.  Q. )
10 0nnq 8428 . . . . . . 7  |-  -.  (/)  e.  Q.
119, 10ndmovrcl 5858 . . . . . 6  |-  ( ( A  .Q  B )  e.  Q.  ->  ( A  e.  Q.  /\  B  e.  Q. ) )
127, 11syl 17 . . . . 5  |-  ( ( A  .Q  B )  =  1Q  ->  ( A  e.  Q.  /\  B  e.  Q. ) )
1312simprd 451 . . . 4  |-  ( ( A  .Q  B )  =  1Q  ->  B  e.  Q. )
14 elex 2735 . . . 4  |-  ( B  e.  Q.  ->  B  e.  _V )
1513, 14syl 17 . . 3  |-  ( ( A  .Q  B )  =  1Q  ->  B  e.  _V )
1615a1i 12 . 2  |-  ( A  e.  Q.  ->  (
( A  .Q  B
)  =  1Q  ->  B  e.  _V ) )
17 oveq1 5717 . . . . 5  |-  ( x  =  A  ->  (
x  .Q  y )  =  ( A  .Q  y ) )
1817eqeq1d 2261 . . . 4  |-  ( x  =  A  ->  (
( x  .Q  y
)  =  1Q  <->  ( A  .Q  y )  =  1Q ) )
19 oveq2 5718 . . . . 5  |-  ( y  =  B  ->  ( A  .Q  y )  =  ( A  .Q  B
) )
2019eqeq1d 2261 . . . 4  |-  ( y  =  B  ->  (
( A  .Q  y
)  =  1Q  <->  ( A  .Q  B )  =  1Q ) )
21 nqerid 8437 . . . . . . . . . 10  |-  ( x  e.  Q.  ->  ( /Q `  x )  =  x )
22 relxp 4701 . . . . . . . . . . . 12  |-  Rel  ( N.  X.  N. )
23 elpqn 8429 . . . . . . . . . . . 12  |-  ( x  e.  Q.  ->  x  e.  ( N.  X.  N. ) )
24 1st2nd 6018 . . . . . . . . . . . 12  |-  ( ( Rel  ( N.  X.  N. )  /\  x  e.  ( N.  X.  N. ) )  ->  x  =  <. ( 1st `  x
) ,  ( 2nd `  x ) >. )
2522, 23, 24sylancr 647 . . . . . . . . . . 11  |-  ( x  e.  Q.  ->  x  =  <. ( 1st `  x
) ,  ( 2nd `  x ) >. )
2625fveq2d 5381 . . . . . . . . . 10  |-  ( x  e.  Q.  ->  ( /Q `  x )  =  ( /Q `  <. ( 1st `  x ) ,  ( 2nd `  x
) >. ) )
2721, 26eqtr3d 2287 . . . . . . . . 9  |-  ( x  e.  Q.  ->  x  =  ( /Q `  <. ( 1st `  x
) ,  ( 2nd `  x ) >. )
)
2827oveq1d 5725 . . . . . . . 8  |-  ( x  e.  Q.  ->  (
x  .Q  ( /Q
`  <. ( 2nd `  x
) ,  ( 1st `  x ) >. )
)  =  ( ( /Q `  <. ( 1st `  x ) ,  ( 2nd `  x
) >. )  .Q  ( /Q `  <. ( 2nd `  x
) ,  ( 1st `  x ) >. )
) )
29 mulerpq 8461 . . . . . . . 8  |-  ( ( /Q `  <. ( 1st `  x ) ,  ( 2nd `  x
) >. )  .Q  ( /Q `  <. ( 2nd `  x
) ,  ( 1st `  x ) >. )
)  =  ( /Q
`  ( <. ( 1st `  x ) ,  ( 2nd `  x
) >.  .pQ  <. ( 2nd `  x ) ,  ( 1st `  x )
>. ) )
3028, 29syl6eq 2301 . . . . . . 7  |-  ( x  e.  Q.  ->  (
x  .Q  ( /Q
`  <. ( 2nd `  x
) ,  ( 1st `  x ) >. )
)  =  ( /Q
`  ( <. ( 1st `  x ) ,  ( 2nd `  x
) >.  .pQ  <. ( 2nd `  x ) ,  ( 1st `  x )
>. ) ) )
31 xp1st 6001 . . . . . . . . . . 11  |-  ( x  e.  ( N.  X.  N. )  ->  ( 1st `  x )  e.  N. )
3223, 31syl 17 . . . . . . . . . 10  |-  ( x  e.  Q.  ->  ( 1st `  x )  e. 
N. )
33 xp2nd 6002 . . . . . . . . . . 11  |-  ( x  e.  ( N.  X.  N. )  ->  ( 2nd `  x )  e.  N. )
3423, 33syl 17 . . . . . . . . . 10  |-  ( x  e.  Q.  ->  ( 2nd `  x )  e. 
N. )
35 mulpipq 8444 . . . . . . . . . 10  |-  ( ( ( ( 1st `  x
)  e.  N.  /\  ( 2nd `  x )  e.  N. )  /\  ( ( 2nd `  x
)  e.  N.  /\  ( 1st `  x )  e.  N. ) )  ->  ( <. ( 1st `  x ) ,  ( 2nd `  x
) >.  .pQ  <. ( 2nd `  x ) ,  ( 1st `  x )
>. )  =  <. ( ( 1st `  x
)  .N  ( 2nd `  x ) ) ,  ( ( 2nd `  x
)  .N  ( 1st `  x ) ) >.
)
3632, 34, 34, 32, 35syl22anc 1188 . . . . . . . . 9  |-  ( x  e.  Q.  ->  ( <. ( 1st `  x
) ,  ( 2nd `  x ) >.  .pQ  <. ( 2nd `  x ) ,  ( 1st `  x
) >. )  =  <. ( ( 1st `  x
)  .N  ( 2nd `  x ) ) ,  ( ( 2nd `  x
)  .N  ( 1st `  x ) ) >.
)
37 mulcompi 8400 . . . . . . . . . 10  |-  ( ( 2nd `  x )  .N  ( 1st `  x
) )  =  ( ( 1st `  x
)  .N  ( 2nd `  x ) )
3837opeq2i 3700 . . . . . . . . 9  |-  <. (
( 1st `  x
)  .N  ( 2nd `  x ) ) ,  ( ( 2nd `  x
)  .N  ( 1st `  x ) ) >.  =  <. ( ( 1st `  x )  .N  ( 2nd `  x ) ) ,  ( ( 1st `  x )  .N  ( 2nd `  x ) )
>.
3936, 38syl6eq 2301 . . . . . . . 8  |-  ( x  e.  Q.  ->  ( <. ( 1st `  x
) ,  ( 2nd `  x ) >.  .pQ  <. ( 2nd `  x ) ,  ( 1st `  x
) >. )  =  <. ( ( 1st `  x
)  .N  ( 2nd `  x ) ) ,  ( ( 1st `  x
)  .N  ( 2nd `  x ) ) >.
)
4039fveq2d 5381 . . . . . . 7  |-  ( x  e.  Q.  ->  ( /Q `  ( <. ( 1st `  x ) ,  ( 2nd `  x
) >.  .pQ  <. ( 2nd `  x ) ,  ( 1st `  x )
>. ) )  =  ( /Q `  <. (
( 1st `  x
)  .N  ( 2nd `  x ) ) ,  ( ( 1st `  x
)  .N  ( 2nd `  x ) ) >.
) )
41 nqerid 8437 . . . . . . . . 9  |-  ( 1Q  e.  Q.  ->  ( /Q `  1Q )  =  1Q )
426, 41ax-mp 10 . . . . . . . 8  |-  ( /Q
`  1Q )  =  1Q
43 mulclpi 8397 . . . . . . . . . . 11  |-  ( ( ( 1st `  x
)  e.  N.  /\  ( 2nd `  x )  e.  N. )  -> 
( ( 1st `  x
)  .N  ( 2nd `  x ) )  e. 
N. )
4432, 34, 43syl2anc 645 . . . . . . . . . 10  |-  ( x  e.  Q.  ->  (
( 1st `  x
)  .N  ( 2nd `  x ) )  e. 
N. )
45 1nqenq 8466 . . . . . . . . . 10  |-  ( ( ( 1st `  x
)  .N  ( 2nd `  x ) )  e. 
N.  ->  1Q  ~Q  <. ( ( 1st `  x
)  .N  ( 2nd `  x ) ) ,  ( ( 1st `  x
)  .N  ( 2nd `  x ) ) >.
)
4644, 45syl 17 . . . . . . . . 9  |-  ( x  e.  Q.  ->  1Q  ~Q 
<. ( ( 1st `  x
)  .N  ( 2nd `  x ) ) ,  ( ( 1st `  x
)  .N  ( 2nd `  x ) ) >.
)
47 elpqn 8429 . . . . . . . . . . 11  |-  ( 1Q  e.  Q.  ->  1Q  e.  ( N.  X.  N. ) )
486, 47ax-mp 10 . . . . . . . . . 10  |-  1Q  e.  ( N.  X.  N. )
49 opelxpi 4628 . . . . . . . . . . 11  |-  ( ( ( ( 1st `  x
)  .N  ( 2nd `  x ) )  e. 
N.  /\  ( ( 1st `  x )  .N  ( 2nd `  x
) )  e.  N. )  ->  <. ( ( 1st `  x )  .N  ( 2nd `  x ) ) ,  ( ( 1st `  x )  .N  ( 2nd `  x ) )
>.  e.  ( N.  X.  N. ) )
5044, 44, 49syl2anc 645 . . . . . . . . . 10  |-  ( x  e.  Q.  ->  <. (
( 1st `  x
)  .N  ( 2nd `  x ) ) ,  ( ( 1st `  x
)  .N  ( 2nd `  x ) ) >.  e.  ( N.  X.  N. ) )
51 nqereq 8439 . . . . . . . . . 10  |-  ( ( 1Q  e.  ( N. 
X.  N. )  /\  <. ( ( 1st `  x
)  .N  ( 2nd `  x ) ) ,  ( ( 1st `  x
)  .N  ( 2nd `  x ) ) >.  e.  ( N.  X.  N. ) )  ->  ( 1Q  ~Q  <. ( ( 1st `  x )  .N  ( 2nd `  x ) ) ,  ( ( 1st `  x )  .N  ( 2nd `  x ) )
>. 
<->  ( /Q `  1Q )  =  ( /Q ` 
<. ( ( 1st `  x
)  .N  ( 2nd `  x ) ) ,  ( ( 1st `  x
)  .N  ( 2nd `  x ) ) >.
) ) )
5248, 50, 51sylancr 647 . . . . . . . . 9  |-  ( x  e.  Q.  ->  ( 1Q  ~Q  <. ( ( 1st `  x )  .N  ( 2nd `  x ) ) ,  ( ( 1st `  x )  .N  ( 2nd `  x ) )
>. 
<->  ( /Q `  1Q )  =  ( /Q ` 
<. ( ( 1st `  x
)  .N  ( 2nd `  x ) ) ,  ( ( 1st `  x
)  .N  ( 2nd `  x ) ) >.
) ) )
5346, 52mpbid 203 . . . . . . . 8  |-  ( x  e.  Q.  ->  ( /Q `  1Q )  =  ( /Q `  <. ( ( 1st `  x
)  .N  ( 2nd `  x ) ) ,  ( ( 1st `  x
)  .N  ( 2nd `  x ) ) >.
) )
5442, 53syl5reqr 2300 . . . . . . 7  |-  ( x  e.  Q.  ->  ( /Q `  <. ( ( 1st `  x )  .N  ( 2nd `  x ) ) ,  ( ( 1st `  x )  .N  ( 2nd `  x ) )
>. )  =  1Q )
5530, 40, 543eqtrd 2289 . . . . . 6  |-  ( x  e.  Q.  ->  (
x  .Q  ( /Q
`  <. ( 2nd `  x
) ,  ( 1st `  x ) >. )
)  =  1Q )
56 fvex 5391 . . . . . . 7  |-  ( /Q
`  <. ( 2nd `  x
) ,  ( 1st `  x ) >. )  e.  _V
57 oveq2 5718 . . . . . . . 8  |-  ( y  =  ( /Q `  <. ( 2nd `  x
) ,  ( 1st `  x ) >. )  ->  ( x  .Q  y
)  =  ( x  .Q  ( /Q `  <. ( 2nd `  x
) ,  ( 1st `  x ) >. )
) )
5857eqeq1d 2261 . . . . . . 7  |-  ( y  =  ( /Q `  <. ( 2nd `  x
) ,  ( 1st `  x ) >. )  ->  ( ( x  .Q  y )  =  1Q  <->  ( x  .Q  ( /Q
`  <. ( 2nd `  x
) ,  ( 1st `  x ) >. )
)  =  1Q ) )
5956, 58cla4ev 2812 . . . . . 6  |-  ( ( x  .Q  ( /Q
`  <. ( 2nd `  x
) ,  ( 1st `  x ) >. )
)  =  1Q  ->  E. y ( x  .Q  y )  =  1Q )
6055, 59syl 17 . . . . 5  |-  ( x  e.  Q.  ->  E. y
( x  .Q  y
)  =  1Q )
61 mulcomnq 8457 . . . . . . 7  |-  ( r  .Q  s )  =  ( s  .Q  r
)
62 mulassnq 8463 . . . . . . 7  |-  ( ( r  .Q  s )  .Q  t )  =  ( r  .Q  (
s  .Q  t ) )
63 mulidnq 8467 . . . . . . 7  |-  ( r  e.  Q.  ->  (
r  .Q  1Q )  =  r )
646, 9, 10, 61, 62, 63caovmo 5909 . . . . . 6  |-  E* y
( x  .Q  y
)  =  1Q
65 eu5 2151 . . . . . 6  |-  ( E! y ( x  .Q  y )  =  1Q  <->  ( E. y ( x  .Q  y )  =  1Q  /\  E* y
( x  .Q  y
)  =  1Q ) )
6664, 65mpbiran2 890 . . . . 5  |-  ( E! y ( x  .Q  y )  =  1Q  <->  E. y ( x  .Q  y )  =  1Q )
6760, 66sylibr 205 . . . 4  |-  ( x  e.  Q.  ->  E! y ( x  .Q  y )  =  1Q )
68 cnvimass 4940 . . . . . . . 8  |-  ( `'  .Q  " { 1Q } )  C_  dom  .Q
69 df-rq 8421 . . . . . . . 8  |-  *Q  =  ( `'  .Q  " { 1Q } )
709eqcomi 2257 . . . . . . . 8  |-  ( Q. 
X.  Q. )  =  dom  .Q
7168, 69, 703sstr4i 3138 . . . . . . 7  |-  *Q  C_  ( Q.  X.  Q. )
72 relxp 4701 . . . . . . 7  |-  Rel  ( Q.  X.  Q. )
73 relss 4682 . . . . . . 7  |-  ( *Q  C_  ( Q.  X.  Q. )  ->  ( Rel  ( Q.  X.  Q. )  ->  Rel  *Q ) )
7471, 72, 73mp2 19 . . . . . 6  |-  Rel  *Q
7569eleq2i 2317 . . . . . . . 8  |-  ( <.
x ,  y >.  e.  *Q  <->  <. x ,  y
>.  e.  ( `'  .Q  " { 1Q } ) )
76 ffn 5246 . . . . . . . . 9  |-  (  .Q  : ( Q.  X.  Q. ) --> Q.  ->  .Q  Fn  ( Q.  X.  Q. )
)
77 fniniseg 5498 . . . . . . . . 9  |-  (  .Q  Fn  ( Q.  X.  Q. )  ->  ( <.
x ,  y >.  e.  ( `'  .Q  " { 1Q } )  <->  ( <. x ,  y >.  e.  ( Q.  X.  Q. )  /\  (  .Q  `  <. x ,  y >. )  =  1Q ) ) )
788, 76, 77mp2b 11 . . . . . . . 8  |-  ( <.
x ,  y >.  e.  ( `'  .Q  " { 1Q } )  <->  ( <. x ,  y >.  e.  ( Q.  X.  Q. )  /\  (  .Q  `  <. x ,  y >. )  =  1Q ) )
79 ancom 439 . . . . . . . . 9  |-  ( (
<. x ,  y >.  e.  ( Q.  X.  Q. )  /\  (  .Q  `  <. x ,  y >.
)  =  1Q )  <-> 
( (  .Q  `  <. x ,  y >.
)  =  1Q  /\  <.
x ,  y >.  e.  ( Q.  X.  Q. ) ) )
80 ancom 439 . . . . . . . . . 10  |-  ( ( x  e.  Q.  /\  ( x  .Q  y
)  =  1Q )  <-> 
( ( x  .Q  y )  =  1Q 
/\  x  e.  Q. ) )
81 eleq1 2313 . . . . . . . . . . . . . . 15  |-  ( ( x  .Q  y )  =  1Q  ->  (
( x  .Q  y
)  e.  Q.  <->  1Q  e.  Q. ) )
826, 81mpbiri 226 . . . . . . . . . . . . . 14  |-  ( ( x  .Q  y )  =  1Q  ->  (
x  .Q  y )  e.  Q. )
839, 10ndmovrcl 5858 . . . . . . . . . . . . . 14  |-  ( ( x  .Q  y )  e.  Q.  ->  (
x  e.  Q.  /\  y  e.  Q. )
)
8482, 83syl 17 . . . . . . . . . . . . 13  |-  ( ( x  .Q  y )  =  1Q  ->  (
x  e.  Q.  /\  y  e.  Q. )
)
85 opelxpi 4628 . . . . . . . . . . . . 13  |-  ( ( x  e.  Q.  /\  y  e.  Q. )  -> 
<. x ,  y >.  e.  ( Q.  X.  Q. ) )
8684, 85syl 17 . . . . . . . . . . . 12  |-  ( ( x  .Q  y )  =  1Q  ->  <. x ,  y >.  e.  ( Q.  X.  Q. )
)
8784simpld 447 . . . . . . . . . . . 12  |-  ( ( x  .Q  y )  =  1Q  ->  x  e.  Q. )
8886, 872thd 233 . . . . . . . . . . 11  |-  ( ( x  .Q  y )  =  1Q  ->  ( <. x ,  y >.  e.  ( Q.  X.  Q. ) 
<->  x  e.  Q. )
)
8988pm5.32i 621 . . . . . . . . . 10  |-  ( ( ( x  .Q  y
)  =  1Q  /\  <.
x ,  y >.  e.  ( Q.  X.  Q. ) )  <->  ( (
x  .Q  y )  =  1Q  /\  x  e.  Q. ) )
90 df-ov 5713 . . . . . . . . . . . 12  |-  ( x  .Q  y )  =  (  .Q  `  <. x ,  y >. )
9190eqeq1i 2260 . . . . . . . . . . 11  |-  ( ( x  .Q  y )  =  1Q  <->  (  .Q  ` 
<. x ,  y >.
)  =  1Q )
9291anbi1i 679 . . . . . . . . . 10  |-  ( ( ( x  .Q  y
)  =  1Q  /\  <.
x ,  y >.  e.  ( Q.  X.  Q. ) )  <->  ( (  .Q  `  <. x ,  y
>. )  =  1Q  /\ 
<. x ,  y >.  e.  ( Q.  X.  Q. ) ) )
9380, 89, 923bitr2ri 267 . . . . . . . . 9  |-  ( ( (  .Q  `  <. x ,  y >. )  =  1Q  /\  <. x ,  y >.  e.  ( Q.  X.  Q. )
)  <->  ( x  e. 
Q.  /\  ( x  .Q  y )  =  1Q ) )
9479, 93bitri 242 . . . . . . . 8  |-  ( (
<. x ,  y >.  e.  ( Q.  X.  Q. )  /\  (  .Q  `  <. x ,  y >.
)  =  1Q )  <-> 
( x  e.  Q.  /\  ( x  .Q  y
)  =  1Q ) )
9575, 78, 943bitri 264 . . . . . . 7  |-  ( <.
x ,  y >.  e.  *Q  <->  ( x  e. 
Q.  /\  ( x  .Q  y )  =  1Q ) )
9695a1i 12 . . . . . 6  |-  (  T. 
->  ( <. x ,  y
>.  e.  *Q  <->  ( x  e.  Q.  /\  ( x  .Q  y )  =  1Q ) ) )
9774, 96opabbi2dv 4740 . . . . 5  |-  (  T. 
->  *Q  =  { <. x ,  y >.  |  ( x  e.  Q.  /\  ( x  .Q  y
)  =  1Q ) } )
9897trud 1320 . . . 4  |-  *Q  =  { <. x ,  y
>.  |  ( x  e.  Q.  /\  ( x  .Q  y )  =  1Q ) }
9918, 20, 67, 98fvopab3g 5450 . . 3  |-  ( ( A  e.  Q.  /\  B  e.  _V )  ->  ( ( *Q `  A )  =  B  <-> 
( A  .Q  B
)  =  1Q ) )
10099ex 425 . 2  |-  ( A  e.  Q.  ->  ( B  e.  _V  ->  ( ( *Q `  A
)  =  B  <->  ( A  .Q  B )  =  1Q ) ) )
1014, 16, 100pm5.21ndd 345 1  |-  ( A  e.  Q.  ->  (
( *Q `  A
)  =  B  <->  ( A  .Q  B )  =  1Q ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    T. wtru 1312   E.wex 1537    = wceq 1619    e. wcel 1621   E!weu 2114   E*wmo 2115   _Vcvv 2727    C_ wss 3078   {csn 3544   <.cop 3547   class class class wbr 3920   {copab 3973    X. cxp 4578   `'ccnv 4579   dom cdm 4580   "cima 4583   Rel wrel 4585    Fn wfn 4587   -->wf 4588   ` cfv 4592  (class class class)co 5710   1stc1st 5972   2ndc2nd 5973   N.cnpi 8346    .N cmi 8348    .pQ cmpq 8351    ~Q ceq 8353   Q.cnq 8354   1Qc1q 8355   /Qcerq 8356    .Q cmq 8358   *Qcrq 8359
This theorem is referenced by:  recidnq  8469  recrecnq  8471  reclem3pr  8553
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-recs 6274  df-rdg 6309  df-1o 6365  df-oadd 6369  df-omul 6370  df-er 6546  df-ni 8376  df-mi 8378  df-lti 8379  df-mpq 8413  df-enq 8415  df-nq 8416  df-erq 8417  df-mq 8419  df-1nq 8420  df-rq 8421
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