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Theorem recmulnq 8774
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
Dummy variables  x  y  s  r  t are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fvex 5682 . . . 4  |-  ( *Q
`  A )  e. 
_V
21a1i 11 . . 3  |-  ( A  e.  Q.  ->  ( *Q `  A )  e. 
_V )
3 eleq1 2447 . . 3  |-  ( ( *Q `  A )  =  B  ->  (
( *Q `  A
)  e.  _V  <->  B  e.  _V ) )
42, 3syl5ibcom 212 . 2  |-  ( A  e.  Q.  ->  (
( *Q `  A
)  =  B  ->  B  e.  _V )
)
5 id 20 . . . . . . 7  |-  ( ( A  .Q  B )  =  1Q  ->  ( A  .Q  B )  =  1Q )
6 1nq 8738 . . . . . . 7  |-  1Q  e.  Q.
75, 6syl6eqel 2475 . . . . . 6  |-  ( ( A  .Q  B )  =  1Q  ->  ( A  .Q  B )  e. 
Q. )
8 mulnqf 8759 . . . . . . . 8  |-  .Q  :
( Q.  X.  Q. )
--> Q.
98fdmi 5536 . . . . . . 7  |-  dom  .Q  =  ( Q.  X.  Q. )
10 0nnq 8734 . . . . . . 7  |-  -.  (/)  e.  Q.
119, 10ndmovrcl 6172 . . . . . 6  |-  ( ( A  .Q  B )  e.  Q.  ->  ( A  e.  Q.  /\  B  e.  Q. ) )
127, 11syl 16 . . . . 5  |-  ( ( A  .Q  B )  =  1Q  ->  ( A  e.  Q.  /\  B  e.  Q. ) )
1312simprd 450 . . . 4  |-  ( ( A  .Q  B )  =  1Q  ->  B  e.  Q. )
14 elex 2907 . . . 4  |-  ( B  e.  Q.  ->  B  e.  _V )
1513, 14syl 16 . . 3  |-  ( ( A  .Q  B )  =  1Q  ->  B  e.  _V )
1615a1i 11 . 2  |-  ( A  e.  Q.  ->  (
( A  .Q  B
)  =  1Q  ->  B  e.  _V ) )
17 oveq1 6027 . . . . 5  |-  ( x  =  A  ->  (
x  .Q  y )  =  ( A  .Q  y ) )
1817eqeq1d 2395 . . . 4  |-  ( x  =  A  ->  (
( x  .Q  y
)  =  1Q  <->  ( A  .Q  y )  =  1Q ) )
19 oveq2 6028 . . . . 5  |-  ( y  =  B  ->  ( A  .Q  y )  =  ( A  .Q  B
) )
2019eqeq1d 2395 . . . 4  |-  ( y  =  B  ->  (
( A  .Q  y
)  =  1Q  <->  ( A  .Q  B )  =  1Q ) )
21 nqerid 8743 . . . . . . . . . 10  |-  ( x  e.  Q.  ->  ( /Q `  x )  =  x )
22 relxp 4923 . . . . . . . . . . . 12  |-  Rel  ( N.  X.  N. )
23 elpqn 8735 . . . . . . . . . . . 12  |-  ( x  e.  Q.  ->  x  e.  ( N.  X.  N. ) )
24 1st2nd 6332 . . . . . . . . . . . 12  |-  ( ( Rel  ( N.  X.  N. )  /\  x  e.  ( N.  X.  N. ) )  ->  x  =  <. ( 1st `  x
) ,  ( 2nd `  x ) >. )
2522, 23, 24sylancr 645 . . . . . . . . . . 11  |-  ( x  e.  Q.  ->  x  =  <. ( 1st `  x
) ,  ( 2nd `  x ) >. )
2625fveq2d 5672 . . . . . . . . . 10  |-  ( x  e.  Q.  ->  ( /Q `  x )  =  ( /Q `  <. ( 1st `  x ) ,  ( 2nd `  x
) >. ) )
2721, 26eqtr3d 2421 . . . . . . . . 9  |-  ( x  e.  Q.  ->  x  =  ( /Q `  <. ( 1st `  x
) ,  ( 2nd `  x ) >. )
)
2827oveq1d 6035 . . . . . . . 8  |-  ( x  e.  Q.  ->  (
x  .Q  ( /Q
`  <. ( 2nd `  x
) ,  ( 1st `  x ) >. )
)  =  ( ( /Q `  <. ( 1st `  x ) ,  ( 2nd `  x
) >. )  .Q  ( /Q `  <. ( 2nd `  x
) ,  ( 1st `  x ) >. )
) )
29 mulerpq 8767 . . . . . . . 8  |-  ( ( /Q `  <. ( 1st `  x ) ,  ( 2nd `  x
) >. )  .Q  ( /Q `  <. ( 2nd `  x
) ,  ( 1st `  x ) >. )
)  =  ( /Q
`  ( <. ( 1st `  x ) ,  ( 2nd `  x
) >.  .pQ  <. ( 2nd `  x ) ,  ( 1st `  x )
>. ) )
3028, 29syl6eq 2435 . . . . . . 7  |-  ( x  e.  Q.  ->  (
x  .Q  ( /Q
`  <. ( 2nd `  x
) ,  ( 1st `  x ) >. )
)  =  ( /Q
`  ( <. ( 1st `  x ) ,  ( 2nd `  x
) >.  .pQ  <. ( 2nd `  x ) ,  ( 1st `  x )
>. ) ) )
31 xp1st 6315 . . . . . . . . . . 11  |-  ( x  e.  ( N.  X.  N. )  ->  ( 1st `  x )  e.  N. )
3223, 31syl 16 . . . . . . . . . 10  |-  ( x  e.  Q.  ->  ( 1st `  x )  e. 
N. )
33 xp2nd 6316 . . . . . . . . . . 11  |-  ( x  e.  ( N.  X.  N. )  ->  ( 2nd `  x )  e.  N. )
3423, 33syl 16 . . . . . . . . . 10  |-  ( x  e.  Q.  ->  ( 2nd `  x )  e. 
N. )
35 mulpipq 8750 . . . . . . . . . 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 1185 . . . . . . . . 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 8706 . . . . . . . . . 10  |-  ( ( 2nd `  x )  .N  ( 1st `  x
) )  =  ( ( 1st `  x
)  .N  ( 2nd `  x ) )
3837opeq2i 3930 . . . . . . . . 9  |-  <. (
( 1st `  x
)  .N  ( 2nd `  x ) ) ,  ( ( 2nd `  x
)  .N  ( 1st `  x ) ) >.  =  <. ( ( 1st `  x )  .N  ( 2nd `  x ) ) ,  ( ( 1st `  x )  .N  ( 2nd `  x ) )
>.
3936, 38syl6eq 2435 . . . . . . . 8  |-  ( x  e.  Q.  ->  ( <. ( 1st `  x
) ,  ( 2nd `  x ) >.  .pQ  <. ( 2nd `  x ) ,  ( 1st `  x
) >. )  =  <. ( ( 1st `  x
)  .N  ( 2nd `  x ) ) ,  ( ( 1st `  x
)  .N  ( 2nd `  x ) ) >.
)
4039fveq2d 5672 . . . . . . 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 8743 . . . . . . . . 9  |-  ( 1Q  e.  Q.  ->  ( /Q `  1Q )  =  1Q )
426, 41ax-mp 8 . . . . . . . 8  |-  ( /Q
`  1Q )  =  1Q
43 mulclpi 8703 . . . . . . . . . . 11  |-  ( ( ( 1st `  x
)  e.  N.  /\  ( 2nd `  x )  e.  N. )  -> 
( ( 1st `  x
)  .N  ( 2nd `  x ) )  e. 
N. )
4432, 34, 43syl2anc 643 . . . . . . . . . 10  |-  ( x  e.  Q.  ->  (
( 1st `  x
)  .N  ( 2nd `  x ) )  e. 
N. )
45 1nqenq 8772 . . . . . . . . . 10  |-  ( ( ( 1st `  x
)  .N  ( 2nd `  x ) )  e. 
N.  ->  1Q  ~Q  <. ( ( 1st `  x
)  .N  ( 2nd `  x ) ) ,  ( ( 1st `  x
)  .N  ( 2nd `  x ) ) >.
)
4644, 45syl 16 . . . . . . . . 9  |-  ( x  e.  Q.  ->  1Q  ~Q 
<. ( ( 1st `  x
)  .N  ( 2nd `  x ) ) ,  ( ( 1st `  x
)  .N  ( 2nd `  x ) ) >.
)
47 elpqn 8735 . . . . . . . . . . 11  |-  ( 1Q  e.  Q.  ->  1Q  e.  ( N.  X.  N. ) )
486, 47ax-mp 8 . . . . . . . . . 10  |-  1Q  e.  ( N.  X.  N. )
49 opelxpi 4850 . . . . . . . . . . 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 643 . . . . . . . . . 10  |-  ( x  e.  Q.  ->  <. (
( 1st `  x
)  .N  ( 2nd `  x ) ) ,  ( ( 1st `  x
)  .N  ( 2nd `  x ) ) >.  e.  ( N.  X.  N. ) )
51 nqereq 8745 . . . . . . . . . 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 645 . . . . . . . . 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 202 . . . . . . . 8  |-  ( x  e.  Q.  ->  ( /Q `  1Q )  =  ( /Q `  <. ( ( 1st `  x
)  .N  ( 2nd `  x ) ) ,  ( ( 1st `  x
)  .N  ( 2nd `  x ) ) >.
) )
5442, 53syl5reqr 2434 . . . . . . 7  |-  ( x  e.  Q.  ->  ( /Q `  <. ( ( 1st `  x )  .N  ( 2nd `  x ) ) ,  ( ( 1st `  x )  .N  ( 2nd `  x ) )
>. )  =  1Q )
5530, 40, 543eqtrd 2423 . . . . . 6  |-  ( x  e.  Q.  ->  (
x  .Q  ( /Q
`  <. ( 2nd `  x
) ,  ( 1st `  x ) >. )
)  =  1Q )
56 fvex 5682 . . . . . . 7  |-  ( /Q
`  <. ( 2nd `  x
) ,  ( 1st `  x ) >. )  e.  _V
57 oveq2 6028 . . . . . . . 8  |-  ( y  =  ( /Q `  <. ( 2nd `  x
) ,  ( 1st `  x ) >. )  ->  ( x  .Q  y
)  =  ( x  .Q  ( /Q `  <. ( 2nd `  x
) ,  ( 1st `  x ) >. )
) )
5857eqeq1d 2395 . . . . . . 7  |-  ( y  =  ( /Q `  <. ( 2nd `  x
) ,  ( 1st `  x ) >. )  ->  ( ( x  .Q  y )  =  1Q  <->  ( x  .Q  ( /Q
`  <. ( 2nd `  x
) ,  ( 1st `  x ) >. )
)  =  1Q ) )
5956, 58spcev 2986 . . . . . 6  |-  ( ( x  .Q  ( /Q
`  <. ( 2nd `  x
) ,  ( 1st `  x ) >. )
)  =  1Q  ->  E. y ( x  .Q  y )  =  1Q )
6055, 59syl 16 . . . . 5  |-  ( x  e.  Q.  ->  E. y
( x  .Q  y
)  =  1Q )
61 mulcomnq 8763 . . . . . . 7  |-  ( r  .Q  s )  =  ( s  .Q  r
)
62 mulassnq 8769 . . . . . . 7  |-  ( ( r  .Q  s )  .Q  t )  =  ( r  .Q  (
s  .Q  t ) )
63 mulidnq 8773 . . . . . . 7  |-  ( r  e.  Q.  ->  (
r  .Q  1Q )  =  r )
646, 9, 10, 61, 62, 63caovmo 6223 . . . . . 6  |-  E* y
( x  .Q  y
)  =  1Q
65 eu5 2276 . . . . . 6  |-  ( E! y ( x  .Q  y )  =  1Q  <->  ( E. y ( x  .Q  y )  =  1Q  /\  E* y
( x  .Q  y
)  =  1Q ) )
6664, 65mpbiran2 886 . . . . 5  |-  ( E! y ( x  .Q  y )  =  1Q  <->  E. y ( x  .Q  y )  =  1Q )
6760, 66sylibr 204 . . . 4  |-  ( x  e.  Q.  ->  E! y ( x  .Q  y )  =  1Q )
68 cnvimass 5164 . . . . . . . 8  |-  ( `'  .Q  " { 1Q } )  C_  dom  .Q
69 df-rq 8727 . . . . . . . 8  |-  *Q  =  ( `'  .Q  " { 1Q } )
709eqcomi 2391 . . . . . . . 8  |-  ( Q. 
X.  Q. )  =  dom  .Q
7168, 69, 703sstr4i 3330 . . . . . . 7  |-  *Q  C_  ( Q.  X.  Q. )
72 relxp 4923 . . . . . . 7  |-  Rel  ( Q.  X.  Q. )
73 relss 4903 . . . . . . 7  |-  ( *Q  C_  ( Q.  X.  Q. )  ->  ( Rel  ( Q.  X.  Q. )  ->  Rel  *Q ) )
7471, 72, 73mp2 9 . . . . . 6  |-  Rel  *Q
7569eleq2i 2451 . . . . . . . 8  |-  ( <.
x ,  y >.  e.  *Q  <->  <. x ,  y
>.  e.  ( `'  .Q  " { 1Q } ) )
76 ffn 5531 . . . . . . . . 9  |-  (  .Q  : ( Q.  X.  Q. ) --> Q.  ->  .Q  Fn  ( Q.  X.  Q. )
)
77 fniniseg 5790 . . . . . . . . 9  |-  (  .Q  Fn  ( Q.  X.  Q. )  ->  ( <.
x ,  y >.  e.  ( `'  .Q  " { 1Q } )  <->  ( <. x ,  y >.  e.  ( Q.  X.  Q. )  /\  (  .Q  `  <. x ,  y >. )  =  1Q ) ) )
788, 76, 77mp2b 10 . . . . . . . 8  |-  ( <.
x ,  y >.  e.  ( `'  .Q  " { 1Q } )  <->  ( <. x ,  y >.  e.  ( Q.  X.  Q. )  /\  (  .Q  `  <. x ,  y >. )  =  1Q ) )
79 ancom 438 . . . . . . . . 9  |-  ( (
<. x ,  y >.  e.  ( Q.  X.  Q. )  /\  (  .Q  `  <. x ,  y >.
)  =  1Q )  <-> 
( (  .Q  `  <. x ,  y >.
)  =  1Q  /\  <.
x ,  y >.  e.  ( Q.  X.  Q. ) ) )
80 ancom 438 . . . . . . . . . 10  |-  ( ( x  e.  Q.  /\  ( x  .Q  y
)  =  1Q )  <-> 
( ( x  .Q  y )  =  1Q 
/\  x  e.  Q. ) )
81 eleq1 2447 . . . . . . . . . . . . . . 15  |-  ( ( x  .Q  y )  =  1Q  ->  (
( x  .Q  y
)  e.  Q.  <->  1Q  e.  Q. ) )
826, 81mpbiri 225 . . . . . . . . . . . . . 14  |-  ( ( x  .Q  y )  =  1Q  ->  (
x  .Q  y )  e.  Q. )
839, 10ndmovrcl 6172 . . . . . . . . . . . . . 14  |-  ( ( x  .Q  y )  e.  Q.  ->  (
x  e.  Q.  /\  y  e.  Q. )
)
8482, 83syl 16 . . . . . . . . . . . . 13  |-  ( ( x  .Q  y )  =  1Q  ->  (
x  e.  Q.  /\  y  e.  Q. )
)
85 opelxpi 4850 . . . . . . . . . . . . 13  |-  ( ( x  e.  Q.  /\  y  e.  Q. )  -> 
<. x ,  y >.  e.  ( Q.  X.  Q. ) )
8684, 85syl 16 . . . . . . . . . . . 12  |-  ( ( x  .Q  y )  =  1Q  ->  <. x ,  y >.  e.  ( Q.  X.  Q. )
)
8784simpld 446 . . . . . . . . . . . 12  |-  ( ( x  .Q  y )  =  1Q  ->  x  e.  Q. )
8886, 872thd 232 . . . . . . . . . . 11  |-  ( ( x  .Q  y )  =  1Q  ->  ( <. x ,  y >.  e.  ( Q.  X.  Q. ) 
<->  x  e.  Q. )
)
8988pm5.32i 619 . . . . . . . . . 10  |-  ( ( ( x  .Q  y
)  =  1Q  /\  <.
x ,  y >.  e.  ( Q.  X.  Q. ) )  <->  ( (
x  .Q  y )  =  1Q  /\  x  e.  Q. ) )
90 df-ov 6023 . . . . . . . . . . . 12  |-  ( x  .Q  y )  =  (  .Q  `  <. x ,  y >. )
9190eqeq1i 2394 . . . . . . . . . . 11  |-  ( ( x  .Q  y )  =  1Q  <->  (  .Q  ` 
<. x ,  y >.
)  =  1Q )
9291anbi1i 677 . . . . . . . . . 10  |-  ( ( ( x  .Q  y
)  =  1Q  /\  <.
x ,  y >.  e.  ( Q.  X.  Q. ) )  <->  ( (  .Q  `  <. x ,  y
>. )  =  1Q  /\ 
<. x ,  y >.  e.  ( Q.  X.  Q. ) ) )
9380, 89, 923bitr2ri 266 . . . . . . . . 9  |-  ( ( (  .Q  `  <. x ,  y >. )  =  1Q  /\  <. x ,  y >.  e.  ( Q.  X.  Q. )
)  <->  ( x  e. 
Q.  /\  ( x  .Q  y )  =  1Q ) )
9479, 93bitri 241 . . . . . . . 8  |-  ( (
<. x ,  y >.  e.  ( Q.  X.  Q. )  /\  (  .Q  `  <. x ,  y >.
)  =  1Q )  <-> 
( x  e.  Q.  /\  ( x  .Q  y
)  =  1Q ) )
9575, 78, 943bitri 263 . . . . . . 7  |-  ( <.
x ,  y >.  e.  *Q  <->  ( x  e. 
Q.  /\  ( x  .Q  y )  =  1Q ) )
9695a1i 11 . . . . . 6  |-  (  T. 
->  ( <. x ,  y
>.  e.  *Q  <->  ( x  e.  Q.  /\  ( x  .Q  y )  =  1Q ) ) )
9774, 96opabbi2dv 4962 . . . . 5  |-  (  T. 
->  *Q  =  { <. x ,  y >.  |  ( x  e.  Q.  /\  ( x  .Q  y
)  =  1Q ) } )
9897trud 1329 . . . 4  |-  *Q  =  { <. x ,  y
>.  |  ( x  e.  Q.  /\  ( x  .Q  y )  =  1Q ) }
9918, 20, 67, 98fvopab3g 5741 . . 3  |-  ( ( A  e.  Q.  /\  B  e.  _V )  ->  ( ( *Q `  A )  =  B  <-> 
( A  .Q  B
)  =  1Q ) )
10099ex 424 . 2  |-  ( A  e.  Q.  ->  ( B  e.  _V  ->  ( ( *Q `  A
)  =  B  <->  ( A  .Q  B )  =  1Q ) ) )
1014, 16, 100pm5.21ndd 344 1  |-  ( A  e.  Q.  ->  (
( *Q `  A
)  =  B  <->  ( A  .Q  B )  =  1Q ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    T. wtru 1322   E.wex 1547    = wceq 1649    e. wcel 1717   E!weu 2238   E*wmo 2239   _Vcvv 2899    C_ wss 3263   {csn 3757   <.cop 3760   class class class wbr 4153   {copab 4206    X. cxp 4816   `'ccnv 4817   dom cdm 4818   "cima 4821   Rel wrel 4823    Fn wfn 5389   -->wf 5390   ` cfv 5394  (class class class)co 6020   1stc1st 6286   2ndc2nd 6287   N.cnpi 8652    .N cmi 8654    .pQ cmpq 8657    ~Q ceq 8659   Q.cnq 8660   1Qc1q 8661   /Qcerq 8662    .Q cmq 8664   *Qcrq 8665
This theorem is referenced by:  recidnq  8775  recrecnq  8777  reclem3pr  8859
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-recs 6569  df-rdg 6604  df-1o 6660  df-oadd 6664  df-omul 6665  df-er 6841  df-ni 8682  df-mi 8684  df-lti 8685  df-mpq 8719  df-enq 8721  df-nq 8722  df-erq 8723  df-mq 8725  df-1nq 8726  df-rq 8727
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