MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  recmulnq Unicode version

Theorem recmulnq 8583
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 ) )
Dummy variables  x  y  s  r  t are mutually distinct and distinct from all other variables.

Proof of Theorem recmulnq
StepHypRef Expression
1 fvex 5499 . . . 4  |-  ( *Q
`  A )  e. 
_V
21a1i 12 . . 3  |-  ( A  e.  Q.  ->  ( *Q `  A )  e. 
_V )
3 eleq1 2344 . . 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 8547 . . . . . . 7  |-  1Q  e.  Q.
75, 6syl6eqel 2372 . . . . . 6  |-  ( ( A  .Q  B )  =  1Q  ->  ( A  .Q  B )  e. 
Q. )
8 mulnqf 8568 . . . . . . . 8  |-  .Q  :
( Q.  X.  Q. )
--> Q.
98fdmi 5359 . . . . . . 7  |-  dom  .Q  =  ( Q.  X.  Q. )
10 0nnq 8543 . . . . . . 7  |-  -.  (/)  e.  Q.
119, 10ndmovrcl 5967 . . . . . 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 2797 . . . 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 5826 . . . . 5  |-  ( x  =  A  ->  (
x  .Q  y )  =  ( A  .Q  y ) )
1817eqeq1d 2292 . . . 4  |-  ( x  =  A  ->  (
( x  .Q  y
)  =  1Q  <->  ( A  .Q  y )  =  1Q ) )
19 oveq2 5827 . . . . 5  |-  ( y  =  B  ->  ( A  .Q  y )  =  ( A  .Q  B
) )
2019eqeq1d 2292 . . . 4  |-  ( y  =  B  ->  (
( A  .Q  y
)  =  1Q  <->  ( A  .Q  B )  =  1Q ) )
21 nqerid 8552 . . . . . . . . . 10  |-  ( x  e.  Q.  ->  ( /Q `  x )  =  x )
22 relxp 4793 . . . . . . . . . . . 12  |-  Rel  ( N.  X.  N. )
23 elpqn 8544 . . . . . . . . . . . 12  |-  ( x  e.  Q.  ->  x  e.  ( N.  X.  N. ) )
24 1st2nd 6127 . . . . . . . . . . . 12  |-  ( ( Rel  ( N.  X.  N. )  /\  x  e.  ( N.  X.  N. ) )  ->  x  =  <. ( 1st `  x
) ,  ( 2nd `  x ) >. )
2522, 23, 24sylancr 646 . . . . . . . . . . 11  |-  ( x  e.  Q.  ->  x  =  <. ( 1st `  x
) ,  ( 2nd `  x ) >. )
2625fveq2d 5489 . . . . . . . . . 10  |-  ( x  e.  Q.  ->  ( /Q `  x )  =  ( /Q `  <. ( 1st `  x ) ,  ( 2nd `  x
) >. ) )
2721, 26eqtr3d 2318 . . . . . . . . 9  |-  ( x  e.  Q.  ->  x  =  ( /Q `  <. ( 1st `  x
) ,  ( 2nd `  x ) >. )
)
2827oveq1d 5834 . . . . . . . 8  |-  ( x  e.  Q.  ->  (
x  .Q  ( /Q
`  <. ( 2nd `  x
) ,  ( 1st `  x ) >. )
)  =  ( ( /Q `  <. ( 1st `  x ) ,  ( 2nd `  x
) >. )  .Q  ( /Q `  <. ( 2nd `  x
) ,  ( 1st `  x ) >. )
) )
29 mulerpq 8576 . . . . . . . 8  |-  ( ( /Q `  <. ( 1st `  x ) ,  ( 2nd `  x
) >. )  .Q  ( /Q `  <. ( 2nd `  x
) ,  ( 1st `  x ) >. )
)  =  ( /Q
`  ( <. ( 1st `  x ) ,  ( 2nd `  x
) >.  .pQ  <. ( 2nd `  x ) ,  ( 1st `  x )
>. ) )
3028, 29syl6eq 2332 . . . . . . 7  |-  ( x  e.  Q.  ->  (
x  .Q  ( /Q
`  <. ( 2nd `  x
) ,  ( 1st `  x ) >. )
)  =  ( /Q
`  ( <. ( 1st `  x ) ,  ( 2nd `  x
) >.  .pQ  <. ( 2nd `  x ) ,  ( 1st `  x )
>. ) ) )
31 xp1st 6110 . . . . . . . . . . 11  |-  ( x  e.  ( N.  X.  N. )  ->  ( 1st `  x )  e.  N. )
3223, 31syl 17 . . . . . . . . . 10  |-  ( x  e.  Q.  ->  ( 1st `  x )  e. 
N. )
33 xp2nd 6111 . . . . . . . . . . 11  |-  ( x  e.  ( N.  X.  N. )  ->  ( 2nd `  x )  e.  N. )
3423, 33syl 17 . . . . . . . . . 10  |-  ( x  e.  Q.  ->  ( 2nd `  x )  e. 
N. )
35 mulpipq 8559 . . . . . . . . . 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 8515 . . . . . . . . . 10  |-  ( ( 2nd `  x )  .N  ( 1st `  x
) )  =  ( ( 1st `  x
)  .N  ( 2nd `  x ) )
3837opeq2i 3801 . . . . . . . . 9  |-  <. (
( 1st `  x
)  .N  ( 2nd `  x ) ) ,  ( ( 2nd `  x
)  .N  ( 1st `  x ) ) >.  =  <. ( ( 1st `  x )  .N  ( 2nd `  x ) ) ,  ( ( 1st `  x )  .N  ( 2nd `  x ) )
>.
3936, 38syl6eq 2332 . . . . . . . 8  |-  ( x  e.  Q.  ->  ( <. ( 1st `  x
) ,  ( 2nd `  x ) >.  .pQ  <. ( 2nd `  x ) ,  ( 1st `  x
) >. )  =  <. ( ( 1st `  x
)  .N  ( 2nd `  x ) ) ,  ( ( 1st `  x
)  .N  ( 2nd `  x ) ) >.
)
4039fveq2d 5489 . . . . . . 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 8552 . . . . . . . . 9  |-  ( 1Q  e.  Q.  ->  ( /Q `  1Q )  =  1Q )
426, 41ax-mp 10 . . . . . . . 8  |-  ( /Q
`  1Q )  =  1Q
43 mulclpi 8512 . . . . . . . . . . 11  |-  ( ( ( 1st `  x
)  e.  N.  /\  ( 2nd `  x )  e.  N. )  -> 
( ( 1st `  x
)  .N  ( 2nd `  x ) )  e. 
N. )
4432, 34, 43syl2anc 644 . . . . . . . . . 10  |-  ( x  e.  Q.  ->  (
( 1st `  x
)  .N  ( 2nd `  x ) )  e. 
N. )
45 1nqenq 8581 . . . . . . . . . 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 8544 . . . . . . . . . . 11  |-  ( 1Q  e.  Q.  ->  1Q  e.  ( N.  X.  N. ) )
486, 47ax-mp 10 . . . . . . . . . 10  |-  1Q  e.  ( N.  X.  N. )
49 opelxpi 4720 . . . . . . . . . . 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 644 . . . . . . . . . 10  |-  ( x  e.  Q.  ->  <. (
( 1st `  x
)  .N  ( 2nd `  x ) ) ,  ( ( 1st `  x
)  .N  ( 2nd `  x ) ) >.  e.  ( N.  X.  N. ) )
51 nqereq 8554 . . . . . . . . . 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 646 . . . . . . . . 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 2331 . . . . . . 7  |-  ( x  e.  Q.  ->  ( /Q `  <. ( ( 1st `  x )  .N  ( 2nd `  x ) ) ,  ( ( 1st `  x )  .N  ( 2nd `  x ) )
>. )  =  1Q )
5530, 40, 543eqtrd 2320 . . . . . 6  |-  ( x  e.  Q.  ->  (
x  .Q  ( /Q
`  <. ( 2nd `  x
) ,  ( 1st `  x ) >. )
)  =  1Q )
56 fvex 5499 . . . . . . 7  |-  ( /Q
`  <. ( 2nd `  x
) ,  ( 1st `  x ) >. )  e.  _V
57 oveq2 5827 . . . . . . . 8  |-  ( y  =  ( /Q `  <. ( 2nd `  x
) ,  ( 1st `  x ) >. )  ->  ( x  .Q  y
)  =  ( x  .Q  ( /Q `  <. ( 2nd `  x
) ,  ( 1st `  x ) >. )
) )
5857eqeq1d 2292 . . . . . . 7  |-  ( y  =  ( /Q `  <. ( 2nd `  x
) ,  ( 1st `  x ) >. )  ->  ( ( x  .Q  y )  =  1Q  <->  ( x  .Q  ( /Q
`  <. ( 2nd `  x
) ,  ( 1st `  x ) >. )
)  =  1Q ) )
5956, 58spcev 2876 . . . . . 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 8572 . . . . . . 7  |-  ( r  .Q  s )  =  ( s  .Q  r
)
62 mulassnq 8578 . . . . . . 7  |-  ( ( r  .Q  s )  .Q  t )  =  ( r  .Q  (
s  .Q  t ) )
63 mulidnq 8582 . . . . . . 7  |-  ( r  e.  Q.  ->  (
r  .Q  1Q )  =  r )
646, 9, 10, 61, 62, 63caovmo 6018 . . . . . 6  |-  E* y
( x  .Q  y
)  =  1Q
65 eu5 2182 . . . . . 6  |-  ( E! y ( x  .Q  y )  =  1Q  <->  ( E. y ( x  .Q  y )  =  1Q  /\  E* y
( x  .Q  y
)  =  1Q ) )
6664, 65mpbiran2 887 . . . . 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 5032 . . . . . . . 8  |-  ( `'  .Q  " { 1Q } )  C_  dom  .Q
69 df-rq 8536 . . . . . . . 8  |-  *Q  =  ( `'  .Q  " { 1Q } )
709eqcomi 2288 . . . . . . . 8  |-  ( Q. 
X.  Q. )  =  dom  .Q
7168, 69, 703sstr4i 3218 . . . . . . 7  |-  *Q  C_  ( Q.  X.  Q. )
72 relxp 4793 . . . . . . 7  |-  Rel  ( Q.  X.  Q. )
73 relss 4774 . . . . . . 7  |-  ( *Q  C_  ( Q.  X.  Q. )  ->  ( Rel  ( Q.  X.  Q. )  ->  Rel  *Q ) )
7471, 72, 73mp2 19 . . . . . 6  |-  Rel  *Q
7569eleq2i 2348 . . . . . . . 8  |-  ( <.
x ,  y >.  e.  *Q  <->  <. x ,  y
>.  e.  ( `'  .Q  " { 1Q } ) )
76 ffn 5354 . . . . . . . . 9  |-  (  .Q  : ( Q.  X.  Q. ) --> Q.  ->  .Q  Fn  ( Q.  X.  Q. )
)
77 fniniseg 5607 . . . . . . . . 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 2344 . . . . . . . . . . . . . . 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 5967 . . . . . . . . . . . . . 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 4720 . . . . . . . . . . . . 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 620 . . . . . . . . . 10  |-  ( ( ( x  .Q  y
)  =  1Q  /\  <.
x ,  y >.  e.  ( Q.  X.  Q. ) )  <->  ( (
x  .Q  y )  =  1Q  /\  x  e.  Q. ) )
90 df-ov 5822 . . . . . . . . . . . 12  |-  ( x  .Q  y )  =  (  .Q  `  <. x ,  y >. )
9190eqeq1i 2291 . . . . . . . . . . 11  |-  ( ( x  .Q  y )  =  1Q  <->  (  .Q  ` 
<. x ,  y >.
)  =  1Q )
9291anbi1i 678 . . . . . . . . . 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 4832 . . . . 5  |-  (  T. 
->  *Q  =  { <. x ,  y >.  |  ( x  e.  Q.  /\  ( x  .Q  y
)  =  1Q ) } )
9897trud 1316 . . . 4  |-  *Q  =  { <. x ,  y
>.  |  ( x  e.  Q.  /\  ( x  .Q  y )  =  1Q ) }
9918, 20, 67, 98fvopab3g 5559 . . 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 1309   E.wex 1529    = wceq 1624    e. wcel 1685   E!weu 2144   E*wmo 2145   _Vcvv 2789    C_ wss 3153   {csn 3641   <.cop 3644   class class class wbr 4024   {copab 4077    X. cxp 4686   `'ccnv 4687   dom cdm 4688   "cima 4691   Rel wrel 4693    Fn wfn 5216   -->wf 5217   ` cfv 5221  (class class class)co 5819   1stc1st 6081   2ndc2nd 6082   N.cnpi 8461    .N cmi 8463    .pQ cmpq 8466    ~Q ceq 8468   Q.cnq 8469   1Qc1q 8470   /Qcerq 8471    .Q cmq 8473   *Qcrq 8474
This theorem is referenced by:  recidnq  8584  recrecnq  8586  reclem3pr  8668
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-1st 6083  df-2nd 6084  df-recs 6383  df-rdg 6418  df-1o 6474  df-oadd 6478  df-omul 6479  df-er 6655  df-ni 8491  df-mi 8493  df-lti 8494  df-mpq 8528  df-enq 8530  df-nq 8531  df-erq 8532  df-mq 8534  df-1nq 8535  df-rq 8536
  Copyright terms: Public domain W3C validator