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

Theorem omeu 6583
Description: The division algorithm for ordinal multiplication. (Contributed by Mario Carneiro, 28-Feb-2013.)
Assertion
Ref Expression
omeu  |-  ( ( A  e.  On  /\  B  e.  On  /\  A  =/=  (/) )  ->  E! z E. x  e.  On  E. y  e.  A  ( z  =  <. x ,  y >.  /\  (
( A  .o  x
)  +o  y )  =  B ) )
Distinct variable groups:    x, A, y, z    x, B, y, z

Proof of Theorem omeu
Dummy variables  r 
s  t are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 omeulem1 6580 . . 3  |-  ( ( A  e.  On  /\  B  e.  On  /\  A  =/=  (/) )  ->  E. x  e.  On  E. y  e.  A  ( ( A  .o  x )  +o  y )  =  B )
2 opex 4237 . . . . . . . . 9  |-  <. x ,  y >.  e.  _V
32isseti 2794 . . . . . . . 8  |-  E. z 
z  =  <. x ,  y >.
4 19.41v 1842 . . . . . . . 8  |-  ( E. z ( z  = 
<. x ,  y >.  /\  ( ( A  .o  x )  +o  y
)  =  B )  <-> 
( E. z  z  =  <. x ,  y
>.  /\  ( ( A  .o  x )  +o  y )  =  B ) )
53, 4mpbiran 884 . . . . . . 7  |-  ( E. z ( z  = 
<. x ,  y >.  /\  ( ( A  .o  x )  +o  y
)  =  B )  <-> 
( ( A  .o  x )  +o  y
)  =  B )
65rexbii 2568 . . . . . 6  |-  ( E. y  e.  A  E. z ( z  = 
<. x ,  y >.  /\  ( ( A  .o  x )  +o  y
)  =  B )  <->  E. y  e.  A  ( ( A  .o  x )  +o  y
)  =  B )
7 rexcom4 2807 . . . . . 6  |-  ( E. y  e.  A  E. z ( z  = 
<. x ,  y >.  /\  ( ( A  .o  x )  +o  y
)  =  B )  <->  E. z E. y  e.  A  ( z  = 
<. x ,  y >.  /\  ( ( A  .o  x )  +o  y
)  =  B ) )
86, 7bitr3i 242 . . . . 5  |-  ( E. y  e.  A  ( ( A  .o  x
)  +o  y )  =  B  <->  E. z E. y  e.  A  ( z  =  <. x ,  y >.  /\  (
( A  .o  x
)  +o  y )  =  B ) )
98rexbii 2568 . . . 4  |-  ( E. x  e.  On  E. y  e.  A  (
( A  .o  x
)  +o  y )  =  B  <->  E. x  e.  On  E. z E. y  e.  A  ( z  =  <. x ,  y >.  /\  (
( A  .o  x
)  +o  y )  =  B ) )
10 rexcom4 2807 . . . 4  |-  ( E. x  e.  On  E. z E. y  e.  A  ( z  =  <. x ,  y >.  /\  (
( A  .o  x
)  +o  y )  =  B )  <->  E. z E. x  e.  On  E. y  e.  A  ( z  =  <. x ,  y >.  /\  (
( A  .o  x
)  +o  y )  =  B ) )
119, 10bitri 240 . . 3  |-  ( E. x  e.  On  E. y  e.  A  (
( A  .o  x
)  +o  y )  =  B  <->  E. z E. x  e.  On  E. y  e.  A  ( z  =  <. x ,  y >.  /\  (
( A  .o  x
)  +o  y )  =  B ) )
121, 11sylib 188 . 2  |-  ( ( A  e.  On  /\  B  e.  On  /\  A  =/=  (/) )  ->  E. z E. x  e.  On  E. y  e.  A  ( z  =  <. x ,  y >.  /\  (
( A  .o  x
)  +o  y )  =  B ) )
13 simp2rl 1024 . . . . . . . . . . 11  |-  ( ( ( A  e.  On  /\  B  e.  On  /\  A  =/=  (/) )  /\  (
( x  e.  On  /\  y  e.  A )  /\  ( z  = 
<. x ,  y >.  /\  ( ( A  .o  x )  +o  y
)  =  B ) )  /\  ( ( r  e.  On  /\  s  e.  A )  /\  ( t  =  <. r ,  s >.  /\  (
( A  .o  r
)  +o  s )  =  B ) ) )  ->  z  =  <. x ,  y >.
)
14 simp3rl 1028 . . . . . . . . . . . 12  |-  ( ( ( A  e.  On  /\  B  e.  On  /\  A  =/=  (/) )  /\  (
( x  e.  On  /\  y  e.  A )  /\  ( z  = 
<. x ,  y >.  /\  ( ( A  .o  x )  +o  y
)  =  B ) )  /\  ( ( r  e.  On  /\  s  e.  A )  /\  ( t  =  <. r ,  s >.  /\  (
( A  .o  r
)  +o  s )  =  B ) ) )  ->  t  =  <. r ,  s >.
)
15 simp2rr 1025 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  On  /\  B  e.  On  /\  A  =/=  (/) )  /\  (
( x  e.  On  /\  y  e.  A )  /\  ( z  = 
<. x ,  y >.  /\  ( ( A  .o  x )  +o  y
)  =  B ) )  /\  ( ( r  e.  On  /\  s  e.  A )  /\  ( t  =  <. r ,  s >.  /\  (
( A  .o  r
)  +o  s )  =  B ) ) )  ->  ( ( A  .o  x )  +o  y )  =  B )
16 simp3rr 1029 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  On  /\  B  e.  On  /\  A  =/=  (/) )  /\  (
( x  e.  On  /\  y  e.  A )  /\  ( z  = 
<. x ,  y >.  /\  ( ( A  .o  x )  +o  y
)  =  B ) )  /\  ( ( r  e.  On  /\  s  e.  A )  /\  ( t  =  <. r ,  s >.  /\  (
( A  .o  r
)  +o  s )  =  B ) ) )  ->  ( ( A  .o  r )  +o  s )  =  B )
1715, 16eqtr4d 2318 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  On  /\  B  e.  On  /\  A  =/=  (/) )  /\  (
( x  e.  On  /\  y  e.  A )  /\  ( z  = 
<. x ,  y >.  /\  ( ( A  .o  x )  +o  y
)  =  B ) )  /\  ( ( r  e.  On  /\  s  e.  A )  /\  ( t  =  <. r ,  s >.  /\  (
( A  .o  r
)  +o  s )  =  B ) ) )  ->  ( ( A  .o  x )  +o  y )  =  ( ( A  .o  r
)  +o  s ) )
18 simp11 985 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  On  /\  B  e.  On  /\  A  =/=  (/) )  /\  (
( x  e.  On  /\  y  e.  A )  /\  ( z  = 
<. x ,  y >.  /\  ( ( A  .o  x )  +o  y
)  =  B ) )  /\  ( ( r  e.  On  /\  s  e.  A )  /\  ( t  =  <. r ,  s >.  /\  (
( A  .o  r
)  +o  s )  =  B ) ) )  ->  A  e.  On )
19 simp13 987 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  On  /\  B  e.  On  /\  A  =/=  (/) )  /\  (
( x  e.  On  /\  y  e.  A )  /\  ( z  = 
<. x ,  y >.  /\  ( ( A  .o  x )  +o  y
)  =  B ) )  /\  ( ( r  e.  On  /\  s  e.  A )  /\  ( t  =  <. r ,  s >.  /\  (
( A  .o  r
)  +o  s )  =  B ) ) )  ->  A  =/=  (/) )
20 simp2ll 1022 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  On  /\  B  e.  On  /\  A  =/=  (/) )  /\  (
( x  e.  On  /\  y  e.  A )  /\  ( z  = 
<. x ,  y >.  /\  ( ( A  .o  x )  +o  y
)  =  B ) )  /\  ( ( r  e.  On  /\  s  e.  A )  /\  ( t  =  <. r ,  s >.  /\  (
( A  .o  r
)  +o  s )  =  B ) ) )  ->  x  e.  On )
21 simp2lr 1023 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  On  /\  B  e.  On  /\  A  =/=  (/) )  /\  (
( x  e.  On  /\  y  e.  A )  /\  ( z  = 
<. x ,  y >.  /\  ( ( A  .o  x )  +o  y
)  =  B ) )  /\  ( ( r  e.  On  /\  s  e.  A )  /\  ( t  =  <. r ,  s >.  /\  (
( A  .o  r
)  +o  s )  =  B ) ) )  ->  y  e.  A )
22 simp3ll 1026 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  On  /\  B  e.  On  /\  A  =/=  (/) )  /\  (
( x  e.  On  /\  y  e.  A )  /\  ( z  = 
<. x ,  y >.  /\  ( ( A  .o  x )  +o  y
)  =  B ) )  /\  ( ( r  e.  On  /\  s  e.  A )  /\  ( t  =  <. r ,  s >.  /\  (
( A  .o  r
)  +o  s )  =  B ) ) )  ->  r  e.  On )
23 simp3lr 1027 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  On  /\  B  e.  On  /\  A  =/=  (/) )  /\  (
( x  e.  On  /\  y  e.  A )  /\  ( z  = 
<. x ,  y >.  /\  ( ( A  .o  x )  +o  y
)  =  B ) )  /\  ( ( r  e.  On  /\  s  e.  A )  /\  ( t  =  <. r ,  s >.  /\  (
( A  .o  r
)  +o  s )  =  B ) ) )  ->  s  e.  A )
24 omopth2 6582 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( x  e.  On  /\  y  e.  A )  /\  ( r  e.  On  /\  s  e.  A ) )  -> 
( ( ( A  .o  x )  +o  y )  =  ( ( A  .o  r
)  +o  s )  <-> 
( x  =  r  /\  y  =  s ) ) )
2518, 19, 20, 21, 22, 23, 24syl222anc 1198 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  On  /\  B  e.  On  /\  A  =/=  (/) )  /\  (
( x  e.  On  /\  y  e.  A )  /\  ( z  = 
<. x ,  y >.  /\  ( ( A  .o  x )  +o  y
)  =  B ) )  /\  ( ( r  e.  On  /\  s  e.  A )  /\  ( t  =  <. r ,  s >.  /\  (
( A  .o  r
)  +o  s )  =  B ) ) )  ->  ( (
( A  .o  x
)  +o  y )  =  ( ( A  .o  r )  +o  s )  <->  ( x  =  r  /\  y  =  s ) ) )
2617, 25mpbid 201 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  On  /\  B  e.  On  /\  A  =/=  (/) )  /\  (
( x  e.  On  /\  y  e.  A )  /\  ( z  = 
<. x ,  y >.  /\  ( ( A  .o  x )  +o  y
)  =  B ) )  /\  ( ( r  e.  On  /\  s  e.  A )  /\  ( t  =  <. r ,  s >.  /\  (
( A  .o  r
)  +o  s )  =  B ) ) )  ->  ( x  =  r  /\  y  =  s ) )
27 opeq12 3798 . . . . . . . . . . . . 13  |-  ( ( x  =  r  /\  y  =  s )  -> 
<. x ,  y >.  =  <. r ,  s
>. )
2826, 27syl 15 . . . . . . . . . . . 12  |-  ( ( ( A  e.  On  /\  B  e.  On  /\  A  =/=  (/) )  /\  (
( x  e.  On  /\  y  e.  A )  /\  ( z  = 
<. x ,  y >.  /\  ( ( A  .o  x )  +o  y
)  =  B ) )  /\  ( ( r  e.  On  /\  s  e.  A )  /\  ( t  =  <. r ,  s >.  /\  (
( A  .o  r
)  +o  s )  =  B ) ) )  ->  <. x ,  y >.  =  <. r ,  s >. )
2914, 28eqtr4d 2318 . . . . . . . . . . 11  |-  ( ( ( A  e.  On  /\  B  e.  On  /\  A  =/=  (/) )  /\  (
( x  e.  On  /\  y  e.  A )  /\  ( z  = 
<. x ,  y >.  /\  ( ( A  .o  x )  +o  y
)  =  B ) )  /\  ( ( r  e.  On  /\  s  e.  A )  /\  ( t  =  <. r ,  s >.  /\  (
( A  .o  r
)  +o  s )  =  B ) ) )  ->  t  =  <. x ,  y >.
)
3013, 29eqtr4d 2318 . . . . . . . . . 10  |-  ( ( ( A  e.  On  /\  B  e.  On  /\  A  =/=  (/) )  /\  (
( x  e.  On  /\  y  e.  A )  /\  ( z  = 
<. x ,  y >.  /\  ( ( A  .o  x )  +o  y
)  =  B ) )  /\  ( ( r  e.  On  /\  s  e.  A )  /\  ( t  =  <. r ,  s >.  /\  (
( A  .o  r
)  +o  s )  =  B ) ) )  ->  z  =  t )
31303expia 1153 . . . . . . . . 9  |-  ( ( ( A  e.  On  /\  B  e.  On  /\  A  =/=  (/) )  /\  (
( x  e.  On  /\  y  e.  A )  /\  ( z  = 
<. x ,  y >.  /\  ( ( A  .o  x )  +o  y
)  =  B ) ) )  ->  (
( ( r  e.  On  /\  s  e.  A )  /\  (
t  =  <. r ,  s >.  /\  (
( A  .o  r
)  +o  s )  =  B ) )  ->  z  =  t ) )
3231exp4b 590 . . . . . . . 8  |-  ( ( A  e.  On  /\  B  e.  On  /\  A  =/=  (/) )  ->  (
( ( x  e.  On  /\  y  e.  A )  /\  (
z  =  <. x ,  y >.  /\  (
( A  .o  x
)  +o  y )  =  B ) )  ->  ( ( r  e.  On  /\  s  e.  A )  ->  (
( t  =  <. r ,  s >.  /\  (
( A  .o  r
)  +o  s )  =  B )  -> 
z  =  t ) ) ) )
3332exp3a 425 . . . . . . 7  |-  ( ( A  e.  On  /\  B  e.  On  /\  A  =/=  (/) )  ->  (
( x  e.  On  /\  y  e.  A )  ->  ( ( z  =  <. x ,  y
>.  /\  ( ( A  .o  x )  +o  y )  =  B )  ->  ( (
r  e.  On  /\  s  e.  A )  ->  ( ( t  = 
<. r ,  s >.  /\  ( ( A  .o  r )  +o  s
)  =  B )  ->  z  =  t ) ) ) ) )
3433rexlimdvv 2673 . . . . . 6  |-  ( ( A  e.  On  /\  B  e.  On  /\  A  =/=  (/) )  ->  ( E. x  e.  On  E. y  e.  A  ( z  =  <. x ,  y >.  /\  (
( A  .o  x
)  +o  y )  =  B )  -> 
( ( r  e.  On  /\  s  e.  A )  ->  (
( t  =  <. r ,  s >.  /\  (
( A  .o  r
)  +o  s )  =  B )  -> 
z  =  t ) ) ) )
3534imp 418 . . . . 5  |-  ( ( ( A  e.  On  /\  B  e.  On  /\  A  =/=  (/) )  /\  E. x  e.  On  E. y  e.  A  ( z  =  <. x ,  y
>.  /\  ( ( A  .o  x )  +o  y )  =  B ) )  ->  (
( r  e.  On  /\  s  e.  A )  ->  ( ( t  =  <. r ,  s
>.  /\  ( ( A  .o  r )  +o  s )  =  B )  ->  z  =  t ) ) )
3635rexlimdvv 2673 . . . 4  |-  ( ( ( A  e.  On  /\  B  e.  On  /\  A  =/=  (/) )  /\  E. x  e.  On  E. y  e.  A  ( z  =  <. x ,  y
>.  /\  ( ( A  .o  x )  +o  y )  =  B ) )  ->  ( E. r  e.  On  E. s  e.  A  ( t  =  <. r ,  s >.  /\  (
( A  .o  r
)  +o  s )  =  B )  -> 
z  =  t ) )
3736expimpd 586 . . 3  |-  ( ( A  e.  On  /\  B  e.  On  /\  A  =/=  (/) )  ->  (
( E. x  e.  On  E. y  e.  A  ( z  = 
<. x ,  y >.  /\  ( ( A  .o  x )  +o  y
)  =  B )  /\  E. r  e.  On  E. s  e.  A  ( t  = 
<. r ,  s >.  /\  ( ( A  .o  r )  +o  s
)  =  B ) )  ->  z  =  t ) )
3837alrimivv 1618 . 2  |-  ( ( A  e.  On  /\  B  e.  On  /\  A  =/=  (/) )  ->  A. z A. t ( ( E. x  e.  On  E. y  e.  A  (
z  =  <. x ,  y >.  /\  (
( A  .o  x
)  +o  y )  =  B )  /\  E. r  e.  On  E. s  e.  A  (
t  =  <. r ,  s >.  /\  (
( A  .o  r
)  +o  s )  =  B ) )  ->  z  =  t ) )
39 opeq1 3796 . . . . . . 7  |-  ( x  =  r  ->  <. x ,  y >.  =  <. r ,  y >. )
4039eqeq2d 2294 . . . . . 6  |-  ( x  =  r  ->  (
z  =  <. x ,  y >.  <->  z  =  <. r ,  y >.
) )
41 oveq2 5866 . . . . . . . 8  |-  ( x  =  r  ->  ( A  .o  x )  =  ( A  .o  r
) )
4241oveq1d 5873 . . . . . . 7  |-  ( x  =  r  ->  (
( A  .o  x
)  +o  y )  =  ( ( A  .o  r )  +o  y ) )
4342eqeq1d 2291 . . . . . 6  |-  ( x  =  r  ->  (
( ( A  .o  x )  +o  y
)  =  B  <->  ( ( A  .o  r )  +o  y )  =  B ) )
4440, 43anbi12d 691 . . . . 5  |-  ( x  =  r  ->  (
( z  =  <. x ,  y >.  /\  (
( A  .o  x
)  +o  y )  =  B )  <->  ( z  =  <. r ,  y
>.  /\  ( ( A  .o  r )  +o  y )  =  B ) ) )
45 opeq2 3797 . . . . . . 7  |-  ( y  =  s  ->  <. r ,  y >.  =  <. r ,  s >. )
4645eqeq2d 2294 . . . . . 6  |-  ( y  =  s  ->  (
z  =  <. r ,  y >.  <->  z  =  <. r ,  s >.
) )
47 oveq2 5866 . . . . . . 7  |-  ( y  =  s  ->  (
( A  .o  r
)  +o  y )  =  ( ( A  .o  r )  +o  s ) )
4847eqeq1d 2291 . . . . . 6  |-  ( y  =  s  ->  (
( ( A  .o  r )  +o  y
)  =  B  <->  ( ( A  .o  r )  +o  s )  =  B ) )
4946, 48anbi12d 691 . . . . 5  |-  ( y  =  s  ->  (
( z  =  <. r ,  y >.  /\  (
( A  .o  r
)  +o  y )  =  B )  <->  ( z  =  <. r ,  s
>.  /\  ( ( A  .o  r )  +o  s )  =  B ) ) )
5044, 49cbvrex2v 2773 . . . 4  |-  ( E. x  e.  On  E. y  e.  A  (
z  =  <. x ,  y >.  /\  (
( A  .o  x
)  +o  y )  =  B )  <->  E. r  e.  On  E. s  e.  A  ( z  = 
<. r ,  s >.  /\  ( ( A  .o  r )  +o  s
)  =  B ) )
51 eqeq1 2289 . . . . . 6  |-  ( z  =  t  ->  (
z  =  <. r ,  s >.  <->  t  =  <. r ,  s >.
) )
5251anbi1d 685 . . . . 5  |-  ( z  =  t  ->  (
( z  =  <. r ,  s >.  /\  (
( A  .o  r
)  +o  s )  =  B )  <->  ( t  =  <. r ,  s
>.  /\  ( ( A  .o  r )  +o  s )  =  B ) ) )
53522rexbidv 2586 . . . 4  |-  ( z  =  t  ->  ( E. r  e.  On  E. s  e.  A  ( z  =  <. r ,  s >.  /\  (
( A  .o  r
)  +o  s )  =  B )  <->  E. r  e.  On  E. s  e.  A  ( t  = 
<. r ,  s >.  /\  ( ( A  .o  r )  +o  s
)  =  B ) ) )
5450, 53syl5bb 248 . . 3  |-  ( z  =  t  ->  ( E. x  e.  On  E. y  e.  A  ( z  =  <. x ,  y >.  /\  (
( A  .o  x
)  +o  y )  =  B )  <->  E. r  e.  On  E. s  e.  A  ( t  = 
<. r ,  s >.  /\  ( ( A  .o  r )  +o  s
)  =  B ) ) )
5554eu4 2182 . 2  |-  ( E! z E. x  e.  On  E. y  e.  A  ( z  = 
<. x ,  y >.  /\  ( ( A  .o  x )  +o  y
)  =  B )  <-> 
( E. z E. x  e.  On  E. y  e.  A  (
z  =  <. x ,  y >.  /\  (
( A  .o  x
)  +o  y )  =  B )  /\  A. z A. t ( ( E. x  e.  On  E. y  e.  A  ( z  = 
<. x ,  y >.  /\  ( ( A  .o  x )  +o  y
)  =  B )  /\  E. r  e.  On  E. s  e.  A  ( t  = 
<. r ,  s >.  /\  ( ( A  .o  r )  +o  s
)  =  B ) )  ->  z  =  t ) ) )
5612, 38, 55sylanbrc 645 1  |-  ( ( A  e.  On  /\  B  e.  On  /\  A  =/=  (/) )  ->  E! z E. x  e.  On  E. y  e.  A  ( z  =  <. x ,  y >.  /\  (
( A  .o  x
)  +o  y )  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934   A.wal 1527   E.wex 1528    = wceq 1623    e. wcel 1684   E!weu 2143    =/= wne 2446   E.wrex 2544   (/)c0 3455   <.cop 3643   Oncon0 4392  (class class class)co 5858    +o coa 6476    .o comu 6477
This theorem is referenced by:  oeeui  6600  omxpenlem  6963
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-omul 6484
  Copyright terms: Public domain W3C validator