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Theorem omeu 6599
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 6596 . . 3  |-  ( ( A  e.  On  /\  B  e.  On  /\  A  =/=  (/) )  ->  E. x  e.  On  E. y  e.  A  ( ( A  .o  x )  +o  y )  =  B )
2 opex 4253 . . . . . . . . 9  |-  <. x ,  y >.  e.  _V
32isseti 2807 . . . . . . . 8  |-  E. z 
z  =  <. x ,  y >.
4 19.41v 1854 . . . . . . . 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 2581 . . . . . 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 2820 . . . . . 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 2581 . . . 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 2820 . . . 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 2331 . . . . . . . . . . . . . 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 6598 . . . . . . . . . . . . . . 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 3814 . . . . . . . . . . . . 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 2331 . . . . . . . . . . 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 2331 . . . . . . . . . 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 2686 . . . . . 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 2686 . . . 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 1622 . 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 3812 . . . . . . 7  |-  ( x  =  r  ->  <. x ,  y >.  =  <. r ,  y >. )
4039eqeq2d 2307 . . . . . 6  |-  ( x  =  r  ->  (
z  =  <. x ,  y >.  <->  z  =  <. r ,  y >.
) )
41 oveq2 5882 . . . . . . . 8  |-  ( x  =  r  ->  ( A  .o  x )  =  ( A  .o  r
) )
4241oveq1d 5889 . . . . . . 7  |-  ( x  =  r  ->  (
( A  .o  x
)  +o  y )  =  ( ( A  .o  r )  +o  y ) )
4342eqeq1d 2304 . . . . . 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 3813 . . . . . . 7  |-  ( y  =  s  ->  <. r ,  y >.  =  <. r ,  s >. )
4645eqeq2d 2307 . . . . . 6  |-  ( y  =  s  ->  (
z  =  <. r ,  y >.  <->  z  =  <. r ,  s >.
) )
47 oveq2 5882 . . . . . . 7  |-  ( y  =  s  ->  (
( A  .o  r
)  +o  y )  =  ( ( A  .o  r )  +o  s ) )
4847eqeq1d 2304 . . . . . 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 2786 . . . 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 2302 . . . . . 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 2599 . . . 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 2195 . 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 1530   E.wex 1531    = wceq 1632    e. wcel 1696   E!weu 2156    =/= wne 2459   E.wrex 2557   (/)c0 3468   <.cop 3656   Oncon0 4408  (class class class)co 5874    +o coa 6492    .o comu 6493
This theorem is referenced by:  oeeui  6616  omxpenlem  6979
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-omul 6500
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