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Theorem omeu 7298
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 7295 . . 3  |-  ( ( A  e.  On  /\  B  e.  On  /\  A  =/=  (/) )  ->  E. x  e.  On  E. y  e.  A  ( ( A  .o  x )  +o  y )  =  B )
2 opex 4685 . . . . . . . . 9  |-  <. x ,  y >.  e.  _V
32isseti 3086 . . . . . . . 8  |-  E. z 
z  =  <. x ,  y >.
4 19.41v 1823 . . . . . . . 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 926 . . . . . . 7  |-  ( E. z ( z  = 
<. x ,  y >.  /\  ( ( A  .o  x )  +o  y
)  =  B )  <-> 
( ( A  .o  x )  +o  y
)  =  B )
65rexbii 2924 . . . . . 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 3101 . . . . . 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 254 . . . . 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 2924 . . . 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 3101 . . . 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 252 . . 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 199 . 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 1074 . . . . . . . . . . 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 1078 . . . . . . . . . . . 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 1075 . . . . . . . . . . . . . . 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 1079 . . . . . . . . . . . . . . 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 2466 . . . . . . . . . . . . . 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 1035 . . . . . . . . . . . . . . 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 1037 . . . . . . . . . . . . . . 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 1072 . . . . . . . . . . . . . . 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 1073 . . . . . . . . . . . . . . 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 1076 . . . . . . . . . . . . . . 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 1077 . . . . . . . . . . . . . . 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 7297 . . . . . . . . . . . . . . 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 1280 . . . . . . . . . . . . . 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 213 . . . . . . . . . . . . 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 4189 . . . . . . . . . . . . 13  |-  ( ( x  =  r  /\  y  =  s )  -> 
<. x ,  y >.  =  <. r ,  s
>. )
2826, 27syl 17 . . . . . . . . . . . 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 2466 . . . . . . . . . . 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 2466 . . . . . . . . . 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 1207 . . . . . . . . 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 610 . . . . . . . 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 ) ) ) )
3332expd 437 . . . . . . 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 2920 . . . . . 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 430 . . . . 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 2920 . . . 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 606 . . 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 1768 . 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 4187 . . . . . . 7  |-  ( x  =  r  ->  <. x ,  y >.  =  <. r ,  y >. )
4039eqeq2d 2436 . . . . . 6  |-  ( x  =  r  ->  (
z  =  <. x ,  y >.  <->  z  =  <. r ,  y >.
) )
41 oveq2 6314 . . . . . . . 8  |-  ( x  =  r  ->  ( A  .o  x )  =  ( A  .o  r
) )
4241oveq1d 6321 . . . . . . 7  |-  ( x  =  r  ->  (
( A  .o  x
)  +o  y )  =  ( ( A  .o  r )  +o  y ) )
4342eqeq1d 2424 . . . . . 6  |-  ( x  =  r  ->  (
( ( A  .o  x )  +o  y
)  =  B  <->  ( ( A  .o  r )  +o  y )  =  B ) )
4440, 43anbi12d 715 . . . . 5  |-  ( x  =  r  ->  (
( z  =  <. x ,  y >.  /\  (
( A  .o  x
)  +o  y )  =  B )  <->  ( z  =  <. r ,  y
>.  /\  ( ( A  .o  r )  +o  y )  =  B ) ) )
45 opeq2 4188 . . . . . . 7  |-  ( y  =  s  ->  <. r ,  y >.  =  <. r ,  s >. )
4645eqeq2d 2436 . . . . . 6  |-  ( y  =  s  ->  (
z  =  <. r ,  y >.  <->  z  =  <. r ,  s >.
) )
47 oveq2 6314 . . . . . . 7  |-  ( y  =  s  ->  (
( A  .o  r
)  +o  y )  =  ( ( A  .o  r )  +o  s ) )
4847eqeq1d 2424 . . . . . 6  |-  ( y  =  s  ->  (
( ( A  .o  r )  +o  y
)  =  B  <->  ( ( A  .o  r )  +o  s )  =  B ) )
4946, 48anbi12d 715 . . . . 5  |-  ( y  =  s  ->  (
( z  =  <. r ,  y >.  /\  (
( A  .o  r
)  +o  y )  =  B )  <->  ( z  =  <. r ,  s
>.  /\  ( ( A  .o  r )  +o  s )  =  B ) ) )
5044, 49cbvrex2v 3063 . . . 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 2426 . . . . . 6  |-  ( z  =  t  ->  (
z  =  <. r ,  s >.  <->  t  =  <. r ,  s >.
) )
5251anbi1d 709 . . . . 5  |-  ( z  =  t  ->  (
( z  =  <. r ,  s >.  /\  (
( A  .o  r
)  +o  s )  =  B )  <->  ( t  =  <. r ,  s
>.  /\  ( ( A  .o  r )  +o  s )  =  B ) ) )
53522rexbidv 2943 . . . 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 260 . . 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 2317 . 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 668 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 setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982   A.wal 1435    = wceq 1437   E.wex 1657    e. wcel 1872   E!weu 2269    =/= wne 2614   E.wrex 2772   (/)c0 3761   <.cop 4004   Oncon0 5442  (class class class)co 6306    +o coa 7191    .o comu 7192
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6598
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-reu 2778  df-rmo 2779  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-int 4256  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6309  df-oprab 6310  df-mpt2 6311  df-om 6708  df-1st 6808  df-2nd 6809  df-wrecs 7040  df-recs 7102  df-rdg 7140  df-1o 7194  df-oadd 7198  df-omul 7199
This theorem is referenced by:  oeeui  7315  omxpenlem  7683
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