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Theorem omeu 7029
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 7026 . . 3  |-  ( ( A  e.  On  /\  B  e.  On  /\  A  =/=  (/) )  ->  E. x  e.  On  E. y  e.  A  ( ( A  .o  x )  +o  y )  =  B )
2 opex 4561 . . . . . . . . 9  |-  <. x ,  y >.  e.  _V
32isseti 2983 . . . . . . . 8  |-  E. z 
z  =  <. x ,  y >.
4 19.41v 1920 . . . . . . . 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 909 . . . . . . 7  |-  ( E. z ( z  = 
<. x ,  y >.  /\  ( ( A  .o  x )  +o  y
)  =  B )  <-> 
( ( A  .o  x )  +o  y
)  =  B )
65rexbii 2745 . . . . . 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 2997 . . . . . 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 251 . . . . 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 2745 . . . 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 2997 . . . 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 249 . . 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 196 . 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 1057 . . . . . . . . . . 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 1061 . . . . . . . . . . . 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 1058 . . . . . . . . . . . . . . 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 1062 . . . . . . . . . . . . . . 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 2478 . . . . . . . . . . . . . 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 1018 . . . . . . . . . . . . . . 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 1020 . . . . . . . . . . . . . . 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 1055 . . . . . . . . . . . . . . 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 1056 . . . . . . . . . . . . . . 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 1059 . . . . . . . . . . . . . . 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 1060 . . . . . . . . . . . . . . 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 7028 . . . . . . . . . . . . . . 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 1234 . . . . . . . . . . . . . 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 210 . . . . . . . . . . . . 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 4066 . . . . . . . . . . . . 13  |-  ( ( x  =  r  /\  y  =  s )  -> 
<. x ,  y >.  =  <. r ,  s
>. )
2826, 27syl 16 . . . . . . . . . . . 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 2478 . . . . . . . . . . 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 2478 . . . . . . . . . 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 1189 . . . . . . . . 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 607 . . . . . . . 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 436 . . . . . . 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 2852 . . . . . 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 429 . . . . 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 2852 . . . 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 603 . . 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 1686 . 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 4064 . . . . . . 7  |-  ( x  =  r  ->  <. x ,  y >.  =  <. r ,  y >. )
4039eqeq2d 2454 . . . . . 6  |-  ( x  =  r  ->  (
z  =  <. x ,  y >.  <->  z  =  <. r ,  y >.
) )
41 oveq2 6104 . . . . . . . 8  |-  ( x  =  r  ->  ( A  .o  x )  =  ( A  .o  r
) )
4241oveq1d 6111 . . . . . . 7  |-  ( x  =  r  ->  (
( A  .o  x
)  +o  y )  =  ( ( A  .o  r )  +o  y ) )
4342eqeq1d 2451 . . . . . 6  |-  ( x  =  r  ->  (
( ( A  .o  x )  +o  y
)  =  B  <->  ( ( A  .o  r )  +o  y )  =  B ) )
4440, 43anbi12d 710 . . . . 5  |-  ( x  =  r  ->  (
( z  =  <. x ,  y >.  /\  (
( A  .o  x
)  +o  y )  =  B )  <->  ( z  =  <. r ,  y
>.  /\  ( ( A  .o  r )  +o  y )  =  B ) ) )
45 opeq2 4065 . . . . . . 7  |-  ( y  =  s  ->  <. r ,  y >.  =  <. r ,  s >. )
4645eqeq2d 2454 . . . . . 6  |-  ( y  =  s  ->  (
z  =  <. r ,  y >.  <->  z  =  <. r ,  s >.
) )
47 oveq2 6104 . . . . . . 7  |-  ( y  =  s  ->  (
( A  .o  r
)  +o  y )  =  ( ( A  .o  r )  +o  s ) )
4847eqeq1d 2451 . . . . . 6  |-  ( y  =  s  ->  (
( ( A  .o  r )  +o  y
)  =  B  <->  ( ( A  .o  r )  +o  s )  =  B ) )
4946, 48anbi12d 710 . . . . 5  |-  ( y  =  s  ->  (
( z  =  <. r ,  y >.  /\  (
( A  .o  r
)  +o  y )  =  B )  <->  ( z  =  <. r ,  s
>.  /\  ( ( A  .o  r )  +o  s )  =  B ) ) )
5044, 49cbvrex2v 2961 . . . 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 2449 . . . . . 6  |-  ( z  =  t  ->  (
z  =  <. r ,  s >.  <->  t  =  <. r ,  s >.
) )
5251anbi1d 704 . . . . 5  |-  ( z  =  t  ->  (
( z  =  <. r ,  s >.  /\  (
( A  .o  r
)  +o  s )  =  B )  <->  ( t  =  <. r ,  s
>.  /\  ( ( A  .o  r )  +o  s )  =  B ) ) )
53522rexbidv 2763 . . . 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 257 . . 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 2318 . 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 664 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 184    /\ wa 369    /\ w3a 965   A.wal 1367    = wceq 1369   E.wex 1586    e. wcel 1756   E!weu 2253    =/= wne 2611   E.wrex 2721   (/)c0 3642   <.cop 3888   Oncon0 4724  (class class class)co 6096    +o coa 6922    .o comu 6923
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-1o 6925  df-oadd 6929  df-omul 6930
This theorem is referenced by:  oeeui  7046  omxpenlem  7417
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