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Theorem omeu 7246
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 7243 . . 3  |-  ( ( A  e.  On  /\  B  e.  On  /\  A  =/=  (/) )  ->  E. x  e.  On  E. y  e.  A  ( ( A  .o  x )  +o  y )  =  B )
2 opex 4717 . . . . . . . . 9  |-  <. x ,  y >.  e.  _V
32isseti 3124 . . . . . . . 8  |-  E. z 
z  =  <. x ,  y >.
4 19.41v 1945 . . . . . . . 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 916 . . . . . . 7  |-  ( E. z ( z  = 
<. x ,  y >.  /\  ( ( A  .o  x )  +o  y
)  =  B )  <-> 
( ( A  .o  x )  +o  y
)  =  B )
65rexbii 2969 . . . . . 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 3138 . . . . . 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 2969 . . . 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 3138 . . . 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 1065 . . . . . . . . . . 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 1069 . . . . . . . . . . . 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 1066 . . . . . . . . . . . . . . 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 1070 . . . . . . . . . . . . . . 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 2511 . . . . . . . . . . . . . 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 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 ) ) )  ->  A  e.  On )
19 simp13 1028 . . . . . . . . . . . . . . 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 1063 . . . . . . . . . . . . . . 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 1064 . . . . . . . . . . . . . . 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 1067 . . . . . . . . . . . . . . 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 1068 . . . . . . . . . . . . . . 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 7245 . . . . . . . . . . . . . . 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 1244 . . . . . . . . . . . . . 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 4221 . . . . . . . . . . . . 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 2511 . . . . . . . . . . 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 2511 . . . . . . . . . 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 1198 . . . . . . . . 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 2965 . . . . . 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 2965 . . . 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 1696 . 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 4219 . . . . . . 7  |-  ( x  =  r  ->  <. x ,  y >.  =  <. r ,  y >. )
4039eqeq2d 2481 . . . . . 6  |-  ( x  =  r  ->  (
z  =  <. x ,  y >.  <->  z  =  <. r ,  y >.
) )
41 oveq2 6303 . . . . . . . 8  |-  ( x  =  r  ->  ( A  .o  x )  =  ( A  .o  r
) )
4241oveq1d 6310 . . . . . . 7  |-  ( x  =  r  ->  (
( A  .o  x
)  +o  y )  =  ( ( A  .o  r )  +o  y ) )
4342eqeq1d 2469 . . . . . 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 4220 . . . . . . 7  |-  ( y  =  s  ->  <. r ,  y >.  =  <. r ,  s >. )
4645eqeq2d 2481 . . . . . 6  |-  ( y  =  s  ->  (
z  =  <. r ,  y >.  <->  z  =  <. r ,  s >.
) )
47 oveq2 6303 . . . . . . 7  |-  ( y  =  s  ->  (
( A  .o  r
)  +o  y )  =  ( ( A  .o  r )  +o  s ) )
4847eqeq1d 2469 . . . . . 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 3102 . . . 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 2471 . . . . . 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 2985 . . . 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 2340 . 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 973   A.wal 1377    = wceq 1379   E.wex 1596    e. wcel 1767   E!weu 2275    =/= wne 2662   E.wrex 2818   (/)c0 3790   <.cop 4039   Oncon0 4884  (class class class)co 6295    +o coa 7139    .o comu 7140
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-omul 7147
This theorem is referenced by:  oeeui  7263  omxpenlem  7630
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