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Theorem omopth2 6598
Description: An ordered pair-like theorem for ordinal multiplication. (Contributed by Mario Carneiro, 29-May-2015.)
Assertion
Ref Expression
omopth2  |-  ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  -> 
( ( ( A  .o  B )  +o  C )  =  ( ( A  .o  D
)  +o  E )  <-> 
( B  =  D  /\  C  =  E ) ) )

Proof of Theorem omopth2
StepHypRef Expression
1 simpl2l 1008 . . . . . . 7  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  B  e.  On )
2 eloni 4418 . . . . . . 7  |-  ( B  e.  On  ->  Ord  B )
31, 2syl 15 . . . . . 6  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  Ord  B )
4 simpl3l 1010 . . . . . . 7  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  D  e.  On )
5 eloni 4418 . . . . . . 7  |-  ( D  e.  On  ->  Ord  D )
64, 5syl 15 . . . . . 6  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  Ord  D )
7 ordtri3or 4440 . . . . . 6  |-  ( ( Ord  B  /\  Ord  D )  ->  ( B  e.  D  \/  B  =  D  \/  D  e.  B ) )
83, 6, 7syl2anc 642 . . . . 5  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  ( B  e.  D  \/  B  =  D  \/  D  e.  B ) )
9 simpl1l 1006 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  A  e.  On )
10 omcl 6551 . . . . . . . . . . . 12  |-  ( ( A  e.  On  /\  D  e.  On )  ->  ( A  .o  D
)  e.  On )
119, 4, 10syl2anc 642 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  ( A  .o  D )  e.  On )
12 simpl3r 1011 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  E  e.  A
)
13 onelon 4433 . . . . . . . . . . . 12  |-  ( ( A  e.  On  /\  E  e.  A )  ->  E  e.  On )
149, 12, 13syl2anc 642 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  E  e.  On )
15 oacl 6550 . . . . . . . . . . 11  |-  ( ( ( A  .o  D
)  e.  On  /\  E  e.  On )  ->  ( ( A  .o  D )  +o  E
)  e.  On )
1611, 14, 15syl2anc 642 . . . . . . . . . 10  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  ( ( A  .o  D )  +o  E )  e.  On )
17 eloni 4418 . . . . . . . . . 10  |-  ( ( ( A  .o  D
)  +o  E )  e.  On  ->  Ord  ( ( A  .o  D )  +o  E
) )
18 ordirr 4426 . . . . . . . . . 10  |-  ( Ord  ( ( A  .o  D )  +o  E
)  ->  -.  (
( A  .o  D
)  +o  E )  e.  ( ( A  .o  D )  +o  E ) )
1916, 17, 183syl 18 . . . . . . . . 9  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  -.  ( ( A  .o  D )  +o  E )  e.  ( ( A  .o  D
)  +o  E ) )
20 simpr 447 . . . . . . . . . 10  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  ( ( A  .o  B )  +o  C )  =  ( ( A  .o  D
)  +o  E ) )
2120eleq1d 2362 . . . . . . . . 9  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  ( ( ( A  .o  B )  +o  C )  e.  ( ( A  .o  D )  +o  E
)  <->  ( ( A  .o  D )  +o  E )  e.  ( ( A  .o  D
)  +o  E ) ) )
2219, 21mtbird 292 . . . . . . . 8  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  -.  ( ( A  .o  B )  +o  C )  e.  ( ( A  .o  D
)  +o  E ) )
23 orc 374 . . . . . . . . 9  |-  ( B  e.  D  ->  ( B  e.  D  \/  ( B  =  D  /\  C  e.  E
) ) )
24 omeulem2 6597 . . . . . . . . . 10  |-  ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  -> 
( ( B  e.  D  \/  ( B  =  D  /\  C  e.  E ) )  -> 
( ( A  .o  B )  +o  C
)  e.  ( ( A  .o  D )  +o  E ) ) )
2524adantr 451 . . . . . . . . 9  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  ( ( B  e.  D  \/  ( B  =  D  /\  C  e.  E )
)  ->  ( ( A  .o  B )  +o  C )  e.  ( ( A  .o  D
)  +o  E ) ) )
2623, 25syl5 28 . . . . . . . 8  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  ( B  e.  D  ->  ( ( A  .o  B )  +o  C )  e.  ( ( A  .o  D
)  +o  E ) ) )
2722, 26mtod 168 . . . . . . 7  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  -.  B  e.  D )
2827pm2.21d 98 . . . . . 6  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  ( B  e.  D  ->  B  =  D ) )
29 idd 21 . . . . . 6  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  ( B  =  D  ->  B  =  D ) )
3020eleq2d 2363 . . . . . . . . 9  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  ( ( ( A  .o  D )  +o  E )  e.  ( ( A  .o  B )  +o  C
)  <->  ( ( A  .o  D )  +o  E )  e.  ( ( A  .o  D
)  +o  E ) ) )
3119, 30mtbird 292 . . . . . . . 8  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  -.  ( ( A  .o  D )  +o  E )  e.  ( ( A  .o  B
)  +o  C ) )
32 orc 374 . . . . . . . . 9  |-  ( D  e.  B  ->  ( D  e.  B  \/  ( D  =  B  /\  E  e.  C
) ) )
33 simpl1r 1007 . . . . . . . . . 10  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  A  =/=  (/) )
34 simpl2r 1009 . . . . . . . . . 10  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  C  e.  A
)
35 omeulem2 6597 . . . . . . . . . 10  |-  ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( D  e.  On  /\  E  e.  A )  /\  ( B  e.  On  /\  C  e.  A ) )  -> 
( ( D  e.  B  \/  ( D  =  B  /\  E  e.  C ) )  -> 
( ( A  .o  D )  +o  E
)  e.  ( ( A  .o  B )  +o  C ) ) )
369, 33, 4, 12, 1, 34, 35syl222anc 1198 . . . . . . . . 9  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  ( ( D  e.  B  \/  ( D  =  B  /\  E  e.  C )
)  ->  ( ( A  .o  D )  +o  E )  e.  ( ( A  .o  B
)  +o  C ) ) )
3732, 36syl5 28 . . . . . . . 8  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  ( D  e.  B  ->  ( ( A  .o  D )  +o  E )  e.  ( ( A  .o  B
)  +o  C ) ) )
3831, 37mtod 168 . . . . . . 7  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  -.  D  e.  B )
3938pm2.21d 98 . . . . . 6  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  ( D  e.  B  ->  B  =  D ) )
4028, 29, 393jaod 1246 . . . . 5  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  ( ( B  e.  D  \/  B  =  D  \/  D  e.  B )  ->  B  =  D ) )
418, 40mpd 14 . . . 4  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  B  =  D )
42 onelon 4433 . . . . . . . 8  |-  ( ( A  e.  On  /\  C  e.  A )  ->  C  e.  On )
43 eloni 4418 . . . . . . . 8  |-  ( C  e.  On  ->  Ord  C )
4442, 43syl 15 . . . . . . 7  |-  ( ( A  e.  On  /\  C  e.  A )  ->  Ord  C )
459, 34, 44syl2anc 642 . . . . . 6  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  Ord  C )
46 eloni 4418 . . . . . . . 8  |-  ( E  e.  On  ->  Ord  E )
4713, 46syl 15 . . . . . . 7  |-  ( ( A  e.  On  /\  E  e.  A )  ->  Ord  E )
489, 12, 47syl2anc 642 . . . . . 6  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  Ord  E )
49 ordtri3or 4440 . . . . . 6  |-  ( ( Ord  C  /\  Ord  E )  ->  ( C  e.  E  \/  C  =  E  \/  E  e.  C ) )
5045, 48, 49syl2anc 642 . . . . 5  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  ( C  e.  E  \/  C  =  E  \/  E  e.  C ) )
51 olc 373 . . . . . . . . . 10  |-  ( ( B  =  D  /\  C  e.  E )  ->  ( B  e.  D  \/  ( B  =  D  /\  C  e.  E
) ) )
5251, 25syl5 28 . . . . . . . . 9  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  ( ( B  =  D  /\  C  e.  E )  ->  (
( A  .o  B
)  +o  C )  e.  ( ( A  .o  D )  +o  E ) ) )
5341, 52mpand 656 . . . . . . . 8  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  ( C  e.  E  ->  ( ( A  .o  B )  +o  C )  e.  ( ( A  .o  D
)  +o  E ) ) )
5422, 53mtod 168 . . . . . . 7  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  -.  C  e.  E )
5554pm2.21d 98 . . . . . 6  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  ( C  e.  E  ->  C  =  E ) )
56 idd 21 . . . . . 6  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  ( C  =  E  ->  C  =  E ) )
5741eqcomd 2301 . . . . . . . . 9  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  D  =  B )
58 olc 373 . . . . . . . . . 10  |-  ( ( D  =  B  /\  E  e.  C )  ->  ( D  e.  B  \/  ( D  =  B  /\  E  e.  C
) ) )
5958, 36syl5 28 . . . . . . . . 9  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  ( ( D  =  B  /\  E  e.  C )  ->  (
( A  .o  D
)  +o  E )  e.  ( ( A  .o  B )  +o  C ) ) )
6057, 59mpand 656 . . . . . . . 8  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  ( E  e.  C  ->  ( ( A  .o  D )  +o  E )  e.  ( ( A  .o  B
)  +o  C ) ) )
6131, 60mtod 168 . . . . . . 7  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  -.  E  e.  C )
6261pm2.21d 98 . . . . . 6  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  ( E  e.  C  ->  C  =  E ) )
6355, 56, 623jaod 1246 . . . . 5  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  ( ( C  e.  E  \/  C  =  E  \/  E  e.  C )  ->  C  =  E ) )
6450, 63mpd 14 . . . 4  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  C  =  E )
6541, 64jca 518 . . 3  |-  ( ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  /\  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )  ->  ( B  =  D  /\  C  =  E ) )
6665ex 423 . 2  |-  ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  -> 
( ( ( A  .o  B )  +o  C )  =  ( ( A  .o  D
)  +o  E )  ->  ( B  =  D  /\  C  =  E ) ) )
67 oveq2 5882 . . 3  |-  ( B  =  D  ->  ( A  .o  B )  =  ( A  .o  D
) )
68 id 19 . . 3  |-  ( C  =  E  ->  C  =  E )
6967, 68oveqan12d 5893 . 2  |-  ( ( B  =  D  /\  C  =  E )  ->  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )
7066, 69impbid1 194 1  |-  ( ( ( A  e.  On  /\  A  =/=  (/) )  /\  ( B  e.  On  /\  C  e.  A )  /\  ( D  e.  On  /\  E  e.  A ) )  -> 
( ( ( A  .o  B )  +o  C )  =  ( ( A  .o  D
)  +o  E )  <-> 
( B  =  D  /\  C  =  E ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    \/ w3o 933    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   (/)c0 3468   Ord word 4407   Oncon0 4408  (class class class)co 5874    +o coa 6492    .o comu 6493
This theorem is referenced by:  omeu  6599  dfac12lem2  7786
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-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-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-recs 6404  df-rdg 6439  df-oadd 6499  df-omul 6500
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