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Theorem omopth2 7035
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 1041 . . . . . . 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 4741 . . . . . . 7  |-  ( B  e.  On  ->  Ord  B )
31, 2syl 16 . . . . . 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 1043 . . . . . . 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 4741 . . . . . . 7  |-  ( D  e.  On  ->  Ord  D )
64, 5syl 16 . . . . . 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 4763 . . . . . 6  |-  ( ( Ord  B  /\  Ord  D )  ->  ( B  e.  D  \/  B  =  D  \/  D  e.  B ) )
83, 6, 7syl2anc 661 . . . . 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 simpr 461 . . . . . . . . 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 )  =  ( ( A  .o  D
)  +o  E ) )
10 simpl1l 1039 . . . . . . . . . . . 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 )
11 omcl 6988 . . . . . . . . . . . 12  |-  ( ( A  e.  On  /\  D  e.  On )  ->  ( A  .o  D
)  e.  On )
1210, 4, 11syl2anc 661 . . . . . . . . . . 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 )
13 simpl3r 1044 . . . . . . . . . . . 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
)
14 onelon 4756 . . . . . . . . . . . 12  |-  ( ( A  e.  On  /\  E  e.  A )  ->  E  e.  On )
1510, 13, 14syl2anc 661 . . . . . . . . . . 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 )
16 oacl 6987 . . . . . . . . . . 11  |-  ( ( ( A  .o  D
)  e.  On  /\  E  e.  On )  ->  ( ( A  .o  D )  +o  E
)  e.  On )
1712, 15, 16syl2anc 661 . . . . . . . . . 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 )
18 eloni 4741 . . . . . . . . . 10  |-  ( ( ( A  .o  D
)  +o  E )  e.  On  ->  Ord  ( ( A  .o  D )  +o  E
) )
19 ordirr 4749 . . . . . . . . . 10  |-  ( Ord  ( ( A  .o  D )  +o  E
)  ->  -.  (
( A  .o  D
)  +o  E )  e.  ( ( A  .o  D )  +o  E ) )
2017, 18, 193syl 20 . . . . . . . . 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 ) )
219, 20eqneltrd 2536 . . . . . . . 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 ) )
22 orc 385 . . . . . . . . 9  |-  ( B  e.  D  ->  ( B  e.  D  \/  ( B  =  D  /\  C  e.  E
) ) )
23 omeulem2 7034 . . . . . . . . . 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 ) ) )
2423adantr 465 . . . . . . . . 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 ) ) )
2522, 24syl5 32 . . . . . . . 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 ) ) )
2621, 25mtod 177 . . . . . . 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 )
2726pm2.21d 106 . . . . . 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 ) )
28 idd 24 . . . . . 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 ) )
2920, 9neleqtrrd 2540 . . . . . . . 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 ) )
30 orc 385 . . . . . . . . 9  |-  ( D  e.  B  ->  ( D  e.  B  \/  ( D  =  B  /\  E  e.  C
) ) )
31 simpl1r 1040 . . . . . . . . . 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  =/=  (/) )
32 simpl2r 1042 . . . . . . . . . 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
)
33 omeulem2 7034 . . . . . . . . . 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 ) ) )
3410, 31, 4, 13, 1, 32, 33syl222anc 1234 . . . . . . . . 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 ) ) )
3530, 34syl5 32 . . . . . . . 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 ) ) )
3629, 35mtod 177 . . . . . . 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 )
3736pm2.21d 106 . . . . . 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 ) )
3827, 28, 373jaod 1282 . . . . 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 ) )
398, 38mpd 15 . . . 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 )
40 onelon 4756 . . . . . . . 8  |-  ( ( A  e.  On  /\  C  e.  A )  ->  C  e.  On )
41 eloni 4741 . . . . . . . 8  |-  ( C  e.  On  ->  Ord  C )
4240, 41syl 16 . . . . . . 7  |-  ( ( A  e.  On  /\  C  e.  A )  ->  Ord  C )
4310, 32, 42syl2anc 661 . . . . . 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 )
44 eloni 4741 . . . . . . . 8  |-  ( E  e.  On  ->  Ord  E )
4514, 44syl 16 . . . . . . 7  |-  ( ( A  e.  On  /\  E  e.  A )  ->  Ord  E )
4610, 13, 45syl2anc 661 . . . . . 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 )
47 ordtri3or 4763 . . . . . 6  |-  ( ( Ord  C  /\  Ord  E )  ->  ( C  e.  E  \/  C  =  E  \/  E  e.  C ) )
4843, 46, 47syl2anc 661 . . . . 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 ) )
49 olc 384 . . . . . . . . . 10  |-  ( ( B  =  D  /\  C  e.  E )  ->  ( B  e.  D  \/  ( B  =  D  /\  C  e.  E
) ) )
5049, 24syl5 32 . . . . . . . . 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 ) ) )
5139, 50mpand 675 . . . . . . . 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 ) ) )
5221, 51mtod 177 . . . . . . 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 )
5352pm2.21d 106 . . . . . 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 ) )
54 idd 24 . . . . . 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 ) )
5539eqcomd 2448 . . . . . . . . 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 )
56 olc 384 . . . . . . . . . 10  |-  ( ( D  =  B  /\  E  e.  C )  ->  ( D  e.  B  \/  ( D  =  B  /\  E  e.  C
) ) )
5756, 34syl5 32 . . . . . . . . 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 ) ) )
5855, 57mpand 675 . . . . . . . 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 ) ) )
5929, 58mtod 177 . . . . . . 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 )
6059pm2.21d 106 . . . . . 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 ) )
6153, 54, 603jaod 1282 . . . . 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 ) )
6248, 61mpd 15 . . . 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 )
6339, 62jca 532 . . 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 ) )
6463ex 434 . 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 ) ) )
65 oveq2 6111 . . 3  |-  ( B  =  D  ->  ( A  .o  B )  =  ( A  .o  D
) )
66 id 22 . . 3  |-  ( C  =  E  ->  C  =  E )
6765, 66oveqan12d 6122 . 2  |-  ( ( B  =  D  /\  C  =  E )  ->  ( ( A  .o  B )  +o  C
)  =  ( ( A  .o  D )  +o  E ) )
6864, 67impbid1 203 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    \/ w3o 964    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2618   (/)c0 3649   Ord word 4730   Oncon0 4731  (class class class)co 6103    +o coa 6929    .o comu 6930
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 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384
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 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-reu 2734  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-om 6489  df-recs 6844  df-rdg 6878  df-oadd 6936  df-omul 6937
This theorem is referenced by:  omeu  7036  dfac12lem2  8325
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