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Theorem omopth2 7245
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 1049 . . . . . . 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 4894 . . . . . . 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 1051 . . . . . . 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 4894 . . . . . . 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 4916 . . . . . 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 1047 . . . . . . . . . . . 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 7198 . . . . . . . . . . . 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 1052 . . . . . . . . . . . 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 4909 . . . . . . . . . . . 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 7197 . . . . . . . . . . 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 4894 . . . . . . . . . 10  |-  ( ( ( A  .o  D
)  +o  E )  e.  On  ->  Ord  ( ( A  .o  D )  +o  E
) )
19 ordirr 4902 . . . . . . . . . 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 2576 . . . . . . . 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 7244 . . . . . . . . . 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 2580 . . . . . . . 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 1048 . . . . . . . . . 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 1050 . . . . . . . . . 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 7244 . . . . . . . . . 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 1244 . . . . . . . . 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 1292 . . . . 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 4909 . . . . . . . 8  |-  ( ( A  e.  On  /\  C  e.  A )  ->  C  e.  On )
41 eloni 4894 . . . . . . . 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 4894 . . . . . . . 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 4916 . . . . . 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 2475 . . . . . . . . 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 1292 . . . . 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 6303 . . 3  |-  ( B  =  D  ->  ( A  .o  B )  =  ( A  .o  D
) )
66 id 22 . . 3  |-  ( C  =  E  ->  C  =  E )
6765, 66oveqan12d 6314 . 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 972    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   (/)c0 3790   Ord word 4883   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-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-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-recs 7054  df-rdg 7088  df-oadd 7146  df-omul 7147
This theorem is referenced by:  omeu  7246  dfac12lem2  8536
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