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Theorem relexp01min 36299
Description: With exponents limited to 0 and 1, the composition of powers of a relation is the relation raised to the minimum of exponents. (Contributed by RP, 12-Jun-2020.)
Assertion
Ref Expression
relexp01min  |-  ( ( ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
)  /\  ( J  e.  { 0 ,  1 }  /\  K  e. 
{ 0 ,  1 } ) )  -> 
( ( R ^r  J ) ^r  K )  =  ( R ^r  I ) )

Proof of Theorem relexp01min
StepHypRef Expression
1 elpri 3984 . . 3  |-  ( J  e.  { 0 ,  1 }  ->  ( J  =  0  \/  J  =  1 ) )
2 elpri 3984 . . 3  |-  ( K  e.  { 0 ,  1 }  ->  ( K  =  0  \/  K  =  1 ) )
3 dmresi 5159 . . . . . . . . . . 11  |-  dom  (  _I  |`  ( dom  R  u.  ran  R ) )  =  ( dom  R  u.  ran  R )
4 rnresi 5180 . . . . . . . . . . 11  |-  ran  (  _I  |`  ( dom  R  u.  ran  R ) )  =  ( dom  R  u.  ran  R )
53, 4uneq12i 3585 . . . . . . . . . 10  |-  ( dom  (  _I  |`  ( dom  R  u.  ran  R
) )  u.  ran  (  _I  |`  ( dom 
R  u.  ran  R
) ) )  =  ( ( dom  R  u.  ran  R )  u.  ( dom  R  u.  ran  R ) )
6 unidm 3576 . . . . . . . . . 10  |-  ( ( dom  R  u.  ran  R )  u.  ( dom 
R  u.  ran  R
) )  =  ( dom  R  u.  ran  R )
75, 6eqtri 2472 . . . . . . . . 9  |-  ( dom  (  _I  |`  ( dom  R  u.  ran  R
) )  u.  ran  (  _I  |`  ( dom 
R  u.  ran  R
) ) )  =  ( dom  R  u.  ran  R )
87reseq2i 5101 . . . . . . . 8  |-  (  _I  |`  ( dom  (  _I  |`  ( dom  R  u.  ran  R ) )  u. 
ran  (  _I  |`  ( dom  R  u.  ran  R
) ) ) )  =  (  _I  |`  ( dom  R  u.  ran  R
) )
9 simp1 1007 . . . . . . . . . . . 12  |-  ( ( J  =  0  /\  K  =  0  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  J  =  0 )
109oveq2d 6304 . . . . . . . . . . 11  |-  ( ( J  =  0  /\  K  =  0  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  ( R ^r  J )  =  ( R ^r  0 ) )
11 simp3l 1035 . . . . . . . . . . . 12  |-  ( ( J  =  0  /\  K  =  0  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  R  e.  V )
12 relexp0g 13078 . . . . . . . . . . . 12  |-  ( R  e.  V  ->  ( R ^r  0 )  =  (  _I  |`  ( dom  R  u.  ran  R
) ) )
1311, 12syl 17 . . . . . . . . . . 11  |-  ( ( J  =  0  /\  K  =  0  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  ( R ^r  0 )  =  (  _I  |`  ( dom  R  u.  ran  R
) ) )
1410, 13eqtrd 2484 . . . . . . . . . 10  |-  ( ( J  =  0  /\  K  =  0  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  ( R ^r  J )  =  (  _I  |`  ( dom  R  u.  ran  R
) ) )
15 simp2 1008 . . . . . . . . . 10  |-  ( ( J  =  0  /\  K  =  0  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  K  =  0 )
1614, 15oveq12d 6306 . . . . . . . . 9  |-  ( ( J  =  0  /\  K  =  0  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  (
( R ^r  J ) ^r  K )  =  ( (  _I  |`  ( dom  R  u.  ran  R
) ) ^r 
0 ) )
17 dmexg 6721 . . . . . . . . . . . 12  |-  ( R  e.  V  ->  dom  R  e.  _V )
18 rnexg 6722 . . . . . . . . . . . 12  |-  ( R  e.  V  ->  ran  R  e.  _V )
19 unexg 6589 . . . . . . . . . . . 12  |-  ( ( dom  R  e.  _V  /\ 
ran  R  e.  _V )  ->  ( dom  R  u.  ran  R )  e. 
_V )
2017, 18, 19syl2anc 666 . . . . . . . . . . 11  |-  ( R  e.  V  ->  ( dom  R  u.  ran  R
)  e.  _V )
2120resiexd 6129 . . . . . . . . . 10  |-  ( R  e.  V  ->  (  _I  |`  ( dom  R  u.  ran  R ) )  e.  _V )
22 relexp0g 13078 . . . . . . . . . 10  |-  ( (  _I  |`  ( dom  R  u.  ran  R ) )  e.  _V  ->  ( (  _I  |`  ( dom  R  u.  ran  R
) ) ^r 
0 )  =  (  _I  |`  ( dom  (  _I  |`  ( dom 
R  u.  ran  R
) )  u.  ran  (  _I  |`  ( dom 
R  u.  ran  R
) ) ) ) )
2311, 21, 223syl 18 . . . . . . . . 9  |-  ( ( J  =  0  /\  K  =  0  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  (
(  _I  |`  ( dom  R  u.  ran  R
) ) ^r 
0 )  =  (  _I  |`  ( dom  (  _I  |`  ( dom 
R  u.  ran  R
) )  u.  ran  (  _I  |`  ( dom 
R  u.  ran  R
) ) ) ) )
2416, 23eqtrd 2484 . . . . . . . 8  |-  ( ( J  =  0  /\  K  =  0  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  (
( R ^r  J ) ^r  K )  =  (  _I  |`  ( dom  (  _I  |`  ( dom 
R  u.  ran  R
) )  u.  ran  (  _I  |`  ( dom 
R  u.  ran  R
) ) ) ) )
25 simp3r 1036 . . . . . . . . . . 11  |-  ( ( J  =  0  /\  K  =  0  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  I  =  if ( J  < 
K ,  J ,  K ) )
26 0re 9640 . . . . . . . . . . . . . 14  |-  0  e.  RR
2726ltnri 9740 . . . . . . . . . . . . 13  |-  -.  0  <  0
289, 15breq12d 4414 . . . . . . . . . . . . 13  |-  ( ( J  =  0  /\  K  =  0  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  ( J  <  K  <->  0  <  0 ) )
2927, 28mtbiri 305 . . . . . . . . . . . 12  |-  ( ( J  =  0  /\  K  =  0  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  -.  J  <  K )
3029iffalsed 3891 . . . . . . . . . . 11  |-  ( ( J  =  0  /\  K  =  0  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  if ( J  <  K ,  J ,  K )  =  K )
3125, 30, 153eqtrd 2488 . . . . . . . . . 10  |-  ( ( J  =  0  /\  K  =  0  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  I  =  0 )
3231oveq2d 6304 . . . . . . . . 9  |-  ( ( J  =  0  /\  K  =  0  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  ( R ^r  I )  =  ( R ^r  0 ) )
3332, 13eqtrd 2484 . . . . . . . 8  |-  ( ( J  =  0  /\  K  =  0  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  ( R ^r  I )  =  (  _I  |`  ( dom  R  u.  ran  R
) ) )
348, 24, 333eqtr4a 2510 . . . . . . 7  |-  ( ( J  =  0  /\  K  =  0  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  (
( R ^r  J ) ^r  K )  =  ( R ^r  I ) )
35343exp 1206 . . . . . 6  |-  ( J  =  0  ->  ( K  =  0  ->  ( ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
)  ->  ( ( R ^r  J ) ^r  K )  =  ( R ^r  I ) ) ) )
36 simp1 1007 . . . . . . . . . . 11  |-  ( ( J  =  1  /\  K  =  0  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  J  =  1 )
3736oveq2d 6304 . . . . . . . . . 10  |-  ( ( J  =  1  /\  K  =  0  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  ( R ^r  J )  =  ( R ^r  1 ) )
38 simp3l 1035 . . . . . . . . . . 11  |-  ( ( J  =  1  /\  K  =  0  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  R  e.  V )
39 relexp1g 13082 . . . . . . . . . . 11  |-  ( R  e.  V  ->  ( R ^r  1 )  =  R )
4038, 39syl 17 . . . . . . . . . 10  |-  ( ( J  =  1  /\  K  =  0  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  ( R ^r  1 )  =  R )
4137, 40eqtrd 2484 . . . . . . . . 9  |-  ( ( J  =  1  /\  K  =  0  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  ( R ^r  J )  =  R )
42 simp2 1008 . . . . . . . . 9  |-  ( ( J  =  1  /\  K  =  0  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  K  =  0 )
4341, 42oveq12d 6306 . . . . . . . 8  |-  ( ( J  =  1  /\  K  =  0  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  (
( R ^r  J ) ^r  K )  =  ( R ^r  0 ) )
44 simp3r 1036 . . . . . . . . . 10  |-  ( ( J  =  1  /\  K  =  0  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  I  =  if ( J  < 
K ,  J ,  K ) )
45 0lt1 10133 . . . . . . . . . . . . 13  |-  0  <  1
46 1re 9639 . . . . . . . . . . . . . 14  |-  1  e.  RR
4726, 46ltnsymi 9750 . . . . . . . . . . . . 13  |-  ( 0  <  1  ->  -.  1  <  0 )
4845, 47mp1i 13 . . . . . . . . . . . 12  |-  ( ( J  =  1  /\  K  =  0  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  -.  1  <  0 )
4936, 42breq12d 4414 . . . . . . . . . . . 12  |-  ( ( J  =  1  /\  K  =  0  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  ( J  <  K  <->  1  <  0 ) )
5048, 49mtbird 303 . . . . . . . . . . 11  |-  ( ( J  =  1  /\  K  =  0  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  -.  J  <  K )
5150iffalsed 3891 . . . . . . . . . 10  |-  ( ( J  =  1  /\  K  =  0  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  if ( J  <  K ,  J ,  K )  =  K )
5244, 51, 423eqtrd 2488 . . . . . . . . 9  |-  ( ( J  =  1  /\  K  =  0  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  I  =  0 )
5352oveq2d 6304 . . . . . . . 8  |-  ( ( J  =  1  /\  K  =  0  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  ( R ^r  I )  =  ( R ^r  0 ) )
5443, 53eqtr4d 2487 . . . . . . 7  |-  ( ( J  =  1  /\  K  =  0  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  (
( R ^r  J ) ^r  K )  =  ( R ^r  I ) )
55543exp 1206 . . . . . 6  |-  ( J  =  1  ->  ( K  =  0  ->  ( ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
)  ->  ( ( R ^r  J ) ^r  K )  =  ( R ^r  I ) ) ) )
5635, 55jaoi 381 . . . . 5  |-  ( ( J  =  0  \/  J  =  1 )  ->  ( K  =  0  ->  ( ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K ) )  -> 
( ( R ^r  J ) ^r  K )  =  ( R ^r  I ) ) ) )
57 ovex 6316 . . . . . . . . 9  |-  ( R ^r  0 )  e.  _V
58 relexp1g 13082 . . . . . . . . 9  |-  ( ( R ^r  0 )  e.  _V  ->  ( ( R ^r 
0 ) ^r 
1 )  =  ( R ^r  0 ) )
5957, 58mp1i 13 . . . . . . . 8  |-  ( ( J  =  0  /\  K  =  1  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  (
( R ^r 
0 ) ^r 
1 )  =  ( R ^r  0 ) )
60 simp1 1007 . . . . . . . . . 10  |-  ( ( J  =  0  /\  K  =  1  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  J  =  0 )
6160oveq2d 6304 . . . . . . . . 9  |-  ( ( J  =  0  /\  K  =  1  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  ( R ^r  J )  =  ( R ^r  0 ) )
62 simp2 1008 . . . . . . . . 9  |-  ( ( J  =  0  /\  K  =  1  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  K  =  1 )
6361, 62oveq12d 6306 . . . . . . . 8  |-  ( ( J  =  0  /\  K  =  1  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  (
( R ^r  J ) ^r  K )  =  ( ( R ^r 
0 ) ^r 
1 ) )
64 simp3r 1036 . . . . . . . . . 10  |-  ( ( J  =  0  /\  K  =  1  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  I  =  if ( J  < 
K ,  J ,  K ) )
6560, 62breq12d 4414 . . . . . . . . . . . 12  |-  ( ( J  =  0  /\  K  =  1  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  ( J  <  K  <->  0  <  1 ) )
6645, 65mpbiri 237 . . . . . . . . . . 11  |-  ( ( J  =  0  /\  K  =  1  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  J  <  K )
6766iftrued 3888 . . . . . . . . . 10  |-  ( ( J  =  0  /\  K  =  1  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  if ( J  <  K ,  J ,  K )  =  J )
6864, 67, 603eqtrd 2488 . . . . . . . . 9  |-  ( ( J  =  0  /\  K  =  1  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  I  =  0 )
6968oveq2d 6304 . . . . . . . 8  |-  ( ( J  =  0  /\  K  =  1  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  ( R ^r  I )  =  ( R ^r  0 ) )
7059, 63, 693eqtr4d 2494 . . . . . . 7  |-  ( ( J  =  0  /\  K  =  1  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  (
( R ^r  J ) ^r  K )  =  ( R ^r  I ) )
71703exp 1206 . . . . . 6  |-  ( J  =  0  ->  ( K  =  1  ->  ( ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
)  ->  ( ( R ^r  J ) ^r  K )  =  ( R ^r  I ) ) ) )
72 simp1 1007 . . . . . . . . . . 11  |-  ( ( J  =  1  /\  K  =  1  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  J  =  1 )
7372oveq2d 6304 . . . . . . . . . 10  |-  ( ( J  =  1  /\  K  =  1  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  ( R ^r  J )  =  ( R ^r  1 ) )
74 simp3l 1035 . . . . . . . . . . 11  |-  ( ( J  =  1  /\  K  =  1  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  R  e.  V )
7574, 39syl 17 . . . . . . . . . 10  |-  ( ( J  =  1  /\  K  =  1  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  ( R ^r  1 )  =  R )
7673, 75eqtrd 2484 . . . . . . . . 9  |-  ( ( J  =  1  /\  K  =  1  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  ( R ^r  J )  =  R )
77 simp2 1008 . . . . . . . . 9  |-  ( ( J  =  1  /\  K  =  1  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  K  =  1 )
7876, 77oveq12d 6306 . . . . . . . 8  |-  ( ( J  =  1  /\  K  =  1  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  (
( R ^r  J ) ^r  K )  =  ( R ^r  1 ) )
79 simp3r 1036 . . . . . . . . . 10  |-  ( ( J  =  1  /\  K  =  1  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  I  =  if ( J  < 
K ,  J ,  K ) )
8046ltnri 9740 . . . . . . . . . . . 12  |-  -.  1  <  1
8172, 77breq12d 4414 . . . . . . . . . . . 12  |-  ( ( J  =  1  /\  K  =  1  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  ( J  <  K  <->  1  <  1 ) )
8280, 81mtbiri 305 . . . . . . . . . . 11  |-  ( ( J  =  1  /\  K  =  1  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  -.  J  <  K )
8382iffalsed 3891 . . . . . . . . . 10  |-  ( ( J  =  1  /\  K  =  1  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  if ( J  <  K ,  J ,  K )  =  K )
8479, 83, 773eqtrd 2488 . . . . . . . . 9  |-  ( ( J  =  1  /\  K  =  1  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  I  =  1 )
8584oveq2d 6304 . . . . . . . 8  |-  ( ( J  =  1  /\  K  =  1  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  ( R ^r  I )  =  ( R ^r  1 ) )
8678, 85eqtr4d 2487 . . . . . . 7  |-  ( ( J  =  1  /\  K  =  1  /\  ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
) )  ->  (
( R ^r  J ) ^r  K )  =  ( R ^r  I ) )
87863exp 1206 . . . . . 6  |-  ( J  =  1  ->  ( K  =  1  ->  ( ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
)  ->  ( ( R ^r  J ) ^r  K )  =  ( R ^r  I ) ) ) )
8871, 87jaoi 381 . . . . 5  |-  ( ( J  =  0  \/  J  =  1 )  ->  ( K  =  1  ->  ( ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K ) )  -> 
( ( R ^r  J ) ^r  K )  =  ( R ^r  I ) ) ) )
8956, 88jaod 382 . . . 4  |-  ( ( J  =  0  \/  J  =  1 )  ->  ( ( K  =  0  \/  K  =  1 )  -> 
( ( R  e.  V  /\  I  =  if ( J  < 
K ,  J ,  K ) )  -> 
( ( R ^r  J ) ^r  K )  =  ( R ^r  I ) ) ) )
9089imp 431 . . 3  |-  ( ( ( J  =  0  \/  J  =  1 )  /\  ( K  =  0  \/  K  =  1 ) )  ->  ( ( R  e.  V  /\  I  =  if ( J  < 
K ,  J ,  K ) )  -> 
( ( R ^r  J ) ^r  K )  =  ( R ^r  I ) ) )
911, 2, 90syl2an 480 . 2  |-  ( ( J  e.  { 0 ,  1 }  /\  K  e.  { 0 ,  1 } )  ->  ( ( R  e.  V  /\  I  =  if ( J  < 
K ,  J ,  K ) )  -> 
( ( R ^r  J ) ^r  K )  =  ( R ^r  I ) ) )
9291impcom 432 1  |-  ( ( ( R  e.  V  /\  I  =  if ( J  <  K ,  J ,  K )
)  /\  ( J  e.  { 0 ,  1 }  /\  K  e. 
{ 0 ,  1 } ) )  -> 
( ( R ^r  J ) ^r  K )  =  ( R ^r  I ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 370    /\ wa 371    /\ w3a 984    = wceq 1443    e. wcel 1886   _Vcvv 3044    u. cun 3401   ifcif 3880   {cpr 3969   class class class wbr 4401    _I cid 4743   dom cdm 4833   ran crn 4834    |` cres 4835  (class class class)co 6288   0cc0 9536   1c1 9537    < clt 9672   ^r crelexp 13076
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-om 6690  df-2nd 6791  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-er 7360  df-en 7567  df-dom 7568  df-sdom 7569  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-nn 10607  df-n0 10867  df-z 10935  df-uz 11157  df-seq 12211  df-relexp 13077
This theorem is referenced by:  relexp1idm  36300  relexp0idm  36301
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