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Theorem relexpxpmin 36321
Description: The composition of powers of a cross-product of non-disjoint sets is the cross product raised to the minimum exponent. (Contributed by RP, 13-Jun-2020.)
Assertion
Ref Expression
relexpxpmin  |-  ( ( ( A  e.  U  /\  B  e.  V  /\  ( A  i^i  B
)  =/=  (/) )  /\  ( I  =  if ( J  <  K ,  J ,  K )  /\  J  e.  NN0  /\  K  e.  NN0 )
)  ->  ( (
( A  X.  B
) ^r  J ) ^r  K )  =  ( ( A  X.  B ) ^r  I ) )

Proof of Theorem relexpxpmin
StepHypRef Expression
1 elnn0 10878 . . . . 5  |-  ( K  e.  NN0  <->  ( K  e.  NN  \/  K  =  0 ) )
2 elnn0 10878 . . . . . . 7  |-  ( J  e.  NN0  <->  ( J  e.  NN  \/  J  =  0 ) )
3 ifeqor 3927 . . . . . . . . . . 11  |-  ( if ( J  <  K ,  J ,  K )  =  J  \/  if ( J  <  K ,  J ,  K )  =  K )
4 andi 879 . . . . . . . . . . . 12  |-  ( ( I  =  if ( J  <  K ,  J ,  K )  /\  ( if ( J  <  K ,  J ,  K )  =  J  \/  if ( J  <  K ,  J ,  K )  =  K ) )  <->  ( (
I  =  if ( J  <  K ,  J ,  K )  /\  if ( J  < 
K ,  J ,  K )  =  J )  \/  ( I  =  if ( J  <  K ,  J ,  K )  /\  if ( J  <  K ,  J ,  K )  =  K ) ) )
54biimpi 198 . . . . . . . . . . 11  |-  ( ( I  =  if ( J  <  K ,  J ,  K )  /\  ( if ( J  <  K ,  J ,  K )  =  J  \/  if ( J  <  K ,  J ,  K )  =  K ) )  ->  (
( I  =  if ( J  <  K ,  J ,  K )  /\  if ( J  <  K ,  J ,  K )  =  J )  \/  ( I  =  if ( J  <  K ,  J ,  K )  /\  if ( J  <  K ,  J ,  K )  =  K ) ) )
63, 5mpan2 678 . . . . . . . . . 10  |-  ( I  =  if ( J  <  K ,  J ,  K )  ->  (
( I  =  if ( J  <  K ,  J ,  K )  /\  if ( J  <  K ,  J ,  K )  =  J )  \/  ( I  =  if ( J  <  K ,  J ,  K )  /\  if ( J  <  K ,  J ,  K )  =  K ) ) )
7 eqtr 2472 . . . . . . . . . . 11  |-  ( ( I  =  if ( J  <  K ,  J ,  K )  /\  if ( J  < 
K ,  J ,  K )  =  J )  ->  I  =  J )
8 eqtr 2472 . . . . . . . . . . 11  |-  ( ( I  =  if ( J  <  K ,  J ,  K )  /\  if ( J  < 
K ,  J ,  K )  =  K )  ->  I  =  K )
97, 8orim12i 519 . . . . . . . . . 10  |-  ( ( ( I  =  if ( J  <  K ,  J ,  K )  /\  if ( J  <  K ,  J ,  K )  =  J )  \/  ( I  =  if ( J  <  K ,  J ,  K )  /\  if ( J  <  K ,  J ,  K )  =  K ) )  -> 
( I  =  J  \/  I  =  K ) )
10 relexpxpnnidm 36307 . . . . . . . . . . . . . . 15  |-  ( K  e.  NN  ->  (
( A  e.  U  /\  B  e.  V  /\  ( A  i^i  B
)  =/=  (/) )  -> 
( ( A  X.  B ) ^r  K )  =  ( A  X.  B ) ) )
1110imp 431 . . . . . . . . . . . . . 14  |-  ( ( K  e.  NN  /\  ( A  e.  U  /\  B  e.  V  /\  ( A  i^i  B
)  =/=  (/) ) )  ->  ( ( A  X.  B ) ^r  K )  =  ( A  X.  B ) )
12113ad2antl3 1173 . . . . . . . . . . . . 13  |-  ( ( ( I  =  J  /\  J  e.  NN  /\  K  e.  NN )  /\  ( A  e.  U  /\  B  e.  V  /\  ( A  i^i  B )  =/=  (/) ) )  ->  (
( A  X.  B
) ^r  K )  =  ( A  X.  B ) )
13 relexpxpnnidm 36307 . . . . . . . . . . . . . . . 16  |-  ( J  e.  NN  ->  (
( A  e.  U  /\  B  e.  V  /\  ( A  i^i  B
)  =/=  (/) )  -> 
( ( A  X.  B ) ^r  J )  =  ( A  X.  B ) ) )
1413imp 431 . . . . . . . . . . . . . . 15  |-  ( ( J  e.  NN  /\  ( A  e.  U  /\  B  e.  V  /\  ( A  i^i  B
)  =/=  (/) ) )  ->  ( ( A  X.  B ) ^r  J )  =  ( A  X.  B ) )
15143ad2antl2 1172 . . . . . . . . . . . . . 14  |-  ( ( ( I  =  J  /\  J  e.  NN  /\  K  e.  NN )  /\  ( A  e.  U  /\  B  e.  V  /\  ( A  i^i  B )  =/=  (/) ) )  ->  (
( A  X.  B
) ^r  J )  =  ( A  X.  B ) )
1615oveq1d 6310 . . . . . . . . . . . . 13  |-  ( ( ( I  =  J  /\  J  e.  NN  /\  K  e.  NN )  /\  ( A  e.  U  /\  B  e.  V  /\  ( A  i^i  B )  =/=  (/) ) )  ->  (
( ( A  X.  B ) ^r  J ) ^r  K )  =  ( ( A  X.  B
) ^r  K ) )
17 simpl1 1012 . . . . . . . . . . . . . . 15  |-  ( ( ( I  =  J  /\  J  e.  NN  /\  K  e.  NN )  /\  ( A  e.  U  /\  B  e.  V  /\  ( A  i^i  B )  =/=  (/) ) )  ->  I  =  J )
1817oveq2d 6311 . . . . . . . . . . . . . 14  |-  ( ( ( I  =  J  /\  J  e.  NN  /\  K  e.  NN )  /\  ( A  e.  U  /\  B  e.  V  /\  ( A  i^i  B )  =/=  (/) ) )  ->  (
( A  X.  B
) ^r  I )  =  ( ( A  X.  B ) ^r  J ) )
1918, 15eqtrd 2487 . . . . . . . . . . . . 13  |-  ( ( ( I  =  J  /\  J  e.  NN  /\  K  e.  NN )  /\  ( A  e.  U  /\  B  e.  V  /\  ( A  i^i  B )  =/=  (/) ) )  ->  (
( A  X.  B
) ^r  I )  =  ( A  X.  B ) )
2012, 16, 193eqtr4d 2497 . . . . . . . . . . . 12  |-  ( ( ( I  =  J  /\  J  e.  NN  /\  K  e.  NN )  /\  ( A  e.  U  /\  B  e.  V  /\  ( A  i^i  B )  =/=  (/) ) )  ->  (
( ( A  X.  B ) ^r  J ) ^r  K )  =  ( ( A  X.  B
) ^r  I ) )
21203exp1 1226 . . . . . . . . . . 11  |-  ( I  =  J  ->  ( J  e.  NN  ->  ( K  e.  NN  ->  ( ( A  e.  U  /\  B  e.  V  /\  ( A  i^i  B
)  =/=  (/) )  -> 
( ( ( A  X.  B ) ^r  J ) ^r  K )  =  ( ( A  X.  B
) ^r  I ) ) ) ) )
22143ad2antl2 1172 . . . . . . . . . . . . 13  |-  ( ( ( I  =  K  /\  J  e.  NN  /\  K  e.  NN )  /\  ( A  e.  U  /\  B  e.  V  /\  ( A  i^i  B )  =/=  (/) ) )  ->  (
( A  X.  B
) ^r  J )  =  ( A  X.  B ) )
23 simpl1 1012 . . . . . . . . . . . . . 14  |-  ( ( ( I  =  K  /\  J  e.  NN  /\  K  e.  NN )  /\  ( A  e.  U  /\  B  e.  V  /\  ( A  i^i  B )  =/=  (/) ) )  ->  I  =  K )
2423eqcomd 2459 . . . . . . . . . . . . 13  |-  ( ( ( I  =  K  /\  J  e.  NN  /\  K  e.  NN )  /\  ( A  e.  U  /\  B  e.  V  /\  ( A  i^i  B )  =/=  (/) ) )  ->  K  =  I )
2522, 24oveq12d 6313 . . . . . . . . . . . 12  |-  ( ( ( I  =  K  /\  J  e.  NN  /\  K  e.  NN )  /\  ( A  e.  U  /\  B  e.  V  /\  ( A  i^i  B )  =/=  (/) ) )  ->  (
( ( A  X.  B ) ^r  J ) ^r  K )  =  ( ( A  X.  B
) ^r  I ) )
26253exp1 1226 . . . . . . . . . . 11  |-  ( I  =  K  ->  ( J  e.  NN  ->  ( K  e.  NN  ->  ( ( A  e.  U  /\  B  e.  V  /\  ( A  i^i  B
)  =/=  (/) )  -> 
( ( ( A  X.  B ) ^r  J ) ^r  K )  =  ( ( A  X.  B
) ^r  I ) ) ) ) )
2721, 26jaoi 381 . . . . . . . . . 10  |-  ( ( I  =  J  \/  I  =  K )  ->  ( J  e.  NN  ->  ( K  e.  NN  ->  ( ( A  e.  U  /\  B  e.  V  /\  ( A  i^i  B )  =/=  (/) )  ->  ( ( ( A  X.  B
) ^r  J ) ^r  K )  =  ( ( A  X.  B ) ^r  I ) ) ) ) )
286, 9, 273syl 18 . . . . . . . . 9  |-  ( I  =  if ( J  <  K ,  J ,  K )  ->  ( J  e.  NN  ->  ( K  e.  NN  ->  ( ( A  e.  U  /\  B  e.  V  /\  ( A  i^i  B
)  =/=  (/) )  -> 
( ( ( A  X.  B ) ^r  J ) ^r  K )  =  ( ( A  X.  B
) ^r  I ) ) ) ) )
2928com13 83 . . . . . . . 8  |-  ( K  e.  NN  ->  ( J  e.  NN  ->  ( I  =  if ( J  <  K ,  J ,  K )  ->  ( ( A  e.  U  /\  B  e.  V  /\  ( A  i^i  B )  =/=  (/) )  ->  ( ( ( A  X.  B
) ^r  J ) ^r  K )  =  ( ( A  X.  B ) ^r  I ) ) ) ) )
30 simp3 1011 . . . . . . . . . . 11  |-  ( ( K  e.  NN  /\  J  =  0  /\  I  =  if ( J  <  K ,  J ,  K ) )  ->  I  =  if ( J  <  K ,  J ,  K ) )
31 simp2 1010 . . . . . . . . . . . . 13  |-  ( ( K  e.  NN  /\  J  =  0  /\  I  =  if ( J  <  K ,  J ,  K ) )  ->  J  =  0 )
32 simp1 1009 . . . . . . . . . . . . . 14  |-  ( ( K  e.  NN  /\  J  =  0  /\  I  =  if ( J  <  K ,  J ,  K ) )  ->  K  e.  NN )
3332nngt0d 10660 . . . . . . . . . . . . 13  |-  ( ( K  e.  NN  /\  J  =  0  /\  I  =  if ( J  <  K ,  J ,  K ) )  -> 
0  <  K )
3431, 33eqbrtrd 4426 . . . . . . . . . . . 12  |-  ( ( K  e.  NN  /\  J  =  0  /\  I  =  if ( J  <  K ,  J ,  K ) )  ->  J  <  K )
3534iftrued 3891 . . . . . . . . . . 11  |-  ( ( K  e.  NN  /\  J  =  0  /\  I  =  if ( J  <  K ,  J ,  K ) )  ->  if ( J  <  K ,  J ,  K )  =  J )
3630, 35, 313eqtrd 2491 . . . . . . . . . 10  |-  ( ( K  e.  NN  /\  J  =  0  /\  I  =  if ( J  <  K ,  J ,  K ) )  ->  I  =  0 )
37 simpr1 1015 . . . . . . . . . . . . . . 15  |-  ( ( ( K  e.  NN  /\  J  =  0  /\  I  =  0 )  /\  ( A  e.  U  /\  B  e.  V  /\  ( A  i^i  B )  =/=  (/) ) )  ->  A  e.  U )
38 simpr2 1016 . . . . . . . . . . . . . . 15  |-  ( ( ( K  e.  NN  /\  J  =  0  /\  I  =  0 )  /\  ( A  e.  U  /\  B  e.  V  /\  ( A  i^i  B )  =/=  (/) ) )  ->  B  e.  V )
39 xpexg 6598 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  U  /\  B  e.  V )  ->  ( A  X.  B
)  e.  _V )
4037, 38, 39syl2anc 667 . . . . . . . . . . . . . 14  |-  ( ( ( K  e.  NN  /\  J  =  0  /\  I  =  0 )  /\  ( A  e.  U  /\  B  e.  V  /\  ( A  i^i  B )  =/=  (/) ) )  ->  ( A  X.  B )  e. 
_V )
41 dmexg 6729 . . . . . . . . . . . . . . 15  |-  ( ( A  X.  B )  e.  _V  ->  dom  ( A  X.  B
)  e.  _V )
42 rnexg 6730 . . . . . . . . . . . . . . 15  |-  ( ( A  X.  B )  e.  _V  ->  ran  ( A  X.  B
)  e.  _V )
4341, 42jca 535 . . . . . . . . . . . . . 14  |-  ( ( A  X.  B )  e.  _V  ->  ( dom  ( A  X.  B
)  e.  _V  /\  ran  ( A  X.  B
)  e.  _V )
)
44 unexg 6597 . . . . . . . . . . . . . 14  |-  ( ( dom  ( A  X.  B )  e.  _V  /\ 
ran  ( A  X.  B )  e.  _V )  ->  ( dom  ( A  X.  B )  u. 
ran  ( A  X.  B ) )  e. 
_V )
4540, 43, 443syl 18 . . . . . . . . . . . . 13  |-  ( ( ( K  e.  NN  /\  J  =  0  /\  I  =  0 )  /\  ( A  e.  U  /\  B  e.  V  /\  ( A  i^i  B )  =/=  (/) ) )  ->  ( dom  ( A  X.  B
)  u.  ran  ( A  X.  B ) )  e.  _V )
46 simpl1 1012 . . . . . . . . . . . . . 14  |-  ( ( ( K  e.  NN  /\  J  =  0  /\  I  =  0 )  /\  ( A  e.  U  /\  B  e.  V  /\  ( A  i^i  B )  =/=  (/) ) )  ->  K  e.  NN )
4746nnnn0d 10932 . . . . . . . . . . . . 13  |-  ( ( ( K  e.  NN  /\  J  =  0  /\  I  =  0 )  /\  ( A  e.  U  /\  B  e.  V  /\  ( A  i^i  B )  =/=  (/) ) )  ->  K  e.  NN0 )
48 relexpiidm 36308 . . . . . . . . . . . . 13  |-  ( ( ( dom  ( A  X.  B )  u. 
ran  ( A  X.  B ) )  e. 
_V  /\  K  e.  NN0 )  ->  ( (  _I  |`  ( dom  ( A  X.  B )  u. 
ran  ( A  X.  B ) ) ) ^r  K )  =  (  _I  |`  ( dom  ( A  X.  B
)  u.  ran  ( A  X.  B ) ) ) )
4945, 47, 48syl2anc 667 . . . . . . . . . . . 12  |-  ( ( ( K  e.  NN  /\  J  =  0  /\  I  =  0 )  /\  ( A  e.  U  /\  B  e.  V  /\  ( A  i^i  B )  =/=  (/) ) )  ->  (
(  _I  |`  ( dom  ( A  X.  B
)  u.  ran  ( A  X.  B ) ) ) ^r  K )  =  (  _I  |`  ( dom  ( A  X.  B )  u. 
ran  ( A  X.  B ) ) ) )
50 simpl2 1013 . . . . . . . . . . . . . . 15  |-  ( ( ( K  e.  NN  /\  J  =  0  /\  I  =  0 )  /\  ( A  e.  U  /\  B  e.  V  /\  ( A  i^i  B )  =/=  (/) ) )  ->  J  =  0 )
5150oveq2d 6311 . . . . . . . . . . . . . 14  |-  ( ( ( K  e.  NN  /\  J  =  0  /\  I  =  0 )  /\  ( A  e.  U  /\  B  e.  V  /\  ( A  i^i  B )  =/=  (/) ) )  ->  (
( A  X.  B
) ^r  J )  =  ( ( A  X.  B ) ^r  0 ) )
52 relexp0g 13097 . . . . . . . . . . . . . . 15  |-  ( ( A  X.  B )  e.  _V  ->  (
( A  X.  B
) ^r  0 )  =  (  _I  |`  ( dom  ( A  X.  B )  u. 
ran  ( A  X.  B ) ) ) )
5340, 52syl 17 . . . . . . . . . . . . . 14  |-  ( ( ( K  e.  NN  /\  J  =  0  /\  I  =  0 )  /\  ( A  e.  U  /\  B  e.  V  /\  ( A  i^i  B )  =/=  (/) ) )  ->  (
( A  X.  B
) ^r  0 )  =  (  _I  |`  ( dom  ( A  X.  B )  u. 
ran  ( A  X.  B ) ) ) )
5451, 53eqtrd 2487 . . . . . . . . . . . . 13  |-  ( ( ( K  e.  NN  /\  J  =  0  /\  I  =  0 )  /\  ( A  e.  U  /\  B  e.  V  /\  ( A  i^i  B )  =/=  (/) ) )  ->  (
( A  X.  B
) ^r  J )  =  (  _I  |`  ( dom  ( A  X.  B )  u. 
ran  ( A  X.  B ) ) ) )
5554oveq1d 6310 . . . . . . . . . . . 12  |-  ( ( ( K  e.  NN  /\  J  =  0  /\  I  =  0 )  /\  ( A  e.  U  /\  B  e.  V  /\  ( A  i^i  B )  =/=  (/) ) )  ->  (
( ( A  X.  B ) ^r  J ) ^r  K )  =  ( (  _I  |`  ( dom  ( A  X.  B
)  u.  ran  ( A  X.  B ) ) ) ^r  K ) )
56 simpl3 1014 . . . . . . . . . . . . . 14  |-  ( ( ( K  e.  NN  /\  J  =  0  /\  I  =  0 )  /\  ( A  e.  U  /\  B  e.  V  /\  ( A  i^i  B )  =/=  (/) ) )  ->  I  =  0 )
5756oveq2d 6311 . . . . . . . . . . . . 13  |-  ( ( ( K  e.  NN  /\  J  =  0  /\  I  =  0 )  /\  ( A  e.  U  /\  B  e.  V  /\  ( A  i^i  B )  =/=  (/) ) )  ->  (
( A  X.  B
) ^r  I )  =  ( ( A  X.  B ) ^r  0 ) )
5857, 53eqtrd 2487 . . . . . . . . . . . 12  |-  ( ( ( K  e.  NN  /\  J  =  0  /\  I  =  0 )  /\  ( A  e.  U  /\  B  e.  V  /\  ( A  i^i  B )  =/=  (/) ) )  ->  (
( A  X.  B
) ^r  I )  =  (  _I  |`  ( dom  ( A  X.  B )  u. 
ran  ( A  X.  B ) ) ) )
5949, 55, 583eqtr4d 2497 . . . . . . . . . . 11  |-  ( ( ( K  e.  NN  /\  J  =  0  /\  I  =  0 )  /\  ( A  e.  U  /\  B  e.  V  /\  ( A  i^i  B )  =/=  (/) ) )  ->  (
( ( A  X.  B ) ^r  J ) ^r  K )  =  ( ( A  X.  B
) ^r  I ) )
6059ex 436 . . . . . . . . . 10  |-  ( ( K  e.  NN  /\  J  =  0  /\  I  =  0 )  ->  ( ( A  e.  U  /\  B  e.  V  /\  ( A  i^i  B )  =/=  (/) )  ->  ( ( ( A  X.  B
) ^r  J ) ^r  K )  =  ( ( A  X.  B ) ^r  I ) ) )
6136, 60syld3an3 1314 . . . . . . . . 9  |-  ( ( K  e.  NN  /\  J  =  0  /\  I  =  if ( J  <  K ,  J ,  K ) )  -> 
( ( A  e.  U  /\  B  e.  V  /\  ( A  i^i  B )  =/=  (/) )  ->  ( ( ( A  X.  B
) ^r  J ) ^r  K )  =  ( ( A  X.  B ) ^r  I ) ) )
62613exp 1208 . . . . . . . 8  |-  ( K  e.  NN  ->  ( J  =  0  ->  ( I  =  if ( J  <  K ,  J ,  K )  ->  ( ( A  e.  U  /\  B  e.  V  /\  ( A  i^i  B )  =/=  (/) )  ->  ( ( ( A  X.  B
) ^r  J ) ^r  K )  =  ( ( A  X.  B ) ^r  I ) ) ) ) )
6329, 62jaod 382 . . . . . . 7  |-  ( K  e.  NN  ->  (
( J  e.  NN  \/  J  =  0
)  ->  ( I  =  if ( J  < 
K ,  J ,  K )  ->  (
( A  e.  U  /\  B  e.  V  /\  ( A  i^i  B
)  =/=  (/) )  -> 
( ( ( A  X.  B ) ^r  J ) ^r  K )  =  ( ( A  X.  B
) ^r  I ) ) ) ) )
642, 63syl5bi 221 . . . . . 6  |-  ( K  e.  NN  ->  ( J  e.  NN0  ->  (
I  =  if ( J  <  K ,  J ,  K )  ->  ( ( A  e.  U  /\  B  e.  V  /\  ( A  i^i  B )  =/=  (/) )  ->  ( ( ( A  X.  B
) ^r  J ) ^r  K )  =  ( ( A  X.  B ) ^r  I ) ) ) ) )
65 simp1 1009 . . . . . . . 8  |-  ( ( K  =  0  /\  J  e.  NN0  /\  I  =  if ( J  <  K ,  J ,  K ) )  ->  K  =  0 )
662biimpi 198 . . . . . . . . 9  |-  ( J  e.  NN0  ->  ( J  e.  NN  \/  J  =  0 ) )
67663ad2ant2 1031 . . . . . . . 8  |-  ( ( K  =  0  /\  J  e.  NN0  /\  I  =  if ( J  <  K ,  J ,  K ) )  -> 
( J  e.  NN  \/  J  =  0
) )
68 simp3 1011 . . . . . . . . 9  |-  ( ( K  =  0  /\  J  e.  NN0  /\  I  =  if ( J  <  K ,  J ,  K ) )  ->  I  =  if ( J  <  K ,  J ,  K ) )
69 nn0nlt0 10903 . . . . . . . . . . . 12  |-  ( J  e.  NN0  ->  -.  J  <  0 )
70693ad2ant2 1031 . . . . . . . . . . 11  |-  ( ( K  =  0  /\  J  e.  NN0  /\  I  =  if ( J  <  K ,  J ,  K ) )  ->  -.  J  <  0
)
7165breq2d 4417 . . . . . . . . . . 11  |-  ( ( K  =  0  /\  J  e.  NN0  /\  I  =  if ( J  <  K ,  J ,  K ) )  -> 
( J  <  K  <->  J  <  0 ) )
7270, 71mtbird 303 . . . . . . . . . 10  |-  ( ( K  =  0  /\  J  e.  NN0  /\  I  =  if ( J  <  K ,  J ,  K ) )  ->  -.  J  <  K )
7372iffalsed 3894 . . . . . . . . 9  |-  ( ( K  =  0  /\  J  e.  NN0  /\  I  =  if ( J  <  K ,  J ,  K ) )  ->  if ( J  <  K ,  J ,  K )  =  K )
7468, 73, 653eqtrd 2491 . . . . . . . 8  |-  ( ( K  =  0  /\  J  e.  NN0  /\  I  =  if ( J  <  K ,  J ,  K ) )  ->  I  =  0 )
75133ad2ant2 1031 . . . . . . . . . . . . 13  |-  ( ( K  =  0  /\  J  e.  NN  /\  I  =  0 )  ->  ( ( A  e.  U  /\  B  e.  V  /\  ( A  i^i  B )  =/=  (/) )  ->  ( ( A  X.  B ) ^r  J )  =  ( A  X.  B ) ) )
7675imp 431 . . . . . . . . . . . 12  |-  ( ( ( K  =  0  /\  J  e.  NN  /\  I  =  0 )  /\  ( A  e.  U  /\  B  e.  V  /\  ( A  i^i  B )  =/=  (/) ) )  ->  (
( A  X.  B
) ^r  J )  =  ( A  X.  B ) )
7776oveq1d 6310 . . . . . . . . . . 11  |-  ( ( ( K  =  0  /\  J  e.  NN  /\  I  =  0 )  /\  ( A  e.  U  /\  B  e.  V  /\  ( A  i^i  B )  =/=  (/) ) )  ->  (
( ( A  X.  B ) ^r  J ) ^r 
0 )  =  ( ( A  X.  B
) ^r  0 ) )
78 simpl1 1012 . . . . . . . . . . . 12  |-  ( ( ( K  =  0  /\  J  e.  NN  /\  I  =  0 )  /\  ( A  e.  U  /\  B  e.  V  /\  ( A  i^i  B )  =/=  (/) ) )  ->  K  =  0 )
7978oveq2d 6311 . . . . . . . . . . 11  |-  ( ( ( K  =  0  /\  J  e.  NN  /\  I  =  0 )  /\  ( A  e.  U  /\  B  e.  V  /\  ( A  i^i  B )  =/=  (/) ) )  ->  (
( ( A  X.  B ) ^r  J ) ^r  K )  =  ( ( ( A  X.  B ) ^r  J ) ^r 
0 ) )
80 simpl3 1014 . . . . . . . . . . . 12  |-  ( ( ( K  =  0  /\  J  e.  NN  /\  I  =  0 )  /\  ( A  e.  U  /\  B  e.  V  /\  ( A  i^i  B )  =/=  (/) ) )  ->  I  =  0 )
8180oveq2d 6311 . . . . . . . . . . 11  |-  ( ( ( K  =  0  /\  J  e.  NN  /\  I  =  0 )  /\  ( A  e.  U  /\  B  e.  V  /\  ( A  i^i  B )  =/=  (/) ) )  ->  (
( A  X.  B
) ^r  I )  =  ( ( A  X.  B ) ^r  0 ) )
8277, 79, 813eqtr4d 2497 . . . . . . . . . 10  |-  ( ( ( K  =  0  /\  J  e.  NN  /\  I  =  0 )  /\  ( A  e.  U  /\  B  e.  V  /\  ( A  i^i  B )  =/=  (/) ) )  ->  (
( ( A  X.  B ) ^r  J ) ^r  K )  =  ( ( A  X.  B
) ^r  I ) )
83823exp1 1226 . . . . . . . . 9  |-  ( K  =  0  ->  ( J  e.  NN  ->  ( I  =  0  -> 
( ( A  e.  U  /\  B  e.  V  /\  ( A  i^i  B )  =/=  (/) )  ->  ( ( ( A  X.  B
) ^r  J ) ^r  K )  =  ( ( A  X.  B ) ^r  I ) ) ) ) )
84 simpr1 1015 . . . . . . . . . . . . 13  |-  ( ( ( K  =  0  /\  J  =  0  /\  I  =  0 )  /\  ( A  e.  U  /\  B  e.  V  /\  ( A  i^i  B )  =/=  (/) ) )  ->  A  e.  U )
85 simpr2 1016 . . . . . . . . . . . . 13  |-  ( ( ( K  =  0  /\  J  =  0  /\  I  =  0 )  /\  ( A  e.  U  /\  B  e.  V  /\  ( A  i^i  B )  =/=  (/) ) )  ->  B  e.  V )
8684, 85, 39syl2anc 667 . . . . . . . . . . . 12  |-  ( ( ( K  =  0  /\  J  =  0  /\  I  =  0 )  /\  ( A  e.  U  /\  B  e.  V  /\  ( A  i^i  B )  =/=  (/) ) )  ->  ( A  X.  B )  e. 
_V )
87 relexp0idm 36319 . . . . . . . . . . . 12  |-  ( ( A  X.  B )  e.  _V  ->  (
( ( A  X.  B ) ^r 
0 ) ^r 
0 )  =  ( ( A  X.  B
) ^r  0 ) )
8886, 87syl 17 . . . . . . . . . . 11  |-  ( ( ( K  =  0  /\  J  =  0  /\  I  =  0 )  /\  ( A  e.  U  /\  B  e.  V  /\  ( A  i^i  B )  =/=  (/) ) )  ->  (
( ( A  X.  B ) ^r 
0 ) ^r 
0 )  =  ( ( A  X.  B
) ^r  0 ) )
89 simpl2 1013 . . . . . . . . . . . . 13  |-  ( ( ( K  =  0  /\  J  =  0  /\  I  =  0 )  /\  ( A  e.  U  /\  B  e.  V  /\  ( A  i^i  B )  =/=  (/) ) )  ->  J  =  0 )
9089oveq2d 6311 . . . . . . . . . . . 12  |-  ( ( ( K  =  0  /\  J  =  0  /\  I  =  0 )  /\  ( A  e.  U  /\  B  e.  V  /\  ( A  i^i  B )  =/=  (/) ) )  ->  (
( A  X.  B
) ^r  J )  =  ( ( A  X.  B ) ^r  0 ) )
91 simpl1 1012 . . . . . . . . . . . 12  |-  ( ( ( K  =  0  /\  J  =  0  /\  I  =  0 )  /\  ( A  e.  U  /\  B  e.  V  /\  ( A  i^i  B )  =/=  (/) ) )  ->  K  =  0 )
9290, 91oveq12d 6313 . . . . . . . . . . 11  |-  ( ( ( K  =  0  /\  J  =  0  /\  I  =  0 )  /\  ( A  e.  U  /\  B  e.  V  /\  ( A  i^i  B )  =/=  (/) ) )  ->  (
( ( A  X.  B ) ^r  J ) ^r  K )  =  ( ( ( A  X.  B ) ^r 
0 ) ^r 
0 ) )
93 simpl3 1014 . . . . . . . . . . . 12  |-  ( ( ( K  =  0  /\  J  =  0  /\  I  =  0 )  /\  ( A  e.  U  /\  B  e.  V  /\  ( A  i^i  B )  =/=  (/) ) )  ->  I  =  0 )
9493oveq2d 6311 . . . . . . . . . . 11  |-  ( ( ( K  =  0  /\  J  =  0  /\  I  =  0 )  /\  ( A  e.  U  /\  B  e.  V  /\  ( A  i^i  B )  =/=  (/) ) )  ->  (
( A  X.  B
) ^r  I )  =  ( ( A  X.  B ) ^r  0 ) )
9588, 92, 943eqtr4d 2497 . . . . . . . . . 10  |-  ( ( ( K  =  0  /\  J  =  0  /\  I  =  0 )  /\  ( A  e.  U  /\  B  e.  V  /\  ( A  i^i  B )  =/=  (/) ) )  ->  (
( ( A  X.  B ) ^r  J ) ^r  K )  =  ( ( A  X.  B
) ^r  I ) )
96953exp1 1226 . . . . . . . . 9  |-  ( K  =  0  ->  ( J  =  0  ->  ( I  =  0  -> 
( ( A  e.  U  /\  B  e.  V  /\  ( A  i^i  B )  =/=  (/) )  ->  ( ( ( A  X.  B
) ^r  J ) ^r  K )  =  ( ( A  X.  B ) ^r  I ) ) ) ) )
9783, 96jaod 382 . . . . . . . 8  |-  ( K  =  0  ->  (
( J  e.  NN  \/  J  =  0
)  ->  ( I  =  0  ->  (
( A  e.  U  /\  B  e.  V  /\  ( A  i^i  B
)  =/=  (/) )  -> 
( ( ( A  X.  B ) ^r  J ) ^r  K )  =  ( ( A  X.  B
) ^r  I ) ) ) ) )
9865, 67, 74, 97syl3c 63 . . . . . . 7  |-  ( ( K  =  0  /\  J  e.  NN0  /\  I  =  if ( J  <  K ,  J ,  K ) )  -> 
( ( A  e.  U  /\  B  e.  V  /\  ( A  i^i  B )  =/=  (/) )  ->  ( ( ( A  X.  B
) ^r  J ) ^r  K )  =  ( ( A  X.  B ) ^r  I ) ) )
99983exp 1208 . . . . . 6  |-  ( K  =  0  ->  ( J  e.  NN0  ->  (
I  =  if ( J  <  K ,  J ,  K )  ->  ( ( A  e.  U  /\  B  e.  V  /\  ( A  i^i  B )  =/=  (/) )  ->  ( ( ( A  X.  B
) ^r  J ) ^r  K )  =  ( ( A  X.  B ) ^r  I ) ) ) ) )
10064, 99jaoi 381 . . . . 5  |-  ( ( K  e.  NN  \/  K  =  0 )  ->  ( J  e. 
NN0  ->  ( I  =  if ( J  < 
K ,  J ,  K )  ->  (
( A  e.  U  /\  B  e.  V  /\  ( A  i^i  B
)  =/=  (/) )  -> 
( ( ( A  X.  B ) ^r  J ) ^r  K )  =  ( ( A  X.  B
) ^r  I ) ) ) ) )
1011, 100sylbi 199 . . . 4  |-  ( K  e.  NN0  ->  ( J  e.  NN0  ->  ( I  =  if ( J  <  K ,  J ,  K )  ->  (
( A  e.  U  /\  B  e.  V  /\  ( A  i^i  B
)  =/=  (/) )  -> 
( ( ( A  X.  B ) ^r  J ) ^r  K )  =  ( ( A  X.  B
) ^r  I ) ) ) ) )
102101com13 83 . . 3  |-  ( I  =  if ( J  <  K ,  J ,  K )  ->  ( J  e.  NN0  ->  ( K  e.  NN0  ->  (
( A  e.  U  /\  B  e.  V  /\  ( A  i^i  B
)  =/=  (/) )  -> 
( ( ( A  X.  B ) ^r  J ) ^r  K )  =  ( ( A  X.  B
) ^r  I ) ) ) ) )
1031023imp 1203 . 2  |-  ( ( I  =  if ( J  <  K ,  J ,  K )  /\  J  e.  NN0  /\  K  e.  NN0 )  ->  ( ( A  e.  U  /\  B  e.  V  /\  ( A  i^i  B )  =/=  (/) )  ->  ( ( ( A  X.  B
) ^r  J ) ^r  K )  =  ( ( A  X.  B ) ^r  I ) ) )
104103impcom 432 1  |-  ( ( ( A  e.  U  /\  B  e.  V  /\  ( A  i^i  B
)  =/=  (/) )  /\  ( I  =  if ( J  <  K ,  J ,  K )  /\  J  e.  NN0  /\  K  e.  NN0 )
)  ->  ( (
( A  X.  B
) ^r  J ) ^r  K )  =  ( ( A  X.  B ) ^r  I ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 370    /\ wa 371    /\ w3a 986    = wceq 1446    e. wcel 1889    =/= wne 2624   _Vcvv 3047    u. cun 3404    i^i cin 3405   (/)c0 3733   ifcif 3883   class class class wbr 4405    _I cid 4747    X. cxp 4835   dom cdm 4837   ran crn 4838    |` cres 4839  (class class class)co 6295   0cc0 9544    < clt 9680   NNcn 10616   NN0cn0 10876   ^r crelexp 13095
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-cnex 9600  ax-resscn 9601  ax-1cn 9602  ax-icn 9603  ax-addcl 9604  ax-addrcl 9605  ax-mulcl 9606  ax-mulrcl 9607  ax-mulcom 9608  ax-addass 9609  ax-mulass 9610  ax-distr 9611  ax-i2m1 9612  ax-1ne0 9613  ax-1rid 9614  ax-rnegex 9615  ax-rrecex 9616  ax-cnre 9617  ax-pre-lttri 9618  ax-pre-lttrn 9619  ax-pre-ltadd 9620  ax-pre-mulgt0 9621
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-reu 2746  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6698  df-2nd 6799  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-er 7368  df-en 7575  df-dom 7576  df-sdom 7577  df-pnf 9682  df-mnf 9683  df-xr 9684  df-ltxr 9685  df-le 9686  df-sub 9867  df-neg 9868  df-nn 10617  df-n0 10877  df-z 10945  df-uz 11167  df-seq 12221  df-relexp 13096
This theorem is referenced by: (None)
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