Metamath Proof Explorer

Theorem relexpxpnnidm

Description: Any positive power of a Cartesian product of non-disjoint sets is itself. (Contributed by RP, 13-Jun-2020)

Ref Expression
Assertion relexpxpnnidm 
|- ( N e. NN -> ( ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) -> ( ( A X. B ) ^r N ) = ( A X. B ) ) )

Proof

Step Hyp Ref Expression
1 oveq2 
 |-  ( x = 1 -> ( ( A X. B ) ^r x ) = ( ( A X. B ) ^r 1 ) )
2 1 eqeq1d 
 |-  ( x = 1 -> ( ( ( A X. B ) ^r x ) = ( A X. B ) <-> ( ( A X. B ) ^r 1 ) = ( A X. B ) ) )
3 2 imbi2d 
 |-  ( x = 1 -> ( ( ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) -> ( ( A X. B ) ^r x ) = ( A X. B ) ) <-> ( ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) -> ( ( A X. B ) ^r 1 ) = ( A X. B ) ) ) )
4 oveq2 
 |-  ( x = y -> ( ( A X. B ) ^r x ) = ( ( A X. B ) ^r y ) )
5 4 eqeq1d 
 |-  ( x = y -> ( ( ( A X. B ) ^r x ) = ( A X. B ) <-> ( ( A X. B ) ^r y ) = ( A X. B ) ) )
6 5 imbi2d 
 |-  ( x = y -> ( ( ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) -> ( ( A X. B ) ^r x ) = ( A X. B ) ) <-> ( ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) -> ( ( A X. B ) ^r y ) = ( A X. B ) ) ) )
7 oveq2 
 |-  ( x = ( y + 1 ) -> ( ( A X. B ) ^r x ) = ( ( A X. B ) ^r ( y + 1 ) ) )
8 7 eqeq1d 
 |-  ( x = ( y + 1 ) -> ( ( ( A X. B ) ^r x ) = ( A X. B ) <-> ( ( A X. B ) ^r ( y + 1 ) ) = ( A X. B ) ) )
9 8 imbi2d 
 |-  ( x = ( y + 1 ) -> ( ( ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) -> ( ( A X. B ) ^r x ) = ( A X. B ) ) <-> ( ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) -> ( ( A X. B ) ^r ( y + 1 ) ) = ( A X. B ) ) ) )
10 oveq2 
 |-  ( x = N -> ( ( A X. B ) ^r x ) = ( ( A X. B ) ^r N ) )
11 10 eqeq1d 
 |-  ( x = N -> ( ( ( A X. B ) ^r x ) = ( A X. B ) <-> ( ( A X. B ) ^r N ) = ( A X. B ) ) )
12 11 imbi2d 
 |-  ( x = N -> ( ( ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) -> ( ( A X. B ) ^r x ) = ( A X. B ) ) <-> ( ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) -> ( ( A X. B ) ^r N ) = ( A X. B ) ) ) )
13 3simpa 
 |-  ( ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) -> ( A e. U /\ B e. V ) )
14 xpexg 
 |-  ( ( A e. U /\ B e. V ) -> ( A X. B ) e. _V )
15 relexp1g 
 |-  ( ( A X. B ) e. _V -> ( ( A X. B ) ^r 1 ) = ( A X. B ) )
16 13 14 15 3syl 
 |-  ( ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) -> ( ( A X. B ) ^r 1 ) = ( A X. B ) )
17 simp2 
 |-  ( ( y e. NN /\ ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) /\ ( ( A X. B ) ^r y ) = ( A X. B ) ) -> ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) )
18 17 13 14 3syl 
 |-  ( ( y e. NN /\ ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) /\ ( ( A X. B ) ^r y ) = ( A X. B ) ) -> ( A X. B ) e. _V )
19 simp1 
 |-  ( ( y e. NN /\ ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) /\ ( ( A X. B ) ^r y ) = ( A X. B ) ) -> y e. NN )
20 relexpsucnnr 
 |-  ( ( ( A X. B ) e. _V /\ y e. NN ) -> ( ( A X. B ) ^r ( y + 1 ) ) = ( ( ( A X. B ) ^r y ) o. ( A X. B ) ) )
21 18 19 20 syl2anc 
 |-  ( ( y e. NN /\ ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) /\ ( ( A X. B ) ^r y ) = ( A X. B ) ) -> ( ( A X. B ) ^r ( y + 1 ) ) = ( ( ( A X. B ) ^r y ) o. ( A X. B ) ) )
22 simp3 
 |-  ( ( y e. NN /\ ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) /\ ( ( A X. B ) ^r y ) = ( A X. B ) ) -> ( ( A X. B ) ^r y ) = ( A X. B ) )
23 22 coeq1d 
 |-  ( ( y e. NN /\ ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) /\ ( ( A X. B ) ^r y ) = ( A X. B ) ) -> ( ( ( A X. B ) ^r y ) o. ( A X. B ) ) = ( ( A X. B ) o. ( A X. B ) ) )
24 simp23 
 |-  ( ( y e. NN /\ ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) /\ ( ( A X. B ) ^r y ) = ( A X. B ) ) -> ( A i^i B ) =/= (/) )
25 24 xpcoidgend 
 |-  ( ( y e. NN /\ ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) /\ ( ( A X. B ) ^r y ) = ( A X. B ) ) -> ( ( A X. B ) o. ( A X. B ) ) = ( A X. B ) )
26 21 23 25 3eqtrd 
 |-  ( ( y e. NN /\ ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) /\ ( ( A X. B ) ^r y ) = ( A X. B ) ) -> ( ( A X. B ) ^r ( y + 1 ) ) = ( A X. B ) )
27 26 3exp 
 |-  ( y e. NN -> ( ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) -> ( ( ( A X. B ) ^r y ) = ( A X. B ) -> ( ( A X. B ) ^r ( y + 1 ) ) = ( A X. B ) ) ) )
28 27 a2d 
 |-  ( y e. NN -> ( ( ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) -> ( ( A X. B ) ^r y ) = ( A X. B ) ) -> ( ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) -> ( ( A X. B ) ^r ( y + 1 ) ) = ( A X. B ) ) ) )
29 3 6 9 12 16 28 nnind 
 |-  ( N e. NN -> ( ( A e. U /\ B e. V /\ ( A i^i B ) =/= (/) ) -> ( ( A X. B ) ^r N ) = ( A X. B ) ) )