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Theorem relexpxpnnidm 36266
Description: Any positive power of a cross product of non-disjoint sets is itself. (Contributed by RP, 13-Jun-2020.)
Assertion
Ref Expression
relexpxpnnidm  |-  ( N  e.  NN  ->  (
( A  e.  U  /\  B  e.  V  /\  ( A  i^i  B
)  =/=  (/) )  -> 
( ( A  X.  B ) ^r  N )  =  ( A  X.  B ) ) )

Proof of Theorem relexpxpnnidm
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 6314 . . . 4  |-  ( x  =  1  ->  (
( A  X.  B
) ^r  x )  =  ( ( A  X.  B ) ^r  1 ) )
21eqeq1d 2424 . . 3  |-  ( x  =  1  ->  (
( ( A  X.  B ) ^r 
x )  =  ( A  X.  B )  <-> 
( ( A  X.  B ) ^r 
1 )  =  ( A  X.  B ) ) )
32imbi2d 317 . 2  |-  ( 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 6314 . . . 4  |-  ( x  =  y  ->  (
( A  X.  B
) ^r  x )  =  ( ( A  X.  B ) ^r  y ) )
54eqeq1d 2424 . . 3  |-  ( x  =  y  ->  (
( ( A  X.  B ) ^r 
x )  =  ( A  X.  B )  <-> 
( ( A  X.  B ) ^r 
y )  =  ( A  X.  B ) ) )
65imbi2d 317 . 2  |-  ( 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 6314 . . . 4  |-  ( x  =  ( y  +  1 )  ->  (
( A  X.  B
) ^r  x )  =  ( ( A  X.  B ) ^r  ( y  +  1 ) ) )
87eqeq1d 2424 . . 3  |-  ( x  =  ( y  +  1 )  ->  (
( ( A  X.  B ) ^r 
x )  =  ( A  X.  B )  <-> 
( ( A  X.  B ) ^r 
( y  +  1 ) )  =  ( A  X.  B ) ) )
98imbi2d 317 . 2  |-  ( 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 6314 . . . 4  |-  ( x  =  N  ->  (
( A  X.  B
) ^r  x )  =  ( ( A  X.  B ) ^r  N ) )
1110eqeq1d 2424 . . 3  |-  ( x  =  N  ->  (
( ( A  X.  B ) ^r 
x )  =  ( A  X.  B )  <-> 
( ( A  X.  B ) ^r  N )  =  ( A  X.  B ) ) )
1211imbi2d 317 . 2  |-  ( 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 1002 . . 3  |-  ( ( A  e.  U  /\  B  e.  V  /\  ( A  i^i  B )  =/=  (/) )  ->  ( A  e.  U  /\  B  e.  V )
)
14 xpexg 6608 . . 3  |-  ( ( A  e.  U  /\  B  e.  V )  ->  ( A  X.  B
)  e.  _V )
15 relexp1g 13090 . . 3  |-  ( ( A  X.  B )  e.  _V  ->  (
( A  X.  B
) ^r  1 )  =  ( A  X.  B ) )
1613, 14, 153syl 18 . 2  |-  ( ( A  e.  U  /\  B  e.  V  /\  ( A  i^i  B )  =/=  (/) )  ->  (
( A  X.  B
) ^r  1 )  =  ( A  X.  B ) )
17 simp2 1006 . . . . . . 7  |-  ( ( 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 )  =/=  (/) ) )
1817, 13, 143syl 18 . . . . . 6  |-  ( ( 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 1005 . . . . . 6  |-  ( ( 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 13089 . . . . . 6  |-  ( ( ( A  X.  B
)  e.  _V  /\  y  e.  NN )  ->  ( ( A  X.  B ) ^r 
( y  +  1 ) )  =  ( ( ( A  X.  B ) ^r 
y )  o.  ( A  X.  B ) ) )
2118, 19, 20syl2anc 665 . . . . 5  |-  ( ( 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 1007 . . . . . 6  |-  ( ( 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
) )
2322coeq1d 5015 . . . . 5  |-  ( ( 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 1040 . . . . . 6  |-  ( ( 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 )  =/=  (/) )
2524xpcoidgend 13040 . . . . 5  |-  ( ( 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 ) )
2621, 23, 253eqtrd 2467 . . . 4  |-  ( ( 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
) )
27263exp 1204 . . 3  |-  ( 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
) ) ) )
2827a2d 29 . 2  |-  ( 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 ) ) ) )
293, 6, 9, 12, 16, 28nnind 10635 1  |-  ( N  e.  NN  ->  (
( A  e.  U  /\  B  e.  V  /\  ( A  i^i  B
)  =/=  (/) )  -> 
( ( A  X.  B ) ^r  N )  =  ( A  X.  B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872    =/= wne 2614   _Vcvv 3080    i^i cin 3435   (/)c0 3761    X. cxp 4851    o. ccom 4857  (class class class)co 6306   1c1 9548    + caddc 9550   NNcn 10617   ^r crelexp 13084
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6598  ax-cnex 9603  ax-resscn 9604  ax-1cn 9605  ax-icn 9606  ax-addcl 9607  ax-addrcl 9608  ax-mulcl 9609  ax-mulrcl 9610  ax-mulcom 9611  ax-addass 9612  ax-mulass 9613  ax-distr 9614  ax-i2m1 9615  ax-1ne0 9616  ax-1rid 9617  ax-rnegex 9618  ax-rrecex 9619  ax-cnre 9620  ax-pre-lttri 9621  ax-pre-lttrn 9622  ax-pre-ltadd 9623  ax-pre-mulgt0 9624
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6268  df-ov 6309  df-oprab 6310  df-mpt2 6311  df-om 6708  df-2nd 6809  df-wrecs 7040  df-recs 7102  df-rdg 7140  df-er 7375  df-en 7582  df-dom 7583  df-sdom 7584  df-pnf 9685  df-mnf 9686  df-xr 9687  df-ltxr 9688  df-le 9689  df-sub 9870  df-neg 9871  df-nn 10618  df-n0 10878  df-z 10946  df-uz 11168  df-seq 12221  df-relexp 13085
This theorem is referenced by:  relexpxpmin  36280
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