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Theorem relexpsucr 13031
Description: A reduction for relation exponentiation to the right. (Contributed by RP, 23-May-2020.)
Assertion
Ref Expression
relexpsucr  |-  ( ( R  e.  V  /\  Rel  R  /\  N  e. 
NN0 )  ->  ( R ^r  ( N  +  1 ) )  =  ( ( R ^r  N )  o.  R ) )

Proof of Theorem relexpsucr
StepHypRef Expression
1 elnn0 10817 . . . 4  |-  ( N  e.  NN0  <->  ( N  e.  NN  \/  N  =  0 ) )
2 simp3 1007 . . . . . . 7  |-  ( ( N  e.  NN  /\  Rel  R  /\  R  e.  V )  ->  R  e.  V )
3 simp1 1005 . . . . . . 7  |-  ( ( N  e.  NN  /\  Rel  R  /\  R  e.  V )  ->  N  e.  NN )
4 relexpsucnnr 13027 . . . . . . 7  |-  ( ( R  e.  V  /\  N  e.  NN )  ->  ( R ^r 
( N  +  1 ) )  =  ( ( R ^r  N )  o.  R
) )
52, 3, 4syl2anc 665 . . . . . 6  |-  ( ( N  e.  NN  /\  Rel  R  /\  R  e.  V )  ->  ( R ^r  ( N  +  1 ) )  =  ( ( R ^r  N )  o.  R ) )
653expib 1208 . . . . 5  |-  ( N  e.  NN  ->  (
( Rel  R  /\  R  e.  V )  ->  ( R ^r 
( N  +  1 ) )  =  ( ( R ^r  N )  o.  R
) ) )
7 simp2 1006 . . . . . . . 8  |-  ( ( N  =  0  /\ 
Rel  R  /\  R  e.  V )  ->  Rel  R )
8 relcoi2 5320 . . . . . . . . 9  |-  ( Rel 
R  ->  ( (  _I  |`  U. U. R
)  o.  R )  =  R )
98eqcomd 2429 . . . . . . . 8  |-  ( Rel 
R  ->  R  =  ( (  _I  |`  U. U. R )  o.  R
) )
107, 9syl 17 . . . . . . 7  |-  ( ( N  =  0  /\ 
Rel  R  /\  R  e.  V )  ->  R  =  ( (  _I  |`  U. U. R )  o.  R ) )
11 simp1 1005 . . . . . . . . . . 11  |-  ( ( N  =  0  /\ 
Rel  R  /\  R  e.  V )  ->  N  =  0 )
1211oveq1d 6259 . . . . . . . . . 10  |-  ( ( N  =  0  /\ 
Rel  R  /\  R  e.  V )  ->  ( N  +  1 )  =  ( 0  +  1 ) )
13 0p1e1 10667 . . . . . . . . . 10  |-  ( 0  +  1 )  =  1
1412, 13syl6eq 2473 . . . . . . . . 9  |-  ( ( N  =  0  /\ 
Rel  R  /\  R  e.  V )  ->  ( N  +  1 )  =  1 )
1514oveq2d 6260 . . . . . . . 8  |-  ( ( N  =  0  /\ 
Rel  R  /\  R  e.  V )  ->  ( R ^r  ( N  +  1 ) )  =  ( R ^r  1 ) )
16 simp3 1007 . . . . . . . . 9  |-  ( ( N  =  0  /\ 
Rel  R  /\  R  e.  V )  ->  R  e.  V )
17 relexp1g 13028 . . . . . . . . 9  |-  ( R  e.  V  ->  ( R ^r  1 )  =  R )
1816, 17syl 17 . . . . . . . 8  |-  ( ( N  =  0  /\ 
Rel  R  /\  R  e.  V )  ->  ( R ^r  1 )  =  R )
1915, 18eqtrd 2457 . . . . . . 7  |-  ( ( N  =  0  /\ 
Rel  R  /\  R  e.  V )  ->  ( R ^r  ( N  +  1 ) )  =  R )
2011oveq2d 6260 . . . . . . . . 9  |-  ( ( N  =  0  /\ 
Rel  R  /\  R  e.  V )  ->  ( R ^r  N )  =  ( R ^r  0 ) )
21 relexp0 13025 . . . . . . . . . 10  |-  ( ( R  e.  V  /\  Rel  R )  ->  ( R ^r  0 )  =  (  _I  |`  U. U. R ) )
2216, 7, 21syl2anc 665 . . . . . . . . 9  |-  ( ( N  =  0  /\ 
Rel  R  /\  R  e.  V )  ->  ( R ^r  0 )  =  (  _I  |`  U. U. R ) )
2320, 22eqtrd 2457 . . . . . . . 8  |-  ( ( N  =  0  /\ 
Rel  R  /\  R  e.  V )  ->  ( R ^r  N )  =  (  _I  |`  U. U. R ) )
2423coeq1d 4953 . . . . . . 7  |-  ( ( N  =  0  /\ 
Rel  R  /\  R  e.  V )  ->  (
( R ^r  N )  o.  R
)  =  ( (  _I  |`  U. U. R
)  o.  R ) )
2510, 19, 243eqtr4d 2467 . . . . . 6  |-  ( ( N  =  0  /\ 
Rel  R  /\  R  e.  V )  ->  ( R ^r  ( N  +  1 ) )  =  ( ( R ^r  N )  o.  R ) )
26253expib 1208 . . . . 5  |-  ( N  =  0  ->  (
( Rel  R  /\  R  e.  V )  ->  ( R ^r 
( N  +  1 ) )  =  ( ( R ^r  N )  o.  R
) ) )
276, 26jaoi 380 . . . 4  |-  ( ( N  e.  NN  \/  N  =  0 )  ->  ( ( Rel 
R  /\  R  e.  V )  ->  ( R ^r  ( N  +  1 ) )  =  ( ( R ^r  N )  o.  R ) ) )
281, 27sylbi 198 . . 3  |-  ( N  e.  NN0  ->  ( ( Rel  R  /\  R  e.  V )  ->  ( R ^r  ( N  +  1 ) )  =  ( ( R ^r  N )  o.  R ) ) )
29283impib 1203 . 2  |-  ( ( N  e.  NN0  /\  Rel  R  /\  R  e.  V )  ->  ( R ^r  ( N  +  1 ) )  =  ( ( R ^r  N )  o.  R ) )
30293com13 1210 1  |-  ( ( R  e.  V  /\  Rel  R  /\  N  e. 
NN0 )  ->  ( R ^r  ( N  +  1 ) )  =  ( ( R ^r  N )  o.  R ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 369    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872   U.cuni 4157    _I cid 4701    |` cres 4793    o. ccom 4795   Rel wrel 4796  (class class class)co 6244   0cc0 9485   1c1 9486    + caddc 9488   NNcn 10555   NN0cn0 10815   ^r crelexp 13022
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 2058  ax-ext 2403  ax-sep 4484  ax-nul 4493  ax-pow 4540  ax-pr 4598  ax-un 6536  ax-cnex 9541  ax-resscn 9542  ax-1cn 9543  ax-icn 9544  ax-addcl 9545  ax-addrcl 9546  ax-mulcl 9547  ax-mulrcl 9548  ax-mulcom 9549  ax-addass 9550  ax-mulass 9551  ax-distr 9552  ax-i2m1 9553  ax-1ne0 9554  ax-1rid 9555  ax-rnegex 9556  ax-rrecex 9557  ax-cnre 9558  ax-pre-lttri 9559  ax-pre-lttrn 9560  ax-pre-ltadd 9561  ax-pre-mulgt0 9562
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 2275  df-mo 2276  df-clab 2410  df-cleq 2416  df-clel 2419  df-nfc 2553  df-ne 2596  df-nel 2597  df-ral 2714  df-rex 2715  df-reu 2716  df-rab 2718  df-v 3019  df-sbc 3238  df-csb 3334  df-dif 3377  df-un 3379  df-in 3381  df-ss 3388  df-pss 3390  df-nul 3700  df-if 3850  df-pw 3921  df-sn 3937  df-pr 3939  df-tp 3941  df-op 3943  df-uni 4158  df-iun 4239  df-br 4362  df-opab 4421  df-mpt 4422  df-tr 4457  df-eprel 4702  df-id 4706  df-po 4712  df-so 4713  df-fr 4750  df-we 4752  df-xp 4797  df-rel 4798  df-cnv 4799  df-co 4800  df-dm 4801  df-rn 4802  df-res 4803  df-ima 4804  df-pred 5337  df-ord 5383  df-on 5384  df-lim 5385  df-suc 5386  df-iota 5503  df-fun 5541  df-fn 5542  df-f 5543  df-f1 5544  df-fo 5545  df-f1o 5546  df-fv 5547  df-riota 6206  df-ov 6247  df-oprab 6248  df-mpt2 6249  df-om 6646  df-2nd 6747  df-wrecs 6978  df-recs 7040  df-rdg 7078  df-er 7313  df-en 7520  df-dom 7521  df-sdom 7522  df-pnf 9623  df-mnf 9624  df-xr 9625  df-ltxr 9626  df-le 9627  df-sub 9808  df-neg 9809  df-nn 10556  df-n0 10816  df-z 10884  df-uz 11106  df-seq 12159  df-relexp 13023
This theorem is referenced by:  relexpsucrd  13032
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