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Theorem relexpsucl 13173
Description: A reduction for relation exponentiation to the left. (Contributed by RP, 23-May-2020.)
Assertion
Ref Expression
relexpsucl  |-  ( ( R  e.  V  /\  Rel  R  /\  N  e. 
NN0 )  ->  ( R ^r  ( N  +  1 ) )  =  ( R  o.  ( R ^r  N ) ) )

Proof of Theorem relexpsucl
StepHypRef Expression
1 elnn0 10895 . . . 4  |-  ( N  e.  NN0  <->  ( N  e.  NN  \/  N  =  0 ) )
2 simp3 1032 . . . . . . 7  |-  ( ( N  e.  NN  /\  Rel  R  /\  R  e.  V )  ->  R  e.  V )
3 simp1 1030 . . . . . . 7  |-  ( ( N  e.  NN  /\  Rel  R  /\  R  e.  V )  ->  N  e.  NN )
4 relexpsucnnl 13172 . . . . . . 7  |-  ( ( R  e.  V  /\  N  e.  NN )  ->  ( R ^r 
( N  +  1 ) )  =  ( R  o.  ( R ^r  N ) ) )
52, 3, 4syl2anc 673 . . . . . 6  |-  ( ( N  e.  NN  /\  Rel  R  /\  R  e.  V )  ->  ( R ^r  ( N  +  1 ) )  =  ( R  o.  ( R ^r  N ) ) )
653expib 1234 . . . . 5  |-  ( N  e.  NN  ->  (
( Rel  R  /\  R  e.  V )  ->  ( R ^r 
( N  +  1 ) )  =  ( R  o.  ( R ^r  N ) ) ) )
7 simp2 1031 . . . . . . . 8  |-  ( ( N  =  0  /\ 
Rel  R  /\  R  e.  V )  ->  Rel  R )
8 relcoi1 5371 . . . . . . . 8  |-  ( Rel 
R  ->  ( R  o.  (  _I  |`  U. U. R ) )  =  R )
97, 8syl 17 . . . . . . 7  |-  ( ( N  =  0  /\ 
Rel  R  /\  R  e.  V )  ->  ( R  o.  (  _I  |` 
U. U. R ) )  =  R )
10 simp1 1030 . . . . . . . . . 10  |-  ( ( N  =  0  /\ 
Rel  R  /\  R  e.  V )  ->  N  =  0 )
1110oveq2d 6324 . . . . . . . . 9  |-  ( ( N  =  0  /\ 
Rel  R  /\  R  e.  V )  ->  ( R ^r  N )  =  ( R ^r  0 ) )
12 simp3 1032 . . . . . . . . . 10  |-  ( ( N  =  0  /\ 
Rel  R  /\  R  e.  V )  ->  R  e.  V )
13 relexp0 13163 . . . . . . . . . 10  |-  ( ( R  e.  V  /\  Rel  R )  ->  ( R ^r  0 )  =  (  _I  |`  U. U. R ) )
1412, 7, 13syl2anc 673 . . . . . . . . 9  |-  ( ( N  =  0  /\ 
Rel  R  /\  R  e.  V )  ->  ( R ^r  0 )  =  (  _I  |`  U. U. R ) )
1511, 14eqtrd 2505 . . . . . . . 8  |-  ( ( N  =  0  /\ 
Rel  R  /\  R  e.  V )  ->  ( R ^r  N )  =  (  _I  |`  U. U. R ) )
1615coeq2d 5002 . . . . . . 7  |-  ( ( N  =  0  /\ 
Rel  R  /\  R  e.  V )  ->  ( R  o.  ( R ^r  N ) )  =  ( R  o.  (  _I  |`  U. U. R ) ) )
1710oveq1d 6323 . . . . . . . . . 10  |-  ( ( N  =  0  /\ 
Rel  R  /\  R  e.  V )  ->  ( N  +  1 )  =  ( 0  +  1 ) )
18 0p1e1 10743 . . . . . . . . . 10  |-  ( 0  +  1 )  =  1
1917, 18syl6eq 2521 . . . . . . . . 9  |-  ( ( N  =  0  /\ 
Rel  R  /\  R  e.  V )  ->  ( N  +  1 )  =  1 )
2019oveq2d 6324 . . . . . . . 8  |-  ( ( N  =  0  /\ 
Rel  R  /\  R  e.  V )  ->  ( R ^r  ( N  +  1 ) )  =  ( R ^r  1 ) )
21 relexp1g 13166 . . . . . . . . 9  |-  ( R  e.  V  ->  ( R ^r  1 )  =  R )
2212, 21syl 17 . . . . . . . 8  |-  ( ( N  =  0  /\ 
Rel  R  /\  R  e.  V )  ->  ( R ^r  1 )  =  R )
2320, 22eqtrd 2505 . . . . . . 7  |-  ( ( N  =  0  /\ 
Rel  R  /\  R  e.  V )  ->  ( R ^r  ( N  +  1 ) )  =  R )
249, 16, 233eqtr4rd 2516 . . . . . 6  |-  ( ( N  =  0  /\ 
Rel  R  /\  R  e.  V )  ->  ( R ^r  ( N  +  1 ) )  =  ( R  o.  ( R ^r  N ) ) )
25243expib 1234 . . . . 5  |-  ( N  =  0  ->  (
( Rel  R  /\  R  e.  V )  ->  ( R ^r 
( N  +  1 ) )  =  ( R  o.  ( R ^r  N ) ) ) )
266, 25jaoi 386 . . . 4  |-  ( ( N  e.  NN  \/  N  =  0 )  ->  ( ( Rel 
R  /\  R  e.  V )  ->  ( R ^r  ( N  +  1 ) )  =  ( R  o.  ( R ^r  N ) ) ) )
271, 26sylbi 200 . . 3  |-  ( N  e.  NN0  ->  ( ( Rel  R  /\  R  e.  V )  ->  ( R ^r  ( N  +  1 ) )  =  ( R  o.  ( R ^r  N ) ) ) )
28273impib 1229 . 2  |-  ( ( N  e.  NN0  /\  Rel  R  /\  R  e.  V )  ->  ( R ^r  ( N  +  1 ) )  =  ( R  o.  ( R ^r  N ) ) )
29283com13 1236 1  |-  ( ( R  e.  V  /\  Rel  R  /\  N  e. 
NN0 )  ->  ( R ^r  ( N  +  1 ) )  =  ( R  o.  ( R ^r  N ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 375    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904   U.cuni 4190    _I cid 4749    |` cres 4841    o. ccom 4843   Rel wrel 4844  (class class class)co 6308   0cc0 9557   1c1 9558    + caddc 9560   NNcn 10631   NN0cn0 10893   ^r crelexp 13160
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-n0 10894  df-z 10962  df-uz 11183  df-seq 12252  df-relexp 13161
This theorem is referenced by:  relexpsucld  13174  relexpindlem  13203
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