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Theorem relexpsucl 12951
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 10793 . . . 4  |-  ( N  e.  NN0  <->  ( N  e.  NN  \/  N  =  0 ) )
2 simp3 996 . . . . . . 7  |-  ( ( N  e.  NN  /\  Rel  R  /\  R  e.  V )  ->  R  e.  V )
3 simp1 994 . . . . . . 7  |-  ( ( N  e.  NN  /\  Rel  R  /\  R  e.  V )  ->  N  e.  NN )
4 relexpsucnnl 12950 . . . . . . 7  |-  ( ( R  e.  V  /\  N  e.  NN )  ->  ( R ^r 
( N  +  1 ) )  =  ( R  o.  ( R ^r  N ) ) )
52, 3, 4syl2anc 659 . . . . . 6  |-  ( ( N  e.  NN  /\  Rel  R  /\  R  e.  V )  ->  ( R ^r  ( N  +  1 ) )  =  ( R  o.  ( R ^r  N ) ) )
653expib 1197 . . . . 5  |-  ( N  e.  NN  ->  (
( Rel  R  /\  R  e.  V )  ->  ( R ^r 
( N  +  1 ) )  =  ( R  o.  ( R ^r  N ) ) ) )
7 simp2 995 . . . . . . . 8  |-  ( ( N  =  0  /\ 
Rel  R  /\  R  e.  V )  ->  Rel  R )
8 relcoi1 5519 . . . . . . . 8  |-  ( Rel 
R  ->  ( R  o.  (  _I  |`  U. U. R ) )  =  R )
97, 8syl 16 . . . . . . 7  |-  ( ( N  =  0  /\ 
Rel  R  /\  R  e.  V )  ->  ( R  o.  (  _I  |` 
U. U. R ) )  =  R )
10 simp1 994 . . . . . . . . . 10  |-  ( ( N  =  0  /\ 
Rel  R  /\  R  e.  V )  ->  N  =  0 )
1110oveq2d 6286 . . . . . . . . 9  |-  ( ( N  =  0  /\ 
Rel  R  /\  R  e.  V )  ->  ( R ^r  N )  =  ( R ^r  0 ) )
12 simp3 996 . . . . . . . . . 10  |-  ( ( N  =  0  /\ 
Rel  R  /\  R  e.  V )  ->  R  e.  V )
13 relexp0 12943 . . . . . . . . . 10  |-  ( ( R  e.  V  /\  Rel  R )  ->  ( R ^r  0 )  =  (  _I  |`  U. U. R ) )
1412, 7, 13syl2anc 659 . . . . . . . . 9  |-  ( ( N  =  0  /\ 
Rel  R  /\  R  e.  V )  ->  ( R ^r  0 )  =  (  _I  |`  U. U. R ) )
1511, 14eqtrd 2495 . . . . . . . 8  |-  ( ( N  =  0  /\ 
Rel  R  /\  R  e.  V )  ->  ( R ^r  N )  =  (  _I  |`  U. U. R ) )
1615coeq2d 5154 . . . . . . 7  |-  ( ( N  =  0  /\ 
Rel  R  /\  R  e.  V )  ->  ( R  o.  ( R ^r  N ) )  =  ( R  o.  (  _I  |`  U. U. R ) ) )
1710oveq1d 6285 . . . . . . . . . 10  |-  ( ( N  =  0  /\ 
Rel  R  /\  R  e.  V )  ->  ( N  +  1 )  =  ( 0  +  1 ) )
18 0p1e1 10643 . . . . . . . . . 10  |-  ( 0  +  1 )  =  1
1917, 18syl6eq 2511 . . . . . . . . 9  |-  ( ( N  =  0  /\ 
Rel  R  /\  R  e.  V )  ->  ( N  +  1 )  =  1 )
2019oveq2d 6286 . . . . . . . 8  |-  ( ( N  =  0  /\ 
Rel  R  /\  R  e.  V )  ->  ( R ^r  ( N  +  1 ) )  =  ( R ^r  1 ) )
21 relexp1g 12946 . . . . . . . . 9  |-  ( R  e.  V  ->  ( R ^r  1 )  =  R )
2212, 21syl 16 . . . . . . . 8  |-  ( ( N  =  0  /\ 
Rel  R  /\  R  e.  V )  ->  ( R ^r  1 )  =  R )
2320, 22eqtrd 2495 . . . . . . 7  |-  ( ( N  =  0  /\ 
Rel  R  /\  R  e.  V )  ->  ( R ^r  ( N  +  1 ) )  =  R )
249, 16, 233eqtr4rd 2506 . . . . . 6  |-  ( ( N  =  0  /\ 
Rel  R  /\  R  e.  V )  ->  ( R ^r  ( N  +  1 ) )  =  ( R  o.  ( R ^r  N ) ) )
25243expib 1197 . . . . 5  |-  ( N  =  0  ->  (
( Rel  R  /\  R  e.  V )  ->  ( R ^r 
( N  +  1 ) )  =  ( R  o.  ( R ^r  N ) ) ) )
266, 25jaoi 377 . . . 4  |-  ( ( N  e.  NN  \/  N  =  0 )  ->  ( ( Rel 
R  /\  R  e.  V )  ->  ( R ^r  ( N  +  1 ) )  =  ( R  o.  ( R ^r  N ) ) ) )
271, 26sylbi 195 . . 3  |-  ( N  e.  NN0  ->  ( ( Rel  R  /\  R  e.  V )  ->  ( R ^r  ( N  +  1 ) )  =  ( R  o.  ( R ^r  N ) ) ) )
28273impib 1192 . 2  |-  ( ( N  e.  NN0  /\  Rel  R  /\  R  e.  V )  ->  ( R ^r  ( N  +  1 ) )  =  ( R  o.  ( R ^r  N ) ) )
29283com13 1199 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 366    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823   U.cuni 4235    _I cid 4779    |` cres 4990    o. ccom 4992   Rel wrel 4993  (class class class)co 6270   0cc0 9481   1c1 9482    + caddc 9484   NNcn 10531   NN0cn0 10791   ^r crelexp 12940
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-2nd 6774  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-n0 10792  df-z 10861  df-uz 11083  df-seq 12093  df-relexp 12941
This theorem is referenced by:  relexpsucld  12952  relexpindlem  12981
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