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Theorem freceq12 6311
 Description: Equality theorem for finite recursive function generator. (Contributed by Scott Fenton, 31-Jul-2019.)
Assertion
Ref Expression
freceq12 ((F = G I = J) → FRec (F, I) = FRec (G, J))

Proof of Theorem freceq12
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 opeq2 4579 . . . . 5 (I = J0c, I = 0c, J)
21sneqd 3746 . . . 4 (I = J → {0c, I} = {0c, J})
3 clos1eq1 5874 . . . 4 ({0c, I} = {0c, J} → Clos1 ({0c, I}, PProd ((x V (x +c 1c)), F)) = Clos1 ({0c, J}, PProd ((x V (x +c 1c)), F)))
42, 3syl 15 . . 3 (I = J Clos1 ({0c, I}, PProd ((x V (x +c 1c)), F)) = Clos1 ({0c, J}, PProd ((x V (x +c 1c)), F)))
5 pprodeq2 5835 . . . 4 (F = GPProd ((x V (x +c 1c)), F) = PProd ((x V (x +c 1c)), G))
6 clos1eq2 5875 . . . 4 ( PProd ((x V (x +c 1c)), F) = PProd ((x V (x +c 1c)), G) → Clos1 ({0c, J}, PProd ((x V (x +c 1c)), F)) = Clos1 ({0c, J}, PProd ((x V (x +c 1c)), G)))
75, 6syl 15 . . 3 (F = G Clos1 ({0c, J}, PProd ((x V (x +c 1c)), F)) = Clos1 ({0c, J}, PProd ((x V (x +c 1c)), G)))
84, 7sylan9eqr 2407 . 2 ((F = G I = J) → Clos1 ({0c, I}, PProd ((x V (x +c 1c)), F)) = Clos1 ({0c, J}, PProd ((x V (x +c 1c)), G)))
9 df-frec 6310 . 2 FRec (F, I) = Clos1 ({0c, I}, PProd ((x V (x +c 1c)), F))
10 df-frec 6310 . 2 FRec (G, J) = Clos1 ({0c, J}, PProd ((x V (x +c 1c)), G))
118, 9, 103eqtr4g 2410 1 ((F = G I = J) → FRec (F, I) = FRec (G, J))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642  Vcvv 2859  {csn 3737  1cc1c 4134  0cc0c 4374   +c cplc 4375  ⟨cop 4561   ↦ cmpt 5651   PProd cpprod 5737   Clos1 cclos1 5872   FRec cfrec 6309 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-addc 4378  df-nnc 4379  df-phi 4565  df-op 4566  df-opab 4623  df-br 4640  df-co 4726  df-ima 4727  df-txp 5736  df-pprod 5738  df-clos1 5873  df-frec 6310 This theorem is referenced by:  frecxp  6314  frecxpg  6315
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