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.

Step	Hyp	Ref	Expression
1		opeq2 4579	. . . . 5 |- (I = J -> <.0c, I>. = <.0c, J>.)
2	1	sneqd 3746	. . . 4 |- (I = J -> {<.0c, I>.} = {<.0c, J>.})
3		clos1eq1 5874	. . . 4 |- ({<.0c, I>.} = {<.0c, J>.} -> Clos1 ({<.0c, I>.}, PProd ((x e. _V |-> (x +o 1c)), F)) = Clos1 ({<.0c, J>.}, PProd ((x e. _V |-> (x +o 1c)), F)))
4	2, 3	syl 15	. . 3 |- (I = J -> Clos1 ({<.0c, I>.}, PProd ((x e. _V |-> (x +o 1c)), F)) = Clos1 ({<.0c, J>.}, PProd ((x e. _V |-> (x +o 1c)), F)))
5		pprodeq2 5835	. . . 4 |- (F = G -> PProd ((x e. _V |-> (x +o 1c)), F) = PProd ((x e. _V |-> (x +o 1c)), G))
6		clos1eq2 5875	. . . 4 |- ( PProd ((x e. _V |-> (x +o 1c)), F) = PProd ((x e. _V |-> (x +o 1c)), G) -> Clos1 ({<.0c, J>.}, PProd ((x e. _V |-> (x +o 1c)), F)) = Clos1 ({<.0c, J>.}, PProd ((x e. _V |-> (x +o 1c)), G)))
7	5, 6	syl 15	. . 3 |- (F = G -> Clos1 ({<.0c, J>.}, PProd ((x e. _V |-> (x +o 1c)), F)) = Clos1 ({<.0c, J>.}, PProd ((x e. _V |-> (x +o 1c)), G)))
8	4, 7	sylan9eqr 2407	. 2 |- ((F = G /\ I = J) -> Clos1 ({<.0c, I>.}, PProd ((x e. _V |-> (x +o 1c)), F)) = Clos1 ({<.0c, J>.}, PProd ((x e. _V |-> (x +o 1c)), G)))
9		df-frec 6310	. 2 |- FRec (F, I) = Clos1 ({<.0c, I>.}, PProd ((x e. _V |-> (x +o 1c)), F))
10		df-frec 6310	. 2 |- FRec (G, J) = Clos1 ({<.0c, J>.}, PProd ((x e. _V |-> (x +o 1c)), G))
11	8, 9, 10	3eqtr4g 2410	1 |- ((F = G /\ I = J) -> FRec (F, I) = FRec (G, J))

Syntax hints:   -> wi 4   /\ wa 358   = wceq 1642  _Vcvv 2859  {csn 3737  1_cc1c 4134  0_cc0c 4374   +o 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