Theorem freceq12 6312

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 4580	. . . . 5 ⊢ (I = J → ⟨0c, I⟩ = ⟨0c, J⟩)
2	1	sneqd 3747	. . . 4 ⊢ (I = J → {⟨0c, I⟩} = {⟨0c, J⟩})
3		clos1eq1 5875	. . . 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)))
4	2, 3	syl 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 5836	. . . 4 ⊢ (F = G → PProd ((x ∈ V ↦ (x +c 1c)), F) = PProd ((x ∈ V ↦ (x +c 1c)), G))
6		clos1eq2 5876	. . . 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)))
7	5, 6	syl 15	. . 3 ⊢ (F = G → Clos1 ({⟨0c, J⟩}, PProd ((x ∈ V ↦ (x +c 1c)), F)) = Clos1 ({⟨0c, J⟩}, PProd ((x ∈ V ↦ (x +c 1c)), G)))
8	4, 7	sylan9eqr 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 6311	. 2 ⊢ FRec (F, I) = Clos1 ({⟨0c, I⟩}, PProd ((x ∈ V ↦ (x +c 1c)), F))
10		df-frec 6311	. 2 ⊢ FRec (G, J) = Clos1 ({⟨0c, J⟩}, PProd ((x ∈ V ↦ (x +c 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 2860  {csn 3738  1_cc1c 4135  0_cc0c 4375   +_c cplc 4376  ⟨cop 4562   ↦ cmpt 5652   PProd cpprod 5738   Clos1 cclos1 5873   FRec cfrec 6310

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 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

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 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-addc 4379  df-nnc 4380  df-phi 4566  df-op 4567  df-opab 4624  df-br 4641  df-co 4727  df-ima 4728  df-txp 5737  df-pprod 5739  df-clos1 5874  df-frec 6311

This theorem is referenced by:  frecxp  6315  frecxpg  6316