Theorem frecuzrdgrrn 10770

Description: The function 𝑅 (used in the definition of the recursive definition generator on upper integers) yields ordered pairs of integers and elements of 𝑆. (Contributed by Jim Kingdon, 28-Mar-2022.)

Hypotheses

Ref	Expression
frec2uz.1	⊢ (𝜑 → 𝐶 ∈ ℤ)
frec2uz.2	⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)
frecuzrdgrrn.a	⊢ (𝜑 → 𝐴 ∈ 𝑆)
frecuzrdgrrn.f	⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝐶) ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)
frecuzrdgrrn.2	⊢ 𝑅 = frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)

Assertion

Ref	Expression
frecuzrdgrrn	⊢ ((𝜑 ∧ 𝐷 ∈ ω) → (𝑅‘𝐷) ∈ ((ℤ≥‘𝐶) × 𝑆))

Distinct variable groups:   𝑦,𝐴   𝑥,𝐶,𝑦   𝑦,𝐺   𝑥,𝐹,𝑦   𝑥,𝑆,𝑦   𝜑,𝑥,𝑦

Allowed substitution hints:   𝐴(𝑥)   𝐷(𝑥,𝑦)   𝑅(𝑥,𝑦)   𝐺(𝑥)

Proof of Theorem frecuzrdgrrn

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		frecuzrdgrrn.2	. . 3 ⊢ 𝑅 = frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)
2	1	fveq1i 5671	. 2 ⊢ (𝑅‘𝐷) = (frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘𝐷)
3		frec2uz.1	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ ℤ)
4		uzid 9868	. . . . . 6 ⊢ (𝐶 ∈ ℤ → 𝐶 ∈ (ℤ≥‘𝐶))
5	3, 4	syl 14	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ (ℤ≥‘𝐶))
6		frecuzrdgrrn.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑆)
7		opelxp 4779	. . . . 5 ⊢ (⟨𝐶, 𝐴⟩ ∈ ((ℤ≥‘𝐶) × 𝑆) ↔ (𝐶 ∈ (ℤ≥‘𝐶) ∧ 𝐴 ∈ 𝑆))
8	5, 6, 7	sylanbrc 417	. . . 4 ⊢ (𝜑 → ⟨𝐶, 𝐴⟩ ∈ ((ℤ≥‘𝐶) × 𝑆))
9	8	adantr 276	. . 3 ⊢ ((𝜑 ∧ 𝐷 ∈ ω) → ⟨𝐶, 𝐴⟩ ∈ ((ℤ≥‘𝐶) × 𝑆))
10		1st2nd2 6369	. . . . . . 7 ⊢ (𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆) → 𝑧 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)
11		fveq2 5670	. . . . . . . 8 ⊢ (𝑧 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩ → ((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘𝑧) = ((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘⟨(1st ‘𝑧), (2nd ‘𝑧)⟩))
12		df-ov 6053	. . . . . . . 8 ⊢ ((1st ‘𝑧)(𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd ‘𝑧)) = ((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)
13	11, 12	eqtr4di 2283	. . . . . . 7 ⊢ (𝑧 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩ → ((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘𝑧) = ((1st ‘𝑧)(𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd ‘𝑧)))
14	10, 13	syl 14	. . . . . 6 ⊢ (𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆) → ((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘𝑧) = ((1st ‘𝑧)(𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd ‘𝑧)))
15	14	adantl 277	. . . . 5 ⊢ (((𝜑 ∧ 𝐷 ∈ ω) ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → ((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘𝑧) = ((1st ‘𝑧)(𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd ‘𝑧)))
16		xp1st 6359	. . . . . . 7 ⊢ (𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆) → (1st ‘𝑧) ∈ (ℤ≥‘𝐶))
17	16	adantl 277	. . . . . 6 ⊢ (((𝜑 ∧ 𝐷 ∈ ω) ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → (1st ‘𝑧) ∈ (ℤ≥‘𝐶))
18		xp2nd 6360	. . . . . . 7 ⊢ (𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆) → (2nd ‘𝑧) ∈ 𝑆)
19	18	adantl 277	. . . . . 6 ⊢ (((𝜑 ∧ 𝐷 ∈ ω) ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → (2nd ‘𝑧) ∈ 𝑆)
20		peano2uz 9915	. . . . . . . 8 ⊢ ((1st ‘𝑧) ∈ (ℤ≥‘𝐶) → ((1st ‘𝑧) + 1) ∈ (ℤ≥‘𝐶))
21	17, 20	syl 14	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐷 ∈ ω) ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → ((1st ‘𝑧) + 1) ∈ (ℤ≥‘𝐶))
22		frecuzrdgrrn.f	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝐶) ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)
23	22	ralrimivva 2624	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑥 ∈ (ℤ≥‘𝐶)∀𝑦 ∈ 𝑆 (𝑥𝐹𝑦) ∈ 𝑆)
24	23	ad2antrr 488	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐷 ∈ ω) ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → ∀𝑥 ∈ (ℤ≥‘𝐶)∀𝑦 ∈ 𝑆 (𝑥𝐹𝑦) ∈ 𝑆)
25		oveq1 6057	. . . . . . . . . . 11 ⊢ (𝑥 = (1st ‘𝑧) → (𝑥𝐹𝑦) = ((1st ‘𝑧)𝐹𝑦))
26	25	eleq1d 2301	. . . . . . . . . 10 ⊢ (𝑥 = (1st ‘𝑧) → ((𝑥𝐹𝑦) ∈ 𝑆 ↔ ((1st ‘𝑧)𝐹𝑦) ∈ 𝑆))
27		oveq2 6058	. . . . . . . . . . 11 ⊢ (𝑦 = (2nd ‘𝑧) → ((1st ‘𝑧)𝐹𝑦) = ((1st ‘𝑧)𝐹(2nd ‘𝑧)))
28	27	eleq1d 2301	. . . . . . . . . 10 ⊢ (𝑦 = (2nd ‘𝑧) → (((1st ‘𝑧)𝐹𝑦) ∈ 𝑆 ↔ ((1st ‘𝑧)𝐹(2nd ‘𝑧)) ∈ 𝑆))
29	26, 28	rspc2v 2934	. . . . . . . . 9 ⊢ (((1st ‘𝑧) ∈ (ℤ≥‘𝐶) ∧ (2nd ‘𝑧) ∈ 𝑆) → (∀𝑥 ∈ (ℤ≥‘𝐶)∀𝑦 ∈ 𝑆 (𝑥𝐹𝑦) ∈ 𝑆 → ((1st ‘𝑧)𝐹(2nd ‘𝑧)) ∈ 𝑆))
30	17, 19, 29	syl2anc 411	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐷 ∈ ω) ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → (∀𝑥 ∈ (ℤ≥‘𝐶)∀𝑦 ∈ 𝑆 (𝑥𝐹𝑦) ∈ 𝑆 → ((1st ‘𝑧)𝐹(2nd ‘𝑧)) ∈ 𝑆))
31	24, 30	mpd 13	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐷 ∈ ω) ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → ((1st ‘𝑧)𝐹(2nd ‘𝑧)) ∈ 𝑆)
32		opelxp 4779	. . . . . . 7 ⊢ (⟨((1st ‘𝑧) + 1), ((1st ‘𝑧)𝐹(2nd ‘𝑧))⟩ ∈ ((ℤ≥‘𝐶) × 𝑆) ↔ (((1st ‘𝑧) + 1) ∈ (ℤ≥‘𝐶) ∧ ((1st ‘𝑧)𝐹(2nd ‘𝑧)) ∈ 𝑆))
33	21, 31, 32	sylanbrc 417	. . . . . 6 ⊢ (((𝜑 ∧ 𝐷 ∈ ω) ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → ⟨((1st ‘𝑧) + 1), ((1st ‘𝑧)𝐹(2nd ‘𝑧))⟩ ∈ ((ℤ≥‘𝐶) × 𝑆))
34		oveq1 6057	. . . . . . . 8 ⊢ (𝑥 = (1st ‘𝑧) → (𝑥 + 1) = ((1st ‘𝑧) + 1))
35	34, 25	opeq12d 3891	. . . . . . 7 ⊢ (𝑥 = (1st ‘𝑧) → ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩ = ⟨((1st ‘𝑧) + 1), ((1st ‘𝑧)𝐹𝑦)⟩)
36	27	opeq2d 3890	. . . . . . 7 ⊢ (𝑦 = (2nd ‘𝑧) → ⟨((1st ‘𝑧) + 1), ((1st ‘𝑧)𝐹𝑦)⟩ = ⟨((1st ‘𝑧) + 1), ((1st ‘𝑧)𝐹(2nd ‘𝑧))⟩)
37		eqid 2232	. . . . . . 7 ⊢ (𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩) = (𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)
38	35, 36, 37	ovmpog 6188	. . . . . 6 ⊢ (((1st ‘𝑧) ∈ (ℤ≥‘𝐶) ∧ (2nd ‘𝑧) ∈ 𝑆 ∧ ⟨((1st ‘𝑧) + 1), ((1st ‘𝑧)𝐹(2nd ‘𝑧))⟩ ∈ ((ℤ≥‘𝐶) × 𝑆)) → ((1st ‘𝑧)(𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd ‘𝑧)) = ⟨((1st ‘𝑧) + 1), ((1st ‘𝑧)𝐹(2nd ‘𝑧))⟩)
39	17, 19, 33, 38	syl3anc 1274	. . . . 5 ⊢ (((𝜑 ∧ 𝐷 ∈ ω) ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → ((1st ‘𝑧)(𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd ‘𝑧)) = ⟨((1st ‘𝑧) + 1), ((1st ‘𝑧)𝐹(2nd ‘𝑧))⟩)
40	15, 39	eqtrd 2265	. . . 4 ⊢ (((𝜑 ∧ 𝐷 ∈ ω) ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → ((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘𝑧) = ⟨((1st ‘𝑧) + 1), ((1st ‘𝑧)𝐹(2nd ‘𝑧))⟩)
41	40, 33	eqeltrd 2309	. . 3 ⊢ (((𝜑 ∧ 𝐷 ∈ ω) ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → ((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘𝑧) ∈ ((ℤ≥‘𝐶) × 𝑆))
42		simpr 110	. . 3 ⊢ ((𝜑 ∧ 𝐷 ∈ ω) → 𝐷 ∈ ω)
43	9, 41, 42	freccl 6634	. 2 ⊢ ((𝜑 ∧ 𝐷 ∈ ω) → (frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘𝐷) ∈ ((ℤ≥‘𝐶) × 𝑆))
44	2, 43	eqeltrid 2319	1 ⊢ ((𝜑 ∧ 𝐷 ∈ ω) → (𝑅‘𝐷) ∈ ((ℤ≥‘𝐶) × 𝑆))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1398   ∈ wcel 2203  ∀wral 2520  ⟨cop 3692   ↦ cmpt 4171  ωcom 4712   × cxp 4747  ‘cfv 5352  (class class class)co 6050   ∈ cmpo 6052  1^st c1st 6332  2^nd c2nd 6333  freccfrec 6621  1c1 8128   + caddc 8130  ℤcz 9577  ℤ_≥cuz 9853

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-addass 8229  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-0id 8235  ax-rnegex 8236  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-ltadd 8243

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-iord 4487  df-on 4489  df-ilim 4490  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-frec 6622  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-inn 9238  df-n0 9497  df-z 9578  df-uz 9854

This theorem is referenced by:  frec2uzrdg  10771  frecuzrdgtcl  10774  frecuzrdgsuc  10776