Theorem frecuzrdgrrn 10343

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 5487	. 2 ⊢ (𝑅‘𝐷) = (frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘𝐷)
3		frec2uz.1	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ ℤ)
4		uzid 9480	. . . . . 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 4634	. . . . 5 ⊢ (⟨𝐶, 𝐴⟩ ∈ ((ℤ≥‘𝐶) × 𝑆) ↔ (𝐶 ∈ (ℤ≥‘𝐶) ∧ 𝐴 ∈ 𝑆))
8	5, 6, 7	sylanbrc 414	. . . 4 ⊢ (𝜑 → ⟨𝐶, 𝐴⟩ ∈ ((ℤ≥‘𝐶) × 𝑆))
9	8	adantr 274	. . 3 ⊢ ((𝜑 ∧ 𝐷 ∈ ω) → ⟨𝐶, 𝐴⟩ ∈ ((ℤ≥‘𝐶) × 𝑆))
10		1st2nd2 6143	. . . . . . 7 ⊢ (𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆) → 𝑧 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)
11		fveq2 5486	. . . . . . . 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 5845	. . . . . . . 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 2217	. . . . . . 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 275	. . . . 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 6133	. . . . . . 7 ⊢ (𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆) → (1st ‘𝑧) ∈ (ℤ≥‘𝐶))
17	16	adantl 275	. . . . . 6 ⊢ (((𝜑 ∧ 𝐷 ∈ ω) ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → (1st ‘𝑧) ∈ (ℤ≥‘𝐶))
18		xp2nd 6134	. . . . . . 7 ⊢ (𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆) → (2nd ‘𝑧) ∈ 𝑆)
19	18	adantl 275	. . . . . 6 ⊢ (((𝜑 ∧ 𝐷 ∈ ω) ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → (2nd ‘𝑧) ∈ 𝑆)
20		peano2uz 9521	. . . . . . . 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 2548	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑥 ∈ (ℤ≥‘𝐶)∀𝑦 ∈ 𝑆 (𝑥𝐹𝑦) ∈ 𝑆)
24	23	ad2antrr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐷 ∈ ω) ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → ∀𝑥 ∈ (ℤ≥‘𝐶)∀𝑦 ∈ 𝑆 (𝑥𝐹𝑦) ∈ 𝑆)
25		oveq1 5849	. . . . . . . . . . 11 ⊢ (𝑥 = (1st ‘𝑧) → (𝑥𝐹𝑦) = ((1st ‘𝑧)𝐹𝑦))
26	25	eleq1d 2235	. . . . . . . . . 10 ⊢ (𝑥 = (1st ‘𝑧) → ((𝑥𝐹𝑦) ∈ 𝑆 ↔ ((1st ‘𝑧)𝐹𝑦) ∈ 𝑆))
27		oveq2 5850	. . . . . . . . . . 11 ⊢ (𝑦 = (2nd ‘𝑧) → ((1st ‘𝑧)𝐹𝑦) = ((1st ‘𝑧)𝐹(2nd ‘𝑧)))
28	27	eleq1d 2235	. . . . . . . . . 10 ⊢ (𝑦 = (2nd ‘𝑧) → (((1st ‘𝑧)𝐹𝑦) ∈ 𝑆 ↔ ((1st ‘𝑧)𝐹(2nd ‘𝑧)) ∈ 𝑆))
29	26, 28	rspc2v 2843	. . . . . . . . 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 409	. . . . . . . 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 4634	. . . . . . 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 414	. . . . . 6 ⊢ (((𝜑 ∧ 𝐷 ∈ ω) ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → ⟨((1st ‘𝑧) + 1), ((1st ‘𝑧)𝐹(2nd ‘𝑧))⟩ ∈ ((ℤ≥‘𝐶) × 𝑆))
34		oveq1 5849	. . . . . . . 8 ⊢ (𝑥 = (1st ‘𝑧) → (𝑥 + 1) = ((1st ‘𝑧) + 1))
35	34, 25	opeq12d 3766	. . . . . . 7 ⊢ (𝑥 = (1st ‘𝑧) → ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩ = ⟨((1st ‘𝑧) + 1), ((1st ‘𝑧)𝐹𝑦)⟩)
36	27	opeq2d 3765	. . . . . . 7 ⊢ (𝑦 = (2nd ‘𝑧) → ⟨((1st ‘𝑧) + 1), ((1st ‘𝑧)𝐹𝑦)⟩ = ⟨((1st ‘𝑧) + 1), ((1st ‘𝑧)𝐹(2nd ‘𝑧))⟩)
37		eqid 2165	. . . . . . 7 ⊢ (𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩) = (𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)
38	35, 36, 37	ovmpog 5976	. . . . . 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 1228	. . . . 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 2198	. . . 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 2243	. . 3 ⊢ (((𝜑 ∧ 𝐷 ∈ ω) ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → ((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘𝑧) ∈ ((ℤ≥‘𝐶) × 𝑆))
42		simpr 109	. . 3 ⊢ ((𝜑 ∧ 𝐷 ∈ ω) → 𝐷 ∈ ω)
43	9, 41, 42	freccl 6371	. 2 ⊢ ((𝜑 ∧ 𝐷 ∈ ω) → (frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘𝐷) ∈ ((ℤ≥‘𝐶) × 𝑆))
44	2, 43	eqeltrid 2253	1 ⊢ ((𝜑 ∧ 𝐷 ∈ ω) → (𝑅‘𝐷) ∈ ((ℤ≥‘𝐶) × 𝑆))

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1343   ∈ wcel 2136  ∀wral 2444  ⟨cop 3579   ↦ cmpt 4043  ωcom 4567   × cxp 4602  ‘cfv 5188  (class class class)co 5842   ∈ cmpo 5844  1^st c1st 6106  2^nd c2nd 6107  freccfrec 6358  1c1 7754   + caddc 7756  ℤcz 9191  ℤ_≥cuz 9466

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-addcom 7853  ax-addass 7855  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-0id 7861  ax-rnegex 7862  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-ltadd 7869

This theorem depends on definitions:  df-bi 116  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-iord 4344  df-on 4346  df-ilim 4347  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-frec 6359  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-inn 8858  df-n0 9115  df-z 9192  df-uz 9467

This theorem is referenced by:  frec2uzrdg  10344  frecuzrdgtcl  10347  frecuzrdgsuc  10349