Theorem frecuzrdgrclt 10486

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. Similar to frecuzrdgrcl 10481 except that 𝑆 and 𝑇 need not be the same. (Contributed by Jim Kingdon, 22-Apr-2022.)

Hypotheses

Ref	Expression
frecuzrdgrclt.c	⊢ (𝜑 → 𝐶 ∈ ℤ)
frecuzrdgrclt.a	⊢ (𝜑 → 𝐴 ∈ 𝑆)
frecuzrdgrclt.t	⊢ (𝜑 → 𝑆 ⊆ 𝑇)
frecuzrdgrclt.f	⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝐶) ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)
frecuzrdgrclt.r	⊢ 𝑅 = frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)

Assertion

Ref	Expression
frecuzrdgrclt	⊢ (𝜑 → 𝑅:ω⟶((ℤ≥‘𝐶) × 𝑆))

Distinct variable groups:   𝑥,𝐶,𝑦   𝑥,𝐹,𝑦   𝑥,𝑆,𝑦   𝑥,𝑇,𝑦   𝜑,𝑥,𝑦

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

Proof of Theorem frecuzrdgrclt

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		1st2nd2 6228	. . . . . . 7 ⊢ (𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆) → 𝑧 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)
2	1	adantl 277	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → 𝑧 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)
3	2	fveq2d 5558	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → ((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘𝑧) = ((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘⟨(1st ‘𝑧), (2nd ‘𝑧)⟩))
4		df-ov 5921	. . . . . . 7 ⊢ ((1st ‘𝑧)(𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd ‘𝑧)) = ((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)
5		xp1st 6218	. . . . . . . . 9 ⊢ (𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆) → (1st ‘𝑧) ∈ (ℤ≥‘𝐶))
6	5	adantl 277	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → (1st ‘𝑧) ∈ (ℤ≥‘𝐶))
7		frecuzrdgrclt.t	. . . . . . . . . 10 ⊢ (𝜑 → 𝑆 ⊆ 𝑇)
8	7	sseld 3178	. . . . . . . . 9 ⊢ (𝜑 → ((2nd ‘𝑧) ∈ 𝑆 → (2nd ‘𝑧) ∈ 𝑇))
9		xp2nd 6219	. . . . . . . . 9 ⊢ (𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆) → (2nd ‘𝑧) ∈ 𝑆)
10	8, 9	impel 280	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → (2nd ‘𝑧) ∈ 𝑇)
11		peano2uz 9648	. . . . . . . . . 10 ⊢ ((1st ‘𝑧) ∈ (ℤ≥‘𝐶) → ((1st ‘𝑧) + 1) ∈ (ℤ≥‘𝐶))
12	6, 11	syl 14	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → ((1st ‘𝑧) + 1) ∈ (ℤ≥‘𝐶))
13		frecuzrdgrclt.f	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝐶) ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)
14	13	ralrimivva 2576	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑥 ∈ (ℤ≥‘𝐶)∀𝑦 ∈ 𝑆 (𝑥𝐹𝑦) ∈ 𝑆)
15	14	adantr 276	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → ∀𝑥 ∈ (ℤ≥‘𝐶)∀𝑦 ∈ 𝑆 (𝑥𝐹𝑦) ∈ 𝑆)
16	9	adantl 277	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → (2nd ‘𝑧) ∈ 𝑆)
17		oveq1 5925	. . . . . . . . . . . . 13 ⊢ (𝑥 = (1st ‘𝑧) → (𝑥𝐹𝑦) = ((1st ‘𝑧)𝐹𝑦))
18	17	eleq1d 2262	. . . . . . . . . . . 12 ⊢ (𝑥 = (1st ‘𝑧) → ((𝑥𝐹𝑦) ∈ 𝑆 ↔ ((1st ‘𝑧)𝐹𝑦) ∈ 𝑆))
19		oveq2 5926	. . . . . . . . . . . . 13 ⊢ (𝑦 = (2nd ‘𝑧) → ((1st ‘𝑧)𝐹𝑦) = ((1st ‘𝑧)𝐹(2nd ‘𝑧)))
20	19	eleq1d 2262	. . . . . . . . . . . 12 ⊢ (𝑦 = (2nd ‘𝑧) → (((1st ‘𝑧)𝐹𝑦) ∈ 𝑆 ↔ ((1st ‘𝑧)𝐹(2nd ‘𝑧)) ∈ 𝑆))
21	18, 20	rspc2v 2877	. . . . . . . . . . 11 ⊢ (((1st ‘𝑧) ∈ (ℤ≥‘𝐶) ∧ (2nd ‘𝑧) ∈ 𝑆) → (∀𝑥 ∈ (ℤ≥‘𝐶)∀𝑦 ∈ 𝑆 (𝑥𝐹𝑦) ∈ 𝑆 → ((1st ‘𝑧)𝐹(2nd ‘𝑧)) ∈ 𝑆))
22	6, 16, 21	syl2anc 411	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → (∀𝑥 ∈ (ℤ≥‘𝐶)∀𝑦 ∈ 𝑆 (𝑥𝐹𝑦) ∈ 𝑆 → ((1st ‘𝑧)𝐹(2nd ‘𝑧)) ∈ 𝑆))
23	15, 22	mpd 13	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → ((1st ‘𝑧)𝐹(2nd ‘𝑧)) ∈ 𝑆)
24		opelxp 4689	. . . . . . . . 9 ⊢ (⟨((1st ‘𝑧) + 1), ((1st ‘𝑧)𝐹(2nd ‘𝑧))⟩ ∈ ((ℤ≥‘𝐶) × 𝑆) ↔ (((1st ‘𝑧) + 1) ∈ (ℤ≥‘𝐶) ∧ ((1st ‘𝑧)𝐹(2nd ‘𝑧)) ∈ 𝑆))
25	12, 23, 24	sylanbrc 417	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → ⟨((1st ‘𝑧) + 1), ((1st ‘𝑧)𝐹(2nd ‘𝑧))⟩ ∈ ((ℤ≥‘𝐶) × 𝑆))
26		oveq1 5925	. . . . . . . . . 10 ⊢ (𝑥 = (1st ‘𝑧) → (𝑥 + 1) = ((1st ‘𝑧) + 1))
27	26, 17	opeq12d 3812	. . . . . . . . 9 ⊢ (𝑥 = (1st ‘𝑧) → ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩ = ⟨((1st ‘𝑧) + 1), ((1st ‘𝑧)𝐹𝑦)⟩)
28	19	opeq2d 3811	. . . . . . . . 9 ⊢ (𝑦 = (2nd ‘𝑧) → ⟨((1st ‘𝑧) + 1), ((1st ‘𝑧)𝐹𝑦)⟩ = ⟨((1st ‘𝑧) + 1), ((1st ‘𝑧)𝐹(2nd ‘𝑧))⟩)
29		eqid 2193	. . . . . . . . 9 ⊢ (𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩) = (𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)
30	27, 28, 29	ovmpog 6053	. . . . . . . 8 ⊢ (((1st ‘𝑧) ∈ (ℤ≥‘𝐶) ∧ (2nd ‘𝑧) ∈ 𝑇 ∧ ⟨((1st ‘𝑧) + 1), ((1st ‘𝑧)𝐹(2nd ‘𝑧))⟩ ∈ ((ℤ≥‘𝐶) × 𝑆)) → ((1st ‘𝑧)(𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd ‘𝑧)) = ⟨((1st ‘𝑧) + 1), ((1st ‘𝑧)𝐹(2nd ‘𝑧))⟩)
31	6, 10, 25, 30	syl3anc 1249	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → ((1st ‘𝑧)(𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd ‘𝑧)) = ⟨((1st ‘𝑧) + 1), ((1st ‘𝑧)𝐹(2nd ‘𝑧))⟩)
32	4, 31	eqtr3id 2240	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → ((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘⟨(1st ‘𝑧), (2nd ‘𝑧)⟩) = ⟨((1st ‘𝑧) + 1), ((1st ‘𝑧)𝐹(2nd ‘𝑧))⟩)
33	32, 25	eqeltrd 2270	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → ((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘⟨(1st ‘𝑧), (2nd ‘𝑧)⟩) ∈ ((ℤ≥‘𝐶) × 𝑆))
34	3, 33	eqeltrd 2270	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → ((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘𝑧) ∈ ((ℤ≥‘𝐶) × 𝑆))
35	34	ralrimiva 2567	. . 3 ⊢ (𝜑 → ∀𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘𝑧) ∈ ((ℤ≥‘𝐶) × 𝑆))
36		frecuzrdgrclt.c	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℤ)
37		uzid 9606	. . . . 5 ⊢ (𝐶 ∈ ℤ → 𝐶 ∈ (ℤ≥‘𝐶))
38	36, 37	syl 14	. . . 4 ⊢ (𝜑 → 𝐶 ∈ (ℤ≥‘𝐶))
39		frecuzrdgrclt.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑆)
40		opelxp 4689	. . . 4 ⊢ (⟨𝐶, 𝐴⟩ ∈ ((ℤ≥‘𝐶) × 𝑆) ↔ (𝐶 ∈ (ℤ≥‘𝐶) ∧ 𝐴 ∈ 𝑆))
41	38, 39, 40	sylanbrc 417	. . 3 ⊢ (𝜑 → ⟨𝐶, 𝐴⟩ ∈ ((ℤ≥‘𝐶) × 𝑆))
42		frecfcl 6458	. . 3 ⊢ ((∀𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘𝑧) ∈ ((ℤ≥‘𝐶) × 𝑆) ∧ ⟨𝐶, 𝐴⟩ ∈ ((ℤ≥‘𝐶) × 𝑆)) → frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩):ω⟶((ℤ≥‘𝐶) × 𝑆))
43	35, 41, 42	syl2anc 411	. 2 ⊢ (𝜑 → frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩):ω⟶((ℤ≥‘𝐶) × 𝑆))
44		frecuzrdgrclt.r	. . 3 ⊢ 𝑅 = frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)
45	44	feq1i 5396	. 2 ⊢ (𝑅:ω⟶((ℤ≥‘𝐶) × 𝑆) ↔ frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩):ω⟶((ℤ≥‘𝐶) × 𝑆))
46	43, 45	sylibr 134	1 ⊢ (𝜑 → 𝑅:ω⟶((ℤ≥‘𝐶) × 𝑆))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2164  ∀wral 2472   ⊆ wss 3153  ⟨cop 3621  ωcom 4622   × cxp 4657  ⟶wf 5250  ‘cfv 5254  (class class class)co 5918   ∈ cmpo 5920  1^st c1st 6191  2^nd c2nd 6192  freccfrec 6443  1c1 7873   + caddc 7875  ℤcz 9317  ℤ_≥cuz 9592

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-addcom 7972  ax-addass 7974  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-0id 7980  ax-rnegex 7981  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-ltadd 7988

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-iord 4397  df-on 4399  df-ilim 4400  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-frec 6444  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-inn 8983  df-n0 9241  df-z 9318  df-uz 9593

This theorem is referenced by:  frecuzrdgg  10487  frecuzrdgdomlem  10488  frecuzrdgfunlem  10490  frecuzrdgtclt  10492  frecuzrdg0t  10493  frecuzrdgsuctlem  10494  seq3val  10531  seqvalcd  10532