Theorem seqovcd 10882

Description: A closure law for the recursive sequence builder. This is a lemma for theorems such as seqf2 10883 and seq1cd 10884 and is unlikely to be needed once such theorems are proved. (Contributed by Jim Kingdon, 20-Jul-2023.)

Hypotheses

Ref	Expression
seqovcd.f	⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘(𝑀 + 1))) → (𝐹‘𝑥) ∈ 𝐷)
seqovcd.pl	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)) → (𝑥 + 𝑦) ∈ 𝐶)

Assertion

Ref	Expression
seqovcd	⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝑀) ∧ 𝑦 ∈ 𝐶)) → (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝐶 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦) ∈ 𝐶)

Distinct variable groups:   𝑥, + ,𝑦,𝑤,𝑧   𝑥,𝐶,𝑦,𝑤,𝑧   𝑥,𝐷,𝑦   𝑥,𝐹,𝑤,𝑧   𝑥,𝑀,𝑤,𝑧   𝜑,𝑥,𝑦

Allowed substitution hints:   𝜑(𝑧,𝑤)   𝐷(𝑧,𝑤)   𝐹(𝑦)   𝑀(𝑦)

Proof of Theorem seqovcd

Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simprl 535	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝑀) ∧ 𝑦 ∈ 𝐶)) → 𝑥 ∈ (ℤ≥‘𝑀))
2		simprr 537	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝑀) ∧ 𝑦 ∈ 𝐶)) → 𝑦 ∈ 𝐶)
3		seqovcd.pl	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)) → (𝑥 + 𝑦) ∈ 𝐶)
4	3	ralrimivva 2632	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐷 (𝑥 + 𝑦) ∈ 𝐶)
5		oveq1 6082	. . . . . . . 8 ⊢ (𝑥 = 𝑎 → (𝑥 + 𝑦) = (𝑎 + 𝑦))
6	5	eleq1d 2307	. . . . . . 7 ⊢ (𝑥 = 𝑎 → ((𝑥 + 𝑦) ∈ 𝐶 ↔ (𝑎 + 𝑦) ∈ 𝐶))
7		oveq2 6083	. . . . . . . 8 ⊢ (𝑦 = 𝑏 → (𝑎 + 𝑦) = (𝑎 + 𝑏))
8	7	eleq1d 2307	. . . . . . 7 ⊢ (𝑦 = 𝑏 → ((𝑎 + 𝑦) ∈ 𝐶 ↔ (𝑎 + 𝑏) ∈ 𝐶))
9	6, 8	cbvral2v 2799	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐷 (𝑥 + 𝑦) ∈ 𝐶 ↔ ∀𝑎 ∈ 𝐶 ∀𝑏 ∈ 𝐷 (𝑎 + 𝑏) ∈ 𝐶)
10	4, 9	sylib 122	. . . . 5 ⊢ (𝜑 → ∀𝑎 ∈ 𝐶 ∀𝑏 ∈ 𝐷 (𝑎 + 𝑏) ∈ 𝐶)
11	10	adantr 276	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝑀) ∧ 𝑦 ∈ 𝐶)) → ∀𝑎 ∈ 𝐶 ∀𝑏 ∈ 𝐷 (𝑎 + 𝑏) ∈ 𝐶)
12		fveq2 5690	. . . . . . 7 ⊢ (𝑎 = (𝑥 + 1) → (𝐹‘𝑎) = (𝐹‘(𝑥 + 1)))
13	12	eleq1d 2307	. . . . . 6 ⊢ (𝑎 = (𝑥 + 1) → ((𝐹‘𝑎) ∈ 𝐷 ↔ (𝐹‘(𝑥 + 1)) ∈ 𝐷))
14		seqovcd.f	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘(𝑀 + 1))) → (𝐹‘𝑥) ∈ 𝐷)
15	14	ralrimiva 2623	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ (ℤ≥‘(𝑀 + 1))(𝐹‘𝑥) ∈ 𝐷)
16		fveq2 5690	. . . . . . . . . 10 ⊢ (𝑥 = 𝑎 → (𝐹‘𝑥) = (𝐹‘𝑎))
17	16	eleq1d 2307	. . . . . . . . 9 ⊢ (𝑥 = 𝑎 → ((𝐹‘𝑥) ∈ 𝐷 ↔ (𝐹‘𝑎) ∈ 𝐷))
18	17	cbvralv 2786	. . . . . . . 8 ⊢ (∀𝑥 ∈ (ℤ≥‘(𝑀 + 1))(𝐹‘𝑥) ∈ 𝐷 ↔ ∀𝑎 ∈ (ℤ≥‘(𝑀 + 1))(𝐹‘𝑎) ∈ 𝐷)
19	15, 18	sylib 122	. . . . . . 7 ⊢ (𝜑 → ∀𝑎 ∈ (ℤ≥‘(𝑀 + 1))(𝐹‘𝑎) ∈ 𝐷)
20	19	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝑀) ∧ 𝑦 ∈ 𝐶)) → ∀𝑎 ∈ (ℤ≥‘(𝑀 + 1))(𝐹‘𝑎) ∈ 𝐷)
21		eluzp1p1 9927	. . . . . . 7 ⊢ (𝑥 ∈ (ℤ≥‘𝑀) → (𝑥 + 1) ∈ (ℤ≥‘(𝑀 + 1)))
22	1, 21	syl 14	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝑀) ∧ 𝑦 ∈ 𝐶)) → (𝑥 + 1) ∈ (ℤ≥‘(𝑀 + 1)))
23	13, 20, 22	rspcdva 2934	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝑀) ∧ 𝑦 ∈ 𝐶)) → (𝐹‘(𝑥 + 1)) ∈ 𝐷)
24		oveq12 6084	. . . . . . 7 ⊢ ((𝑎 = 𝑦 ∧ 𝑏 = (𝐹‘(𝑥 + 1))) → (𝑎 + 𝑏) = (𝑦 + (𝐹‘(𝑥 + 1))))
25	24	eleq1d 2307	. . . . . 6 ⊢ ((𝑎 = 𝑦 ∧ 𝑏 = (𝐹‘(𝑥 + 1))) → ((𝑎 + 𝑏) ∈ 𝐶 ↔ (𝑦 + (𝐹‘(𝑥 + 1))) ∈ 𝐶))
26	25	rspc2gv 2942	. . . . 5 ⊢ ((𝑦 ∈ 𝐶 ∧ (𝐹‘(𝑥 + 1)) ∈ 𝐷) → (∀𝑎 ∈ 𝐶 ∀𝑏 ∈ 𝐷 (𝑎 + 𝑏) ∈ 𝐶 → (𝑦 + (𝐹‘(𝑥 + 1))) ∈ 𝐶))
27	2, 23, 26	syl2anc 415	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝑀) ∧ 𝑦 ∈ 𝐶)) → (∀𝑎 ∈ 𝐶 ∀𝑏 ∈ 𝐷 (𝑎 + 𝑏) ∈ 𝐶 → (𝑦 + (𝐹‘(𝑥 + 1))) ∈ 𝐶))
28	11, 27	mpd 13	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝑀) ∧ 𝑦 ∈ 𝐶)) → (𝑦 + (𝐹‘(𝑥 + 1))) ∈ 𝐶)
29		fvoveq1 6098	. . . . 5 ⊢ (𝑧 = 𝑥 → (𝐹‘(𝑧 + 1)) = (𝐹‘(𝑥 + 1)))
30	29	oveq2d 6091	. . . 4 ⊢ (𝑧 = 𝑥 → (𝑤 + (𝐹‘(𝑧 + 1))) = (𝑤 + (𝐹‘(𝑥 + 1))))
31		oveq1 6082	. . . 4 ⊢ (𝑤 = 𝑦 → (𝑤 + (𝐹‘(𝑥 + 1))) = (𝑦 + (𝐹‘(𝑥 + 1))))
32		eqid 2238	. . . 4 ⊢ (𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝐶 ↦ (𝑤 + (𝐹‘(𝑧 + 1)))) = (𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝐶 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))
33	30, 31, 32	ovmpog 6213	. . 3 ⊢ ((𝑥 ∈ (ℤ≥‘𝑀) ∧ 𝑦 ∈ 𝐶 ∧ (𝑦 + (𝐹‘(𝑥 + 1))) ∈ 𝐶) → (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝐶 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦) = (𝑦 + (𝐹‘(𝑥 + 1))))
34	1, 2, 28, 33	syl3anc 1278	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝑀) ∧ 𝑦 ∈ 𝐶)) → (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝐶 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦) = (𝑦 + (𝐹‘(𝑥 + 1))))
35	34, 28	eqeltrd 2315	1 ⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝑀) ∧ 𝑦 ∈ 𝐶)) → (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝐶 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦) ∈ 𝐶)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1402   ∈ wcel 2209  ∀wral 2528  ‘cfv 5372  (class class class)co 6075   ∈ cmpo 6077  1c1 8170   + caddc 8172  ℤ_≥cuz 9900

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-distr 8273  ax-i2m1 8274  ax-0id 8277  ax-rnegex 8278  ax-cnre 8280  ax-pre-ltadd 8285

This theorem depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-inn 9284  df-n0 9543  df-z 9624  df-uz 9901

This theorem is referenced by:  seqf2  10883  seq1cd  10884  seqp1cd  10885