Theorem seqexw 13737

Description: Weak version of seqex 13723 that holds without ax-rep 5209. A sequence builder exists when its binary operation input exists and its starting index is an integer. (Contributed by Rohan Ridenour, 14-Aug-2023.)

Hypotheses

Ref	Expression
seqexw.1	⊢ + ∈ V
seqexw.2	⊢ 𝑀 ∈ ℤ

Assertion

Ref	Expression
seqexw	⊢ seq𝑀( + , 𝐹) ∈ V

Proof of Theorem seqexw

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		seqexw.2	. . . 4 ⊢ 𝑀 ∈ ℤ
2		seqfn 13733	. . . 4 ⊢ (𝑀 ∈ ℤ → seq𝑀( + , 𝐹) Fn (ℤ≥‘𝑀))
3	1, 2	ax-mp 5	. . 3 ⊢ seq𝑀( + , 𝐹) Fn (ℤ≥‘𝑀)
4		fnfun 6533	. . 3 ⊢ (seq𝑀( + , 𝐹) Fn (ℤ≥‘𝑀) → Fun seq𝑀( + , 𝐹))
5	3, 4	ax-mp 5	. 2 ⊢ Fun seq𝑀( + , 𝐹)
6	3	fndmi 6537	. . 3 ⊢ dom seq𝑀( + , 𝐹) = (ℤ≥‘𝑀)
7		fvex 6787	. . 3 ⊢ (ℤ≥‘𝑀) ∈ V
8	6, 7	eqeltri 2835	. 2 ⊢ dom seq𝑀( + , 𝐹) ∈ V
9		seqexw.1	. . . . 5 ⊢ + ∈ V
10	9	rnex 7759	. . . 4 ⊢ ran + ∈ V
11		prex 5355	. . . 4 ⊢ {∅, (𝐹‘𝑀)} ∈ V
12	10, 11	unex 7596	. . 3 ⊢ (ran + ∪ {∅, (𝐹‘𝑀)}) ∈ V
13		fveq2 6774	. . . . . . 7 ⊢ (𝑦 = 𝑀 → (seq𝑀( + , 𝐹)‘𝑦) = (seq𝑀( + , 𝐹)‘𝑀))
14	13	eleq1d 2823	. . . . . 6 ⊢ (𝑦 = 𝑀 → ((seq𝑀( + , 𝐹)‘𝑦) ∈ (ran + ∪ {∅, (𝐹‘𝑀)}) ↔ (seq𝑀( + , 𝐹)‘𝑀) ∈ (ran + ∪ {∅, (𝐹‘𝑀)})))
15		fveq2 6774	. . . . . . 7 ⊢ (𝑦 = 𝑧 → (seq𝑀( + , 𝐹)‘𝑦) = (seq𝑀( + , 𝐹)‘𝑧))
16	15	eleq1d 2823	. . . . . 6 ⊢ (𝑦 = 𝑧 → ((seq𝑀( + , 𝐹)‘𝑦) ∈ (ran + ∪ {∅, (𝐹‘𝑀)}) ↔ (seq𝑀( + , 𝐹)‘𝑧) ∈ (ran + ∪ {∅, (𝐹‘𝑀)})))
17		fveq2 6774	. . . . . . 7 ⊢ (𝑦 = (𝑧 + 1) → (seq𝑀( + , 𝐹)‘𝑦) = (seq𝑀( + , 𝐹)‘(𝑧 + 1)))
18	17	eleq1d 2823	. . . . . 6 ⊢ (𝑦 = (𝑧 + 1) → ((seq𝑀( + , 𝐹)‘𝑦) ∈ (ran + ∪ {∅, (𝐹‘𝑀)}) ↔ (seq𝑀( + , 𝐹)‘(𝑧 + 1)) ∈ (ran + ∪ {∅, (𝐹‘𝑀)})))
19		fveq2 6774	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (seq𝑀( + , 𝐹)‘𝑦) = (seq𝑀( + , 𝐹)‘𝑥))
20	19	eleq1d 2823	. . . . . 6 ⊢ (𝑦 = 𝑥 → ((seq𝑀( + , 𝐹)‘𝑦) ∈ (ran + ∪ {∅, (𝐹‘𝑀)}) ↔ (seq𝑀( + , 𝐹)‘𝑥) ∈ (ran + ∪ {∅, (𝐹‘𝑀)})))
21		seq1 13734	. . . . . . 7 ⊢ (𝑀 ∈ ℤ → (seq𝑀( + , 𝐹)‘𝑀) = (𝐹‘𝑀))
22		ssun2 4107	. . . . . . . 8 ⊢ {∅, (𝐹‘𝑀)} ⊆ (ran + ∪ {∅, (𝐹‘𝑀)})
23		fvex 6787	. . . . . . . . 9 ⊢ (𝐹‘𝑀) ∈ V
24	23	prid2 4699	. . . . . . . 8 ⊢ (𝐹‘𝑀) ∈ {∅, (𝐹‘𝑀)}
25	22, 24	sselii 3918	. . . . . . 7 ⊢ (𝐹‘𝑀) ∈ (ran + ∪ {∅, (𝐹‘𝑀)})
26	21, 25	eqeltrdi 2847	. . . . . 6 ⊢ (𝑀 ∈ ℤ → (seq𝑀( + , 𝐹)‘𝑀) ∈ (ran + ∪ {∅, (𝐹‘𝑀)}))
27		seqp1 13736	. . . . . . . . 9 ⊢ (𝑧 ∈ (ℤ≥‘𝑀) → (seq𝑀( + , 𝐹)‘(𝑧 + 1)) = ((seq𝑀( + , 𝐹)‘𝑧) + (𝐹‘(𝑧 + 1))))
28	27	adantr 481	. . . . . . . 8 ⊢ ((𝑧 ∈ (ℤ≥‘𝑀) ∧ (seq𝑀( + , 𝐹)‘𝑧) ∈ (ran + ∪ {∅, (𝐹‘𝑀)})) → (seq𝑀( + , 𝐹)‘(𝑧 + 1)) = ((seq𝑀( + , 𝐹)‘𝑧) + (𝐹‘(𝑧 + 1))))
29		df-ov 7278	. . . . . . . . . 10 ⊢ ((seq𝑀( + , 𝐹)‘𝑧) + (𝐹‘(𝑧 + 1))) = ( + ‘⟨(seq𝑀( + , 𝐹)‘𝑧), (𝐹‘(𝑧 + 1))⟩)
30		snsspr1 4747	. . . . . . . . . . . 12 ⊢ {∅} ⊆ {∅, (𝐹‘𝑀)}
31		unss2 4115	. . . . . . . . . . . 12 ⊢ ({∅} ⊆ {∅, (𝐹‘𝑀)} → (ran + ∪ {∅}) ⊆ (ran + ∪ {∅, (𝐹‘𝑀)}))
32	30, 31	ax-mp 5	. . . . . . . . . . 11 ⊢ (ran + ∪ {∅}) ⊆ (ran + ∪ {∅, (𝐹‘𝑀)})
33		fvrn0 6802	. . . . . . . . . . 11 ⊢ ( + ‘⟨(seq𝑀( + , 𝐹)‘𝑧), (𝐹‘(𝑧 + 1))⟩) ∈ (ran + ∪ {∅})
34	32, 33	sselii 3918	. . . . . . . . . 10 ⊢ ( + ‘⟨(seq𝑀( + , 𝐹)‘𝑧), (𝐹‘(𝑧 + 1))⟩) ∈ (ran + ∪ {∅, (𝐹‘𝑀)})
35	29, 34	eqeltri 2835	. . . . . . . . 9 ⊢ ((seq𝑀( + , 𝐹)‘𝑧) + (𝐹‘(𝑧 + 1))) ∈ (ran + ∪ {∅, (𝐹‘𝑀)})
36	35	a1i 11	. . . . . . . 8 ⊢ ((𝑧 ∈ (ℤ≥‘𝑀) ∧ (seq𝑀( + , 𝐹)‘𝑧) ∈ (ran + ∪ {∅, (𝐹‘𝑀)})) → ((seq𝑀( + , 𝐹)‘𝑧) + (𝐹‘(𝑧 + 1))) ∈ (ran + ∪ {∅, (𝐹‘𝑀)}))
37	28, 36	eqeltrd 2839	. . . . . . 7 ⊢ ((𝑧 ∈ (ℤ≥‘𝑀) ∧ (seq𝑀( + , 𝐹)‘𝑧) ∈ (ran + ∪ {∅, (𝐹‘𝑀)})) → (seq𝑀( + , 𝐹)‘(𝑧 + 1)) ∈ (ran + ∪ {∅, (𝐹‘𝑀)}))
38	37	ex 413	. . . . . 6 ⊢ (𝑧 ∈ (ℤ≥‘𝑀) → ((seq𝑀( + , 𝐹)‘𝑧) ∈ (ran + ∪ {∅, (𝐹‘𝑀)}) → (seq𝑀( + , 𝐹)‘(𝑧 + 1)) ∈ (ran + ∪ {∅, (𝐹‘𝑀)})))
39	14, 16, 18, 20, 26, 38	uzind4 12646	. . . . 5 ⊢ (𝑥 ∈ (ℤ≥‘𝑀) → (seq𝑀( + , 𝐹)‘𝑥) ∈ (ran + ∪ {∅, (𝐹‘𝑀)}))
40	39	rgen 3074	. . . 4 ⊢ ∀𝑥 ∈ (ℤ≥‘𝑀)(seq𝑀( + , 𝐹)‘𝑥) ∈ (ran + ∪ {∅, (𝐹‘𝑀)})
41		fnfvrnss 6994	. . . 4 ⊢ ((seq𝑀( + , 𝐹) Fn (ℤ≥‘𝑀) ∧ ∀𝑥 ∈ (ℤ≥‘𝑀)(seq𝑀( + , 𝐹)‘𝑥) ∈ (ran + ∪ {∅, (𝐹‘𝑀)})) → ran seq𝑀( + , 𝐹) ⊆ (ran + ∪ {∅, (𝐹‘𝑀)}))
42	3, 40, 41	mp2an 689	. . 3 ⊢ ran seq𝑀( + , 𝐹) ⊆ (ran + ∪ {∅, (𝐹‘𝑀)})
43	12, 42	ssexi 5246	. 2 ⊢ ran seq𝑀( + , 𝐹) ∈ V
44		funexw 7794	. 2 ⊢ ((Fun seq𝑀( + , 𝐹) ∧ dom seq𝑀( + , 𝐹) ∈ V ∧ ran seq𝑀( + , 𝐹) ∈ V) → seq𝑀( + , 𝐹) ∈ V)
45	5, 8, 43, 44	mp3an 1460	1 ⊢ seq𝑀( + , 𝐹) ∈ V

Syntax hints:   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∀wral 3064  Vcvv 3432   ∪ cun 3885   ⊆ wss 3887  ∅c0 4256  {csn 4561  {cpr 4563  ⟨cop 4567  dom cdm 5589  ran crn 5590  Fun wfun 6427   Fn wfn 6428  ‘cfv 6433  (class class class)co 7275  1c1 10872   + caddc 10874  ℤcz 12319  ℤ_≥cuz 12582  seqcseq 13721

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-er 8498  df-en 8734  df-dom 8735  df-sdom 8736  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-n0 12234  df-z 12320  df-uz 12583  df-seq 13722

This theorem is referenced by:  mulgfval  18702