Theorem seqexw 13970

Description: Weak version of seqex 13956 that holds without ax-rep 5199. 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 13966	. . . 4 ⊢ (𝑀 ∈ ℤ → seq𝑀( + , 𝐹) Fn (ℤ≥‘𝑀))
3	1, 2	ax-mp 5	. . 3 ⊢ seq𝑀( + , 𝐹) Fn (ℤ≥‘𝑀)
4		fnfun 6585	. . 3 ⊢ (seq𝑀( + , 𝐹) Fn (ℤ≥‘𝑀) → Fun seq𝑀( + , 𝐹))
5	3, 4	ax-mp 5	. 2 ⊢ Fun seq𝑀( + , 𝐹)
6	3	fndmi 6589	. . 3 ⊢ dom seq𝑀( + , 𝐹) = (ℤ≥‘𝑀)
7		fvex 6840	. . 3 ⊢ (ℤ≥‘𝑀) ∈ V
8	6, 7	eqeltri 2835	. 2 ⊢ dom seq𝑀( + , 𝐹) ∈ V
9		seqexw.1	. . . . 5 ⊢ + ∈ V
10	9	rnex 7850	. . . 4 ⊢ ran + ∈ V
11		prex 5367	. . . 4 ⊢ {∅, (𝐹‘𝑀)} ∈ V
12	10, 11	unex 7687	. . 3 ⊢ (ran + ∪ {∅, (𝐹‘𝑀)}) ∈ V
13		fveq2 6827	. . . . . . 7 ⊢ (𝑦 = 𝑀 → (seq𝑀( + , 𝐹)‘𝑦) = (seq𝑀( + , 𝐹)‘𝑀))
14	13	eleq1d 2824	. . . . . 6 ⊢ (𝑦 = 𝑀 → ((seq𝑀( + , 𝐹)‘𝑦) ∈ (ran + ∪ {∅, (𝐹‘𝑀)}) ↔ (seq𝑀( + , 𝐹)‘𝑀) ∈ (ran + ∪ {∅, (𝐹‘𝑀)})))
15		fveq2 6827	. . . . . . 7 ⊢ (𝑦 = 𝑧 → (seq𝑀( + , 𝐹)‘𝑦) = (seq𝑀( + , 𝐹)‘𝑧))
16	15	eleq1d 2824	. . . . . 6 ⊢ (𝑦 = 𝑧 → ((seq𝑀( + , 𝐹)‘𝑦) ∈ (ran + ∪ {∅, (𝐹‘𝑀)}) ↔ (seq𝑀( + , 𝐹)‘𝑧) ∈ (ran + ∪ {∅, (𝐹‘𝑀)})))
17		fveq2 6827	. . . . . . 7 ⊢ (𝑦 = (𝑧 + 1) → (seq𝑀( + , 𝐹)‘𝑦) = (seq𝑀( + , 𝐹)‘(𝑧 + 1)))
18	17	eleq1d 2824	. . . . . 6 ⊢ (𝑦 = (𝑧 + 1) → ((seq𝑀( + , 𝐹)‘𝑦) ∈ (ran + ∪ {∅, (𝐹‘𝑀)}) ↔ (seq𝑀( + , 𝐹)‘(𝑧 + 1)) ∈ (ran + ∪ {∅, (𝐹‘𝑀)})))
19		fveq2 6827	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (seq𝑀( + , 𝐹)‘𝑦) = (seq𝑀( + , 𝐹)‘𝑥))
20	19	eleq1d 2824	. . . . . 6 ⊢ (𝑦 = 𝑥 → ((seq𝑀( + , 𝐹)‘𝑦) ∈ (ran + ∪ {∅, (𝐹‘𝑀)}) ↔ (seq𝑀( + , 𝐹)‘𝑥) ∈ (ran + ∪ {∅, (𝐹‘𝑀)})))
21		seq1 13967	. . . . . . 7 ⊢ (𝑀 ∈ ℤ → (seq𝑀( + , 𝐹)‘𝑀) = (𝐹‘𝑀))
22		ssun2 4108	. . . . . . . 8 ⊢ {∅, (𝐹‘𝑀)} ⊆ (ran + ∪ {∅, (𝐹‘𝑀)})
23		fvex 6840	. . . . . . . . 9 ⊢ (𝐹‘𝑀) ∈ V
24	23	prid2 4695	. . . . . . . 8 ⊢ (𝐹‘𝑀) ∈ {∅, (𝐹‘𝑀)}
25	22, 24	sselii 3912	. . . . . . 7 ⊢ (𝐹‘𝑀) ∈ (ran + ∪ {∅, (𝐹‘𝑀)})
26	21, 25	eqeltrdi 2847	. . . . . 6 ⊢ (𝑀 ∈ ℤ → (seq𝑀( + , 𝐹)‘𝑀) ∈ (ran + ∪ {∅, (𝐹‘𝑀)}))
27		seqp1 13969	. . . . . . . . 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 7359	. . . . . . . . . 10 ⊢ ((seq𝑀( + , 𝐹)‘𝑧) + (𝐹‘(𝑧 + 1))) = ( + ‘⟨(seq𝑀( + , 𝐹)‘𝑧), (𝐹‘(𝑧 + 1))⟩)
30		snsspr1 4745	. . . . . . . . . . . 12 ⊢ {∅} ⊆ {∅, (𝐹‘𝑀)}
31		unss2 4116	. . . . . . . . . . . 12 ⊢ ({∅} ⊆ {∅, (𝐹‘𝑀)} → (ran + ∪ {∅}) ⊆ (ran + ∪ {∅, (𝐹‘𝑀)}))
32	30, 31	ax-mp 5	. . . . . . . . . . 11 ⊢ (ran + ∪ {∅}) ⊆ (ran + ∪ {∅, (𝐹‘𝑀)})
33		fvrn0 6855	. . . . . . . . . . 11 ⊢ ( + ‘⟨(seq𝑀( + , 𝐹)‘𝑧), (𝐹‘(𝑧 + 1))⟩) ∈ (ran + ∪ {∅})
34	32, 33	sselii 3912	. . . . . . . . . 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 12847	. . . . 5 ⊢ (𝑥 ∈ (ℤ≥‘𝑀) → (seq𝑀( + , 𝐹)‘𝑥) ∈ (ran + ∪ {∅, (𝐹‘𝑀)}))
40	39	rgen 3055	. . . 4 ⊢ ∀𝑥 ∈ (ℤ≥‘𝑀)(seq𝑀( + , 𝐹)‘𝑥) ∈ (ran + ∪ {∅, (𝐹‘𝑀)})
41		fnfvrnss 7062	. . . 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 698	. . 3 ⊢ ran seq𝑀( + , 𝐹) ⊆ (ran + ∪ {∅, (𝐹‘𝑀)})
43	12, 42	ssexi 5250	. 2 ⊢ ran seq𝑀( + , 𝐹) ∈ V
44		funexw 7894	. 2 ⊢ ((Fun seq𝑀( + , 𝐹) ∧ dom seq𝑀( + , 𝐹) ∈ V ∧ ran seq𝑀( + , 𝐹) ∈ V) → seq𝑀( + , 𝐹) ∈ V)
45	5, 8, 43, 44	mp3an 1469	1 ⊢ seq𝑀( + , 𝐹) ∈ V

Syntax hints:   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ∀wral 3053  Vcvv 3431   ∪ cun 3881   ⊆ wss 3883  ∅c0 4261  {csn 4555  {cpr 4557  ⟨cop 4561  dom cdm 5618  ran crn 5619  Fun wfun 6479   Fn wfn 6480  ‘cfv 6485  (class class class)co 7356  1c1 11030   + caddc 11032  ℤcz 12515  ℤ_≥cuz 12779  seqcseq 13954

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-er 8633  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12166  df-n0 12429  df-z 12516  df-uz 12780  df-seq 13955

This theorem is referenced by:  mulgfval  19036