Theorem seqexw 14049

Description: Weak version of seqex 14035 that holds without ax-rep 5239. 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 14045	. . . 4 ⊢ (𝑀 ∈ ℤ → seq𝑀( + , 𝐹) Fn (ℤ≥‘𝑀))
3	1, 2	ax-mp 5	. . 3 ⊢ seq𝑀( + , 𝐹) Fn (ℤ≥‘𝑀)
4		fnfun 6633	. . 3 ⊢ (seq𝑀( + , 𝐹) Fn (ℤ≥‘𝑀) → Fun seq𝑀( + , 𝐹))
5	3, 4	ax-mp 5	. 2 ⊢ Fun seq𝑀( + , 𝐹)
6	3	fndmi 6637	. . 3 ⊢ dom seq𝑀( + , 𝐹) = (ℤ≥‘𝑀)
7		fvex 6892	. . 3 ⊢ (ℤ≥‘𝑀) ∈ V
8	6, 7	eqeltri 2865	. 2 ⊢ dom seq𝑀( + , 𝐹) ∈ V
9		seqexw.1	. . . . 5 ⊢ + ∈ V
10	9	rnex 7903	. . . 4 ⊢ ran + ∈ V
11		prex 5407	. . . 4 ⊢ {∅, (𝐹‘𝑀)} ∈ V
12	10, 11	unex 7739	. . 3 ⊢ (ran + ∪ {∅, (𝐹‘𝑀)}) ∈ V
13		fveq2 6879	. . . . . . 7 ⊢ (𝑦 = 𝑀 → (seq𝑀( + , 𝐹)‘𝑦) = (seq𝑀( + , 𝐹)‘𝑀))
14	13	eleq1d 2854	. . . . . 6 ⊢ (𝑦 = 𝑀 → ((seq𝑀( + , 𝐹)‘𝑦) ∈ (ran + ∪ {∅, (𝐹‘𝑀)}) ↔ (seq𝑀( + , 𝐹)‘𝑀) ∈ (ran + ∪ {∅, (𝐹‘𝑀)})))
15		fveq2 6879	. . . . . . 7 ⊢ (𝑦 = 𝑧 → (seq𝑀( + , 𝐹)‘𝑦) = (seq𝑀( + , 𝐹)‘𝑧))
16	15	eleq1d 2854	. . . . . 6 ⊢ (𝑦 = 𝑧 → ((seq𝑀( + , 𝐹)‘𝑦) ∈ (ran + ∪ {∅, (𝐹‘𝑀)}) ↔ (seq𝑀( + , 𝐹)‘𝑧) ∈ (ran + ∪ {∅, (𝐹‘𝑀)})))
17		fveq2 6879	. . . . . . 7 ⊢ (𝑦 = (𝑧 + 1) → (seq𝑀( + , 𝐹)‘𝑦) = (seq𝑀( + , 𝐹)‘(𝑧 + 1)))
18	17	eleq1d 2854	. . . . . 6 ⊢ (𝑦 = (𝑧 + 1) → ((seq𝑀( + , 𝐹)‘𝑦) ∈ (ran + ∪ {∅, (𝐹‘𝑀)}) ↔ (seq𝑀( + , 𝐹)‘(𝑧 + 1)) ∈ (ran + ∪ {∅, (𝐹‘𝑀)})))
19		fveq2 6879	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (seq𝑀( + , 𝐹)‘𝑦) = (seq𝑀( + , 𝐹)‘𝑥))
20	19	eleq1d 2854	. . . . . 6 ⊢ (𝑦 = 𝑥 → ((seq𝑀( + , 𝐹)‘𝑦) ∈ (ran + ∪ {∅, (𝐹‘𝑀)}) ↔ (seq𝑀( + , 𝐹)‘𝑥) ∈ (ran + ∪ {∅, (𝐹‘𝑀)})))
21		seq1 14046	. . . . . . 7 ⊢ (𝑀 ∈ ℤ → (seq𝑀( + , 𝐹)‘𝑀) = (𝐹‘𝑀))
22		ssun2 4140	. . . . . . . 8 ⊢ {∅, (𝐹‘𝑀)} ⊆ (ran + ∪ {∅, (𝐹‘𝑀)})
23		fvex 6892	. . . . . . . . 9 ⊢ (𝐹‘𝑀) ∈ V
24	23	prid2 4731	. . . . . . . 8 ⊢ (𝐹‘𝑀) ∈ {∅, (𝐹‘𝑀)}
25	22, 24	sselii 3942	. . . . . . 7 ⊢ (𝐹‘𝑀) ∈ (ran + ∪ {∅, (𝐹‘𝑀)})
26	21, 25	eqeltrdi 2877	. . . . . 6 ⊢ (𝑀 ∈ ℤ → (seq𝑀( + , 𝐹)‘𝑀) ∈ (ran + ∪ {∅, (𝐹‘𝑀)}))
27		seqp1 14048	. . . . . . . . 9 ⊢ (𝑧 ∈ (ℤ≥‘𝑀) → (seq𝑀( + , 𝐹)‘(𝑧 + 1)) = ((seq𝑀( + , 𝐹)‘𝑧) + (𝐹‘(𝑧 + 1))))
28	27	adantr 485	. . . . . . . 8 ⊢ ((𝑧 ∈ (ℤ≥‘𝑀) ∧ (seq𝑀( + , 𝐹)‘𝑧) ∈ (ran + ∪ {∅, (𝐹‘𝑀)})) → (seq𝑀( + , 𝐹)‘(𝑧 + 1)) = ((seq𝑀( + , 𝐹)‘𝑧) + (𝐹‘(𝑧 + 1))))
29		df-ov 7411	. . . . . . . . . 10 ⊢ ((seq𝑀( + , 𝐹)‘𝑧) + (𝐹‘(𝑧 + 1))) = ( + ‘⟨(seq𝑀( + , 𝐹)‘𝑧), (𝐹‘(𝑧 + 1))⟩)
30		snsspr1 4781	. . . . . . . . . . . 12 ⊢ {∅} ⊆ {∅, (𝐹‘𝑀)}
31		unss2 4148	. . . . . . . . . . . 12 ⊢ ({∅} ⊆ {∅, (𝐹‘𝑀)} → (ran + ∪ {∅}) ⊆ (ran + ∪ {∅, (𝐹‘𝑀)}))
32	30, 31	ax-mp 5	. . . . . . . . . . 11 ⊢ (ran + ∪ {∅}) ⊆ (ran + ∪ {∅, (𝐹‘𝑀)})
33		fvrn0 6907	. . . . . . . . . . 11 ⊢ ( + ‘⟨(seq𝑀( + , 𝐹)‘𝑧), (𝐹‘(𝑧 + 1))⟩) ∈ (ran + ∪ {∅})
34	32, 33	sselii 3942	. . . . . . . . . 10 ⊢ ( + ‘⟨(seq𝑀( + , 𝐹)‘𝑧), (𝐹‘(𝑧 + 1))⟩) ∈ (ran + ∪ {∅, (𝐹‘𝑀)})
35	29, 34	eqeltri 2865	. . . . . . . . 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 2869	. . . . . . 7 ⊢ ((𝑧 ∈ (ℤ≥‘𝑀) ∧ (seq𝑀( + , 𝐹)‘𝑧) ∈ (ran + ∪ {∅, (𝐹‘𝑀)})) → (seq𝑀( + , 𝐹)‘(𝑧 + 1)) ∈ (ran + ∪ {∅, (𝐹‘𝑀)}))
38	37	ex 417	. . . . . 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 12926	. . . . 5 ⊢ (𝑥 ∈ (ℤ≥‘𝑀) → (seq𝑀( + , 𝐹)‘𝑥) ∈ (ran + ∪ {∅, (𝐹‘𝑀)}))
40	39	rgen 3087	. . . 4 ⊢ ∀𝑥 ∈ (ℤ≥‘𝑀)(seq𝑀( + , 𝐹)‘𝑥) ∈ (ran + ∪ {∅, (𝐹‘𝑀)})
41		fnfvrnss 7114	. . . 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 704	. . 3 ⊢ ran seq𝑀( + , 𝐹) ⊆ (ran + ∪ {∅, (𝐹‘𝑀)})
43	12, 42	ssexi 5290	. 2 ⊢ ran seq𝑀( + , 𝐹) ∈ V
44		funexw 7945	. 2 ⊢ ((Fun seq𝑀( + , 𝐹) ∧ dom seq𝑀( + , 𝐹) ∈ V ∧ ran seq𝑀( + , 𝐹) ∈ V) → seq𝑀( + , 𝐹) ∈ V)
45	5, 8, 43, 44	mp3an 1487	1 ⊢ seq𝑀( + , 𝐹) ∈ V

Syntax hints:   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ∀wral 3085  Vcvv 3463   ∪ cun 3911   ⊆ wss 3913  ∅c0 4294  {csn 4591  {cpr 4593  ⟨cop 4597  dom cdm 5659  ran crn 5660  Fun wfun 6528   Fn wfn 6529  ‘cfv 6534  (class class class)co 7408  1c1 11097   + caddc 11099  ℤcz 12587  ℤ_≥cuz 12858  seqcseq 14033

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5258  ax-nul 5268  ax-pow 5334  ax-pr 5402  ax-un 7730  ax-cnex 11152  ax-resscn 11153  ax-1cn 11154  ax-icn 11155  ax-addcl 11156  ax-addrcl 11157  ax-mulcl 11158  ax-mulrcl 11159  ax-mulcom 11160  ax-addass 11161  ax-mulass 11162  ax-distr 11163  ax-i2m1 11164  ax-1ne0 11165  ax-1rid 11166  ax-rnegex 11167  ax-rrecex 11168  ax-cnre 11169  ax-pre-lttri 11170  ax-pre-lttrn 11171  ax-pre-ltadd 11172  ax-pre-mulgt0 11173

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6300  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6490  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-riota 7365  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7859  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-er 8690  df-en 8940  df-dom 8941  df-sdom 8942  df-pnf 11241  df-mnf 11242  df-xr 11243  df-ltxr 11244  df-le 11245  df-sub 11439  df-neg 11440  df-nn 12230  df-n0 12501  df-z 12588  df-uz 12859  df-seq 14034

This theorem is referenced by:  mulgfval  19131