Theorem seqexw 14015

Description: Weak version of seqex 14001 that holds without ax-rep 5285. 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 14011	. . . 4 ⊢ (𝑀 ∈ ℤ → seq𝑀( + , 𝐹) Fn (ℤ≥‘𝑀))
3	1, 2	ax-mp 5	. . 3 ⊢ seq𝑀( + , 𝐹) Fn (ℤ≥‘𝑀)
4		fnfun 6654	. . 3 ⊢ (seq𝑀( + , 𝐹) Fn (ℤ≥‘𝑀) → Fun seq𝑀( + , 𝐹))
5	3, 4	ax-mp 5	. 2 ⊢ Fun seq𝑀( + , 𝐹)
6	3	fndmi 6658	. . 3 ⊢ dom seq𝑀( + , 𝐹) = (ℤ≥‘𝑀)
7		fvex 6910	. . 3 ⊢ (ℤ≥‘𝑀) ∈ V
8	6, 7	eqeltri 2825	. 2 ⊢ dom seq𝑀( + , 𝐹) ∈ V
9		seqexw.1	. . . . 5 ⊢ + ∈ V
10	9	rnex 7918	. . . 4 ⊢ ran + ∈ V
11		prex 5434	. . . 4 ⊢ {∅, (𝐹‘𝑀)} ∈ V
12	10, 11	unex 7748	. . 3 ⊢ (ran + ∪ {∅, (𝐹‘𝑀)}) ∈ V
13		fveq2 6897	. . . . . . 7 ⊢ (𝑦 = 𝑀 → (seq𝑀( + , 𝐹)‘𝑦) = (seq𝑀( + , 𝐹)‘𝑀))
14	13	eleq1d 2814	. . . . . 6 ⊢ (𝑦 = 𝑀 → ((seq𝑀( + , 𝐹)‘𝑦) ∈ (ran + ∪ {∅, (𝐹‘𝑀)}) ↔ (seq𝑀( + , 𝐹)‘𝑀) ∈ (ran + ∪ {∅, (𝐹‘𝑀)})))
15		fveq2 6897	. . . . . . 7 ⊢ (𝑦 = 𝑧 → (seq𝑀( + , 𝐹)‘𝑦) = (seq𝑀( + , 𝐹)‘𝑧))
16	15	eleq1d 2814	. . . . . 6 ⊢ (𝑦 = 𝑧 → ((seq𝑀( + , 𝐹)‘𝑦) ∈ (ran + ∪ {∅, (𝐹‘𝑀)}) ↔ (seq𝑀( + , 𝐹)‘𝑧) ∈ (ran + ∪ {∅, (𝐹‘𝑀)})))
17		fveq2 6897	. . . . . . 7 ⊢ (𝑦 = (𝑧 + 1) → (seq𝑀( + , 𝐹)‘𝑦) = (seq𝑀( + , 𝐹)‘(𝑧 + 1)))
18	17	eleq1d 2814	. . . . . 6 ⊢ (𝑦 = (𝑧 + 1) → ((seq𝑀( + , 𝐹)‘𝑦) ∈ (ran + ∪ {∅, (𝐹‘𝑀)}) ↔ (seq𝑀( + , 𝐹)‘(𝑧 + 1)) ∈ (ran + ∪ {∅, (𝐹‘𝑀)})))
19		fveq2 6897	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (seq𝑀( + , 𝐹)‘𝑦) = (seq𝑀( + , 𝐹)‘𝑥))
20	19	eleq1d 2814	. . . . . 6 ⊢ (𝑦 = 𝑥 → ((seq𝑀( + , 𝐹)‘𝑦) ∈ (ran + ∪ {∅, (𝐹‘𝑀)}) ↔ (seq𝑀( + , 𝐹)‘𝑥) ∈ (ran + ∪ {∅, (𝐹‘𝑀)})))
21		seq1 14012	. . . . . . 7 ⊢ (𝑀 ∈ ℤ → (seq𝑀( + , 𝐹)‘𝑀) = (𝐹‘𝑀))
22		ssun2 4173	. . . . . . . 8 ⊢ {∅, (𝐹‘𝑀)} ⊆ (ran + ∪ {∅, (𝐹‘𝑀)})
23		fvex 6910	. . . . . . . . 9 ⊢ (𝐹‘𝑀) ∈ V
24	23	prid2 4768	. . . . . . . 8 ⊢ (𝐹‘𝑀) ∈ {∅, (𝐹‘𝑀)}
25	22, 24	sselii 3977	. . . . . . 7 ⊢ (𝐹‘𝑀) ∈ (ran + ∪ {∅, (𝐹‘𝑀)})
26	21, 25	eqeltrdi 2837	. . . . . 6 ⊢ (𝑀 ∈ ℤ → (seq𝑀( + , 𝐹)‘𝑀) ∈ (ran + ∪ {∅, (𝐹‘𝑀)}))
27		seqp1 14014	. . . . . . . . 9 ⊢ (𝑧 ∈ (ℤ≥‘𝑀) → (seq𝑀( + , 𝐹)‘(𝑧 + 1)) = ((seq𝑀( + , 𝐹)‘𝑧) + (𝐹‘(𝑧 + 1))))
28	27	adantr 480	. . . . . . . 8 ⊢ ((𝑧 ∈ (ℤ≥‘𝑀) ∧ (seq𝑀( + , 𝐹)‘𝑧) ∈ (ran + ∪ {∅, (𝐹‘𝑀)})) → (seq𝑀( + , 𝐹)‘(𝑧 + 1)) = ((seq𝑀( + , 𝐹)‘𝑧) + (𝐹‘(𝑧 + 1))))
29		df-ov 7423	. . . . . . . . . 10 ⊢ ((seq𝑀( + , 𝐹)‘𝑧) + (𝐹‘(𝑧 + 1))) = ( + ‘⟨(seq𝑀( + , 𝐹)‘𝑧), (𝐹‘(𝑧 + 1))⟩)
30		snsspr1 4818	. . . . . . . . . . . 12 ⊢ {∅} ⊆ {∅, (𝐹‘𝑀)}
31		unss2 4181	. . . . . . . . . . . 12 ⊢ ({∅} ⊆ {∅, (𝐹‘𝑀)} → (ran + ∪ {∅}) ⊆ (ran + ∪ {∅, (𝐹‘𝑀)}))
32	30, 31	ax-mp 5	. . . . . . . . . . 11 ⊢ (ran + ∪ {∅}) ⊆ (ran + ∪ {∅, (𝐹‘𝑀)})
33		fvrn0 6927	. . . . . . . . . . 11 ⊢ ( + ‘⟨(seq𝑀( + , 𝐹)‘𝑧), (𝐹‘(𝑧 + 1))⟩) ∈ (ran + ∪ {∅})
34	32, 33	sselii 3977	. . . . . . . . . 10 ⊢ ( + ‘⟨(seq𝑀( + , 𝐹)‘𝑧), (𝐹‘(𝑧 + 1))⟩) ∈ (ran + ∪ {∅, (𝐹‘𝑀)})
35	29, 34	eqeltri 2825	. . . . . . . . 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 2829	. . . . . . 7 ⊢ ((𝑧 ∈ (ℤ≥‘𝑀) ∧ (seq𝑀( + , 𝐹)‘𝑧) ∈ (ran + ∪ {∅, (𝐹‘𝑀)})) → (seq𝑀( + , 𝐹)‘(𝑧 + 1)) ∈ (ran + ∪ {∅, (𝐹‘𝑀)}))
38	37	ex 412	. . . . . 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 12921	. . . . 5 ⊢ (𝑥 ∈ (ℤ≥‘𝑀) → (seq𝑀( + , 𝐹)‘𝑥) ∈ (ran + ∪ {∅, (𝐹‘𝑀)}))
40	39	rgen 3060	. . . 4 ⊢ ∀𝑥 ∈ (ℤ≥‘𝑀)(seq𝑀( + , 𝐹)‘𝑥) ∈ (ran + ∪ {∅, (𝐹‘𝑀)})
41		fnfvrnss 7131	. . . 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 691	. . 3 ⊢ ran seq𝑀( + , 𝐹) ⊆ (ran + ∪ {∅, (𝐹‘𝑀)})
43	12, 42	ssexi 5322	. 2 ⊢ ran seq𝑀( + , 𝐹) ∈ V
44		funexw 7955	. 2 ⊢ ((Fun seq𝑀( + , 𝐹) ∧ dom seq𝑀( + , 𝐹) ∈ V ∧ ran seq𝑀( + , 𝐹) ∈ V) → seq𝑀( + , 𝐹) ∈ V)
45	5, 8, 43, 44	mp3an 1458	1 ⊢ seq𝑀( + , 𝐹) ∈ V

Syntax hints:   ∧ wa 395   = wceq 1534   ∈ wcel 2099  ∀wral 3058  Vcvv 3471   ∪ cun 3945   ⊆ wss 3947  ∅c0 4323  {csn 4629  {cpr 4631  ⟨cop 4635  dom cdm 5678  ran crn 5679  Fun wfun 6542   Fn wfn 6543  ‘cfv 6548  (class class class)co 7420  1c1 11140   + caddc 11142  ℤcz 12589  ℤ_≥cuz 12853  seqcseq 13999

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740  ax-cnex 11195  ax-resscn 11196  ax-1cn 11197  ax-icn 11198  ax-addcl 11199  ax-addrcl 11200  ax-mulcl 11201  ax-mulrcl 11202  ax-mulcom 11203  ax-addass 11204  ax-mulass 11205  ax-distr 11206  ax-i2m1 11207  ax-1ne0 11208  ax-1rid 11209  ax-rnegex 11210  ax-rrecex 11211  ax-cnre 11212  ax-pre-lttri 11213  ax-pre-lttrn 11214  ax-pre-ltadd 11215  ax-pre-mulgt0 11216

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6305  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-riota 7376  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7871  df-2nd 7994  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-er 8725  df-en 8965  df-dom 8966  df-sdom 8967  df-pnf 11281  df-mnf 11282  df-xr 11283  df-ltxr 11284  df-le 11285  df-sub 11477  df-neg 11478  df-nn 12244  df-n0 12504  df-z 12590  df-uz 12854  df-seq 14000

This theorem is referenced by:  mulgfval  19025

Otomatik - 173.255.232.114 CloudFlare DNS Türk Telekom DNS Google DNS Open DNS

OSZAR »