Theorem seqcl2 13739

Description: Closure properties of the recursive sequence builder. (Contributed by Mario Carneiro, 2-Jul-2013.) (Revised by Mario Carneiro, 27-May-2014.)

Hypotheses

Ref	Expression
seqcl2.1	⊢ (𝜑 → (𝐹‘𝑀) ∈ 𝐶)
seqcl2.2	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)) → (𝑥 + 𝑦) ∈ 𝐶)
seqcl2.3	⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
seqcl2.4	⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑀 + 1)...𝑁)) → (𝐹‘𝑥) ∈ 𝐷)

Assertion

Ref	Expression
seqcl2	⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) ∈ 𝐶)

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

Allowed substitution hint:   𝑁(𝑦)

Proof of Theorem seqcl2

Dummy variable 𝑛 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		seqcl2.3	. . 3 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
2		eluzfz2 13263	. . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁))
3	1, 2	syl 17	. 2 ⊢ (𝜑 → 𝑁 ∈ (𝑀...𝑁))
4		eleq1 2828	. . . . . 6 ⊢ (𝑥 = 𝑀 → (𝑥 ∈ (𝑀...𝑁) ↔ 𝑀 ∈ (𝑀...𝑁)))
5		fveq2 6771	. . . . . . 7 ⊢ (𝑥 = 𝑀 → (seq𝑀( + , 𝐹)‘𝑥) = (seq𝑀( + , 𝐹)‘𝑀))
6	5	eleq1d 2825	. . . . . 6 ⊢ (𝑥 = 𝑀 → ((seq𝑀( + , 𝐹)‘𝑥) ∈ 𝐶 ↔ (seq𝑀( + , 𝐹)‘𝑀) ∈ 𝐶))
7	4, 6	imbi12d 345	. . . . 5 ⊢ (𝑥 = 𝑀 → ((𝑥 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑥) ∈ 𝐶) ↔ (𝑀 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑀) ∈ 𝐶)))
8	7	imbi2d 341	. . . 4 ⊢ (𝑥 = 𝑀 → ((𝜑 → (𝑥 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑥) ∈ 𝐶)) ↔ (𝜑 → (𝑀 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑀) ∈ 𝐶))))
9		eleq1 2828	. . . . . 6 ⊢ (𝑥 = 𝑛 → (𝑥 ∈ (𝑀...𝑁) ↔ 𝑛 ∈ (𝑀...𝑁)))
10		fveq2 6771	. . . . . . 7 ⊢ (𝑥 = 𝑛 → (seq𝑀( + , 𝐹)‘𝑥) = (seq𝑀( + , 𝐹)‘𝑛))
11	10	eleq1d 2825	. . . . . 6 ⊢ (𝑥 = 𝑛 → ((seq𝑀( + , 𝐹)‘𝑥) ∈ 𝐶 ↔ (seq𝑀( + , 𝐹)‘𝑛) ∈ 𝐶))
12	9, 11	imbi12d 345	. . . . 5 ⊢ (𝑥 = 𝑛 → ((𝑥 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑥) ∈ 𝐶) ↔ (𝑛 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑛) ∈ 𝐶)))
13	12	imbi2d 341	. . . 4 ⊢ (𝑥 = 𝑛 → ((𝜑 → (𝑥 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑥) ∈ 𝐶)) ↔ (𝜑 → (𝑛 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑛) ∈ 𝐶))))
14		eleq1 2828	. . . . . 6 ⊢ (𝑥 = (𝑛 + 1) → (𝑥 ∈ (𝑀...𝑁) ↔ (𝑛 + 1) ∈ (𝑀...𝑁)))
15		fveq2 6771	. . . . . . 7 ⊢ (𝑥 = (𝑛 + 1) → (seq𝑀( + , 𝐹)‘𝑥) = (seq𝑀( + , 𝐹)‘(𝑛 + 1)))
16	15	eleq1d 2825	. . . . . 6 ⊢ (𝑥 = (𝑛 + 1) → ((seq𝑀( + , 𝐹)‘𝑥) ∈ 𝐶 ↔ (seq𝑀( + , 𝐹)‘(𝑛 + 1)) ∈ 𝐶))
17	14, 16	imbi12d 345	. . . . 5 ⊢ (𝑥 = (𝑛 + 1) → ((𝑥 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑥) ∈ 𝐶) ↔ ((𝑛 + 1) ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘(𝑛 + 1)) ∈ 𝐶)))
18	17	imbi2d 341	. . . 4 ⊢ (𝑥 = (𝑛 + 1) → ((𝜑 → (𝑥 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑥) ∈ 𝐶)) ↔ (𝜑 → ((𝑛 + 1) ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘(𝑛 + 1)) ∈ 𝐶))))
19		eleq1 2828	. . . . . 6 ⊢ (𝑥 = 𝑁 → (𝑥 ∈ (𝑀...𝑁) ↔ 𝑁 ∈ (𝑀...𝑁)))
20		fveq2 6771	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (seq𝑀( + , 𝐹)‘𝑥) = (seq𝑀( + , 𝐹)‘𝑁))
21	20	eleq1d 2825	. . . . . 6 ⊢ (𝑥 = 𝑁 → ((seq𝑀( + , 𝐹)‘𝑥) ∈ 𝐶 ↔ (seq𝑀( + , 𝐹)‘𝑁) ∈ 𝐶))
22	19, 21	imbi12d 345	. . . . 5 ⊢ (𝑥 = 𝑁 → ((𝑥 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑥) ∈ 𝐶) ↔ (𝑁 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑁) ∈ 𝐶)))
23	22	imbi2d 341	. . . 4 ⊢ (𝑥 = 𝑁 → ((𝜑 → (𝑥 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑥) ∈ 𝐶)) ↔ (𝜑 → (𝑁 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑁) ∈ 𝐶))))
24		seqcl2.1	. . . . . 6 ⊢ (𝜑 → (𝐹‘𝑀) ∈ 𝐶)
25		seq1 13732	. . . . . . 7 ⊢ (𝑀 ∈ ℤ → (seq𝑀( + , 𝐹)‘𝑀) = (𝐹‘𝑀))
26	25	eleq1d 2825	. . . . . 6 ⊢ (𝑀 ∈ ℤ → ((seq𝑀( + , 𝐹)‘𝑀) ∈ 𝐶 ↔ (𝐹‘𝑀) ∈ 𝐶))
27	24, 26	syl5ibr 245	. . . . 5 ⊢ (𝑀 ∈ ℤ → (𝜑 → (seq𝑀( + , 𝐹)‘𝑀) ∈ 𝐶))
28	27	a1dd 50	. . . 4 ⊢ (𝑀 ∈ ℤ → (𝜑 → (𝑀 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑀) ∈ 𝐶)))
29		peano2fzr 13268	. . . . . . . 8 ⊢ ((𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → 𝑛 ∈ (𝑀...𝑁))
30	29	adantl 482	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → 𝑛 ∈ (𝑀...𝑁))
31	30	expr 457	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → ((𝑛 + 1) ∈ (𝑀...𝑁) → 𝑛 ∈ (𝑀...𝑁)))
32	31	imim1d 82	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → ((𝑛 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑛) ∈ 𝐶) → ((𝑛 + 1) ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑛) ∈ 𝐶)))
33		fveq2 6771	. . . . . . . . 9 ⊢ (𝑥 = (𝑛 + 1) → (𝐹‘𝑥) = (𝐹‘(𝑛 + 1)))
34	33	eleq1d 2825	. . . . . . . 8 ⊢ (𝑥 = (𝑛 + 1) → ((𝐹‘𝑥) ∈ 𝐷 ↔ (𝐹‘(𝑛 + 1)) ∈ 𝐷))
35		seqcl2.4	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑀 + 1)...𝑁)) → (𝐹‘𝑥) ∈ 𝐷)
36	35	ralrimiva 3110	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑥 ∈ ((𝑀 + 1)...𝑁)(𝐹‘𝑥) ∈ 𝐷)
37	36	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → ∀𝑥 ∈ ((𝑀 + 1)...𝑁)(𝐹‘𝑥) ∈ 𝐷)
38		eluzp1p1 12609	. . . . . . . . . 10 ⊢ (𝑛 ∈ (ℤ≥‘𝑀) → (𝑛 + 1) ∈ (ℤ≥‘(𝑀 + 1)))
39	38	ad2antrl 725	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝑛 + 1) ∈ (ℤ≥‘(𝑀 + 1)))
40		elfzuz3 13252	. . . . . . . . . 10 ⊢ ((𝑛 + 1) ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ≥‘(𝑛 + 1)))
41	40	ad2antll 726	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → 𝑁 ∈ (ℤ≥‘(𝑛 + 1)))
42		elfzuzb 13249	. . . . . . . . 9 ⊢ ((𝑛 + 1) ∈ ((𝑀 + 1)...𝑁) ↔ ((𝑛 + 1) ∈ (ℤ≥‘(𝑀 + 1)) ∧ 𝑁 ∈ (ℤ≥‘(𝑛 + 1))))
43	39, 41, 42	sylanbrc 583	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))
44	34, 37, 43	rspcdva 3563	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝐹‘(𝑛 + 1)) ∈ 𝐷)
45		seqcl2.2	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)) → (𝑥 + 𝑦) ∈ 𝐶)
46	45	caovclg 7458	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((seq𝑀( + , 𝐹)‘𝑛) ∈ 𝐶 ∧ (𝐹‘(𝑛 + 1)) ∈ 𝐷)) → ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))) ∈ 𝐶)
47	46	ex 413	. . . . . . . 8 ⊢ (𝜑 → (((seq𝑀( + , 𝐹)‘𝑛) ∈ 𝐶 ∧ (𝐹‘(𝑛 + 1)) ∈ 𝐷) → ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))) ∈ 𝐶))
48	47	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (((seq𝑀( + , 𝐹)‘𝑛) ∈ 𝐶 ∧ (𝐹‘(𝑛 + 1)) ∈ 𝐷) → ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))) ∈ 𝐶))
49	44, 48	mpan2d 691	. . . . . 6 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → ((seq𝑀( + , 𝐹)‘𝑛) ∈ 𝐶 → ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))) ∈ 𝐶))
50		seqp1 13734	. . . . . . . 8 ⊢ (𝑛 ∈ (ℤ≥‘𝑀) → (seq𝑀( + , 𝐹)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))))
51	50	ad2antrl 725	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (seq𝑀( + , 𝐹)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))))
52	51	eleq1d 2825	. . . . . 6 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → ((seq𝑀( + , 𝐹)‘(𝑛 + 1)) ∈ 𝐶 ↔ ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))) ∈ 𝐶))
53	49, 52	sylibrd 258	. . . . 5 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → ((seq𝑀( + , 𝐹)‘𝑛) ∈ 𝐶 → (seq𝑀( + , 𝐹)‘(𝑛 + 1)) ∈ 𝐶))
54	32, 53	animpimp2impd 843	. . . 4 ⊢ (𝑛 ∈ (ℤ≥‘𝑀) → ((𝜑 → (𝑛 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑛) ∈ 𝐶)) → (𝜑 → ((𝑛 + 1) ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘(𝑛 + 1)) ∈ 𝐶))))
55	8, 13, 18, 23, 28, 54	uzind4 12645	. . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝜑 → (𝑁 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑁) ∈ 𝐶)))
56	1, 55	mpcom 38	. 2 ⊢ (𝜑 → (𝑁 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑁) ∈ 𝐶))
57	3, 56	mpd 15	1 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) ∈ 𝐶)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1542   ∈ wcel 2110  ∀wral 3066  ‘cfv 6432  (class class class)co 7271  1c1 10873   + caddc 10875  ℤcz 12319  ℤ_≥cuz 12581  ...cfz 13238  seqcseq 13719

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 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7582  ax-cnex 10928  ax-resscn 10929  ax-1cn 10930  ax-icn 10931  ax-addcl 10932  ax-addrcl 10933  ax-mulcl 10934  ax-mulrcl 10935  ax-mulcom 10936  ax-addass 10937  ax-mulass 10938  ax-distr 10939  ax-i2m1 10940  ax-1ne0 10941  ax-1rid 10942  ax-rnegex 10943  ax-rrecex 10944  ax-cnre 10945  ax-pre-lttri 10946  ax-pre-lttrn 10947  ax-pre-ltadd 10948  ax-pre-mulgt0 10949

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-nel 3052  df-ral 3071  df-rex 3072  df-reu 3073  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-iun 4932  df-br 5080  df-opab 5142  df-mpt 5163  df-tr 5197  df-id 5490  df-eprel 5496  df-po 5504  df-so 5505  df-fr 5545  df-we 5547  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-pred 6201  df-ord 6268  df-on 6269  df-lim 6270  df-suc 6271  df-iota 6390  df-fun 6434  df-fn 6435  df-f 6436  df-f1 6437  df-fo 6438  df-f1o 6439  df-fv 6440  df-riota 7228  df-ov 7274  df-oprab 7275  df-mpo 7276  df-om 7707  df-1st 7824  df-2nd 7825  df-frecs 8088  df-wrecs 8119  df-recs 8193  df-rdg 8232  df-er 8481  df-en 8717  df-dom 8718  df-sdom 8719  df-pnf 11012  df-mnf 11013  df-xr 11014  df-ltxr 11015  df-le 11016  df-sub 11207  df-neg 11208  df-nn 11974  df-n0 12234  df-z 12320  df-uz 12582  df-fz 13239  df-seq 13720

This theorem is referenced by:  seqf2  13740  seqcl  13741  seqz  13769