Theorem seqcl2 14087

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 13589	. . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁))
3	1, 2	syl 18	. 2 ⊢ (𝜑 → 𝑁 ∈ (𝑀...𝑁))
4		eleq1 2848	. . . . . 6 ⊢ (𝑥 = 𝑀 → (𝑥 ∈ (𝑀...𝑁) ↔ 𝑀 ∈ (𝑀...𝑁)))
5		fveq2 6879	. . . . . . 7 ⊢ (𝑥 = 𝑀 → (seq𝑀( + , 𝐹)‘𝑥) = (seq𝑀( + , 𝐹)‘𝑀))
6	5	eleq1d 2845	. . . . . 6 ⊢ (𝑥 = 𝑀 → ((seq𝑀( + , 𝐹)‘𝑥) ∈ 𝐶 ↔ (seq𝑀( + , 𝐹)‘𝑀) ∈ 𝐶))
7	4, 6	imbi12d 347	. . . . 5 ⊢ (𝑥 = 𝑀 → ((𝑥 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑥) ∈ 𝐶) ↔ (𝑀 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑀) ∈ 𝐶)))
8	7	imbi2d 343	. . . 4 ⊢ (𝑥 = 𝑀 → ((𝜑 → (𝑥 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑥) ∈ 𝐶)) ↔ (𝜑 → (𝑀 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑀) ∈ 𝐶))))
9		eleq1 2848	. . . . . 6 ⊢ (𝑥 = 𝑛 → (𝑥 ∈ (𝑀...𝑁) ↔ 𝑛 ∈ (𝑀...𝑁)))
10		fveq2 6879	. . . . . . 7 ⊢ (𝑥 = 𝑛 → (seq𝑀( + , 𝐹)‘𝑥) = (seq𝑀( + , 𝐹)‘𝑛))
11	10	eleq1d 2845	. . . . . 6 ⊢ (𝑥 = 𝑛 → ((seq𝑀( + , 𝐹)‘𝑥) ∈ 𝐶 ↔ (seq𝑀( + , 𝐹)‘𝑛) ∈ 𝐶))
12	9, 11	imbi12d 347	. . . . 5 ⊢ (𝑥 = 𝑛 → ((𝑥 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑥) ∈ 𝐶) ↔ (𝑛 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑛) ∈ 𝐶)))
13	12	imbi2d 343	. . . 4 ⊢ (𝑥 = 𝑛 → ((𝜑 → (𝑥 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑥) ∈ 𝐶)) ↔ (𝜑 → (𝑛 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑛) ∈ 𝐶))))
14		eleq1 2848	. . . . . 6 ⊢ (𝑥 = (𝑛 + 1) → (𝑥 ∈ (𝑀...𝑁) ↔ (𝑛 + 1) ∈ (𝑀...𝑁)))
15		fveq2 6879	. . . . . . 7 ⊢ (𝑥 = (𝑛 + 1) → (seq𝑀( + , 𝐹)‘𝑥) = (seq𝑀( + , 𝐹)‘(𝑛 + 1)))
16	15	eleq1d 2845	. . . . . 6 ⊢ (𝑥 = (𝑛 + 1) → ((seq𝑀( + , 𝐹)‘𝑥) ∈ 𝐶 ↔ (seq𝑀( + , 𝐹)‘(𝑛 + 1)) ∈ 𝐶))
17	14, 16	imbi12d 347	. . . . 5 ⊢ (𝑥 = (𝑛 + 1) → ((𝑥 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑥) ∈ 𝐶) ↔ ((𝑛 + 1) ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘(𝑛 + 1)) ∈ 𝐶)))
18	17	imbi2d 343	. . . 4 ⊢ (𝑥 = (𝑛 + 1) → ((𝜑 → (𝑥 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑥) ∈ 𝐶)) ↔ (𝜑 → ((𝑛 + 1) ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘(𝑛 + 1)) ∈ 𝐶))))
19		eleq1 2848	. . . . . 6 ⊢ (𝑥 = 𝑁 → (𝑥 ∈ (𝑀...𝑁) ↔ 𝑁 ∈ (𝑀...𝑁)))
20		fveq2 6879	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (seq𝑀( + , 𝐹)‘𝑥) = (seq𝑀( + , 𝐹)‘𝑁))
21	20	eleq1d 2845	. . . . . 6 ⊢ (𝑥 = 𝑁 → ((seq𝑀( + , 𝐹)‘𝑥) ∈ 𝐶 ↔ (seq𝑀( + , 𝐹)‘𝑁) ∈ 𝐶))
22	19, 21	imbi12d 347	. . . . 5 ⊢ (𝑥 = 𝑁 → ((𝑥 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑥) ∈ 𝐶) ↔ (𝑁 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑁) ∈ 𝐶)))
23	22	imbi2d 343	. . . 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 14081	. . . . . . 7 ⊢ (𝑀 ∈ ℤ → (seq𝑀( + , 𝐹)‘𝑀) = (𝐹‘𝑀))
26	25	eleq1d 2845	. . . . . 6 ⊢ (𝑀 ∈ ℤ → ((seq𝑀( + , 𝐹)‘𝑀) ∈ 𝐶 ↔ (𝐹‘𝑀) ∈ 𝐶))
27	24, 26	imbitrrid 249	. . . . 5 ⊢ (𝑀 ∈ ℤ → (𝜑 → (seq𝑀( + , 𝐹)‘𝑀) ∈ 𝐶))
28	27	a1dd 51	. . . 4 ⊢ (𝑀 ∈ ℤ → (𝜑 → (𝑀 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑀) ∈ 𝐶)))
29		peano2fzr 13594	. . . . . . . 8 ⊢ ((𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → 𝑛 ∈ (𝑀...𝑁))
30	29	adantl 487	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → 𝑛 ∈ (𝑀...𝑁))
31	30	expr 462	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → ((𝑛 + 1) ∈ (𝑀...𝑁) → 𝑛 ∈ (𝑀...𝑁)))
32	31	imim1d 83	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → ((𝑛 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑛) ∈ 𝐶) → ((𝑛 + 1) ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑛) ∈ 𝐶)))
33		fveq2 6879	. . . . . . . . 9 ⊢ (𝑥 = (𝑛 + 1) → (𝐹‘𝑥) = (𝐹‘(𝑛 + 1)))
34	33	eleq1d 2845	. . . . . . . 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 3154	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑥 ∈ ((𝑀 + 1)...𝑁)(𝐹‘𝑥) ∈ 𝐷)
37	36	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → ∀𝑥 ∈ ((𝑀 + 1)...𝑁)(𝐹‘𝑥) ∈ 𝐷)
38		eluzp1p1 12918	. . . . . . . . . 10 ⊢ (𝑛 ∈ (ℤ≥‘𝑀) → (𝑛 + 1) ∈ (ℤ≥‘(𝑀 + 1)))
39	38	ad2antrl 741	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝑛 + 1) ∈ (ℤ≥‘(𝑀 + 1)))
40		elfzuz3 13578	. . . . . . . . . 10 ⊢ ((𝑛 + 1) ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ≥‘(𝑛 + 1)))
41	40	ad2antll 742	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → 𝑁 ∈ (ℤ≥‘(𝑛 + 1)))
42		elfzuzb 13575	. . . . . . . . 9 ⊢ ((𝑛 + 1) ∈ ((𝑀 + 1)...𝑁) ↔ ((𝑛 + 1) ∈ (ℤ≥‘(𝑀 + 1)) ∧ 𝑁 ∈ (ℤ≥‘(𝑛 + 1))))
43	39, 41, 42	sylanbrc 595	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))
44	34, 37, 43	rspcdva 3577	. . . . . . 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 7607	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((seq𝑀( + , 𝐹)‘𝑛) ∈ 𝐶 ∧ (𝐹‘(𝑛 + 1)) ∈ 𝐷)) → ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))) ∈ 𝐶)
47	46	ex 418	. . . . . . . 8 ⊢ (𝜑 → (((seq𝑀( + , 𝐹)‘𝑛) ∈ 𝐶 ∧ (𝐹‘(𝑛 + 1)) ∈ 𝐷) → ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))) ∈ 𝐶))
48	47	adantr 486	. . . . . . 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 707	. . . . . 6 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → ((seq𝑀( + , 𝐹)‘𝑛) ∈ 𝐶 → ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))) ∈ 𝐶))
50		seqp1 14083	. . . . . . . 8 ⊢ (𝑛 ∈ (ℤ≥‘𝑀) → (seq𝑀( + , 𝐹)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))))
51	50	ad2antrl 741	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (seq𝑀( + , 𝐹)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))))
52	51	eleq1d 2845	. . . . . 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 262	. . . . 5 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → ((seq𝑀( + , 𝐹)‘𝑛) ∈ 𝐶 → (seq𝑀( + , 𝐹)‘(𝑛 + 1)) ∈ 𝐶))
54	32, 53	animpimp2impd 860	. . . 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 12958	. . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝜑 → (𝑁 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑁) ∈ 𝐶)))
56	1, 55	mpcom 39	. 2 ⊢ (𝜑 → (𝑁 ∈ (𝑀...𝑁) → (seq𝑀( + , 𝐹)‘𝑁) ∈ 𝐶))
57	3, 56	mpd 16	1 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) ∈ 𝐶)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3076  ‘cfv 6533  (class class class)co 7414  1c1 11128   + caddc 11130  ℤcz 12618  ℤ_≥cuz 12890  ...cfz 13564  seqcseq 14068

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7737  ax-cnex 11183  ax-resscn 11184  ax-1cn 11185  ax-icn 11186  ax-addcl 11187  ax-addrcl 11188  ax-mulcl 11189  ax-mulrcl 11190  ax-mulcom 11191  ax-addass 11192  ax-mulass 11193  ax-distr 11194  ax-i2m1 11195  ax-1ne0 11196  ax-1rid 11197  ax-rnegex 11198  ax-rrecex 11199  ax-cnre 11200  ax-pre-lttri 11201  ax-pre-lttrn 11202  ax-pre-ltadd 11203  ax-pre-mulgt0 11204

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7371  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7864  df-1st 7987  df-2nd 7988  df-frecs 8281  df-wrecs 8312  df-recs 8361  df-rdg 8400  df-er 8699  df-en 8956  df-dom 8957  df-sdom 8958  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11470  df-neg 11471  df-nn 12261  df-n0 12532  df-z 12619  df-uz 12891  df-fz 13565  df-seq 14069

This theorem is used by:  seqf2  14088  seqcl  14089  seqz  14117