Theorem seqcl2 14058

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 13561	. . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁))
3	1, 2	syl 18	. 2 ⊢ (𝜑 → 𝑁 ∈ (𝑀...𝑁))
4		eleq1 2851	. . . . . 6 ⊢ (𝑥 = 𝑀 → (𝑥 ∈ (𝑀...𝑁) ↔ 𝑀 ∈ (𝑀...𝑁)))
5		fveq2 6883	. . . . . . 7 ⊢ (𝑥 = 𝑀 → (seq𝑀( + , 𝐹)‘𝑥) = (seq𝑀( + , 𝐹)‘𝑀))
6	5	eleq1d 2848	. . . . . 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 2851	. . . . . 6 ⊢ (𝑥 = 𝑛 → (𝑥 ∈ (𝑀...𝑁) ↔ 𝑛 ∈ (𝑀...𝑁)))
10		fveq2 6883	. . . . . . 7 ⊢ (𝑥 = 𝑛 → (seq𝑀( + , 𝐹)‘𝑥) = (seq𝑀( + , 𝐹)‘𝑛))
11	10	eleq1d 2848	. . . . . 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 2851	. . . . . 6 ⊢ (𝑥 = (𝑛 + 1) → (𝑥 ∈ (𝑀...𝑁) ↔ (𝑛 + 1) ∈ (𝑀...𝑁)))
15		fveq2 6883	. . . . . . 7 ⊢ (𝑥 = (𝑛 + 1) → (seq𝑀( + , 𝐹)‘𝑥) = (seq𝑀( + , 𝐹)‘(𝑛 + 1)))
16	15	eleq1d 2848	. . . . . 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 2851	. . . . . 6 ⊢ (𝑥 = 𝑁 → (𝑥 ∈ (𝑀...𝑁) ↔ 𝑁 ∈ (𝑀...𝑁)))
20		fveq2 6883	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (seq𝑀( + , 𝐹)‘𝑥) = (seq𝑀( + , 𝐹)‘𝑁))
21	20	eleq1d 2848	. . . . . 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 14052	. . . . . . 7 ⊢ (𝑀 ∈ ℤ → (seq𝑀( + , 𝐹)‘𝑀) = (𝐹‘𝑀))
26	25	eleq1d 2848	. . . . . 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 13566	. . . . . . . 8 ⊢ ((𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → 𝑛 ∈ (𝑀...𝑁))
30	29	adantl 486	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → 𝑛 ∈ (𝑀...𝑁))
31	30	expr 461	. . . . . 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 6883	. . . . . . . . 9 ⊢ (𝑥 = (𝑛 + 1) → (𝐹‘𝑥) = (𝐹‘(𝑛 + 1)))
34	33	eleq1d 2848	. . . . . . . 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 3157	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑥 ∈ ((𝑀 + 1)...𝑁)(𝐹‘𝑥) ∈ 𝐷)
37	36	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → ∀𝑥 ∈ ((𝑀 + 1)...𝑁)(𝐹‘𝑥) ∈ 𝐷)
38		eluzp1p1 12891	. . . . . . . . . 10 ⊢ (𝑛 ∈ (ℤ≥‘𝑀) → (𝑛 + 1) ∈ (ℤ≥‘(𝑀 + 1)))
39	38	ad2antrl 740	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝑛 + 1) ∈ (ℤ≥‘(𝑀 + 1)))
40		elfzuz3 13550	. . . . . . . . . 10 ⊢ ((𝑛 + 1) ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ≥‘(𝑛 + 1)))
41	40	ad2antll 741	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → 𝑁 ∈ (ℤ≥‘(𝑛 + 1)))
42		elfzuzb 13547	. . . . . . . . 9 ⊢ ((𝑛 + 1) ∈ ((𝑀 + 1)...𝑁) ↔ ((𝑛 + 1) ∈ (ℤ≥‘(𝑀 + 1)) ∧ 𝑁 ∈ (ℤ≥‘(𝑛 + 1))))
43	39, 41, 42	sylanbrc 594	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝑛 + 1) ∈ ((𝑀 + 1)...𝑁))
44	34, 37, 43	rspcdva 3583	. . . . . . 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 7604	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((seq𝑀( + , 𝐹)‘𝑛) ∈ 𝐶 ∧ (𝐹‘(𝑛 + 1)) ∈ 𝐷)) → ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))) ∈ 𝐶)
47	46	ex 417	. . . . . . . 8 ⊢ (𝜑 → (((seq𝑀( + , 𝐹)‘𝑛) ∈ 𝐶 ∧ (𝐹‘(𝑛 + 1)) ∈ 𝐷) → ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))) ∈ 𝐶))
48	47	adantr 485	. . . . . . 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 706	. . . . . 6 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → ((seq𝑀( + , 𝐹)‘𝑛) ∈ 𝐶 → ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))) ∈ 𝐶))
50		seqp1 14054	. . . . . . . 8 ⊢ (𝑛 ∈ (ℤ≥‘𝑀) → (seq𝑀( + , 𝐹)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))))
51	50	ad2antrl 740	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (seq𝑀( + , 𝐹)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))))
52	51	eleq1d 2848	. . . . . 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 859	. . . 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 12931	. . 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)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ∀wral 3079  ‘cfv 6538  (class class class)co 7412  1c1 11102   + caddc 11104  ℤcz 12592  ℤ_≥cuz 12863  ...cfz 13536  seqcseq 14039

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734  ax-cnex 11157  ax-resscn 11158  ax-1cn 11159  ax-icn 11160  ax-addcl 11161  ax-addrcl 11162  ax-mulcl 11163  ax-mulrcl 11164  ax-mulcom 11165  ax-addass 11166  ax-mulass 11167  ax-distr 11168  ax-i2m1 11169  ax-1ne0 11170  ax-1rid 11171  ax-rnegex 11172  ax-rrecex 11173  ax-cnre 11174  ax-pre-lttri 11175  ax-pre-lttrn 11176  ax-pre-ltadd 11177  ax-pre-mulgt0 11178

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7864  df-1st 7987  df-2nd 7988  df-frecs 8279  df-wrecs 8310  df-recs 8359  df-rdg 8398  df-er 8695  df-en 8945  df-dom 8946  df-sdom 8947  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11444  df-neg 11445  df-nn 12235  df-n0 12506  df-z 12593  df-uz 12864  df-fz 13537  df-seq 14040

This theorem is referenced by:  seqf2  14059  seqcl  14060  seqz  14088