Theorem efnnfsumcl 24829

Description: Finite sum closure in the log-integers. (Contributed by Mario Carneiro, 7-Apr-2016.)

Hypotheses

Ref	Expression
efnnfsumcl.1	⊢ (𝜑 → 𝐴 ∈ Fin)
efnnfsumcl.2	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ)
efnnfsumcl.3	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (exp‘𝐵) ∈ ℕ)

Assertion

Ref	Expression
efnnfsumcl	⊢ (𝜑 → (exp‘Σ𝑘 ∈ 𝐴 𝐵) ∈ ℕ)

Distinct variable groups:   𝐴,𝑘   𝜑,𝑘

Allowed substitution hint:   𝐵(𝑘)

Proof of Theorem efnnfsumcl

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssrab2 3687	. . . . 5 ⊢ {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ} ⊆ ℝ
2		ax-resscn 9993	. . . . 5 ⊢ ℝ ⊆ ℂ
3	1, 2	sstri 3612	. . . 4 ⊢ {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ} ⊆ ℂ
4	3	a1i 11	. . 3 ⊢ (𝜑 → {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ} ⊆ ℂ)
5		fveq2 6191	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (exp‘𝑥) = (exp‘𝑦))
6	5	eleq1d 2686	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((exp‘𝑥) ∈ ℕ ↔ (exp‘𝑦) ∈ ℕ))
7	6	elrab 3363	. . . . 5 ⊢ (𝑦 ∈ {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ} ↔ (𝑦 ∈ ℝ ∧ (exp‘𝑦) ∈ ℕ))
8		fveq2 6191	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (exp‘𝑥) = (exp‘𝑧))
9	8	eleq1d 2686	. . . . . 6 ⊢ (𝑥 = 𝑧 → ((exp‘𝑥) ∈ ℕ ↔ (exp‘𝑧) ∈ ℕ))
10	9	elrab 3363	. . . . 5 ⊢ (𝑧 ∈ {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ} ↔ (𝑧 ∈ ℝ ∧ (exp‘𝑧) ∈ ℕ))
11		simpll 790	. . . . . . 7 ⊢ (((𝑦 ∈ ℝ ∧ (exp‘𝑦) ∈ ℕ) ∧ (𝑧 ∈ ℝ ∧ (exp‘𝑧) ∈ ℕ)) → 𝑦 ∈ ℝ)
12		simprl 794	. . . . . . 7 ⊢ (((𝑦 ∈ ℝ ∧ (exp‘𝑦) ∈ ℕ) ∧ (𝑧 ∈ ℝ ∧ (exp‘𝑧) ∈ ℕ)) → 𝑧 ∈ ℝ)
13	11, 12	readdcld 10069	. . . . . 6 ⊢ (((𝑦 ∈ ℝ ∧ (exp‘𝑦) ∈ ℕ) ∧ (𝑧 ∈ ℝ ∧ (exp‘𝑧) ∈ ℕ)) → (𝑦 + 𝑧) ∈ ℝ)
14	11	recnd 10068	. . . . . . . 8 ⊢ (((𝑦 ∈ ℝ ∧ (exp‘𝑦) ∈ ℕ) ∧ (𝑧 ∈ ℝ ∧ (exp‘𝑧) ∈ ℕ)) → 𝑦 ∈ ℂ)
15	12	recnd 10068	. . . . . . . 8 ⊢ (((𝑦 ∈ ℝ ∧ (exp‘𝑦) ∈ ℕ) ∧ (𝑧 ∈ ℝ ∧ (exp‘𝑧) ∈ ℕ)) → 𝑧 ∈ ℂ)
16		efadd 14824	. . . . . . . 8 ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (exp‘(𝑦 + 𝑧)) = ((exp‘𝑦) · (exp‘𝑧)))
17	14, 15, 16	syl2anc 693	. . . . . . 7 ⊢ (((𝑦 ∈ ℝ ∧ (exp‘𝑦) ∈ ℕ) ∧ (𝑧 ∈ ℝ ∧ (exp‘𝑧) ∈ ℕ)) → (exp‘(𝑦 + 𝑧)) = ((exp‘𝑦) · (exp‘𝑧)))
18		nnmulcl 11043	. . . . . . . 8 ⊢ (((exp‘𝑦) ∈ ℕ ∧ (exp‘𝑧) ∈ ℕ) → ((exp‘𝑦) · (exp‘𝑧)) ∈ ℕ)
19	18	ad2ant2l 782	. . . . . . 7 ⊢ (((𝑦 ∈ ℝ ∧ (exp‘𝑦) ∈ ℕ) ∧ (𝑧 ∈ ℝ ∧ (exp‘𝑧) ∈ ℕ)) → ((exp‘𝑦) · (exp‘𝑧)) ∈ ℕ)
20	17, 19	eqeltrd 2701	. . . . . 6 ⊢ (((𝑦 ∈ ℝ ∧ (exp‘𝑦) ∈ ℕ) ∧ (𝑧 ∈ ℝ ∧ (exp‘𝑧) ∈ ℕ)) → (exp‘(𝑦 + 𝑧)) ∈ ℕ)
21		fveq2 6191	. . . . . . . 8 ⊢ (𝑥 = (𝑦 + 𝑧) → (exp‘𝑥) = (exp‘(𝑦 + 𝑧)))
22	21	eleq1d 2686	. . . . . . 7 ⊢ (𝑥 = (𝑦 + 𝑧) → ((exp‘𝑥) ∈ ℕ ↔ (exp‘(𝑦 + 𝑧)) ∈ ℕ))
23	22	elrab 3363	. . . . . 6 ⊢ ((𝑦 + 𝑧) ∈ {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ} ↔ ((𝑦 + 𝑧) ∈ ℝ ∧ (exp‘(𝑦 + 𝑧)) ∈ ℕ))
24	13, 20, 23	sylanbrc 698	. . . . 5 ⊢ (((𝑦 ∈ ℝ ∧ (exp‘𝑦) ∈ ℕ) ∧ (𝑧 ∈ ℝ ∧ (exp‘𝑧) ∈ ℕ)) → (𝑦 + 𝑧) ∈ {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ})
25	7, 10, 24	syl2anb 496	. . . 4 ⊢ ((𝑦 ∈ {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ} ∧ 𝑧 ∈ {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ}) → (𝑦 + 𝑧) ∈ {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ})
26	25	adantl 482	. . 3 ⊢ ((𝜑 ∧ (𝑦 ∈ {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ} ∧ 𝑧 ∈ {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ})) → (𝑦 + 𝑧) ∈ {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ})
27		efnnfsumcl.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ Fin)
28		efnnfsumcl.2	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ)
29		efnnfsumcl.3	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (exp‘𝐵) ∈ ℕ)
30		fveq2 6191	. . . . . 6 ⊢ (𝑥 = 𝐵 → (exp‘𝑥) = (exp‘𝐵))
31	30	eleq1d 2686	. . . . 5 ⊢ (𝑥 = 𝐵 → ((exp‘𝑥) ∈ ℕ ↔ (exp‘𝐵) ∈ ℕ))
32	31	elrab 3363	. . . 4 ⊢ (𝐵 ∈ {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ} ↔ (𝐵 ∈ ℝ ∧ (exp‘𝐵) ∈ ℕ))
33	28, 29, 32	sylanbrc 698	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ})
34		0re 10040	. . . . 5 ⊢ 0 ∈ ℝ
35		1nn 11031	. . . . 5 ⊢ 1 ∈ ℕ
36		fveq2 6191	. . . . . . . 8 ⊢ (𝑥 = 0 → (exp‘𝑥) = (exp‘0))
37		ef0 14821	. . . . . . . 8 ⊢ (exp‘0) = 1
38	36, 37	syl6eq 2672	. . . . . . 7 ⊢ (𝑥 = 0 → (exp‘𝑥) = 1)
39	38	eleq1d 2686	. . . . . 6 ⊢ (𝑥 = 0 → ((exp‘𝑥) ∈ ℕ ↔ 1 ∈ ℕ))
40	39	elrab 3363	. . . . 5 ⊢ (0 ∈ {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ} ↔ (0 ∈ ℝ ∧ 1 ∈ ℕ))
41	34, 35, 40	mpbir2an 955	. . . 4 ⊢ 0 ∈ {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ}
42	41	a1i 11	. . 3 ⊢ (𝜑 → 0 ∈ {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ})
43	4, 26, 27, 33, 42	fsumcllem 14463	. 2 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ∈ {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ})
44		fveq2 6191	. . . . 5 ⊢ (𝑥 = Σ𝑘 ∈ 𝐴 𝐵 → (exp‘𝑥) = (exp‘Σ𝑘 ∈ 𝐴 𝐵))
45	44	eleq1d 2686	. . . 4 ⊢ (𝑥 = Σ𝑘 ∈ 𝐴 𝐵 → ((exp‘𝑥) ∈ ℕ ↔ (exp‘Σ𝑘 ∈ 𝐴 𝐵) ∈ ℕ))
46	45	elrab 3363	. . 3 ⊢ (Σ𝑘 ∈ 𝐴 𝐵 ∈ {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ} ↔ (Σ𝑘 ∈ 𝐴 𝐵 ∈ ℝ ∧ (exp‘Σ𝑘 ∈ 𝐴 𝐵) ∈ ℕ))
47	46	simprbi 480	. 2 ⊢ (Σ𝑘 ∈ 𝐴 𝐵 ∈ {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ} → (exp‘Σ𝑘 ∈ 𝐴 𝐵) ∈ ℕ)
48	43, 47	syl 17	1 ⊢ (𝜑 → (exp‘Σ𝑘 ∈ 𝐴 𝐵) ∈ ℕ)

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  {crab 2916   ⊆ wss 3574  ‘cfv 5888  (class class class)co 6650  Fincfn 7955  ℂcc 9934  ℝcr 9935  0cc0 9936  1c1 9937   + caddc 9939   · cmul 9941  ℕcn 11020  Σcsu 14416  expce 14792

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-ico 12181  df-fz 12327  df-fzo 12466  df-fl 12593  df-seq 12802  df-exp 12861  df-fac 13061  df-bc 13090  df-hash 13118  df-shft 13807  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-limsup 14202  df-clim 14219  df-rlim 14220  df-sum 14417  df-ef 14798

This theorem is referenced by:  efchtcl  24837  efchpcl  24851