Theorem efnnfsumcl 27091

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 4018	. . . . 5 ⊢ {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ} ⊆ ℝ
2		ax-resscn 11093	. . . . 5 ⊢ ℝ ⊆ ℂ
3	1, 2	sstri 3931	. . . 4 ⊢ {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ} ⊆ ℂ
4	3	a1i 11	. . 3 ⊢ (𝜑 → {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ} ⊆ ℂ)
5		fveq2 6834	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (exp‘𝑥) = (exp‘𝑦))
6	5	eleq1d 2825	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((exp‘𝑥) ∈ ℕ ↔ (exp‘𝑦) ∈ ℕ))
7	6	elrab 3636	. . . . 5 ⊢ (𝑦 ∈ {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ} ↔ (𝑦 ∈ ℝ ∧ (exp‘𝑦) ∈ ℕ))
8		fveq2 6834	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (exp‘𝑥) = (exp‘𝑧))
9	8	eleq1d 2825	. . . . . 6 ⊢ (𝑥 = 𝑧 → ((exp‘𝑥) ∈ ℕ ↔ (exp‘𝑧) ∈ ℕ))
10	9	elrab 3636	. . . . 5 ⊢ (𝑧 ∈ {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ} ↔ (𝑧 ∈ ℝ ∧ (exp‘𝑧) ∈ ℕ))
11		fveq2 6834	. . . . . . 7 ⊢ (𝑥 = (𝑦 + 𝑧) → (exp‘𝑥) = (exp‘(𝑦 + 𝑧)))
12	11	eleq1d 2825	. . . . . 6 ⊢ (𝑥 = (𝑦 + 𝑧) → ((exp‘𝑥) ∈ ℕ ↔ (exp‘(𝑦 + 𝑧)) ∈ ℕ))
13		simpll 772	. . . . . . 7 ⊢ (((𝑦 ∈ ℝ ∧ (exp‘𝑦) ∈ ℕ) ∧ (𝑧 ∈ ℝ ∧ (exp‘𝑧) ∈ ℕ)) → 𝑦 ∈ ℝ)
14		simprl 776	. . . . . . 7 ⊢ (((𝑦 ∈ ℝ ∧ (exp‘𝑦) ∈ ℕ) ∧ (𝑧 ∈ ℝ ∧ (exp‘𝑧) ∈ ℕ)) → 𝑧 ∈ ℝ)
15	13, 14	readdcld 11172	. . . . . 6 ⊢ (((𝑦 ∈ ℝ ∧ (exp‘𝑦) ∈ ℕ) ∧ (𝑧 ∈ ℝ ∧ (exp‘𝑧) ∈ ℕ)) → (𝑦 + 𝑧) ∈ ℝ)
16	13	recnd 11171	. . . . . . . 8 ⊢ (((𝑦 ∈ ℝ ∧ (exp‘𝑦) ∈ ℕ) ∧ (𝑧 ∈ ℝ ∧ (exp‘𝑧) ∈ ℕ)) → 𝑦 ∈ ℂ)
17	14	recnd 11171	. . . . . . . 8 ⊢ (((𝑦 ∈ ℝ ∧ (exp‘𝑦) ∈ ℕ) ∧ (𝑧 ∈ ℝ ∧ (exp‘𝑧) ∈ ℕ)) → 𝑧 ∈ ℂ)
18		efadd 16057	. . . . . . . 8 ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (exp‘(𝑦 + 𝑧)) = ((exp‘𝑦) · (exp‘𝑧)))
19	16, 17, 18	syl2anc 590	. . . . . . 7 ⊢ (((𝑦 ∈ ℝ ∧ (exp‘𝑦) ∈ ℕ) ∧ (𝑧 ∈ ℝ ∧ (exp‘𝑧) ∈ ℕ)) → (exp‘(𝑦 + 𝑧)) = ((exp‘𝑦) · (exp‘𝑧)))
20		nnmulcl 12196	. . . . . . . 8 ⊢ (((exp‘𝑦) ∈ ℕ ∧ (exp‘𝑧) ∈ ℕ) → ((exp‘𝑦) · (exp‘𝑧)) ∈ ℕ)
21	20	ad2ant2l 752	. . . . . . 7 ⊢ (((𝑦 ∈ ℝ ∧ (exp‘𝑦) ∈ ℕ) ∧ (𝑧 ∈ ℝ ∧ (exp‘𝑧) ∈ ℕ)) → ((exp‘𝑦) · (exp‘𝑧)) ∈ ℕ)
22	19, 21	eqeltrd 2840	. . . . . 6 ⊢ (((𝑦 ∈ ℝ ∧ (exp‘𝑦) ∈ ℕ) ∧ (𝑧 ∈ ℝ ∧ (exp‘𝑧) ∈ ℕ)) → (exp‘(𝑦 + 𝑧)) ∈ ℕ)
23	12, 15, 22	elrabd 3638	. . . . 5 ⊢ (((𝑦 ∈ ℝ ∧ (exp‘𝑦) ∈ ℕ) ∧ (𝑧 ∈ ℝ ∧ (exp‘𝑧) ∈ ℕ)) → (𝑦 + 𝑧) ∈ {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ})
24	7, 10, 23	syl2anb 604	. . . 4 ⊢ ((𝑦 ∈ {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ} ∧ 𝑧 ∈ {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ}) → (𝑦 + 𝑧) ∈ {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ})
25	24	adantl 482	. . 3 ⊢ ((𝜑 ∧ (𝑦 ∈ {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ} ∧ 𝑧 ∈ {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ})) → (𝑦 + 𝑧) ∈ {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ})
26		efnnfsumcl.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ Fin)
27		fveq2 6834	. . . . 5 ⊢ (𝑥 = 𝐵 → (exp‘𝑥) = (exp‘𝐵))
28	27	eleq1d 2825	. . . 4 ⊢ (𝑥 = 𝐵 → ((exp‘𝑥) ∈ ℕ ↔ (exp‘𝐵) ∈ ℕ))
29		efnnfsumcl.2	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ)
30		efnnfsumcl.3	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (exp‘𝐵) ∈ ℕ)
31	28, 29, 30	elrabd 3638	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ})
32		0re 11144	. . . . 5 ⊢ 0 ∈ ℝ
33		1nn 12183	. . . . 5 ⊢ 1 ∈ ℕ
34		fveq2 6834	. . . . . . . 8 ⊢ (𝑥 = 0 → (exp‘𝑥) = (exp‘0))
35		ef0 16054	. . . . . . . 8 ⊢ (exp‘0) = 1
36	34, 35	eqtrdi 2791	. . . . . . 7 ⊢ (𝑥 = 0 → (exp‘𝑥) = 1)
37	36	eleq1d 2825	. . . . . 6 ⊢ (𝑥 = 0 → ((exp‘𝑥) ∈ ℕ ↔ 1 ∈ ℕ))
38	37	elrab 3636	. . . . 5 ⊢ (0 ∈ {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ} ↔ (0 ∈ ℝ ∧ 1 ∈ ℕ))
39	32, 33, 38	mpbir2an 717	. . . 4 ⊢ 0 ∈ {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ}
40	39	a1i 11	. . 3 ⊢ (𝜑 → 0 ∈ {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ})
41	4, 25, 26, 31, 40	fsumcllem 15692	. 2 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ∈ {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ})
42		fveq2 6834	. . . . 5 ⊢ (𝑥 = Σ𝑘 ∈ 𝐴 𝐵 → (exp‘𝑥) = (exp‘Σ𝑘 ∈ 𝐴 𝐵))
43	42	eleq1d 2825	. . . 4 ⊢ (𝑥 = Σ𝑘 ∈ 𝐴 𝐵 → ((exp‘𝑥) ∈ ℕ ↔ (exp‘Σ𝑘 ∈ 𝐴 𝐵) ∈ ℕ))
44	43	elrab 3636	. . 3 ⊢ (Σ𝑘 ∈ 𝐴 𝐵 ∈ {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ} ↔ (Σ𝑘 ∈ 𝐴 𝐵 ∈ ℝ ∧ (exp‘Σ𝑘 ∈ 𝐴 𝐵) ∈ ℕ))
45	44	simprbi 498	. 2 ⊢ (Σ𝑘 ∈ 𝐴 𝐵 ∈ {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ} → (exp‘Σ𝑘 ∈ 𝐴 𝐵) ∈ ℕ)
46	41, 45	syl 17	1 ⊢ (𝜑 → (exp‘Σ𝑘 ∈ 𝐴 𝐵) ∈ ℕ)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  {crab 3392   ⊆ wss 3890  ‘cfv 6492  (class class class)co 7363  Fincfn 8890  ℂcc 11034  ℝcr 11035  0cc0 11036  1c1 11037   + caddc 11039   · cmul 11041  ℕcn 12172  Σcsu 15646  expce 16024

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 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685  ax-inf2 9560  ax-cnex 11092  ax-resscn 11093  ax-1cn 11094  ax-icn 11095  ax-addcl 11096  ax-addrcl 11097  ax-mulcl 11098  ax-mulrcl 11099  ax-mulcom 11100  ax-addass 11101  ax-mulass 11102  ax-distr 11103  ax-i2m1 11104  ax-1ne0 11105  ax-1rid 11106  ax-rnegex 11107  ax-rrecex 11108  ax-cnre 11109  ax-pre-lttri 11110  ax-pre-lttrn 11111  ax-pre-ltadd 11112  ax-pre-mulgt0 11113  ax-pre-sup 11114

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-nel 3040  df-ral 3055  df-rex 3065  df-rmo 3345  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-int 4885  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-se 5579  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-isom 6501  df-riota 7320  df-ov 7366  df-oprab 7367  df-mpo 7368  df-om 7814  df-1st 7938  df-2nd 7939  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-rdg 8346  df-1o 8402  df-er 8640  df-pm 8773  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-sup 9352  df-inf 9353  df-oi 9422  df-card 9861  df-pnf 11179  df-mnf 11180  df-xr 11181  df-ltxr 11182  df-le 11183  df-sub 11377  df-neg 11378  df-div 11806  df-nn 12173  df-2 12242  df-3 12243  df-n0 12436  df-z 12523  df-uz 12787  df-rp 12941  df-ico 13302  df-fz 13460  df-fzo 13607  df-fl 13749  df-seq 13962  df-exp 14022  df-fac 14234  df-bc 14263  df-hash 14291  df-shft 15027  df-cj 15059  df-re 15060  df-im 15061  df-sqrt 15195  df-abs 15196  df-limsup 15431  df-clim 15448  df-rlim 15449  df-sum 15647  df-ef 16030

This theorem is referenced by:  efchtcl  27099  efchpcl  27113