Theorem efnnfsumcl 24629

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 3650	. . . . 5 ⊢ {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ} ⊆ ℝ
2		ax-resscn 9872	. . . . 5 ⊢ ℝ ⊆ ℂ
3	1, 2	sstri 3577	. . . 4 ⊢ {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ} ⊆ ℂ
4	3	a1i 11	. . 3 ⊢ (𝜑 → {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ} ⊆ ℂ)
5		fveq2 6103	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (exp‘𝑥) = (exp‘𝑦))
6	5	eleq1d 2672	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((exp‘𝑥) ∈ ℕ ↔ (exp‘𝑦) ∈ ℕ))
7	6	elrab 3331	. . . . 5 ⊢ (𝑦 ∈ {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ} ↔ (𝑦 ∈ ℝ ∧ (exp‘𝑦) ∈ ℕ))
8		fveq2 6103	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (exp‘𝑥) = (exp‘𝑧))
9	8	eleq1d 2672	. . . . . 6 ⊢ (𝑥 = 𝑧 → ((exp‘𝑥) ∈ ℕ ↔ (exp‘𝑧) ∈ ℕ))
10	9	elrab 3331	. . . . 5 ⊢ (𝑧 ∈ {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ} ↔ (𝑧 ∈ ℝ ∧ (exp‘𝑧) ∈ ℕ))
11		simpll 786	. . . . . . 7 ⊢ (((𝑦 ∈ ℝ ∧ (exp‘𝑦) ∈ ℕ) ∧ (𝑧 ∈ ℝ ∧ (exp‘𝑧) ∈ ℕ)) → 𝑦 ∈ ℝ)
12		simprl 790	. . . . . . 7 ⊢ (((𝑦 ∈ ℝ ∧ (exp‘𝑦) ∈ ℕ) ∧ (𝑧 ∈ ℝ ∧ (exp‘𝑧) ∈ ℕ)) → 𝑧 ∈ ℝ)
13	11, 12	readdcld 9948	. . . . . 6 ⊢ (((𝑦 ∈ ℝ ∧ (exp‘𝑦) ∈ ℕ) ∧ (𝑧 ∈ ℝ ∧ (exp‘𝑧) ∈ ℕ)) → (𝑦 + 𝑧) ∈ ℝ)
14	11	recnd 9947	. . . . . . . 8 ⊢ (((𝑦 ∈ ℝ ∧ (exp‘𝑦) ∈ ℕ) ∧ (𝑧 ∈ ℝ ∧ (exp‘𝑧) ∈ ℕ)) → 𝑦 ∈ ℂ)
15	12	recnd 9947	. . . . . . . 8 ⊢ (((𝑦 ∈ ℝ ∧ (exp‘𝑦) ∈ ℕ) ∧ (𝑧 ∈ ℝ ∧ (exp‘𝑧) ∈ ℕ)) → 𝑧 ∈ ℂ)
16		efadd 14663	. . . . . . . 8 ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (exp‘(𝑦 + 𝑧)) = ((exp‘𝑦) · (exp‘𝑧)))
17	14, 15, 16	syl2anc 691	. . . . . . 7 ⊢ (((𝑦 ∈ ℝ ∧ (exp‘𝑦) ∈ ℕ) ∧ (𝑧 ∈ ℝ ∧ (exp‘𝑧) ∈ ℕ)) → (exp‘(𝑦 + 𝑧)) = ((exp‘𝑦) · (exp‘𝑧)))
18		nnmulcl 10920	. . . . . . . 8 ⊢ (((exp‘𝑦) ∈ ℕ ∧ (exp‘𝑧) ∈ ℕ) → ((exp‘𝑦) · (exp‘𝑧)) ∈ ℕ)
19	18	ad2ant2l 778	. . . . . . 7 ⊢ (((𝑦 ∈ ℝ ∧ (exp‘𝑦) ∈ ℕ) ∧ (𝑧 ∈ ℝ ∧ (exp‘𝑧) ∈ ℕ)) → ((exp‘𝑦) · (exp‘𝑧)) ∈ ℕ)
20	17, 19	eqeltrd 2688	. . . . . 6 ⊢ (((𝑦 ∈ ℝ ∧ (exp‘𝑦) ∈ ℕ) ∧ (𝑧 ∈ ℝ ∧ (exp‘𝑧) ∈ ℕ)) → (exp‘(𝑦 + 𝑧)) ∈ ℕ)
21		fveq2 6103	. . . . . . . 8 ⊢ (𝑥 = (𝑦 + 𝑧) → (exp‘𝑥) = (exp‘(𝑦 + 𝑧)))
22	21	eleq1d 2672	. . . . . . 7 ⊢ (𝑥 = (𝑦 + 𝑧) → ((exp‘𝑥) ∈ ℕ ↔ (exp‘(𝑦 + 𝑧)) ∈ ℕ))
23	22	elrab 3331	. . . . . 6 ⊢ ((𝑦 + 𝑧) ∈ {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ} ↔ ((𝑦 + 𝑧) ∈ ℝ ∧ (exp‘(𝑦 + 𝑧)) ∈ ℕ))
24	13, 20, 23	sylanbrc 695	. . . . 5 ⊢ (((𝑦 ∈ ℝ ∧ (exp‘𝑦) ∈ ℕ) ∧ (𝑧 ∈ ℝ ∧ (exp‘𝑧) ∈ ℕ)) → (𝑦 + 𝑧) ∈ {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ})
25	7, 10, 24	syl2anb 495	. . . 4 ⊢ ((𝑦 ∈ {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ} ∧ 𝑧 ∈ {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ}) → (𝑦 + 𝑧) ∈ {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ})
26	25	adantl 481	. . 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 6103	. . . . . 6 ⊢ (𝑥 = 𝐵 → (exp‘𝑥) = (exp‘𝐵))
31	30	eleq1d 2672	. . . . 5 ⊢ (𝑥 = 𝐵 → ((exp‘𝑥) ∈ ℕ ↔ (exp‘𝐵) ∈ ℕ))
32	31	elrab 3331	. . . 4 ⊢ (𝐵 ∈ {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ} ↔ (𝐵 ∈ ℝ ∧ (exp‘𝐵) ∈ ℕ))
33	28, 29, 32	sylanbrc 695	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ})
34		0re 9919	. . . . 5 ⊢ 0 ∈ ℝ
35		1nn 10908	. . . . 5 ⊢ 1 ∈ ℕ
36		fveq2 6103	. . . . . . . 8 ⊢ (𝑥 = 0 → (exp‘𝑥) = (exp‘0))
37		ef0 14660	. . . . . . . 8 ⊢ (exp‘0) = 1
38	36, 37	syl6eq 2660	. . . . . . 7 ⊢ (𝑥 = 0 → (exp‘𝑥) = 1)
39	38	eleq1d 2672	. . . . . 6 ⊢ (𝑥 = 0 → ((exp‘𝑥) ∈ ℕ ↔ 1 ∈ ℕ))
40	39	elrab 3331	. . . . 5 ⊢ (0 ∈ {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ} ↔ (0 ∈ ℝ ∧ 1 ∈ ℕ))
41	34, 35, 40	mpbir2an 957	. . . 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 14310	. 2 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ∈ {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ})
44		fveq2 6103	. . . . 5 ⊢ (𝑥 = Σ𝑘 ∈ 𝐴 𝐵 → (exp‘𝑥) = (exp‘Σ𝑘 ∈ 𝐴 𝐵))
45	44	eleq1d 2672	. . . 4 ⊢ (𝑥 = Σ𝑘 ∈ 𝐴 𝐵 → ((exp‘𝑥) ∈ ℕ ↔ (exp‘Σ𝑘 ∈ 𝐴 𝐵) ∈ ℕ))
46	45	elrab 3331	. . 3 ⊢ (Σ𝑘 ∈ 𝐴 𝐵 ∈ {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ} ↔ (Σ𝑘 ∈ 𝐴 𝐵 ∈ ℝ ∧ (exp‘Σ𝑘 ∈ 𝐴 𝐵) ∈ ℕ))
47	46	simprbi 479	. 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 383   = wceq 1475   ∈ wcel 1977  {crab 2900   ⊆ wss 3540  ‘cfv 5804  (class class class)co 6549  Fincfn 7841  ℂcc 9813  ℝcr 9814  0cc0 9815  1c1 9816   + caddc 9818   · cmul 9820  ℕcn 10897  Σcsu 14264  expce 14631

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893  ax-addf 9894  ax-mulf 9895

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-rp 11709  df-ico 12052  df-fz 12198  df-fzo 12335  df-fl 12455  df-seq 12664  df-exp 12723  df-fac 12923  df-bc 12952  df-hash 12980  df-shft 13655  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-limsup 14050  df-clim 14067  df-rlim 14068  df-sum 14265  df-ef 14637

This theorem is referenced by:  efchtcl  24637  efchpcl  24651