Theorem efnnfsumcl 27164

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 4103	. . . . 5 ⊢ {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ} ⊆ ℝ
2		ax-resscn 11241	. . . . 5 ⊢ ℝ ⊆ ℂ
3	1, 2	sstri 4018	. . . 4 ⊢ {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ} ⊆ ℂ
4	3	a1i 11	. . 3 ⊢ (𝜑 → {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ} ⊆ ℂ)
5		fveq2 6920	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (exp‘𝑥) = (exp‘𝑦))
6	5	eleq1d 2829	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((exp‘𝑥) ∈ ℕ ↔ (exp‘𝑦) ∈ ℕ))
7	6	elrab 3708	. . . . 5 ⊢ (𝑦 ∈ {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ} ↔ (𝑦 ∈ ℝ ∧ (exp‘𝑦) ∈ ℕ))
8		fveq2 6920	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (exp‘𝑥) = (exp‘𝑧))
9	8	eleq1d 2829	. . . . . 6 ⊢ (𝑥 = 𝑧 → ((exp‘𝑥) ∈ ℕ ↔ (exp‘𝑧) ∈ ℕ))
10	9	elrab 3708	. . . . 5 ⊢ (𝑧 ∈ {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ} ↔ (𝑧 ∈ ℝ ∧ (exp‘𝑧) ∈ ℕ))
11		fveq2 6920	. . . . . . 7 ⊢ (𝑥 = (𝑦 + 𝑧) → (exp‘𝑥) = (exp‘(𝑦 + 𝑧)))
12	11	eleq1d 2829	. . . . . 6 ⊢ (𝑥 = (𝑦 + 𝑧) → ((exp‘𝑥) ∈ ℕ ↔ (exp‘(𝑦 + 𝑧)) ∈ ℕ))
13		simpll 766	. . . . . . 7 ⊢ (((𝑦 ∈ ℝ ∧ (exp‘𝑦) ∈ ℕ) ∧ (𝑧 ∈ ℝ ∧ (exp‘𝑧) ∈ ℕ)) → 𝑦 ∈ ℝ)
14		simprl 770	. . . . . . 7 ⊢ (((𝑦 ∈ ℝ ∧ (exp‘𝑦) ∈ ℕ) ∧ (𝑧 ∈ ℝ ∧ (exp‘𝑧) ∈ ℕ)) → 𝑧 ∈ ℝ)
15	13, 14	readdcld 11319	. . . . . 6 ⊢ (((𝑦 ∈ ℝ ∧ (exp‘𝑦) ∈ ℕ) ∧ (𝑧 ∈ ℝ ∧ (exp‘𝑧) ∈ ℕ)) → (𝑦 + 𝑧) ∈ ℝ)
16	13	recnd 11318	. . . . . . . 8 ⊢ (((𝑦 ∈ ℝ ∧ (exp‘𝑦) ∈ ℕ) ∧ (𝑧 ∈ ℝ ∧ (exp‘𝑧) ∈ ℕ)) → 𝑦 ∈ ℂ)
17	14	recnd 11318	. . . . . . . 8 ⊢ (((𝑦 ∈ ℝ ∧ (exp‘𝑦) ∈ ℕ) ∧ (𝑧 ∈ ℝ ∧ (exp‘𝑧) ∈ ℕ)) → 𝑧 ∈ ℂ)
18		efadd 16142	. . . . . . . 8 ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (exp‘(𝑦 + 𝑧)) = ((exp‘𝑦) · (exp‘𝑧)))
19	16, 17, 18	syl2anc 583	. . . . . . 7 ⊢ (((𝑦 ∈ ℝ ∧ (exp‘𝑦) ∈ ℕ) ∧ (𝑧 ∈ ℝ ∧ (exp‘𝑧) ∈ ℕ)) → (exp‘(𝑦 + 𝑧)) = ((exp‘𝑦) · (exp‘𝑧)))
20		nnmulcl 12317	. . . . . . . 8 ⊢ (((exp‘𝑦) ∈ ℕ ∧ (exp‘𝑧) ∈ ℕ) → ((exp‘𝑦) · (exp‘𝑧)) ∈ ℕ)
21	20	ad2ant2l 745	. . . . . . 7 ⊢ (((𝑦 ∈ ℝ ∧ (exp‘𝑦) ∈ ℕ) ∧ (𝑧 ∈ ℝ ∧ (exp‘𝑧) ∈ ℕ)) → ((exp‘𝑦) · (exp‘𝑧)) ∈ ℕ)
22	19, 21	eqeltrd 2844	. . . . . 6 ⊢ (((𝑦 ∈ ℝ ∧ (exp‘𝑦) ∈ ℕ) ∧ (𝑧 ∈ ℝ ∧ (exp‘𝑧) ∈ ℕ)) → (exp‘(𝑦 + 𝑧)) ∈ ℕ)
23	12, 15, 22	elrabd 3710	. . . . 5 ⊢ (((𝑦 ∈ ℝ ∧ (exp‘𝑦) ∈ ℕ) ∧ (𝑧 ∈ ℝ ∧ (exp‘𝑧) ∈ ℕ)) → (𝑦 + 𝑧) ∈ {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ})
24	7, 10, 23	syl2anb 597	. . . 4 ⊢ ((𝑦 ∈ {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ} ∧ 𝑧 ∈ {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ}) → (𝑦 + 𝑧) ∈ {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ})
25	24	adantl 481	. . 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 6920	. . . . 5 ⊢ (𝑥 = 𝐵 → (exp‘𝑥) = (exp‘𝐵))
28	27	eleq1d 2829	. . . 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 3710	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ})
32		0re 11292	. . . . 5 ⊢ 0 ∈ ℝ
33		1nn 12304	. . . . 5 ⊢ 1 ∈ ℕ
34		fveq2 6920	. . . . . . . 8 ⊢ (𝑥 = 0 → (exp‘𝑥) = (exp‘0))
35		ef0 16139	. . . . . . . 8 ⊢ (exp‘0) = 1
36	34, 35	eqtrdi 2796	. . . . . . 7 ⊢ (𝑥 = 0 → (exp‘𝑥) = 1)
37	36	eleq1d 2829	. . . . . 6 ⊢ (𝑥 = 0 → ((exp‘𝑥) ∈ ℕ ↔ 1 ∈ ℕ))
38	37	elrab 3708	. . . . 5 ⊢ (0 ∈ {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ} ↔ (0 ∈ ℝ ∧ 1 ∈ ℕ))
39	32, 33, 38	mpbir2an 710	. . . 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 15780	. 2 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ∈ {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ})
42		fveq2 6920	. . . . 5 ⊢ (𝑥 = Σ𝑘 ∈ 𝐴 𝐵 → (exp‘𝑥) = (exp‘Σ𝑘 ∈ 𝐴 𝐵))
43	42	eleq1d 2829	. . . 4 ⊢ (𝑥 = Σ𝑘 ∈ 𝐴 𝐵 → ((exp‘𝑥) ∈ ℕ ↔ (exp‘Σ𝑘 ∈ 𝐴 𝐵) ∈ ℕ))
44	43	elrab 3708	. . 3 ⊢ (Σ𝑘 ∈ 𝐴 𝐵 ∈ {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ} ↔ (Σ𝑘 ∈ 𝐴 𝐵 ∈ ℝ ∧ (exp‘Σ𝑘 ∈ 𝐴 𝐵) ∈ ℕ))
45	44	simprbi 496	. 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 395   = wceq 1537   ∈ wcel 2108  {crab 3443   ⊆ wss 3976  ‘cfv 6573  (class class class)co 7448  Fincfn 9003  ℂcc 11182  ℝcr 11183  0cc0 11184  1c1 11185   + caddc 11187   · cmul 11189  ℕcn 12293  Σcsu 15734  expce 16109

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-inf2 9710  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261  ax-pre-sup 11262

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-isom 6582  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-er 8763  df-pm 8887  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-sup 9511  df-inf 9512  df-oi 9579  df-card 10008  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-div 11948  df-nn 12294  df-2 12356  df-3 12357  df-n0 12554  df-z 12640  df-uz 12904  df-rp 13058  df-ico 13413  df-fz 13568  df-fzo 13712  df-fl 13843  df-seq 14053  df-exp 14113  df-fac 14323  df-bc 14352  df-hash 14380  df-shft 15116  df-cj 15148  df-re 15149  df-im 15150  df-sqrt 15284  df-abs 15285  df-limsup 15517  df-clim 15534  df-rlim 15535  df-sum 15735  df-ef 16115

This theorem is referenced by:  efchtcl  27172  efchpcl  27186