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Theorem efnnfsumcl 25688
Description: Finite sum closure in the log-integers. (Contributed by Mario Carneiro, 7-Apr-2016.)
Hypotheses
Ref Expression
efnnfsumcl.1 (𝜑𝐴 ∈ Fin)
efnnfsumcl.2 ((𝜑𝑘𝐴) → 𝐵 ∈ ℝ)
efnnfsumcl.3 ((𝜑𝑘𝐴) → (exp‘𝐵) ∈ ℕ)
Assertion
Ref Expression
efnnfsumcl (𝜑 → (exp‘Σ𝑘𝐴 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.
StepHypRef Expression
1 ssrab2 4007 . . . . 5 {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ} ⊆ ℝ
2 ax-resscn 10583 . . . . 5 ℝ ⊆ ℂ
31, 2sstri 3924 . . . 4 {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ} ⊆ ℂ
43a1i 11 . . 3 (𝜑 → {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ} ⊆ ℂ)
5 fveq2 6645 . . . . . . 7 (𝑥 = 𝑦 → (exp‘𝑥) = (exp‘𝑦))
65eleq1d 2874 . . . . . 6 (𝑥 = 𝑦 → ((exp‘𝑥) ∈ ℕ ↔ (exp‘𝑦) ∈ ℕ))
76elrab 3628 . . . . 5 (𝑦 ∈ {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ} ↔ (𝑦 ∈ ℝ ∧ (exp‘𝑦) ∈ ℕ))
8 fveq2 6645 . . . . . . 7 (𝑥 = 𝑧 → (exp‘𝑥) = (exp‘𝑧))
98eleq1d 2874 . . . . . 6 (𝑥 = 𝑧 → ((exp‘𝑥) ∈ ℕ ↔ (exp‘𝑧) ∈ ℕ))
109elrab 3628 . . . . 5 (𝑧 ∈ {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ} ↔ (𝑧 ∈ ℝ ∧ (exp‘𝑧) ∈ ℕ))
11 fveq2 6645 . . . . . . 7 (𝑥 = (𝑦 + 𝑧) → (exp‘𝑥) = (exp‘(𝑦 + 𝑧)))
1211eleq1d 2874 . . . . . 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)
1513, 14readdcld 10659 . . . . . 6 (((𝑦 ∈ ℝ ∧ (exp‘𝑦) ∈ ℕ) ∧ (𝑧 ∈ ℝ ∧ (exp‘𝑧) ∈ ℕ)) → (𝑦 + 𝑧) ∈ ℝ)
1613recnd 10658 . . . . . . . 8 (((𝑦 ∈ ℝ ∧ (exp‘𝑦) ∈ ℕ) ∧ (𝑧 ∈ ℝ ∧ (exp‘𝑧) ∈ ℕ)) → 𝑦 ∈ ℂ)
1714recnd 10658 . . . . . . . 8 (((𝑦 ∈ ℝ ∧ (exp‘𝑦) ∈ ℕ) ∧ (𝑧 ∈ ℝ ∧ (exp‘𝑧) ∈ ℕ)) → 𝑧 ∈ ℂ)
18 efadd 15439 . . . . . . . 8 ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (exp‘(𝑦 + 𝑧)) = ((exp‘𝑦) · (exp‘𝑧)))
1916, 17, 18syl2anc 587 . . . . . . 7 (((𝑦 ∈ ℝ ∧ (exp‘𝑦) ∈ ℕ) ∧ (𝑧 ∈ ℝ ∧ (exp‘𝑧) ∈ ℕ)) → (exp‘(𝑦 + 𝑧)) = ((exp‘𝑦) · (exp‘𝑧)))
20 nnmulcl 11649 . . . . . . . 8 (((exp‘𝑦) ∈ ℕ ∧ (exp‘𝑧) ∈ ℕ) → ((exp‘𝑦) · (exp‘𝑧)) ∈ ℕ)
2120ad2ant2l 745 . . . . . . 7 (((𝑦 ∈ ℝ ∧ (exp‘𝑦) ∈ ℕ) ∧ (𝑧 ∈ ℝ ∧ (exp‘𝑧) ∈ ℕ)) → ((exp‘𝑦) · (exp‘𝑧)) ∈ ℕ)
2219, 21eqeltrd 2890 . . . . . 6 (((𝑦 ∈ ℝ ∧ (exp‘𝑦) ∈ ℕ) ∧ (𝑧 ∈ ℝ ∧ (exp‘𝑧) ∈ ℕ)) → (exp‘(𝑦 + 𝑧)) ∈ ℕ)
2312, 15, 22elrabd 3630 . . . . 5 (((𝑦 ∈ ℝ ∧ (exp‘𝑦) ∈ ℕ) ∧ (𝑧 ∈ ℝ ∧ (exp‘𝑧) ∈ ℕ)) → (𝑦 + 𝑧) ∈ {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ})
247, 10, 23syl2anb 600 . . . 4 ((𝑦 ∈ {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ} ∧ 𝑧 ∈ {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ}) → (𝑦 + 𝑧) ∈ {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ})
2524adantl 485 . . 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 (𝜑𝐴 ∈ Fin)
27 fveq2 6645 . . . . 5 (𝑥 = 𝐵 → (exp‘𝑥) = (exp‘𝐵))
2827eleq1d 2874 . . . 4 (𝑥 = 𝐵 → ((exp‘𝑥) ∈ ℕ ↔ (exp‘𝐵) ∈ ℕ))
29 efnnfsumcl.2 . . . 4 ((𝜑𝑘𝐴) → 𝐵 ∈ ℝ)
30 efnnfsumcl.3 . . . 4 ((𝜑𝑘𝐴) → (exp‘𝐵) ∈ ℕ)
3128, 29, 30elrabd 3630 . . 3 ((𝜑𝑘𝐴) → 𝐵 ∈ {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ})
32 0re 10632 . . . . 5 0 ∈ ℝ
33 1nn 11636 . . . . 5 1 ∈ ℕ
34 fveq2 6645 . . . . . . . 8 (𝑥 = 0 → (exp‘𝑥) = (exp‘0))
35 ef0 15436 . . . . . . . 8 (exp‘0) = 1
3634, 35eqtrdi 2849 . . . . . . 7 (𝑥 = 0 → (exp‘𝑥) = 1)
3736eleq1d 2874 . . . . . 6 (𝑥 = 0 → ((exp‘𝑥) ∈ ℕ ↔ 1 ∈ ℕ))
3837elrab 3628 . . . . 5 (0 ∈ {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ} ↔ (0 ∈ ℝ ∧ 1 ∈ ℕ))
3932, 33, 38mpbir2an 710 . . . 4 0 ∈ {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ}
4039a1i 11 . . 3 (𝜑 → 0 ∈ {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ})
414, 25, 26, 31, 40fsumcllem 15081 . 2 (𝜑 → Σ𝑘𝐴 𝐵 ∈ {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ})
42 fveq2 6645 . . . . 5 (𝑥 = Σ𝑘𝐴 𝐵 → (exp‘𝑥) = (exp‘Σ𝑘𝐴 𝐵))
4342eleq1d 2874 . . . 4 (𝑥 = Σ𝑘𝐴 𝐵 → ((exp‘𝑥) ∈ ℕ ↔ (exp‘Σ𝑘𝐴 𝐵) ∈ ℕ))
4443elrab 3628 . . 3 𝑘𝐴 𝐵 ∈ {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ} ↔ (Σ𝑘𝐴 𝐵 ∈ ℝ ∧ (exp‘Σ𝑘𝐴 𝐵) ∈ ℕ))
4544simprbi 500 . 2 𝑘𝐴 𝐵 ∈ {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ} → (exp‘Σ𝑘𝐴 𝐵) ∈ ℕ)
4641, 45syl 17 1 (𝜑 → (exp‘Σ𝑘𝐴 𝐵) ∈ ℕ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2111  {crab 3110  wss 3881  cfv 6324  (class class class)co 7135  Fincfn 8492  cc 10524  cr 10525  0cc0 10526  1c1 10527   + caddc 10529   · cmul 10531  cn 11625  Σcsu 15034  expce 15407
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-inf2 9088  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603  ax-pre-sup 10604  ax-addf 10605  ax-mulf 10606
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-se 5479  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-isom 6333  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-oadd 8089  df-er 8272  df-pm 8392  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-sup 8890  df-inf 8891  df-oi 8958  df-card 9352  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-div 11287  df-nn 11626  df-2 11688  df-3 11689  df-n0 11886  df-z 11970  df-uz 12232  df-rp 12378  df-ico 12732  df-fz 12886  df-fzo 13029  df-fl 13157  df-seq 13365  df-exp 13426  df-fac 13630  df-bc 13659  df-hash 13687  df-shft 14418  df-cj 14450  df-re 14451  df-im 14452  df-sqrt 14586  df-abs 14587  df-limsup 14820  df-clim 14837  df-rlim 14838  df-sum 15035  df-ef 15413
This theorem is referenced by:  efchtcl  25696  efchpcl  25710
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