MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  efnnfsumcl Structured version   Visualization version   GIF version

Theorem efnnfsumcl 26452
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 4037 . . . . 5 {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ} ⊆ ℝ
2 ax-resscn 11108 . . . . 5 ℝ ⊆ ℂ
31, 2sstri 3953 . . . 4 {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ} ⊆ ℂ
43a1i 11 . . 3 (𝜑 → {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ} ⊆ ℂ)
5 fveq2 6842 . . . . . . 7 (𝑥 = 𝑦 → (exp‘𝑥) = (exp‘𝑦))
65eleq1d 2822 . . . . . 6 (𝑥 = 𝑦 → ((exp‘𝑥) ∈ ℕ ↔ (exp‘𝑦) ∈ ℕ))
76elrab 3645 . . . . 5 (𝑦 ∈ {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ} ↔ (𝑦 ∈ ℝ ∧ (exp‘𝑦) ∈ ℕ))
8 fveq2 6842 . . . . . . 7 (𝑥 = 𝑧 → (exp‘𝑥) = (exp‘𝑧))
98eleq1d 2822 . . . . . 6 (𝑥 = 𝑧 → ((exp‘𝑥) ∈ ℕ ↔ (exp‘𝑧) ∈ ℕ))
109elrab 3645 . . . . 5 (𝑧 ∈ {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ} ↔ (𝑧 ∈ ℝ ∧ (exp‘𝑧) ∈ ℕ))
11 fveq2 6842 . . . . . . 7 (𝑥 = (𝑦 + 𝑧) → (exp‘𝑥) = (exp‘(𝑦 + 𝑧)))
1211eleq1d 2822 . . . . . 6 (𝑥 = (𝑦 + 𝑧) → ((exp‘𝑥) ∈ ℕ ↔ (exp‘(𝑦 + 𝑧)) ∈ ℕ))
13 simpll 765 . . . . . . 7 (((𝑦 ∈ ℝ ∧ (exp‘𝑦) ∈ ℕ) ∧ (𝑧 ∈ ℝ ∧ (exp‘𝑧) ∈ ℕ)) → 𝑦 ∈ ℝ)
14 simprl 769 . . . . . . 7 (((𝑦 ∈ ℝ ∧ (exp‘𝑦) ∈ ℕ) ∧ (𝑧 ∈ ℝ ∧ (exp‘𝑧) ∈ ℕ)) → 𝑧 ∈ ℝ)
1513, 14readdcld 11184 . . . . . 6 (((𝑦 ∈ ℝ ∧ (exp‘𝑦) ∈ ℕ) ∧ (𝑧 ∈ ℝ ∧ (exp‘𝑧) ∈ ℕ)) → (𝑦 + 𝑧) ∈ ℝ)
1613recnd 11183 . . . . . . . 8 (((𝑦 ∈ ℝ ∧ (exp‘𝑦) ∈ ℕ) ∧ (𝑧 ∈ ℝ ∧ (exp‘𝑧) ∈ ℕ)) → 𝑦 ∈ ℂ)
1714recnd 11183 . . . . . . . 8 (((𝑦 ∈ ℝ ∧ (exp‘𝑦) ∈ ℕ) ∧ (𝑧 ∈ ℝ ∧ (exp‘𝑧) ∈ ℕ)) → 𝑧 ∈ ℂ)
18 efadd 15976 . . . . . . . 8 ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (exp‘(𝑦 + 𝑧)) = ((exp‘𝑦) · (exp‘𝑧)))
1916, 17, 18syl2anc 584 . . . . . . 7 (((𝑦 ∈ ℝ ∧ (exp‘𝑦) ∈ ℕ) ∧ (𝑧 ∈ ℝ ∧ (exp‘𝑧) ∈ ℕ)) → (exp‘(𝑦 + 𝑧)) = ((exp‘𝑦) · (exp‘𝑧)))
20 nnmulcl 12177 . . . . . . . 8 (((exp‘𝑦) ∈ ℕ ∧ (exp‘𝑧) ∈ ℕ) → ((exp‘𝑦) · (exp‘𝑧)) ∈ ℕ)
2120ad2ant2l 744 . . . . . . 7 (((𝑦 ∈ ℝ ∧ (exp‘𝑦) ∈ ℕ) ∧ (𝑧 ∈ ℝ ∧ (exp‘𝑧) ∈ ℕ)) → ((exp‘𝑦) · (exp‘𝑧)) ∈ ℕ)
2219, 21eqeltrd 2838 . . . . . 6 (((𝑦 ∈ ℝ ∧ (exp‘𝑦) ∈ ℕ) ∧ (𝑧 ∈ ℝ ∧ (exp‘𝑧) ∈ ℕ)) → (exp‘(𝑦 + 𝑧)) ∈ ℕ)
2312, 15, 22elrabd 3647 . . . . 5 (((𝑦 ∈ ℝ ∧ (exp‘𝑦) ∈ ℕ) ∧ (𝑧 ∈ ℝ ∧ (exp‘𝑧) ∈ ℕ)) → (𝑦 + 𝑧) ∈ {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ})
247, 10, 23syl2anb 598 . . . 4 ((𝑦 ∈ {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ} ∧ 𝑧 ∈ {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ}) → (𝑦 + 𝑧) ∈ {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ})
2524adantl 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 (𝜑𝐴 ∈ Fin)
27 fveq2 6842 . . . . 5 (𝑥 = 𝐵 → (exp‘𝑥) = (exp‘𝐵))
2827eleq1d 2822 . . . 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 3647 . . 3 ((𝜑𝑘𝐴) → 𝐵 ∈ {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ})
32 0re 11157 . . . . 5 0 ∈ ℝ
33 1nn 12164 . . . . 5 1 ∈ ℕ
34 fveq2 6842 . . . . . . . 8 (𝑥 = 0 → (exp‘𝑥) = (exp‘0))
35 ef0 15973 . . . . . . . 8 (exp‘0) = 1
3634, 35eqtrdi 2792 . . . . . . 7 (𝑥 = 0 → (exp‘𝑥) = 1)
3736eleq1d 2822 . . . . . 6 (𝑥 = 0 → ((exp‘𝑥) ∈ ℕ ↔ 1 ∈ ℕ))
3837elrab 3645 . . . . 5 (0 ∈ {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ} ↔ (0 ∈ ℝ ∧ 1 ∈ ℕ))
3932, 33, 38mpbir2an 709 . . . 4 0 ∈ {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ}
4039a1i 11 . . 3 (𝜑 → 0 ∈ {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ})
414, 25, 26, 31, 40fsumcllem 15617 . 2 (𝜑 → Σ𝑘𝐴 𝐵 ∈ {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ})
42 fveq2 6842 . . . . 5 (𝑥 = Σ𝑘𝐴 𝐵 → (exp‘𝑥) = (exp‘Σ𝑘𝐴 𝐵))
4342eleq1d 2822 . . . 4 (𝑥 = Σ𝑘𝐴 𝐵 → ((exp‘𝑥) ∈ ℕ ↔ (exp‘Σ𝑘𝐴 𝐵) ∈ ℕ))
4443elrab 3645 . . 3 𝑘𝐴 𝐵 ∈ {𝑥 ∈ ℝ ∣ (exp‘𝑥) ∈ ℕ} ↔ (Σ𝑘𝐴 𝐵 ∈ ℝ ∧ (exp‘Σ𝑘𝐴 𝐵) ∈ ℕ))
4544simprbi 497 . 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 396   = wceq 1541  wcel 2106  {crab 3407  wss 3910  cfv 6496  (class class class)co 7357  Fincfn 8883  cc 11049  cr 11050  0cc0 11051  1c1 11052   + caddc 11054   · cmul 11056  cn 12153  Σcsu 15570  expce 15944
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-pm 8768  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9378  df-inf 9379  df-oi 9446  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-n0 12414  df-z 12500  df-uz 12764  df-rp 12916  df-ico 13270  df-fz 13425  df-fzo 13568  df-fl 13697  df-seq 13907  df-exp 13968  df-fac 14174  df-bc 14203  df-hash 14231  df-shft 14952  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-limsup 15353  df-clim 15370  df-rlim 15371  df-sum 15571  df-ef 15950
This theorem is referenced by:  efchtcl  26460  efchpcl  26474
  Copyright terms: Public domain W3C validator