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

Theorem trireciplem 15844
Description: Lemma for trirecip 15845. Show that the sum converges. (Contributed by Scott Fenton, 22-Apr-2014.) (Revised by Mario Carneiro, 22-May-2014.)
Hypothesis
Ref Expression
trireciplem.1 𝐹 = (𝑛 ∈ ℕ ↦ (1 / (𝑛 · (𝑛 + 1))))
Assertion
Ref Expression
trireciplem seq1( + , 𝐹) ⇝ 1

Proof of Theorem trireciplem
Dummy variables 𝑗 𝑘 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nnuz 12898 . . . 4 ℕ = (ℤ‘1)
2 1zzd 12626 . . . 4 (⊤ → 1 ∈ ℤ)
3 1cnd 11241 . . . . 5 (⊤ → 1 ∈ ℂ)
4 nnex 12251 . . . . . . 7 ℕ ∈ V
54mptex 7235 . . . . . 6 (𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1))) ∈ V
65a1i 11 . . . . 5 (⊤ → (𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1))) ∈ V)
7 oveq1 7426 . . . . . . . 8 (𝑛 = 𝑘 → (𝑛 + 1) = (𝑘 + 1))
87oveq2d 7435 . . . . . . 7 (𝑛 = 𝑘 → (1 / (𝑛 + 1)) = (1 / (𝑘 + 1)))
9 eqid 2725 . . . . . . 7 (𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1))) = (𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1)))
10 ovex 7452 . . . . . . 7 (1 / (𝑘 + 1)) ∈ V
118, 9, 10fvmpt 7004 . . . . . 6 (𝑘 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1)))‘𝑘) = (1 / (𝑘 + 1)))
1211adantl 480 . . . . 5 ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1)))‘𝑘) = (1 / (𝑘 + 1)))
131, 2, 3, 2, 6, 12divcnvshft 15837 . . . 4 (⊤ → (𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1))) ⇝ 0)
14 seqex 14004 . . . . 5 seq1( + , 𝐹) ∈ V
1514a1i 11 . . . 4 (⊤ → seq1( + , 𝐹) ∈ V)
16 peano2nn 12257 . . . . . . . 8 (𝑘 ∈ ℕ → (𝑘 + 1) ∈ ℕ)
1716adantl 480 . . . . . . 7 ((⊤ ∧ 𝑘 ∈ ℕ) → (𝑘 + 1) ∈ ℕ)
1817nnrecred 12296 . . . . . 6 ((⊤ ∧ 𝑘 ∈ ℕ) → (1 / (𝑘 + 1)) ∈ ℝ)
1918recnd 11274 . . . . 5 ((⊤ ∧ 𝑘 ∈ ℕ) → (1 / (𝑘 + 1)) ∈ ℂ)
2012, 19eqeltrd 2825 . . . 4 ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1)))‘𝑘) ∈ ℂ)
2112oveq2d 7435 . . . . 5 ((⊤ ∧ 𝑘 ∈ ℕ) → (1 − ((𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1)))‘𝑘)) = (1 − (1 / (𝑘 + 1))))
22 elfznn 13565 . . . . . . . . . . . 12 (𝑗 ∈ (1...𝑘) → 𝑗 ∈ ℕ)
2322adantl 480 . . . . . . . . . . 11 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → 𝑗 ∈ ℕ)
2423nncnd 12261 . . . . . . . . . 10 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → 𝑗 ∈ ℂ)
25 peano2cn 11418 . . . . . . . . . 10 (𝑗 ∈ ℂ → (𝑗 + 1) ∈ ℂ)
2624, 25syl 17 . . . . . . . . 9 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (𝑗 + 1) ∈ ℂ)
27 peano2nn 12257 . . . . . . . . . . . 12 (𝑗 ∈ ℕ → (𝑗 + 1) ∈ ℕ)
2823, 27syl 17 . . . . . . . . . . 11 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (𝑗 + 1) ∈ ℕ)
2923, 28nnmulcld 12298 . . . . . . . . . 10 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (𝑗 · (𝑗 + 1)) ∈ ℕ)
3029nncnd 12261 . . . . . . . . 9 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (𝑗 · (𝑗 + 1)) ∈ ℂ)
3129nnne0d 12295 . . . . . . . . 9 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (𝑗 · (𝑗 + 1)) ≠ 0)
3226, 24, 30, 31divsubdird 12062 . . . . . . . 8 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (((𝑗 + 1) − 𝑗) / (𝑗 · (𝑗 + 1))) = (((𝑗 + 1) / (𝑗 · (𝑗 + 1))) − (𝑗 / (𝑗 · (𝑗 + 1)))))
33 ax-1cn 11198 . . . . . . . . . 10 1 ∈ ℂ
34 pncan2 11499 . . . . . . . . . 10 ((𝑗 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑗 + 1) − 𝑗) = 1)
3524, 33, 34sylancl 584 . . . . . . . . 9 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → ((𝑗 + 1) − 𝑗) = 1)
3635oveq1d 7434 . . . . . . . 8 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (((𝑗 + 1) − 𝑗) / (𝑗 · (𝑗 + 1))) = (1 / (𝑗 · (𝑗 + 1))))
3726mulridd 11263 . . . . . . . . . . 11 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → ((𝑗 + 1) · 1) = (𝑗 + 1))
3826, 24mulcomd 11267 . . . . . . . . . . 11 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → ((𝑗 + 1) · 𝑗) = (𝑗 · (𝑗 + 1)))
3937, 38oveq12d 7437 . . . . . . . . . 10 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (((𝑗 + 1) · 1) / ((𝑗 + 1) · 𝑗)) = ((𝑗 + 1) / (𝑗 · (𝑗 + 1))))
40 1cnd 11241 . . . . . . . . . . 11 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → 1 ∈ ℂ)
4123nnne0d 12295 . . . . . . . . . . 11 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → 𝑗 ≠ 0)
4228nnne0d 12295 . . . . . . . . . . 11 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (𝑗 + 1) ≠ 0)
4340, 24, 26, 41, 42divcan5d 12049 . . . . . . . . . 10 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (((𝑗 + 1) · 1) / ((𝑗 + 1) · 𝑗)) = (1 / 𝑗))
4439, 43eqtr3d 2767 . . . . . . . . 9 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → ((𝑗 + 1) / (𝑗 · (𝑗 + 1))) = (1 / 𝑗))
4524mulridd 11263 . . . . . . . . . . 11 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (𝑗 · 1) = 𝑗)
4645oveq1d 7434 . . . . . . . . . 10 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → ((𝑗 · 1) / (𝑗 · (𝑗 + 1))) = (𝑗 / (𝑗 · (𝑗 + 1))))
4740, 26, 24, 42, 41divcan5d 12049 . . . . . . . . . 10 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → ((𝑗 · 1) / (𝑗 · (𝑗 + 1))) = (1 / (𝑗 + 1)))
4846, 47eqtr3d 2767 . . . . . . . . 9 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (𝑗 / (𝑗 · (𝑗 + 1))) = (1 / (𝑗 + 1)))
4944, 48oveq12d 7437 . . . . . . . 8 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (((𝑗 + 1) / (𝑗 · (𝑗 + 1))) − (𝑗 / (𝑗 · (𝑗 + 1)))) = ((1 / 𝑗) − (1 / (𝑗 + 1))))
5032, 36, 493eqtr3d 2773 . . . . . . 7 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (1 / (𝑗 · (𝑗 + 1))) = ((1 / 𝑗) − (1 / (𝑗 + 1))))
5150sumeq2dv 15685 . . . . . 6 ((⊤ ∧ 𝑘 ∈ ℕ) → Σ𝑗 ∈ (1...𝑘)(1 / (𝑗 · (𝑗 + 1))) = Σ𝑗 ∈ (1...𝑘)((1 / 𝑗) − (1 / (𝑗 + 1))))
52 oveq2 7427 . . . . . . 7 (𝑛 = 𝑗 → (1 / 𝑛) = (1 / 𝑗))
53 oveq2 7427 . . . . . . 7 (𝑛 = (𝑗 + 1) → (1 / 𝑛) = (1 / (𝑗 + 1)))
54 oveq2 7427 . . . . . . . 8 (𝑛 = 1 → (1 / 𝑛) = (1 / 1))
55 1div1e1 11937 . . . . . . . 8 (1 / 1) = 1
5654, 55eqtrdi 2781 . . . . . . 7 (𝑛 = 1 → (1 / 𝑛) = 1)
57 oveq2 7427 . . . . . . 7 (𝑛 = (𝑘 + 1) → (1 / 𝑛) = (1 / (𝑘 + 1)))
58 nnz 12612 . . . . . . . 8 (𝑘 ∈ ℕ → 𝑘 ∈ ℤ)
5958adantl 480 . . . . . . 7 ((⊤ ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℤ)
6017, 1eleqtrdi 2835 . . . . . . 7 ((⊤ ∧ 𝑘 ∈ ℕ) → (𝑘 + 1) ∈ (ℤ‘1))
61 elfznn 13565 . . . . . . . . . 10 (𝑛 ∈ (1...(𝑘 + 1)) → 𝑛 ∈ ℕ)
6261adantl 480 . . . . . . . . 9 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...(𝑘 + 1))) → 𝑛 ∈ ℕ)
6362nnrecred 12296 . . . . . . . 8 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...(𝑘 + 1))) → (1 / 𝑛) ∈ ℝ)
6463recnd 11274 . . . . . . 7 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...(𝑘 + 1))) → (1 / 𝑛) ∈ ℂ)
6552, 53, 56, 57, 59, 60, 64telfsum 15786 . . . . . 6 ((⊤ ∧ 𝑘 ∈ ℕ) → Σ𝑗 ∈ (1...𝑘)((1 / 𝑗) − (1 / (𝑗 + 1))) = (1 − (1 / (𝑘 + 1))))
6651, 65eqtrd 2765 . . . . 5 ((⊤ ∧ 𝑘 ∈ ℕ) → Σ𝑗 ∈ (1...𝑘)(1 / (𝑗 · (𝑗 + 1))) = (1 − (1 / (𝑘 + 1))))
67 id 22 . . . . . . . . . 10 (𝑛 = 𝑗𝑛 = 𝑗)
68 oveq1 7426 . . . . . . . . . 10 (𝑛 = 𝑗 → (𝑛 + 1) = (𝑗 + 1))
6967, 68oveq12d 7437 . . . . . . . . 9 (𝑛 = 𝑗 → (𝑛 · (𝑛 + 1)) = (𝑗 · (𝑗 + 1)))
7069oveq2d 7435 . . . . . . . 8 (𝑛 = 𝑗 → (1 / (𝑛 · (𝑛 + 1))) = (1 / (𝑗 · (𝑗 + 1))))
71 trireciplem.1 . . . . . . . 8 𝐹 = (𝑛 ∈ ℕ ↦ (1 / (𝑛 · (𝑛 + 1))))
72 ovex 7452 . . . . . . . 8 (1 / (𝑗 · (𝑗 + 1))) ∈ V
7370, 71, 72fvmpt 7004 . . . . . . 7 (𝑗 ∈ ℕ → (𝐹𝑗) = (1 / (𝑗 · (𝑗 + 1))))
7423, 73syl 17 . . . . . 6 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (𝐹𝑗) = (1 / (𝑗 · (𝑗 + 1))))
75 simpr 483 . . . . . . 7 ((⊤ ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
7675, 1eleqtrdi 2835 . . . . . 6 ((⊤ ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ (ℤ‘1))
7729nnrecred 12296 . . . . . . 7 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (1 / (𝑗 · (𝑗 + 1))) ∈ ℝ)
7877recnd 11274 . . . . . 6 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (1 / (𝑗 · (𝑗 + 1))) ∈ ℂ)
7974, 76, 78fsumser 15712 . . . . 5 ((⊤ ∧ 𝑘 ∈ ℕ) → Σ𝑗 ∈ (1...𝑘)(1 / (𝑗 · (𝑗 + 1))) = (seq1( + , 𝐹)‘𝑘))
8021, 66, 793eqtr2rd 2772 . . . 4 ((⊤ ∧ 𝑘 ∈ ℕ) → (seq1( + , 𝐹)‘𝑘) = (1 − ((𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1)))‘𝑘)))
811, 2, 13, 3, 15, 20, 80climsubc2 15619 . . 3 (⊤ → seq1( + , 𝐹) ⇝ (1 − 0))
8281mptru 1540 . 2 seq1( + , 𝐹) ⇝ (1 − 0)
83 1m0e1 12366 . 2 (1 − 0) = 1
8482, 83breqtri 5174 1 seq1( + , 𝐹) ⇝ 1
Colors of variables: wff setvar class
Syntax hints:  wa 394   = wceq 1533  wtru 1534  wcel 2098  Vcvv 3461   class class class wbr 5149  cmpt 5232  cfv 6549  (class class class)co 7419  cc 11138  0cc0 11140  1c1 11141   + caddc 11143   · cmul 11145  cmin 11476   / cdiv 11903  cn 12245  cz 12591  cuz 12855  ...cfz 13519  seqcseq 14002  cli 15464  Σcsu 15668
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-inf2 9666  ax-cnex 11196  ax-resscn 11197  ax-1cn 11198  ax-icn 11199  ax-addcl 11200  ax-addrcl 11201  ax-mulcl 11202  ax-mulrcl 11203  ax-mulcom 11204  ax-addass 11205  ax-mulass 11206  ax-distr 11207  ax-i2m1 11208  ax-1ne0 11209  ax-1rid 11210  ax-rnegex 11211  ax-rrecex 11212  ax-cnre 11213  ax-pre-lttri 11214  ax-pre-lttrn 11215  ax-pre-ltadd 11216  ax-pre-mulgt0 11217  ax-pre-sup 11218
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-int 4951  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-se 5634  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-isom 6558  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7872  df-1st 7994  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-er 8725  df-pm 8848  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-sup 9467  df-inf 9468  df-oi 9535  df-card 9964  df-pnf 11282  df-mnf 11283  df-xr 11284  df-ltxr 11285  df-le 11286  df-sub 11478  df-neg 11479  df-div 11904  df-nn 12246  df-2 12308  df-3 12309  df-n0 12506  df-z 12592  df-uz 12856  df-rp 13010  df-fz 13520  df-fzo 13663  df-fl 13793  df-seq 14003  df-exp 14063  df-hash 14326  df-shft 15050  df-cj 15082  df-re 15083  df-im 15084  df-sqrt 15218  df-abs 15219  df-clim 15468  df-rlim 15469  df-sum 15669
This theorem is referenced by:  trirecip  15845  stirlinglem12  45611
  Copyright terms: Public domain W3C validator