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Theorem trireciplem 15747
Description: Lemma for trirecip 15748. 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 12806 . . . 4 ℕ = (ℤ‘1)
2 1zzd 12534 . . . 4 (⊤ → 1 ∈ ℤ)
3 1cnd 11150 . . . . 5 (⊤ → 1 ∈ ℂ)
4 nnex 12159 . . . . . . 7 ℕ ∈ V
54mptex 7173 . . . . . 6 (𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1))) ∈ V
65a1i 11 . . . . 5 (⊤ → (𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1))) ∈ V)
7 oveq1 7364 . . . . . . . 8 (𝑛 = 𝑘 → (𝑛 + 1) = (𝑘 + 1))
87oveq2d 7373 . . . . . . 7 (𝑛 = 𝑘 → (1 / (𝑛 + 1)) = (1 / (𝑘 + 1)))
9 eqid 2736 . . . . . . 7 (𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1))) = (𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1)))
10 ovex 7390 . . . . . . 7 (1 / (𝑘 + 1)) ∈ V
118, 9, 10fvmpt 6948 . . . . . 6 (𝑘 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1)))‘𝑘) = (1 / (𝑘 + 1)))
1211adantl 482 . . . . 5 ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1)))‘𝑘) = (1 / (𝑘 + 1)))
131, 2, 3, 2, 6, 12divcnvshft 15740 . . . 4 (⊤ → (𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1))) ⇝ 0)
14 seqex 13908 . . . . 5 seq1( + , 𝐹) ∈ V
1514a1i 11 . . . 4 (⊤ → seq1( + , 𝐹) ∈ V)
16 peano2nn 12165 . . . . . . . 8 (𝑘 ∈ ℕ → (𝑘 + 1) ∈ ℕ)
1716adantl 482 . . . . . . 7 ((⊤ ∧ 𝑘 ∈ ℕ) → (𝑘 + 1) ∈ ℕ)
1817nnrecred 12204 . . . . . 6 ((⊤ ∧ 𝑘 ∈ ℕ) → (1 / (𝑘 + 1)) ∈ ℝ)
1918recnd 11183 . . . . 5 ((⊤ ∧ 𝑘 ∈ ℕ) → (1 / (𝑘 + 1)) ∈ ℂ)
2012, 19eqeltrd 2838 . . . 4 ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1)))‘𝑘) ∈ ℂ)
2112oveq2d 7373 . . . . 5 ((⊤ ∧ 𝑘 ∈ ℕ) → (1 − ((𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1)))‘𝑘)) = (1 − (1 / (𝑘 + 1))))
22 elfznn 13470 . . . . . . . . . . . 12 (𝑗 ∈ (1...𝑘) → 𝑗 ∈ ℕ)
2322adantl 482 . . . . . . . . . . 11 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → 𝑗 ∈ ℕ)
2423nncnd 12169 . . . . . . . . . 10 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → 𝑗 ∈ ℂ)
25 peano2cn 11327 . . . . . . . . . 10 (𝑗 ∈ ℂ → (𝑗 + 1) ∈ ℂ)
2624, 25syl 17 . . . . . . . . 9 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (𝑗 + 1) ∈ ℂ)
27 peano2nn 12165 . . . . . . . . . . . 12 (𝑗 ∈ ℕ → (𝑗 + 1) ∈ ℕ)
2823, 27syl 17 . . . . . . . . . . 11 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (𝑗 + 1) ∈ ℕ)
2923, 28nnmulcld 12206 . . . . . . . . . 10 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (𝑗 · (𝑗 + 1)) ∈ ℕ)
3029nncnd 12169 . . . . . . . . 9 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (𝑗 · (𝑗 + 1)) ∈ ℂ)
3129nnne0d 12203 . . . . . . . . 9 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (𝑗 · (𝑗 + 1)) ≠ 0)
3226, 24, 30, 31divsubdird 11970 . . . . . . . 8 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (((𝑗 + 1) − 𝑗) / (𝑗 · (𝑗 + 1))) = (((𝑗 + 1) / (𝑗 · (𝑗 + 1))) − (𝑗 / (𝑗 · (𝑗 + 1)))))
33 ax-1cn 11109 . . . . . . . . . 10 1 ∈ ℂ
34 pncan2 11408 . . . . . . . . . 10 ((𝑗 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑗 + 1) − 𝑗) = 1)
3524, 33, 34sylancl 586 . . . . . . . . 9 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → ((𝑗 + 1) − 𝑗) = 1)
3635oveq1d 7372 . . . . . . . 8 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (((𝑗 + 1) − 𝑗) / (𝑗 · (𝑗 + 1))) = (1 / (𝑗 · (𝑗 + 1))))
3726mulid1d 11172 . . . . . . . . . . 11 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → ((𝑗 + 1) · 1) = (𝑗 + 1))
3826, 24mulcomd 11176 . . . . . . . . . . 11 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → ((𝑗 + 1) · 𝑗) = (𝑗 · (𝑗 + 1)))
3937, 38oveq12d 7375 . . . . . . . . . 10 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (((𝑗 + 1) · 1) / ((𝑗 + 1) · 𝑗)) = ((𝑗 + 1) / (𝑗 · (𝑗 + 1))))
40 1cnd 11150 . . . . . . . . . . 11 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → 1 ∈ ℂ)
4123nnne0d 12203 . . . . . . . . . . 11 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → 𝑗 ≠ 0)
4228nnne0d 12203 . . . . . . . . . . 11 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (𝑗 + 1) ≠ 0)
4340, 24, 26, 41, 42divcan5d 11957 . . . . . . . . . 10 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (((𝑗 + 1) · 1) / ((𝑗 + 1) · 𝑗)) = (1 / 𝑗))
4439, 43eqtr3d 2778 . . . . . . . . 9 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → ((𝑗 + 1) / (𝑗 · (𝑗 + 1))) = (1 / 𝑗))
4524mulid1d 11172 . . . . . . . . . . 11 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (𝑗 · 1) = 𝑗)
4645oveq1d 7372 . . . . . . . . . 10 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → ((𝑗 · 1) / (𝑗 · (𝑗 + 1))) = (𝑗 / (𝑗 · (𝑗 + 1))))
4740, 26, 24, 42, 41divcan5d 11957 . . . . . . . . . 10 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → ((𝑗 · 1) / (𝑗 · (𝑗 + 1))) = (1 / (𝑗 + 1)))
4846, 47eqtr3d 2778 . . . . . . . . 9 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (𝑗 / (𝑗 · (𝑗 + 1))) = (1 / (𝑗 + 1)))
4944, 48oveq12d 7375 . . . . . . . 8 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (((𝑗 + 1) / (𝑗 · (𝑗 + 1))) − (𝑗 / (𝑗 · (𝑗 + 1)))) = ((1 / 𝑗) − (1 / (𝑗 + 1))))
5032, 36, 493eqtr3d 2784 . . . . . . 7 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (1 / (𝑗 · (𝑗 + 1))) = ((1 / 𝑗) − (1 / (𝑗 + 1))))
5150sumeq2dv 15588 . . . . . 6 ((⊤ ∧ 𝑘 ∈ ℕ) → Σ𝑗 ∈ (1...𝑘)(1 / (𝑗 · (𝑗 + 1))) = Σ𝑗 ∈ (1...𝑘)((1 / 𝑗) − (1 / (𝑗 + 1))))
52 oveq2 7365 . . . . . . 7 (𝑛 = 𝑗 → (1 / 𝑛) = (1 / 𝑗))
53 oveq2 7365 . . . . . . 7 (𝑛 = (𝑗 + 1) → (1 / 𝑛) = (1 / (𝑗 + 1)))
54 oveq2 7365 . . . . . . . 8 (𝑛 = 1 → (1 / 𝑛) = (1 / 1))
55 1div1e1 11845 . . . . . . . 8 (1 / 1) = 1
5654, 55eqtrdi 2792 . . . . . . 7 (𝑛 = 1 → (1 / 𝑛) = 1)
57 oveq2 7365 . . . . . . 7 (𝑛 = (𝑘 + 1) → (1 / 𝑛) = (1 / (𝑘 + 1)))
58 nnz 12520 . . . . . . . 8 (𝑘 ∈ ℕ → 𝑘 ∈ ℤ)
5958adantl 482 . . . . . . 7 ((⊤ ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℤ)
6017, 1eleqtrdi 2848 . . . . . . 7 ((⊤ ∧ 𝑘 ∈ ℕ) → (𝑘 + 1) ∈ (ℤ‘1))
61 elfznn 13470 . . . . . . . . . 10 (𝑛 ∈ (1...(𝑘 + 1)) → 𝑛 ∈ ℕ)
6261adantl 482 . . . . . . . . 9 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...(𝑘 + 1))) → 𝑛 ∈ ℕ)
6362nnrecred 12204 . . . . . . . 8 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...(𝑘 + 1))) → (1 / 𝑛) ∈ ℝ)
6463recnd 11183 . . . . . . 7 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...(𝑘 + 1))) → (1 / 𝑛) ∈ ℂ)
6552, 53, 56, 57, 59, 60, 64telfsum 15689 . . . . . 6 ((⊤ ∧ 𝑘 ∈ ℕ) → Σ𝑗 ∈ (1...𝑘)((1 / 𝑗) − (1 / (𝑗 + 1))) = (1 − (1 / (𝑘 + 1))))
6651, 65eqtrd 2776 . . . . 5 ((⊤ ∧ 𝑘 ∈ ℕ) → Σ𝑗 ∈ (1...𝑘)(1 / (𝑗 · (𝑗 + 1))) = (1 − (1 / (𝑘 + 1))))
67 id 22 . . . . . . . . . 10 (𝑛 = 𝑗𝑛 = 𝑗)
68 oveq1 7364 . . . . . . . . . 10 (𝑛 = 𝑗 → (𝑛 + 1) = (𝑗 + 1))
6967, 68oveq12d 7375 . . . . . . . . 9 (𝑛 = 𝑗 → (𝑛 · (𝑛 + 1)) = (𝑗 · (𝑗 + 1)))
7069oveq2d 7373 . . . . . . . 8 (𝑛 = 𝑗 → (1 / (𝑛 · (𝑛 + 1))) = (1 / (𝑗 · (𝑗 + 1))))
71 trireciplem.1 . . . . . . . 8 𝐹 = (𝑛 ∈ ℕ ↦ (1 / (𝑛 · (𝑛 + 1))))
72 ovex 7390 . . . . . . . 8 (1 / (𝑗 · (𝑗 + 1))) ∈ V
7370, 71, 72fvmpt 6948 . . . . . . 7 (𝑗 ∈ ℕ → (𝐹𝑗) = (1 / (𝑗 · (𝑗 + 1))))
7423, 73syl 17 . . . . . 6 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (𝐹𝑗) = (1 / (𝑗 · (𝑗 + 1))))
75 simpr 485 . . . . . . 7 ((⊤ ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
7675, 1eleqtrdi 2848 . . . . . 6 ((⊤ ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ (ℤ‘1))
7729nnrecred 12204 . . . . . . 7 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (1 / (𝑗 · (𝑗 + 1))) ∈ ℝ)
7877recnd 11183 . . . . . 6 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (1 / (𝑗 · (𝑗 + 1))) ∈ ℂ)
7974, 76, 78fsumser 15615 . . . . 5 ((⊤ ∧ 𝑘 ∈ ℕ) → Σ𝑗 ∈ (1...𝑘)(1 / (𝑗 · (𝑗 + 1))) = (seq1( + , 𝐹)‘𝑘))
8021, 66, 793eqtr2rd 2783 . . . 4 ((⊤ ∧ 𝑘 ∈ ℕ) → (seq1( + , 𝐹)‘𝑘) = (1 − ((𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1)))‘𝑘)))
811, 2, 13, 3, 15, 20, 80climsubc2 15521 . . 3 (⊤ → seq1( + , 𝐹) ⇝ (1 − 0))
8281mptru 1548 . 2 seq1( + , 𝐹) ⇝ (1 − 0)
83 1m0e1 12274 . 2 (1 − 0) = 1
8482, 83breqtri 5130 1 seq1( + , 𝐹) ⇝ 1
Colors of variables: wff setvar class
Syntax hints:  wa 396   = wceq 1541  wtru 1542  wcel 2106  Vcvv 3445   class class class wbr 5105  cmpt 5188  cfv 6496  (class class class)co 7357  cc 11049  0cc0 11051  1c1 11052   + caddc 11054   · cmul 11056  cmin 11385   / cdiv 11812  cn 12153  cz 12499  cuz 12763  ...cfz 13424  seqcseq 13906  cli 15366  Σcsu 15570
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-fz 13425  df-fzo 13568  df-fl 13697  df-seq 13907  df-exp 13968  df-hash 14231  df-shft 14952  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-clim 15370  df-rlim 15371  df-sum 15571
This theorem is referenced by:  trirecip  15748  stirlinglem12  44316
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