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Theorem chtnprm 26647
Description: The Chebyshev function at a non-prime. (Contributed by Mario Carneiro, 19-Sep-2014.)
Assertion
Ref Expression
chtnprm ((𝐴 ∈ β„€ ∧ Β¬ (𝐴 + 1) ∈ β„™) β†’ (ΞΈβ€˜(𝐴 + 1)) = (ΞΈβ€˜π΄))
Proof of Theorem chtnprm
Dummy variables 𝑝 π‘₯ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simprr 771 . . . . . . . . . . . . 13 ((𝐴 ∈ β„€ ∧ (Β¬ (𝐴 + 1) ∈ β„™ ∧ π‘₯ ∈ ((2...(𝐴 + 1)) ∩ β„™))) β†’ π‘₯ ∈ ((2...(𝐴 + 1)) ∩ β„™))
21elin2d 4198 . . . . . . . . . . . 12 ((𝐴 ∈ β„€ ∧ (Β¬ (𝐴 + 1) ∈ β„™ ∧ π‘₯ ∈ ((2...(𝐴 + 1)) ∩ β„™))) β†’ π‘₯ ∈ β„™)
3 simprl 769 . . . . . . . . . . . 12 ((𝐴 ∈ β„€ ∧ (Β¬ (𝐴 + 1) ∈ β„™ ∧ π‘₯ ∈ ((2...(𝐴 + 1)) ∩ β„™))) β†’ Β¬ (𝐴 + 1) ∈ β„™)
4 nelne2 3040 . . . . . . . . . . . 12 ((π‘₯ ∈ β„™ ∧ Β¬ (𝐴 + 1) ∈ β„™) β†’ π‘₯ β‰  (𝐴 + 1))
52, 3, 4syl2anc 584 . . . . . . . . . . 11 ((𝐴 ∈ β„€ ∧ (Β¬ (𝐴 + 1) ∈ β„™ ∧ π‘₯ ∈ ((2...(𝐴 + 1)) ∩ β„™))) β†’ π‘₯ β‰  (𝐴 + 1))
6 velsn 4643 . . . . . . . . . . . 12 (π‘₯ ∈ {(𝐴 + 1)} ↔ π‘₯ = (𝐴 + 1))
76necon3bbii 2988 . . . . . . . . . . 11 (Β¬ π‘₯ ∈ {(𝐴 + 1)} ↔ π‘₯ β‰  (𝐴 + 1))
85, 7sylibr 233 . . . . . . . . . 10 ((𝐴 ∈ β„€ ∧ (Β¬ (𝐴 + 1) ∈ β„™ ∧ π‘₯ ∈ ((2...(𝐴 + 1)) ∩ β„™))) β†’ Β¬ π‘₯ ∈ {(𝐴 + 1)})
91elin1d 4197 . . . . . . . . . . . . 13 ((𝐴 ∈ β„€ ∧ (Β¬ (𝐴 + 1) ∈ β„™ ∧ π‘₯ ∈ ((2...(𝐴 + 1)) ∩ β„™))) β†’ π‘₯ ∈ (2...(𝐴 + 1)))
10 2z 12590 . . . . . . . . . . . . . 14 2 ∈ β„€
11 zcn 12559 . . . . . . . . . . . . . . . . . 18 (𝐴 ∈ β„€ β†’ 𝐴 ∈ β„‚)
1211adantr 481 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ β„€ ∧ (Β¬ (𝐴 + 1) ∈ β„™ ∧ π‘₯ ∈ ((2...(𝐴 + 1)) ∩ β„™))) β†’ 𝐴 ∈ β„‚)
13 ax-1cn 11164 . . . . . . . . . . . . . . . . 17 1 ∈ β„‚
14 pncan 11462 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ β„‚ ∧ 1 ∈ β„‚) β†’ ((𝐴 + 1) βˆ’ 1) = 𝐴)
1512, 13, 14sylancl 586 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ β„€ ∧ (Β¬ (𝐴 + 1) ∈ β„™ ∧ π‘₯ ∈ ((2...(𝐴 + 1)) ∩ β„™))) β†’ ((𝐴 + 1) βˆ’ 1) = 𝐴)
16 elfzuz2 13502 . . . . . . . . . . . . . . . . 17 (π‘₯ ∈ (2...(𝐴 + 1)) β†’ (𝐴 + 1) ∈ (β„€β‰₯β€˜2))
17 uz2m1nn 12903 . . . . . . . . . . . . . . . . 17 ((𝐴 + 1) ∈ (β„€β‰₯β€˜2) β†’ ((𝐴 + 1) βˆ’ 1) ∈ β„•)
189, 16, 173syl 18 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ β„€ ∧ (Β¬ (𝐴 + 1) ∈ β„™ ∧ π‘₯ ∈ ((2...(𝐴 + 1)) ∩ β„™))) β†’ ((𝐴 + 1) βˆ’ 1) ∈ β„•)
1915, 18eqeltrrd 2834 . . . . . . . . . . . . . . 15 ((𝐴 ∈ β„€ ∧ (Β¬ (𝐴 + 1) ∈ β„™ ∧ π‘₯ ∈ ((2...(𝐴 + 1)) ∩ β„™))) β†’ 𝐴 ∈ β„•)
20 nnuz 12861 . . . . . . . . . . . . . . . 16 β„• = (β„€β‰₯β€˜1)
21 2m1e1 12334 . . . . . . . . . . . . . . . . 17 (2 βˆ’ 1) = 1
2221fveq2i 6891 . . . . . . . . . . . . . . . 16 (β„€β‰₯β€˜(2 βˆ’ 1)) = (β„€β‰₯β€˜1)
2320, 22eqtr4i 2763 . . . . . . . . . . . . . . 15 β„• = (β„€β‰₯β€˜(2 βˆ’ 1))
2419, 23eleqtrdi 2843 . . . . . . . . . . . . . 14 ((𝐴 ∈ β„€ ∧ (Β¬ (𝐴 + 1) ∈ β„™ ∧ π‘₯ ∈ ((2...(𝐴 + 1)) ∩ β„™))) β†’ 𝐴 ∈ (β„€β‰₯β€˜(2 βˆ’ 1)))
25 fzsuc2 13555 . . . . . . . . . . . . . 14 ((2 ∈ β„€ ∧ 𝐴 ∈ (β„€β‰₯β€˜(2 βˆ’ 1))) β†’ (2...(𝐴 + 1)) = ((2...𝐴) βˆͺ {(𝐴 + 1)}))
2610, 24, 25sylancr 587 . . . . . . . . . . . . 13 ((𝐴 ∈ β„€ ∧ (Β¬ (𝐴 + 1) ∈ β„™ ∧ π‘₯ ∈ ((2...(𝐴 + 1)) ∩ β„™))) β†’ (2...(𝐴 + 1)) = ((2...𝐴) βˆͺ {(𝐴 + 1)}))
279, 26eleqtrd 2835 . . . . . . . . . . . 12 ((𝐴 ∈ β„€ ∧ (Β¬ (𝐴 + 1) ∈ β„™ ∧ π‘₯ ∈ ((2...(𝐴 + 1)) ∩ β„™))) β†’ π‘₯ ∈ ((2...𝐴) βˆͺ {(𝐴 + 1)}))
28 elun 4147 . . . . . . . . . . . 12 (π‘₯ ∈ ((2...𝐴) βˆͺ {(𝐴 + 1)}) ↔ (π‘₯ ∈ (2...𝐴) ∨ π‘₯ ∈ {(𝐴 + 1)}))
2927, 28sylib 217 . . . . . . . . . . 11 ((𝐴 ∈ β„€ ∧ (Β¬ (𝐴 + 1) ∈ β„™ ∧ π‘₯ ∈ ((2...(𝐴 + 1)) ∩ β„™))) β†’ (π‘₯ ∈ (2...𝐴) ∨ π‘₯ ∈ {(𝐴 + 1)}))
3029ord 862 . . . . . . . . . 10 ((𝐴 ∈ β„€ ∧ (Β¬ (𝐴 + 1) ∈ β„™ ∧ π‘₯ ∈ ((2...(𝐴 + 1)) ∩ β„™))) β†’ (Β¬ π‘₯ ∈ (2...𝐴) β†’ π‘₯ ∈ {(𝐴 + 1)}))
318, 30mt3d 148 . . . . . . . . 9 ((𝐴 ∈ β„€ ∧ (Β¬ (𝐴 + 1) ∈ β„™ ∧ π‘₯ ∈ ((2...(𝐴 + 1)) ∩ β„™))) β†’ π‘₯ ∈ (2...𝐴))
3231, 2elind 4193 . . . . . . . 8 ((𝐴 ∈ β„€ ∧ (Β¬ (𝐴 + 1) ∈ β„™ ∧ π‘₯ ∈ ((2...(𝐴 + 1)) ∩ β„™))) β†’ π‘₯ ∈ ((2...𝐴) ∩ β„™))
3332expr 457 . . . . . . 7 ((𝐴 ∈ β„€ ∧ Β¬ (𝐴 + 1) ∈ β„™) β†’ (π‘₯ ∈ ((2...(𝐴 + 1)) ∩ β„™) β†’ π‘₯ ∈ ((2...𝐴) ∩ β„™)))
3433ssrdv 3987 . . . . . 6 ((𝐴 ∈ β„€ ∧ Β¬ (𝐴 + 1) ∈ β„™) β†’ ((2...(𝐴 + 1)) ∩ β„™) βŠ† ((2...𝐴) ∩ β„™))
35 uzid 12833 . . . . . . . 8 (𝐴 ∈ β„€ β†’ 𝐴 ∈ (β„€β‰₯β€˜π΄))
3635adantr 481 . . . . . . 7 ((𝐴 ∈ β„€ ∧ Β¬ (𝐴 + 1) ∈ β„™) β†’ 𝐴 ∈ (β„€β‰₯β€˜π΄))
37 peano2uz 12881 . . . . . . 7 (𝐴 ∈ (β„€β‰₯β€˜π΄) β†’ (𝐴 + 1) ∈ (β„€β‰₯β€˜π΄))
38 fzss2 13537 . . . . . . 7 ((𝐴 + 1) ∈ (β„€β‰₯β€˜π΄) β†’ (2...𝐴) βŠ† (2...(𝐴 + 1)))
39 ssrin 4232 . . . . . . 7 ((2...𝐴) βŠ† (2...(𝐴 + 1)) β†’ ((2...𝐴) ∩ β„™) βŠ† ((2...(𝐴 + 1)) ∩ β„™))
4036, 37, 38, 394syl 19 . . . . . 6 ((𝐴 ∈ β„€ ∧ Β¬ (𝐴 + 1) ∈ β„™) β†’ ((2...𝐴) ∩ β„™) βŠ† ((2...(𝐴 + 1)) ∩ β„™))
4134, 40eqssd 3998 . . . . 5 ((𝐴 ∈ β„€ ∧ Β¬ (𝐴 + 1) ∈ β„™) β†’ ((2...(𝐴 + 1)) ∩ β„™) = ((2...𝐴) ∩ β„™))
42 peano2z 12599 . . . . . . . . 9 (𝐴 ∈ β„€ β†’ (𝐴 + 1) ∈ β„€)
4342adantr 481 . . . . . . . 8 ((𝐴 ∈ β„€ ∧ Β¬ (𝐴 + 1) ∈ β„™) β†’ (𝐴 + 1) ∈ β„€)
44 flid 13769 . . . . . . . 8 ((𝐴 + 1) ∈ β„€ β†’ (βŒŠβ€˜(𝐴 + 1)) = (𝐴 + 1))
4543, 44syl 17 . . . . . . 7 ((𝐴 ∈ β„€ ∧ Β¬ (𝐴 + 1) ∈ β„™) β†’ (βŒŠβ€˜(𝐴 + 1)) = (𝐴 + 1))
4645oveq2d 7421 . . . . . 6 ((𝐴 ∈ β„€ ∧ Β¬ (𝐴 + 1) ∈ β„™) β†’ (2...(βŒŠβ€˜(𝐴 + 1))) = (2...(𝐴 + 1)))
4746ineq1d 4210 . . . . 5 ((𝐴 ∈ β„€ ∧ Β¬ (𝐴 + 1) ∈ β„™) β†’ ((2...(βŒŠβ€˜(𝐴 + 1))) ∩ β„™) = ((2...(𝐴 + 1)) ∩ β„™))
48 flid 13769 . . . . . . . 8 (𝐴 ∈ β„€ β†’ (βŒŠβ€˜π΄) = 𝐴)
4948adantr 481 . . . . . . 7 ((𝐴 ∈ β„€ ∧ Β¬ (𝐴 + 1) ∈ β„™) β†’ (βŒŠβ€˜π΄) = 𝐴)
5049oveq2d 7421 . . . . . 6 ((𝐴 ∈ β„€ ∧ Β¬ (𝐴 + 1) ∈ β„™) β†’ (2...(βŒŠβ€˜π΄)) = (2...𝐴))
5150ineq1d 4210 . . . . 5 ((𝐴 ∈ β„€ ∧ Β¬ (𝐴 + 1) ∈ β„™) β†’ ((2...(βŒŠβ€˜π΄)) ∩ β„™) = ((2...𝐴) ∩ β„™))
5241, 47, 513eqtr4d 2782 . . . 4 ((𝐴 ∈ β„€ ∧ Β¬ (𝐴 + 1) ∈ β„™) β†’ ((2...(βŒŠβ€˜(𝐴 + 1))) ∩ β„™) = ((2...(βŒŠβ€˜π΄)) ∩ β„™))
53 zre 12558 . . . . . 6 (𝐴 ∈ β„€ β†’ 𝐴 ∈ ℝ)
5453adantr 481 . . . . 5 ((𝐴 ∈ β„€ ∧ Β¬ (𝐴 + 1) ∈ β„™) β†’ 𝐴 ∈ ℝ)
55 peano2re 11383 . . . . 5 (𝐴 ∈ ℝ β†’ (𝐴 + 1) ∈ ℝ)
56 ppisval 26597 . . . . 5 ((𝐴 + 1) ∈ ℝ β†’ ((0[,](𝐴 + 1)) ∩ β„™) = ((2...(βŒŠβ€˜(𝐴 + 1))) ∩ β„™))
5754, 55, 563syl 18 . . . 4 ((𝐴 ∈ β„€ ∧ Β¬ (𝐴 + 1) ∈ β„™) β†’ ((0[,](𝐴 + 1)) ∩ β„™) = ((2...(βŒŠβ€˜(𝐴 + 1))) ∩ β„™))
58 ppisval 26597 . . . . 5 (𝐴 ∈ ℝ β†’ ((0[,]𝐴) ∩ β„™) = ((2...(βŒŠβ€˜π΄)) ∩ β„™))
5954, 58syl 17 . . . 4 ((𝐴 ∈ β„€ ∧ Β¬ (𝐴 + 1) ∈ β„™) β†’ ((0[,]𝐴) ∩ β„™) = ((2...(βŒŠβ€˜π΄)) ∩ β„™))
6052, 57, 593eqtr4d 2782 . . 3 ((𝐴 ∈ β„€ ∧ Β¬ (𝐴 + 1) ∈ β„™) β†’ ((0[,](𝐴 + 1)) ∩ β„™) = ((0[,]𝐴) ∩ β„™))
6160sumeq1d 15643 . 2 ((𝐴 ∈ β„€ ∧ Β¬ (𝐴 + 1) ∈ β„™) β†’ Σ𝑝 ∈ ((0[,](𝐴 + 1)) ∩ β„™)(logβ€˜π‘) = Σ𝑝 ∈ ((0[,]𝐴) ∩ β„™)(logβ€˜π‘))
62 chtval 26603 . . 3 ((𝐴 + 1) ∈ ℝ β†’ (ΞΈβ€˜(𝐴 + 1)) = Σ𝑝 ∈ ((0[,](𝐴 + 1)) ∩ β„™)(logβ€˜π‘))
6354, 55, 623syl 18 . 2 ((𝐴 ∈ β„€ ∧ Β¬ (𝐴 + 1) ∈ β„™) β†’ (ΞΈβ€˜(𝐴 + 1)) = Σ𝑝 ∈ ((0[,](𝐴 + 1)) ∩ β„™)(logβ€˜π‘))
64 chtval 26603 . . 3 (𝐴 ∈ ℝ β†’ (ΞΈβ€˜π΄) = Σ𝑝 ∈ ((0[,]𝐴) ∩ β„™)(logβ€˜π‘))
6554, 64syl 17 . 2 ((𝐴 ∈ β„€ ∧ Β¬ (𝐴 + 1) ∈ β„™) β†’ (ΞΈβ€˜π΄) = Σ𝑝 ∈ ((0[,]𝐴) ∩ β„™)(logβ€˜π‘))
6661, 63, 653eqtr4d 2782 1 ((𝐴 ∈ β„€ ∧ Β¬ (𝐴 + 1) ∈ β„™) β†’ (ΞΈβ€˜(𝐴 + 1)) = (ΞΈβ€˜π΄))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 396   ∨ wo 845   = wceq 1541   ∈ wcel 2106   β‰  wne 2940   βˆͺ cun 3945   ∩ cin 3946   βŠ† wss 3947  {csn 4627  β€˜cfv 6540  (class class class)co 7405  β„‚cc 11104  β„cr 11105  0cc0 11106  1c1 11107   + caddc 11109   βˆ’ cmin 11440  β„•cn 12208  2c2 12263  β„€cz 12554  β„€β‰₯cuz 12818  [,]cicc 13323  ...cfz 13480  βŒŠcfl 13751  Ξ£csu 15628  β„™cprime 16604  logclog 26054  ΞΈccht 26584
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 2703  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183  ax-pre-sup 11184
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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-2o 8463  df-er 8699  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-sup 9433  df-inf 9434  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-div 11868  df-nn 12209  df-2 12271  df-3 12272  df-n0 12469  df-z 12555  df-uz 12819  df-rp 12971  df-icc 13327  df-fz 13481  df-fl 13753  df-seq 13963  df-exp 14024  df-cj 15042  df-re 15043  df-im 15044  df-sqrt 15178  df-abs 15179  df-sum 15629  df-dvds 16194  df-prm 16605  df-cht 26590
This theorem is referenced by:  chtub  26704
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