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Theorem chtnprm 27106
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 4201 . . . . . . . . . . . 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 3037 . . . . . . . . . . . 12 ((π‘₯ ∈ β„™ ∧ Β¬ (𝐴 + 1) ∈ β„™) β†’ π‘₯ β‰  (𝐴 + 1))
52, 3, 4syl2anc 582 . . . . . . . . . . 11 ((𝐴 ∈ β„€ ∧ (Β¬ (𝐴 + 1) ∈ β„™ ∧ π‘₯ ∈ ((2...(𝐴 + 1)) ∩ β„™))) β†’ π‘₯ β‰  (𝐴 + 1))
6 velsn 4648 . . . . . . . . . . . 12 (π‘₯ ∈ {(𝐴 + 1)} ↔ π‘₯ = (𝐴 + 1))
76necon3bbii 2985 . . . . . . . . . . 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 4200 . . . . . . . . . . . . 13 ((𝐴 ∈ β„€ ∧ (Β¬ (𝐴 + 1) ∈ β„™ ∧ π‘₯ ∈ ((2...(𝐴 + 1)) ∩ β„™))) β†’ π‘₯ ∈ (2...(𝐴 + 1)))
10 2z 12632 . . . . . . . . . . . . . 14 2 ∈ β„€
11 zcn 12601 . . . . . . . . . . . . . . . . . 18 (𝐴 ∈ β„€ β†’ 𝐴 ∈ β„‚)
1211adantr 479 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ β„€ ∧ (Β¬ (𝐴 + 1) ∈ β„™ ∧ π‘₯ ∈ ((2...(𝐴 + 1)) ∩ β„™))) β†’ 𝐴 ∈ β„‚)
13 ax-1cn 11204 . . . . . . . . . . . . . . . . 17 1 ∈ β„‚
14 pncan 11504 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ β„‚ ∧ 1 ∈ β„‚) β†’ ((𝐴 + 1) βˆ’ 1) = 𝐴)
1512, 13, 14sylancl 584 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ β„€ ∧ (Β¬ (𝐴 + 1) ∈ β„™ ∧ π‘₯ ∈ ((2...(𝐴 + 1)) ∩ β„™))) β†’ ((𝐴 + 1) βˆ’ 1) = 𝐴)
16 elfzuz2 13546 . . . . . . . . . . . . . . . . 17 (π‘₯ ∈ (2...(𝐴 + 1)) β†’ (𝐴 + 1) ∈ (β„€β‰₯β€˜2))
17 uz2m1nn 12945 . . . . . . . . . . . . . . . . 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 2830 . . . . . . . . . . . . . . 15 ((𝐴 ∈ β„€ ∧ (Β¬ (𝐴 + 1) ∈ β„™ ∧ π‘₯ ∈ ((2...(𝐴 + 1)) ∩ β„™))) β†’ 𝐴 ∈ β„•)
20 nnuz 12903 . . . . . . . . . . . . . . . 16 β„• = (β„€β‰₯β€˜1)
21 2m1e1 12376 . . . . . . . . . . . . . . . . 17 (2 βˆ’ 1) = 1
2221fveq2i 6905 . . . . . . . . . . . . . . . 16 (β„€β‰₯β€˜(2 βˆ’ 1)) = (β„€β‰₯β€˜1)
2320, 22eqtr4i 2759 . . . . . . . . . . . . . . 15 β„• = (β„€β‰₯β€˜(2 βˆ’ 1))
2419, 23eleqtrdi 2839 . . . . . . . . . . . . . 14 ((𝐴 ∈ β„€ ∧ (Β¬ (𝐴 + 1) ∈ β„™ ∧ π‘₯ ∈ ((2...(𝐴 + 1)) ∩ β„™))) β†’ 𝐴 ∈ (β„€β‰₯β€˜(2 βˆ’ 1)))
25 fzsuc2 13599 . . . . . . . . . . . . . 14 ((2 ∈ β„€ ∧ 𝐴 ∈ (β„€β‰₯β€˜(2 βˆ’ 1))) β†’ (2...(𝐴 + 1)) = ((2...𝐴) βˆͺ {(𝐴 + 1)}))
2610, 24, 25sylancr 585 . . . . . . . . . . . . 13 ((𝐴 ∈ β„€ ∧ (Β¬ (𝐴 + 1) ∈ β„™ ∧ π‘₯ ∈ ((2...(𝐴 + 1)) ∩ β„™))) β†’ (2...(𝐴 + 1)) = ((2...𝐴) βˆͺ {(𝐴 + 1)}))
279, 26eleqtrd 2831 . . . . . . . . . . . 12 ((𝐴 ∈ β„€ ∧ (Β¬ (𝐴 + 1) ∈ β„™ ∧ π‘₯ ∈ ((2...(𝐴 + 1)) ∩ β„™))) β†’ π‘₯ ∈ ((2...𝐴) βˆͺ {(𝐴 + 1)}))
28 elun 4149 . . . . . . . . . . . 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 4196 . . . . . . . 8 ((𝐴 ∈ β„€ ∧ (Β¬ (𝐴 + 1) ∈ β„™ ∧ π‘₯ ∈ ((2...(𝐴 + 1)) ∩ β„™))) β†’ π‘₯ ∈ ((2...𝐴) ∩ β„™))
3332expr 455 . . . . . . 7 ((𝐴 ∈ β„€ ∧ Β¬ (𝐴 + 1) ∈ β„™) β†’ (π‘₯ ∈ ((2...(𝐴 + 1)) ∩ β„™) β†’ π‘₯ ∈ ((2...𝐴) ∩ β„™)))
3433ssrdv 3988 . . . . . 6 ((𝐴 ∈ β„€ ∧ Β¬ (𝐴 + 1) ∈ β„™) β†’ ((2...(𝐴 + 1)) ∩ β„™) βŠ† ((2...𝐴) ∩ β„™))
35 uzid 12875 . . . . . . . 8 (𝐴 ∈ β„€ β†’ 𝐴 ∈ (β„€β‰₯β€˜π΄))
3635adantr 479 . . . . . . 7 ((𝐴 ∈ β„€ ∧ Β¬ (𝐴 + 1) ∈ β„™) β†’ 𝐴 ∈ (β„€β‰₯β€˜π΄))
37 peano2uz 12923 . . . . . . 7 (𝐴 ∈ (β„€β‰₯β€˜π΄) β†’ (𝐴 + 1) ∈ (β„€β‰₯β€˜π΄))
38 fzss2 13581 . . . . . . 7 ((𝐴 + 1) ∈ (β„€β‰₯β€˜π΄) β†’ (2...𝐴) βŠ† (2...(𝐴 + 1)))
39 ssrin 4236 . . . . . . 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 3999 . . . . 5 ((𝐴 ∈ β„€ ∧ Β¬ (𝐴 + 1) ∈ β„™) β†’ ((2...(𝐴 + 1)) ∩ β„™) = ((2...𝐴) ∩ β„™))
42 peano2z 12641 . . . . . . . . 9 (𝐴 ∈ β„€ β†’ (𝐴 + 1) ∈ β„€)
4342adantr 479 . . . . . . . 8 ((𝐴 ∈ β„€ ∧ Β¬ (𝐴 + 1) ∈ β„™) β†’ (𝐴 + 1) ∈ β„€)
44 flid 13813 . . . . . . . 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 7442 . . . . . 6 ((𝐴 ∈ β„€ ∧ Β¬ (𝐴 + 1) ∈ β„™) β†’ (2...(βŒŠβ€˜(𝐴 + 1))) = (2...(𝐴 + 1)))
4746ineq1d 4213 . . . . 5 ((𝐴 ∈ β„€ ∧ Β¬ (𝐴 + 1) ∈ β„™) β†’ ((2...(βŒŠβ€˜(𝐴 + 1))) ∩ β„™) = ((2...(𝐴 + 1)) ∩ β„™))
48 flid 13813 . . . . . . . 8 (𝐴 ∈ β„€ β†’ (βŒŠβ€˜π΄) = 𝐴)
4948adantr 479 . . . . . . 7 ((𝐴 ∈ β„€ ∧ Β¬ (𝐴 + 1) ∈ β„™) β†’ (βŒŠβ€˜π΄) = 𝐴)
5049oveq2d 7442 . . . . . 6 ((𝐴 ∈ β„€ ∧ Β¬ (𝐴 + 1) ∈ β„™) β†’ (2...(βŒŠβ€˜π΄)) = (2...𝐴))
5150ineq1d 4213 . . . . 5 ((𝐴 ∈ β„€ ∧ Β¬ (𝐴 + 1) ∈ β„™) β†’ ((2...(βŒŠβ€˜π΄)) ∩ β„™) = ((2...𝐴) ∩ β„™))
5241, 47, 513eqtr4d 2778 . . . 4 ((𝐴 ∈ β„€ ∧ Β¬ (𝐴 + 1) ∈ β„™) β†’ ((2...(βŒŠβ€˜(𝐴 + 1))) ∩ β„™) = ((2...(βŒŠβ€˜π΄)) ∩ β„™))
53 zre 12600 . . . . . 6 (𝐴 ∈ β„€ β†’ 𝐴 ∈ ℝ)
5453adantr 479 . . . . 5 ((𝐴 ∈ β„€ ∧ Β¬ (𝐴 + 1) ∈ β„™) β†’ 𝐴 ∈ ℝ)
55 peano2re 11425 . . . . 5 (𝐴 ∈ ℝ β†’ (𝐴 + 1) ∈ ℝ)
56 ppisval 27056 . . . . 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 27056 . . . . 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 2778 . . 3 ((𝐴 ∈ β„€ ∧ Β¬ (𝐴 + 1) ∈ β„™) β†’ ((0[,](𝐴 + 1)) ∩ β„™) = ((0[,]𝐴) ∩ β„™))
6160sumeq1d 15687 . 2 ((𝐴 ∈ β„€ ∧ Β¬ (𝐴 + 1) ∈ β„™) β†’ Σ𝑝 ∈ ((0[,](𝐴 + 1)) ∩ β„™)(logβ€˜π‘) = Σ𝑝 ∈ ((0[,]𝐴) ∩ β„™)(logβ€˜π‘))
62 chtval 27062 . . 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 27062 . . 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 2778 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 394   ∨ wo 845   = wceq 1533   ∈ wcel 2098   β‰  wne 2937   βˆͺ cun 3947   ∩ cin 3948   βŠ† wss 3949  {csn 4632  β€˜cfv 6553  (class class class)co 7426  β„‚cc 11144  β„cr 11145  0cc0 11146  1c1 11147   + caddc 11149   βˆ’ cmin 11482  β„•cn 12250  2c2 12305  β„€cz 12596  β„€β‰₯cuz 12860  [,]cicc 13367  ...cfz 13524  βŒŠcfl 13795  Ξ£csu 15672  β„™cprime 16649  logclog 26508  ΞΈccht 27043
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 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11202  ax-resscn 11203  ax-1cn 11204  ax-icn 11205  ax-addcl 11206  ax-addrcl 11207  ax-mulcl 11208  ax-mulrcl 11209  ax-mulcom 11210  ax-addass 11211  ax-mulass 11212  ax-distr 11213  ax-i2m1 11214  ax-1ne0 11215  ax-1rid 11216  ax-rnegex 11217  ax-rrecex 11218  ax-cnre 11219  ax-pre-lttri 11220  ax-pre-lttrn 11221  ax-pre-ltadd 11222  ax-pre-mulgt0 11223  ax-pre-sup 11224
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 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7877  df-1st 7999  df-2nd 8000  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-rdg 8437  df-1o 8493  df-2o 8494  df-er 8731  df-en 8971  df-dom 8972  df-sdom 8973  df-fin 8974  df-sup 9473  df-inf 9474  df-pnf 11288  df-mnf 11289  df-xr 11290  df-ltxr 11291  df-le 11292  df-sub 11484  df-neg 11485  df-div 11910  df-nn 12251  df-2 12313  df-3 12314  df-n0 12511  df-z 12597  df-uz 12861  df-rp 13015  df-icc 13371  df-fz 13525  df-fl 13797  df-seq 14007  df-exp 14067  df-cj 15086  df-re 15087  df-im 15088  df-sqrt 15222  df-abs 15223  df-sum 15673  df-dvds 16239  df-prm 16650  df-cht 27049
This theorem is referenced by:  chtub  27165
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