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Theorem chtnprm 27036
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 770 . . . . . . . . . . . . 13 ((𝐴 ∈ β„€ ∧ (Β¬ (𝐴 + 1) ∈ β„™ ∧ π‘₯ ∈ ((2...(𝐴 + 1)) ∩ β„™))) β†’ π‘₯ ∈ ((2...(𝐴 + 1)) ∩ β„™))
21elin2d 4194 . . . . . . . . . . . 12 ((𝐴 ∈ β„€ ∧ (Β¬ (𝐴 + 1) ∈ β„™ ∧ π‘₯ ∈ ((2...(𝐴 + 1)) ∩ β„™))) β†’ π‘₯ ∈ β„™)
3 simprl 768 . . . . . . . . . . . 12 ((𝐴 ∈ β„€ ∧ (Β¬ (𝐴 + 1) ∈ β„™ ∧ π‘₯ ∈ ((2...(𝐴 + 1)) ∩ β„™))) β†’ Β¬ (𝐴 + 1) ∈ β„™)
4 nelne2 3034 . . . . . . . . . . . 12 ((π‘₯ ∈ β„™ ∧ Β¬ (𝐴 + 1) ∈ β„™) β†’ π‘₯ β‰  (𝐴 + 1))
52, 3, 4syl2anc 583 . . . . . . . . . . 11 ((𝐴 ∈ β„€ ∧ (Β¬ (𝐴 + 1) ∈ β„™ ∧ π‘₯ ∈ ((2...(𝐴 + 1)) ∩ β„™))) β†’ π‘₯ β‰  (𝐴 + 1))
6 velsn 4639 . . . . . . . . . . . 12 (π‘₯ ∈ {(𝐴 + 1)} ↔ π‘₯ = (𝐴 + 1))
76necon3bbii 2982 . . . . . . . . . . 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 4193 . . . . . . . . . . . . 13 ((𝐴 ∈ β„€ ∧ (Β¬ (𝐴 + 1) ∈ β„™ ∧ π‘₯ ∈ ((2...(𝐴 + 1)) ∩ β„™))) β†’ π‘₯ ∈ (2...(𝐴 + 1)))
10 2z 12595 . . . . . . . . . . . . . 14 2 ∈ β„€
11 zcn 12564 . . . . . . . . . . . . . . . . . 18 (𝐴 ∈ β„€ β†’ 𝐴 ∈ β„‚)
1211adantr 480 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ β„€ ∧ (Β¬ (𝐴 + 1) ∈ β„™ ∧ π‘₯ ∈ ((2...(𝐴 + 1)) ∩ β„™))) β†’ 𝐴 ∈ β„‚)
13 ax-1cn 11167 . . . . . . . . . . . . . . . . 17 1 ∈ β„‚
14 pncan 11467 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ β„‚ ∧ 1 ∈ β„‚) β†’ ((𝐴 + 1) βˆ’ 1) = 𝐴)
1512, 13, 14sylancl 585 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ β„€ ∧ (Β¬ (𝐴 + 1) ∈ β„™ ∧ π‘₯ ∈ ((2...(𝐴 + 1)) ∩ β„™))) β†’ ((𝐴 + 1) βˆ’ 1) = 𝐴)
16 elfzuz2 13509 . . . . . . . . . . . . . . . . 17 (π‘₯ ∈ (2...(𝐴 + 1)) β†’ (𝐴 + 1) ∈ (β„€β‰₯β€˜2))
17 uz2m1nn 12908 . . . . . . . . . . . . . . . . 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 2828 . . . . . . . . . . . . . . 15 ((𝐴 ∈ β„€ ∧ (Β¬ (𝐴 + 1) ∈ β„™ ∧ π‘₯ ∈ ((2...(𝐴 + 1)) ∩ β„™))) β†’ 𝐴 ∈ β„•)
20 nnuz 12866 . . . . . . . . . . . . . . . 16 β„• = (β„€β‰₯β€˜1)
21 2m1e1 12339 . . . . . . . . . . . . . . . . 17 (2 βˆ’ 1) = 1
2221fveq2i 6887 . . . . . . . . . . . . . . . 16 (β„€β‰₯β€˜(2 βˆ’ 1)) = (β„€β‰₯β€˜1)
2320, 22eqtr4i 2757 . . . . . . . . . . . . . . 15 β„• = (β„€β‰₯β€˜(2 βˆ’ 1))
2419, 23eleqtrdi 2837 . . . . . . . . . . . . . 14 ((𝐴 ∈ β„€ ∧ (Β¬ (𝐴 + 1) ∈ β„™ ∧ π‘₯ ∈ ((2...(𝐴 + 1)) ∩ β„™))) β†’ 𝐴 ∈ (β„€β‰₯β€˜(2 βˆ’ 1)))
25 fzsuc2 13562 . . . . . . . . . . . . . 14 ((2 ∈ β„€ ∧ 𝐴 ∈ (β„€β‰₯β€˜(2 βˆ’ 1))) β†’ (2...(𝐴 + 1)) = ((2...𝐴) βˆͺ {(𝐴 + 1)}))
2610, 24, 25sylancr 586 . . . . . . . . . . . . 13 ((𝐴 ∈ β„€ ∧ (Β¬ (𝐴 + 1) ∈ β„™ ∧ π‘₯ ∈ ((2...(𝐴 + 1)) ∩ β„™))) β†’ (2...(𝐴 + 1)) = ((2...𝐴) βˆͺ {(𝐴 + 1)}))
279, 26eleqtrd 2829 . . . . . . . . . . . 12 ((𝐴 ∈ β„€ ∧ (Β¬ (𝐴 + 1) ∈ β„™ ∧ π‘₯ ∈ ((2...(𝐴 + 1)) ∩ β„™))) β†’ π‘₯ ∈ ((2...𝐴) βˆͺ {(𝐴 + 1)}))
28 elun 4143 . . . . . . . . . . . 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 861 . . . . . . . . . 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 4189 . . . . . . . 8 ((𝐴 ∈ β„€ ∧ (Β¬ (𝐴 + 1) ∈ β„™ ∧ π‘₯ ∈ ((2...(𝐴 + 1)) ∩ β„™))) β†’ π‘₯ ∈ ((2...𝐴) ∩ β„™))
3332expr 456 . . . . . . 7 ((𝐴 ∈ β„€ ∧ Β¬ (𝐴 + 1) ∈ β„™) β†’ (π‘₯ ∈ ((2...(𝐴 + 1)) ∩ β„™) β†’ π‘₯ ∈ ((2...𝐴) ∩ β„™)))
3433ssrdv 3983 . . . . . 6 ((𝐴 ∈ β„€ ∧ Β¬ (𝐴 + 1) ∈ β„™) β†’ ((2...(𝐴 + 1)) ∩ β„™) βŠ† ((2...𝐴) ∩ β„™))
35 uzid 12838 . . . . . . . 8 (𝐴 ∈ β„€ β†’ 𝐴 ∈ (β„€β‰₯β€˜π΄))
3635adantr 480 . . . . . . 7 ((𝐴 ∈ β„€ ∧ Β¬ (𝐴 + 1) ∈ β„™) β†’ 𝐴 ∈ (β„€β‰₯β€˜π΄))
37 peano2uz 12886 . . . . . . 7 (𝐴 ∈ (β„€β‰₯β€˜π΄) β†’ (𝐴 + 1) ∈ (β„€β‰₯β€˜π΄))
38 fzss2 13544 . . . . . . 7 ((𝐴 + 1) ∈ (β„€β‰₯β€˜π΄) β†’ (2...𝐴) βŠ† (2...(𝐴 + 1)))
39 ssrin 4228 . . . . . . 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 3994 . . . . 5 ((𝐴 ∈ β„€ ∧ Β¬ (𝐴 + 1) ∈ β„™) β†’ ((2...(𝐴 + 1)) ∩ β„™) = ((2...𝐴) ∩ β„™))
42 peano2z 12604 . . . . . . . . 9 (𝐴 ∈ β„€ β†’ (𝐴 + 1) ∈ β„€)
4342adantr 480 . . . . . . . 8 ((𝐴 ∈ β„€ ∧ Β¬ (𝐴 + 1) ∈ β„™) β†’ (𝐴 + 1) ∈ β„€)
44 flid 13776 . . . . . . . 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 7420 . . . . . 6 ((𝐴 ∈ β„€ ∧ Β¬ (𝐴 + 1) ∈ β„™) β†’ (2...(βŒŠβ€˜(𝐴 + 1))) = (2...(𝐴 + 1)))
4746ineq1d 4206 . . . . 5 ((𝐴 ∈ β„€ ∧ Β¬ (𝐴 + 1) ∈ β„™) β†’ ((2...(βŒŠβ€˜(𝐴 + 1))) ∩ β„™) = ((2...(𝐴 + 1)) ∩ β„™))
48 flid 13776 . . . . . . . 8 (𝐴 ∈ β„€ β†’ (βŒŠβ€˜π΄) = 𝐴)
4948adantr 480 . . . . . . 7 ((𝐴 ∈ β„€ ∧ Β¬ (𝐴 + 1) ∈ β„™) β†’ (βŒŠβ€˜π΄) = 𝐴)
5049oveq2d 7420 . . . . . 6 ((𝐴 ∈ β„€ ∧ Β¬ (𝐴 + 1) ∈ β„™) β†’ (2...(βŒŠβ€˜π΄)) = (2...𝐴))
5150ineq1d 4206 . . . . 5 ((𝐴 ∈ β„€ ∧ Β¬ (𝐴 + 1) ∈ β„™) β†’ ((2...(βŒŠβ€˜π΄)) ∩ β„™) = ((2...𝐴) ∩ β„™))
5241, 47, 513eqtr4d 2776 . . . 4 ((𝐴 ∈ β„€ ∧ Β¬ (𝐴 + 1) ∈ β„™) β†’ ((2...(βŒŠβ€˜(𝐴 + 1))) ∩ β„™) = ((2...(βŒŠβ€˜π΄)) ∩ β„™))
53 zre 12563 . . . . . 6 (𝐴 ∈ β„€ β†’ 𝐴 ∈ ℝ)
5453adantr 480 . . . . 5 ((𝐴 ∈ β„€ ∧ Β¬ (𝐴 + 1) ∈ β„™) β†’ 𝐴 ∈ ℝ)
55 peano2re 11388 . . . . 5 (𝐴 ∈ ℝ β†’ (𝐴 + 1) ∈ ℝ)
56 ppisval 26986 . . . . 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 26986 . . . . 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 2776 . . 3 ((𝐴 ∈ β„€ ∧ Β¬ (𝐴 + 1) ∈ β„™) β†’ ((0[,](𝐴 + 1)) ∩ β„™) = ((0[,]𝐴) ∩ β„™))
6160sumeq1d 15650 . 2 ((𝐴 ∈ β„€ ∧ Β¬ (𝐴 + 1) ∈ β„™) β†’ Σ𝑝 ∈ ((0[,](𝐴 + 1)) ∩ β„™)(logβ€˜π‘) = Σ𝑝 ∈ ((0[,]𝐴) ∩ β„™)(logβ€˜π‘))
62 chtval 26992 . . 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 26992 . . 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 2776 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 395   ∨ wo 844   = wceq 1533   ∈ wcel 2098   β‰  wne 2934   βˆͺ cun 3941   ∩ cin 3942   βŠ† wss 3943  {csn 4623  β€˜cfv 6536  (class class class)co 7404  β„‚cc 11107  β„cr 11108  0cc0 11109  1c1 11110   + caddc 11112   βˆ’ cmin 11445  β„•cn 12213  2c2 12268  β„€cz 12559  β„€β‰₯cuz 12823  [,]cicc 13330  ...cfz 13487  βŒŠcfl 13758  Ξ£csu 15635  β„™cprime 16612  logclog 26438  ΞΈccht 26973
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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721  ax-cnex 11165  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185  ax-pre-mulgt0 11186  ax-pre-sup 11187
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6293  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7360  df-ov 7407  df-oprab 7408  df-mpo 7409  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8264  df-wrecs 8295  df-recs 8369  df-rdg 8408  df-1o 8464  df-2o 8465  df-er 8702  df-en 8939  df-dom 8940  df-sdom 8941  df-fin 8942  df-sup 9436  df-inf 9437  df-pnf 11251  df-mnf 11252  df-xr 11253  df-ltxr 11254  df-le 11255  df-sub 11447  df-neg 11448  df-div 11873  df-nn 12214  df-2 12276  df-3 12277  df-n0 12474  df-z 12560  df-uz 12824  df-rp 12978  df-icc 13334  df-fz 13488  df-fl 13760  df-seq 13970  df-exp 14030  df-cj 15049  df-re 15050  df-im 15051  df-sqrt 15185  df-abs 15186  df-sum 15636  df-dvds 16202  df-prm 16613  df-cht 26979
This theorem is referenced by:  chtub  27095
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