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Theorem chtnprm 26519
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 772 . . . . . . . . . . . . 13 ((𝐴 ∈ β„€ ∧ (Β¬ (𝐴 + 1) ∈ β„™ ∧ π‘₯ ∈ ((2...(𝐴 + 1)) ∩ β„™))) β†’ π‘₯ ∈ ((2...(𝐴 + 1)) ∩ β„™))
21elin2d 4160 . . . . . . . . . . . 12 ((𝐴 ∈ β„€ ∧ (Β¬ (𝐴 + 1) ∈ β„™ ∧ π‘₯ ∈ ((2...(𝐴 + 1)) ∩ β„™))) β†’ π‘₯ ∈ β„™)
3 simprl 770 . . . . . . . . . . . 12 ((𝐴 ∈ β„€ ∧ (Β¬ (𝐴 + 1) ∈ β„™ ∧ π‘₯ ∈ ((2...(𝐴 + 1)) ∩ β„™))) β†’ Β¬ (𝐴 + 1) ∈ β„™)
4 nelne2 3039 . . . . . . . . . . . 12 ((π‘₯ ∈ β„™ ∧ Β¬ (𝐴 + 1) ∈ β„™) β†’ π‘₯ β‰  (𝐴 + 1))
52, 3, 4syl2anc 585 . . . . . . . . . . 11 ((𝐴 ∈ β„€ ∧ (Β¬ (𝐴 + 1) ∈ β„™ ∧ π‘₯ ∈ ((2...(𝐴 + 1)) ∩ β„™))) β†’ π‘₯ β‰  (𝐴 + 1))
6 velsn 4603 . . . . . . . . . . . 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 4159 . . . . . . . . . . . . 13 ((𝐴 ∈ β„€ ∧ (Β¬ (𝐴 + 1) ∈ β„™ ∧ π‘₯ ∈ ((2...(𝐴 + 1)) ∩ β„™))) β†’ π‘₯ ∈ (2...(𝐴 + 1)))
10 2z 12540 . . . . . . . . . . . . . 14 2 ∈ β„€
11 zcn 12509 . . . . . . . . . . . . . . . . . 18 (𝐴 ∈ β„€ β†’ 𝐴 ∈ β„‚)
1211adantr 482 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ β„€ ∧ (Β¬ (𝐴 + 1) ∈ β„™ ∧ π‘₯ ∈ ((2...(𝐴 + 1)) ∩ β„™))) β†’ 𝐴 ∈ β„‚)
13 ax-1cn 11114 . . . . . . . . . . . . . . . . 17 1 ∈ β„‚
14 pncan 11412 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ β„‚ ∧ 1 ∈ β„‚) β†’ ((𝐴 + 1) βˆ’ 1) = 𝐴)
1512, 13, 14sylancl 587 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ β„€ ∧ (Β¬ (𝐴 + 1) ∈ β„™ ∧ π‘₯ ∈ ((2...(𝐴 + 1)) ∩ β„™))) β†’ ((𝐴 + 1) βˆ’ 1) = 𝐴)
16 elfzuz2 13452 . . . . . . . . . . . . . . . . 17 (π‘₯ ∈ (2...(𝐴 + 1)) β†’ (𝐴 + 1) ∈ (β„€β‰₯β€˜2))
17 uz2m1nn 12853 . . . . . . . . . . . . . . . . 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 2835 . . . . . . . . . . . . . . 15 ((𝐴 ∈ β„€ ∧ (Β¬ (𝐴 + 1) ∈ β„™ ∧ π‘₯ ∈ ((2...(𝐴 + 1)) ∩ β„™))) β†’ 𝐴 ∈ β„•)
20 nnuz 12811 . . . . . . . . . . . . . . . 16 β„• = (β„€β‰₯β€˜1)
21 2m1e1 12284 . . . . . . . . . . . . . . . . 17 (2 βˆ’ 1) = 1
2221fveq2i 6846 . . . . . . . . . . . . . . . 16 (β„€β‰₯β€˜(2 βˆ’ 1)) = (β„€β‰₯β€˜1)
2320, 22eqtr4i 2764 . . . . . . . . . . . . . . 15 β„• = (β„€β‰₯β€˜(2 βˆ’ 1))
2419, 23eleqtrdi 2844 . . . . . . . . . . . . . 14 ((𝐴 ∈ β„€ ∧ (Β¬ (𝐴 + 1) ∈ β„™ ∧ π‘₯ ∈ ((2...(𝐴 + 1)) ∩ β„™))) β†’ 𝐴 ∈ (β„€β‰₯β€˜(2 βˆ’ 1)))
25 fzsuc2 13505 . . . . . . . . . . . . . 14 ((2 ∈ β„€ ∧ 𝐴 ∈ (β„€β‰₯β€˜(2 βˆ’ 1))) β†’ (2...(𝐴 + 1)) = ((2...𝐴) βˆͺ {(𝐴 + 1)}))
2610, 24, 25sylancr 588 . . . . . . . . . . . . 13 ((𝐴 ∈ β„€ ∧ (Β¬ (𝐴 + 1) ∈ β„™ ∧ π‘₯ ∈ ((2...(𝐴 + 1)) ∩ β„™))) β†’ (2...(𝐴 + 1)) = ((2...𝐴) βˆͺ {(𝐴 + 1)}))
279, 26eleqtrd 2836 . . . . . . . . . . . 12 ((𝐴 ∈ β„€ ∧ (Β¬ (𝐴 + 1) ∈ β„™ ∧ π‘₯ ∈ ((2...(𝐴 + 1)) ∩ β„™))) β†’ π‘₯ ∈ ((2...𝐴) βˆͺ {(𝐴 + 1)}))
28 elun 4109 . . . . . . . . . . . 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 863 . . . . . . . . . 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 4155 . . . . . . . 8 ((𝐴 ∈ β„€ ∧ (Β¬ (𝐴 + 1) ∈ β„™ ∧ π‘₯ ∈ ((2...(𝐴 + 1)) ∩ β„™))) β†’ π‘₯ ∈ ((2...𝐴) ∩ β„™))
3332expr 458 . . . . . . 7 ((𝐴 ∈ β„€ ∧ Β¬ (𝐴 + 1) ∈ β„™) β†’ (π‘₯ ∈ ((2...(𝐴 + 1)) ∩ β„™) β†’ π‘₯ ∈ ((2...𝐴) ∩ β„™)))
3433ssrdv 3951 . . . . . 6 ((𝐴 ∈ β„€ ∧ Β¬ (𝐴 + 1) ∈ β„™) β†’ ((2...(𝐴 + 1)) ∩ β„™) βŠ† ((2...𝐴) ∩ β„™))
35 uzid 12783 . . . . . . . 8 (𝐴 ∈ β„€ β†’ 𝐴 ∈ (β„€β‰₯β€˜π΄))
3635adantr 482 . . . . . . 7 ((𝐴 ∈ β„€ ∧ Β¬ (𝐴 + 1) ∈ β„™) β†’ 𝐴 ∈ (β„€β‰₯β€˜π΄))
37 peano2uz 12831 . . . . . . 7 (𝐴 ∈ (β„€β‰₯β€˜π΄) β†’ (𝐴 + 1) ∈ (β„€β‰₯β€˜π΄))
38 fzss2 13487 . . . . . . 7 ((𝐴 + 1) ∈ (β„€β‰₯β€˜π΄) β†’ (2...𝐴) βŠ† (2...(𝐴 + 1)))
39 ssrin 4194 . . . . . . 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 3962 . . . . 5 ((𝐴 ∈ β„€ ∧ Β¬ (𝐴 + 1) ∈ β„™) β†’ ((2...(𝐴 + 1)) ∩ β„™) = ((2...𝐴) ∩ β„™))
42 peano2z 12549 . . . . . . . . 9 (𝐴 ∈ β„€ β†’ (𝐴 + 1) ∈ β„€)
4342adantr 482 . . . . . . . 8 ((𝐴 ∈ β„€ ∧ Β¬ (𝐴 + 1) ∈ β„™) β†’ (𝐴 + 1) ∈ β„€)
44 flid 13719 . . . . . . . 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 7374 . . . . . 6 ((𝐴 ∈ β„€ ∧ Β¬ (𝐴 + 1) ∈ β„™) β†’ (2...(βŒŠβ€˜(𝐴 + 1))) = (2...(𝐴 + 1)))
4746ineq1d 4172 . . . . 5 ((𝐴 ∈ β„€ ∧ Β¬ (𝐴 + 1) ∈ β„™) β†’ ((2...(βŒŠβ€˜(𝐴 + 1))) ∩ β„™) = ((2...(𝐴 + 1)) ∩ β„™))
48 flid 13719 . . . . . . . 8 (𝐴 ∈ β„€ β†’ (βŒŠβ€˜π΄) = 𝐴)
4948adantr 482 . . . . . . 7 ((𝐴 ∈ β„€ ∧ Β¬ (𝐴 + 1) ∈ β„™) β†’ (βŒŠβ€˜π΄) = 𝐴)
5049oveq2d 7374 . . . . . 6 ((𝐴 ∈ β„€ ∧ Β¬ (𝐴 + 1) ∈ β„™) β†’ (2...(βŒŠβ€˜π΄)) = (2...𝐴))
5150ineq1d 4172 . . . . 5 ((𝐴 ∈ β„€ ∧ Β¬ (𝐴 + 1) ∈ β„™) β†’ ((2...(βŒŠβ€˜π΄)) ∩ β„™) = ((2...𝐴) ∩ β„™))
5241, 47, 513eqtr4d 2783 . . . 4 ((𝐴 ∈ β„€ ∧ Β¬ (𝐴 + 1) ∈ β„™) β†’ ((2...(βŒŠβ€˜(𝐴 + 1))) ∩ β„™) = ((2...(βŒŠβ€˜π΄)) ∩ β„™))
53 zre 12508 . . . . . 6 (𝐴 ∈ β„€ β†’ 𝐴 ∈ ℝ)
5453adantr 482 . . . . 5 ((𝐴 ∈ β„€ ∧ Β¬ (𝐴 + 1) ∈ β„™) β†’ 𝐴 ∈ ℝ)
55 peano2re 11333 . . . . 5 (𝐴 ∈ ℝ β†’ (𝐴 + 1) ∈ ℝ)
56 ppisval 26469 . . . . 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 26469 . . . . 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 2783 . . 3 ((𝐴 ∈ β„€ ∧ Β¬ (𝐴 + 1) ∈ β„™) β†’ ((0[,](𝐴 + 1)) ∩ β„™) = ((0[,]𝐴) ∩ β„™))
6160sumeq1d 15591 . 2 ((𝐴 ∈ β„€ ∧ Β¬ (𝐴 + 1) ∈ β„™) β†’ Σ𝑝 ∈ ((0[,](𝐴 + 1)) ∩ β„™)(logβ€˜π‘) = Σ𝑝 ∈ ((0[,]𝐴) ∩ β„™)(logβ€˜π‘))
62 chtval 26475 . . 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 26475 . . 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 2783 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 397   ∨ wo 846   = wceq 1542   ∈ wcel 2107   β‰  wne 2940   βˆͺ cun 3909   ∩ cin 3910   βŠ† wss 3911  {csn 4587  β€˜cfv 6497  (class class class)co 7358  β„‚cc 11054  β„cr 11055  0cc0 11056  1c1 11057   + caddc 11059   βˆ’ cmin 11390  β„•cn 12158  2c2 12213  β„€cz 12504  β„€β‰₯cuz 12768  [,]cicc 13273  ...cfz 13430  βŒŠcfl 13701  Ξ£csu 15576  β„™cprime 16552  logclog 25926  ΞΈccht 26456
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-cnex 11112  ax-resscn 11113  ax-1cn 11114  ax-icn 11115  ax-addcl 11116  ax-addrcl 11117  ax-mulcl 11118  ax-mulrcl 11119  ax-mulcom 11120  ax-addass 11121  ax-mulass 11122  ax-distr 11123  ax-i2m1 11124  ax-1ne0 11125  ax-1rid 11126  ax-rnegex 11127  ax-rrecex 11128  ax-cnre 11129  ax-pre-lttri 11130  ax-pre-lttrn 11131  ax-pre-ltadd 11132  ax-pre-mulgt0 11133  ax-pre-sup 11134
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7804  df-1st 7922  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-1o 8413  df-2o 8414  df-er 8651  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-sup 9383  df-inf 9384  df-pnf 11196  df-mnf 11197  df-xr 11198  df-ltxr 11199  df-le 11200  df-sub 11392  df-neg 11393  df-div 11818  df-nn 12159  df-2 12221  df-3 12222  df-n0 12419  df-z 12505  df-uz 12769  df-rp 12921  df-icc 13277  df-fz 13431  df-fl 13703  df-seq 13913  df-exp 13974  df-cj 14990  df-re 14991  df-im 14992  df-sqrt 15126  df-abs 15127  df-sum 15577  df-dvds 16142  df-prm 16553  df-cht 26462
This theorem is referenced by:  chtub  26576
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