ppiltx - Metamath Proof Explorer

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Theorem ppiltx 25753

Description: The prime-counting function π is strictly less than the identity. (Contributed by Mario Carneiro, 22-Sep-2014.)

**Assertion**
Ref	Expression
ppiltx	⊢ (𝐴 ∈ ℝ⁺ → (π‘𝐴) < 𝐴)

**Proof of Theorem ppiltx**
Step	Hyp	Ref	Expression
1		rpre 12396	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℝ)
2		ppicl 25707	. . . . . 6 ⊢ (𝐴 ∈ ℝ → (π‘𝐴) ∈ ℕ₀)
3	1, 2	syl 17	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (π‘𝐴) ∈ ℕ₀)
4	3	nn0red 11955	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → (π‘𝐴) ∈ ℝ)
5	4	adantr 483	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) ∈ ℕ) → (π‘𝐴) ∈ ℝ)
6		reflcl 13165	. . . . 5 ⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ∈ ℝ)
7	1, 6	syl 17	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → (⌊‘𝐴) ∈ ℝ)
8	7	adantr 483	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) ∈ ℕ) → (⌊‘𝐴) ∈ ℝ)
9	1	adantr 483	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) ∈ ℕ) → 𝐴 ∈ ℝ)
10		fzfi 13339	. . . . . 6 ⊢ (1...(⌊‘𝐴)) ∈ Fin
11		inss1 4204	. . . . . . 7 ⊢ ((2...(⌊‘𝐴)) ∩ ℙ) ⊆ (2...(⌊‘𝐴))
12		2eluzge1 12293	. . . . . . . . 9 ⊢ 2 ∈ (ℤ_≥‘1)
13		fzss1 12945	. . . . . . . . 9 ⊢ (2 ∈ (ℤ_≥‘1) → (2...(⌊‘𝐴)) ⊆ (1...(⌊‘𝐴)))
14	12, 13	mp1i 13	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) ∈ ℕ) → (2...(⌊‘𝐴)) ⊆ (1...(⌊‘𝐴)))
15		simpr 487	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) ∈ ℕ) → (⌊‘𝐴) ∈ ℕ)
16		nnuz 12280	. . . . . . . . . . . 12 ⊢ ℕ = (ℤ_≥‘1)
17	15, 16	eleqtrdi 2923	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) ∈ ℕ) → (⌊‘𝐴) ∈ (ℤ_≥‘1))
18		eluzfz1 12913	. . . . . . . . . . 11 ⊢ ((⌊‘𝐴) ∈ (ℤ_≥‘1) → 1 ∈ (1...(⌊‘𝐴)))
19	17, 18	syl 17	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) ∈ ℕ) → 1 ∈ (1...(⌊‘𝐴)))
20		1lt2 11807	. . . . . . . . . . . 12 ⊢ 1 < 2
21		1re 10640	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℝ
22		2re 11710	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℝ
23	21, 22	ltnlei 10760	. . . . . . . . . . . 12 ⊢ (1 < 2 ↔ ¬ 2 ≤ 1)
24	20, 23	mpbi 232	. . . . . . . . . . 11 ⊢ ¬ 2 ≤ 1
25		elfzle1 12909	. . . . . . . . . . 11 ⊢ (1 ∈ (2...(⌊‘𝐴)) → 2 ≤ 1)
26	24, 25	mto 199	. . . . . . . . . 10 ⊢ ¬ 1 ∈ (2...(⌊‘𝐴))
27		nelne1 3113	. . . . . . . . . 10 ⊢ ((1 ∈ (1...(⌊‘𝐴)) ∧ ¬ 1 ∈ (2...(⌊‘𝐴))) → (1...(⌊‘𝐴)) ≠ (2...(⌊‘𝐴)))
28	19, 26, 27	sylancl 588	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) ∈ ℕ) → (1...(⌊‘𝐴)) ≠ (2...(⌊‘𝐴)))
29	28	necomd 3071	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) ∈ ℕ) → (2...(⌊‘𝐴)) ≠ (1...(⌊‘𝐴)))
30		df-pss 3953	. . . . . . . 8 ⊢ ((2...(⌊‘𝐴)) ⊊ (1...(⌊‘𝐴)) ↔ ((2...(⌊‘𝐴)) ⊆ (1...(⌊‘𝐴)) ∧ (2...(⌊‘𝐴)) ≠ (1...(⌊‘𝐴))))
31	14, 29, 30	sylanbrc 585	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) ∈ ℕ) → (2...(⌊‘𝐴)) ⊊ (1...(⌊‘𝐴)))
32		sspsstr 4081	. . . . . . 7 ⊢ ((((2...(⌊‘𝐴)) ∩ ℙ) ⊆ (2...(⌊‘𝐴)) ∧ (2...(⌊‘𝐴)) ⊊ (1...(⌊‘𝐴))) → ((2...(⌊‘𝐴)) ∩ ℙ) ⊊ (1...(⌊‘𝐴)))
33	11, 31, 32	sylancr 589	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) ∈ ℕ) → ((2...(⌊‘𝐴)) ∩ ℙ) ⊊ (1...(⌊‘𝐴)))
34		php3 8702	. . . . . 6 ⊢ (((1...(⌊‘𝐴)) ∈ Fin ∧ ((2...(⌊‘𝐴)) ∩ ℙ) ⊊ (1...(⌊‘𝐴))) → ((2...(⌊‘𝐴)) ∩ ℙ) ≺ (1...(⌊‘𝐴)))
35	10, 33, 34	sylancr 589	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) ∈ ℕ) → ((2...(⌊‘𝐴)) ∩ ℙ) ≺ (1...(⌊‘𝐴)))
36		fzfi 13339	. . . . . . 7 ⊢ (2...(⌊‘𝐴)) ∈ Fin
37		ssfi 8737	. . . . . . 7 ⊢ (((2...(⌊‘𝐴)) ∈ Fin ∧ ((2...(⌊‘𝐴)) ∩ ℙ) ⊆ (2...(⌊‘𝐴))) → ((2...(⌊‘𝐴)) ∩ ℙ) ∈ Fin)
38	36, 11, 37	mp2an 690	. . . . . 6 ⊢ ((2...(⌊‘𝐴)) ∩ ℙ) ∈ Fin
39		hashsdom 13741	. . . . . 6 ⊢ ((((2...(⌊‘𝐴)) ∩ ℙ) ∈ Fin ∧ (1...(⌊‘𝐴)) ∈ Fin) → ((♯‘((2...(⌊‘𝐴)) ∩ ℙ)) < (♯‘(1...(⌊‘𝐴))) ↔ ((2...(⌊‘𝐴)) ∩ ℙ) ≺ (1...(⌊‘𝐴))))
40	38, 10, 39	mp2an 690	. . . . 5 ⊢ ((♯‘((2...(⌊‘𝐴)) ∩ ℙ)) < (♯‘(1...(⌊‘𝐴))) ↔ ((2...(⌊‘𝐴)) ∩ ℙ) ≺ (1...(⌊‘𝐴)))
41	35, 40	sylibr 236	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) ∈ ℕ) → (♯‘((2...(⌊‘𝐴)) ∩ ℙ)) < (♯‘(1...(⌊‘𝐴))))
42	1	flcld 13167	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → (⌊‘𝐴) ∈ ℤ)
43		ppival2 25704	. . . . . . 7 ⊢ ((⌊‘𝐴) ∈ ℤ → (π‘(⌊‘𝐴)) = (♯‘((2...(⌊‘𝐴)) ∩ ℙ)))
44	42, 43	syl 17	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (π‘(⌊‘𝐴)) = (♯‘((2...(⌊‘𝐴)) ∩ ℙ)))
45		ppifl 25736	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (π‘(⌊‘𝐴)) = (π‘𝐴))
46	1, 45	syl 17	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (π‘(⌊‘𝐴)) = (π‘𝐴))
47	44, 46	eqtr3d 2858	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (♯‘((2...(⌊‘𝐴)) ∩ ℙ)) = (π‘𝐴))
48	47	adantr 483	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) ∈ ℕ) → (♯‘((2...(⌊‘𝐴)) ∩ ℙ)) = (π‘𝐴))
49		rpge0 12401	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → 0 ≤ 𝐴)
50		flge0nn0 13189	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (⌊‘𝐴) ∈ ℕ₀)
51	1, 49, 50	syl2anc 586	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (⌊‘𝐴) ∈ ℕ₀)
52		hashfz1 13705	. . . . . 6 ⊢ ((⌊‘𝐴) ∈ ℕ₀ → (♯‘(1...(⌊‘𝐴))) = (⌊‘𝐴))
53	51, 52	syl 17	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (♯‘(1...(⌊‘𝐴))) = (⌊‘𝐴))
54	53	adantr 483	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) ∈ ℕ) → (♯‘(1...(⌊‘𝐴))) = (⌊‘𝐴))
55	41, 48, 54	3brtr3d 5096	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) ∈ ℕ) → (π‘𝐴) < (⌊‘𝐴))
56		flle 13168	. . . 4 ⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ≤ 𝐴)
57	9, 56	syl 17	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) ∈ ℕ) → (⌊‘𝐴) ≤ 𝐴)
58	5, 8, 9, 55, 57	ltletrd 10799	. 2 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) ∈ ℕ) → (π‘𝐴) < 𝐴)
59	46	adantr 483	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) = 0) → (π‘(⌊‘𝐴)) = (π‘𝐴))
60		simpr 487	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) = 0) → (⌊‘𝐴) = 0)
61	60	fveq2d 6673	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) = 0) → (π‘(⌊‘𝐴)) = (π‘0))
62		2pos 11739	. . . . . 6 ⊢ 0 < 2
63		0re 10642	. . . . . . 7 ⊢ 0 ∈ ℝ
64		ppieq0 25752	. . . . . . 7 ⊢ (0 ∈ ℝ → ((π‘0) = 0 ↔ 0 < 2))
65	63, 64	ax-mp 5	. . . . . 6 ⊢ ((π‘0) = 0 ↔ 0 < 2)
66	62, 65	mpbir 233	. . . . 5 ⊢ (π‘0) = 0
67	61, 66	syl6eq 2872	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) = 0) → (π‘(⌊‘𝐴)) = 0)
68	59, 67	eqtr3d 2858	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) = 0) → (π‘𝐴) = 0)
69		rpgt0 12400	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → 0 < 𝐴)
70	69	adantr 483	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) = 0) → 0 < 𝐴)
71	68, 70	eqbrtrd 5087	. 2 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) = 0) → (π‘𝐴) < 𝐴)
72		elnn0 11898	. . 3 ⊢ ((⌊‘𝐴) ∈ ℕ₀ ↔ ((⌊‘𝐴) ∈ ℕ ∨ (⌊‘𝐴) = 0))
73	51, 72	sylib 220	. 2 ⊢ (𝐴 ∈ ℝ⁺ → ((⌊‘𝐴) ∈ ℕ ∨ (⌊‘𝐴) = 0))
74	58, 71, 73	mpjaodan 955	1 ⊢ (𝐴 ∈ ℝ⁺ → (π‘𝐴) < 𝐴)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∨ wo 843 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ∩ cin 3934 ⊆ wss 3935 ⊊ wpss 3936 class class class wbr 5065 ‘cfv 6354 (class class class)co 7155 ≺ csdm 8507 Fincfn 8508 ℝcr 10535 0cc0 10536 1c1 10537 < clt 10674 ≤ cle 10675 ℕcn 11637 2c2 11691 ℕ₀cn0 11896 ℤcz 11980 ℤ_≥cuz 12242 ℝ⁺crp 12388 ...cfz 12891 ⌊cfl 13159 ♯chash 13689 ℙcprime 16014 πcppi 25670

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 ax-pre-sup 10614

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-1st 7688 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-2o 8102 df-oadd 8105 df-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-sup 8905 df-inf 8906 df-dju 9329 df-card 9367 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-div 11297 df-nn 11638 df-2 11699 df-3 11700 df-n0 11897 df-xnn0 11967 df-z 11981 df-uz 12243 df-rp 12389 df-icc 12744 df-fz 12892 df-fl 13161 df-seq 13369 df-exp 13429 df-hash 13690 df-cj 14457 df-re 14458 df-im 14459 df-sqrt 14593 df-abs 14594 df-dvds 15607 df-prm 16015 df-ppi 25676

This theorem is referenced by: chtppilimlem1 26048

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