ppiltx - Metamath Proof Explorer

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Theorem ppiltx 27372

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 13037	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℝ)
2		ppicl 27326	. . . . . 6 ⊢ (𝐴 ∈ ℝ → (π‘𝐴) ∈ ℕ₀)
3	1, 2	syl 18	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (π‘𝐴) ∈ ℕ₀)
4	3	nn0red 12577	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → (π‘𝐴) ∈ ℝ)
5	4	adantr 486	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) ∈ ℕ) → (π‘𝐴) ∈ ℝ)
6		reflcl 13843	. . . . 5 ⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ∈ ℝ)
7	1, 6	syl 18	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → (⌊‘𝐴) ∈ ℝ)
8	7	adantr 486	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) ∈ ℕ) → (⌊‘𝐴) ∈ ℝ)
9	1	adantr 486	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) ∈ ℕ) → 𝐴 ∈ ℝ)
10		fzfi 14022	. . . . . 6 ⊢ (1...(⌊‘𝐴)) ∈ Fin
11		inss1 4189	. . . . . . 7 ⊢ ((2...(⌊‘𝐴)) ∩ ℙ) ⊆ (2...(⌊‘𝐴))
12		2eluzge1 12918	. . . . . . . . 9 ⊢ 2 ∈ (ℤ_≥‘1)
13		fzss1 13604	. . . . . . . . 9 ⊢ (2 ∈ (ℤ_≥‘1) → (2...(⌊‘𝐴)) ⊆ (1...(⌊‘𝐴)))
14	12, 13	mp1i 14	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) ∈ ℕ) → (2...(⌊‘𝐴)) ⊆ (1...(⌊‘𝐴)))
15		simpr 490	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) ∈ ℕ) → (⌊‘𝐴) ∈ ℕ)
16		nnuz 12913	. . . . . . . . . . . 12 ⊢ ℕ = (ℤ_≥‘1)
17	15, 16	eleqtrdi 2875	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) ∈ ℕ) → (⌊‘𝐴) ∈ (ℤ_≥‘1))
18		eluzfz1 13571	. . . . . . . . . . 11 ⊢ ((⌊‘𝐴) ∈ (ℤ_≥‘1) → 1 ∈ (1...(⌊‘𝐴)))
19	17, 18	syl 18	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) ∈ ℕ) → 1 ∈ (1...(⌊‘𝐴)))
20		1lt2 12424	. . . . . . . . . . . 12 ⊢ 1 < 2
21		1re 11219	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℝ
22		2re 12326	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℝ
23	21, 22	ltnlei 11342	. . . . . . . . . . . 12 ⊢ (1 < 2 ↔ ¬ 2 ≤ 1)
24	20, 23	mpbi 233	. . . . . . . . . . 11 ⊢ ¬ 2 ≤ 1
25		elfzle1 13567	. . . . . . . . . . 11 ⊢ (1 ∈ (2...(⌊‘𝐴)) → 2 ≤ 1)
26	24, 25	mto 200	. . . . . . . . . 10 ⊢ ¬ 1 ∈ (2...(⌊‘𝐴))
27		nelne1 3057	. . . . . . . . . 10 ⊢ ((1 ∈ (1...(⌊‘𝐴)) ∧ ¬ 1 ∈ (2...(⌊‘𝐴))) → (1...(⌊‘𝐴)) ≠ (2...(⌊‘𝐴)))
28	19, 26, 27	sylancl 598	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) ∈ ℕ) → (1...(⌊‘𝐴)) ≠ (2...(⌊‘𝐴)))
29	28	necomd 3015	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) ∈ ℕ) → (2...(⌊‘𝐴)) ≠ (1...(⌊‘𝐴)))
30		df-pss 3926	. . . . . . . 8 ⊢ ((2...(⌊‘𝐴)) ⊊ (1...(⌊‘𝐴)) ↔ ((2...(⌊‘𝐴)) ⊆ (1...(⌊‘𝐴)) ∧ (2...(⌊‘𝐴)) ≠ (1...(⌊‘𝐴))))
31	14, 29, 30	sylanbrc 595	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) ∈ ℕ) → (2...(⌊‘𝐴)) ⊊ (1...(⌊‘𝐴)))
32		sspsstr 4064	. . . . . . 7 ⊢ ((((2...(⌊‘𝐴)) ∩ ℙ) ⊆ (2...(⌊‘𝐴)) ∧ (2...(⌊‘𝐴)) ⊊ (1...(⌊‘𝐴))) → ((2...(⌊‘𝐴)) ∩ ℙ) ⊊ (1...(⌊‘𝐴)))
33	11, 31, 32	sylancr 599	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) ∈ ℕ) → ((2...(⌊‘𝐴)) ∩ ℙ) ⊊ (1...(⌊‘𝐴)))
34		php3 9196	. . . . . 6 ⊢ (((1...(⌊‘𝐴)) ∈ Fin ∧ ((2...(⌊‘𝐴)) ∩ ℙ) ⊊ (1...(⌊‘𝐴))) → ((2...(⌊‘𝐴)) ∩ ℙ) ≺ (1...(⌊‘𝐴)))
35	10, 33, 34	sylancr 599	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) ∈ ℕ) → ((2...(⌊‘𝐴)) ∩ ℙ) ≺ (1...(⌊‘𝐴)))
36		fzfi 14022	. . . . . . 7 ⊢ (2...(⌊‘𝐴)) ∈ Fin
37		ssfi 9160	. . . . . . 7 ⊢ (((2...(⌊‘𝐴)) ∈ Fin ∧ ((2...(⌊‘𝐴)) ∩ ℙ) ⊆ (2...(⌊‘𝐴))) → ((2...(⌊‘𝐴)) ∩ ℙ) ∈ Fin)
38	36, 11, 37	mp2an 705	. . . . . 6 ⊢ ((2...(⌊‘𝐴)) ∩ ℙ) ∈ Fin
39		hashsdom 14431	. . . . . 6 ⊢ ((((2...(⌊‘𝐴)) ∩ ℙ) ∈ Fin ∧ (1...(⌊‘𝐴)) ∈ Fin) → ((♯‘((2...(⌊‘𝐴)) ∩ ℙ)) < (♯‘(1...(⌊‘𝐴))) ↔ ((2...(⌊‘𝐴)) ∩ ℙ) ≺ (1...(⌊‘𝐴))))
40	38, 10, 39	mp2an 705	. . . . 5 ⊢ ((♯‘((2...(⌊‘𝐴)) ∩ ℙ)) < (♯‘(1...(⌊‘𝐴))) ↔ ((2...(⌊‘𝐴)) ∩ ℙ) ≺ (1...(⌊‘𝐴)))
41	35, 40	sylibr 237	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) ∈ ℕ) → (♯‘((2...(⌊‘𝐴)) ∩ ℙ)) < (♯‘(1...(⌊‘𝐴))))
42	1	flcld 13845	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → (⌊‘𝐴) ∈ ℤ)
43		ppival2 27323	. . . . . . 7 ⊢ ((⌊‘𝐴) ∈ ℤ → (π‘(⌊‘𝐴)) = (♯‘((2...(⌊‘𝐴)) ∩ ℙ)))
44	42, 43	syl 18	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (π‘(⌊‘𝐴)) = (♯‘((2...(⌊‘𝐴)) ∩ ℙ)))
45		ppifl 27355	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (π‘(⌊‘𝐴)) = (π‘𝐴))
46	1, 45	syl 18	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (π‘(⌊‘𝐴)) = (π‘𝐴))
47	44, 46	eqtr3d 2802	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (♯‘((2...(⌊‘𝐴)) ∩ ℙ)) = (π‘𝐴))
48	47	adantr 486	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) ∈ ℕ) → (♯‘((2...(⌊‘𝐴)) ∩ ℙ)) = (π‘𝐴))
49		rpge0 13042	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → 0 ≤ 𝐴)
50		flge0nn0 13867	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (⌊‘𝐴) ∈ ℕ₀)
51	1, 49, 50	syl2anc 596	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (⌊‘𝐴) ∈ ℕ₀)
52		hashfz1 14396	. . . . . 6 ⊢ ((⌊‘𝐴) ∈ ℕ₀ → (♯‘(1...(⌊‘𝐴))) = (⌊‘𝐴))
53	51, 52	syl 18	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (♯‘(1...(⌊‘𝐴))) = (⌊‘𝐴))
54	53	adantr 486	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) ∈ ℕ) → (♯‘(1...(⌊‘𝐴))) = (⌊‘𝐴))
55	41, 48, 54	3brtr3d 5144	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) ∈ ℕ) → (π‘𝐴) < (⌊‘𝐴))
56		flle 13846	. . . 4 ⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ≤ 𝐴)
57	9, 56	syl 18	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) ∈ ℕ) → (⌊‘𝐴) ≤ 𝐴)
58	5, 8, 9, 55, 57	ltletrd 11381	. 2 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) ∈ ℕ) → (π‘𝐴) < 𝐴)
59	46	adantr 486	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) = 0) → (π‘(⌊‘𝐴)) = (π‘𝐴))
60		simpr 490	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) = 0) → (⌊‘𝐴) = 0)
61	60	fveq2d 6889	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) = 0) → (π‘(⌊‘𝐴)) = (π‘0))
62		2pos 12356	. . . . . 6 ⊢ 0 < 2
63		0re 11221	. . . . . . 7 ⊢ 0 ∈ ℝ
64		ppieq0 27371	. . . . . . 7 ⊢ (0 ∈ ℝ → ((π‘0) = 0 ↔ 0 < 2))
65	63, 64	ax-mp 5	. . . . . 6 ⊢ ((π‘0) = 0 ↔ 0 < 2)
66	62, 65	mpbir 234	. . . . 5 ⊢ (π‘0) = 0
67	61, 66	eqtrdi 2816	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) = 0) → (π‘(⌊‘𝐴)) = 0)
68	59, 67	eqtr3d 2802	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) = 0) → (π‘𝐴) = 0)
69		rpgt0 13041	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → 0 < 𝐴)
70	69	adantr 486	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) = 0) → 0 < 𝐴)
71	68, 70	eqbrtrd 5135	. 2 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) = 0) → (π‘𝐴) < 𝐴)
72		elnn0 12517	. . 3 ⊢ ((⌊‘𝐴) ∈ ℕ₀ ↔ ((⌊‘𝐴) ∈ ℕ ∨ (⌊‘𝐴) = 0))
73	51, 72	sylib 221	. 2 ⊢ (𝐴 ∈ ℝ⁺ → ((⌊‘𝐴) ∈ ℕ ∨ (⌊‘𝐴) = 0))
74	58, 71, 73	mpjaodan 973	1 ⊢ (𝐴 ∈ ℝ⁺ → (π‘𝐴) < 𝐴)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 = wceq 1570 ∈ wcel 2146 ≠ wne 2960 ∩ cin 3905 ⊆ wss 3906 ⊊ wpss 3907 class class class wbr 5111 ‘cfv 6540 (class class class)co 7416 ≺ csdm 8944 Fincfn 8945 ℝcr 11110 0cc0 11111 1c1 11112 < clt 11254 ≤ cle 11255 ℕcn 12244 2c2 12306 ℕ₀cn0 12515 ℤcz 12602 ℤ_≥cuz 12874 ℝ⁺crp 13028 ...cfz 13547 ⌊cfl 13837 ♯chash 14380 ℙcprime 16747 πcppi 27289

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7738 ax-cnex 11167 ax-resscn 11168 ax-1cn 11169 ax-icn 11170 ax-addcl 11171 ax-addrcl 11172 ax-mulcl 11173 ax-mulrcl 11174 ax-mulcom 11175 ax-addass 11176 ax-mulass 11177 ax-distr 11178 ax-i2m1 11179 ax-1ne0 11180 ax-1rid 11181 ax-rnegex 11182 ax-rrecex 11183 ax-cnre 11184 ax-pre-lttri 11185 ax-pre-lttrn 11186 ax-pre-ltadd 11187 ax-pre-mulgt0 11188 ax-pre-sup 11189

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4875 df-int 4915 df-iun 4960 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7865 df-1st 7988 df-2nd 7989 df-frecs 8280 df-wrecs 8311 df-recs 8360 df-rdg 8399 df-1o 8455 df-2o 8456 df-oadd 8459 df-er 8696 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-sup 9405 df-inf 9406 df-dju 9899 df-card 9937 df-pnf 11256 df-mnf 11257 df-xr 11258 df-ltxr 11259 df-le 11260 df-sub 11454 df-neg 11455 df-div 11883 df-nn 12245 df-2 12314 df-3 12315 df-n0 12516 df-xnn0 12589 df-z 12603 df-uz 12875 df-rp 13029 df-icc 13391 df-fz 13548 df-fl 13839 df-seq 14052 df-exp 14112 df-hash 14381 df-cj 15170 df-re 15171 df-im 15172 df-sqrt 15306 df-abs 15307 df-dvds 16329 df-prm 16748 df-ppi 27295

This theorem is used by: chtppilimlem1 27668

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