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Theorem ppiltx 26670

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 12978	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℝ)
2		ppicl 26624	. . . . . 6 ⊢ (𝐴 ∈ ℝ → (π‘𝐴) ∈ ℕ₀)
3	1, 2	syl 17	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (π‘𝐴) ∈ ℕ₀)
4	3	nn0red 12529	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → (π‘𝐴) ∈ ℝ)
5	4	adantr 481	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) ∈ ℕ) → (π‘𝐴) ∈ ℝ)
6		reflcl 13757	. . . . 5 ⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ∈ ℝ)
7	1, 6	syl 17	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → (⌊‘𝐴) ∈ ℝ)
8	7	adantr 481	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) ∈ ℕ) → (⌊‘𝐴) ∈ ℝ)
9	1	adantr 481	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) ∈ ℕ) → 𝐴 ∈ ℝ)
10		fzfi 13933	. . . . . 6 ⊢ (1...(⌊‘𝐴)) ∈ Fin
11		inss1 4227	. . . . . . 7 ⊢ ((2...(⌊‘𝐴)) ∩ ℙ) ⊆ (2...(⌊‘𝐴))
12		2eluzge1 12874	. . . . . . . . 9 ⊢ 2 ∈ (ℤ_≥‘1)
13		fzss1 13536	. . . . . . . . 9 ⊢ (2 ∈ (ℤ_≥‘1) → (2...(⌊‘𝐴)) ⊆ (1...(⌊‘𝐴)))
14	12, 13	mp1i 13	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) ∈ ℕ) → (2...(⌊‘𝐴)) ⊆ (1...(⌊‘𝐴)))
15		simpr 485	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) ∈ ℕ) → (⌊‘𝐴) ∈ ℕ)
16		nnuz 12861	. . . . . . . . . . . 12 ⊢ ℕ = (ℤ_≥‘1)
17	15, 16	eleqtrdi 2843	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) ∈ ℕ) → (⌊‘𝐴) ∈ (ℤ_≥‘1))
18		eluzfz1 13504	. . . . . . . . . . 11 ⊢ ((⌊‘𝐴) ∈ (ℤ_≥‘1) → 1 ∈ (1...(⌊‘𝐴)))
19	17, 18	syl 17	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) ∈ ℕ) → 1 ∈ (1...(⌊‘𝐴)))
20		1lt2 12379	. . . . . . . . . . . 12 ⊢ 1 < 2
21		1re 11210	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℝ
22		2re 12282	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℝ
23	21, 22	ltnlei 11331	. . . . . . . . . . . 12 ⊢ (1 < 2 ↔ ¬ 2 ≤ 1)
24	20, 23	mpbi 229	. . . . . . . . . . 11 ⊢ ¬ 2 ≤ 1
25		elfzle1 13500	. . . . . . . . . . 11 ⊢ (1 ∈ (2...(⌊‘𝐴)) → 2 ≤ 1)
26	24, 25	mto 196	. . . . . . . . . 10 ⊢ ¬ 1 ∈ (2...(⌊‘𝐴))
27		nelne1 3039	. . . . . . . . . 10 ⊢ ((1 ∈ (1...(⌊‘𝐴)) ∧ ¬ 1 ∈ (2...(⌊‘𝐴))) → (1...(⌊‘𝐴)) ≠ (2...(⌊‘𝐴)))
28	19, 26, 27	sylancl 586	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) ∈ ℕ) → (1...(⌊‘𝐴)) ≠ (2...(⌊‘𝐴)))
29	28	necomd 2996	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) ∈ ℕ) → (2...(⌊‘𝐴)) ≠ (1...(⌊‘𝐴)))
30		df-pss 3966	. . . . . . . 8 ⊢ ((2...(⌊‘𝐴)) ⊊ (1...(⌊‘𝐴)) ↔ ((2...(⌊‘𝐴)) ⊆ (1...(⌊‘𝐴)) ∧ (2...(⌊‘𝐴)) ≠ (1...(⌊‘𝐴))))
31	14, 29, 30	sylanbrc 583	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) ∈ ℕ) → (2...(⌊‘𝐴)) ⊊ (1...(⌊‘𝐴)))
32		sspsstr 4104	. . . . . . 7 ⊢ ((((2...(⌊‘𝐴)) ∩ ℙ) ⊆ (2...(⌊‘𝐴)) ∧ (2...(⌊‘𝐴)) ⊊ (1...(⌊‘𝐴))) → ((2...(⌊‘𝐴)) ∩ ℙ) ⊊ (1...(⌊‘𝐴)))
33	11, 31, 32	sylancr 587	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) ∈ ℕ) → ((2...(⌊‘𝐴)) ∩ ℙ) ⊊ (1...(⌊‘𝐴)))
34		php3 9208	. . . . . 6 ⊢ (((1...(⌊‘𝐴)) ∈ Fin ∧ ((2...(⌊‘𝐴)) ∩ ℙ) ⊊ (1...(⌊‘𝐴))) → ((2...(⌊‘𝐴)) ∩ ℙ) ≺ (1...(⌊‘𝐴)))
35	10, 33, 34	sylancr 587	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) ∈ ℕ) → ((2...(⌊‘𝐴)) ∩ ℙ) ≺ (1...(⌊‘𝐴)))
36		fzfi 13933	. . . . . . 7 ⊢ (2...(⌊‘𝐴)) ∈ Fin
37		ssfi 9169	. . . . . . 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 14337	. . . . . 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 233	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) ∈ ℕ) → (♯‘((2...(⌊‘𝐴)) ∩ ℙ)) < (♯‘(1...(⌊‘𝐴))))
42	1	flcld 13759	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → (⌊‘𝐴) ∈ ℤ)
43		ppival2 26621	. . . . . . 7 ⊢ ((⌊‘𝐴) ∈ ℤ → (π‘(⌊‘𝐴)) = (♯‘((2...(⌊‘𝐴)) ∩ ℙ)))
44	42, 43	syl 17	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (π‘(⌊‘𝐴)) = (♯‘((2...(⌊‘𝐴)) ∩ ℙ)))
45		ppifl 26653	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (π‘(⌊‘𝐴)) = (π‘𝐴))
46	1, 45	syl 17	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (π‘(⌊‘𝐴)) = (π‘𝐴))
47	44, 46	eqtr3d 2774	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (♯‘((2...(⌊‘𝐴)) ∩ ℙ)) = (π‘𝐴))
48	47	adantr 481	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) ∈ ℕ) → (♯‘((2...(⌊‘𝐴)) ∩ ℙ)) = (π‘𝐴))
49		rpge0 12983	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → 0 ≤ 𝐴)
50		flge0nn0 13781	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (⌊‘𝐴) ∈ ℕ₀)
51	1, 49, 50	syl2anc 584	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (⌊‘𝐴) ∈ ℕ₀)
52		hashfz1 14302	. . . . . 6 ⊢ ((⌊‘𝐴) ∈ ℕ₀ → (♯‘(1...(⌊‘𝐴))) = (⌊‘𝐴))
53	51, 52	syl 17	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (♯‘(1...(⌊‘𝐴))) = (⌊‘𝐴))
54	53	adantr 481	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) ∈ ℕ) → (♯‘(1...(⌊‘𝐴))) = (⌊‘𝐴))
55	41, 48, 54	3brtr3d 5178	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) ∈ ℕ) → (π‘𝐴) < (⌊‘𝐴))
56		flle 13760	. . . 4 ⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ≤ 𝐴)
57	9, 56	syl 17	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) ∈ ℕ) → (⌊‘𝐴) ≤ 𝐴)
58	5, 8, 9, 55, 57	ltletrd 11370	. 2 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) ∈ ℕ) → (π‘𝐴) < 𝐴)
59	46	adantr 481	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) = 0) → (π‘(⌊‘𝐴)) = (π‘𝐴))
60		simpr 485	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) = 0) → (⌊‘𝐴) = 0)
61	60	fveq2d 6892	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) = 0) → (π‘(⌊‘𝐴)) = (π‘0))
62		2pos 12311	. . . . . 6 ⊢ 0 < 2
63		0re 11212	. . . . . . 7 ⊢ 0 ∈ ℝ
64		ppieq0 26669	. . . . . . 7 ⊢ (0 ∈ ℝ → ((π‘0) = 0 ↔ 0 < 2))
65	63, 64	ax-mp 5	. . . . . 6 ⊢ ((π‘0) = 0 ↔ 0 < 2)
66	62, 65	mpbir 230	. . . . 5 ⊢ (π‘0) = 0
67	61, 66	eqtrdi 2788	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) = 0) → (π‘(⌊‘𝐴)) = 0)
68	59, 67	eqtr3d 2774	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) = 0) → (π‘𝐴) = 0)
69		rpgt0 12982	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → 0 < 𝐴)
70	69	adantr 481	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) = 0) → 0 < 𝐴)
71	68, 70	eqbrtrd 5169	. 2 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) = 0) → (π‘𝐴) < 𝐴)
72		elnn0 12470	. . 3 ⊢ ((⌊‘𝐴) ∈ ℕ₀ ↔ ((⌊‘𝐴) ∈ ℕ ∨ (⌊‘𝐴) = 0))
73	51, 72	sylib 217	. 2 ⊢ (𝐴 ∈ ℝ⁺ → ((⌊‘𝐴) ∈ ℕ ∨ (⌊‘𝐴) = 0))
74	58, 71, 73	mpjaodan 957	1 ⊢ (𝐴 ∈ ℝ⁺ → (π‘𝐴) < 𝐴)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∨ wo 845 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ∩ cin 3946 ⊆ wss 3947 ⊊ wpss 3948 class class class wbr 5147 ‘cfv 6540 (class class class)co 7405 ≺ csdm 8934 Fincfn 8935 ℝcr 11105 0cc0 11106 1c1 11107 < clt 11244 ≤ cle 11245 ℕcn 12208 2c2 12263 ℕ₀cn0 12468 ℤcz 12554 ℤ_≥cuz 12818 ℝ⁺crp 12970 ...cfz 13480 ⌊cfl 13751 ♯chash 14286 ℙcprime 16604 πcppi 26587

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-oadd 8466 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-sup 9433 df-inf 9434 df-dju 9892 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-n0 12469 df-xnn0 12541 df-z 12555 df-uz 12819 df-rp 12971 df-icc 13327 df-fz 13481 df-fl 13753 df-seq 13963 df-exp 14024 df-hash 14287 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-dvds 16194 df-prm 16605 df-ppi 26593

This theorem is referenced by: chtppilimlem1 26965

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