ppiltx - Metamath Proof Explorer

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Theorem ppiltx 27158

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 12942	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℝ)
2		ppicl 27112	. . . . . 6 ⊢ (𝐴 ∈ ℝ → (π‘𝐴) ∈ ℕ₀)
3	1, 2	syl 17	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (π‘𝐴) ∈ ℕ₀)
4	3	nn0red 12490	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → (π‘𝐴) ∈ ℝ)
5	4	adantr 481	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) ∈ ℕ) → (π‘𝐴) ∈ ℝ)
6		reflcl 13746	. . . . 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 13925	. . . . . 6 ⊢ (1...(⌊‘𝐴)) ∈ Fin
11		inss1 4165	. . . . . . 7 ⊢ ((2...(⌊‘𝐴)) ∩ ℙ) ⊆ (2...(⌊‘𝐴))
12		2eluzge1 12823	. . . . . . . . 9 ⊢ 2 ∈ (ℤ_≥‘1)
13		fzss1 13508	. . . . . . . . 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 12818	. . . . . . . . . . . 12 ⊢ ℕ = (ℤ_≥‘1)
17	15, 16	eleqtrdi 2849	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) ∈ ℕ) → (⌊‘𝐴) ∈ (ℤ_≥‘1))
18		eluzfz1 13476	. . . . . . . . . . 11 ⊢ ((⌊‘𝐴) ∈ (ℤ_≥‘1) → 1 ∈ (1...(⌊‘𝐴)))
19	17, 18	syl 17	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) ∈ ℕ) → 1 ∈ (1...(⌊‘𝐴)))
20		1lt2 12338	. . . . . . . . . . . 12 ⊢ 1 < 2
21		1re 11135	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℝ
22		2re 12246	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℝ
23	21, 22	ltnlei 11258	. . . . . . . . . . . 12 ⊢ (1 < 2 ↔ ¬ 2 ≤ 1)
24	20, 23	mpbi 231	. . . . . . . . . . 11 ⊢ ¬ 2 ≤ 1
25		elfzle1 13472	. . . . . . . . . . 11 ⊢ (1 ∈ (2...(⌊‘𝐴)) → 2 ≤ 1)
26	24, 25	mto 198	. . . . . . . . . 10 ⊢ ¬ 1 ∈ (2...(⌊‘𝐴))
27		nelne1 3031	. . . . . . . . . 10 ⊢ ((1 ∈ (1...(⌊‘𝐴)) ∧ ¬ 1 ∈ (2...(⌊‘𝐴))) → (1...(⌊‘𝐴)) ≠ (2...(⌊‘𝐴)))
28	19, 26, 27	sylancl 592	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) ∈ ℕ) → (1...(⌊‘𝐴)) ≠ (2...(⌊‘𝐴)))
29	28	necomd 2989	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) ∈ ℕ) → (2...(⌊‘𝐴)) ≠ (1...(⌊‘𝐴)))
30		df-pss 3903	. . . . . . . 8 ⊢ ((2...(⌊‘𝐴)) ⊊ (1...(⌊‘𝐴)) ↔ ((2...(⌊‘𝐴)) ⊆ (1...(⌊‘𝐴)) ∧ (2...(⌊‘𝐴)) ≠ (1...(⌊‘𝐴))))
31	14, 29, 30	sylanbrc 589	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) ∈ ℕ) → (2...(⌊‘𝐴)) ⊊ (1...(⌊‘𝐴)))
32		sspsstr 4039	. . . . . . 7 ⊢ ((((2...(⌊‘𝐴)) ∩ ℙ) ⊆ (2...(⌊‘𝐴)) ∧ (2...(⌊‘𝐴)) ⊊ (1...(⌊‘𝐴))) → ((2...(⌊‘𝐴)) ∩ ℙ) ⊊ (1...(⌊‘𝐴)))
33	11, 31, 32	sylancr 593	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) ∈ ℕ) → ((2...(⌊‘𝐴)) ∩ ℙ) ⊊ (1...(⌊‘𝐴)))
34		php3 9133	. . . . . 6 ⊢ (((1...(⌊‘𝐴)) ∈ Fin ∧ ((2...(⌊‘𝐴)) ∩ ℙ) ⊊ (1...(⌊‘𝐴))) → ((2...(⌊‘𝐴)) ∩ ℙ) ≺ (1...(⌊‘𝐴)))
35	10, 33, 34	sylancr 593	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) ∈ ℕ) → ((2...(⌊‘𝐴)) ∩ ℙ) ≺ (1...(⌊‘𝐴)))
36		fzfi 13925	. . . . . . 7 ⊢ (2...(⌊‘𝐴)) ∈ Fin
37		ssfi 9097	. . . . . . 7 ⊢ (((2...(⌊‘𝐴)) ∈ Fin ∧ ((2...(⌊‘𝐴)) ∩ ℙ) ⊆ (2...(⌊‘𝐴))) → ((2...(⌊‘𝐴)) ∩ ℙ) ∈ Fin)
38	36, 11, 37	mp2an 698	. . . . . 6 ⊢ ((2...(⌊‘𝐴)) ∩ ℙ) ∈ Fin
39		hashsdom 14334	. . . . . 6 ⊢ ((((2...(⌊‘𝐴)) ∩ ℙ) ∈ Fin ∧ (1...(⌊‘𝐴)) ∈ Fin) → ((♯‘((2...(⌊‘𝐴)) ∩ ℙ)) < (♯‘(1...(⌊‘𝐴))) ↔ ((2...(⌊‘𝐴)) ∩ ℙ) ≺ (1...(⌊‘𝐴))))
40	38, 10, 39	mp2an 698	. . . . 5 ⊢ ((♯‘((2...(⌊‘𝐴)) ∩ ℙ)) < (♯‘(1...(⌊‘𝐴))) ↔ ((2...(⌊‘𝐴)) ∩ ℙ) ≺ (1...(⌊‘𝐴)))
41	35, 40	sylibr 235	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) ∈ ℕ) → (♯‘((2...(⌊‘𝐴)) ∩ ℙ)) < (♯‘(1...(⌊‘𝐴))))
42	1	flcld 13748	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → (⌊‘𝐴) ∈ ℤ)
43		ppival2 27109	. . . . . . 7 ⊢ ((⌊‘𝐴) ∈ ℤ → (π‘(⌊‘𝐴)) = (♯‘((2...(⌊‘𝐴)) ∩ ℙ)))
44	42, 43	syl 17	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (π‘(⌊‘𝐴)) = (♯‘((2...(⌊‘𝐴)) ∩ ℙ)))
45		ppifl 27141	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (π‘(⌊‘𝐴)) = (π‘𝐴))
46	1, 45	syl 17	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (π‘(⌊‘𝐴)) = (π‘𝐴))
47	44, 46	eqtr3d 2776	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (♯‘((2...(⌊‘𝐴)) ∩ ℙ)) = (π‘𝐴))
48	47	adantr 481	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) ∈ ℕ) → (♯‘((2...(⌊‘𝐴)) ∩ ℙ)) = (π‘𝐴))
49		rpge0 12947	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → 0 ≤ 𝐴)
50		flge0nn0 13770	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (⌊‘𝐴) ∈ ℕ₀)
51	1, 49, 50	syl2anc 590	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (⌊‘𝐴) ∈ ℕ₀)
52		hashfz1 14299	. . . . . 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 5103	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) ∈ ℕ) → (π‘𝐴) < (⌊‘𝐴))
56		flle 13749	. . . 4 ⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ≤ 𝐴)
57	9, 56	syl 17	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) ∈ ℕ) → (⌊‘𝐴) ≤ 𝐴)
58	5, 8, 9, 55, 57	ltletrd 11297	. 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 6831	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) = 0) → (π‘(⌊‘𝐴)) = (π‘0))
62		2pos 12275	. . . . . 6 ⊢ 0 < 2
63		0re 11137	. . . . . . 7 ⊢ 0 ∈ ℝ
64		ppieq0 27157	. . . . . . 7 ⊢ (0 ∈ ℝ → ((π‘0) = 0 ↔ 0 < 2))
65	63, 64	ax-mp 5	. . . . . 6 ⊢ ((π‘0) = 0 ↔ 0 < 2)
66	62, 65	mpbir 232	. . . . 5 ⊢ (π‘0) = 0
67	61, 66	eqtrdi 2790	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) = 0) → (π‘(⌊‘𝐴)) = 0)
68	59, 67	eqtr3d 2776	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) = 0) → (π‘𝐴) = 0)
69		rpgt0 12946	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → 0 < 𝐴)
70	69	adantr 481	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) = 0) → 0 < 𝐴)
71	68, 70	eqbrtrd 5094	. 2 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) = 0) → (π‘𝐴) < 𝐴)
72		elnn0 12430	. . 3 ⊢ ((⌊‘𝐴) ∈ ℕ₀ ↔ ((⌊‘𝐴) ∈ ℕ ∨ (⌊‘𝐴) = 0))
73	51, 72	sylib 219	. 2 ⊢ (𝐴 ∈ ℝ⁺ → ((⌊‘𝐴) ∈ ℕ ∨ (⌊‘𝐴) = 0))
74	58, 71, 73	mpjaodan 966	1 ⊢ (𝐴 ∈ ℝ⁺ → (π‘𝐴) < 𝐴)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∧ wa 396 ∨ wo 853 = wceq 1547 ∈ wcel 2119 ≠ wne 2934 ∩ cin 3882 ⊆ wss 3883 ⊊ wpss 3884 class class class wbr 5072 ‘cfv 6485 (class class class)co 7356 ≺ csdm 8882 Fincfn 8883 ℝcr 11028 0cc0 11029 1c1 11030 < clt 11170 ≤ cle 11171 ℕcn 12165 2c2 12227 ℕ₀cn0 12428 ℤcz 12515 ℤ_≥cuz 12779 ℝ⁺crp 12933 ...cfz 13452 ⌊cfl 13740 ♯chash 14283 ℙcprime 16631 πcppi 27075

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-pre-sup 11107

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-op 4562 df-uni 4839 df-int 4878 df-iun 4923 df-br 5073 df-opab 5135 df-mpt 5154 df-tr 5180 df-id 5513 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5571 df-we 5573 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-pred 6252 df-ord 6313 df-on 6314 df-lim 6315 df-suc 6316 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-oadd 8399 df-er 8633 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-sup 9345 df-inf 9346 df-dju 9816 df-card 9854 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12166 df-2 12235 df-3 12236 df-n0 12429 df-xnn0 12502 df-z 12516 df-uz 12780 df-rp 12934 df-icc 13296 df-fz 13453 df-fl 13742 df-seq 13955 df-exp 14015 df-hash 14284 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-dvds 16213 df-prm 16632 df-ppi 27081

This theorem is referenced by: chtppilimlem1 27454

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