ppiltx - Metamath Proof Explorer

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Theorem ppiltx 27550

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 13167	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℝ)
2		ppicl 27504	. . . . . 6 ⊢ (𝐴 ∈ ℝ → (π‘𝐴) ∈ ℕ₀)
3	1, 2	syl 18	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (π‘𝐴) ∈ ℕ₀)
4	3	nn0red 12706	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → (π‘𝐴) ∈ ℝ)
5	4	adantr 486	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) ∈ ℕ) → (π‘𝐴) ∈ ℝ)
6		reflcl 13974	. . . . 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 14153	. . . . . 6 ⊢ (1...(⌊‘𝐴)) ∈ Fin
11		inss1 4136	. . . . . . 7 ⊢ ((2...(⌊‘𝐴)) ∩ ℙ) ⊆ (2...(⌊‘𝐴))
12		2eluzge1 13047	. . . . . . . . 9 ⊢ 2 ∈ (ℤ_≥‘1)
13		fzss1 13735	. . . . . . . . 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 13042	. . . . . . . . . . . 12 ⊢ ℕ = (ℤ_≥‘1)
17	15, 16	eleqtrdi 2874	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) ∈ ℕ) → (⌊‘𝐴) ∈ (ℤ_≥‘1))
18		eluzfz1 13702	. . . . . . . . . . 11 ⊢ ((⌊‘𝐴) ∈ (ℤ_≥‘1) → 1 ∈ (1...(⌊‘𝐴)))
19	17, 18	syl 18	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) ∈ ℕ) → 1 ∈ (1...(⌊‘𝐴)))
20		1lt2 12553	. . . . . . . . . . . 12 ⊢ 1 < 2
21		1re 11345	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℝ
22		2re 12455	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℝ
23	21, 22	ltnlei 11468	. . . . . . . . . . . 12 ⊢ (1 < 2 ↔ ¬ 2 ≤ 1)
24	20, 23	mpbi 233	. . . . . . . . . . 11 ⊢ ¬ 2 ≤ 1
25		elfzle1 13698	. . . . . . . . . . 11 ⊢ (1 ∈ (2...(⌊‘𝐴)) → 2 ≤ 1)
26	24, 25	mto 200	. . . . . . . . . 10 ⊢ ¬ 1 ∈ (2...(⌊‘𝐴))
27		nelne1 3056	. . . . . . . . . 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 3014	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) ∈ ℕ) → (2...(⌊‘𝐴)) ≠ (1...(⌊‘𝐴)))
30		df-pss 4017	. . . . . . . 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 4039	. . . . . . 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 9260	. . . . . 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 14153	. . . . . . 7 ⊢ (2...(⌊‘𝐴)) ∈ Fin
37		ssfi 9224	. . . . . . 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 14564	. . . . . 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 13976	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → (⌊‘𝐴) ∈ ℤ)
43		ppival2 27501	. . . . . . 7 ⊢ ((⌊‘𝐴) ∈ ℤ → (π‘(⌊‘𝐴)) = (♯‘((2...(⌊‘𝐴)) ∩ ℙ)))
44	42, 43	syl 18	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (π‘(⌊‘𝐴)) = (♯‘((2...(⌊‘𝐴)) ∩ ℙ)))
45		ppifl 27533	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (π‘(⌊‘𝐴)) = (π‘𝐴))
46	1, 45	syl 18	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (π‘(⌊‘𝐴)) = (π‘𝐴))
47	44, 46	eqtr3d 2800	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (♯‘((2...(⌊‘𝐴)) ∩ ℙ)) = (π‘𝐴))
48	47	adantr 486	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) ∈ ℕ) → (♯‘((2...(⌊‘𝐴)) ∩ ℙ)) = (π‘𝐴))
49		rpge0 13172	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → 0 ≤ 𝐴)
50		flge0nn0 13998	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (⌊‘𝐴) ∈ ℕ₀)
51	1, 49, 50	syl2anc 596	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (⌊‘𝐴) ∈ ℕ₀)
52		hashfz1 14529	. . . . . 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 5140	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) ∈ ℕ) → (π‘𝐴) < (⌊‘𝐴))
56		flle 13977	. . . 4 ⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ≤ 𝐴)
57	9, 56	syl 18	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) ∈ ℕ) → (⌊‘𝐴) ≤ 𝐴)
58	5, 8, 9, 55, 57	ltletrd 11507	. 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 6918	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) = 0) → (π‘(⌊‘𝐴)) = (π‘0))
62		2pos 12485	. . . . . 6 ⊢ 0 < 2
63		0re 11347	. . . . . . 7 ⊢ 0 ∈ ℝ
64		ppieq0 27549	. . . . . . 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 2814	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) = 0) → (π‘(⌊‘𝐴)) = 0)
68	59, 67	eqtr3d 2800	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) = 0) → (π‘𝐴) = 0)
69		rpgt0 13171	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → 0 < 𝐴)
70	69	adantr 486	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) = 0) → 0 < 𝐴)
71	68, 70	eqbrtrd 5131	. 2 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (⌊‘𝐴) = 0) → (π‘𝐴) < 𝐴)
72		elnn0 12646	. . 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 2147 ≠ wne 2959 ⊆ wss 3900 ⊊ wpss 4016 ∩ cin 4092 class class class wbr 5106 ‘cfv 6564 (class class class)co 7449 ≺ csdm 9008 Fincfn 9009 ℝcr 11236 0cc0 11237 1c1 11238 < clt 11380 ≤ cle 11381 ℕcn 12373 2c2 12435 ℕ₀cn0 12644 ℤcz 12731 ℤ_≥cuz 13003 ℝ⁺crp 13158 ...cfz 13677 ⌊cfl 13968 ♯chash 14513 ℙcprime 16887 πcppi 27467

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 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-sep 5253 ax-nul 5264 ax-pow 5332 ax-pr 5396 ax-un 7781 ax-cnex 11293 ax-resscn 11294 ax-1cn 11295 ax-icn 11296 ax-addcl 11297 ax-addrcl 11298 ax-mulcl 11299 ax-mulrcl 11300 ax-mulcom 11301 ax-addass 11302 ax-mulass 11303 ax-distr 11304 ax-i2m1 11305 ax-1ne0 11306 ax-1rid 11307 ax-rnegex 11308 ax-rrecex 11309 ax-cnre 11310 ax-pre-lttri 11311 ax-pre-lttrn 11312 ax-pre-ltadd 11313 ax-pre-mulgt0 11314 ax-pre-sup 11315

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 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4839 df-uni 4871 df-int 4911 df-iun 4956 df-br 5107 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5551 df-eprel 5556 df-po 5564 df-so 5565 df-fr 5609 df-we 5611 df-xp 5662 df-rel 5663 df-cnv 5664 df-co 5665 df-dm 5666 df-rn 5667 df-res 5668 df-ima 5669 df-pred 6328 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6518 df-fun 6566 df-fn 6567 df-f 6568 df-f1 6569 df-fo 6570 df-f1o 6571 df-fv 6572 df-riota 7406 df-ov 7452 df-oprab 7453 df-mpo 7454 df-om 7910 df-1st 8033 df-2nd 8034 df-frecs 8335 df-wrecs 8366 df-recs 8415 df-rdg 8454 df-1o 8512 df-2o 8513 df-oadd 8516 df-er 8753 df-en 9010 df-dom 9011 df-sdom 9012 df-fin 9013 df-sup 9469 df-inf 9470 df-dju 10017 df-card 10055 df-pnf 11382 df-mnf 11383 df-xr 11384 df-ltxr 11385 df-le 11386 df-sub 11580 df-neg 11581 df-div 12012 df-nn 12374 df-2 12443 df-3 12444 df-n0 12645 df-xnn0 12718 df-z 12732 df-uz 13004 df-rp 13159 df-icc 13521 df-fz 13678 df-fl 13970 df-seq 14183 df-exp 14243 df-hash 14514 df-cj 15307 df-re 15308 df-im 15309 df-sqrt 15443 df-abs 15444 df-dvds 16464 df-prm 16888 df-ppi 27473

This theorem is used by: chtppilimlem1 27846

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