chpeq0 - Metamath Proof Explorer

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Theorem chpeq0 27117

Description: The second Chebyshev function is zero iff its argument is less than 2. (Contributed by Mario Carneiro, 9-Apr-2016.)

**Assertion**
Ref	Expression
chpeq0	⊢ (𝐴 ∈ ℝ → ((ψ‘𝐴) = 0 ↔ 𝐴 < 2))

**Proof of Theorem chpeq0**
Step	Hyp	Ref	Expression
1		2re 12202	. . . . 5 ⊢ 2 ∈ ℝ
2		lenlt 11194	. . . . 5 ⊢ ((2 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (2 ≤ 𝐴 ↔ ¬ 𝐴 < 2))
3	1, 2	mpan 690	. . . 4 ⊢ (𝐴 ∈ ℝ → (2 ≤ 𝐴 ↔ ¬ 𝐴 < 2))
4		chprpcl 27116	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 2 ≤ 𝐴) → (ψ‘𝐴) ∈ ℝ⁺)
5	4	rpne0d 12942	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 2 ≤ 𝐴) → (ψ‘𝐴) ≠ 0)
6	5	ex 412	. . . 4 ⊢ (𝐴 ∈ ℝ → (2 ≤ 𝐴 → (ψ‘𝐴) ≠ 0))
7	3, 6	sylbird 260	. . 3 ⊢ (𝐴 ∈ ℝ → (¬ 𝐴 < 2 → (ψ‘𝐴) ≠ 0))
8	7	necon4bd 2945	. 2 ⊢ (𝐴 ∈ ℝ → ((ψ‘𝐴) = 0 → 𝐴 < 2))
9		reflcl 13700	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ∈ ℝ)
10	9	adantr 480	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 2) → (⌊‘𝐴) ∈ ℝ)
11		1red 11116	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 2) → 1 ∈ ℝ)
12		2z 12507	. . . . . . . . . 10 ⊢ 2 ∈ ℤ
13		fllt 13710	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 2 ∈ ℤ) → (𝐴 < 2 ↔ (⌊‘𝐴) < 2))
14	12, 13	mpan2 691	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → (𝐴 < 2 ↔ (⌊‘𝐴) < 2))
15	14	biimpa 476	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 2) → (⌊‘𝐴) < 2)
16		df-2 12191	. . . . . . . 8 ⊢ 2 = (1 + 1)
17	15, 16	breqtrdi 5133	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 2) → (⌊‘𝐴) < (1 + 1))
18		flcl 13699	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ∈ ℤ)
19	18	adantr 480	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 2) → (⌊‘𝐴) ∈ ℤ)
20		1z 12505	. . . . . . . 8 ⊢ 1 ∈ ℤ
21		zleltp1 12526	. . . . . . . 8 ⊢ (((⌊‘𝐴) ∈ ℤ ∧ 1 ∈ ℤ) → ((⌊‘𝐴) ≤ 1 ↔ (⌊‘𝐴) < (1 + 1)))
22	19, 20, 21	sylancl 586	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 2) → ((⌊‘𝐴) ≤ 1 ↔ (⌊‘𝐴) < (1 + 1)))
23	17, 22	mpbird 257	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 2) → (⌊‘𝐴) ≤ 1)
24		chpwordi 27065	. . . . . 6 ⊢ (((⌊‘𝐴) ∈ ℝ ∧ 1 ∈ ℝ ∧ (⌊‘𝐴) ≤ 1) → (ψ‘(⌊‘𝐴)) ≤ (ψ‘1))
25	10, 11, 23, 24	syl3anc 1373	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 2) → (ψ‘(⌊‘𝐴)) ≤ (ψ‘1))
26		chpfl 27058	. . . . . 6 ⊢ (𝐴 ∈ ℝ → (ψ‘(⌊‘𝐴)) = (ψ‘𝐴))
27	26	adantr 480	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 2) → (ψ‘(⌊‘𝐴)) = (ψ‘𝐴))
28		chp1 27075	. . . . . 6 ⊢ (ψ‘1) = 0
29	28	a1i 11	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 2) → (ψ‘1) = 0)
30	25, 27, 29	3brtr3d 5123	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 2) → (ψ‘𝐴) ≤ 0)
31		chpge0 27034	. . . . 5 ⊢ (𝐴 ∈ ℝ → 0 ≤ (ψ‘𝐴))
32	31	adantr 480	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 2) → 0 ≤ (ψ‘𝐴))
33		chpcl 27032	. . . . . 6 ⊢ (𝐴 ∈ ℝ → (ψ‘𝐴) ∈ ℝ)
34	33	adantr 480	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 2) → (ψ‘𝐴) ∈ ℝ)
35		0re 11117	. . . . 5 ⊢ 0 ∈ ℝ
36		letri3 11201	. . . . 5 ⊢ (((ψ‘𝐴) ∈ ℝ ∧ 0 ∈ ℝ) → ((ψ‘𝐴) = 0 ↔ ((ψ‘𝐴) ≤ 0 ∧ 0 ≤ (ψ‘𝐴))))
37	34, 35, 36	sylancl 586	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 2) → ((ψ‘𝐴) = 0 ↔ ((ψ‘𝐴) ≤ 0 ∧ 0 ≤ (ψ‘𝐴))))
38	30, 32, 37	mpbir2and 713	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 2) → (ψ‘𝐴) = 0)
39	38	ex 412	. 2 ⊢ (𝐴 ∈ ℝ → (𝐴 < 2 → (ψ‘𝐴) = 0))
40	8, 39	impbid 212	1 ⊢ (𝐴 ∈ ℝ → ((ψ‘𝐴) = 0 ↔ 𝐴 < 2))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 class class class wbr 5092 ‘cfv 6482 (class class class)co 7349 ℝcr 11008 0cc0 11009 1c1 11010 + caddc 11012 < clt 11149 ≤ cle 11150 2c2 12183 ℤcz 12471 ⌊cfl 13694 ψcchp 27001

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5218 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-inf2 9537 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-pre-sup 11087 ax-addf 11088

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3343 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-tp 4582 df-op 4584 df-uni 4859 df-int 4897 df-iun 4943 df-iin 4944 df-br 5093 df-opab 5155 df-mpt 5174 df-tr 5200 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-se 5573 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6249 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-isom 6491 df-riota 7306 df-ov 7352 df-oprab 7353 df-mpo 7354 df-of 7613 df-om 7800 df-1st 7924 df-2nd 7925 df-supp 8094 df-frecs 8214 df-wrecs 8245 df-recs 8294 df-rdg 8332 df-1o 8388 df-2o 8389 df-oadd 8392 df-er 8625 df-map 8755 df-pm 8756 df-ixp 8825 df-en 8873 df-dom 8874 df-sdom 8875 df-fin 8876 df-fsupp 9252 df-fi 9301 df-sup 9332 df-inf 9333 df-oi 9402 df-dju 9797 df-card 9835 df-pnf 11151 df-mnf 11152 df-xr 11153 df-ltxr 11154 df-le 11155 df-sub 11349 df-neg 11350 df-div 11778 df-nn 12129 df-2 12191 df-3 12192 df-4 12193 df-5 12194 df-6 12195 df-7 12196 df-8 12197 df-9 12198 df-n0 12385 df-z 12472 df-dec 12592 df-uz 12736 df-q 12850 df-rp 12894 df-xneg 13014 df-xadd 13015 df-xmul 13016 df-ioo 13252 df-ioc 13253 df-ico 13254 df-icc 13255 df-fz 13411 df-fzo 13558 df-fl 13696 df-mod 13774 df-seq 13909 df-exp 13969 df-fac 14181 df-bc 14210 df-hash 14238 df-shft 14974 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-limsup 15378 df-clim 15395 df-rlim 15396 df-sum 15594 df-ef 15974 df-sin 15976 df-cos 15977 df-pi 15979 df-dvds 16164 df-gcd 16406 df-prm 16583 df-pc 16749 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-starv 17176 df-sca 17177 df-vsca 17178 df-ip 17179 df-tset 17180 df-ple 17181 df-ds 17183 df-unif 17184 df-hom 17185 df-cco 17186 df-rest 17326 df-topn 17327 df-0g 17345 df-gsum 17346 df-topgen 17347 df-pt 17348 df-prds 17351 df-xrs 17406 df-qtop 17411 df-imas 17412 df-xps 17414 df-mre 17488 df-mrc 17489 df-acs 17491 df-mgm 18514 df-sgrp 18593 df-mnd 18609 df-submnd 18658 df-mulg 18947 df-cntz 19196 df-cmn 19661 df-psmet 21253 df-xmet 21254 df-met 21255 df-bl 21256 df-mopn 21257 df-fbas 21258 df-fg 21259 df-cnfld 21262 df-top 22779 df-topon 22796 df-topsp 22818 df-bases 22831 df-cld 22904 df-ntr 22905 df-cls 22906 df-nei 22983 df-lp 23021 df-perf 23022 df-cn 23112 df-cnp 23113 df-haus 23200 df-tx 23447 df-hmeo 23640 df-fil 23731 df-fm 23823 df-flim 23824 df-flf 23825 df-xms 24206 df-ms 24207 df-tms 24208 df-cncf 24769 df-limc 25765 df-dv 25766 df-log 26463 df-cht 27005 df-vma 27006 df-chp 27007

This theorem is referenced by: chteq0 27118 chpo1ubb 27390 selberg2lem 27459 pntrmax 27473 pntrsumo1 27474 pntrlog2bndlem2 27487 pntrlog2bndlem4 27489

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