chpeq0 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > chpeq0		Structured version Visualization version GIF version

Theorem chpeq0 27119

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 12260	. . . . 5 ⊢ 2 ∈ ℝ
2		lenlt 11252	. . . . 5 ⊢ ((2 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (2 ≤ 𝐴 ↔ ¬ 𝐴 < 2))
3	1, 2	mpan 690	. . . 4 ⊢ (𝐴 ∈ ℝ → (2 ≤ 𝐴 ↔ ¬ 𝐴 < 2))
4		chprpcl 27118	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 2 ≤ 𝐴) → (ψ‘𝐴) ∈ ℝ⁺)
5	4	rpne0d 13000	. . . . 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 13758	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ∈ ℝ)
10	9	adantr 480	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 2) → (⌊‘𝐴) ∈ ℝ)
11		1red 11175	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 2) → 1 ∈ ℝ)
12		2z 12565	. . . . . . . . . 10 ⊢ 2 ∈ ℤ
13		fllt 13768	. . . . . . . . . 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 12249	. . . . . . . 8 ⊢ 2 = (1 + 1)
17	15, 16	breqtrdi 5148	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 2) → (⌊‘𝐴) < (1 + 1))
18		flcl 13757	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ∈ ℤ)
19	18	adantr 480	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 2) → (⌊‘𝐴) ∈ ℤ)
20		1z 12563	. . . . . . . 8 ⊢ 1 ∈ ℤ
21		zleltp1 12584	. . . . . . . 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 27067	. . . . . 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 27060	. . . . . 6 ⊢ (𝐴 ∈ ℝ → (ψ‘(⌊‘𝐴)) = (ψ‘𝐴))
27	26	adantr 480	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 2) → (ψ‘(⌊‘𝐴)) = (ψ‘𝐴))
28		chp1 27077	. . . . . 6 ⊢ (ψ‘1) = 0
29	28	a1i 11	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 2) → (ψ‘1) = 0)
30	25, 27, 29	3brtr3d 5138	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 2) → (ψ‘𝐴) ≤ 0)
31		chpge0 27036	. . . . 5 ⊢ (𝐴 ∈ ℝ → 0 ≤ (ψ‘𝐴))
32	31	adantr 480	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 2) → 0 ≤ (ψ‘𝐴))
33		chpcl 27034	. . . . . 6 ⊢ (𝐴 ∈ ℝ → (ψ‘𝐴) ∈ ℝ)
34	33	adantr 480	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 2) → (ψ‘𝐴) ∈ ℝ)
35		0re 11176	. . . . 5 ⊢ 0 ∈ ℝ
36		letri3 11259	. . . . 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 5107 ‘cfv 6511 (class class class)co 7387 ℝcr 11067 0cc0 11068 1c1 11069 + caddc 11071 < clt 11208 ≤ cle 11209 2c2 12241 ℤcz 12529 ⌊cfl 13752 ψcchp 27003

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 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-inf2 9594 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-pre-sup 11146 ax-addf 11147

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 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-iin 4958 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-se 5592 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-isom 6520 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-of 7653 df-om 7843 df-1st 7968 df-2nd 7969 df-supp 8140 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-2o 8435 df-oadd 8438 df-er 8671 df-map 8801 df-pm 8802 df-ixp 8871 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-fsupp 9313 df-fi 9362 df-sup 9393 df-inf 9394 df-oi 9463 df-dju 9854 df-card 9892 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-div 11836 df-nn 12187 df-2 12249 df-3 12250 df-4 12251 df-5 12252 df-6 12253 df-7 12254 df-8 12255 df-9 12256 df-n0 12443 df-z 12530 df-dec 12650 df-uz 12794 df-q 12908 df-rp 12952 df-xneg 13072 df-xadd 13073 df-xmul 13074 df-ioo 13310 df-ioc 13311 df-ico 13312 df-icc 13313 df-fz 13469 df-fzo 13616 df-fl 13754 df-mod 13832 df-seq 13967 df-exp 14027 df-fac 14239 df-bc 14268 df-hash 14296 df-shft 15033 df-cj 15065 df-re 15066 df-im 15067 df-sqrt 15201 df-abs 15202 df-limsup 15437 df-clim 15454 df-rlim 15455 df-sum 15653 df-ef 16033 df-sin 16035 df-cos 16036 df-pi 16038 df-dvds 16223 df-gcd 16465 df-prm 16642 df-pc 16808 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-hom 17244 df-cco 17245 df-rest 17385 df-topn 17386 df-0g 17404 df-gsum 17405 df-topgen 17406 df-pt 17407 df-prds 17410 df-xrs 17465 df-qtop 17470 df-imas 17471 df-xps 17473 df-mre 17547 df-mrc 17548 df-acs 17550 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-submnd 18711 df-mulg 19000 df-cntz 19249 df-cmn 19712 df-psmet 21256 df-xmet 21257 df-met 21258 df-bl 21259 df-mopn 21260 df-fbas 21261 df-fg 21262 df-cnfld 21265 df-top 22781 df-topon 22798 df-topsp 22820 df-bases 22833 df-cld 22906 df-ntr 22907 df-cls 22908 df-nei 22985 df-lp 23023 df-perf 23024 df-cn 23114 df-cnp 23115 df-haus 23202 df-tx 23449 df-hmeo 23642 df-fil 23733 df-fm 23825 df-flim 23826 df-flf 23827 df-xms 24208 df-ms 24209 df-tms 24210 df-cncf 24771 df-limc 25767 df-dv 25768 df-log 26465 df-cht 27007 df-vma 27008 df-chp 27009

This theorem is referenced by: chteq0 27120 chpo1ubb 27392 selberg2lem 27461 pntrmax 27475 pntrsumo1 27476 pntrlog2bndlem2 27489 pntrlog2bndlem4 27491

W3C validator