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Theorem chpeq0 25147

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 11290	. . . . 5 ⊢ 2 ∈ ℝ
2		lenlt 10316	. . . . 5 ⊢ ((2 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (2 ≤ 𝐴 ↔ ¬ 𝐴 < 2))
3	1, 2	mpan 670	. . . 4 ⊢ (𝐴 ∈ ℝ → (2 ≤ 𝐴 ↔ ¬ 𝐴 < 2))
4		chprpcl 25146	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 2 ≤ 𝐴) → (ψ‘𝐴) ∈ ℝ⁺)
5	4	rpne0d 12073	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 2 ≤ 𝐴) → (ψ‘𝐴) ≠ 0)
6	5	ex 397	. . . 4 ⊢ (𝐴 ∈ ℝ → (2 ≤ 𝐴 → (ψ‘𝐴) ≠ 0))
7	3, 6	sylbird 250	. . 3 ⊢ (𝐴 ∈ ℝ → (¬ 𝐴 < 2 → (ψ‘𝐴) ≠ 0))
8	7	necon4bd 2963	. 2 ⊢ (𝐴 ∈ ℝ → ((ψ‘𝐴) = 0 → 𝐴 < 2))
9		reflcl 12798	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ∈ ℝ)
10	9	adantr 466	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 2) → (⌊‘𝐴) ∈ ℝ)
11		1red 10255	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 2) → 1 ∈ ℝ)
12		2z 11609	. . . . . . . . . 10 ⊢ 2 ∈ ℤ
13		fllt 12808	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 2 ∈ ℤ) → (𝐴 < 2 ↔ (⌊‘𝐴) < 2))
14	12, 13	mpan2 671	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → (𝐴 < 2 ↔ (⌊‘𝐴) < 2))
15	14	biimpa 462	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 2) → (⌊‘𝐴) < 2)
16		df-2 11279	. . . . . . . 8 ⊢ 2 = (1 + 1)
17	15, 16	syl6breq 4827	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 2) → (⌊‘𝐴) < (1 + 1))
18		flcl 12797	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ∈ ℤ)
19	18	adantr 466	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 2) → (⌊‘𝐴) ∈ ℤ)
20		1z 11607	. . . . . . . 8 ⊢ 1 ∈ ℤ
21		zleltp1 11628	. . . . . . . 8 ⊢ (((⌊‘𝐴) ∈ ℤ ∧ 1 ∈ ℤ) → ((⌊‘𝐴) ≤ 1 ↔ (⌊‘𝐴) < (1 + 1)))
22	19, 20, 21	sylancl 574	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 2) → ((⌊‘𝐴) ≤ 1 ↔ (⌊‘𝐴) < (1 + 1)))
23	17, 22	mpbird 247	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 2) → (⌊‘𝐴) ≤ 1)
24		chpwordi 25097	. . . . . 6 ⊢ (((⌊‘𝐴) ∈ ℝ ∧ 1 ∈ ℝ ∧ (⌊‘𝐴) ≤ 1) → (ψ‘(⌊‘𝐴)) ≤ (ψ‘1))
25	10, 11, 23, 24	syl3anc 1476	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 2) → (ψ‘(⌊‘𝐴)) ≤ (ψ‘1))
26		chpfl 25090	. . . . . 6 ⊢ (𝐴 ∈ ℝ → (ψ‘(⌊‘𝐴)) = (ψ‘𝐴))
27	26	adantr 466	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 2) → (ψ‘(⌊‘𝐴)) = (ψ‘𝐴))
28		chp1 25107	. . . . . 6 ⊢ (ψ‘1) = 0
29	28	a1i 11	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 2) → (ψ‘1) = 0)
30	25, 27, 29	3brtr3d 4817	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 2) → (ψ‘𝐴) ≤ 0)
31		chpge0 25066	. . . . 5 ⊢ (𝐴 ∈ ℝ → 0 ≤ (ψ‘𝐴))
32	31	adantr 466	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 2) → 0 ≤ (ψ‘𝐴))
33		chpcl 25064	. . . . . 6 ⊢ (𝐴 ∈ ℝ → (ψ‘𝐴) ∈ ℝ)
34	33	adantr 466	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 2) → (ψ‘𝐴) ∈ ℝ)
35		0re 10240	. . . . 5 ⊢ 0 ∈ ℝ
36		letri3 10323	. . . . 5 ⊢ (((ψ‘𝐴) ∈ ℝ ∧ 0 ∈ ℝ) → ((ψ‘𝐴) = 0 ↔ ((ψ‘𝐴) ≤ 0 ∧ 0 ≤ (ψ‘𝐴))))
37	34, 35, 36	sylancl 574	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 2) → ((ψ‘𝐴) = 0 ↔ ((ψ‘𝐴) ≤ 0 ∧ 0 ≤ (ψ‘𝐴))))
38	30, 32, 37	mpbir2and 692	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 2) → (ψ‘𝐴) = 0)
39	38	ex 397	. 2 ⊢ (𝐴 ∈ ℝ → (𝐴 < 2 → (ψ‘𝐴) = 0))
40	8, 39	impbid 202	1 ⊢ (𝐴 ∈ ℝ → ((ψ‘𝐴) = 0 ↔ 𝐴 < 2))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 382 = wceq 1631 ∈ wcel 2145 ≠ wne 2943 class class class wbr 4786 ‘cfv 6029 (class class class)co 6791 ℝcr 10135 0cc0 10136 1c1 10137 + caddc 10139 < clt 10274 ≤ cle 10275 2c2 11270 ℤcz 11577 ⌊cfl 12792 ψcchp 25033

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7094 ax-inf2 8700 ax-cnex 10192 ax-resscn 10193 ax-1cn 10194 ax-icn 10195 ax-addcl 10196 ax-addrcl 10197 ax-mulcl 10198 ax-mulrcl 10199 ax-mulcom 10200 ax-addass 10201 ax-mulass 10202 ax-distr 10203 ax-i2m1 10204 ax-1ne0 10205 ax-1rid 10206 ax-rnegex 10207 ax-rrecex 10208 ax-cnre 10209 ax-pre-lttri 10210 ax-pre-lttrn 10211 ax-pre-ltadd 10212 ax-pre-mulgt0 10213 ax-pre-sup 10214 ax-addf 10215 ax-mulf 10216

This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-fal 1637 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-int 4612 df-iun 4656 df-iin 4657 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-se 5209 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5821 df-ord 5867 df-on 5868 df-lim 5869 df-suc 5870 df-iota 5992 df-fun 6031 df-fn 6032 df-f 6033 df-f1 6034 df-fo 6035 df-f1o 6036 df-fv 6037 df-isom 6038 df-riota 6752 df-ov 6794 df-oprab 6795 df-mpt2 6796 df-of 7042 df-om 7211 df-1st 7313 df-2nd 7314 df-supp 7445 df-wrecs 7557 df-recs 7619 df-rdg 7657 df-1o 7711 df-2o 7712 df-oadd 7715 df-er 7894 df-map 8009 df-pm 8010 df-ixp 8061 df-en 8108 df-dom 8109 df-sdom 8110 df-fin 8111 df-fsupp 8430 df-fi 8471 df-sup 8502 df-inf 8503 df-oi 8569 df-card 8963 df-cda 9190 df-pnf 10276 df-mnf 10277 df-xr 10278 df-ltxr 10279 df-le 10280 df-sub 10468 df-neg 10469 df-div 10885 df-nn 11221 df-2 11279 df-3 11280 df-4 11281 df-5 11282 df-6 11283 df-7 11284 df-8 11285 df-9 11286 df-n0 11493 df-z 11578 df-dec 11694 df-uz 11887 df-q 11990 df-rp 12029 df-xneg 12144 df-xadd 12145 df-xmul 12146 df-ioo 12377 df-ioc 12378 df-ico 12379 df-icc 12380 df-fz 12527 df-fzo 12667 df-fl 12794 df-mod 12870 df-seq 13002 df-exp 13061 df-fac 13258 df-bc 13287 df-hash 13315 df-shft 14008 df-cj 14040 df-re 14041 df-im 14042 df-sqrt 14176 df-abs 14177 df-limsup 14403 df-clim 14420 df-rlim 14421 df-sum 14618 df-ef 14997 df-sin 14999 df-cos 15000 df-pi 15002 df-dvds 15183 df-gcd 15418 df-prm 15586 df-pc 15742 df-struct 16059 df-ndx 16060 df-slot 16061 df-base 16063 df-sets 16064 df-ress 16065 df-plusg 16155 df-mulr 16156 df-starv 16157 df-sca 16158 df-vsca 16159 df-ip 16160 df-tset 16161 df-ple 16162 df-ds 16165 df-unif 16166 df-hom 16167 df-cco 16168 df-rest 16284 df-topn 16285 df-0g 16303 df-gsum 16304 df-topgen 16305 df-pt 16306 df-prds 16309 df-xrs 16363 df-qtop 16368 df-imas 16369 df-xps 16371 df-mre 16447 df-mrc 16448 df-acs 16450 df-mgm 17443 df-sgrp 17485 df-mnd 17496 df-submnd 17537 df-mulg 17742 df-cntz 17950 df-cmn 18395 df-psmet 19946 df-xmet 19947 df-met 19948 df-bl 19949 df-mopn 19950 df-fbas 19951 df-fg 19952 df-cnfld 19955 df-top 20912 df-topon 20929 df-topsp 20951 df-bases 20964 df-cld 21037 df-ntr 21038 df-cls 21039 df-nei 21116 df-lp 21154 df-perf 21155 df-cn 21245 df-cnp 21246 df-haus 21333 df-tx 21579 df-hmeo 21772 df-fil 21863 df-fm 21955 df-flim 21956 df-flf 21957 df-xms 22338 df-ms 22339 df-tms 22340 df-cncf 22894 df-limc 23843 df-dv 23844 df-log 24517 df-cht 25037 df-vma 25038 df-chp 25039

This theorem is referenced by: chteq0 25148 chpo1ubb 25384 selberg2lem 25453 pntrmax 25467 pntrsumo1 25468 pntrlog2bndlem2 25481 pntrlog2bndlem4 25483

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