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Theorem chpeq0 26935

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 12290	. . . . 5 ⊢ 2 ∈ ℝ
2		lenlt 11296	. . . . 5 ⊢ ((2 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (2 ≤ 𝐴 ↔ ¬ 𝐴 < 2))
3	1, 2	mpan 688	. . . 4 ⊢ (𝐴 ∈ ℝ → (2 ≤ 𝐴 ↔ ¬ 𝐴 < 2))
4		chprpcl 26934	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 2 ≤ 𝐴) → (ψ‘𝐴) ∈ ℝ⁺)
5	4	rpne0d 13025	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 2 ≤ 𝐴) → (ψ‘𝐴) ≠ 0)
6	5	ex 413	. . . 4 ⊢ (𝐴 ∈ ℝ → (2 ≤ 𝐴 → (ψ‘𝐴) ≠ 0))
7	3, 6	sylbird 259	. . 3 ⊢ (𝐴 ∈ ℝ → (¬ 𝐴 < 2 → (ψ‘𝐴) ≠ 0))
8	7	necon4bd 2960	. 2 ⊢ (𝐴 ∈ ℝ → ((ψ‘𝐴) = 0 → 𝐴 < 2))
9		reflcl 13765	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ∈ ℝ)
10	9	adantr 481	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 2) → (⌊‘𝐴) ∈ ℝ)
11		1red 11219	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 2) → 1 ∈ ℝ)
12		2z 12598	. . . . . . . . . 10 ⊢ 2 ∈ ℤ
13		fllt 13775	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 2 ∈ ℤ) → (𝐴 < 2 ↔ (⌊‘𝐴) < 2))
14	12, 13	mpan2 689	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → (𝐴 < 2 ↔ (⌊‘𝐴) < 2))
15	14	biimpa 477	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 2) → (⌊‘𝐴) < 2)
16		df-2 12279	. . . . . . . 8 ⊢ 2 = (1 + 1)
17	15, 16	breqtrdi 5189	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 2) → (⌊‘𝐴) < (1 + 1))
18		flcl 13764	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ∈ ℤ)
19	18	adantr 481	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 2) → (⌊‘𝐴) ∈ ℤ)
20		1z 12596	. . . . . . . 8 ⊢ 1 ∈ ℤ
21		zleltp1 12617	. . . . . . . 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 256	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 2) → (⌊‘𝐴) ≤ 1)
24		chpwordi 26885	. . . . . 6 ⊢ (((⌊‘𝐴) ∈ ℝ ∧ 1 ∈ ℝ ∧ (⌊‘𝐴) ≤ 1) → (ψ‘(⌊‘𝐴)) ≤ (ψ‘1))
25	10, 11, 23, 24	syl3anc 1371	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 2) → (ψ‘(⌊‘𝐴)) ≤ (ψ‘1))
26		chpfl 26878	. . . . . 6 ⊢ (𝐴 ∈ ℝ → (ψ‘(⌊‘𝐴)) = (ψ‘𝐴))
27	26	adantr 481	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 2) → (ψ‘(⌊‘𝐴)) = (ψ‘𝐴))
28		chp1 26895	. . . . . 6 ⊢ (ψ‘1) = 0
29	28	a1i 11	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 2) → (ψ‘1) = 0)
30	25, 27, 29	3brtr3d 5179	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 2) → (ψ‘𝐴) ≤ 0)
31		chpge0 26854	. . . . 5 ⊢ (𝐴 ∈ ℝ → 0 ≤ (ψ‘𝐴))
32	31	adantr 481	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 2) → 0 ≤ (ψ‘𝐴))
33		chpcl 26852	. . . . . 6 ⊢ (𝐴 ∈ ℝ → (ψ‘𝐴) ∈ ℝ)
34	33	adantr 481	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 2) → (ψ‘𝐴) ∈ ℝ)
35		0re 11220	. . . . 5 ⊢ 0 ∈ ℝ
36		letri3 11303	. . . . 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 711	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 2) → (ψ‘𝐴) = 0)
39	38	ex 413	. 2 ⊢ (𝐴 ∈ ℝ → (𝐴 < 2 → (ψ‘𝐴) = 0))
40	8, 39	impbid 211	1 ⊢ (𝐴 ∈ ℝ → ((ψ‘𝐴) = 0 ↔ 𝐴 < 2))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 class class class wbr 5148 ‘cfv 6543 (class class class)co 7411 ℝcr 11111 0cc0 11112 1c1 11113 + caddc 11115 < clt 11252 ≤ cle 11253 2c2 12271 ℤcz 12562 ⌊cfl 13759 ψcchp 26821

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-inf2 9638 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 ax-addf 11191 ax-mulf 11192

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7672 df-om 7858 df-1st 7977 df-2nd 7978 df-supp 8149 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-2o 8469 df-oadd 8472 df-er 8705 df-map 8824 df-pm 8825 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-fi 9408 df-sup 9439 df-inf 9440 df-oi 9507 df-dju 9898 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-q 12937 df-rp 12979 df-xneg 13096 df-xadd 13097 df-xmul 13098 df-ioo 13332 df-ioc 13333 df-ico 13334 df-icc 13335 df-fz 13489 df-fzo 13632 df-fl 13761 df-mod 13839 df-seq 13971 df-exp 14032 df-fac 14238 df-bc 14267 df-hash 14295 df-shft 15018 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-limsup 15419 df-clim 15436 df-rlim 15437 df-sum 15637 df-ef 16015 df-sin 16017 df-cos 16018 df-pi 16020 df-dvds 16202 df-gcd 16440 df-prm 16613 df-pc 16774 df-struct 17084 df-sets 17101 df-slot 17119 df-ndx 17131 df-base 17149 df-ress 17178 df-plusg 17214 df-mulr 17215 df-starv 17216 df-sca 17217 df-vsca 17218 df-ip 17219 df-tset 17220 df-ple 17221 df-ds 17223 df-unif 17224 df-hom 17225 df-cco 17226 df-rest 17372 df-topn 17373 df-0g 17391 df-gsum 17392 df-topgen 17393 df-pt 17394 df-prds 17397 df-xrs 17452 df-qtop 17457 df-imas 17458 df-xps 17460 df-mre 17534 df-mrc 17535 df-acs 17537 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-submnd 18706 df-mulg 18987 df-cntz 19222 df-cmn 19691 df-psmet 21136 df-xmet 21137 df-met 21138 df-bl 21139 df-mopn 21140 df-fbas 21141 df-fg 21142 df-cnfld 21145 df-top 22616 df-topon 22633 df-topsp 22655 df-bases 22669 df-cld 22743 df-ntr 22744 df-cls 22745 df-nei 22822 df-lp 22860 df-perf 22861 df-cn 22951 df-cnp 22952 df-haus 23039 df-tx 23286 df-hmeo 23479 df-fil 23570 df-fm 23662 df-flim 23663 df-flf 23664 df-xms 24046 df-ms 24047 df-tms 24048 df-cncf 24618 df-limc 25607 df-dv 25608 df-log 26289 df-cht 26825 df-vma 26826 df-chp 26827

This theorem is referenced by: chteq0 26936 chpo1ubb 27208 selberg2lem 27277 pntrmax 27291 pntrsumo1 27292 pntrlog2bndlem2 27305 pntrlog2bndlem4 27307

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