tgoldbachgtda - Mathbox for Thierry Arnoux

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Theorem tgoldbachgtda 34645

Description: Lemma for tgoldbachgtd 34646. (Contributed by Thierry Arnoux, 15-Dec-2021.)

**Hypotheses**
Ref	Expression
tgoldbachgtda.o	⊢ 𝑂 = {𝑧 ∈ ℤ ∣ ¬ 2 ∥ 𝑧}
tgoldbachgtda.n	⊢ (𝜑 → 𝑁 ∈ 𝑂)
tgoldbachgtda.0	⊢ (𝜑 → (;10↑;27) ≤ 𝑁)
tgoldbachgtda.h	⊢ (𝜑 → 𝐻:ℕ⟶(0[,)+∞))
tgoldbachgtda.k	⊢ (𝜑 → 𝐾:ℕ⟶(0[,)+∞))
tgoldbachgtda.1	⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐾‘𝑚) ≤ (1._0_7_9_9_55))
tgoldbachgtda.2	⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐻‘𝑚) ≤ (1._4_14))
tgoldbachgtda.3	⊢ (𝜑 → ((0._0_0_0_4_2_2_48) · (𝑁↑2)) ≤ ∫(0(,)1)(((((Λ ∘_f · 𝐻)vts𝑁)‘𝑥) · ((((Λ ∘_f · 𝐾)vts𝑁)‘𝑥)↑2)) · (exp‘((i · (2 · π)) · (-𝑁 · 𝑥)))) d𝑥)

**Assertion**
Ref	Expression
tgoldbachgtda	⊢ (𝜑 → 0 < (♯‘((𝑂 ∩ ℙ)(repr‘3)𝑁)))

Distinct variable groups: 𝑚,𝐻,𝑥 𝑚,𝐾,𝑥 𝑚,𝑁,𝑥,𝑧 𝑚,𝑂,𝑧 𝜑,𝑚,𝑥 Allowed substitution hints: 𝜑(𝑧) 𝐻(𝑧) 𝐾(𝑧) 𝑂(𝑥)

Proof of Theorem tgoldbachgtda Dummy variable 𝑛 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		tgoldbachgtda.o	. . . . . 6 ⊢ 𝑂 = {𝑧 ∈ ℤ ∣ ¬ 2 ∥ 𝑧}
2		tgoldbachgtda.n	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ 𝑂)
3		tgoldbachgtda.0	. . . . . 6 ⊢ (𝜑 → (;10↑;27) ≤ 𝑁)
4	1, 2, 3	tgoldbachgnn 34643	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ)
5	4	nnnn0d 12479	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
6		3nn0 12436	. . . . 5 ⊢ 3 ∈ ℕ₀
7	6	a1i 11	. . . 4 ⊢ (𝜑 → 3 ∈ ℕ₀)
8		inss2 4197	. . . . . 6 ⊢ (𝑂 ∩ ℙ) ⊆ ℙ
9		prmssnn 16622	. . . . . 6 ⊢ ℙ ⊆ ℕ
10	8, 9	sstri 3953	. . . . 5 ⊢ (𝑂 ∩ ℙ) ⊆ ℕ
11	10	a1i 11	. . . 4 ⊢ (𝜑 → (𝑂 ∩ ℙ) ⊆ ℕ)
12	5, 7, 11	reprfi2 34607	. . 3 ⊢ (𝜑 → ((𝑂 ∩ ℙ)(repr‘3)𝑁) ∈ Fin)
13		tgoldbachgtda.h	. . . . . . . 8 ⊢ (𝜑 → 𝐻:ℕ⟶(0[,)+∞))
14		tgoldbachgtda.k	. . . . . . . 8 ⊢ (𝜑 → 𝐾:ℕ⟶(0[,)+∞))
15		tgoldbachgtda.1	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐾‘𝑚) ≤ (1._0_7_9_9_55))
16		tgoldbachgtda.2	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐻‘𝑚) ≤ (1._4_14))
17		tgoldbachgtda.3	. . . . . . . 8 ⊢ (𝜑 → ((0._0_0_0_4_2_2_48) · (𝑁↑2)) ≤ ∫(0(,)1)(((((Λ ∘_f · 𝐻)vts𝑁)‘𝑥) · ((((Λ ∘_f · 𝐾)vts𝑁)‘𝑥)↑2)) · (exp‘((i · (2 · π)) · (-𝑁 · 𝑥)))) d𝑥)
18	1, 2, 3, 13, 14, 15, 16, 17	tgoldbachgtde 34644	. . . . . . 7 ⊢ (𝜑 → 0 < Σ𝑛 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑁)(((Λ‘(𝑛‘0)) · (𝐻‘(𝑛‘0))) · (((Λ‘(𝑛‘1)) · (𝐾‘(𝑛‘1))) · ((Λ‘(𝑛‘2)) · (𝐾‘(𝑛‘2))))))
19	18	gt0ne0d 11718	. . . . . 6 ⊢ (𝜑 → Σ𝑛 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑁)(((Λ‘(𝑛‘0)) · (𝐻‘(𝑛‘0))) · (((Λ‘(𝑛‘1)) · (𝐾‘(𝑛‘1))) · ((Λ‘(𝑛‘2)) · (𝐾‘(𝑛‘2))))) ≠ 0)
20	19	neneqd 2930	. . . . 5 ⊢ (𝜑 → ¬ Σ𝑛 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑁)(((Λ‘(𝑛‘0)) · (𝐻‘(𝑛‘0))) · (((Λ‘(𝑛‘1)) · (𝐾‘(𝑛‘1))) · ((Λ‘(𝑛‘2)) · (𝐾‘(𝑛‘2))))) = 0)
21		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑂 ∩ ℙ)(repr‘3)𝑁) = ∅) → ((𝑂 ∩ ℙ)(repr‘3)𝑁) = ∅)
22	21	sumeq1d 15642	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑂 ∩ ℙ)(repr‘3)𝑁) = ∅) → Σ𝑛 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑁)(((Λ‘(𝑛‘0)) · (𝐻‘(𝑛‘0))) · (((Λ‘(𝑛‘1)) · (𝐾‘(𝑛‘1))) · ((Λ‘(𝑛‘2)) · (𝐾‘(𝑛‘2))))) = Σ𝑛 ∈ ∅ (((Λ‘(𝑛‘0)) · (𝐻‘(𝑛‘0))) · (((Λ‘(𝑛‘1)) · (𝐾‘(𝑛‘1))) · ((Λ‘(𝑛‘2)) · (𝐾‘(𝑛‘2))))))
23		sum0 15663	. . . . . 6 ⊢ Σ𝑛 ∈ ∅ (((Λ‘(𝑛‘0)) · (𝐻‘(𝑛‘0))) · (((Λ‘(𝑛‘1)) · (𝐾‘(𝑛‘1))) · ((Λ‘(𝑛‘2)) · (𝐾‘(𝑛‘2))))) = 0
24	22, 23	eqtrdi 2780	. . . . 5 ⊢ ((𝜑 ∧ ((𝑂 ∩ ℙ)(repr‘3)𝑁) = ∅) → Σ𝑛 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑁)(((Λ‘(𝑛‘0)) · (𝐻‘(𝑛‘0))) · (((Λ‘(𝑛‘1)) · (𝐾‘(𝑛‘1))) · ((Λ‘(𝑛‘2)) · (𝐾‘(𝑛‘2))))) = 0)
25	20, 24	mtand 815	. . . 4 ⊢ (𝜑 → ¬ ((𝑂 ∩ ℙ)(repr‘3)𝑁) = ∅)
26	25	neqned 2932	. . 3 ⊢ (𝜑 → ((𝑂 ∩ ℙ)(repr‘3)𝑁) ≠ ∅)
27		hashnncl 14307	. . . 4 ⊢ (((𝑂 ∩ ℙ)(repr‘3)𝑁) ∈ Fin → ((♯‘((𝑂 ∩ ℙ)(repr‘3)𝑁)) ∈ ℕ ↔ ((𝑂 ∩ ℙ)(repr‘3)𝑁) ≠ ∅))
28	27	biimpar 477	. . 3 ⊢ ((((𝑂 ∩ ℙ)(repr‘3)𝑁) ∈ Fin ∧ ((𝑂 ∩ ℙ)(repr‘3)𝑁) ≠ ∅) → (♯‘((𝑂 ∩ ℙ)(repr‘3)𝑁)) ∈ ℕ)
29	12, 26, 28	syl2anc 584	. 2 ⊢ (𝜑 → (♯‘((𝑂 ∩ ℙ)(repr‘3)𝑁)) ∈ ℕ)
30		nngt0 12193	. 2 ⊢ ((♯‘((𝑂 ∩ ℙ)(repr‘3)𝑁)) ∈ ℕ → 0 < (♯‘((𝑂 ∩ ℙ)(repr‘3)𝑁)))
31	29, 30	syl 17	1 ⊢ (𝜑 → 0 < (♯‘((𝑂 ∩ ℙ)(repr‘3)𝑁)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 {crab 3402 ∩ cin 3910 ⊆ wss 3911 ∅c0 4292 class class class wbr 5102 ⟶wf 6495 ‘cfv 6499 (class class class)co 7369 ∘_f cof 7631 Fincfn 8895 0cc0 11044 1c1 11045 ici 11046 · cmul 11049 +∞cpnf 11181 < clt 11184 ≤ cle 11185 -cneg 11382 ℕcn 12162 2c2 12217 3c3 12218 4c4 12219 5c5 12220 7c7 12222 8c8 12223 9c9 12224 ℕ₀cn0 12418 ℤcz 12505 cdc 12625 (,)cioo 13282 [,)cico 13284 ↑cexp 14002 ♯chash 14271 Σcsu 15628 expce 16003 πcpi 16008 ∥ cdvds 16198 ℙcprime 16617 ∫citg 25552 Λcvma 27035 cdp2 32841 .cdp 32858 reprcrepr 34592 vtscvts 34619

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 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-reg 9521 ax-inf2 9570 ax-cc 10364 ax-ac2 10392 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121 ax-pre-sup 11122 ax-addf 11123 ax-ros335 34629 ax-ros336 34630

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 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-symdif 4212 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4868 df-int 4907 df-iun 4953 df-iin 4954 df-disj 5070 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-isom 6508 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-of 7633 df-ofr 7634 df-om 7823 df-1st 7947 df-2nd 7948 df-supp 8117 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-1o 8411 df-2o 8412 df-oadd 8415 df-omul 8416 df-er 8648 df-map 8778 df-pm 8779 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9289 df-fi 9338 df-sup 9369 df-inf 9370 df-oi 9439 df-r1 9693 df-rank 9694 df-dju 9830 df-card 9868 df-acn 9871 df-ac 10045 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-div 11812 df-nn 12163 df-2 12225 df-3 12226 df-4 12227 df-5 12228 df-6 12229 df-7 12230 df-8 12231 df-9 12232 df-n0 12419 df-xnn0 12492 df-z 12506 df-dec 12626 df-uz 12770 df-q 12884 df-rp 12928 df-xneg 13048 df-xadd 13049 df-xmul 13050 df-ioo 13286 df-ioc 13287 df-ico 13288 df-icc 13289 df-fz 13445 df-fzo 13592 df-fl 13730 df-mod 13808 df-seq 13943 df-exp 14003 df-fac 14215 df-bc 14244 df-hash 14272 df-word 14455 df-concat 14512 df-s1 14537 df-s2 14790 df-s3 14791 df-shft 15009 df-cj 15041 df-re 15042 df-im 15043 df-sqrt 15177 df-abs 15178 df-limsup 15413 df-clim 15430 df-rlim 15431 df-sum 15629 df-prod 15846 df-ef 16009 df-e 16010 df-sin 16011 df-cos 16012 df-tan 16013 df-pi 16014 df-dvds 16199 df-gcd 16441 df-prm 16618 df-pc 16784 df-struct 17093 df-sets 17110 df-slot 17128 df-ndx 17140 df-base 17156 df-ress 17177 df-plusg 17209 df-mulr 17210 df-starv 17211 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-hom 17220 df-cco 17221 df-rest 17361 df-topn 17362 df-0g 17380 df-gsum 17381 df-topgen 17382 df-pt 17383 df-prds 17386 df-xrs 17441 df-qtop 17446 df-imas 17447 df-xps 17449 df-mre 17523 df-mrc 17524 df-acs 17526 df-mgm 18549 df-sgrp 18628 df-mnd 18644 df-submnd 18693 df-mulg 18982 df-cntz 19231 df-pmtr 19356 df-cmn 19696 df-psmet 21288 df-xmet 21289 df-met 21290 df-bl 21291 df-mopn 21292 df-fbas 21293 df-fg 21294 df-cnfld 21297 df-top 22814 df-topon 22831 df-topsp 22853 df-bases 22866 df-cld 22939 df-ntr 22940 df-cls 22941 df-nei 23018 df-lp 23056 df-perf 23057 df-cn 23147 df-cnp 23148 df-haus 23235 df-cmp 23307 df-tx 23482 df-hmeo 23675 df-fil 23766 df-fm 23858 df-flim 23859 df-flf 23860 df-xms 24241 df-ms 24242 df-tms 24243 df-cncf 24804 df-ovol 25398 df-vol 25399 df-mbf 25553 df-itg1 25554 df-itg2 25555 df-ibl 25556 df-itg 25557 df-0p 25604 df-limc 25800 df-dv 25801 df-ulm 26319 df-log 26498 df-cxp 26499 df-atan 26810 df-cht 27040 df-vma 27041 df-chp 27042 df-dp2 32842 df-dp 32859 df-repr 34593 df-vts 34620

This theorem is referenced by: tgoldbachgtd 34646

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