tgoldbachgtda - Mathbox for Thierry Arnoux

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Theorem tgoldbachgtda 34639

Description: Lemma for tgoldbachgtd 34640. (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 34637	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ)
5	4	nnnn0d 12560	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
6		3nn0 12517	. . . . 5 ⊢ 3 ∈ ℕ₀
7	6	a1i 11	. . . 4 ⊢ (𝜑 → 3 ∈ ℕ₀)
8		inss2 4213	. . . . . 6 ⊢ (𝑂 ∩ ℙ) ⊆ ℙ
9		prmssnn 16693	. . . . . 6 ⊢ ℙ ⊆ ℕ
10	8, 9	sstri 3968	. . . . 5 ⊢ (𝑂 ∩ ℙ) ⊆ ℕ
11	10	a1i 11	. . . 4 ⊢ (𝜑 → (𝑂 ∩ ℙ) ⊆ ℕ)
12	5, 7, 11	reprfi2 34601	. . 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 34638	. . . . . . 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 11799	. . . . . 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 2937	. . . . 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 15714	. . . . . 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 15735	. . . . . 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 2786	. . . . 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 2939	. . 3 ⊢ (𝜑 → ((𝑂 ∩ ℙ)(repr‘3)𝑁) ≠ ∅)
27		hashnncl 14382	. . . 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 12269	. 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 2108 ≠ wne 2932 {crab 3415 ∩ cin 3925 ⊆ wss 3926 ∅c0 4308 class class class wbr 5119 ⟶wf 6526 ‘cfv 6530 (class class class)co 7403 ∘_f cof 7667 Fincfn 8957 0cc0 11127 1c1 11128 ici 11129 · cmul 11132 +∞cpnf 11264 < clt 11267 ≤ cle 11268 -cneg 11465 ℕcn 12238 2c2 12293 3c3 12294 4c4 12295 5c5 12296 7c7 12298 8c8 12299 9c9 12300 ℕ₀cn0 12499 ℤcz 12586 cdc 12706 (,)cioo 13360 [,)cico 13362 ↑cexp 14077 ♯chash 14346 Σcsu 15700 expce 16075 πcpi 16080 ∥ cdvds 16270 ℙcprime 16688 ∫citg 25569 Λcvma 27052 cdp2 32791 .cdp 32808 reprcrepr 34586 vtscvts 34613

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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7727 ax-reg 9604 ax-inf2 9653 ax-cc 10447 ax-ac2 10475 ax-cnex 11183 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204 ax-pre-sup 11205 ax-addf 11206 ax-ros335 34623 ax-ros336 34624

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 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-symdif 4228 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-tp 4606 df-op 4608 df-uni 4884 df-int 4923 df-iun 4969 df-iin 4970 df-disj 5087 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-se 5607 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6483 df-fun 6532 df-fn 6533 df-f 6534 df-f1 6535 df-fo 6536 df-f1o 6537 df-fv 6538 df-isom 6539 df-riota 7360 df-ov 7406 df-oprab 7407 df-mpo 7408 df-of 7669 df-ofr 7670 df-om 7860 df-1st 7986 df-2nd 7987 df-supp 8158 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-1o 8478 df-2o 8479 df-oadd 8482 df-omul 8483 df-er 8717 df-map 8840 df-pm 8841 df-ixp 8910 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-fsupp 9372 df-fi 9421 df-sup 9452 df-inf 9453 df-oi 9522 df-r1 9776 df-rank 9777 df-dju 9913 df-card 9951 df-acn 9954 df-ac 10128 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273 df-sub 11466 df-neg 11467 df-div 11893 df-nn 12239 df-2 12301 df-3 12302 df-4 12303 df-5 12304 df-6 12305 df-7 12306 df-8 12307 df-9 12308 df-n0 12500 df-xnn0 12573 df-z 12587 df-dec 12707 df-uz 12851 df-q 12963 df-rp 13007 df-xneg 13126 df-xadd 13127 df-xmul 13128 df-ioo 13364 df-ioc 13365 df-ico 13366 df-icc 13367 df-fz 13523 df-fzo 13670 df-fl 13807 df-mod 13885 df-seq 14018 df-exp 14078 df-fac 14290 df-bc 14319 df-hash 14347 df-word 14530 df-concat 14587 df-s1 14612 df-s2 14865 df-s3 14866 df-shft 15084 df-cj 15116 df-re 15117 df-im 15118 df-sqrt 15252 df-abs 15253 df-limsup 15485 df-clim 15502 df-rlim 15503 df-sum 15701 df-prod 15918 df-ef 16081 df-e 16082 df-sin 16083 df-cos 16084 df-tan 16085 df-pi 16086 df-dvds 16271 df-gcd 16512 df-prm 16689 df-pc 16855 df-struct 17164 df-sets 17181 df-slot 17199 df-ndx 17211 df-base 17227 df-ress 17250 df-plusg 17282 df-mulr 17283 df-starv 17284 df-sca 17285 df-vsca 17286 df-ip 17287 df-tset 17288 df-ple 17289 df-ds 17291 df-unif 17292 df-hom 17293 df-cco 17294 df-rest 17434 df-topn 17435 df-0g 17453 df-gsum 17454 df-topgen 17455 df-pt 17456 df-prds 17459 df-xrs 17514 df-qtop 17519 df-imas 17520 df-xps 17522 df-mre 17596 df-mrc 17597 df-acs 17599 df-mgm 18616 df-sgrp 18695 df-mnd 18711 df-submnd 18760 df-mulg 19049 df-cntz 19298 df-pmtr 19421 df-cmn 19761 df-psmet 21305 df-xmet 21306 df-met 21307 df-bl 21308 df-mopn 21309 df-fbas 21310 df-fg 21311 df-cnfld 21314 df-top 22830 df-topon 22847 df-topsp 22869 df-bases 22882 df-cld 22955 df-ntr 22956 df-cls 22957 df-nei 23034 df-lp 23072 df-perf 23073 df-cn 23163 df-cnp 23164 df-haus 23251 df-cmp 23323 df-tx 23498 df-hmeo 23691 df-fil 23782 df-fm 23874 df-flim 23875 df-flf 23876 df-xms 24257 df-ms 24258 df-tms 24259 df-cncf 24820 df-ovol 25415 df-vol 25416 df-mbf 25570 df-itg1 25571 df-itg2 25572 df-ibl 25573 df-itg 25574 df-0p 25621 df-limc 25817 df-dv 25818 df-ulm 26336 df-log 26515 df-cxp 26516 df-atan 26827 df-cht 27057 df-vma 27058 df-chp 27059 df-dp2 32792 df-dp 32809 df-repr 34587 df-vts 34614

This theorem is referenced by: tgoldbachgtd 34640

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