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Mirrors > Home > MPE Home > Th. List > Mathboxes > tgoldbachgtda | Structured version Visualization version GIF version |
Description: Lemma for tgoldbachgtd 32225. (Contributed by Thierry Arnoux, 15-Dec-2021.) |
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) |
Ref | Expression |
---|---|
tgoldbachgtda | ⊢ (𝜑 → 0 < (♯‘((𝑂 ∩ ℙ)(repr‘3)𝑁))) |
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 32222 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ) |
5 | 4 | nnnn0d 12049 | . . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
6 | 3nn0 12007 | . . . . 5 ⊢ 3 ∈ ℕ0 | |
7 | 6 | a1i 11 | . . . 4 ⊢ (𝜑 → 3 ∈ ℕ0) |
8 | inss2 4130 | . . . . . 6 ⊢ (𝑂 ∩ ℙ) ⊆ ℙ | |
9 | prmssnn 16130 | . . . . . 6 ⊢ ℙ ⊆ ℕ | |
10 | 8, 9 | sstri 3896 | . . . . 5 ⊢ (𝑂 ∩ ℙ) ⊆ ℕ |
11 | 10 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝑂 ∩ ℙ) ⊆ ℕ) |
12 | 5, 7, 11 | reprfi2 32186 | . . 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 32223 | . . . . . . 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 11295 | . . . . . 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 2940 | . . . . 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 488 | . . . . . . 7 ⊢ ((𝜑 ∧ ((𝑂 ∩ ℙ)(repr‘3)𝑁) = ∅) → ((𝑂 ∩ ℙ)(repr‘3)𝑁) = ∅) | |
22 | 21 | sumeq1d 15164 | . . . . . 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 15184 | . . . . . 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 2790 | . . . . 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 816 | . . . 4 ⊢ (𝜑 → ¬ ((𝑂 ∩ ℙ)(repr‘3)𝑁) = ∅) |
26 | 25 | neqned 2942 | . . 3 ⊢ (𝜑 → ((𝑂 ∩ ℙ)(repr‘3)𝑁) ≠ ∅) |
27 | hashnncl 13832 | . . . 4 ⊢ (((𝑂 ∩ ℙ)(repr‘3)𝑁) ∈ Fin → ((♯‘((𝑂 ∩ ℙ)(repr‘3)𝑁)) ∈ ℕ ↔ ((𝑂 ∩ ℙ)(repr‘3)𝑁) ≠ ∅)) | |
28 | 27 | biimpar 481 | . . 3 ⊢ ((((𝑂 ∩ ℙ)(repr‘3)𝑁) ∈ Fin ∧ ((𝑂 ∩ ℙ)(repr‘3)𝑁) ≠ ∅) → (♯‘((𝑂 ∩ ℙ)(repr‘3)𝑁)) ∈ ℕ) |
29 | 12, 26, 28 | syl2anc 587 | . 2 ⊢ (𝜑 → (♯‘((𝑂 ∩ ℙ)(repr‘3)𝑁)) ∈ ℕ) |
30 | nngt0 11760 | . 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 399 = wceq 1542 ∈ wcel 2114 ≠ wne 2935 {crab 3058 ∩ cin 3852 ⊆ wss 3853 ∅c0 4221 class class class wbr 5040 ⟶wf 6346 ‘cfv 6350 (class class class)co 7183 ∘f cof 7436 Fincfn 8568 0cc0 10628 1c1 10629 ici 10630 · cmul 10633 +∞cpnf 10763 < clt 10766 ≤ cle 10767 -cneg 10962 ℕcn 11729 2c2 11784 3c3 11785 4c4 11786 5c5 11787 7c7 11789 8c8 11790 9c9 11791 ℕ0cn0 11989 ℤcz 12075 ;cdc 12192 (,)cioo 12834 [,)cico 12836 ↑cexp 13534 ♯chash 13795 Σcsu 15148 expce 15520 πcpi 15525 ∥ cdvds 15712 ℙcprime 16125 ∫citg 24383 Λcvma 25842 _cdp2 30733 .cdp 30750 reprcrepr 32171 vtscvts 32198 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7492 ax-reg 9142 ax-inf2 9190 ax-cc 9948 ax-ac2 9976 ax-cnex 10684 ax-resscn 10685 ax-1cn 10686 ax-icn 10687 ax-addcl 10688 ax-addrcl 10689 ax-mulcl 10690 ax-mulrcl 10691 ax-mulcom 10692 ax-addass 10693 ax-mulass 10694 ax-distr 10695 ax-i2m1 10696 ax-1ne0 10697 ax-1rid 10698 ax-rnegex 10699 ax-rrecex 10700 ax-cnre 10701 ax-pre-lttri 10702 ax-pre-lttrn 10703 ax-pre-ltadd 10704 ax-pre-mulgt0 10705 ax-pre-sup 10706 ax-addf 10707 ax-mulf 10708 ax-ros335 32208 ax-ros336 32209 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-symdif 4143 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-tp 4531 df-op 4533 df-uni 4807 df-int 4847 df-iun 4893 df-iin 4894 df-disj 5006 df-br 5041 df-opab 5103 df-mpt 5121 df-tr 5147 df-id 5439 df-eprel 5444 df-po 5452 df-so 5453 df-fr 5493 df-se 5494 df-we 5495 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-pred 6139 df-ord 6186 df-on 6187 df-lim 6188 df-suc 6189 df-iota 6308 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-isom 6359 df-riota 7140 df-ov 7186 df-oprab 7187 df-mpo 7188 df-of 7438 df-ofr 7439 df-om 7613 df-1st 7727 df-2nd 7728 df-supp 7870 df-wrecs 7989 df-recs 8050 df-rdg 8088 df-1o 8144 df-2o 8145 df-oadd 8148 df-omul 8149 df-er 8333 df-map 8452 df-pm 8453 df-ixp 8521 df-en 8569 df-dom 8570 df-sdom 8571 df-fin 8572 df-fsupp 8920 df-fi 8961 df-sup 8992 df-inf 8993 df-oi 9060 df-r1 9279 df-rank 9280 df-dju 9416 df-card 9454 df-acn 9457 df-ac 9629 df-pnf 10768 df-mnf 10769 df-xr 10770 df-ltxr 10771 df-le 10772 df-sub 10963 df-neg 10964 df-div 11389 df-nn 11730 df-2 11792 df-3 11793 df-4 11794 df-5 11795 df-6 11796 df-7 11797 df-8 11798 df-9 11799 df-n0 11990 df-xnn0 12062 df-z 12076 df-dec 12193 df-uz 12338 df-q 12444 df-rp 12486 df-xneg 12603 df-xadd 12604 df-xmul 12605 df-ioo 12838 df-ioc 12839 df-ico 12840 df-icc 12841 df-fz 12995 df-fzo 13138 df-fl 13266 df-mod 13342 df-seq 13474 df-exp 13535 df-fac 13739 df-bc 13768 df-hash 13796 df-word 13969 df-concat 14025 df-s1 14052 df-s2 14312 df-s3 14313 df-shft 14529 df-cj 14561 df-re 14562 df-im 14563 df-sqrt 14697 df-abs 14698 df-limsup 14931 df-clim 14948 df-rlim 14949 df-sum 15149 df-prod 15365 df-ef 15526 df-e 15527 df-sin 15528 df-cos 15529 df-tan 15530 df-pi 15531 df-dvds 15713 df-gcd 15951 df-prm 16126 df-pc 16287 df-struct 16601 df-ndx 16602 df-slot 16603 df-base 16605 df-sets 16606 df-ress 16607 df-plusg 16694 df-mulr 16695 df-starv 16696 df-sca 16697 df-vsca 16698 df-ip 16699 df-tset 16700 df-ple 16701 df-ds 16703 df-unif 16704 df-hom 16705 df-cco 16706 df-rest 16812 df-topn 16813 df-0g 16831 df-gsum 16832 df-topgen 16833 df-pt 16834 df-prds 16837 df-xrs 16891 df-qtop 16896 df-imas 16897 df-xps 16899 df-mre 16973 df-mrc 16974 df-acs 16976 df-mgm 17981 df-sgrp 18030 df-mnd 18041 df-submnd 18086 df-mulg 18356 df-cntz 18578 df-pmtr 18701 df-cmn 19039 df-psmet 20222 df-xmet 20223 df-met 20224 df-bl 20225 df-mopn 20226 df-fbas 20227 df-fg 20228 df-cnfld 20231 df-top 21658 df-topon 21675 df-topsp 21697 df-bases 21710 df-cld 21783 df-ntr 21784 df-cls 21785 df-nei 21862 df-lp 21900 df-perf 21901 df-cn 21991 df-cnp 21992 df-haus 22079 df-cmp 22151 df-tx 22326 df-hmeo 22519 df-fil 22610 df-fm 22702 df-flim 22703 df-flf 22704 df-xms 23086 df-ms 23087 df-tms 23088 df-cncf 23643 df-ovol 24229 df-vol 24230 df-mbf 24384 df-itg1 24385 df-itg2 24386 df-ibl 24387 df-itg 24388 df-0p 24435 df-limc 24631 df-dv 24632 df-ulm 25137 df-log 25313 df-cxp 25314 df-atan 25618 df-cht 25847 df-vma 25848 df-chp 25849 df-dp2 30734 df-dp 30751 df-repr 32172 df-vts 32199 |
This theorem is referenced by: tgoldbachgtd 32225 |
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