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Mirrors > Home > MPE Home > Th. List > Mathboxes > tgoldbachgtda | Structured version Visualization version GIF version |
Description: Lemma for tgoldbachgtd 30740. (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 30737 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ) |
5 | 4 | nnnn0d 11351 | . . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
6 | 3nn0 11310 | . . . . 5 ⊢ 3 ∈ ℕ0 | |
7 | 6 | a1i 11 | . . . 4 ⊢ (𝜑 → 3 ∈ ℕ0) |
8 | inss2 3834 | . . . . . 6 ⊢ (𝑂 ∩ ℙ) ⊆ ℙ | |
9 | prmssnn 15390 | . . . . . 6 ⊢ ℙ ⊆ ℕ | |
10 | 8, 9 | sstri 3612 | . . . . 5 ⊢ (𝑂 ∩ ℙ) ⊆ ℕ |
11 | 10 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝑂 ∩ ℙ) ⊆ ℕ) |
12 | 5, 7, 11 | reprfi2 30701 | . . 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 30738 | . . . . . . 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 10592 | . . . . . 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 2799 | . . . . 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 477 | . . . . . . 7 ⊢ ((𝜑 ∧ ((𝑂 ∩ ℙ)(repr‘3)𝑁) = ∅) → ((𝑂 ∩ ℙ)(repr‘3)𝑁) = ∅) | |
22 | 21 | sumeq1d 14431 | . . . . . 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 14452 | . . . . . 6 ⊢ Σ𝑛 ∈ ∅ (((Λ‘(𝑛‘0)) · (𝐻‘(𝑛‘0))) · (((Λ‘(𝑛‘1)) · (𝐾‘(𝑛‘1))) · ((Λ‘(𝑛‘2)) · (𝐾‘(𝑛‘2))))) = 0 | |
24 | 22, 23 | syl6eq 2672 | . . . . 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 691 | . . . 4 ⊢ (𝜑 → ¬ ((𝑂 ∩ ℙ)(repr‘3)𝑁) = ∅) |
26 | 25 | neqned 2801 | . . 3 ⊢ (𝜑 → ((𝑂 ∩ ℙ)(repr‘3)𝑁) ≠ ∅) |
27 | hashnncl 13157 | . . . 4 ⊢ (((𝑂 ∩ ℙ)(repr‘3)𝑁) ∈ Fin → ((#‘((𝑂 ∩ ℙ)(repr‘3)𝑁)) ∈ ℕ ↔ ((𝑂 ∩ ℙ)(repr‘3)𝑁) ≠ ∅)) | |
28 | 27 | biimpar 502 | . . 3 ⊢ ((((𝑂 ∩ ℙ)(repr‘3)𝑁) ∈ Fin ∧ ((𝑂 ∩ ℙ)(repr‘3)𝑁) ≠ ∅) → (#‘((𝑂 ∩ ℙ)(repr‘3)𝑁)) ∈ ℕ) |
29 | 12, 26, 28 | syl2anc 693 | . 2 ⊢ (𝜑 → (#‘((𝑂 ∩ ℙ)(repr‘3)𝑁)) ∈ ℕ) |
30 | nngt0 11049 | . 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 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 {crab 2916 ∩ cin 3573 ⊆ wss 3574 ∅c0 3915 class class class wbr 4653 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ∘𝑓 cof 6895 Fincfn 7955 0cc0 9936 1c1 9937 ici 9938 · cmul 9941 +∞cpnf 10071 < clt 10074 ≤ cle 10075 -cneg 10267 ℕcn 11020 2c2 11070 3c3 11071 4c4 11072 5c5 11073 7c7 11075 8c8 11076 9c9 11077 ℕ0cn0 11292 ℤcz 11377 ;cdc 11493 (,)cioo 12175 [,)cico 12177 ↑cexp 12860 #chash 13117 Σcsu 14416 expce 14792 πcpi 14797 ∥ cdvds 14983 ℙcprime 15385 ∫citg 23387 Λcvma 24818 _cdp2 29577 .cdp 29595 reprcrepr 30686 vtscvts 30713 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-reg 8497 ax-inf2 8538 ax-cc 9257 ax-ac2 9285 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016 ax-ros335 30723 ax-ros336 30724 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-disj 4621 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-ofr 6898 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-omul 7565 df-er 7742 df-map 7859 df-pm 7860 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-fi 8317 df-sup 8348 df-inf 8349 df-oi 8415 df-r1 8627 df-rank 8628 df-card 8765 df-acn 8768 df-ac 8939 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-xnn0 11364 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ioo 12179 df-ioc 12180 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-fl 12593 df-mod 12669 df-seq 12802 df-exp 12861 df-fac 13061 df-bc 13090 df-hash 13118 df-word 13299 df-concat 13301 df-s1 13302 df-s2 13593 df-s3 13594 df-shft 13807 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-limsup 14202 df-clim 14219 df-rlim 14220 df-sum 14417 df-prod 14636 df-ef 14798 df-e 14799 df-sin 14800 df-cos 14801 df-tan 14802 df-pi 14803 df-dvds 14984 df-gcd 15217 df-prm 15386 df-pc 15542 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-hom 15966 df-cco 15967 df-rest 16083 df-topn 16084 df-0g 16102 df-gsum 16103 df-topgen 16104 df-pt 16105 df-prds 16108 df-xrs 16162 df-qtop 16167 df-imas 16168 df-xps 16170 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-mulg 17541 df-cntz 17750 df-pmtr 17862 df-cmn 18195 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-fbas 19743 df-fg 19744 df-cnfld 19747 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-cld 20823 df-ntr 20824 df-cls 20825 df-nei 20902 df-lp 20940 df-perf 20941 df-cn 21031 df-cnp 21032 df-haus 21119 df-cmp 21190 df-tx 21365 df-hmeo 21558 df-fil 21650 df-fm 21742 df-flim 21743 df-flf 21744 df-xms 22125 df-ms 22126 df-tms 22127 df-cncf 22681 df-ovol 23233 df-vol 23234 df-mbf 23388 df-itg1 23389 df-itg2 23390 df-ibl 23391 df-itg 23392 df-0p 23437 df-limc 23630 df-dv 23631 df-ulm 24131 df-log 24303 df-cxp 24304 df-atan 24594 df-cht 24823 df-vma 24824 df-chp 24825 df-dp2 29578 df-dp 29596 df-repr 30687 df-vts 30714 |
This theorem is referenced by: tgoldbachgtd 30740 |
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