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Mirrors > Home > MPE Home > Th. List > Mathboxes > tgoldbachgtda | Structured version Visualization version GIF version |
Description: Lemma for tgoldbachgtd 34655. (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 34652 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ) |
5 | 4 | nnnn0d 12584 | . . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
6 | 3nn0 12541 | . . . . 5 ⊢ 3 ∈ ℕ0 | |
7 | 6 | a1i 11 | . . . 4 ⊢ (𝜑 → 3 ∈ ℕ0) |
8 | inss2 4245 | . . . . . 6 ⊢ (𝑂 ∩ ℙ) ⊆ ℙ | |
9 | prmssnn 16709 | . . . . . 6 ⊢ ℙ ⊆ ℕ | |
10 | 8, 9 | sstri 4004 | . . . . 5 ⊢ (𝑂 ∩ ℙ) ⊆ ℕ |
11 | 10 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝑂 ∩ ℙ) ⊆ ℕ) |
12 | 5, 7, 11 | reprfi2 34616 | . . 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 34653 | . . . . . . 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 11824 | . . . . . 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 2942 | . . . . 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 15732 | . . . . . 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 15753 | . . . . . 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 2944 | . . 3 ⊢ (𝜑 → ((𝑂 ∩ ℙ)(repr‘3)𝑁) ≠ ∅) |
27 | hashnncl 14401 | . . . 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 12294 | . 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 1536 ∈ wcel 2105 ≠ wne 2937 {crab 3432 ∩ cin 3961 ⊆ wss 3962 ∅c0 4338 class class class wbr 5147 ⟶wf 6558 ‘cfv 6562 (class class class)co 7430 ∘f cof 7694 Fincfn 8983 0cc0 11152 1c1 11153 ici 11154 · cmul 11157 +∞cpnf 11289 < clt 11292 ≤ cle 11293 -cneg 11490 ℕcn 12263 2c2 12318 3c3 12319 4c4 12320 5c5 12321 7c7 12323 8c8 12324 9c9 12325 ℕ0cn0 12523 ℤcz 12610 ;cdc 12730 (,)cioo 13383 [,)cico 13385 ↑cexp 14098 ♯chash 14365 Σcsu 15718 expce 16093 πcpi 16098 ∥ cdvds 16286 ℙcprime 16704 ∫citg 25666 Λcvma 27149 _cdp2 32837 .cdp 32854 reprcrepr 34601 vtscvts 34628 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-reg 9629 ax-inf2 9678 ax-cc 10472 ax-ac2 10500 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 ax-addf 11231 ax-ros335 34638 ax-ros336 34639 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-symdif 4258 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-iin 4998 df-disj 5115 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-se 5641 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-isom 6571 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-of 7696 df-ofr 7697 df-om 7887 df-1st 8012 df-2nd 8013 df-supp 8184 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-2o 8505 df-oadd 8508 df-omul 8509 df-er 8743 df-map 8866 df-pm 8867 df-ixp 8936 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-fsupp 9399 df-fi 9448 df-sup 9479 df-inf 9480 df-oi 9547 df-r1 9801 df-rank 9802 df-dju 9938 df-card 9976 df-acn 9979 df-ac 10153 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-xnn0 12597 df-z 12611 df-dec 12731 df-uz 12876 df-q 12988 df-rp 13032 df-xneg 13151 df-xadd 13152 df-xmul 13153 df-ioo 13387 df-ioc 13388 df-ico 13389 df-icc 13390 df-fz 13544 df-fzo 13691 df-fl 13828 df-mod 13906 df-seq 14039 df-exp 14099 df-fac 14309 df-bc 14338 df-hash 14366 df-word 14549 df-concat 14605 df-s1 14630 df-s2 14883 df-s3 14884 df-shft 15102 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-limsup 15503 df-clim 15520 df-rlim 15521 df-sum 15719 df-prod 15936 df-ef 16099 df-e 16100 df-sin 16101 df-cos 16102 df-tan 16103 df-pi 16104 df-dvds 16287 df-gcd 16528 df-prm 16705 df-pc 16870 df-struct 17180 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-ress 17274 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-rest 17468 df-topn 17469 df-0g 17487 df-gsum 17488 df-topgen 17489 df-pt 17490 df-prds 17493 df-xrs 17548 df-qtop 17553 df-imas 17554 df-xps 17556 df-mre 17630 df-mrc 17631 df-acs 17633 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-submnd 18809 df-mulg 19098 df-cntz 19347 df-pmtr 19474 df-cmn 19814 df-psmet 21373 df-xmet 21374 df-met 21375 df-bl 21376 df-mopn 21377 df-fbas 21378 df-fg 21379 df-cnfld 21382 df-top 22915 df-topon 22932 df-topsp 22954 df-bases 22968 df-cld 23042 df-ntr 23043 df-cls 23044 df-nei 23121 df-lp 23159 df-perf 23160 df-cn 23250 df-cnp 23251 df-haus 23338 df-cmp 23410 df-tx 23585 df-hmeo 23778 df-fil 23869 df-fm 23961 df-flim 23962 df-flf 23963 df-xms 24345 df-ms 24346 df-tms 24347 df-cncf 24917 df-ovol 25512 df-vol 25513 df-mbf 25667 df-itg1 25668 df-itg2 25669 df-ibl 25670 df-itg 25671 df-0p 25718 df-limc 25915 df-dv 25916 df-ulm 26434 df-log 26612 df-cxp 26613 df-atan 26924 df-cht 27154 df-vma 27155 df-chp 27156 df-dp2 32838 df-dp 32855 df-repr 34602 df-vts 34629 |
This theorem is referenced by: tgoldbachgtd 34655 |
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