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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > tgoldbachgtda | Structured version Visualization version GIF version |
Description: Lemma for tgoldbachgtd 33972. (Contributed by Thierry Arnoux, 15-Dec-2021.) |
Ref | Expression |
---|---|
tgoldbachgtda.o | β’ π = {π§ β β€ β£ Β¬ 2 β₯ π§} |
tgoldbachgtda.n | β’ (π β π β π) |
tgoldbachgtda.0 | β’ (π β (;10β;27) β€ π) |
tgoldbachgtda.h | β’ (π β π»:ββΆ(0[,)+β)) |
tgoldbachgtda.k | β’ (π β πΎ:ββΆ(0[,)+β)) |
tgoldbachgtda.1 | β’ ((π β§ π β β) β (πΎβπ) β€ (1._0_7_9_9_55)) |
tgoldbachgtda.2 | β’ ((π β§ π β β) β (π»βπ) β€ (1._4_14)) |
tgoldbachgtda.3 | β’ (π β ((0._0_0_0_4_2_2_48) Β· (πβ2)) β€ β«(0(,)1)(((((Ξ βf Β· π»)vtsπ)βπ₯) Β· ((((Ξ βf Β· πΎ)vtsπ)βπ₯)β2)) Β· (expβ((i Β· (2 Β· Ο)) Β· (-π Β· π₯)))) dπ₯) |
Ref | Expression |
---|---|
tgoldbachgtda | β’ (π β 0 < (β―β((π β© β)(reprβ3)π))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tgoldbachgtda.o | . . . . . 6 β’ π = {π§ β β€ β£ Β¬ 2 β₯ π§} | |
2 | tgoldbachgtda.n | . . . . . 6 β’ (π β π β π) | |
3 | tgoldbachgtda.0 | . . . . . 6 β’ (π β (;10β;27) β€ π) | |
4 | 1, 2, 3 | tgoldbachgnn 33969 | . . . . 5 β’ (π β π β β) |
5 | 4 | nnnn0d 12536 | . . . 4 β’ (π β π β β0) |
6 | 3nn0 12494 | . . . . 5 β’ 3 β β0 | |
7 | 6 | a1i 11 | . . . 4 β’ (π β 3 β β0) |
8 | inss2 4228 | . . . . . 6 β’ (π β© β) β β | |
9 | prmssnn 16617 | . . . . . 6 β’ β β β | |
10 | 8, 9 | sstri 3990 | . . . . 5 β’ (π β© β) β β |
11 | 10 | a1i 11 | . . . 4 β’ (π β (π β© β) β β) |
12 | 5, 7, 11 | reprfi2 33933 | . . 3 β’ (π β ((π β© β)(reprβ3)π) β Fin) |
13 | tgoldbachgtda.h | . . . . . . . 8 β’ (π β π»:ββΆ(0[,)+β)) | |
14 | tgoldbachgtda.k | . . . . . . . 8 β’ (π β πΎ:ββΆ(0[,)+β)) | |
15 | tgoldbachgtda.1 | . . . . . . . 8 β’ ((π β§ π β β) β (πΎβπ) β€ (1._0_7_9_9_55)) | |
16 | tgoldbachgtda.2 | . . . . . . . 8 β’ ((π β§ π β β) β (π»βπ) β€ (1._4_14)) | |
17 | tgoldbachgtda.3 | . . . . . . . 8 β’ (π β ((0._0_0_0_4_2_2_48) Β· (πβ2)) β€ β«(0(,)1)(((((Ξ βf Β· π»)vtsπ)βπ₯) Β· ((((Ξ βf Β· πΎ)vtsπ)βπ₯)β2)) Β· (expβ((i Β· (2 Β· Ο)) Β· (-π Β· π₯)))) dπ₯) | |
18 | 1, 2, 3, 13, 14, 15, 16, 17 | tgoldbachgtde 33970 | . . . . . . 7 β’ (π β 0 < Ξ£π β ((π β© β)(reprβ3)π)(((Ξβ(πβ0)) Β· (π»β(πβ0))) Β· (((Ξβ(πβ1)) Β· (πΎβ(πβ1))) Β· ((Ξβ(πβ2)) Β· (πΎβ(πβ2)))))) |
19 | 18 | gt0ne0d 11782 | . . . . . 6 β’ (π β Ξ£π β ((π β© β)(reprβ3)π)(((Ξβ(πβ0)) Β· (π»β(πβ0))) Β· (((Ξβ(πβ1)) Β· (πΎβ(πβ1))) Β· ((Ξβ(πβ2)) Β· (πΎβ(πβ2))))) β 0) |
20 | 19 | neneqd 2943 | . . . . 5 β’ (π β Β¬ Ξ£π β ((π β© β)(reprβ3)π)(((Ξβ(πβ0)) Β· (π»β(πβ0))) Β· (((Ξβ(πβ1)) Β· (πΎβ(πβ1))) Β· ((Ξβ(πβ2)) Β· (πΎβ(πβ2))))) = 0) |
21 | simpr 483 | . . . . . . 7 β’ ((π β§ ((π β© β)(reprβ3)π) = β ) β ((π β© β)(reprβ3)π) = β ) | |
22 | 21 | sumeq1d 15651 | . . . . . 6 β’ ((π β§ ((π β© β)(reprβ3)π) = β ) β Ξ£π β ((π β© β)(reprβ3)π)(((Ξβ(πβ0)) Β· (π»β(πβ0))) Β· (((Ξβ(πβ1)) Β· (πΎβ(πβ1))) Β· ((Ξβ(πβ2)) Β· (πΎβ(πβ2))))) = Ξ£π β β (((Ξβ(πβ0)) Β· (π»β(πβ0))) Β· (((Ξβ(πβ1)) Β· (πΎβ(πβ1))) Β· ((Ξβ(πβ2)) Β· (πΎβ(πβ2)))))) |
23 | sum0 15671 | . . . . . 6 β’ Ξ£π β β (((Ξβ(πβ0)) Β· (π»β(πβ0))) Β· (((Ξβ(πβ1)) Β· (πΎβ(πβ1))) Β· ((Ξβ(πβ2)) Β· (πΎβ(πβ2))))) = 0 | |
24 | 22, 23 | eqtrdi 2786 | . . . . 5 β’ ((π β§ ((π β© β)(reprβ3)π) = β ) β Ξ£π β ((π β© β)(reprβ3)π)(((Ξβ(πβ0)) Β· (π»β(πβ0))) Β· (((Ξβ(πβ1)) Β· (πΎβ(πβ1))) Β· ((Ξβ(πβ2)) Β· (πΎβ(πβ2))))) = 0) |
25 | 20, 24 | mtand 812 | . . . 4 β’ (π β Β¬ ((π β© β)(reprβ3)π) = β ) |
26 | 25 | neqned 2945 | . . 3 β’ (π β ((π β© β)(reprβ3)π) β β ) |
27 | hashnncl 14330 | . . . 4 β’ (((π β© β)(reprβ3)π) β Fin β ((β―β((π β© β)(reprβ3)π)) β β β ((π β© β)(reprβ3)π) β β )) | |
28 | 27 | biimpar 476 | . . 3 β’ ((((π β© β)(reprβ3)π) β Fin β§ ((π β© β)(reprβ3)π) β β ) β (β―β((π β© β)(reprβ3)π)) β β) |
29 | 12, 26, 28 | syl2anc 582 | . 2 β’ (π β (β―β((π β© β)(reprβ3)π)) β β) |
30 | nngt0 12247 | . 2 β’ ((β―β((π β© β)(reprβ3)π)) β β β 0 < (β―β((π β© β)(reprβ3)π))) | |
31 | 29, 30 | syl 17 | 1 β’ (π β 0 < (β―β((π β© β)(reprβ3)π))) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 394 = wceq 1539 β wcel 2104 β wne 2938 {crab 3430 β© cin 3946 β wss 3947 β c0 4321 class class class wbr 5147 βΆwf 6538 βcfv 6542 (class class class)co 7411 βf cof 7670 Fincfn 8941 0cc0 11112 1c1 11113 ici 11114 Β· cmul 11117 +βcpnf 11249 < clt 11252 β€ cle 11253 -cneg 11449 βcn 12216 2c2 12271 3c3 12272 4c4 12273 5c5 12274 7c7 12276 8c8 12277 9c9 12278 β0cn0 12476 β€cz 12562 ;cdc 12681 (,)cioo 13328 [,)cico 13330 βcexp 14031 β―chash 14294 Ξ£csu 15636 expce 16009 Οcpi 16014 β₯ cdvds 16201 βcprime 16612 β«citg 25367 Ξcvma 26832 _cdp2 32304 .cdp 32321 reprcrepr 33918 vtscvts 33945 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-reg 9589 ax-inf2 9638 ax-cc 10432 ax-ac2 10460 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 ax-addf 11191 ax-mulf 11192 ax-ros335 33955 ax-ros336 33956 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-symdif 4241 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-disj 5113 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7672 df-ofr 7673 df-om 7858 df-1st 7977 df-2nd 7978 df-supp 8149 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-2o 8469 df-oadd 8472 df-omul 8473 df-er 8705 df-map 8824 df-pm 8825 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-fi 9408 df-sup 9439 df-inf 9440 df-oi 9507 df-r1 9761 df-rank 9762 df-dju 9898 df-card 9936 df-acn 9939 df-ac 10113 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-xnn0 12549 df-z 12563 df-dec 12682 df-uz 12827 df-q 12937 df-rp 12979 df-xneg 13096 df-xadd 13097 df-xmul 13098 df-ioo 13332 df-ioc 13333 df-ico 13334 df-icc 13335 df-fz 13489 df-fzo 13632 df-fl 13761 df-mod 13839 df-seq 13971 df-exp 14032 df-fac 14238 df-bc 14267 df-hash 14295 df-word 14469 df-concat 14525 df-s1 14550 df-s2 14803 df-s3 14804 df-shft 15018 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-limsup 15419 df-clim 15436 df-rlim 15437 df-sum 15637 df-prod 15854 df-ef 16015 df-e 16016 df-sin 16017 df-cos 16018 df-tan 16019 df-pi 16020 df-dvds 16202 df-gcd 16440 df-prm 16613 df-pc 16774 df-struct 17084 df-sets 17101 df-slot 17119 df-ndx 17131 df-base 17149 df-ress 17178 df-plusg 17214 df-mulr 17215 df-starv 17216 df-sca 17217 df-vsca 17218 df-ip 17219 df-tset 17220 df-ple 17221 df-ds 17223 df-unif 17224 df-hom 17225 df-cco 17226 df-rest 17372 df-topn 17373 df-0g 17391 df-gsum 17392 df-topgen 17393 df-pt 17394 df-prds 17397 df-xrs 17452 df-qtop 17457 df-imas 17458 df-xps 17460 df-mre 17534 df-mrc 17535 df-acs 17537 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-submnd 18706 df-mulg 18987 df-cntz 19222 df-pmtr 19351 df-cmn 19691 df-psmet 21136 df-xmet 21137 df-met 21138 df-bl 21139 df-mopn 21140 df-fbas 21141 df-fg 21142 df-cnfld 21145 df-top 22616 df-topon 22633 df-topsp 22655 df-bases 22669 df-cld 22743 df-ntr 22744 df-cls 22745 df-nei 22822 df-lp 22860 df-perf 22861 df-cn 22951 df-cnp 22952 df-haus 23039 df-cmp 23111 df-tx 23286 df-hmeo 23479 df-fil 23570 df-fm 23662 df-flim 23663 df-flf 23664 df-xms 24046 df-ms 24047 df-tms 24048 df-cncf 24618 df-ovol 25213 df-vol 25214 df-mbf 25368 df-itg1 25369 df-itg2 25370 df-ibl 25371 df-itg 25372 df-0p 25419 df-limc 25615 df-dv 25616 df-ulm 26125 df-log 26301 df-cxp 26302 df-atan 26608 df-cht 26837 df-vma 26838 df-chp 26839 df-dp2 32305 df-dp 32322 df-repr 33919 df-vts 33946 |
This theorem is referenced by: tgoldbachgtd 33972 |
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