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| Mirrors > Home > MPE Home > Th. List > Mathboxes > tgoldbachgtda | Structured version Visualization version GIF version | ||
| Description: Lemma for tgoldbachgtd 34677. (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 34674 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ) |
| 5 | 4 | nnnn0d 12587 | . . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
| 6 | 3nn0 12544 | . . . . 5 ⊢ 3 ∈ ℕ0 | |
| 7 | 6 | a1i 11 | . . . 4 ⊢ (𝜑 → 3 ∈ ℕ0) |
| 8 | inss2 4238 | . . . . . 6 ⊢ (𝑂 ∩ ℙ) ⊆ ℙ | |
| 9 | prmssnn 16713 | . . . . . 6 ⊢ ℙ ⊆ ℕ | |
| 10 | 8, 9 | sstri 3993 | . . . . 5 ⊢ (𝑂 ∩ ℙ) ⊆ ℕ |
| 11 | 10 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝑂 ∩ ℙ) ⊆ ℕ) |
| 12 | 5, 7, 11 | reprfi2 34638 | . . 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 34675 | . . . . . . 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 11827 | . . . . . 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 2945 | . . . . 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 15736 | . . . . . 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 15757 | . . . . . 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 2793 | . . . . 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 2947 | . . 3 ⊢ (𝜑 → ((𝑂 ∩ ℙ)(repr‘3)𝑁) ≠ ∅) |
| 27 | hashnncl 14405 | . . . 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 12297 | . 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 2940 {crab 3436 ∩ cin 3950 ⊆ wss 3951 ∅c0 4333 class class class wbr 5143 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ∘f cof 7695 Fincfn 8985 0cc0 11155 1c1 11156 ici 11157 · cmul 11160 +∞cpnf 11292 < clt 11295 ≤ cle 11296 -cneg 11493 ℕcn 12266 2c2 12321 3c3 12322 4c4 12323 5c5 12324 7c7 12326 8c8 12327 9c9 12328 ℕ0cn0 12526 ℤcz 12613 ;cdc 12733 (,)cioo 13387 [,)cico 13389 ↑cexp 14102 ♯chash 14369 Σcsu 15722 expce 16097 πcpi 16102 ∥ cdvds 16290 ℙcprime 16708 ∫citg 25653 Λcvma 27135 _cdp2 32853 .cdp 32870 reprcrepr 34623 vtscvts 34650 |
| 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 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-reg 9632 ax-inf2 9681 ax-cc 10475 ax-ac2 10503 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234 ax-ros335 34660 ax-ros336 34661 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-symdif 4253 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-disj 5111 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-ofr 7698 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-oadd 8510 df-omul 8511 df-er 8745 df-map 8868 df-pm 8869 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-fi 9451 df-sup 9482 df-inf 9483 df-oi 9550 df-r1 9804 df-rank 9805 df-dju 9941 df-card 9979 df-acn 9982 df-ac 10156 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-xnn0 12600 df-z 12614 df-dec 12734 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-ioo 13391 df-ioc 13392 df-ico 13393 df-icc 13394 df-fz 13548 df-fzo 13695 df-fl 13832 df-mod 13910 df-seq 14043 df-exp 14103 df-fac 14313 df-bc 14342 df-hash 14370 df-word 14553 df-concat 14609 df-s1 14634 df-s2 14887 df-s3 14888 df-shft 15106 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-limsup 15507 df-clim 15524 df-rlim 15525 df-sum 15723 df-prod 15940 df-ef 16103 df-e 16104 df-sin 16105 df-cos 16106 df-tan 16107 df-pi 16108 df-dvds 16291 df-gcd 16532 df-prm 16709 df-pc 16875 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 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 17467 df-topn 17468 df-0g 17486 df-gsum 17487 df-topgen 17488 df-pt 17489 df-prds 17492 df-xrs 17547 df-qtop 17552 df-imas 17553 df-xps 17555 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-mulg 19086 df-cntz 19335 df-pmtr 19460 df-cmn 19800 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-fbas 21361 df-fg 21362 df-cnfld 21365 df-top 22900 df-topon 22917 df-topsp 22939 df-bases 22953 df-cld 23027 df-ntr 23028 df-cls 23029 df-nei 23106 df-lp 23144 df-perf 23145 df-cn 23235 df-cnp 23236 df-haus 23323 df-cmp 23395 df-tx 23570 df-hmeo 23763 df-fil 23854 df-fm 23946 df-flim 23947 df-flf 23948 df-xms 24330 df-ms 24331 df-tms 24332 df-cncf 24904 df-ovol 25499 df-vol 25500 df-mbf 25654 df-itg1 25655 df-itg2 25656 df-ibl 25657 df-itg 25658 df-0p 25705 df-limc 25901 df-dv 25902 df-ulm 26420 df-log 26598 df-cxp 26599 df-atan 26910 df-cht 27140 df-vma 27141 df-chp 27142 df-dp2 32854 df-dp 32871 df-repr 34624 df-vts 34651 |
| This theorem is referenced by: tgoldbachgtd 34677 |
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