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Mirrors > Home > MPE Home > Th. List > Mathboxes > nnsum3primesprm | Structured version Visualization version GIF version |
Description: Every prime is "the sum of at most 3" (actually one - the prime itself) primes. (Contributed by AV, 2-Aug-2020.) (Proof shortened by AV, 17-Apr-2021.) |
Ref | Expression |
---|---|
nnsum3primesprm | ⊢ (𝑃 ∈ ℙ → ∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑m (1...𝑑))(𝑑 ≤ 3 ∧ 𝑃 = Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1nn 11824 | . 2 ⊢ 1 ∈ ℕ | |
2 | 1zzd 12191 | . . . . 5 ⊢ (𝑃 ∈ ℙ → 1 ∈ ℤ) | |
3 | id 22 | . . . . 5 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℙ) | |
4 | 2, 3 | fsnd 6692 | . . . 4 ⊢ (𝑃 ∈ ℙ → {〈1, 𝑃〉}:{1}⟶ℙ) |
5 | prmex 16215 | . . . . 5 ⊢ ℙ ∈ V | |
6 | snex 5313 | . . . . 5 ⊢ {1} ∈ V | |
7 | 5, 6 | elmap 8541 | . . . 4 ⊢ ({〈1, 𝑃〉} ∈ (ℙ ↑m {1}) ↔ {〈1, 𝑃〉}:{1}⟶ℙ) |
8 | 4, 7 | sylibr 237 | . . 3 ⊢ (𝑃 ∈ ℙ → {〈1, 𝑃〉} ∈ (ℙ ↑m {1})) |
9 | 1re 10816 | . . . . . . 7 ⊢ 1 ∈ ℝ | |
10 | simpl 486 | . . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝑘 ∈ {1}) → 𝑃 ∈ ℙ) | |
11 | fvsng 6984 | . . . . . . 7 ⊢ ((1 ∈ ℝ ∧ 𝑃 ∈ ℙ) → ({〈1, 𝑃〉}‘1) = 𝑃) | |
12 | 9, 10, 11 | sylancr 590 | . . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝑘 ∈ {1}) → ({〈1, 𝑃〉}‘1) = 𝑃) |
13 | 12 | sumeq2dv 15250 | . . . . 5 ⊢ (𝑃 ∈ ℙ → Σ𝑘 ∈ {1} ({〈1, 𝑃〉}‘1) = Σ𝑘 ∈ {1}𝑃) |
14 | prmz 16213 | . . . . . . 7 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ) | |
15 | 14 | zcnd 12266 | . . . . . 6 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℂ) |
16 | eqidd 2735 | . . . . . . 7 ⊢ (𝑘 = 1 → 𝑃 = 𝑃) | |
17 | 16 | sumsn 15291 | . . . . . 6 ⊢ ((1 ∈ ℝ ∧ 𝑃 ∈ ℂ) → Σ𝑘 ∈ {1}𝑃 = 𝑃) |
18 | 9, 15, 17 | sylancr 590 | . . . . 5 ⊢ (𝑃 ∈ ℙ → Σ𝑘 ∈ {1}𝑃 = 𝑃) |
19 | 13, 18 | eqtr2d 2775 | . . . 4 ⊢ (𝑃 ∈ ℙ → 𝑃 = Σ𝑘 ∈ {1} ({〈1, 𝑃〉}‘1)) |
20 | 1le3 12025 | . . . 4 ⊢ 1 ≤ 3 | |
21 | 19, 20 | jctil 523 | . . 3 ⊢ (𝑃 ∈ ℙ → (1 ≤ 3 ∧ 𝑃 = Σ𝑘 ∈ {1} ({〈1, 𝑃〉}‘1))) |
22 | simpl 486 | . . . . . . . 8 ⊢ ((𝑓 = {〈1, 𝑃〉} ∧ 𝑘 ∈ {1}) → 𝑓 = {〈1, 𝑃〉}) | |
23 | elsni 4548 | . . . . . . . . 9 ⊢ (𝑘 ∈ {1} → 𝑘 = 1) | |
24 | 23 | adantl 485 | . . . . . . . 8 ⊢ ((𝑓 = {〈1, 𝑃〉} ∧ 𝑘 ∈ {1}) → 𝑘 = 1) |
25 | 22, 24 | fveq12d 6713 | . . . . . . 7 ⊢ ((𝑓 = {〈1, 𝑃〉} ∧ 𝑘 ∈ {1}) → (𝑓‘𝑘) = ({〈1, 𝑃〉}‘1)) |
26 | 25 | sumeq2dv 15250 | . . . . . 6 ⊢ (𝑓 = {〈1, 𝑃〉} → Σ𝑘 ∈ {1} (𝑓‘𝑘) = Σ𝑘 ∈ {1} ({〈1, 𝑃〉}‘1)) |
27 | 26 | eqeq2d 2745 | . . . . 5 ⊢ (𝑓 = {〈1, 𝑃〉} → (𝑃 = Σ𝑘 ∈ {1} (𝑓‘𝑘) ↔ 𝑃 = Σ𝑘 ∈ {1} ({〈1, 𝑃〉}‘1))) |
28 | 27 | anbi2d 632 | . . . 4 ⊢ (𝑓 = {〈1, 𝑃〉} → ((1 ≤ 3 ∧ 𝑃 = Σ𝑘 ∈ {1} (𝑓‘𝑘)) ↔ (1 ≤ 3 ∧ 𝑃 = Σ𝑘 ∈ {1} ({〈1, 𝑃〉}‘1)))) |
29 | 28 | rspcev 3530 | . . 3 ⊢ (({〈1, 𝑃〉} ∈ (ℙ ↑m {1}) ∧ (1 ≤ 3 ∧ 𝑃 = Σ𝑘 ∈ {1} ({〈1, 𝑃〉}‘1))) → ∃𝑓 ∈ (ℙ ↑m {1})(1 ≤ 3 ∧ 𝑃 = Σ𝑘 ∈ {1} (𝑓‘𝑘))) |
30 | 8, 21, 29 | syl2anc 587 | . 2 ⊢ (𝑃 ∈ ℙ → ∃𝑓 ∈ (ℙ ↑m {1})(1 ≤ 3 ∧ 𝑃 = Σ𝑘 ∈ {1} (𝑓‘𝑘))) |
31 | oveq2 7210 | . . . . . 6 ⊢ (𝑑 = 1 → (1...𝑑) = (1...1)) | |
32 | 1z 12190 | . . . . . . 7 ⊢ 1 ∈ ℤ | |
33 | fzsn 13137 | . . . . . . 7 ⊢ (1 ∈ ℤ → (1...1) = {1}) | |
34 | 32, 33 | ax-mp 5 | . . . . . 6 ⊢ (1...1) = {1} |
35 | 31, 34 | eqtrdi 2790 | . . . . 5 ⊢ (𝑑 = 1 → (1...𝑑) = {1}) |
36 | 35 | oveq2d 7218 | . . . 4 ⊢ (𝑑 = 1 → (ℙ ↑m (1...𝑑)) = (ℙ ↑m {1})) |
37 | breq1 5046 | . . . . 5 ⊢ (𝑑 = 1 → (𝑑 ≤ 3 ↔ 1 ≤ 3)) | |
38 | 35 | sumeq1d 15248 | . . . . . 6 ⊢ (𝑑 = 1 → Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘) = Σ𝑘 ∈ {1} (𝑓‘𝑘)) |
39 | 38 | eqeq2d 2745 | . . . . 5 ⊢ (𝑑 = 1 → (𝑃 = Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘) ↔ 𝑃 = Σ𝑘 ∈ {1} (𝑓‘𝑘))) |
40 | 37, 39 | anbi12d 634 | . . . 4 ⊢ (𝑑 = 1 → ((𝑑 ≤ 3 ∧ 𝑃 = Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘)) ↔ (1 ≤ 3 ∧ 𝑃 = Σ𝑘 ∈ {1} (𝑓‘𝑘)))) |
41 | 36, 40 | rexeqbidv 3307 | . . 3 ⊢ (𝑑 = 1 → (∃𝑓 ∈ (ℙ ↑m (1...𝑑))(𝑑 ≤ 3 ∧ 𝑃 = Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘)) ↔ ∃𝑓 ∈ (ℙ ↑m {1})(1 ≤ 3 ∧ 𝑃 = Σ𝑘 ∈ {1} (𝑓‘𝑘)))) |
42 | 41 | rspcev 3530 | . 2 ⊢ ((1 ∈ ℕ ∧ ∃𝑓 ∈ (ℙ ↑m {1})(1 ≤ 3 ∧ 𝑃 = Σ𝑘 ∈ {1} (𝑓‘𝑘))) → ∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑m (1...𝑑))(𝑑 ≤ 3 ∧ 𝑃 = Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘))) |
43 | 1, 30, 42 | sylancr 590 | 1 ⊢ (𝑃 ∈ ℙ → ∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑m (1...𝑑))(𝑑 ≤ 3 ∧ 𝑃 = Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ∃wrex 3055 {csn 4531 〈cop 4537 class class class wbr 5043 ⟶wf 6365 ‘cfv 6369 (class class class)co 7202 ↑m cmap 8497 ℂcc 10710 ℝcr 10711 1c1 10713 ≤ cle 10851 ℕcn 11813 3c3 11869 ℤcz 12159 ...cfz 13078 Σcsu 15232 ℙcprime 16209 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-inf2 9245 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 ax-pre-sup 10790 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 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 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-int 4850 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-se 5499 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-isom 6378 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-om 7634 df-1st 7750 df-2nd 7751 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-1o 8191 df-er 8380 df-map 8499 df-en 8616 df-dom 8617 df-sdom 8618 df-fin 8619 df-sup 9047 df-oi 9115 df-card 9538 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-div 11473 df-nn 11814 df-2 11876 df-3 11877 df-n0 12074 df-z 12160 df-uz 12422 df-rp 12570 df-fz 13079 df-fzo 13222 df-seq 13558 df-exp 13619 df-hash 13880 df-cj 14645 df-re 14646 df-im 14647 df-sqrt 14781 df-abs 14782 df-clim 15032 df-sum 15233 df-prm 16210 |
This theorem is referenced by: nnsum4primesprm 44870 nnsum3primesle9 44873 |
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