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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 12245 | . 2 ⊢ 1 ∈ ℕ | |
2 | 1zzd 12615 | . . . . 5 ⊢ (𝑃 ∈ ℙ → 1 ∈ ℤ) | |
3 | id 22 | . . . . 5 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℙ) | |
4 | 2, 3 | fsnd 6876 | . . . 4 ⊢ (𝑃 ∈ ℙ → {〈1, 𝑃〉}:{1}⟶ℙ) |
5 | prmex 16639 | . . . . 5 ⊢ ℙ ∈ V | |
6 | snex 5427 | . . . . 5 ⊢ {1} ∈ V | |
7 | 5, 6 | elmap 8881 | . . . 4 ⊢ ({〈1, 𝑃〉} ∈ (ℙ ↑m {1}) ↔ {〈1, 𝑃〉}:{1}⟶ℙ) |
8 | 4, 7 | sylibr 233 | . . 3 ⊢ (𝑃 ∈ ℙ → {〈1, 𝑃〉} ∈ (ℙ ↑m {1})) |
9 | 1re 11236 | . . . . . . 7 ⊢ 1 ∈ ℝ | |
10 | simpl 482 | . . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝑘 ∈ {1}) → 𝑃 ∈ ℙ) | |
11 | fvsng 7183 | . . . . . . 7 ⊢ ((1 ∈ ℝ ∧ 𝑃 ∈ ℙ) → ({〈1, 𝑃〉}‘1) = 𝑃) | |
12 | 9, 10, 11 | sylancr 586 | . . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝑘 ∈ {1}) → ({〈1, 𝑃〉}‘1) = 𝑃) |
13 | 12 | sumeq2dv 15673 | . . . . 5 ⊢ (𝑃 ∈ ℙ → Σ𝑘 ∈ {1} ({〈1, 𝑃〉}‘1) = Σ𝑘 ∈ {1}𝑃) |
14 | prmz 16637 | . . . . . . 7 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ) | |
15 | 14 | zcnd 12689 | . . . . . 6 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℂ) |
16 | eqidd 2728 | . . . . . . 7 ⊢ (𝑘 = 1 → 𝑃 = 𝑃) | |
17 | 16 | sumsn 15716 | . . . . . 6 ⊢ ((1 ∈ ℝ ∧ 𝑃 ∈ ℂ) → Σ𝑘 ∈ {1}𝑃 = 𝑃) |
18 | 9, 15, 17 | sylancr 586 | . . . . 5 ⊢ (𝑃 ∈ ℙ → Σ𝑘 ∈ {1}𝑃 = 𝑃) |
19 | 13, 18 | eqtr2d 2768 | . . . 4 ⊢ (𝑃 ∈ ℙ → 𝑃 = Σ𝑘 ∈ {1} ({〈1, 𝑃〉}‘1)) |
20 | 1le3 12446 | . . . 4 ⊢ 1 ≤ 3 | |
21 | 19, 20 | jctil 519 | . . 3 ⊢ (𝑃 ∈ ℙ → (1 ≤ 3 ∧ 𝑃 = Σ𝑘 ∈ {1} ({〈1, 𝑃〉}‘1))) |
22 | simpl 482 | . . . . . . . 8 ⊢ ((𝑓 = {〈1, 𝑃〉} ∧ 𝑘 ∈ {1}) → 𝑓 = {〈1, 𝑃〉}) | |
23 | elsni 4641 | . . . . . . . . 9 ⊢ (𝑘 ∈ {1} → 𝑘 = 1) | |
24 | 23 | adantl 481 | . . . . . . . 8 ⊢ ((𝑓 = {〈1, 𝑃〉} ∧ 𝑘 ∈ {1}) → 𝑘 = 1) |
25 | 22, 24 | fveq12d 6898 | . . . . . . 7 ⊢ ((𝑓 = {〈1, 𝑃〉} ∧ 𝑘 ∈ {1}) → (𝑓‘𝑘) = ({〈1, 𝑃〉}‘1)) |
26 | 25 | sumeq2dv 15673 | . . . . . 6 ⊢ (𝑓 = {〈1, 𝑃〉} → Σ𝑘 ∈ {1} (𝑓‘𝑘) = Σ𝑘 ∈ {1} ({〈1, 𝑃〉}‘1)) |
27 | 26 | eqeq2d 2738 | . . . . 5 ⊢ (𝑓 = {〈1, 𝑃〉} → (𝑃 = Σ𝑘 ∈ {1} (𝑓‘𝑘) ↔ 𝑃 = Σ𝑘 ∈ {1} ({〈1, 𝑃〉}‘1))) |
28 | 27 | anbi2d 628 | . . . 4 ⊢ (𝑓 = {〈1, 𝑃〉} → ((1 ≤ 3 ∧ 𝑃 = Σ𝑘 ∈ {1} (𝑓‘𝑘)) ↔ (1 ≤ 3 ∧ 𝑃 = Σ𝑘 ∈ {1} ({〈1, 𝑃〉}‘1)))) |
29 | 28 | rspcev 3607 | . . 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 583 | . 2 ⊢ (𝑃 ∈ ℙ → ∃𝑓 ∈ (ℙ ↑m {1})(1 ≤ 3 ∧ 𝑃 = Σ𝑘 ∈ {1} (𝑓‘𝑘))) |
31 | oveq2 7422 | . . . . . 6 ⊢ (𝑑 = 1 → (1...𝑑) = (1...1)) | |
32 | 1z 12614 | . . . . . . 7 ⊢ 1 ∈ ℤ | |
33 | fzsn 13567 | . . . . . . 7 ⊢ (1 ∈ ℤ → (1...1) = {1}) | |
34 | 32, 33 | ax-mp 5 | . . . . . 6 ⊢ (1...1) = {1} |
35 | 31, 34 | eqtrdi 2783 | . . . . 5 ⊢ (𝑑 = 1 → (1...𝑑) = {1}) |
36 | 35 | oveq2d 7430 | . . . 4 ⊢ (𝑑 = 1 → (ℙ ↑m (1...𝑑)) = (ℙ ↑m {1})) |
37 | breq1 5145 | . . . . 5 ⊢ (𝑑 = 1 → (𝑑 ≤ 3 ↔ 1 ≤ 3)) | |
38 | 35 | sumeq1d 15671 | . . . . . 6 ⊢ (𝑑 = 1 → Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘) = Σ𝑘 ∈ {1} (𝑓‘𝑘)) |
39 | 38 | eqeq2d 2738 | . . . . 5 ⊢ (𝑑 = 1 → (𝑃 = Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘) ↔ 𝑃 = Σ𝑘 ∈ {1} (𝑓‘𝑘))) |
40 | 37, 39 | anbi12d 630 | . . . 4 ⊢ (𝑑 = 1 → ((𝑑 ≤ 3 ∧ 𝑃 = Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘)) ↔ (1 ≤ 3 ∧ 𝑃 = Σ𝑘 ∈ {1} (𝑓‘𝑘)))) |
41 | 36, 40 | rexeqbidv 3338 | . . 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 3607 | . 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 586 | 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 395 = wceq 1534 ∈ wcel 2099 ∃wrex 3065 {csn 4624 〈cop 4630 class class class wbr 5142 ⟶wf 6538 ‘cfv 6542 (class class class)co 7414 ↑m cmap 8836 ℂcc 11128 ℝcr 11129 1c1 11131 ≤ cle 11271 ℕcn 12234 3c3 12290 ℤcz 12580 ...cfz 13508 Σcsu 15656 ℙcprime 16633 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-inf2 9656 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207 ax-pre-sup 11208 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 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 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-map 8838 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-sup 9457 df-oi 9525 df-card 9954 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-div 11894 df-nn 12235 df-2 12297 df-3 12298 df-n0 12495 df-z 12581 df-uz 12845 df-rp 12999 df-fz 13509 df-fzo 13652 df-seq 13991 df-exp 14051 df-hash 14314 df-cj 15070 df-re 15071 df-im 15072 df-sqrt 15206 df-abs 15207 df-clim 15456 df-sum 15657 df-prm 16634 |
This theorem is referenced by: nnsum4primesprm 47054 nnsum3primesle9 47057 |
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