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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 11651 | . 2 ⊢ 1 ∈ ℕ | |
2 | 1zzd 12016 | . . . . 5 ⊢ (𝑃 ∈ ℙ → 1 ∈ ℤ) | |
3 | id 22 | . . . . 5 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℙ) | |
4 | 2, 3 | fsnd 6659 | . . . 4 ⊢ (𝑃 ∈ ℙ → {〈1, 𝑃〉}:{1}⟶ℙ) |
5 | prmex 16023 | . . . . 5 ⊢ ℙ ∈ V | |
6 | snex 5334 | . . . . 5 ⊢ {1} ∈ V | |
7 | 5, 6 | elmap 8437 | . . . 4 ⊢ ({〈1, 𝑃〉} ∈ (ℙ ↑m {1}) ↔ {〈1, 𝑃〉}:{1}⟶ℙ) |
8 | 4, 7 | sylibr 236 | . . 3 ⊢ (𝑃 ∈ ℙ → {〈1, 𝑃〉} ∈ (ℙ ↑m {1})) |
9 | 1re 10643 | . . . . . . 7 ⊢ 1 ∈ ℝ | |
10 | simpl 485 | . . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝑘 ∈ {1}) → 𝑃 ∈ ℙ) | |
11 | fvsng 6944 | . . . . . . 7 ⊢ ((1 ∈ ℝ ∧ 𝑃 ∈ ℙ) → ({〈1, 𝑃〉}‘1) = 𝑃) | |
12 | 9, 10, 11 | sylancr 589 | . . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝑘 ∈ {1}) → ({〈1, 𝑃〉}‘1) = 𝑃) |
13 | 12 | sumeq2dv 15062 | . . . . 5 ⊢ (𝑃 ∈ ℙ → Σ𝑘 ∈ {1} ({〈1, 𝑃〉}‘1) = Σ𝑘 ∈ {1}𝑃) |
14 | prmz 16021 | . . . . . . 7 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ) | |
15 | 14 | zcnd 12091 | . . . . . 6 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℂ) |
16 | eqidd 2824 | . . . . . . 7 ⊢ (𝑘 = 1 → 𝑃 = 𝑃) | |
17 | 16 | sumsn 15103 | . . . . . 6 ⊢ ((1 ∈ ℝ ∧ 𝑃 ∈ ℂ) → Σ𝑘 ∈ {1}𝑃 = 𝑃) |
18 | 9, 15, 17 | sylancr 589 | . . . . 5 ⊢ (𝑃 ∈ ℙ → Σ𝑘 ∈ {1}𝑃 = 𝑃) |
19 | 13, 18 | eqtr2d 2859 | . . . 4 ⊢ (𝑃 ∈ ℙ → 𝑃 = Σ𝑘 ∈ {1} ({〈1, 𝑃〉}‘1)) |
20 | 1le3 11852 | . . . 4 ⊢ 1 ≤ 3 | |
21 | 19, 20 | jctil 522 | . . 3 ⊢ (𝑃 ∈ ℙ → (1 ≤ 3 ∧ 𝑃 = Σ𝑘 ∈ {1} ({〈1, 𝑃〉}‘1))) |
22 | simpl 485 | . . . . . . . 8 ⊢ ((𝑓 = {〈1, 𝑃〉} ∧ 𝑘 ∈ {1}) → 𝑓 = {〈1, 𝑃〉}) | |
23 | elsni 4586 | . . . . . . . . 9 ⊢ (𝑘 ∈ {1} → 𝑘 = 1) | |
24 | 23 | adantl 484 | . . . . . . . 8 ⊢ ((𝑓 = {〈1, 𝑃〉} ∧ 𝑘 ∈ {1}) → 𝑘 = 1) |
25 | 22, 24 | fveq12d 6679 | . . . . . . 7 ⊢ ((𝑓 = {〈1, 𝑃〉} ∧ 𝑘 ∈ {1}) → (𝑓‘𝑘) = ({〈1, 𝑃〉}‘1)) |
26 | 25 | sumeq2dv 15062 | . . . . . 6 ⊢ (𝑓 = {〈1, 𝑃〉} → Σ𝑘 ∈ {1} (𝑓‘𝑘) = Σ𝑘 ∈ {1} ({〈1, 𝑃〉}‘1)) |
27 | 26 | eqeq2d 2834 | . . . . 5 ⊢ (𝑓 = {〈1, 𝑃〉} → (𝑃 = Σ𝑘 ∈ {1} (𝑓‘𝑘) ↔ 𝑃 = Σ𝑘 ∈ {1} ({〈1, 𝑃〉}‘1))) |
28 | 27 | anbi2d 630 | . . . 4 ⊢ (𝑓 = {〈1, 𝑃〉} → ((1 ≤ 3 ∧ 𝑃 = Σ𝑘 ∈ {1} (𝑓‘𝑘)) ↔ (1 ≤ 3 ∧ 𝑃 = Σ𝑘 ∈ {1} ({〈1, 𝑃〉}‘1)))) |
29 | 28 | rspcev 3625 | . . 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 586 | . 2 ⊢ (𝑃 ∈ ℙ → ∃𝑓 ∈ (ℙ ↑m {1})(1 ≤ 3 ∧ 𝑃 = Σ𝑘 ∈ {1} (𝑓‘𝑘))) |
31 | oveq2 7166 | . . . . . 6 ⊢ (𝑑 = 1 → (1...𝑑) = (1...1)) | |
32 | 1z 12015 | . . . . . . 7 ⊢ 1 ∈ ℤ | |
33 | fzsn 12952 | . . . . . . 7 ⊢ (1 ∈ ℤ → (1...1) = {1}) | |
34 | 32, 33 | ax-mp 5 | . . . . . 6 ⊢ (1...1) = {1} |
35 | 31, 34 | syl6eq 2874 | . . . . 5 ⊢ (𝑑 = 1 → (1...𝑑) = {1}) |
36 | 35 | oveq2d 7174 | . . . 4 ⊢ (𝑑 = 1 → (ℙ ↑m (1...𝑑)) = (ℙ ↑m {1})) |
37 | breq1 5071 | . . . . 5 ⊢ (𝑑 = 1 → (𝑑 ≤ 3 ↔ 1 ≤ 3)) | |
38 | 35 | sumeq1d 15060 | . . . . . 6 ⊢ (𝑑 = 1 → Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘) = Σ𝑘 ∈ {1} (𝑓‘𝑘)) |
39 | 38 | eqeq2d 2834 | . . . . 5 ⊢ (𝑑 = 1 → (𝑃 = Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘) ↔ 𝑃 = Σ𝑘 ∈ {1} (𝑓‘𝑘))) |
40 | 37, 39 | anbi12d 632 | . . . 4 ⊢ (𝑑 = 1 → ((𝑑 ≤ 3 ∧ 𝑃 = Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘)) ↔ (1 ≤ 3 ∧ 𝑃 = Σ𝑘 ∈ {1} (𝑓‘𝑘)))) |
41 | 36, 40 | rexeqbidv 3404 | . . 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 3625 | . 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 589 | 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 398 = wceq 1537 ∈ wcel 2114 ∃wrex 3141 {csn 4569 〈cop 4575 class class class wbr 5068 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 ↑m cmap 8408 ℂcc 10537 ℝcr 10538 1c1 10540 ≤ cle 10678 ℕcn 11640 3c3 11696 ℤcz 11984 ...cfz 12895 Σcsu 15044 ℙcprime 16017 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-sup 8908 df-oi 8976 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-n0 11901 df-z 11985 df-uz 12247 df-rp 12393 df-fz 12896 df-fzo 13037 df-seq 13373 df-exp 13433 df-hash 13694 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-clim 14847 df-sum 15045 df-prm 16018 |
This theorem is referenced by: nnsum4primesprm 43963 nnsum3primesle9 43966 |
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