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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 11643 | . 2 ⊢ 1 ∈ ℕ | |
2 | 1zzd 12007 | . . . . 5 ⊢ (𝑃 ∈ ℙ → 1 ∈ ℤ) | |
3 | id 22 | . . . . 5 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℙ) | |
4 | 2, 3 | fsnd 6652 | . . . 4 ⊢ (𝑃 ∈ ℙ → {〈1, 𝑃〉}:{1}⟶ℙ) |
5 | prmex 16015 | . . . . 5 ⊢ ℙ ∈ V | |
6 | snex 5324 | . . . . 5 ⊢ {1} ∈ V | |
7 | 5, 6 | elmap 8429 | . . . 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 10635 | . . . . . . 7 ⊢ 1 ∈ ℝ | |
10 | simpl 485 | . . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝑘 ∈ {1}) → 𝑃 ∈ ℙ) | |
11 | fvsng 6937 | . . . . . . 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 15054 | . . . . 5 ⊢ (𝑃 ∈ ℙ → Σ𝑘 ∈ {1} ({〈1, 𝑃〉}‘1) = Σ𝑘 ∈ {1}𝑃) |
14 | prmz 16013 | . . . . . . 7 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ) | |
15 | 14 | zcnd 12082 | . . . . . 6 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℂ) |
16 | eqidd 2822 | . . . . . . 7 ⊢ (𝑘 = 1 → 𝑃 = 𝑃) | |
17 | 16 | sumsn 15095 | . . . . . 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 2857 | . . . 4 ⊢ (𝑃 ∈ ℙ → 𝑃 = Σ𝑘 ∈ {1} ({〈1, 𝑃〉}‘1)) |
20 | 1le3 11843 | . . . 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 4578 | . . . . . . . . 9 ⊢ (𝑘 ∈ {1} → 𝑘 = 1) | |
24 | 23 | adantl 484 | . . . . . . . 8 ⊢ ((𝑓 = {〈1, 𝑃〉} ∧ 𝑘 ∈ {1}) → 𝑘 = 1) |
25 | 22, 24 | fveq12d 6672 | . . . . . . 7 ⊢ ((𝑓 = {〈1, 𝑃〉} ∧ 𝑘 ∈ {1}) → (𝑓‘𝑘) = ({〈1, 𝑃〉}‘1)) |
26 | 25 | sumeq2dv 15054 | . . . . . 6 ⊢ (𝑓 = {〈1, 𝑃〉} → Σ𝑘 ∈ {1} (𝑓‘𝑘) = Σ𝑘 ∈ {1} ({〈1, 𝑃〉}‘1)) |
27 | 26 | eqeq2d 2832 | . . . . 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 3623 | . . 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 7158 | . . . . . 6 ⊢ (𝑑 = 1 → (1...𝑑) = (1...1)) | |
32 | 1z 12006 | . . . . . . 7 ⊢ 1 ∈ ℤ | |
33 | fzsn 12943 | . . . . . . 7 ⊢ (1 ∈ ℤ → (1...1) = {1}) | |
34 | 32, 33 | ax-mp 5 | . . . . . 6 ⊢ (1...1) = {1} |
35 | 31, 34 | syl6eq 2872 | . . . . 5 ⊢ (𝑑 = 1 → (1...𝑑) = {1}) |
36 | 35 | oveq2d 7166 | . . . 4 ⊢ (𝑑 = 1 → (ℙ ↑m (1...𝑑)) = (ℙ ↑m {1})) |
37 | breq1 5062 | . . . . 5 ⊢ (𝑑 = 1 → (𝑑 ≤ 3 ↔ 1 ≤ 3)) | |
38 | 35 | sumeq1d 15052 | . . . . . 6 ⊢ (𝑑 = 1 → Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘) = Σ𝑘 ∈ {1} (𝑓‘𝑘)) |
39 | 38 | eqeq2d 2832 | . . . . 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 3403 | . . 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 3623 | . 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 1533 ∈ wcel 2110 ∃wrex 3139 {csn 4561 〈cop 4567 class class class wbr 5059 ⟶wf 6346 ‘cfv 6350 (class class class)co 7150 ↑m cmap 8400 ℂcc 10529 ℝcr 10530 1c1 10532 ≤ cle 10670 ℕcn 11632 3c3 11687 ℤcz 11975 ...cfz 12886 Σcsu 15036 ℙcprime 16009 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-inf2 9098 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-se 5510 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-isom 6359 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-sup 8900 df-oi 8968 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-n0 11892 df-z 11976 df-uz 12238 df-rp 12384 df-fz 12887 df-fzo 13028 df-seq 13364 df-exp 13424 df-hash 13685 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-clim 14839 df-sum 15037 df-prm 16010 |
This theorem is referenced by: nnsum4primesprm 43949 nnsum3primesle9 43952 |
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