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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 12275 | . 2 ⊢ 1 ∈ ℕ | |
2 | 1zzd 12646 | . . . . 5 ⊢ (𝑃 ∈ ℙ → 1 ∈ ℤ) | |
3 | id 22 | . . . . 5 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℙ) | |
4 | 2, 3 | fsnd 6892 | . . . 4 ⊢ (𝑃 ∈ ℙ → {〈1, 𝑃〉}:{1}⟶ℙ) |
5 | prmex 16711 | . . . . 5 ⊢ ℙ ∈ V | |
6 | snex 5442 | . . . . 5 ⊢ {1} ∈ V | |
7 | 5, 6 | elmap 8910 | . . . 4 ⊢ ({〈1, 𝑃〉} ∈ (ℙ ↑m {1}) ↔ {〈1, 𝑃〉}:{1}⟶ℙ) |
8 | 4, 7 | sylibr 234 | . . 3 ⊢ (𝑃 ∈ ℙ → {〈1, 𝑃〉} ∈ (ℙ ↑m {1})) |
9 | 1re 11259 | . . . . . . 7 ⊢ 1 ∈ ℝ | |
10 | simpl 482 | . . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝑘 ∈ {1}) → 𝑃 ∈ ℙ) | |
11 | fvsng 7200 | . . . . . . 7 ⊢ ((1 ∈ ℝ ∧ 𝑃 ∈ ℙ) → ({〈1, 𝑃〉}‘1) = 𝑃) | |
12 | 9, 10, 11 | sylancr 587 | . . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝑘 ∈ {1}) → ({〈1, 𝑃〉}‘1) = 𝑃) |
13 | 12 | sumeq2dv 15735 | . . . . 5 ⊢ (𝑃 ∈ ℙ → Σ𝑘 ∈ {1} ({〈1, 𝑃〉}‘1) = Σ𝑘 ∈ {1}𝑃) |
14 | prmz 16709 | . . . . . . 7 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ) | |
15 | 14 | zcnd 12721 | . . . . . 6 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℂ) |
16 | eqidd 2736 | . . . . . . 7 ⊢ (𝑘 = 1 → 𝑃 = 𝑃) | |
17 | 16 | sumsn 15779 | . . . . . 6 ⊢ ((1 ∈ ℝ ∧ 𝑃 ∈ ℂ) → Σ𝑘 ∈ {1}𝑃 = 𝑃) |
18 | 9, 15, 17 | sylancr 587 | . . . . 5 ⊢ (𝑃 ∈ ℙ → Σ𝑘 ∈ {1}𝑃 = 𝑃) |
19 | 13, 18 | eqtr2d 2776 | . . . 4 ⊢ (𝑃 ∈ ℙ → 𝑃 = Σ𝑘 ∈ {1} ({〈1, 𝑃〉}‘1)) |
20 | 1le3 12476 | . . . 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 4648 | . . . . . . . . 9 ⊢ (𝑘 ∈ {1} → 𝑘 = 1) | |
24 | 23 | adantl 481 | . . . . . . . 8 ⊢ ((𝑓 = {〈1, 𝑃〉} ∧ 𝑘 ∈ {1}) → 𝑘 = 1) |
25 | 22, 24 | fveq12d 6914 | . . . . . . 7 ⊢ ((𝑓 = {〈1, 𝑃〉} ∧ 𝑘 ∈ {1}) → (𝑓‘𝑘) = ({〈1, 𝑃〉}‘1)) |
26 | 25 | sumeq2dv 15735 | . . . . . 6 ⊢ (𝑓 = {〈1, 𝑃〉} → Σ𝑘 ∈ {1} (𝑓‘𝑘) = Σ𝑘 ∈ {1} ({〈1, 𝑃〉}‘1)) |
27 | 26 | eqeq2d 2746 | . . . . 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 3622 | . . 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 584 | . 2 ⊢ (𝑃 ∈ ℙ → ∃𝑓 ∈ (ℙ ↑m {1})(1 ≤ 3 ∧ 𝑃 = Σ𝑘 ∈ {1} (𝑓‘𝑘))) |
31 | oveq2 7439 | . . . . . 6 ⊢ (𝑑 = 1 → (1...𝑑) = (1...1)) | |
32 | 1z 12645 | . . . . . . 7 ⊢ 1 ∈ ℤ | |
33 | fzsn 13603 | . . . . . . 7 ⊢ (1 ∈ ℤ → (1...1) = {1}) | |
34 | 32, 33 | ax-mp 5 | . . . . . 6 ⊢ (1...1) = {1} |
35 | 31, 34 | eqtrdi 2791 | . . . . 5 ⊢ (𝑑 = 1 → (1...𝑑) = {1}) |
36 | 35 | oveq2d 7447 | . . . 4 ⊢ (𝑑 = 1 → (ℙ ↑m (1...𝑑)) = (ℙ ↑m {1})) |
37 | breq1 5151 | . . . . 5 ⊢ (𝑑 = 1 → (𝑑 ≤ 3 ↔ 1 ≤ 3)) | |
38 | 35 | sumeq1d 15733 | . . . . . 6 ⊢ (𝑑 = 1 → Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘) = Σ𝑘 ∈ {1} (𝑓‘𝑘)) |
39 | 38 | eqeq2d 2746 | . . . . 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 3345 | . . 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 3622 | . 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 587 | 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 1537 ∈ wcel 2106 ∃wrex 3068 {csn 4631 〈cop 4637 class class class wbr 5148 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ↑m cmap 8865 ℂcc 11151 ℝcr 11152 1c1 11154 ≤ cle 11294 ℕcn 12264 3c3 12320 ℤcz 12611 ...cfz 13544 Σcsu 15719 ℙcprime 16705 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12612 df-uz 12877 df-rp 13033 df-fz 13545 df-fzo 13692 df-seq 14040 df-exp 14100 df-hash 14367 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-clim 15521 df-sum 15720 df-prm 16706 |
This theorem is referenced by: nnsum4primesprm 47716 nnsum3primesle9 47719 |
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