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Mirrors > Home > MPE Home > Th. List > Mathboxes > nnsum4primesle9 | Structured version Visualization version GIF version |
Description: Every integer greater than 1 and less than or equal to 8 is the sum of at most 4 primes. (Contributed by AV, 24-Jul-2020.) (Proof shortened by AV, 2-Aug-2020.) |
Ref | Expression |
---|---|
nnsum4primesle9 | ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑁 ≤ 8) → ∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑m (1...𝑑))(𝑑 ≤ 4 ∧ 𝑁 = Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnsum3primesle9 47193 | . 2 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑁 ≤ 8) → ∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑m (1...𝑑))(𝑑 ≤ 3 ∧ 𝑁 = Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘))) | |
2 | 3lt4 12411 | . . . . . 6 ⊢ 3 < 4 | |
3 | nnre 12244 | . . . . . . 7 ⊢ (𝑑 ∈ ℕ → 𝑑 ∈ ℝ) | |
4 | 3re 12317 | . . . . . . . 8 ⊢ 3 ∈ ℝ | |
5 | 4 | a1i 11 | . . . . . . 7 ⊢ (𝑑 ∈ ℕ → 3 ∈ ℝ) |
6 | 4re 12321 | . . . . . . . 8 ⊢ 4 ∈ ℝ | |
7 | 6 | a1i 11 | . . . . . . 7 ⊢ (𝑑 ∈ ℕ → 4 ∈ ℝ) |
8 | leltletr 11330 | . . . . . . 7 ⊢ ((𝑑 ∈ ℝ ∧ 3 ∈ ℝ ∧ 4 ∈ ℝ) → ((𝑑 ≤ 3 ∧ 3 < 4) → 𝑑 ≤ 4)) | |
9 | 3, 5, 7, 8 | syl3anc 1368 | . . . . . 6 ⊢ (𝑑 ∈ ℕ → ((𝑑 ≤ 3 ∧ 3 < 4) → 𝑑 ≤ 4)) |
10 | 2, 9 | mpan2i 695 | . . . . 5 ⊢ (𝑑 ∈ ℕ → (𝑑 ≤ 3 → 𝑑 ≤ 4)) |
11 | 10 | anim1d 609 | . . . 4 ⊢ (𝑑 ∈ ℕ → ((𝑑 ≤ 3 ∧ 𝑁 = Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘)) → (𝑑 ≤ 4 ∧ 𝑁 = Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘)))) |
12 | 11 | reximdv 3160 | . . 3 ⊢ (𝑑 ∈ ℕ → (∃𝑓 ∈ (ℙ ↑m (1...𝑑))(𝑑 ≤ 3 ∧ 𝑁 = Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘)) → ∃𝑓 ∈ (ℙ ↑m (1...𝑑))(𝑑 ≤ 4 ∧ 𝑁 = Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘)))) |
13 | 12 | reximia 3071 | . 2 ⊢ (∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑m (1...𝑑))(𝑑 ≤ 3 ∧ 𝑁 = Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘)) → ∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑m (1...𝑑))(𝑑 ≤ 4 ∧ 𝑁 = Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘))) |
14 | 1, 13 | syl 17 | 1 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑁 ≤ 8) → ∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑m (1...𝑑))(𝑑 ≤ 4 ∧ 𝑁 = Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∃wrex 3060 class class class wbr 5144 ‘cfv 6543 (class class class)co 7413 ↑m cmap 8838 ℝcr 11132 1c1 11134 < clt 11273 ≤ cle 11274 ℕcn 12237 2c2 12292 3c3 12293 4c4 12294 8c8 12298 ℤ≥cuz 12847 ...cfz 13511 Σcsu 15659 ℙcprime 16636 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-inf2 9659 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 ax-pre-sup 11211 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3961 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-int 4946 df-iun 4994 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-se 5629 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7866 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-2o 8481 df-er 8718 df-map 8840 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-sup 9460 df-inf 9461 df-oi 9528 df-card 9957 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-div 11897 df-nn 12238 df-2 12300 df-3 12301 df-4 12302 df-5 12303 df-6 12304 df-7 12305 df-8 12306 df-9 12307 df-n0 12498 df-z 12584 df-dec 12703 df-uz 12848 df-rp 13002 df-fz 13512 df-fzo 13655 df-seq 13994 df-exp 14054 df-hash 14317 df-cj 15073 df-re 15074 df-im 15075 df-sqrt 15209 df-abs 15210 df-clim 15459 df-sum 15660 df-dvds 16226 df-prm 16637 df-even 47025 df-odd 47026 df-gbe 47147 |
This theorem is referenced by: wtgoldbnnsum4prm 47201 |
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