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Mirrors > Home > MPE Home > Th. List > prmolefac | Structured version Visualization version GIF version |
Description: The primorial of a positive integer is less than or equal to the factorial of the integer. (Contributed by AV, 15-Aug-2020.) (Revised by AV, 29-Aug-2020.) |
Ref | Expression |
---|---|
prmolefac | ⊢ (𝑁 ∈ ℕ0 → (#p‘𝑁) ≤ (!‘𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 2015 | . . 3 ⊢ Ⅎ𝑘 𝑁 ∈ ℕ0 | |
2 | fzfid 13066 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (1...𝑁) ∈ Fin) | |
3 | elfznn 12662 | . . . . . 6 ⊢ (𝑘 ∈ (1...𝑁) → 𝑘 ∈ ℕ) | |
4 | 3 | adantl 475 | . . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (1...𝑁)) → 𝑘 ∈ ℕ) |
5 | 1nn 11362 | . . . . . 6 ⊢ 1 ∈ ℕ | |
6 | 5 | a1i 11 | . . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (1...𝑁)) → 1 ∈ ℕ) |
7 | 4, 6 | ifcld 4350 | . . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (1...𝑁)) → if(𝑘 ∈ ℙ, 𝑘, 1) ∈ ℕ) |
8 | 7 | nnred 11366 | . . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (1...𝑁)) → if(𝑘 ∈ ℙ, 𝑘, 1) ∈ ℝ) |
9 | ifeqor 4354 | . . . 4 ⊢ (if(𝑘 ∈ ℙ, 𝑘, 1) = 𝑘 ∨ if(𝑘 ∈ ℙ, 𝑘, 1) = 1) | |
10 | nnnn0 11625 | . . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℕ0) | |
11 | 10 | nn0ge0d 11680 | . . . . . . . 8 ⊢ (𝑘 ∈ ℕ → 0 ≤ 𝑘) |
12 | 3, 11 | syl 17 | . . . . . . 7 ⊢ (𝑘 ∈ (1...𝑁) → 0 ≤ 𝑘) |
13 | 12 | adantl 475 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (1...𝑁)) → 0 ≤ 𝑘) |
14 | breq2 4876 | . . . . . 6 ⊢ (if(𝑘 ∈ ℙ, 𝑘, 1) = 𝑘 → (0 ≤ if(𝑘 ∈ ℙ, 𝑘, 1) ↔ 0 ≤ 𝑘)) | |
15 | 13, 14 | syl5ibr 238 | . . . . 5 ⊢ (if(𝑘 ∈ ℙ, 𝑘, 1) = 𝑘 → ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (1...𝑁)) → 0 ≤ if(𝑘 ∈ ℙ, 𝑘, 1))) |
16 | 0le1 10874 | . . . . . . 7 ⊢ 0 ≤ 1 | |
17 | breq2 4876 | . . . . . . . 8 ⊢ (if(𝑘 ∈ ℙ, 𝑘, 1) = 1 → (0 ≤ if(𝑘 ∈ ℙ, 𝑘, 1) ↔ 0 ≤ 1)) | |
18 | 17 | adantr 474 | . . . . . . 7 ⊢ ((if(𝑘 ∈ ℙ, 𝑘, 1) = 1 ∧ (𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (1...𝑁))) → (0 ≤ if(𝑘 ∈ ℙ, 𝑘, 1) ↔ 0 ≤ 1)) |
19 | 16, 18 | mpbiri 250 | . . . . . 6 ⊢ ((if(𝑘 ∈ ℙ, 𝑘, 1) = 1 ∧ (𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (1...𝑁))) → 0 ≤ if(𝑘 ∈ ℙ, 𝑘, 1)) |
20 | 19 | ex 403 | . . . . 5 ⊢ (if(𝑘 ∈ ℙ, 𝑘, 1) = 1 → ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (1...𝑁)) → 0 ≤ if(𝑘 ∈ ℙ, 𝑘, 1))) |
21 | 15, 20 | jaoi 890 | . . . 4 ⊢ ((if(𝑘 ∈ ℙ, 𝑘, 1) = 𝑘 ∨ if(𝑘 ∈ ℙ, 𝑘, 1) = 1) → ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (1...𝑁)) → 0 ≤ if(𝑘 ∈ ℙ, 𝑘, 1))) |
22 | 9, 21 | ax-mp 5 | . . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (1...𝑁)) → 0 ≤ if(𝑘 ∈ ℙ, 𝑘, 1)) |
23 | 4 | nnred 11366 | . . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (1...𝑁)) → 𝑘 ∈ ℝ) |
24 | 23 | leidd 10917 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (1...𝑁)) → 𝑘 ≤ 𝑘) |
25 | breq1 4875 | . . . . . 6 ⊢ (if(𝑘 ∈ ℙ, 𝑘, 1) = 𝑘 → (if(𝑘 ∈ ℙ, 𝑘, 1) ≤ 𝑘 ↔ 𝑘 ≤ 𝑘)) | |
26 | 24, 25 | syl5ibr 238 | . . . . 5 ⊢ (if(𝑘 ∈ ℙ, 𝑘, 1) = 𝑘 → ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (1...𝑁)) → if(𝑘 ∈ ℙ, 𝑘, 1) ≤ 𝑘)) |
27 | 4 | nnge1d 11398 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (1...𝑁)) → 1 ≤ 𝑘) |
28 | breq1 4875 | . . . . . 6 ⊢ (if(𝑘 ∈ ℙ, 𝑘, 1) = 1 → (if(𝑘 ∈ ℙ, 𝑘, 1) ≤ 𝑘 ↔ 1 ≤ 𝑘)) | |
29 | 27, 28 | syl5ibr 238 | . . . . 5 ⊢ (if(𝑘 ∈ ℙ, 𝑘, 1) = 1 → ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (1...𝑁)) → if(𝑘 ∈ ℙ, 𝑘, 1) ≤ 𝑘)) |
30 | 26, 29 | jaoi 890 | . . . 4 ⊢ ((if(𝑘 ∈ ℙ, 𝑘, 1) = 𝑘 ∨ if(𝑘 ∈ ℙ, 𝑘, 1) = 1) → ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (1...𝑁)) → if(𝑘 ∈ ℙ, 𝑘, 1) ≤ 𝑘)) |
31 | 9, 30 | ax-mp 5 | . . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (1...𝑁)) → if(𝑘 ∈ ℙ, 𝑘, 1) ≤ 𝑘) |
32 | 1, 2, 8, 22, 23, 31 | fprodle 15098 | . 2 ⊢ (𝑁 ∈ ℕ0 → ∏𝑘 ∈ (1...𝑁)if(𝑘 ∈ ℙ, 𝑘, 1) ≤ ∏𝑘 ∈ (1...𝑁)𝑘) |
33 | prmoval 16107 | . 2 ⊢ (𝑁 ∈ ℕ0 → (#p‘𝑁) = ∏𝑘 ∈ (1...𝑁)if(𝑘 ∈ ℙ, 𝑘, 1)) | |
34 | fprodfac 15075 | . 2 ⊢ (𝑁 ∈ ℕ0 → (!‘𝑁) = ∏𝑘 ∈ (1...𝑁)𝑘) | |
35 | 32, 33, 34 | 3brtr4d 4904 | 1 ⊢ (𝑁 ∈ ℕ0 → (#p‘𝑁) ≤ (!‘𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 ∨ wo 880 = wceq 1658 ∈ wcel 2166 ifcif 4305 class class class wbr 4872 ‘cfv 6122 (class class class)co 6904 0cc0 10251 1c1 10252 ≤ cle 10391 ℕcn 11349 ℕ0cn0 11617 ...cfz 12618 !cfa 13352 ∏cprod 15007 ℙcprime 15756 #pcprmo 16105 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2390 ax-ext 2802 ax-rep 4993 ax-sep 5004 ax-nul 5012 ax-pow 5064 ax-pr 5126 ax-un 7208 ax-inf2 8814 ax-cnex 10307 ax-resscn 10308 ax-1cn 10309 ax-icn 10310 ax-addcl 10311 ax-addrcl 10312 ax-mulcl 10313 ax-mulrcl 10314 ax-mulcom 10315 ax-addass 10316 ax-mulass 10317 ax-distr 10318 ax-i2m1 10319 ax-1ne0 10320 ax-1rid 10321 ax-rnegex 10322 ax-rrecex 10323 ax-cnre 10324 ax-pre-lttri 10325 ax-pre-lttrn 10326 ax-pre-ltadd 10327 ax-pre-mulgt0 10328 ax-pre-sup 10329 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-fal 1672 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2604 df-eu 2639 df-clab 2811 df-cleq 2817 df-clel 2820 df-nfc 2957 df-ne 2999 df-nel 3102 df-ral 3121 df-rex 3122 df-reu 3123 df-rmo 3124 df-rab 3125 df-v 3415 df-sbc 3662 df-csb 3757 df-dif 3800 df-un 3802 df-in 3804 df-ss 3811 df-pss 3813 df-nul 4144 df-if 4306 df-pw 4379 df-sn 4397 df-pr 4399 df-tp 4401 df-op 4403 df-uni 4658 df-int 4697 df-iun 4741 df-br 4873 df-opab 4935 df-mpt 4952 df-tr 4975 df-id 5249 df-eprel 5254 df-po 5262 df-so 5263 df-fr 5300 df-se 5301 df-we 5302 df-xp 5347 df-rel 5348 df-cnv 5349 df-co 5350 df-dm 5351 df-rn 5352 df-res 5353 df-ima 5354 df-pred 5919 df-ord 5965 df-on 5966 df-lim 5967 df-suc 5968 df-iota 6085 df-fun 6124 df-fn 6125 df-f 6126 df-f1 6127 df-fo 6128 df-f1o 6129 df-fv 6130 df-isom 6131 df-riota 6865 df-ov 6907 df-oprab 6908 df-mpt2 6909 df-om 7326 df-1st 7427 df-2nd 7428 df-wrecs 7671 df-recs 7733 df-rdg 7771 df-1o 7825 df-oadd 7829 df-er 8008 df-en 8222 df-dom 8223 df-sdom 8224 df-fin 8225 df-sup 8616 df-oi 8683 df-card 9077 df-pnf 10392 df-mnf 10393 df-xr 10394 df-ltxr 10395 df-le 10396 df-sub 10586 df-neg 10587 df-div 11009 df-nn 11350 df-2 11413 df-3 11414 df-n0 11618 df-z 11704 df-uz 11968 df-rp 12112 df-ico 12468 df-fz 12619 df-fzo 12760 df-seq 13095 df-exp 13154 df-fac 13353 df-hash 13410 df-cj 14215 df-re 14216 df-im 14217 df-sqrt 14351 df-abs 14352 df-clim 14595 df-prod 15008 df-prmo 16106 |
This theorem is referenced by: (None) |
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