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Mirrors > Home > MPE Home > Th. List > ppisval2 | Structured version Visualization version GIF version |
Description: The set of primes less than 𝐴 expressed using a finite set of integers. (Contributed by Mario Carneiro, 22-Sep-2014.) |
Ref | Expression |
---|---|
ppisval2 | ⊢ ((𝐴 ∈ ℝ ∧ 2 ∈ (ℤ≥‘𝑀)) → ((0[,]𝐴) ∩ ℙ) = ((𝑀...(⌊‘𝐴)) ∩ ℙ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ppisval 25243 | . . 3 ⊢ (𝐴 ∈ ℝ → ((0[,]𝐴) ∩ ℙ) = ((2...(⌊‘𝐴)) ∩ ℙ)) | |
2 | 1 | adantr 474 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 2 ∈ (ℤ≥‘𝑀)) → ((0[,]𝐴) ∩ ℙ) = ((2...(⌊‘𝐴)) ∩ ℙ)) |
3 | fzss1 12673 | . . . . 5 ⊢ (2 ∈ (ℤ≥‘𝑀) → (2...(⌊‘𝐴)) ⊆ (𝑀...(⌊‘𝐴))) | |
4 | 3 | adantl 475 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 2 ∈ (ℤ≥‘𝑀)) → (2...(⌊‘𝐴)) ⊆ (𝑀...(⌊‘𝐴))) |
5 | 4 | ssrind 4064 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 2 ∈ (ℤ≥‘𝑀)) → ((2...(⌊‘𝐴)) ∩ ℙ) ⊆ ((𝑀...(⌊‘𝐴)) ∩ ℙ)) |
6 | simpr 479 | . . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 2 ∈ (ℤ≥‘𝑀)) ∧ 𝑥 ∈ ((𝑀...(⌊‘𝐴)) ∩ ℙ)) → 𝑥 ∈ ((𝑀...(⌊‘𝐴)) ∩ ℙ)) | |
7 | elin 4023 | . . . . . . . 8 ⊢ (𝑥 ∈ ((𝑀...(⌊‘𝐴)) ∩ ℙ) ↔ (𝑥 ∈ (𝑀...(⌊‘𝐴)) ∧ 𝑥 ∈ ℙ)) | |
8 | 6, 7 | sylib 210 | . . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 2 ∈ (ℤ≥‘𝑀)) ∧ 𝑥 ∈ ((𝑀...(⌊‘𝐴)) ∩ ℙ)) → (𝑥 ∈ (𝑀...(⌊‘𝐴)) ∧ 𝑥 ∈ ℙ)) |
9 | 8 | simprd 491 | . . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 2 ∈ (ℤ≥‘𝑀)) ∧ 𝑥 ∈ ((𝑀...(⌊‘𝐴)) ∩ ℙ)) → 𝑥 ∈ ℙ) |
10 | prmuz2 15780 | . . . . . 6 ⊢ (𝑥 ∈ ℙ → 𝑥 ∈ (ℤ≥‘2)) | |
11 | 9, 10 | syl 17 | . . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 2 ∈ (ℤ≥‘𝑀)) ∧ 𝑥 ∈ ((𝑀...(⌊‘𝐴)) ∩ ℙ)) → 𝑥 ∈ (ℤ≥‘2)) |
12 | 8 | simpld 490 | . . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 2 ∈ (ℤ≥‘𝑀)) ∧ 𝑥 ∈ ((𝑀...(⌊‘𝐴)) ∩ ℙ)) → 𝑥 ∈ (𝑀...(⌊‘𝐴))) |
13 | elfzuz3 12632 | . . . . . 6 ⊢ (𝑥 ∈ (𝑀...(⌊‘𝐴)) → (⌊‘𝐴) ∈ (ℤ≥‘𝑥)) | |
14 | 12, 13 | syl 17 | . . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 2 ∈ (ℤ≥‘𝑀)) ∧ 𝑥 ∈ ((𝑀...(⌊‘𝐴)) ∩ ℙ)) → (⌊‘𝐴) ∈ (ℤ≥‘𝑥)) |
15 | elfzuzb 12629 | . . . . 5 ⊢ (𝑥 ∈ (2...(⌊‘𝐴)) ↔ (𝑥 ∈ (ℤ≥‘2) ∧ (⌊‘𝐴) ∈ (ℤ≥‘𝑥))) | |
16 | 11, 14, 15 | sylanbrc 578 | . . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 2 ∈ (ℤ≥‘𝑀)) ∧ 𝑥 ∈ ((𝑀...(⌊‘𝐴)) ∩ ℙ)) → 𝑥 ∈ (2...(⌊‘𝐴))) |
17 | 16, 9 | elind 4025 | . . 3 ⊢ (((𝐴 ∈ ℝ ∧ 2 ∈ (ℤ≥‘𝑀)) ∧ 𝑥 ∈ ((𝑀...(⌊‘𝐴)) ∩ ℙ)) → 𝑥 ∈ ((2...(⌊‘𝐴)) ∩ ℙ)) |
18 | 5, 17 | eqelssd 3847 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 2 ∈ (ℤ≥‘𝑀)) → ((2...(⌊‘𝐴)) ∩ ℙ) = ((𝑀...(⌊‘𝐴)) ∩ ℙ)) |
19 | 2, 18 | eqtrd 2861 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 2 ∈ (ℤ≥‘𝑀)) → ((0[,]𝐴) ∩ ℙ) = ((𝑀...(⌊‘𝐴)) ∩ ℙ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1656 ∈ wcel 2164 ∩ cin 3797 ⊆ wss 3798 ‘cfv 6123 (class class class)co 6905 ℝcr 10251 0cc0 10252 2c2 11406 ℤ≥cuz 11968 [,]cicc 12466 ...cfz 12619 ⌊cfl 12886 ℙcprime 15757 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 ax-pre-sup 10330 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-1st 7428 df-2nd 7429 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-1o 7826 df-2o 7827 df-er 8009 df-en 8223 df-dom 8224 df-sdom 8225 df-fin 8226 df-sup 8617 df-inf 8618 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-div 11010 df-nn 11351 df-2 11414 df-3 11415 df-n0 11619 df-z 11705 df-uz 11969 df-rp 12113 df-icc 12470 df-fz 12620 df-fl 12888 df-seq 13096 df-exp 13155 df-cj 14216 df-re 14217 df-im 14218 df-sqrt 14352 df-abs 14353 df-dvds 15358 df-prm 15758 |
This theorem is referenced by: ppival2g 25268 chtdif 25297 prmorcht 25317 chtppilimlem1 25575 |
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