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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 25244 | . . 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 12674 | . . . . 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 4065 | . . 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 4024 | . . . . . . . 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 15781 | . . . . . 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 12633 | . . . . . 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 12630 | . . . . 5 ⊢ (𝑥 ∈ (2...(⌊‘𝐴)) ↔ (𝑥 ∈ (ℤ≥‘2) ∧ (⌊‘𝐴) ∈ (ℤ≥‘𝑥))) | |
16 | 11, 14, 15 | sylanbrc 580 | . . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 2 ∈ (ℤ≥‘𝑀)) ∧ 𝑥 ∈ ((𝑀...(⌊‘𝐴)) ∩ ℙ)) → 𝑥 ∈ (2...(⌊‘𝐴))) |
17 | 16, 9 | elind 4026 | . . 3 ⊢ (((𝐴 ∈ ℝ ∧ 2 ∈ (ℤ≥‘𝑀)) ∧ 𝑥 ∈ ((𝑀...(⌊‘𝐴)) ∩ ℙ)) → 𝑥 ∈ ((2...(⌊‘𝐴)) ∩ ℙ)) |
18 | 5, 17 | eqelssd 3848 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 2 ∈ (ℤ≥‘𝑀)) → ((2...(⌊‘𝐴)) ∩ ℙ) = ((𝑀...(⌊‘𝐴)) ∩ ℙ)) |
19 | 2, 18 | eqtrd 2862 | 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 1658 ∈ wcel 2166 ∩ cin 3798 ⊆ wss 3799 ‘cfv 6124 (class class class)co 6906 ℝcr 10252 0cc0 10253 2c2 11407 ℤ≥cuz 11969 [,]cicc 12467 ...cfz 12620 ⌊cfl 12887 ℙcprime 15758 |
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 2391 ax-ext 2804 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210 ax-cnex 10309 ax-resscn 10310 ax-1cn 10311 ax-icn 10312 ax-addcl 10313 ax-addrcl 10314 ax-mulcl 10315 ax-mulrcl 10316 ax-mulcom 10317 ax-addass 10318 ax-mulass 10319 ax-distr 10320 ax-i2m1 10321 ax-1ne0 10322 ax-1rid 10323 ax-rnegex 10324 ax-rrecex 10325 ax-cnre 10326 ax-pre-lttri 10327 ax-pre-lttrn 10328 ax-pre-ltadd 10329 ax-pre-mulgt0 10330 ax-pre-sup 10331 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-nel 3104 df-ral 3123 df-rex 3124 df-reu 3125 df-rmo 3126 df-rab 3127 df-v 3417 df-sbc 3664 df-csb 3759 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-pss 3815 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4660 df-iun 4743 df-br 4875 df-opab 4937 df-mpt 4954 df-tr 4977 df-id 5251 df-eprel 5256 df-po 5264 df-so 5265 df-fr 5302 df-we 5304 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 df-pred 5921 df-ord 5967 df-on 5968 df-lim 5969 df-suc 5970 df-iota 6087 df-fun 6126 df-fn 6127 df-f 6128 df-f1 6129 df-fo 6130 df-f1o 6131 df-fv 6132 df-riota 6867 df-ov 6909 df-oprab 6910 df-mpt2 6911 df-om 7328 df-1st 7429 df-2nd 7430 df-wrecs 7673 df-recs 7735 df-rdg 7773 df-1o 7827 df-2o 7828 df-er 8010 df-en 8224 df-dom 8225 df-sdom 8226 df-fin 8227 df-sup 8618 df-inf 8619 df-pnf 10394 df-mnf 10395 df-xr 10396 df-ltxr 10397 df-le 10398 df-sub 10588 df-neg 10589 df-div 11011 df-nn 11352 df-2 11415 df-3 11416 df-n0 11620 df-z 11706 df-uz 11970 df-rp 12114 df-icc 12471 df-fz 12621 df-fl 12889 df-seq 13097 df-exp 13156 df-cj 14217 df-re 14218 df-im 14219 df-sqrt 14353 df-abs 14354 df-dvds 15359 df-prm 15759 |
This theorem is referenced by: ppival2g 25269 chtdif 25298 prmorcht 25318 chtppilimlem1 25576 |
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