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Mirrors > Home > MPE Home > Th. List > ppisval | 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 |
---|---|
ppisval | ⊢ (𝐴 ∈ ℝ → ((0[,]𝐴) ∩ ℙ) = ((2...(⌊‘𝐴)) ∩ ℙ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inss2 3834 | . . . . . . . 8 ⊢ ((0[,]𝐴) ∩ ℙ) ⊆ ℙ | |
2 | simpr 477 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑥 ∈ ((0[,]𝐴) ∩ ℙ)) | |
3 | 1, 2 | sseldi 3601 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑥 ∈ ℙ) |
4 | prmuz2 15408 | . . . . . . 7 ⊢ (𝑥 ∈ ℙ → 𝑥 ∈ (ℤ≥‘2)) | |
5 | 3, 4 | syl 17 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑥 ∈ (ℤ≥‘2)) |
6 | prmz 15389 | . . . . . . . 8 ⊢ (𝑥 ∈ ℙ → 𝑥 ∈ ℤ) | |
7 | 3, 6 | syl 17 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑥 ∈ ℤ) |
8 | flcl 12596 | . . . . . . . 8 ⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ∈ ℤ) | |
9 | 8 | adantr 481 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ ((0[,]𝐴) ∩ ℙ)) → (⌊‘𝐴) ∈ ℤ) |
10 | inss1 3833 | . . . . . . . . . . 11 ⊢ ((0[,]𝐴) ∩ ℙ) ⊆ (0[,]𝐴) | |
11 | 10, 2 | sseldi 3601 | . . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑥 ∈ (0[,]𝐴)) |
12 | 0re 10040 | . . . . . . . . . . 11 ⊢ 0 ∈ ℝ | |
13 | simpl 473 | . . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝐴 ∈ ℝ) | |
14 | elicc2 12238 | . . . . . . . . . . 11 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝑥 ∈ (0[,]𝐴) ↔ (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥 ∧ 𝑥 ≤ 𝐴))) | |
15 | 12, 13, 14 | sylancr 695 | . . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ ((0[,]𝐴) ∩ ℙ)) → (𝑥 ∈ (0[,]𝐴) ↔ (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥 ∧ 𝑥 ≤ 𝐴))) |
16 | 11, 15 | mpbid 222 | . . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ ((0[,]𝐴) ∩ ℙ)) → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥 ∧ 𝑥 ≤ 𝐴)) |
17 | 16 | simp3d 1075 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑥 ≤ 𝐴) |
18 | flge 12606 | . . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ ℤ) → (𝑥 ≤ 𝐴 ↔ 𝑥 ≤ (⌊‘𝐴))) | |
19 | 7, 18 | syldan 487 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ ((0[,]𝐴) ∩ ℙ)) → (𝑥 ≤ 𝐴 ↔ 𝑥 ≤ (⌊‘𝐴))) |
20 | 17, 19 | mpbid 222 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑥 ≤ (⌊‘𝐴)) |
21 | eluz2 11693 | . . . . . . 7 ⊢ ((⌊‘𝐴) ∈ (ℤ≥‘𝑥) ↔ (𝑥 ∈ ℤ ∧ (⌊‘𝐴) ∈ ℤ ∧ 𝑥 ≤ (⌊‘𝐴))) | |
22 | 7, 9, 20, 21 | syl3anbrc 1246 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ ((0[,]𝐴) ∩ ℙ)) → (⌊‘𝐴) ∈ (ℤ≥‘𝑥)) |
23 | elfzuzb 12336 | . . . . . 6 ⊢ (𝑥 ∈ (2...(⌊‘𝐴)) ↔ (𝑥 ∈ (ℤ≥‘2) ∧ (⌊‘𝐴) ∈ (ℤ≥‘𝑥))) | |
24 | 5, 22, 23 | sylanbrc 698 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑥 ∈ (2...(⌊‘𝐴))) |
25 | 24, 3 | elind 3798 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑥 ∈ ((2...(⌊‘𝐴)) ∩ ℙ)) |
26 | 25 | ex 450 | . . 3 ⊢ (𝐴 ∈ ℝ → (𝑥 ∈ ((0[,]𝐴) ∩ ℙ) → 𝑥 ∈ ((2...(⌊‘𝐴)) ∩ ℙ))) |
27 | 26 | ssrdv 3609 | . 2 ⊢ (𝐴 ∈ ℝ → ((0[,]𝐴) ∩ ℙ) ⊆ ((2...(⌊‘𝐴)) ∩ ℙ)) |
28 | 2z 11409 | . . . . 5 ⊢ 2 ∈ ℤ | |
29 | fzval2 12329 | . . . . 5 ⊢ ((2 ∈ ℤ ∧ (⌊‘𝐴) ∈ ℤ) → (2...(⌊‘𝐴)) = ((2[,](⌊‘𝐴)) ∩ ℤ)) | |
30 | 28, 8, 29 | sylancr 695 | . . . 4 ⊢ (𝐴 ∈ ℝ → (2...(⌊‘𝐴)) = ((2[,](⌊‘𝐴)) ∩ ℤ)) |
31 | inss1 3833 | . . . . 5 ⊢ ((2[,](⌊‘𝐴)) ∩ ℤ) ⊆ (2[,](⌊‘𝐴)) | |
32 | 12 | a1i 11 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → 0 ∈ ℝ) |
33 | id 22 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ) | |
34 | 0le2 11111 | . . . . . . 7 ⊢ 0 ≤ 2 | |
35 | 34 | a1i 11 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → 0 ≤ 2) |
36 | flle 12600 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ≤ 𝐴) | |
37 | iccss 12241 | . . . . . 6 ⊢ (((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) ∧ (0 ≤ 2 ∧ (⌊‘𝐴) ≤ 𝐴)) → (2[,](⌊‘𝐴)) ⊆ (0[,]𝐴)) | |
38 | 32, 33, 35, 36, 37 | syl22anc 1327 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (2[,](⌊‘𝐴)) ⊆ (0[,]𝐴)) |
39 | 31, 38 | syl5ss 3614 | . . . 4 ⊢ (𝐴 ∈ ℝ → ((2[,](⌊‘𝐴)) ∩ ℤ) ⊆ (0[,]𝐴)) |
40 | 30, 39 | eqsstrd 3639 | . . 3 ⊢ (𝐴 ∈ ℝ → (2...(⌊‘𝐴)) ⊆ (0[,]𝐴)) |
41 | ssrin 3838 | . . 3 ⊢ ((2...(⌊‘𝐴)) ⊆ (0[,]𝐴) → ((2...(⌊‘𝐴)) ∩ ℙ) ⊆ ((0[,]𝐴) ∩ ℙ)) | |
42 | 40, 41 | syl 17 | . 2 ⊢ (𝐴 ∈ ℝ → ((2...(⌊‘𝐴)) ∩ ℙ) ⊆ ((0[,]𝐴) ∩ ℙ)) |
43 | 27, 42 | eqssd 3620 | 1 ⊢ (𝐴 ∈ ℝ → ((0[,]𝐴) ∩ ℙ) = ((2...(⌊‘𝐴)) ∩ ℙ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ∩ cin 3573 ⊆ wss 3574 class class class wbr 4653 ‘cfv 5888 (class class class)co 6650 ℝcr 9935 0cc0 9936 ≤ cle 10075 2c2 11070 ℤcz 11377 ℤ≥cuz 11687 [,]cicc 12178 ...cfz 12326 ⌊cfl 12591 ℙcprime 15385 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-inf 8349 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-icc 12182 df-fz 12327 df-fl 12593 df-seq 12802 df-exp 12861 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-dvds 14984 df-prm 15386 |
This theorem is referenced by: ppisval2 24831 ppifi 24832 ppival2 24854 chtfl 24875 chtprm 24879 chtnprm 24880 ppifl 24886 cht1 24891 chtlepsi 24931 chpval2 24943 chpub 24945 chtvalz 30707 |
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