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Mirrors > Home > MPE Home > Th. List > dfprm2 | Structured version Visualization version GIF version |
Description: The positive irreducible elements of ℤ are the prime numbers. This is an alternative way to define ℙ. (Contributed by Mario Carneiro, 5-Dec-2014.) (Revised by AV, 10-Jun-2019.) |
Ref | Expression |
---|---|
prmirred.i | ⊢ 𝐼 = (Irred‘ℤring) |
Ref | Expression |
---|---|
dfprm2 | ⊢ ℙ = (ℕ ∩ 𝐼) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prmnn 15594 | . . . 4 ⊢ (𝑥 ∈ ℙ → 𝑥 ∈ ℕ) | |
2 | prmirred.i | . . . . . 6 ⊢ 𝐼 = (Irred‘ℤring) | |
3 | 2 | prmirredlem 20055 | . . . . 5 ⊢ (𝑥 ∈ ℕ → (𝑥 ∈ 𝐼 ↔ 𝑥 ∈ ℙ)) |
4 | 3 | bicomd 213 | . . . 4 ⊢ (𝑥 ∈ ℕ → (𝑥 ∈ ℙ ↔ 𝑥 ∈ 𝐼)) |
5 | 1, 4 | biadan2 802 | . . 3 ⊢ (𝑥 ∈ ℙ ↔ (𝑥 ∈ ℕ ∧ 𝑥 ∈ 𝐼)) |
6 | elin 3947 | . . 3 ⊢ (𝑥 ∈ (ℕ ∩ 𝐼) ↔ (𝑥 ∈ ℕ ∧ 𝑥 ∈ 𝐼)) | |
7 | 5, 6 | bitr4i 267 | . 2 ⊢ (𝑥 ∈ ℙ ↔ 𝑥 ∈ (ℕ ∩ 𝐼)) |
8 | 7 | eqriv 2768 | 1 ⊢ ℙ = (ℕ ∩ 𝐼) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 382 = wceq 1631 ∈ wcel 2145 ∩ cin 3722 ‘cfv 6031 ℕcn 11221 ℙcprime 15591 Irredcir 18847 ℤringzring 20032 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7095 ax-cnex 10193 ax-resscn 10194 ax-1cn 10195 ax-icn 10196 ax-addcl 10197 ax-addrcl 10198 ax-mulcl 10199 ax-mulrcl 10200 ax-mulcom 10201 ax-addass 10202 ax-mulass 10203 ax-distr 10204 ax-i2m1 10205 ax-1ne0 10206 ax-1rid 10207 ax-rnegex 10208 ax-rrecex 10209 ax-cnre 10210 ax-pre-lttri 10211 ax-pre-lttrn 10212 ax-pre-ltadd 10213 ax-pre-mulgt0 10214 ax-pre-sup 10215 ax-addf 10216 ax-mulf 10217 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 827 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-int 4612 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-riota 6753 df-ov 6795 df-oprab 6796 df-mpt2 6797 df-om 7212 df-1st 7314 df-2nd 7315 df-tpos 7503 df-wrecs 7558 df-recs 7620 df-rdg 7658 df-1o 7712 df-2o 7713 df-oadd 7716 df-er 7895 df-en 8109 df-dom 8110 df-sdom 8111 df-fin 8112 df-sup 8503 df-pnf 10277 df-mnf 10278 df-xr 10279 df-ltxr 10280 df-le 10281 df-sub 10469 df-neg 10470 df-div 10886 df-nn 11222 df-2 11280 df-3 11281 df-4 11282 df-5 11283 df-6 11284 df-7 11285 df-8 11286 df-9 11287 df-n0 11494 df-z 11579 df-dec 11695 df-uz 11888 df-rp 12035 df-fz 12533 df-seq 13008 df-exp 13067 df-cj 14046 df-re 14047 df-im 14048 df-sqrt 14182 df-abs 14183 df-dvds 15189 df-prm 15592 df-gz 15840 df-struct 16065 df-ndx 16066 df-slot 16067 df-base 16069 df-sets 16070 df-ress 16071 df-plusg 16161 df-mulr 16162 df-starv 16163 df-tset 16167 df-ple 16168 df-ds 16171 df-unif 16172 df-0g 16309 df-mgm 17449 df-sgrp 17491 df-mnd 17502 df-grp 17632 df-minusg 17633 df-subg 17798 df-cmn 18401 df-mgp 18697 df-ur 18709 df-ring 18756 df-cring 18757 df-oppr 18830 df-dvdsr 18848 df-unit 18849 df-irred 18850 df-invr 18879 df-dvr 18890 df-drng 18958 df-subrg 18987 df-cnfld 19961 df-zring 20033 |
This theorem is referenced by: (None) |
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