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Mirrors > Home > MPE Home > Th. List > prmgt1 | Structured version Visualization version GIF version |
Description: A prime number is an integer greater than 1. (Contributed by Alexander van der Vekens, 17-May-2018.) |
Ref | Expression |
---|---|
prmgt1 | ⊢ (𝑃 ∈ ℙ → 1 < 𝑃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prmuz2 15615 | . 2 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ≥‘2)) | |
2 | eluz2 11894 | . . 3 ⊢ (𝑃 ∈ (ℤ≥‘2) ↔ (2 ∈ ℤ ∧ 𝑃 ∈ ℤ ∧ 2 ≤ 𝑃)) | |
3 | df-2 11281 | . . . . . . . 8 ⊢ 2 = (1 + 1) | |
4 | 3 | a1i 11 | . . . . . . 7 ⊢ (𝑃 ∈ ℤ → 2 = (1 + 1)) |
5 | 4 | breq1d 4796 | . . . . . 6 ⊢ (𝑃 ∈ ℤ → (2 ≤ 𝑃 ↔ (1 + 1) ≤ 𝑃)) |
6 | 1z 11609 | . . . . . . . 8 ⊢ 1 ∈ ℤ | |
7 | zltp1le 11629 | . . . . . . . 8 ⊢ ((1 ∈ ℤ ∧ 𝑃 ∈ ℤ) → (1 < 𝑃 ↔ (1 + 1) ≤ 𝑃)) | |
8 | 6, 7 | mpan 670 | . . . . . . 7 ⊢ (𝑃 ∈ ℤ → (1 < 𝑃 ↔ (1 + 1) ≤ 𝑃)) |
9 | 8 | biimprd 238 | . . . . . 6 ⊢ (𝑃 ∈ ℤ → ((1 + 1) ≤ 𝑃 → 1 < 𝑃)) |
10 | 5, 9 | sylbid 230 | . . . . 5 ⊢ (𝑃 ∈ ℤ → (2 ≤ 𝑃 → 1 < 𝑃)) |
11 | 10 | imp 393 | . . . 4 ⊢ ((𝑃 ∈ ℤ ∧ 2 ≤ 𝑃) → 1 < 𝑃) |
12 | 11 | 3adant1 1124 | . . 3 ⊢ ((2 ∈ ℤ ∧ 𝑃 ∈ ℤ ∧ 2 ≤ 𝑃) → 1 < 𝑃) |
13 | 2, 12 | sylbi 207 | . 2 ⊢ (𝑃 ∈ (ℤ≥‘2) → 1 < 𝑃) |
14 | 1, 13 | syl 17 | 1 ⊢ (𝑃 ∈ ℙ → 1 < 𝑃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ w3a 1071 = wceq 1631 ∈ wcel 2145 class class class wbr 4786 ‘cfv 6031 (class class class)co 6793 1c1 10139 + caddc 10141 < clt 10276 ≤ cle 10277 2c2 11272 ℤcz 11579 ℤ≥cuz 11888 ℙcprime 15592 |
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-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 ax-cnex 10194 ax-resscn 10195 ax-1cn 10196 ax-icn 10197 ax-addcl 10198 ax-addrcl 10199 ax-mulcl 10200 ax-mulrcl 10201 ax-mulcom 10202 ax-addass 10203 ax-mulass 10204 ax-distr 10205 ax-i2m1 10206 ax-1ne0 10207 ax-1rid 10208 ax-rnegex 10209 ax-rrecex 10210 ax-cnre 10211 ax-pre-lttri 10212 ax-pre-lttrn 10213 ax-pre-ltadd 10214 ax-pre-mulgt0 10215 ax-pre-sup 10216 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 835 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-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 6754 df-ov 6796 df-oprab 6797 df-mpt2 6798 df-om 7213 df-2nd 7316 df-wrecs 7559 df-recs 7621 df-rdg 7659 df-1o 7713 df-2o 7714 df-er 7896 df-en 8110 df-dom 8111 df-sdom 8112 df-fin 8113 df-sup 8504 df-pnf 10278 df-mnf 10279 df-xr 10280 df-ltxr 10281 df-le 10282 df-sub 10470 df-neg 10471 df-div 10887 df-nn 11223 df-2 11281 df-3 11282 df-n0 11495 df-z 11580 df-uz 11889 df-rp 12036 df-seq 13009 df-exp 13068 df-cj 14047 df-re 14048 df-im 14049 df-sqrt 14183 df-abs 14184 df-dvds 15190 df-prm 15593 |
This theorem is referenced by: isprm7 15627 coprm 15630 prmexpb 15637 modprm1div 15709 vfermltl 15713 dvdsprmpweqle 15797 pcmpt 15803 bposlem6 25235 lgslem4 25246 gausslemma2dlem0i 25310 gausslemma2dlem7 25319 gausslemma2d 25320 rplogsumlem2 25395 padicabvf 25541 numclwwlk5 27587 numclwwlk7 27590 nn0prpwlem 32654 fmtnoprmfac1lem 42004 2pwp1prm 42031 sfprmdvdsmersenne 42048 lighneallem2 42051 |
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