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Mirrors > Home > MPE Home > Th. List > nprmi | Structured version Visualization version GIF version |
Description: An inference for compositeness. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Jun-2015.) |
Ref | Expression |
---|---|
nprmi.1 | ⊢ 𝐴 ∈ ℕ |
nprmi.2 | ⊢ 𝐵 ∈ ℕ |
nprmi.3 | ⊢ 1 < 𝐴 |
nprmi.4 | ⊢ 1 < 𝐵 |
nprmi.5 | ⊢ (𝐴 · 𝐵) = 𝑁 |
Ref | Expression |
---|---|
nprmi | ⊢ ¬ 𝑁 ∈ ℙ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nprmi.1 | . . 3 ⊢ 𝐴 ∈ ℕ | |
2 | nprmi.3 | . . 3 ⊢ 1 < 𝐴 | |
3 | nprmi.2 | . . 3 ⊢ 𝐵 ∈ ℕ | |
4 | nprmi.4 | . . 3 ⊢ 1 < 𝐵 | |
5 | eluz2b2 12310 | . . . 4 ⊢ (𝐴 ∈ (ℤ≥‘2) ↔ (𝐴 ∈ ℕ ∧ 1 < 𝐴)) | |
6 | eluz2b2 12310 | . . . 4 ⊢ (𝐵 ∈ (ℤ≥‘2) ↔ (𝐵 ∈ ℕ ∧ 1 < 𝐵)) | |
7 | nprm 16022 | . . . 4 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝐵 ∈ (ℤ≥‘2)) → ¬ (𝐴 · 𝐵) ∈ ℙ) | |
8 | 5, 6, 7 | syl2anbr 598 | . . 3 ⊢ (((𝐴 ∈ ℕ ∧ 1 < 𝐴) ∧ (𝐵 ∈ ℕ ∧ 1 < 𝐵)) → ¬ (𝐴 · 𝐵) ∈ ℙ) |
9 | 1, 2, 3, 4, 8 | mp4an 689 | . 2 ⊢ ¬ (𝐴 · 𝐵) ∈ ℙ |
10 | nprmi.5 | . . 3 ⊢ (𝐴 · 𝐵) = 𝑁 | |
11 | 10 | eleq1i 2903 | . 2 ⊢ ((𝐴 · 𝐵) ∈ ℙ ↔ 𝑁 ∈ ℙ) |
12 | 9, 11 | mtbi 323 | 1 ⊢ ¬ 𝑁 ∈ ℙ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 396 = wceq 1528 ∈ wcel 2105 class class class wbr 5058 ‘cfv 6349 (class class class)co 7145 1c1 10527 · cmul 10531 < clt 10664 ℕcn 11627 2c2 11681 ℤ≥cuz 12232 ℙcprime 16005 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4833 df-iun 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7569 df-2nd 7681 df-wrecs 7938 df-recs 7999 df-rdg 8037 df-1o 8093 df-2o 8094 df-er 8279 df-en 8499 df-dom 8500 df-sdom 8501 df-fin 8502 df-sup 8895 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11628 df-2 11689 df-3 11690 df-n0 11887 df-z 11971 df-uz 12233 df-rp 12380 df-seq 13360 df-exp 13420 df-cj 14448 df-re 14449 df-im 14450 df-sqrt 14584 df-abs 14585 df-dvds 15598 df-prm 16006 |
This theorem is referenced by: 4nprm 16029 dec5nprm 16392 dec2nprm 16393 6nprm 16433 8nprm 16435 9nprm 16436 10nprm 16437 prmlem2 16443 fmtno4prmfac193 43582 fmtno5nprm 43592 |
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