![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > nprm | GIF version |
Description: A product of two integers greater than one is composite. (Contributed by Mario Carneiro, 20-Jun-2015.) |
Ref | Expression |
---|---|
nprm | ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝐵 ∈ (ℤ≥‘2)) → ¬ (𝐴 · 𝐵) ∈ ℙ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluzelz 9523 | . . . . 5 ⊢ (𝐴 ∈ (ℤ≥‘2) → 𝐴 ∈ ℤ) | |
2 | 1 | adantr 276 | . . . 4 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝐵 ∈ (ℤ≥‘2)) → 𝐴 ∈ ℤ) |
3 | 2 | zred 9361 | . . 3 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝐵 ∈ (ℤ≥‘2)) → 𝐴 ∈ ℝ) |
4 | eluz2b2 9589 | . . . . . 6 ⊢ (𝐵 ∈ (ℤ≥‘2) ↔ (𝐵 ∈ ℕ ∧ 1 < 𝐵)) | |
5 | 4 | simprbi 275 | . . . . 5 ⊢ (𝐵 ∈ (ℤ≥‘2) → 1 < 𝐵) |
6 | 5 | adantl 277 | . . . 4 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝐵 ∈ (ℤ≥‘2)) → 1 < 𝐵) |
7 | eluzelz 9523 | . . . . . . 7 ⊢ (𝐵 ∈ (ℤ≥‘2) → 𝐵 ∈ ℤ) | |
8 | 7 | adantl 277 | . . . . . 6 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝐵 ∈ (ℤ≥‘2)) → 𝐵 ∈ ℤ) |
9 | 8 | zred 9361 | . . . . 5 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝐵 ∈ (ℤ≥‘2)) → 𝐵 ∈ ℝ) |
10 | eluz2nn 9552 | . . . . . . 7 ⊢ (𝐴 ∈ (ℤ≥‘2) → 𝐴 ∈ ℕ) | |
11 | 10 | adantr 276 | . . . . . 6 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝐵 ∈ (ℤ≥‘2)) → 𝐴 ∈ ℕ) |
12 | 11 | nngt0d 8949 | . . . . 5 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝐵 ∈ (ℤ≥‘2)) → 0 < 𝐴) |
13 | ltmulgt11 8807 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 0 < 𝐴) → (1 < 𝐵 ↔ 𝐴 < (𝐴 · 𝐵))) | |
14 | 3, 9, 12, 13 | syl3anc 1238 | . . . 4 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝐵 ∈ (ℤ≥‘2)) → (1 < 𝐵 ↔ 𝐴 < (𝐴 · 𝐵))) |
15 | 6, 14 | mpbid 147 | . . 3 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝐵 ∈ (ℤ≥‘2)) → 𝐴 < (𝐴 · 𝐵)) |
16 | 3, 15 | ltned 8058 | . 2 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝐵 ∈ (ℤ≥‘2)) → 𝐴 ≠ (𝐴 · 𝐵)) |
17 | dvdsmul1 11801 | . . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 𝐴 ∥ (𝐴 · 𝐵)) | |
18 | 1, 7, 17 | syl2an 289 | . . . 4 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝐵 ∈ (ℤ≥‘2)) → 𝐴 ∥ (𝐴 · 𝐵)) |
19 | isprm4 12099 | . . . . . . 7 ⊢ ((𝐴 · 𝐵) ∈ ℙ ↔ ((𝐴 · 𝐵) ∈ (ℤ≥‘2) ∧ ∀𝑥 ∈ (ℤ≥‘2)(𝑥 ∥ (𝐴 · 𝐵) → 𝑥 = (𝐴 · 𝐵)))) | |
20 | 19 | simprbi 275 | . . . . . 6 ⊢ ((𝐴 · 𝐵) ∈ ℙ → ∀𝑥 ∈ (ℤ≥‘2)(𝑥 ∥ (𝐴 · 𝐵) → 𝑥 = (𝐴 · 𝐵))) |
21 | breq1 4003 | . . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑥 ∥ (𝐴 · 𝐵) ↔ 𝐴 ∥ (𝐴 · 𝐵))) | |
22 | eqeq1 2184 | . . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑥 = (𝐴 · 𝐵) ↔ 𝐴 = (𝐴 · 𝐵))) | |
23 | 21, 22 | imbi12d 234 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → ((𝑥 ∥ (𝐴 · 𝐵) → 𝑥 = (𝐴 · 𝐵)) ↔ (𝐴 ∥ (𝐴 · 𝐵) → 𝐴 = (𝐴 · 𝐵)))) |
24 | 23 | rspcv 2837 | . . . . . 6 ⊢ (𝐴 ∈ (ℤ≥‘2) → (∀𝑥 ∈ (ℤ≥‘2)(𝑥 ∥ (𝐴 · 𝐵) → 𝑥 = (𝐴 · 𝐵)) → (𝐴 ∥ (𝐴 · 𝐵) → 𝐴 = (𝐴 · 𝐵)))) |
25 | 20, 24 | syl5 32 | . . . . 5 ⊢ (𝐴 ∈ (ℤ≥‘2) → ((𝐴 · 𝐵) ∈ ℙ → (𝐴 ∥ (𝐴 · 𝐵) → 𝐴 = (𝐴 · 𝐵)))) |
26 | 25 | adantr 276 | . . . 4 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝐵 ∈ (ℤ≥‘2)) → ((𝐴 · 𝐵) ∈ ℙ → (𝐴 ∥ (𝐴 · 𝐵) → 𝐴 = (𝐴 · 𝐵)))) |
27 | 18, 26 | mpid 42 | . . 3 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝐵 ∈ (ℤ≥‘2)) → ((𝐴 · 𝐵) ∈ ℙ → 𝐴 = (𝐴 · 𝐵))) |
28 | 27 | necon3ad 2389 | . 2 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝐵 ∈ (ℤ≥‘2)) → (𝐴 ≠ (𝐴 · 𝐵) → ¬ (𝐴 · 𝐵) ∈ ℙ)) |
29 | 16, 28 | mpd 13 | 1 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝐵 ∈ (ℤ≥‘2)) → ¬ (𝐴 · 𝐵) ∈ ℙ) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1353 ∈ wcel 2148 ≠ wne 2347 ∀wral 2455 class class class wbr 4000 ‘cfv 5212 (class class class)co 5869 ℝcr 7798 0cc0 7799 1c1 7800 · cmul 7804 < clt 7979 ℕcn 8905 2c2 8956 ℤcz 9239 ℤ≥cuz 9514 ∥ cdvds 11775 ℙcprime 12087 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4115 ax-sep 4118 ax-nul 4126 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-setind 4533 ax-iinf 4584 ax-cnex 7890 ax-resscn 7891 ax-1cn 7892 ax-1re 7893 ax-icn 7894 ax-addcl 7895 ax-addrcl 7896 ax-mulcl 7897 ax-mulrcl 7898 ax-addcom 7899 ax-mulcom 7900 ax-addass 7901 ax-mulass 7902 ax-distr 7903 ax-i2m1 7904 ax-0lt1 7905 ax-1rid 7906 ax-0id 7907 ax-rnegex 7908 ax-precex 7909 ax-cnre 7910 ax-pre-ltirr 7911 ax-pre-ltwlin 7912 ax-pre-lttrn 7913 ax-pre-apti 7914 ax-pre-ltadd 7915 ax-pre-mulgt0 7916 ax-pre-mulext 7917 ax-arch 7918 ax-caucvg 7919 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-if 3535 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-tr 4099 df-id 4290 df-po 4293 df-iso 4294 df-iord 4363 df-on 4365 df-ilim 4366 df-suc 4368 df-iom 4587 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-rn 4634 df-res 4635 df-ima 4636 df-iota 5174 df-fun 5214 df-fn 5215 df-f 5216 df-f1 5217 df-fo 5218 df-f1o 5219 df-fv 5220 df-riota 5825 df-ov 5872 df-oprab 5873 df-mpo 5874 df-1st 6135 df-2nd 6136 df-recs 6300 df-frec 6386 df-1o 6411 df-2o 6412 df-er 6529 df-en 6735 df-pnf 7981 df-mnf 7982 df-xr 7983 df-ltxr 7984 df-le 7985 df-sub 8117 df-neg 8118 df-reap 8519 df-ap 8526 df-div 8616 df-inn 8906 df-2 8964 df-3 8965 df-4 8966 df-n0 9163 df-z 9240 df-uz 9515 df-q 9606 df-rp 9638 df-seqfrec 10429 df-exp 10503 df-cj 10832 df-re 10833 df-im 10834 df-rsqrt 10988 df-abs 10989 df-dvds 11776 df-prm 12088 |
This theorem is referenced by: nprmi 12104 dvdsnprmd 12105 sqnprm 12116 |
Copyright terms: Public domain | W3C validator |