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Mirrors > Home > ILE Home > Th. List > infpn2 | GIF version |
Description: There exist infinitely many prime numbers: the set of all primes 𝑆 is unbounded by infpn 12499, so by unbendc 12611 it is infinite. This is Metamath 100 proof #11. (Contributed by NM, 5-May-2005.) |
Ref | Expression |
---|---|
infpn2.1 | ⊢ 𝑆 = {𝑛 ∈ ℕ ∣ (1 < 𝑛 ∧ ∀𝑚 ∈ ℕ ((𝑛 / 𝑚) ∈ ℕ → (𝑚 = 1 ∨ 𝑚 = 𝑛)))} |
Ref | Expression |
---|---|
infpn2 | ⊢ 𝑆 ≈ ℕ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluz2nn 9631 | . . . . . . 7 ⊢ (𝑟 ∈ (ℤ≥‘2) → 𝑟 ∈ ℕ) | |
2 | 1 | adantr 276 | . . . . . 6 ⊢ ((𝑟 ∈ (ℤ≥‘2) ∧ ∀𝑚 ∈ ℕ (𝑚 ∥ 𝑟 → (𝑚 = 1 ∨ 𝑚 = 𝑟))) → 𝑟 ∈ ℕ) |
3 | simpll 527 | . . . . . 6 ⊢ (((𝑟 ∈ ℕ ∧ 1 < 𝑟) ∧ ∀𝑚 ∈ ℕ ((𝑟 / 𝑚) ∈ ℕ → (𝑚 = 1 ∨ 𝑚 = 𝑟))) → 𝑟 ∈ ℕ) | |
4 | eluz2b2 9668 | . . . . . . . 8 ⊢ (𝑟 ∈ (ℤ≥‘2) ↔ (𝑟 ∈ ℕ ∧ 1 < 𝑟)) | |
5 | 4 | a1i 9 | . . . . . . 7 ⊢ (𝑟 ∈ ℕ → (𝑟 ∈ (ℤ≥‘2) ↔ (𝑟 ∈ ℕ ∧ 1 < 𝑟))) |
6 | nndivdvds 11939 | . . . . . . . . 9 ⊢ ((𝑟 ∈ ℕ ∧ 𝑚 ∈ ℕ) → (𝑚 ∥ 𝑟 ↔ (𝑟 / 𝑚) ∈ ℕ)) | |
7 | 6 | imbi1d 231 | . . . . . . . 8 ⊢ ((𝑟 ∈ ℕ ∧ 𝑚 ∈ ℕ) → ((𝑚 ∥ 𝑟 → (𝑚 = 1 ∨ 𝑚 = 𝑟)) ↔ ((𝑟 / 𝑚) ∈ ℕ → (𝑚 = 1 ∨ 𝑚 = 𝑟)))) |
8 | 7 | ralbidva 2490 | . . . . . . 7 ⊢ (𝑟 ∈ ℕ → (∀𝑚 ∈ ℕ (𝑚 ∥ 𝑟 → (𝑚 = 1 ∨ 𝑚 = 𝑟)) ↔ ∀𝑚 ∈ ℕ ((𝑟 / 𝑚) ∈ ℕ → (𝑚 = 1 ∨ 𝑚 = 𝑟)))) |
9 | 5, 8 | anbi12d 473 | . . . . . 6 ⊢ (𝑟 ∈ ℕ → ((𝑟 ∈ (ℤ≥‘2) ∧ ∀𝑚 ∈ ℕ (𝑚 ∥ 𝑟 → (𝑚 = 1 ∨ 𝑚 = 𝑟))) ↔ ((𝑟 ∈ ℕ ∧ 1 < 𝑟) ∧ ∀𝑚 ∈ ℕ ((𝑟 / 𝑚) ∈ ℕ → (𝑚 = 1 ∨ 𝑚 = 𝑟))))) |
10 | 2, 3, 9 | pm5.21nii 705 | . . . . 5 ⊢ ((𝑟 ∈ (ℤ≥‘2) ∧ ∀𝑚 ∈ ℕ (𝑚 ∥ 𝑟 → (𝑚 = 1 ∨ 𝑚 = 𝑟))) ↔ ((𝑟 ∈ ℕ ∧ 1 < 𝑟) ∧ ∀𝑚 ∈ ℕ ((𝑟 / 𝑚) ∈ ℕ → (𝑚 = 1 ∨ 𝑚 = 𝑟)))) |
11 | anass 401 | . . . . 5 ⊢ (((𝑟 ∈ ℕ ∧ 1 < 𝑟) ∧ ∀𝑚 ∈ ℕ ((𝑟 / 𝑚) ∈ ℕ → (𝑚 = 1 ∨ 𝑚 = 𝑟))) ↔ (𝑟 ∈ ℕ ∧ (1 < 𝑟 ∧ ∀𝑚 ∈ ℕ ((𝑟 / 𝑚) ∈ ℕ → (𝑚 = 1 ∨ 𝑚 = 𝑟))))) | |
12 | 10, 11 | bitri 184 | . . . 4 ⊢ ((𝑟 ∈ (ℤ≥‘2) ∧ ∀𝑚 ∈ ℕ (𝑚 ∥ 𝑟 → (𝑚 = 1 ∨ 𝑚 = 𝑟))) ↔ (𝑟 ∈ ℕ ∧ (1 < 𝑟 ∧ ∀𝑚 ∈ ℕ ((𝑟 / 𝑚) ∈ ℕ → (𝑚 = 1 ∨ 𝑚 = 𝑟))))) |
13 | isprm2 12255 | . . . 4 ⊢ (𝑟 ∈ ℙ ↔ (𝑟 ∈ (ℤ≥‘2) ∧ ∀𝑚 ∈ ℕ (𝑚 ∥ 𝑟 → (𝑚 = 1 ∨ 𝑚 = 𝑟)))) | |
14 | breq2 4033 | . . . . . 6 ⊢ (𝑛 = 𝑟 → (1 < 𝑛 ↔ 1 < 𝑟)) | |
15 | oveq1 5925 | . . . . . . . . 9 ⊢ (𝑛 = 𝑟 → (𝑛 / 𝑚) = (𝑟 / 𝑚)) | |
16 | 15 | eleq1d 2262 | . . . . . . . 8 ⊢ (𝑛 = 𝑟 → ((𝑛 / 𝑚) ∈ ℕ ↔ (𝑟 / 𝑚) ∈ ℕ)) |
17 | equequ2 1724 | . . . . . . . . 9 ⊢ (𝑛 = 𝑟 → (𝑚 = 𝑛 ↔ 𝑚 = 𝑟)) | |
18 | 17 | orbi2d 791 | . . . . . . . 8 ⊢ (𝑛 = 𝑟 → ((𝑚 = 1 ∨ 𝑚 = 𝑛) ↔ (𝑚 = 1 ∨ 𝑚 = 𝑟))) |
19 | 16, 18 | imbi12d 234 | . . . . . . 7 ⊢ (𝑛 = 𝑟 → (((𝑛 / 𝑚) ∈ ℕ → (𝑚 = 1 ∨ 𝑚 = 𝑛)) ↔ ((𝑟 / 𝑚) ∈ ℕ → (𝑚 = 1 ∨ 𝑚 = 𝑟)))) |
20 | 19 | ralbidv 2494 | . . . . . 6 ⊢ (𝑛 = 𝑟 → (∀𝑚 ∈ ℕ ((𝑛 / 𝑚) ∈ ℕ → (𝑚 = 1 ∨ 𝑚 = 𝑛)) ↔ ∀𝑚 ∈ ℕ ((𝑟 / 𝑚) ∈ ℕ → (𝑚 = 1 ∨ 𝑚 = 𝑟)))) |
21 | 14, 20 | anbi12d 473 | . . . . 5 ⊢ (𝑛 = 𝑟 → ((1 < 𝑛 ∧ ∀𝑚 ∈ ℕ ((𝑛 / 𝑚) ∈ ℕ → (𝑚 = 1 ∨ 𝑚 = 𝑛))) ↔ (1 < 𝑟 ∧ ∀𝑚 ∈ ℕ ((𝑟 / 𝑚) ∈ ℕ → (𝑚 = 1 ∨ 𝑚 = 𝑟))))) |
22 | infpn2.1 | . . . . 5 ⊢ 𝑆 = {𝑛 ∈ ℕ ∣ (1 < 𝑛 ∧ ∀𝑚 ∈ ℕ ((𝑛 / 𝑚) ∈ ℕ → (𝑚 = 1 ∨ 𝑚 = 𝑛)))} | |
23 | 21, 22 | elrab2 2919 | . . . 4 ⊢ (𝑟 ∈ 𝑆 ↔ (𝑟 ∈ ℕ ∧ (1 < 𝑟 ∧ ∀𝑚 ∈ ℕ ((𝑟 / 𝑚) ∈ ℕ → (𝑚 = 1 ∨ 𝑚 = 𝑟))))) |
24 | 12, 13, 23 | 3bitr4i 212 | . . 3 ⊢ (𝑟 ∈ ℙ ↔ 𝑟 ∈ 𝑆) |
25 | 24 | eqriv 2190 | . 2 ⊢ ℙ = 𝑆 |
26 | prminf 12612 | . 2 ⊢ ℙ ≈ ℕ | |
27 | 25, 26 | eqbrtrri 4052 | 1 ⊢ 𝑆 ≈ ℕ |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 709 = wceq 1364 ∈ wcel 2164 ∀wral 2472 {crab 2476 class class class wbr 4029 ‘cfv 5254 (class class class)co 5918 ≈ cen 6792 1c1 7873 < clt 8054 / cdiv 8691 ℕcn 8982 2c2 9033 ℤ≥cuz 9592 ∥ cdvds 11930 ℙcprime 12245 |
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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-coll 4144 ax-sep 4147 ax-nul 4155 ax-pow 4203 ax-pr 4238 ax-un 4464 ax-setind 4569 ax-iinf 4620 ax-cnex 7963 ax-resscn 7964 ax-1cn 7965 ax-1re 7966 ax-icn 7967 ax-addcl 7968 ax-addrcl 7969 ax-mulcl 7970 ax-mulrcl 7971 ax-addcom 7972 ax-mulcom 7973 ax-addass 7974 ax-mulass 7975 ax-distr 7976 ax-i2m1 7977 ax-0lt1 7978 ax-1rid 7979 ax-0id 7980 ax-rnegex 7981 ax-precex 7982 ax-cnre 7983 ax-pre-ltirr 7984 ax-pre-ltwlin 7985 ax-pre-lttrn 7986 ax-pre-apti 7987 ax-pre-ltadd 7988 ax-pre-mulgt0 7989 ax-pre-mulext 7990 ax-arch 7991 ax-caucvg 7992 |
This theorem depends on definitions: df-bi 117 df-stab 832 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rmo 2480 df-rab 2481 df-v 2762 df-sbc 2986 df-csb 3081 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166 df-nul 3447 df-if 3558 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-int 3871 df-iun 3914 df-br 4030 df-opab 4091 df-mpt 4092 df-tr 4128 df-id 4324 df-po 4327 df-iso 4328 df-iord 4397 df-on 4399 df-ilim 4400 df-suc 4402 df-iom 4623 df-xp 4665 df-rel 4666 df-cnv 4667 df-co 4668 df-dm 4669 df-rn 4670 df-res 4671 df-ima 4672 df-iota 5215 df-fun 5256 df-fn 5257 df-f 5258 df-f1 5259 df-fo 5260 df-f1o 5261 df-fv 5262 df-isom 5263 df-riota 5873 df-ov 5921 df-oprab 5922 df-mpo 5923 df-1st 6193 df-2nd 6194 df-recs 6358 df-frec 6444 df-1o 6469 df-2o 6470 df-er 6587 df-pm 6705 df-en 6795 df-dom 6796 df-fin 6797 df-sup 7043 df-inf 7044 df-dju 7097 df-inl 7106 df-inr 7107 df-case 7143 df-pnf 8056 df-mnf 8057 df-xr 8058 df-ltxr 8059 df-le 8060 df-sub 8192 df-neg 8193 df-reap 8594 df-ap 8601 df-div 8692 df-inn 8983 df-2 9041 df-3 9042 df-4 9043 df-n0 9241 df-z 9318 df-uz 9593 df-q 9685 df-rp 9720 df-fz 10075 df-fzo 10209 df-fl 10339 df-mod 10394 df-seqfrec 10519 df-exp 10610 df-fac 10797 df-cj 10986 df-re 10987 df-im 10988 df-rsqrt 11142 df-abs 11143 df-dvds 11931 df-prm 12246 |
This theorem is referenced by: (None) |
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