![]() |
Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > prmdvdsfmtnof | Structured version Visualization version GIF version |
Description: The mapping of a Fermat number to its smallest prime factor is a function. (Contributed by AV, 4-Aug-2021.) (Proof shortened by II, 16-Feb-2023.) |
Ref | Expression |
---|---|
prmdvdsfmtnof.1 | ⊢ 𝐹 = (𝑓 ∈ ran FermatNo ↦ inf({𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑓}, ℝ, < )) |
Ref | Expression |
---|---|
prmdvdsfmtnof | ⊢ 𝐹:ran FermatNo⟶ℙ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prmdvdsfmtnof.1 | . 2 ⊢ 𝐹 = (𝑓 ∈ ran FermatNo ↦ inf({𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑓}, ℝ, < )) | |
2 | fmtnorn 45621 | . . 3 ⊢ (𝑓 ∈ ran FermatNo ↔ ∃𝑛 ∈ ℕ0 (FermatNo‘𝑛) = 𝑓) | |
3 | ltso 11193 | . . . . . 6 ⊢ < Or ℝ | |
4 | 3 | a1i 11 | . . . . 5 ⊢ ((𝑛 ∈ ℕ0 ∧ (FermatNo‘𝑛) = 𝑓) → < Or ℝ) |
5 | fmtnoge3 45617 | . . . . . . . . 9 ⊢ (𝑛 ∈ ℕ0 → (FermatNo‘𝑛) ∈ (ℤ≥‘3)) | |
6 | 5 | adantr 481 | . . . . . . . 8 ⊢ ((𝑛 ∈ ℕ0 ∧ (FermatNo‘𝑛) = 𝑓) → (FermatNo‘𝑛) ∈ (ℤ≥‘3)) |
7 | eleq1 2825 | . . . . . . . . 9 ⊢ ((FermatNo‘𝑛) = 𝑓 → ((FermatNo‘𝑛) ∈ (ℤ≥‘3) ↔ 𝑓 ∈ (ℤ≥‘3))) | |
8 | 7 | adantl 482 | . . . . . . . 8 ⊢ ((𝑛 ∈ ℕ0 ∧ (FermatNo‘𝑛) = 𝑓) → ((FermatNo‘𝑛) ∈ (ℤ≥‘3) ↔ 𝑓 ∈ (ℤ≥‘3))) |
9 | 6, 8 | mpbid 231 | . . . . . . 7 ⊢ ((𝑛 ∈ ℕ0 ∧ (FermatNo‘𝑛) = 𝑓) → 𝑓 ∈ (ℤ≥‘3)) |
10 | uzuzle23 12768 | . . . . . . 7 ⊢ (𝑓 ∈ (ℤ≥‘3) → 𝑓 ∈ (ℤ≥‘2)) | |
11 | 9, 10 | syl 17 | . . . . . 6 ⊢ ((𝑛 ∈ ℕ0 ∧ (FermatNo‘𝑛) = 𝑓) → 𝑓 ∈ (ℤ≥‘2)) |
12 | eluz2nn 12763 | . . . . . 6 ⊢ (𝑓 ∈ (ℤ≥‘2) → 𝑓 ∈ ℕ) | |
13 | prmdvdsfi 26402 | . . . . . 6 ⊢ (𝑓 ∈ ℕ → {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑓} ∈ Fin) | |
14 | 11, 12, 13 | 3syl 18 | . . . . 5 ⊢ ((𝑛 ∈ ℕ0 ∧ (FermatNo‘𝑛) = 𝑓) → {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑓} ∈ Fin) |
15 | exprmfct 16534 | . . . . . . 7 ⊢ (𝑓 ∈ (ℤ≥‘2) → ∃𝑝 ∈ ℙ 𝑝 ∥ 𝑓) | |
16 | 11, 15 | syl 17 | . . . . . 6 ⊢ ((𝑛 ∈ ℕ0 ∧ (FermatNo‘𝑛) = 𝑓) → ∃𝑝 ∈ ℙ 𝑝 ∥ 𝑓) |
17 | rabn0 4343 | . . . . . 6 ⊢ ({𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑓} ≠ ∅ ↔ ∃𝑝 ∈ ℙ 𝑝 ∥ 𝑓) | |
18 | 16, 17 | sylibr 233 | . . . . 5 ⊢ ((𝑛 ∈ ℕ0 ∧ (FermatNo‘𝑛) = 𝑓) → {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑓} ≠ ∅) |
19 | ssrab2 4035 | . . . . . . 7 ⊢ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑓} ⊆ ℙ | |
20 | prmssnn 16506 | . . . . . . . 8 ⊢ ℙ ⊆ ℕ | |
21 | nnssre 12115 | . . . . . . . 8 ⊢ ℕ ⊆ ℝ | |
22 | 20, 21 | sstri 3951 | . . . . . . 7 ⊢ ℙ ⊆ ℝ |
23 | 19, 22 | sstri 3951 | . . . . . 6 ⊢ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑓} ⊆ ℝ |
24 | 23 | a1i 11 | . . . . 5 ⊢ ((𝑛 ∈ ℕ0 ∧ (FermatNo‘𝑛) = 𝑓) → {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑓} ⊆ ℝ) |
25 | fiinfcl 9395 | . . . . . 6 ⊢ (( < Or ℝ ∧ ({𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑓} ∈ Fin ∧ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑓} ≠ ∅ ∧ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑓} ⊆ ℝ)) → inf({𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑓}, ℝ, < ) ∈ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑓}) | |
26 | 19, 25 | sselid 3940 | . . . . 5 ⊢ (( < Or ℝ ∧ ({𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑓} ∈ Fin ∧ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑓} ≠ ∅ ∧ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑓} ⊆ ℝ)) → inf({𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑓}, ℝ, < ) ∈ ℙ) |
27 | 4, 14, 18, 24, 26 | syl13anc 1372 | . . . 4 ⊢ ((𝑛 ∈ ℕ0 ∧ (FermatNo‘𝑛) = 𝑓) → inf({𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑓}, ℝ, < ) ∈ ℙ) |
28 | 27 | rexlimiva 3142 | . . 3 ⊢ (∃𝑛 ∈ ℕ0 (FermatNo‘𝑛) = 𝑓 → inf({𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑓}, ℝ, < ) ∈ ℙ) |
29 | 2, 28 | sylbi 216 | . 2 ⊢ (𝑓 ∈ ran FermatNo → inf({𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑓}, ℝ, < ) ∈ ℙ) |
30 | 1, 29 | fmpti 7056 | 1 ⊢ 𝐹:ran FermatNo⟶ℙ |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2941 ∃wrex 3071 {crab 3405 ⊆ wss 3908 ∅c0 4280 class class class wbr 5103 ↦ cmpt 5186 Or wor 5542 ran crn 5632 ⟶wf 6489 ‘cfv 6493 Fincfn 8841 infcinf 9335 ℝcr 11008 < clt 11147 ℕcn 12111 2c2 12166 3c3 12167 ℕ0cn0 12371 ℤ≥cuz 12721 ∥ cdvds 16090 ℙcprime 16501 FermatNocfmtno 45614 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-pre-sup 11087 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-om 7795 df-1st 7913 df-2nd 7914 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-rdg 8348 df-1o 8404 df-2o 8405 df-er 8606 df-en 8842 df-dom 8843 df-sdom 8844 df-fin 8845 df-sup 9336 df-inf 9337 df-pnf 11149 df-mnf 11150 df-xr 11151 df-ltxr 11152 df-le 11153 df-sub 11345 df-neg 11346 df-div 11771 df-nn 12112 df-2 12174 df-3 12175 df-n0 12372 df-z 12458 df-uz 12722 df-rp 12870 df-fz 13379 df-seq 13861 df-exp 13922 df-cj 14938 df-re 14939 df-im 14940 df-sqrt 15074 df-abs 15075 df-dvds 16091 df-prm 16502 df-fmtno 45615 |
This theorem is referenced by: prmdvdsfmtnof1 45674 |
Copyright terms: Public domain | W3C validator |