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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > fmtnofz04prm | Structured version Visualization version GIF version |
Description: The first five Fermat numbers are prime, see remark in [ApostolNT] p. 7. (Contributed by AV, 28-Jul-2021.) |
Ref | Expression |
---|---|
fmtnofz04prm | ⊢ (𝑁 ∈ (0...4) → (FermatNo‘𝑁) ∈ ℙ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 4nn0 12439 | . . 3 ⊢ 4 ∈ ℕ0 | |
2 | el1fzopredsuc 45631 | . . 3 ⊢ (4 ∈ ℕ0 → (𝑁 ∈ (0...4) ↔ (𝑁 = 0 ∨ 𝑁 ∈ (1..^4) ∨ 𝑁 = 4))) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ (𝑁 ∈ (0...4) ↔ (𝑁 = 0 ∨ 𝑁 ∈ (1..^4) ∨ 𝑁 = 4)) |
4 | fveq2 6847 | . . . 4 ⊢ (𝑁 = 0 → (FermatNo‘𝑁) = (FermatNo‘0)) | |
5 | fmtno0prm 45824 | . . . 4 ⊢ (FermatNo‘0) ∈ ℙ | |
6 | 4, 5 | eqeltrdi 2846 | . . 3 ⊢ (𝑁 = 0 → (FermatNo‘𝑁) ∈ ℙ) |
7 | eltpi 4653 | . . . . 5 ⊢ (𝑁 ∈ {1, 2, 3} → (𝑁 = 1 ∨ 𝑁 = 2 ∨ 𝑁 = 3)) | |
8 | fveq2 6847 | . . . . . . 7 ⊢ (𝑁 = 1 → (FermatNo‘𝑁) = (FermatNo‘1)) | |
9 | fmtno1prm 45825 | . . . . . . 7 ⊢ (FermatNo‘1) ∈ ℙ | |
10 | 8, 9 | eqeltrdi 2846 | . . . . . 6 ⊢ (𝑁 = 1 → (FermatNo‘𝑁) ∈ ℙ) |
11 | fveq2 6847 | . . . . . . 7 ⊢ (𝑁 = 2 → (FermatNo‘𝑁) = (FermatNo‘2)) | |
12 | fmtno2prm 45826 | . . . . . . 7 ⊢ (FermatNo‘2) ∈ ℙ | |
13 | 11, 12 | eqeltrdi 2846 | . . . . . 6 ⊢ (𝑁 = 2 → (FermatNo‘𝑁) ∈ ℙ) |
14 | fveq2 6847 | . . . . . . 7 ⊢ (𝑁 = 3 → (FermatNo‘𝑁) = (FermatNo‘3)) | |
15 | fmtno3prm 45828 | . . . . . . 7 ⊢ (FermatNo‘3) ∈ ℙ | |
16 | 14, 15 | eqeltrdi 2846 | . . . . . 6 ⊢ (𝑁 = 3 → (FermatNo‘𝑁) ∈ ℙ) |
17 | 10, 13, 16 | 3jaoi 1428 | . . . . 5 ⊢ ((𝑁 = 1 ∨ 𝑁 = 2 ∨ 𝑁 = 3) → (FermatNo‘𝑁) ∈ ℙ) |
18 | 7, 17 | syl 17 | . . . 4 ⊢ (𝑁 ∈ {1, 2, 3} → (FermatNo‘𝑁) ∈ ℙ) |
19 | fzo1to4tp 13667 | . . . 4 ⊢ (1..^4) = {1, 2, 3} | |
20 | 18, 19 | eleq2s 2856 | . . 3 ⊢ (𝑁 ∈ (1..^4) → (FermatNo‘𝑁) ∈ ℙ) |
21 | fveq2 6847 | . . . 4 ⊢ (𝑁 = 4 → (FermatNo‘𝑁) = (FermatNo‘4)) | |
22 | fmtno4prm 45841 | . . . 4 ⊢ (FermatNo‘4) ∈ ℙ | |
23 | 21, 22 | eqeltrdi 2846 | . . 3 ⊢ (𝑁 = 4 → (FermatNo‘𝑁) ∈ ℙ) |
24 | 6, 20, 23 | 3jaoi 1428 | . 2 ⊢ ((𝑁 = 0 ∨ 𝑁 ∈ (1..^4) ∨ 𝑁 = 4) → (FermatNo‘𝑁) ∈ ℙ) |
25 | 3, 24 | sylbi 216 | 1 ⊢ (𝑁 ∈ (0...4) → (FermatNo‘𝑁) ∈ ℙ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∨ w3o 1087 = wceq 1542 ∈ wcel 2107 {ctp 4595 ‘cfv 6501 (class class class)co 7362 0cc0 11058 1c1 11059 2c2 12215 3c3 12216 4c4 12217 ℕ0cn0 12420 ...cfz 13431 ..^cfzo 13574 ℙcprime 16554 FermatNocfmtno 45793 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-inf2 9584 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 ax-pre-sup 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-se 5594 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-isom 6510 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-1st 7926 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-2o 8418 df-oadd 8421 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-sup 9385 df-inf 9386 df-oi 9453 df-dju 9844 df-card 9882 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-div 11820 df-nn 12161 df-2 12223 df-3 12224 df-4 12225 df-5 12226 df-6 12227 df-7 12228 df-8 12229 df-9 12230 df-n0 12421 df-xnn0 12493 df-z 12507 df-dec 12626 df-uz 12771 df-q 12881 df-rp 12923 df-ioo 13275 df-ico 13277 df-fz 13432 df-fzo 13575 df-fl 13704 df-mod 13782 df-seq 13914 df-exp 13975 df-fac 14181 df-hash 14238 df-cj 14991 df-re 14992 df-im 14993 df-sqrt 15127 df-abs 15128 df-clim 15377 df-prod 15796 df-dvds 16144 df-gcd 16382 df-prm 16555 df-odz 16644 df-phi 16645 df-pc 16716 df-lgs 26659 df-fmtno 45794 |
This theorem is referenced by: fmtnole4prm 45844 |
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