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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 12507 | . . 3 ⊢ 4 ∈ ℕ0 | |
2 | el1fzopredsuc 46618 | . . 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 6891 | . . . 4 ⊢ (𝑁 = 0 → (FermatNo‘𝑁) = (FermatNo‘0)) | |
5 | fmtno0prm 46811 | . . . 4 ⊢ (FermatNo‘0) ∈ ℙ | |
6 | 4, 5 | eqeltrdi 2836 | . . 3 ⊢ (𝑁 = 0 → (FermatNo‘𝑁) ∈ ℙ) |
7 | eltpi 4687 | . . . . 5 ⊢ (𝑁 ∈ {1, 2, 3} → (𝑁 = 1 ∨ 𝑁 = 2 ∨ 𝑁 = 3)) | |
8 | fveq2 6891 | . . . . . . 7 ⊢ (𝑁 = 1 → (FermatNo‘𝑁) = (FermatNo‘1)) | |
9 | fmtno1prm 46812 | . . . . . . 7 ⊢ (FermatNo‘1) ∈ ℙ | |
10 | 8, 9 | eqeltrdi 2836 | . . . . . 6 ⊢ (𝑁 = 1 → (FermatNo‘𝑁) ∈ ℙ) |
11 | fveq2 6891 | . . . . . . 7 ⊢ (𝑁 = 2 → (FermatNo‘𝑁) = (FermatNo‘2)) | |
12 | fmtno2prm 46813 | . . . . . . 7 ⊢ (FermatNo‘2) ∈ ℙ | |
13 | 11, 12 | eqeltrdi 2836 | . . . . . 6 ⊢ (𝑁 = 2 → (FermatNo‘𝑁) ∈ ℙ) |
14 | fveq2 6891 | . . . . . . 7 ⊢ (𝑁 = 3 → (FermatNo‘𝑁) = (FermatNo‘3)) | |
15 | fmtno3prm 46815 | . . . . . . 7 ⊢ (FermatNo‘3) ∈ ℙ | |
16 | 14, 15 | eqeltrdi 2836 | . . . . . 6 ⊢ (𝑁 = 3 → (FermatNo‘𝑁) ∈ ℙ) |
17 | 10, 13, 16 | 3jaoi 1425 | . . . . 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 13738 | . . . 4 ⊢ (1..^4) = {1, 2, 3} | |
20 | 18, 19 | eleq2s 2846 | . . 3 ⊢ (𝑁 ∈ (1..^4) → (FermatNo‘𝑁) ∈ ℙ) |
21 | fveq2 6891 | . . . 4 ⊢ (𝑁 = 4 → (FermatNo‘𝑁) = (FermatNo‘4)) | |
22 | fmtno4prm 46828 | . . . 4 ⊢ (FermatNo‘4) ∈ ℙ | |
23 | 21, 22 | eqeltrdi 2836 | . . 3 ⊢ (𝑁 = 4 → (FermatNo‘𝑁) ∈ ℙ) |
24 | 6, 20, 23 | 3jaoi 1425 | . 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 1084 = wceq 1534 ∈ wcel 2099 {ctp 4628 ‘cfv 6542 (class class class)co 7414 0cc0 11124 1c1 11125 2c2 12283 3c3 12284 4c4 12285 ℕ0cn0 12488 ...cfz 13502 ..^cfzo 13645 ℙcprime 16627 FermatNocfmtno 46780 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-inf2 9650 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 ax-pre-sup 11202 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7863 df-1st 7985 df-2nd 7986 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-1o 8478 df-2o 8479 df-oadd 8482 df-er 8716 df-en 8954 df-dom 8955 df-sdom 8956 df-fin 8957 df-sup 9451 df-inf 9452 df-oi 9519 df-dju 9910 df-card 9948 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 df-sub 11462 df-neg 11463 df-div 11888 df-nn 12229 df-2 12291 df-3 12292 df-4 12293 df-5 12294 df-6 12295 df-7 12296 df-8 12297 df-9 12298 df-n0 12489 df-xnn0 12561 df-z 12575 df-dec 12694 df-uz 12839 df-q 12949 df-rp 12993 df-ioo 13346 df-ico 13348 df-fz 13503 df-fzo 13646 df-fl 13775 df-mod 13853 df-seq 13985 df-exp 14045 df-fac 14251 df-hash 14308 df-cj 15064 df-re 15065 df-im 15066 df-sqrt 15200 df-abs 15201 df-clim 15450 df-prod 15868 df-dvds 16217 df-gcd 16455 df-prm 16628 df-odz 16719 df-phi 16720 df-pc 16791 df-lgs 27202 df-fmtno 46781 |
This theorem is referenced by: fmtnole4prm 46831 |
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