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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 11639 | . . 3 ⊢ 4 ∈ ℕ0 | |
2 | el1fzopredsuc 42223 | . . 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 6433 | . . . 4 ⊢ (𝑁 = 0 → (FermatNo‘𝑁) = (FermatNo‘0)) | |
5 | fmtno0prm 42300 | . . . 4 ⊢ (FermatNo‘0) ∈ ℙ | |
6 | 4, 5 | syl6eqel 2914 | . . 3 ⊢ (𝑁 = 0 → (FermatNo‘𝑁) ∈ ℙ) |
7 | eltpi 4448 | . . . . 5 ⊢ (𝑁 ∈ {1, 2, 3} → (𝑁 = 1 ∨ 𝑁 = 2 ∨ 𝑁 = 3)) | |
8 | fveq2 6433 | . . . . . . 7 ⊢ (𝑁 = 1 → (FermatNo‘𝑁) = (FermatNo‘1)) | |
9 | fmtno1prm 42301 | . . . . . . 7 ⊢ (FermatNo‘1) ∈ ℙ | |
10 | 8, 9 | syl6eqel 2914 | . . . . . 6 ⊢ (𝑁 = 1 → (FermatNo‘𝑁) ∈ ℙ) |
11 | fveq2 6433 | . . . . . . 7 ⊢ (𝑁 = 2 → (FermatNo‘𝑁) = (FermatNo‘2)) | |
12 | fmtno2prm 42302 | . . . . . . 7 ⊢ (FermatNo‘2) ∈ ℙ | |
13 | 11, 12 | syl6eqel 2914 | . . . . . 6 ⊢ (𝑁 = 2 → (FermatNo‘𝑁) ∈ ℙ) |
14 | fveq2 6433 | . . . . . . 7 ⊢ (𝑁 = 3 → (FermatNo‘𝑁) = (FermatNo‘3)) | |
15 | fmtno3prm 42304 | . . . . . . 7 ⊢ (FermatNo‘3) ∈ ℙ | |
16 | 14, 15 | syl6eqel 2914 | . . . . . 6 ⊢ (𝑁 = 3 → (FermatNo‘𝑁) ∈ ℙ) |
17 | 10, 13, 16 | 3jaoi 1558 | . . . . 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 12851 | . . . 4 ⊢ (1..^4) = {1, 2, 3} | |
20 | 18, 19 | eleq2s 2924 | . . 3 ⊢ (𝑁 ∈ (1..^4) → (FermatNo‘𝑁) ∈ ℙ) |
21 | fveq2 6433 | . . . 4 ⊢ (𝑁 = 4 → (FermatNo‘𝑁) = (FermatNo‘4)) | |
22 | fmtno4prm 42317 | . . . 4 ⊢ (FermatNo‘4) ∈ ℙ | |
23 | 21, 22 | syl6eqel 2914 | . . 3 ⊢ (𝑁 = 4 → (FermatNo‘𝑁) ∈ ℙ) |
24 | 6, 20, 23 | 3jaoi 1558 | . 2 ⊢ ((𝑁 = 0 ∨ 𝑁 ∈ (1..^4) ∨ 𝑁 = 4) → (FermatNo‘𝑁) ∈ ℙ) |
25 | 3, 24 | sylbi 209 | 1 ⊢ (𝑁 ∈ (0...4) → (FermatNo‘𝑁) ∈ ℙ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∨ w3o 1112 = wceq 1658 ∈ wcel 2166 {ctp 4401 ‘cfv 6123 (class class class)co 6905 0cc0 10252 1c1 10253 2c2 11406 3c3 11407 4c4 11408 ℕ0cn0 11618 ...cfz 12619 ..^cfzo 12760 ℙcprime 15757 FermatNocfmtno 42269 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-inf2 8815 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 ax-pre-sup 10330 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-fal 1672 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-int 4698 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-se 5302 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-isom 6132 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-1st 7428 df-2nd 7429 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-1o 7826 df-2o 7827 df-oadd 7830 df-er 8009 df-map 8124 df-en 8223 df-dom 8224 df-sdom 8225 df-fin 8226 df-sup 8617 df-inf 8618 df-oi 8684 df-card 9078 df-cda 9305 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-div 11010 df-nn 11351 df-2 11414 df-3 11415 df-4 11416 df-5 11417 df-6 11418 df-7 11419 df-8 11420 df-9 11421 df-n0 11619 df-xnn0 11691 df-z 11705 df-dec 11822 df-uz 11969 df-q 12072 df-rp 12113 df-ioo 12467 df-ico 12469 df-fz 12620 df-fzo 12761 df-fl 12888 df-mod 12964 df-seq 13096 df-exp 13155 df-fac 13354 df-hash 13411 df-cj 14216 df-re 14217 df-im 14218 df-sqrt 14352 df-abs 14353 df-clim 14596 df-prod 15009 df-dvds 15358 df-gcd 15590 df-prm 15758 df-odz 15841 df-phi 15842 df-pc 15913 df-lgs 25433 df-fmtno 42270 |
This theorem is referenced by: fmtnole4prm 42320 |
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