Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
||
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 12240 | . . 3 ⊢ 4 ∈ ℕ0 | |
2 | el1fzopredsuc 44773 | . . 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 6767 | . . . 4 ⊢ (𝑁 = 0 → (FermatNo‘𝑁) = (FermatNo‘0)) | |
5 | fmtno0prm 44966 | . . . 4 ⊢ (FermatNo‘0) ∈ ℙ | |
6 | 4, 5 | eqeltrdi 2847 | . . 3 ⊢ (𝑁 = 0 → (FermatNo‘𝑁) ∈ ℙ) |
7 | eltpi 4624 | . . . . 5 ⊢ (𝑁 ∈ {1, 2, 3} → (𝑁 = 1 ∨ 𝑁 = 2 ∨ 𝑁 = 3)) | |
8 | fveq2 6767 | . . . . . . 7 ⊢ (𝑁 = 1 → (FermatNo‘𝑁) = (FermatNo‘1)) | |
9 | fmtno1prm 44967 | . . . . . . 7 ⊢ (FermatNo‘1) ∈ ℙ | |
10 | 8, 9 | eqeltrdi 2847 | . . . . . 6 ⊢ (𝑁 = 1 → (FermatNo‘𝑁) ∈ ℙ) |
11 | fveq2 6767 | . . . . . . 7 ⊢ (𝑁 = 2 → (FermatNo‘𝑁) = (FermatNo‘2)) | |
12 | fmtno2prm 44968 | . . . . . . 7 ⊢ (FermatNo‘2) ∈ ℙ | |
13 | 11, 12 | eqeltrdi 2847 | . . . . . 6 ⊢ (𝑁 = 2 → (FermatNo‘𝑁) ∈ ℙ) |
14 | fveq2 6767 | . . . . . . 7 ⊢ (𝑁 = 3 → (FermatNo‘𝑁) = (FermatNo‘3)) | |
15 | fmtno3prm 44970 | . . . . . . 7 ⊢ (FermatNo‘3) ∈ ℙ | |
16 | 14, 15 | eqeltrdi 2847 | . . . . . 6 ⊢ (𝑁 = 3 → (FermatNo‘𝑁) ∈ ℙ) |
17 | 10, 13, 16 | 3jaoi 1426 | . . . . 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 13463 | . . . 4 ⊢ (1..^4) = {1, 2, 3} | |
20 | 18, 19 | eleq2s 2857 | . . 3 ⊢ (𝑁 ∈ (1..^4) → (FermatNo‘𝑁) ∈ ℙ) |
21 | fveq2 6767 | . . . 4 ⊢ (𝑁 = 4 → (FermatNo‘𝑁) = (FermatNo‘4)) | |
22 | fmtno4prm 44983 | . . . 4 ⊢ (FermatNo‘4) ∈ ℙ | |
23 | 21, 22 | eqeltrdi 2847 | . . 3 ⊢ (𝑁 = 4 → (FermatNo‘𝑁) ∈ ℙ) |
24 | 6, 20, 23 | 3jaoi 1426 | . 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 1085 = wceq 1539 ∈ wcel 2106 {ctp 4566 ‘cfv 6427 (class class class)co 7268 0cc0 10859 1c1 10860 2c2 12016 3c3 12017 4c4 12018 ℕ0cn0 12221 ...cfz 13227 ..^cfzo 13370 ℙcprime 16364 FermatNocfmtno 44935 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5222 ax-nul 5229 ax-pow 5287 ax-pr 5351 ax-un 7579 ax-inf2 9387 ax-cnex 10915 ax-resscn 10916 ax-1cn 10917 ax-icn 10918 ax-addcl 10919 ax-addrcl 10920 ax-mulcl 10921 ax-mulrcl 10922 ax-mulcom 10923 ax-addass 10924 ax-mulass 10925 ax-distr 10926 ax-i2m1 10927 ax-1ne0 10928 ax-1rid 10929 ax-rnegex 10930 ax-rrecex 10931 ax-cnre 10932 ax-pre-lttri 10933 ax-pre-lttrn 10934 ax-pre-ltadd 10935 ax-pre-mulgt0 10936 ax-pre-sup 10937 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3071 df-rmo 3072 df-rab 3073 df-v 3432 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4258 df-if 4461 df-pw 4536 df-sn 4563 df-pr 4565 df-tp 4567 df-op 4569 df-uni 4841 df-int 4881 df-iun 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5485 df-eprel 5491 df-po 5499 df-so 5500 df-fr 5540 df-se 5541 df-we 5542 df-xp 5591 df-rel 5592 df-cnv 5593 df-co 5594 df-dm 5595 df-rn 5596 df-res 5597 df-ima 5598 df-pred 6196 df-ord 6263 df-on 6264 df-lim 6265 df-suc 6266 df-iota 6385 df-fun 6429 df-fn 6430 df-f 6431 df-f1 6432 df-fo 6433 df-f1o 6434 df-fv 6435 df-isom 6436 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-om 7704 df-1st 7821 df-2nd 7822 df-frecs 8085 df-wrecs 8116 df-recs 8190 df-rdg 8229 df-1o 8285 df-2o 8286 df-oadd 8289 df-er 8486 df-map 8605 df-en 8722 df-dom 8723 df-sdom 8724 df-fin 8725 df-sup 9189 df-inf 9190 df-oi 9257 df-dju 9647 df-card 9685 df-pnf 10999 df-mnf 11000 df-xr 11001 df-ltxr 11002 df-le 11003 df-sub 11195 df-neg 11196 df-div 11621 df-nn 11962 df-2 12024 df-3 12025 df-4 12026 df-5 12027 df-6 12028 df-7 12029 df-8 12030 df-9 12031 df-n0 12222 df-xnn0 12294 df-z 12308 df-dec 12426 df-uz 12571 df-q 12677 df-rp 12719 df-ioo 13071 df-ico 13073 df-fz 13228 df-fzo 13371 df-fl 13500 df-mod 13578 df-seq 13710 df-exp 13771 df-fac 13976 df-hash 14033 df-cj 14798 df-re 14799 df-im 14800 df-sqrt 14934 df-abs 14935 df-clim 15185 df-prod 15604 df-dvds 15952 df-gcd 16190 df-prm 16365 df-odz 16454 df-phi 16455 df-pc 16526 df-lgs 26431 df-fmtno 44936 |
This theorem is referenced by: fmtnole4prm 44986 |
Copyright terms: Public domain | W3C validator |