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 11188 | . . 3 ⊢ 4 ∈ ℕ0 | |
2 | el1fzopredsuc 39948 | . . 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 6103 | . . . 4 ⊢ (𝑁 = 0 → (FermatNo‘𝑁) = (FermatNo‘0)) | |
5 | fmtno0prm 40008 | . . . 4 ⊢ (FermatNo‘0) ∈ ℙ | |
6 | 4, 5 | syl6eqel 2696 | . . 3 ⊢ (𝑁 = 0 → (FermatNo‘𝑁) ∈ ℙ) |
7 | eltpi 4176 | . . . . 5 ⊢ (𝑁 ∈ {1, 2, 3} → (𝑁 = 1 ∨ 𝑁 = 2 ∨ 𝑁 = 3)) | |
8 | fveq2 6103 | . . . . . . 7 ⊢ (𝑁 = 1 → (FermatNo‘𝑁) = (FermatNo‘1)) | |
9 | fmtno1prm 40009 | . . . . . . 7 ⊢ (FermatNo‘1) ∈ ℙ | |
10 | 8, 9 | syl6eqel 2696 | . . . . . 6 ⊢ (𝑁 = 1 → (FermatNo‘𝑁) ∈ ℙ) |
11 | fveq2 6103 | . . . . . . 7 ⊢ (𝑁 = 2 → (FermatNo‘𝑁) = (FermatNo‘2)) | |
12 | fmtno2prm 40010 | . . . . . . 7 ⊢ (FermatNo‘2) ∈ ℙ | |
13 | 11, 12 | syl6eqel 2696 | . . . . . 6 ⊢ (𝑁 = 2 → (FermatNo‘𝑁) ∈ ℙ) |
14 | fveq2 6103 | . . . . . . 7 ⊢ (𝑁 = 3 → (FermatNo‘𝑁) = (FermatNo‘3)) | |
15 | fmtno3prm 40012 | . . . . . . 7 ⊢ (FermatNo‘3) ∈ ℙ | |
16 | 14, 15 | syl6eqel 2696 | . . . . . 6 ⊢ (𝑁 = 3 → (FermatNo‘𝑁) ∈ ℙ) |
17 | 10, 13, 16 | 3jaoi 1383 | . . . . 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 12423 | . . . 4 ⊢ (1..^4) = {1, 2, 3} | |
20 | 18, 19 | eleq2s 2706 | . . 3 ⊢ (𝑁 ∈ (1..^4) → (FermatNo‘𝑁) ∈ ℙ) |
21 | fveq2 6103 | . . . 4 ⊢ (𝑁 = 4 → (FermatNo‘𝑁) = (FermatNo‘4)) | |
22 | fmtno4prm 40025 | . . . 4 ⊢ (FermatNo‘4) ∈ ℙ | |
23 | 21, 22 | syl6eqel 2696 | . . 3 ⊢ (𝑁 = 4 → (FermatNo‘𝑁) ∈ ℙ) |
24 | 6, 20, 23 | 3jaoi 1383 | . 2 ⊢ ((𝑁 = 0 ∨ 𝑁 ∈ (1..^4) ∨ 𝑁 = 4) → (FermatNo‘𝑁) ∈ ℙ) |
25 | 3, 24 | sylbi 206 | 1 ⊢ (𝑁 ∈ (0...4) → (FermatNo‘𝑁) ∈ ℙ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∨ w3o 1030 = wceq 1475 ∈ wcel 1977 {ctp 4129 ‘cfv 5804 (class class class)co 6549 0cc0 9815 1c1 9816 2c2 10947 3c3 10948 4c4 10949 ℕ0cn0 11169 ...cfz 12197 ..^cfzo 12334 ℙcprime 15223 FermatNocfmtno 39977 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-xnn0 11241 df-z 11255 df-dec 11370 df-uz 11564 df-q 11665 df-rp 11709 df-ioo 12050 df-ico 12052 df-fz 12198 df-fzo 12335 df-fl 12455 df-mod 12531 df-seq 12664 df-exp 12723 df-fac 12923 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-clim 14067 df-prod 14475 df-dvds 14822 df-gcd 15055 df-prm 15224 df-odz 15308 df-phi 15309 df-pc 15380 df-lgs 24820 df-fmtno 39978 |
This theorem is referenced by: fmtnole4prm 40028 |
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