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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > fmtno3 | Structured version Visualization version GIF version |
Description: The 3 rd Fermat number, see remark in [ApostolNT] p. 7. (Contributed by AV, 13-Jun-2021.) |
Ref | Expression |
---|---|
fmtno3 | ⊢ (FermatNo‘3) = ;;257 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3nn0 12486 | . . 3 ⊢ 3 ∈ ℕ0 | |
2 | fmtno 46132 | . . 3 ⊢ (3 ∈ ℕ0 → (FermatNo‘3) = ((2↑(2↑3)) + 1)) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ (FermatNo‘3) = ((2↑(2↑3)) + 1) |
4 | cu2 14160 | . . . . 5 ⊢ (2↑3) = 8 | |
5 | 4 | oveq2i 7415 | . . . 4 ⊢ (2↑(2↑3)) = (2↑8) |
6 | 5 | oveq1i 7414 | . . 3 ⊢ ((2↑(2↑3)) + 1) = ((2↑8) + 1) |
7 | 2exp8 17018 | . . . 4 ⊢ (2↑8) = ;;256 | |
8 | 7 | oveq1i 7414 | . . 3 ⊢ ((2↑8) + 1) = (;;256 + 1) |
9 | 2nn0 12485 | . . . . 5 ⊢ 2 ∈ ℕ0 | |
10 | 5nn0 12488 | . . . . 5 ⊢ 5 ∈ ℕ0 | |
11 | 9, 10 | deccl 12688 | . . . 4 ⊢ ;25 ∈ ℕ0 |
12 | 6nn0 12489 | . . . 4 ⊢ 6 ∈ ℕ0 | |
13 | 6p1e7 12356 | . . . 4 ⊢ (6 + 1) = 7 | |
14 | eqid 2733 | . . . 4 ⊢ ;;256 = ;;256 | |
15 | 11, 12, 13, 14 | decsuc 12704 | . . 3 ⊢ (;;256 + 1) = ;;257 |
16 | 6, 8, 15 | 3eqtri 2765 | . 2 ⊢ ((2↑(2↑3)) + 1) = ;;257 |
17 | 3, 16 | eqtri 2761 | 1 ⊢ (FermatNo‘3) = ;;257 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2107 ‘cfv 6540 (class class class)co 7404 1c1 11107 + caddc 11109 2c2 12263 3c3 12264 5c5 12266 6c6 12267 7c7 12268 8c8 12269 ℕ0cn0 12468 ;cdc 12673 ↑cexp 14023 FermatNocfmtno 46130 |
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 2704 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7851 df-2nd 7971 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-seq 13963 df-exp 14024 df-fmtno 46131 |
This theorem is referenced by: fmtno3prm 46165 |
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