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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > fmtno2 | Structured version Visualization version GIF version |
Description: The 2 nd Fermat number, see remark in [ApostolNT] p. 7. (Contributed by AV, 13-Jun-2021.) |
Ref | Expression |
---|---|
fmtno2 | ⊢ (FermatNo‘2) = ;17 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2nn0 12485 | . . 3 ⊢ 2 ∈ ℕ0 | |
2 | fmtno 46648 | . . 3 ⊢ (2 ∈ ℕ0 → (FermatNo‘2) = ((2↑(2↑2)) + 1)) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ (FermatNo‘2) = ((2↑(2↑2)) + 1) |
4 | sq2 14157 | . . . . 5 ⊢ (2↑2) = 4 | |
5 | 4 | oveq2i 7412 | . . . 4 ⊢ (2↑(2↑2)) = (2↑4) |
6 | 5 | oveq1i 7411 | . . 3 ⊢ ((2↑(2↑2)) + 1) = ((2↑4) + 1) |
7 | 2exp4 17014 | . . . 4 ⊢ (2↑4) = ;16 | |
8 | 7 | oveq1i 7411 | . . 3 ⊢ ((2↑4) + 1) = (;16 + 1) |
9 | 1nn0 12484 | . . . 4 ⊢ 1 ∈ ℕ0 | |
10 | 6nn0 12489 | . . . 4 ⊢ 6 ∈ ℕ0 | |
11 | 6p1e7 12356 | . . . 4 ⊢ (6 + 1) = 7 | |
12 | eqid 2724 | . . . 4 ⊢ ;16 = ;16 | |
13 | 9, 10, 11, 12 | decsuc 12704 | . . 3 ⊢ (;16 + 1) = ;17 |
14 | 6, 8, 13 | 3eqtri 2756 | . 2 ⊢ ((2↑(2↑2)) + 1) = ;17 |
15 | 3, 14 | eqtri 2752 | 1 ⊢ (FermatNo‘2) = ;17 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2098 ‘cfv 6533 (class class class)co 7401 1c1 11106 + caddc 11108 2c2 12263 4c4 12265 6c6 12267 7c7 12268 ℕ0cn0 12468 ;cdc 12673 ↑cexp 14023 FermatNocfmtno 46646 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-er 8698 df-en 8935 df-dom 8936 df-sdom 8937 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 46647 |
This theorem is referenced by: fmtno2prm 46679 |
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