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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 12542 | . . 3 ⊢ 3 ∈ ℕ0 | |
2 | fmtno 47101 | . . 3 ⊢ (3 ∈ ℕ0 → (FermatNo‘3) = ((2↑(2↑3)) + 1)) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ (FermatNo‘3) = ((2↑(2↑3)) + 1) |
4 | cu2 14218 | . . . . 5 ⊢ (2↑3) = 8 | |
5 | 4 | oveq2i 7435 | . . . 4 ⊢ (2↑(2↑3)) = (2↑8) |
6 | 5 | oveq1i 7434 | . . 3 ⊢ ((2↑(2↑3)) + 1) = ((2↑8) + 1) |
7 | 2exp8 17091 | . . . 4 ⊢ (2↑8) = ;;256 | |
8 | 7 | oveq1i 7434 | . . 3 ⊢ ((2↑8) + 1) = (;;256 + 1) |
9 | 2nn0 12541 | . . . . 5 ⊢ 2 ∈ ℕ0 | |
10 | 5nn0 12544 | . . . . 5 ⊢ 5 ∈ ℕ0 | |
11 | 9, 10 | deccl 12744 | . . . 4 ⊢ ;25 ∈ ℕ0 |
12 | 6nn0 12545 | . . . 4 ⊢ 6 ∈ ℕ0 | |
13 | 6p1e7 12412 | . . . 4 ⊢ (6 + 1) = 7 | |
14 | eqid 2726 | . . . 4 ⊢ ;;256 = ;;256 | |
15 | 11, 12, 13, 14 | decsuc 12760 | . . 3 ⊢ (;;256 + 1) = ;;257 |
16 | 6, 8, 15 | 3eqtri 2758 | . 2 ⊢ ((2↑(2↑3)) + 1) = ;;257 |
17 | 3, 16 | eqtri 2754 | 1 ⊢ (FermatNo‘3) = ;;257 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 ∈ wcel 2099 ‘cfv 6554 (class class class)co 7424 1c1 11159 + caddc 11161 2c2 12319 3c3 12320 5c5 12322 6c6 12323 7c7 12324 8c8 12325 ℕ0cn0 12524 ;cdc 12729 ↑cexp 14081 FermatNocfmtno 47099 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-er 8734 df-en 8975 df-dom 8976 df-sdom 8977 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12611 df-dec 12730 df-uz 12875 df-seq 14022 df-exp 14082 df-fmtno 47100 |
This theorem is referenced by: fmtno3prm 47134 |
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