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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 11348 | . . 3 ⊢ 3 ∈ ℕ0 | |
2 | fmtno 41766 | . . 3 ⊢ (3 ∈ ℕ0 → (FermatNo‘3) = ((2↑(2↑3)) + 1)) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ (FermatNo‘3) = ((2↑(2↑3)) + 1) |
4 | cu2 13003 | . . . . 5 ⊢ (2↑3) = 8 | |
5 | 4 | oveq2i 6701 | . . . 4 ⊢ (2↑(2↑3)) = (2↑8) |
6 | 5 | oveq1i 6700 | . . 3 ⊢ ((2↑(2↑3)) + 1) = ((2↑8) + 1) |
7 | 2exp8 15843 | . . . 4 ⊢ (2↑8) = ;;256 | |
8 | 7 | oveq1i 6700 | . . 3 ⊢ ((2↑8) + 1) = (;;256 + 1) |
9 | 2nn0 11347 | . . . . 5 ⊢ 2 ∈ ℕ0 | |
10 | 5nn0 11350 | . . . . 5 ⊢ 5 ∈ ℕ0 | |
11 | 9, 10 | deccl 11550 | . . . 4 ⊢ ;25 ∈ ℕ0 |
12 | 6nn0 11351 | . . . 4 ⊢ 6 ∈ ℕ0 | |
13 | 6p1e7 11194 | . . . 4 ⊢ (6 + 1) = 7 | |
14 | eqid 2651 | . . . 4 ⊢ ;;256 = ;;256 | |
15 | 11, 12, 13, 14 | decsuc 11573 | . . 3 ⊢ (;;256 + 1) = ;;257 |
16 | 6, 8, 15 | 3eqtri 2677 | . 2 ⊢ ((2↑(2↑3)) + 1) = ;;257 |
17 | 3, 16 | eqtri 2673 | 1 ⊢ (FermatNo‘3) = ;;257 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1523 ∈ wcel 2030 ‘cfv 5926 (class class class)co 6690 1c1 9975 + caddc 9977 2c2 11108 3c3 11109 5c5 11111 6c6 11112 7c7 11113 8c8 11114 ℕ0cn0 11330 ;cdc 11531 ↑cexp 12900 FermatNocfmtno 41764 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-2nd 7211 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-nn 11059 df-2 11117 df-3 11118 df-4 11119 df-5 11120 df-6 11121 df-7 11122 df-8 11123 df-9 11124 df-n0 11331 df-z 11416 df-dec 11532 df-uz 11726 df-seq 12842 df-exp 12901 df-fmtno 41765 |
This theorem is referenced by: fmtno3prm 41799 |
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