![]() |
Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > fmtno1 | Structured version Visualization version GIF version |
Description: The 1 st Fermat number, see remark in [ApostolNT] p. 7. (Contributed by AV, 13-Jun-2021.) |
Ref | Expression |
---|---|
fmtno1 | ⊢ (FermatNo‘1) = 5 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1nn0 12531 | . . 3 ⊢ 1 ∈ ℕ0 | |
2 | fmtno 47134 | . . 3 ⊢ (1 ∈ ℕ0 → (FermatNo‘1) = ((2↑(2↑1)) + 1)) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ (FermatNo‘1) = ((2↑(2↑1)) + 1) |
4 | 2cn 12330 | . . . . . 6 ⊢ 2 ∈ ℂ | |
5 | exp1 14078 | . . . . . 6 ⊢ (2 ∈ ℂ → (2↑1) = 2) | |
6 | 4, 5 | ax-mp 5 | . . . . 5 ⊢ (2↑1) = 2 |
7 | 6 | oveq2i 7424 | . . . 4 ⊢ (2↑(2↑1)) = (2↑2) |
8 | 7 | oveq1i 7423 | . . 3 ⊢ ((2↑(2↑1)) + 1) = ((2↑2) + 1) |
9 | sq2 14206 | . . . 4 ⊢ (2↑2) = 4 | |
10 | 9 | oveq1i 7423 | . . 3 ⊢ ((2↑2) + 1) = (4 + 1) |
11 | 4p1e5 12401 | . . 3 ⊢ (4 + 1) = 5 | |
12 | 8, 10, 11 | 3eqtri 2758 | . 2 ⊢ ((2↑(2↑1)) + 1) = 5 |
13 | 3, 12 | eqtri 2754 | 1 ⊢ (FermatNo‘1) = 5 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 ∈ wcel 2099 ‘cfv 6543 (class class class)co 7413 ℂcc 11144 1c1 11147 + caddc 11149 2c2 12310 4c4 12312 5c5 12313 ℕ0cn0 12515 ↑cexp 14072 FermatNocfmtno 47132 |
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 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7735 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
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 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-iun 4995 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6302 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7866 df-2nd 7993 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-er 8723 df-en 8964 df-dom 8965 df-sdom 8966 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12256 df-2 12318 df-3 12319 df-4 12320 df-5 12321 df-n0 12516 df-z 12602 df-uz 12866 df-seq 14013 df-exp 14073 df-fmtno 47133 |
This theorem is referenced by: fmtnorec2 47148 fmtno1prm 47164 fmtnofac1 47175 |
Copyright terms: Public domain | W3C validator |