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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > fmtnoge3 | Structured version Visualization version GIF version |
Description: Each Fermat number is greater than or equal to 3. (Contributed by AV, 4-Aug-2021.) |
Ref | Expression |
---|---|
fmtnoge3 | β’ (π β β0 β (FermatNoβπ) β (β€β₯β3)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fmtno 46769 | . 2 β’ (π β β0 β (FermatNoβπ) = ((2β(2βπ)) + 1)) | |
2 | 3z 12599 | . . . 4 β’ 3 β β€ | |
3 | 2 | a1i 11 | . . 3 β’ (π β β0 β 3 β β€) |
4 | 2nn0 12493 | . . . . . . 7 β’ 2 β β0 | |
5 | 4 | a1i 11 | . . . . . 6 β’ (π β β0 β 2 β β0) |
6 | id 22 | . . . . . . 7 β’ (π β β0 β π β β0) | |
7 | 5, 6 | nn0expcld 14214 | . . . . . 6 β’ (π β β0 β (2βπ) β β0) |
8 | 5, 7 | nn0expcld 14214 | . . . . 5 β’ (π β β0 β (2β(2βπ)) β β0) |
9 | peano2nn0 12516 | . . . . 5 β’ ((2β(2βπ)) β β0 β ((2β(2βπ)) + 1) β β0) | |
10 | 8, 9 | syl 17 | . . . 4 β’ (π β β0 β ((2β(2βπ)) + 1) β β0) |
11 | 10 | nn0zd 12588 | . . 3 β’ (π β β0 β ((2β(2βπ)) + 1) β β€) |
12 | 3m1e2 12344 | . . . . 5 β’ (3 β 1) = 2 | |
13 | 2cn 12291 | . . . . . . 7 β’ 2 β β | |
14 | exp1 14038 | . . . . . . 7 β’ (2 β β β (2β1) = 2) | |
15 | 13, 14 | ax-mp 5 | . . . . . 6 β’ (2β1) = 2 |
16 | 2re 12290 | . . . . . . . . 9 β’ 2 β β | |
17 | 16 | a1i 11 | . . . . . . . 8 β’ (π β β0 β 2 β β) |
18 | 1le2 12425 | . . . . . . . . 9 β’ 1 β€ 2 | |
19 | 18 | a1i 11 | . . . . . . . 8 β’ (π β β0 β 1 β€ 2) |
20 | 17, 6, 19 | expge1d 14135 | . . . . . . 7 β’ (π β β0 β 1 β€ (2βπ)) |
21 | 1zzd 12597 | . . . . . . . 8 β’ (π β β0 β 1 β β€) | |
22 | 7 | nn0zd 12588 | . . . . . . . 8 β’ (π β β0 β (2βπ) β β€) |
23 | 1lt2 12387 | . . . . . . . . 9 β’ 1 < 2 | |
24 | 23 | a1i 11 | . . . . . . . 8 β’ (π β β0 β 1 < 2) |
25 | 17, 21, 22, 24 | leexp2d 14220 | . . . . . . 7 β’ (π β β0 β (1 β€ (2βπ) β (2β1) β€ (2β(2βπ)))) |
26 | 20, 25 | mpbid 231 | . . . . . 6 β’ (π β β0 β (2β1) β€ (2β(2βπ))) |
27 | 15, 26 | eqbrtrrid 5177 | . . . . 5 β’ (π β β0 β 2 β€ (2β(2βπ))) |
28 | 12, 27 | eqbrtrid 5176 | . . . 4 β’ (π β β0 β (3 β 1) β€ (2β(2βπ))) |
29 | 3re 12296 | . . . . . 6 β’ 3 β β | |
30 | 29 | a1i 11 | . . . . 5 β’ (π β β0 β 3 β β) |
31 | 1red 11219 | . . . . 5 β’ (π β β0 β 1 β β) | |
32 | 8 | nn0red 12537 | . . . . 5 β’ (π β β0 β (2β(2βπ)) β β) |
33 | 30, 31, 32 | lesubaddd 11815 | . . . 4 β’ (π β β0 β ((3 β 1) β€ (2β(2βπ)) β 3 β€ ((2β(2βπ)) + 1))) |
34 | 28, 33 | mpbid 231 | . . 3 β’ (π β β0 β 3 β€ ((2β(2βπ)) + 1)) |
35 | eluz2 12832 | . . 3 β’ (((2β(2βπ)) + 1) β (β€β₯β3) β (3 β β€ β§ ((2β(2βπ)) + 1) β β€ β§ 3 β€ ((2β(2βπ)) + 1))) | |
36 | 3, 11, 34, 35 | syl3anbrc 1340 | . 2 β’ (π β β0 β ((2β(2βπ)) + 1) β (β€β₯β3)) |
37 | 1, 36 | eqeltrd 2827 | 1 β’ (π β β0 β (FermatNoβπ) β (β€β₯β3)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 class class class wbr 5141 βcfv 6537 (class class class)co 7405 βcc 11110 βcr 11111 1c1 11113 + caddc 11115 < clt 11252 β€ cle 11253 β cmin 11448 2c2 12271 3c3 12272 β0cn0 12476 β€cz 12562 β€β₯cuz 12826 βcexp 14032 FermatNocfmtno 46767 |
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 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
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 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-n0 12477 df-z 12563 df-uz 12827 df-rp 12981 df-seq 13973 df-exp 14033 df-fmtno 46768 |
This theorem is referenced by: fmtnonn 46771 prmdvdsfmtnof 46826 prmdvdsfmtnof1 46827 |
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