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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 46916 | . 2 β’ (π β β0 β (FermatNoβπ) = ((2β(2βπ)) + 1)) | |
2 | 3z 12635 | . . . 4 β’ 3 β β€ | |
3 | 2 | a1i 11 | . . 3 β’ (π β β0 β 3 β β€) |
4 | 2nn0 12529 | . . . . . . 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 14250 | . . . . . 6 β’ (π β β0 β (2βπ) β β0) |
8 | 5, 7 | nn0expcld 14250 | . . . . 5 β’ (π β β0 β (2β(2βπ)) β β0) |
9 | peano2nn0 12552 | . . . . 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 12624 | . . 3 β’ (π β β0 β ((2β(2βπ)) + 1) β β€) |
12 | 3m1e2 12380 | . . . . 5 β’ (3 β 1) = 2 | |
13 | 2cn 12327 | . . . . . . 7 β’ 2 β β | |
14 | exp1 14074 | . . . . . . 7 β’ (2 β β β (2β1) = 2) | |
15 | 13, 14 | ax-mp 5 | . . . . . 6 β’ (2β1) = 2 |
16 | 2re 12326 | . . . . . . . . 9 β’ 2 β β | |
17 | 16 | a1i 11 | . . . . . . . 8 β’ (π β β0 β 2 β β) |
18 | 1le2 12461 | . . . . . . . . 9 β’ 1 β€ 2 | |
19 | 18 | a1i 11 | . . . . . . . 8 β’ (π β β0 β 1 β€ 2) |
20 | 17, 6, 19 | expge1d 14171 | . . . . . . 7 β’ (π β β0 β 1 β€ (2βπ)) |
21 | 1zzd 12633 | . . . . . . . 8 β’ (π β β0 β 1 β β€) | |
22 | 7 | nn0zd 12624 | . . . . . . . 8 β’ (π β β0 β (2βπ) β β€) |
23 | 1lt2 12423 | . . . . . . . . 9 β’ 1 < 2 | |
24 | 23 | a1i 11 | . . . . . . . 8 β’ (π β β0 β 1 < 2) |
25 | 17, 21, 22, 24 | leexp2d 14256 | . . . . . . 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 5188 | . . . . 5 β’ (π β β0 β 2 β€ (2β(2βπ))) |
28 | 12, 27 | eqbrtrid 5187 | . . . 4 β’ (π β β0 β (3 β 1) β€ (2β(2βπ))) |
29 | 3re 12332 | . . . . . 6 β’ 3 β β | |
30 | 29 | a1i 11 | . . . . 5 β’ (π β β0 β 3 β β) |
31 | 1red 11255 | . . . . 5 β’ (π β β0 β 1 β β) | |
32 | 8 | nn0red 12573 | . . . . 5 β’ (π β β0 β (2β(2βπ)) β β) |
33 | 30, 31, 32 | lesubaddd 11851 | . . . 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 12868 | . . 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 2829 | 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 5152 βcfv 6553 (class class class)co 7426 βcc 11146 βcr 11147 1c1 11149 + caddc 11151 < clt 11288 β€ cle 11289 β cmin 11484 2c2 12307 3c3 12308 β0cn0 12512 β€cz 12598 β€β₯cuz 12862 βcexp 14068 FermatNocfmtno 46914 |
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 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 ax-cnex 11204 ax-resscn 11205 ax-1cn 11206 ax-icn 11207 ax-addcl 11208 ax-addrcl 11209 ax-mulcl 11210 ax-mulrcl 11211 ax-mulcom 11212 ax-addass 11213 ax-mulass 11214 ax-distr 11215 ax-i2m1 11216 ax-1ne0 11217 ax-1rid 11218 ax-rnegex 11219 ax-rrecex 11220 ax-cnre 11221 ax-pre-lttri 11222 ax-pre-lttrn 11223 ax-pre-ltadd 11224 ax-pre-mulgt0 11225 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 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 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7879 df-2nd 8002 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-er 8733 df-en 8973 df-dom 8974 df-sdom 8975 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-le 11294 df-sub 11486 df-neg 11487 df-div 11912 df-nn 12253 df-2 12315 df-3 12316 df-n0 12513 df-z 12599 df-uz 12863 df-rp 13017 df-seq 14009 df-exp 14069 df-fmtno 46915 |
This theorem is referenced by: fmtnonn 46918 prmdvdsfmtnof 46973 prmdvdsfmtnof1 46974 |
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