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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > fmtnosqrt | Structured version Visualization version GIF version |
Description: The floor of the square root of a Fermat number. (Contributed by AV, 28-Jul-2021.) |
Ref | Expression |
---|---|
fmtnosqrt | β’ (π β β β (ββ(ββ(FermatNoβπ))) = (2β(2β(π β 1)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnnn0 12509 | . . . . 5 β’ (π β β β π β β0) | |
2 | fmtno 46932 | . . . . 5 β’ (π β β0 β (FermatNoβπ) = ((2β(2βπ)) + 1)) | |
3 | 1, 2 | syl 17 | . . . 4 β’ (π β β β (FermatNoβπ) = ((2β(2βπ)) + 1)) |
4 | 3 | fveq2d 6896 | . . 3 β’ (π β β β (ββ(FermatNoβπ)) = (ββ((2β(2βπ)) + 1))) |
5 | 4 | fveq2d 6896 | . 2 β’ (π β β β (ββ(ββ(FermatNoβπ))) = (ββ(ββ((2β(2βπ)) + 1)))) |
6 | id 22 | . . . 4 β’ (π β β β π β β) | |
7 | 1nn0 12518 | . . . . 5 β’ 1 β β0 | |
8 | 7 | a1i 11 | . . . 4 β’ (π β β β 1 β β0) |
9 | 2nn 12315 | . . . . . . . 8 β’ 2 β β | |
10 | 9 | a1i 11 | . . . . . . 7 β’ (π β β β 2 β β) |
11 | 2nn0 12519 | . . . . . . . . . 10 β’ 2 β β0 | |
12 | 11 | a1i 11 | . . . . . . . . 9 β’ (π β β β 2 β β0) |
13 | nnm1nn0 12543 | . . . . . . . . 9 β’ (π β β β (π β 1) β β0) | |
14 | 12, 13 | nn0expcld 14240 | . . . . . . . 8 β’ (π β β β (2β(π β 1)) β β0) |
15 | peano2nn0 12542 | . . . . . . . 8 β’ ((2β(π β 1)) β β0 β ((2β(π β 1)) + 1) β β0) | |
16 | 14, 15 | syl 17 | . . . . . . 7 β’ (π β β β ((2β(π β 1)) + 1) β β0) |
17 | 10, 16 | nnexpcld 14239 | . . . . . 6 β’ (π β β β (2β((2β(π β 1)) + 1)) β β) |
18 | nngt0 12273 | . . . . . 6 β’ ((2β((2β(π β 1)) + 1)) β β β 0 < (2β((2β(π β 1)) + 1))) | |
19 | 17, 18 | syl 17 | . . . . 5 β’ (π β β β 0 < (2β((2β(π β 1)) + 1))) |
20 | 12, 16 | nn0expcld 14240 | . . . . . . . 8 β’ (π β β β (2β((2β(π β 1)) + 1)) β β0) |
21 | 20 | nn0red 12563 | . . . . . . 7 β’ (π β β β (2β((2β(π β 1)) + 1)) β β) |
22 | 1re 11244 | . . . . . . . 8 β’ 1 β β | |
23 | 22 | a1i 11 | . . . . . . 7 β’ (π β β β 1 β β) |
24 | 21, 23 | jca 510 | . . . . . 6 β’ (π β β β ((2β((2β(π β 1)) + 1)) β β β§ 1 β β)) |
25 | ltaddpos2 11735 | . . . . . 6 β’ (((2β((2β(π β 1)) + 1)) β β β§ 1 β β) β (0 < (2β((2β(π β 1)) + 1)) β 1 < ((2β((2β(π β 1)) + 1)) + 1))) | |
26 | 24, 25 | syl 17 | . . . . 5 β’ (π β β β (0 < (2β((2β(π β 1)) + 1)) β 1 < ((2β((2β(π β 1)) + 1)) + 1))) |
27 | 19, 26 | mpbid 231 | . . . 4 β’ (π β β β 1 < ((2β((2β(π β 1)) + 1)) + 1)) |
28 | 6, 8, 27 | 3jca 1125 | . . 3 β’ (π β β β (π β β β§ 1 β β0 β§ 1 < ((2β((2β(π β 1)) + 1)) + 1))) |
29 | sqrtpwpw2p 46941 | . . 3 β’ ((π β β β§ 1 β β0 β§ 1 < ((2β((2β(π β 1)) + 1)) + 1)) β (ββ(ββ((2β(2βπ)) + 1))) = (2β(2β(π β 1)))) | |
30 | 28, 29 | syl 17 | . 2 β’ (π β β β (ββ(ββ((2β(2βπ)) + 1))) = (2β(2β(π β 1)))) |
31 | 5, 30 | eqtrd 2765 | 1 β’ (π β β β (ββ(ββ(FermatNoβπ))) = (2β(2β(π β 1)))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 394 β§ w3a 1084 = wceq 1533 β wcel 2098 class class class wbr 5143 βcfv 6543 (class class class)co 7416 βcr 11137 0cc0 11138 1c1 11139 + caddc 11141 < clt 11278 β cmin 11474 βcn 12242 2c2 12297 β0cn0 12502 βcfl 13787 βcexp 14058 βcsqrt 15212 FermatNocfmtno 46930 |
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 2696 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-pre-sup 11216 |
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 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 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 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 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 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-er 8723 df-en 8963 df-dom 8964 df-sdom 8965 df-sup 9465 df-inf 9466 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-2 12305 df-3 12306 df-n0 12503 df-z 12589 df-uz 12853 df-rp 13007 df-fl 13789 df-seq 13999 df-exp 14059 df-cj 15078 df-re 15079 df-im 15080 df-sqrt 15214 df-fmtno 46931 |
This theorem is referenced by: fmtno4sqrt 46974 |
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