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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 12475 | . . . . 5 β’ (π β β β π β β0) | |
2 | fmtno 46183 | . . . . 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 6892 | . . 3 β’ (π β β β (ββ(FermatNoβπ)) = (ββ((2β(2βπ)) + 1))) |
5 | 4 | fveq2d 6892 | . 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 12484 | . . . . 5 β’ 1 β β0 | |
8 | 7 | a1i 11 | . . . 4 β’ (π β β β 1 β β0) |
9 | 2nn 12281 | . . . . . . . 8 β’ 2 β β | |
10 | 9 | a1i 11 | . . . . . . 7 β’ (π β β β 2 β β) |
11 | 2nn0 12485 | . . . . . . . . . 10 β’ 2 β β0 | |
12 | 11 | a1i 11 | . . . . . . . . 9 β’ (π β β β 2 β β0) |
13 | nnm1nn0 12509 | . . . . . . . . 9 β’ (π β β β (π β 1) β β0) | |
14 | 12, 13 | nn0expcld 14205 | . . . . . . . 8 β’ (π β β β (2β(π β 1)) β β0) |
15 | peano2nn0 12508 | . . . . . . . 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 14204 | . . . . . 6 β’ (π β β β (2β((2β(π β 1)) + 1)) β β) |
18 | nngt0 12239 | . . . . . 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 14205 | . . . . . . . 8 β’ (π β β β (2β((2β(π β 1)) + 1)) β β0) |
21 | 20 | nn0red 12529 | . . . . . . 7 β’ (π β β β (2β((2β(π β 1)) + 1)) β β) |
22 | 1re 11210 | . . . . . . . 8 β’ 1 β β | |
23 | 22 | a1i 11 | . . . . . . 7 β’ (π β β β 1 β β) |
24 | 21, 23 | jca 512 | . . . . . 6 β’ (π β β β ((2β((2β(π β 1)) + 1)) β β β§ 1 β β)) |
25 | ltaddpos2 11701 | . . . . . 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 1128 | . . 3 β’ (π β β β (π β β β§ 1 β β0 β§ 1 < ((2β((2β(π β 1)) + 1)) + 1))) |
29 | sqrtpwpw2p 46192 | . . 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 2772 | 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 396 β§ w3a 1087 = wceq 1541 β wcel 2106 class class class wbr 5147 βcfv 6540 (class class class)co 7405 βcr 11105 0cc0 11106 1c1 11107 + caddc 11109 < clt 11244 β cmin 11440 βcn 12208 2c2 12263 β0cn0 12468 βcfl 13751 βcexp 14023 βcsqrt 15176 FermatNocfmtno 46181 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-sup 9433 df-inf 9434 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-n0 12469 df-z 12555 df-uz 12819 df-rp 12971 df-fl 13753 df-seq 13963 df-exp 14024 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-fmtno 46182 |
This theorem is referenced by: fmtno4sqrt 46225 |
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