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| Mirrors > Home > MPE Home > Th. List > Mathboxes > fmtno4prmfac193 | Structured version Visualization version GIF version | ||
| Description: If P was a (prime) factor of the fourth Fermat number, it would be 193. (Contributed by AV, 28-Jul-2021.) |
| Ref | Expression |
|---|---|
| fmtno4prmfac193 | β’ ((π β β β§ π β₯ (FermatNoβ4) β§ π β€ (ββ(ββ(FermatNoβ4)))) β π = ;;193) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | fmtno4prmfac 46226 | . 2 β’ ((π β β β§ π β₯ (FermatNoβ4) β§ π β€ (ββ(ββ(FermatNoβ4)))) β (π = ;65 β¨ π = ;;129 β¨ π = ;;193)) | |
| 2 | 5nn 12294 | . . . . . . . 8 β’ 5 β β | |
| 3 | 1nn0 12484 | . . . . . . . . 9 β’ 1 β β0 | |
| 4 | 3nn 12287 | . . . . . . . . 9 β’ 3 β β | |
| 5 | 3, 4 | decnncl 12693 | . . . . . . . 8 β’ ;13 β β |
| 6 | 1lt5 12388 | . . . . . . . 8 β’ 1 < 5 | |
| 7 | 1nn 12219 | . . . . . . . . 9 β’ 1 β β | |
| 8 | 3nn0 12486 | . . . . . . . . 9 β’ 3 β β0 | |
| 9 | 1lt10 12812 | . . . . . . . . 9 β’ 1 < ;10 | |
| 10 | 7, 8, 3, 9 | declti 12711 | . . . . . . . 8 β’ 1 < ;13 |
| 11 | eqid 2732 | . . . . . . . 8 β’ (5 Β· ;13) = (5 Β· ;13) | |
| 12 | 2, 5, 6, 10, 11 | nprmi 16622 | . . . . . . 7 β’ Β¬ (5 Β· ;13) β β |
| 13 | id 22 | . . . . . . . . 9 β’ (π = ;65 β π = ;65) | |
| 14 | 5nn0 12488 | . . . . . . . . . 10 β’ 5 β β0 | |
| 15 | eqid 2732 | . . . . . . . . . 10 β’ ;13 = ;13 | |
| 16 | 5cn 12296 | . . . . . . . . . . . . 13 β’ 5 β β | |
| 17 | 16 | mulridi 11214 | . . . . . . . . . . . 12 β’ (5 Β· 1) = 5 |
| 18 | 17 | oveq1i 7415 | . . . . . . . . . . 11 β’ ((5 Β· 1) + 1) = (5 + 1) |
| 19 | 5p1e6 12355 | . . . . . . . . . . 11 β’ (5 + 1) = 6 | |
| 20 | 18, 19 | eqtri 2760 | . . . . . . . . . 10 β’ ((5 Β· 1) + 1) = 6 |
| 21 | 5t3e15 12774 | . . . . . . . . . 10 β’ (5 Β· 3) = ;15 | |
| 22 | 14, 3, 8, 15, 14, 3, 20, 21 | decmul2c 12739 | . . . . . . . . 9 β’ (5 Β· ;13) = ;65 |
| 23 | 13, 22 | eqtr4di 2790 | . . . . . . . 8 β’ (π = ;65 β π = (5 Β· ;13)) |
| 24 | 23 | eleq1d 2818 | . . . . . . 7 β’ (π = ;65 β (π β β β (5 Β· ;13) β β)) |
| 25 | 12, 24 | mtbiri 326 | . . . . . 6 β’ (π = ;65 β Β¬ π β β) |
| 26 | 25 | pm2.21d 121 | . . . . 5 β’ (π = ;65 β (π β β β π = ;;193)) |
| 27 | 4nn0 12487 | . . . . . . . . 9 β’ 4 β β0 | |
| 28 | 27, 4 | decnncl 12693 | . . . . . . . 8 β’ ;43 β β |
| 29 | 4nn 12291 | . . . . . . . . 9 β’ 4 β β | |
| 30 | 29, 8, 3, 9 | declti 12711 | . . . . . . . 8 β’ 1 < ;43 |
| 31 | 1lt3 12381 | . . . . . . . 8 β’ 1 < 3 | |
| 32 | eqid 2732 | . . . . . . . 8 β’ (;43 Β· 3) = (;43 Β· 3) | |
| 33 | 28, 4, 30, 31, 32 | nprmi 16622 | . . . . . . 7 β’ Β¬ (;43 Β· 3) β β |
| 34 | id 22 | . . . . . . . . 9 β’ (π = ;;129 β π = ;;129) | |
| 35 | eqid 2732 | . . . . . . . . . 10 β’ ;43 = ;43 | |
| 36 | 4t3e12 12771 | . . . . . . . . . 10 β’ (4 Β· 3) = ;12 | |
| 37 | 3t3e9 12375 | . . . . . . . . . 10 β’ (3 Β· 3) = 9 | |
| 38 | 8, 27, 8, 35, 36, 37 | decmul1 12737 | . . . . . . . . 9 β’ (;43 Β· 3) = ;;129 |
| 39 | 34, 38 | eqtr4di 2790 | . . . . . . . 8 β’ (π = ;;129 β π = (;43 Β· 3)) |
| 40 | 39 | eleq1d 2818 | . . . . . . 7 β’ (π = ;;129 β (π β β β (;43 Β· 3) β β)) |
| 41 | 33, 40 | mtbiri 326 | . . . . . 6 β’ (π = ;;129 β Β¬ π β β) |
| 42 | 41 | pm2.21d 121 | . . . . 5 β’ (π = ;;129 β (π β β β π = ;;193)) |
| 43 | ax-1 6 | . . . . 5 β’ (π = ;;193 β (π β β β π = ;;193)) | |
| 44 | 26, 42, 43 | 3jaoi 1427 | . . . 4 β’ ((π = ;65 β¨ π = ;;129 β¨ π = ;;193) β (π β β β π = ;;193)) |
| 45 | 44 | com12 32 | . . 3 β’ (π β β β ((π = ;65 β¨ π = ;;129 β¨ π = ;;193) β π = ;;193)) |
| 46 | 45 | 3ad2ant1 1133 | . 2 β’ ((π β β β§ π β₯ (FermatNoβ4) β§ π β€ (ββ(ββ(FermatNoβ4)))) β ((π = ;65 β¨ π = ;;129 β¨ π = ;;193) β π = ;;193)) |
| 47 | 1, 46 | mpd 15 | 1 β’ ((π β β β§ π β₯ (FermatNoβ4) β§ π β€ (ββ(ββ(FermatNoβ4)))) β π = ;;193) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β¨ w3o 1086 β§ w3a 1087 = wceq 1541 β wcel 2106 class class class wbr 5147 βcfv 6540 (class class class)co 7405 1c1 11107 + caddc 11109 Β· cmul 11111 β€ cle 11245 2c2 12263 3c3 12264 4c4 12265 5c5 12266 6c6 12267 9c9 12270 ;cdc 12673 βcfl 13751 βcsqrt 15176 β₯ cdvds 16193 βcprime 16604 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-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-inf2 9632 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-tp 4632 df-op 4634 df-uni 4908 df-int 4950 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-se 5631 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-isom 6549 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-oadd 8466 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-sup 9433 df-inf 9434 df-oi 9501 df-dju 9892 df-card 9930 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-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-xnn0 12541 df-z 12555 df-dec 12674 df-uz 12819 df-q 12929 df-rp 12971 df-ioo 13324 df-ico 13326 df-fz 13481 df-fzo 13624 df-fl 13753 df-mod 13831 df-seq 13963 df-exp 14024 df-fac 14230 df-hash 14287 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-clim 15428 df-prod 15846 df-dvds 16194 df-gcd 16432 df-prm 16605 df-odz 16694 df-phi 16695 df-pc 16766 df-lgs 26787 df-fmtno 46182 |
| This theorem is referenced by: fmtno4prm 46229 |
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