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Mathbox for Alexander van der Vekens |
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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 45838 | . 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 12246 | . . . . . . . 8 β’ 5 β β | |
3 | 1nn0 12436 | . . . . . . . . 9 β’ 1 β β0 | |
4 | 3nn 12239 | . . . . . . . . 9 β’ 3 β β | |
5 | 3, 4 | decnncl 12645 | . . . . . . . 8 β’ ;13 β β |
6 | 1lt5 12340 | . . . . . . . 8 β’ 1 < 5 | |
7 | 1nn 12171 | . . . . . . . . 9 β’ 1 β β | |
8 | 3nn0 12438 | . . . . . . . . 9 β’ 3 β β0 | |
9 | 1lt10 12764 | . . . . . . . . 9 β’ 1 < ;10 | |
10 | 7, 8, 3, 9 | declti 12663 | . . . . . . . 8 β’ 1 < ;13 |
11 | eqid 2737 | . . . . . . . 8 β’ (5 Β· ;13) = (5 Β· ;13) | |
12 | 2, 5, 6, 10, 11 | nprmi 16572 | . . . . . . 7 β’ Β¬ (5 Β· ;13) β β |
13 | id 22 | . . . . . . . . 9 β’ (π = ;65 β π = ;65) | |
14 | 5nn0 12440 | . . . . . . . . . 10 β’ 5 β β0 | |
15 | eqid 2737 | . . . . . . . . . 10 β’ ;13 = ;13 | |
16 | 5cn 12248 | . . . . . . . . . . . . 13 β’ 5 β β | |
17 | 16 | mulid1i 11166 | . . . . . . . . . . . 12 β’ (5 Β· 1) = 5 |
18 | 17 | oveq1i 7372 | . . . . . . . . . . 11 β’ ((5 Β· 1) + 1) = (5 + 1) |
19 | 5p1e6 12307 | . . . . . . . . . . 11 β’ (5 + 1) = 6 | |
20 | 18, 19 | eqtri 2765 | . . . . . . . . . 10 β’ ((5 Β· 1) + 1) = 6 |
21 | 5t3e15 12726 | . . . . . . . . . 10 β’ (5 Β· 3) = ;15 | |
22 | 14, 3, 8, 15, 14, 3, 20, 21 | decmul2c 12691 | . . . . . . . . 9 β’ (5 Β· ;13) = ;65 |
23 | 13, 22 | eqtr4di 2795 | . . . . . . . 8 β’ (π = ;65 β π = (5 Β· ;13)) |
24 | 23 | eleq1d 2823 | . . . . . . 7 β’ (π = ;65 β (π β β β (5 Β· ;13) β β)) |
25 | 12, 24 | mtbiri 327 | . . . . . 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 12439 | . . . . . . . . 9 β’ 4 β β0 | |
28 | 27, 4 | decnncl 12645 | . . . . . . . 8 β’ ;43 β β |
29 | 4nn 12243 | . . . . . . . . 9 β’ 4 β β | |
30 | 29, 8, 3, 9 | declti 12663 | . . . . . . . 8 β’ 1 < ;43 |
31 | 1lt3 12333 | . . . . . . . 8 β’ 1 < 3 | |
32 | eqid 2737 | . . . . . . . 8 β’ (;43 Β· 3) = (;43 Β· 3) | |
33 | 28, 4, 30, 31, 32 | nprmi 16572 | . . . . . . 7 β’ Β¬ (;43 Β· 3) β β |
34 | id 22 | . . . . . . . . 9 β’ (π = ;;129 β π = ;;129) | |
35 | eqid 2737 | . . . . . . . . . 10 β’ ;43 = ;43 | |
36 | 4t3e12 12723 | . . . . . . . . . 10 β’ (4 Β· 3) = ;12 | |
37 | 3t3e9 12327 | . . . . . . . . . 10 β’ (3 Β· 3) = 9 | |
38 | 8, 27, 8, 35, 36, 37 | decmul1 12689 | . . . . . . . . 9 β’ (;43 Β· 3) = ;;129 |
39 | 34, 38 | eqtr4di 2795 | . . . . . . . 8 β’ (π = ;;129 β π = (;43 Β· 3)) |
40 | 39 | eleq1d 2823 | . . . . . . 7 β’ (π = ;;129 β (π β β β (;43 Β· 3) β β)) |
41 | 33, 40 | mtbiri 327 | . . . . . 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 1428 | . . . 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 1134 | . 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 1087 β§ w3a 1088 = wceq 1542 β wcel 2107 class class class wbr 5110 βcfv 6501 (class class class)co 7362 1c1 11059 + caddc 11061 Β· cmul 11063 β€ cle 11197 2c2 12215 3c3 12216 4c4 12217 5c5 12218 6c6 12219 9c9 12222 ;cdc 12625 βcfl 13702 βcsqrt 15125 β₯ cdvds 16143 βcprime 16554 FermatNocfmtno 45793 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-inf2 9584 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 ax-pre-sup 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-se 5594 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-isom 6510 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-1st 7926 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-2o 8418 df-oadd 8421 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-sup 9385 df-inf 9386 df-oi 9453 df-dju 9844 df-card 9882 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-div 11820 df-nn 12161 df-2 12223 df-3 12224 df-4 12225 df-5 12226 df-6 12227 df-7 12228 df-8 12229 df-9 12230 df-n0 12421 df-xnn0 12493 df-z 12507 df-dec 12626 df-uz 12771 df-q 12881 df-rp 12923 df-ioo 13275 df-ico 13277 df-fz 13432 df-fzo 13575 df-fl 13704 df-mod 13782 df-seq 13914 df-exp 13975 df-fac 14181 df-hash 14238 df-cj 14991 df-re 14992 df-im 14993 df-sqrt 15127 df-abs 15128 df-clim 15377 df-prod 15796 df-dvds 16144 df-gcd 16382 df-prm 16555 df-odz 16644 df-phi 16645 df-pc 16716 df-lgs 26659 df-fmtno 45794 |
This theorem is referenced by: fmtno4prm 45841 |
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