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| Mirrors > Home > MPE Home > Th. List > Mathboxes > goldbachth | Structured version Visualization version GIF version | ||
| Description: Goldbach's theorem: Two different Fermat numbers are coprime. See ProofWiki "Goldbach's theorem", 31-Jul-2021, https://proofwiki.org/wiki/Goldbach%27s_Theorem or Wikipedia "Fermat number", 31-Jul-2021, https://en.wikipedia.org/wiki/Fermat_number#Basic_properties. (Contributed by AV, 1-Aug-2021.) |
| Ref | Expression |
|---|---|
| goldbachth | β’ ((π β β0 β§ π β β0 β§ π β π) β ((FermatNoβπ) gcd (FermatNoβπ)) = 1) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | nn0re 12485 | . . . 4 β’ (π β β0 β π β β) | |
| 2 | nn0re 12485 | . . . 4 β’ (π β β0 β π β β) | |
| 3 | lttri4 11302 | . . . 4 β’ ((π β β β§ π β β) β (π < π β¨ π = π β¨ π < π)) | |
| 4 | 1, 2, 3 | syl2an 594 | . . 3 β’ ((π β β0 β§ π β β0) β (π < π β¨ π = π β¨ π < π)) |
| 5 | 4 | 3adant3 1130 | . 2 β’ ((π β β0 β§ π β β0 β§ π β π) β (π < π β¨ π = π β¨ π < π)) |
| 6 | fmtnonn 46497 | . . . . . . . . . 10 β’ (π β β0 β (FermatNoβπ) β β) | |
| 7 | 6 | nnzd 12589 | . . . . . . . . 9 β’ (π β β0 β (FermatNoβπ) β β€) |
| 8 | fmtnonn 46497 | . . . . . . . . . 10 β’ (π β β0 β (FermatNoβπ) β β) | |
| 9 | 8 | nnzd 12589 | . . . . . . . . 9 β’ (π β β0 β (FermatNoβπ) β β€) |
| 10 | gcdcom 16458 | . . . . . . . . 9 β’ (((FermatNoβπ) β β€ β§ (FermatNoβπ) β β€) β ((FermatNoβπ) gcd (FermatNoβπ)) = ((FermatNoβπ) gcd (FermatNoβπ))) | |
| 11 | 7, 9, 10 | syl2anr 595 | . . . . . . . 8 β’ ((π β β0 β§ π β β0) β ((FermatNoβπ) gcd (FermatNoβπ)) = ((FermatNoβπ) gcd (FermatNoβπ))) |
| 12 | 11 | 3adant3 1130 | . . . . . . 7 β’ ((π β β0 β§ π β β0 β§ π < π) β ((FermatNoβπ) gcd (FermatNoβπ)) = ((FermatNoβπ) gcd (FermatNoβπ))) |
| 13 | goldbachthlem2 46512 | . . . . . . 7 β’ ((π β β0 β§ π β β0 β§ π < π) β ((FermatNoβπ) gcd (FermatNoβπ)) = 1) | |
| 14 | 12, 13 | eqtrd 2770 | . . . . . 6 β’ ((π β β0 β§ π β β0 β§ π < π) β ((FermatNoβπ) gcd (FermatNoβπ)) = 1) |
| 15 | 14 | 3exp 1117 | . . . . 5 β’ (π β β0 β (π β β0 β (π < π β ((FermatNoβπ) gcd (FermatNoβπ)) = 1))) |
| 16 | 15 | impcom 406 | . . . 4 β’ ((π β β0 β§ π β β0) β (π < π β ((FermatNoβπ) gcd (FermatNoβπ)) = 1)) |
| 17 | 16 | 3adant3 1130 | . . 3 β’ ((π β β0 β§ π β β0 β§ π β π) β (π < π β ((FermatNoβπ) gcd (FermatNoβπ)) = 1)) |
| 18 | eqneqall 2949 | . . . . 5 β’ (π = π β (π β π β ((FermatNoβπ) gcd (FermatNoβπ)) = 1)) | |
| 19 | 18 | com12 32 | . . . 4 β’ (π β π β (π = π β ((FermatNoβπ) gcd (FermatNoβπ)) = 1)) |
| 20 | 19 | 3ad2ant3 1133 | . . 3 β’ ((π β β0 β§ π β β0 β§ π β π) β (π = π β ((FermatNoβπ) gcd (FermatNoβπ)) = 1)) |
| 21 | goldbachthlem2 46512 | . . . . 5 β’ ((π β β0 β§ π β β0 β§ π < π) β ((FermatNoβπ) gcd (FermatNoβπ)) = 1) | |
| 22 | 21 | 3expia 1119 | . . . 4 β’ ((π β β0 β§ π β β0) β (π < π β ((FermatNoβπ) gcd (FermatNoβπ)) = 1)) |
| 23 | 22 | 3adant3 1130 | . . 3 β’ ((π β β0 β§ π β β0 β§ π β π) β (π < π β ((FermatNoβπ) gcd (FermatNoβπ)) = 1)) |
| 24 | 17, 20, 23 | 3jaod 1426 | . 2 β’ ((π β β0 β§ π β β0 β§ π β π) β ((π < π β¨ π = π β¨ π < π) β ((FermatNoβπ) gcd (FermatNoβπ)) = 1)) |
| 25 | 5, 24 | mpd 15 | 1 β’ ((π β β0 β§ π β β0 β§ π β π) β ((FermatNoβπ) gcd (FermatNoβπ)) = 1) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β¨ w3o 1084 β§ w3a 1085 = wceq 1539 β wcel 2104 β wne 2938 class class class wbr 5147 βcfv 6542 (class class class)co 7411 βcr 11111 1c1 11113 < clt 11252 β0cn0 12476 β€cz 12562 gcd cgcd 16439 FermatNocfmtno 46493 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-inf2 9638 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 |
| This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 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-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 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-2o 8469 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-sup 9439 df-inf 9440 df-oi 9507 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-n0 12477 df-z 12563 df-uz 12827 df-rp 12979 df-fz 13489 df-fzo 13632 df-seq 13971 df-exp 14032 df-hash 14295 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-clim 15436 df-prod 15854 df-dvds 16202 df-gcd 16440 df-prm 16613 df-fmtno 46494 |
| This theorem is referenced by: prmdvdsfmtnof1lem2 46551 |
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