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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 12446 | . . . 4 β’ (π β β0 β π β β) | |
| 2 | nn0re 12446 | . . . 4 β’ (π β β0 β π β β) | |
| 3 | lttri4 11263 | . . . 4 β’ ((π β β β§ π β β) β (π < π β¨ π = π β¨ π < π)) | |
| 4 | 1, 2, 3 | syl2an 596 | . . 3 β’ ((π β β0 β§ π β β0) β (π < π β¨ π = π β¨ π < π)) |
| 5 | 4 | 3adant3 1132 | . 2 β’ ((π β β0 β§ π β β0 β§ π β π) β (π < π β¨ π = π β¨ π < π)) |
| 6 | fmtnonn 45876 | . . . . . . . . . 10 β’ (π β β0 β (FermatNoβπ) β β) | |
| 7 | 6 | nnzd 12550 | . . . . . . . . 9 β’ (π β β0 β (FermatNoβπ) β β€) |
| 8 | fmtnonn 45876 | . . . . . . . . . 10 β’ (π β β0 β (FermatNoβπ) β β) | |
| 9 | 8 | nnzd 12550 | . . . . . . . . 9 β’ (π β β0 β (FermatNoβπ) β β€) |
| 10 | gcdcom 16419 | . . . . . . . . 9 β’ (((FermatNoβπ) β β€ β§ (FermatNoβπ) β β€) β ((FermatNoβπ) gcd (FermatNoβπ)) = ((FermatNoβπ) gcd (FermatNoβπ))) | |
| 11 | 7, 9, 10 | syl2anr 597 | . . . . . . . 8 β’ ((π β β0 β§ π β β0) β ((FermatNoβπ) gcd (FermatNoβπ)) = ((FermatNoβπ) gcd (FermatNoβπ))) |
| 12 | 11 | 3adant3 1132 | . . . . . . 7 β’ ((π β β0 β§ π β β0 β§ π < π) β ((FermatNoβπ) gcd (FermatNoβπ)) = ((FermatNoβπ) gcd (FermatNoβπ))) |
| 13 | goldbachthlem2 45891 | . . . . . . 7 β’ ((π β β0 β§ π β β0 β§ π < π) β ((FermatNoβπ) gcd (FermatNoβπ)) = 1) | |
| 14 | 12, 13 | eqtrd 2771 | . . . . . 6 β’ ((π β β0 β§ π β β0 β§ π < π) β ((FermatNoβπ) gcd (FermatNoβπ)) = 1) |
| 15 | 14 | 3exp 1119 | . . . . 5 β’ (π β β0 β (π β β0 β (π < π β ((FermatNoβπ) gcd (FermatNoβπ)) = 1))) |
| 16 | 15 | impcom 408 | . . . 4 β’ ((π β β0 β§ π β β0) β (π < π β ((FermatNoβπ) gcd (FermatNoβπ)) = 1)) |
| 17 | 16 | 3adant3 1132 | . . 3 β’ ((π β β0 β§ π β β0 β§ π β π) β (π < π β ((FermatNoβπ) gcd (FermatNoβπ)) = 1)) |
| 18 | eqneqall 2950 | . . . . 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 1135 | . . 3 β’ ((π β β0 β§ π β β0 β§ π β π) β (π = π β ((FermatNoβπ) gcd (FermatNoβπ)) = 1)) |
| 21 | goldbachthlem2 45891 | . . . . 5 β’ ((π β β0 β§ π β β0 β§ π < π) β ((FermatNoβπ) gcd (FermatNoβπ)) = 1) | |
| 22 | 21 | 3expia 1121 | . . . 4 β’ ((π β β0 β§ π β β0) β (π < π β ((FermatNoβπ) gcd (FermatNoβπ)) = 1)) |
| 23 | 22 | 3adant3 1132 | . . 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 1428 | . 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 1086 β§ w3a 1087 = wceq 1541 β wcel 2106 β wne 2939 class class class wbr 5125 βcfv 6516 (class class class)co 7377 βcr 11074 1c1 11076 < clt 11213 β0cn0 12437 β€cz 12523 gcd cgcd 16400 FermatNocfmtno 45872 |
| 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 2702 ax-rep 5262 ax-sep 5276 ax-nul 5283 ax-pow 5340 ax-pr 5404 ax-un 7692 ax-inf2 9601 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-pre-sup 11153 |
| 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 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3364 df-reu 3365 df-rab 3419 df-v 3461 df-sbc 3758 df-csb 3874 df-dif 3931 df-un 3933 df-in 3935 df-ss 3945 df-pss 3947 df-nul 4303 df-if 4507 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4886 df-int 4928 df-iun 4976 df-br 5126 df-opab 5188 df-mpt 5209 df-tr 5243 df-id 5551 df-eprel 5557 df-po 5565 df-so 5566 df-fr 5608 df-se 5609 df-we 5610 df-xp 5659 df-rel 5660 df-cnv 5661 df-co 5662 df-dm 5663 df-rn 5664 df-res 5665 df-ima 5666 df-pred 6273 df-ord 6340 df-on 6341 df-lim 6342 df-suc 6343 df-iota 6468 df-fun 6518 df-fn 6519 df-f 6520 df-f1 6521 df-fo 6522 df-f1o 6523 df-fv 6524 df-isom 6525 df-riota 7333 df-ov 7380 df-oprab 7381 df-mpo 7382 df-om 7823 df-1st 7941 df-2nd 7942 df-frecs 8232 df-wrecs 8263 df-recs 8337 df-rdg 8376 df-1o 8432 df-2o 8433 df-er 8670 df-en 8906 df-dom 8907 df-sdom 8908 df-fin 8909 df-sup 9402 df-inf 9403 df-oi 9470 df-card 9899 df-pnf 11215 df-mnf 11216 df-xr 11217 df-ltxr 11218 df-le 11219 df-sub 11411 df-neg 11412 df-div 11837 df-nn 12178 df-2 12240 df-3 12241 df-4 12242 df-5 12243 df-n0 12438 df-z 12524 df-uz 12788 df-rp 12940 df-fz 13450 df-fzo 13593 df-seq 13932 df-exp 13993 df-hash 14256 df-cj 15011 df-re 15012 df-im 15013 df-sqrt 15147 df-abs 15148 df-clim 15397 df-prod 15815 df-dvds 16163 df-gcd 16401 df-prm 16574 df-fmtno 45873 |
| This theorem is referenced by: prmdvdsfmtnof1lem2 45930 |
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