Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
||
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 | ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0 ∧ 𝑁 ≠ 𝑀) → ((FermatNo‘𝑁) gcd (FermatNo‘𝑀)) = 1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0re 12225 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ) | |
2 | nn0re 12225 | . . . 4 ⊢ (𝑀 ∈ ℕ0 → 𝑀 ∈ ℝ) | |
3 | lttri4 11043 | . . . 4 ⊢ ((𝑁 ∈ ℝ ∧ 𝑀 ∈ ℝ) → (𝑁 < 𝑀 ∨ 𝑁 = 𝑀 ∨ 𝑀 < 𝑁)) | |
4 | 1, 2, 3 | syl2an 595 | . . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0) → (𝑁 < 𝑀 ∨ 𝑁 = 𝑀 ∨ 𝑀 < 𝑁)) |
5 | 4 | 3adant3 1130 | . 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0 ∧ 𝑁 ≠ 𝑀) → (𝑁 < 𝑀 ∨ 𝑁 = 𝑀 ∨ 𝑀 < 𝑁)) |
6 | fmtnonn 44935 | . . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ0 → (FermatNo‘𝑁) ∈ ℕ) | |
7 | 6 | nnzd 12407 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ0 → (FermatNo‘𝑁) ∈ ℤ) |
8 | fmtnonn 44935 | . . . . . . . . . 10 ⊢ (𝑀 ∈ ℕ0 → (FermatNo‘𝑀) ∈ ℕ) | |
9 | 8 | nnzd 12407 | . . . . . . . . 9 ⊢ (𝑀 ∈ ℕ0 → (FermatNo‘𝑀) ∈ ℤ) |
10 | gcdcom 16201 | . . . . . . . . 9 ⊢ (((FermatNo‘𝑁) ∈ ℤ ∧ (FermatNo‘𝑀) ∈ ℤ) → ((FermatNo‘𝑁) gcd (FermatNo‘𝑀)) = ((FermatNo‘𝑀) gcd (FermatNo‘𝑁))) | |
11 | 7, 9, 10 | syl2anr 596 | . . . . . . . 8 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ((FermatNo‘𝑁) gcd (FermatNo‘𝑀)) = ((FermatNo‘𝑀) gcd (FermatNo‘𝑁))) |
12 | 11 | 3adant3 1130 | . . . . . . 7 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑁 < 𝑀) → ((FermatNo‘𝑁) gcd (FermatNo‘𝑀)) = ((FermatNo‘𝑀) gcd (FermatNo‘𝑁))) |
13 | goldbachthlem2 44950 | . . . . . . 7 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑁 < 𝑀) → ((FermatNo‘𝑀) gcd (FermatNo‘𝑁)) = 1) | |
14 | 12, 13 | eqtrd 2779 | . . . . . 6 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑁 < 𝑀) → ((FermatNo‘𝑁) gcd (FermatNo‘𝑀)) = 1) |
15 | 14 | 3exp 1117 | . . . . 5 ⊢ (𝑀 ∈ ℕ0 → (𝑁 ∈ ℕ0 → (𝑁 < 𝑀 → ((FermatNo‘𝑁) gcd (FermatNo‘𝑀)) = 1))) |
16 | 15 | impcom 407 | . . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0) → (𝑁 < 𝑀 → ((FermatNo‘𝑁) gcd (FermatNo‘𝑀)) = 1)) |
17 | 16 | 3adant3 1130 | . . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0 ∧ 𝑁 ≠ 𝑀) → (𝑁 < 𝑀 → ((FermatNo‘𝑁) gcd (FermatNo‘𝑀)) = 1)) |
18 | eqneqall 2955 | . . . . 5 ⊢ (𝑁 = 𝑀 → (𝑁 ≠ 𝑀 → ((FermatNo‘𝑁) gcd (FermatNo‘𝑀)) = 1)) | |
19 | 18 | com12 32 | . . . 4 ⊢ (𝑁 ≠ 𝑀 → (𝑁 = 𝑀 → ((FermatNo‘𝑁) gcd (FermatNo‘𝑀)) = 1)) |
20 | 19 | 3ad2ant3 1133 | . . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0 ∧ 𝑁 ≠ 𝑀) → (𝑁 = 𝑀 → ((FermatNo‘𝑁) gcd (FermatNo‘𝑀)) = 1)) |
21 | goldbachthlem2 44950 | . . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0 ∧ 𝑀 < 𝑁) → ((FermatNo‘𝑁) gcd (FermatNo‘𝑀)) = 1) | |
22 | 21 | 3expia 1119 | . . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0) → (𝑀 < 𝑁 → ((FermatNo‘𝑁) gcd (FermatNo‘𝑀)) = 1)) |
23 | 22 | 3adant3 1130 | . . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0 ∧ 𝑁 ≠ 𝑀) → (𝑀 < 𝑁 → ((FermatNo‘𝑁) gcd (FermatNo‘𝑀)) = 1)) |
24 | 17, 20, 23 | 3jaod 1426 | . 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0 ∧ 𝑁 ≠ 𝑀) → ((𝑁 < 𝑀 ∨ 𝑁 = 𝑀 ∨ 𝑀 < 𝑁) → ((FermatNo‘𝑁) gcd (FermatNo‘𝑀)) = 1)) |
25 | 5, 24 | mpd 15 | 1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0 ∧ 𝑁 ≠ 𝑀) → ((FermatNo‘𝑁) gcd (FermatNo‘𝑀)) = 1) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ w3o 1084 ∧ w3a 1085 = wceq 1541 ∈ wcel 2109 ≠ wne 2944 class class class wbr 5078 ‘cfv 6430 (class class class)co 7268 ℝcr 10854 1c1 10856 < clt 10993 ℕ0cn0 12216 ℤcz 12302 gcd cgcd 16182 FermatNocfmtno 44931 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-rep 5213 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7579 ax-inf2 9360 ax-cnex 10911 ax-resscn 10912 ax-1cn 10913 ax-icn 10914 ax-addcl 10915 ax-addrcl 10916 ax-mulcl 10917 ax-mulrcl 10918 ax-mulcom 10919 ax-addass 10920 ax-mulass 10921 ax-distr 10922 ax-i2m1 10923 ax-1ne0 10924 ax-1rid 10925 ax-rnegex 10926 ax-rrecex 10927 ax-cnre 10928 ax-pre-lttri 10929 ax-pre-lttrn 10930 ax-pre-ltadd 10931 ax-pre-mulgt0 10932 ax-pre-sup 10933 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3070 df-rex 3071 df-reu 3072 df-rmo 3073 df-rab 3074 df-v 3432 df-sbc 3720 df-csb 3837 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-pss 3910 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4845 df-int 4885 df-iun 4931 df-br 5079 df-opab 5141 df-mpt 5162 df-tr 5196 df-id 5488 df-eprel 5494 df-po 5502 df-so 5503 df-fr 5543 df-se 5544 df-we 5545 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-pred 6199 df-ord 6266 df-on 6267 df-lim 6268 df-suc 6269 df-iota 6388 df-fun 6432 df-fn 6433 df-f 6434 df-f1 6435 df-fo 6436 df-f1o 6437 df-fv 6438 df-isom 6439 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-om 7701 df-1st 7817 df-2nd 7818 df-frecs 8081 df-wrecs 8112 df-recs 8186 df-rdg 8225 df-1o 8281 df-2o 8282 df-er 8472 df-en 8708 df-dom 8709 df-sdom 8710 df-fin 8711 df-sup 9162 df-inf 9163 df-oi 9230 df-card 9681 df-pnf 10995 df-mnf 10996 df-xr 10997 df-ltxr 10998 df-le 10999 df-sub 11190 df-neg 11191 df-div 11616 df-nn 11957 df-2 12019 df-3 12020 df-4 12021 df-5 12022 df-n0 12217 df-z 12303 df-uz 12565 df-rp 12713 df-fz 13222 df-fzo 13365 df-seq 13703 df-exp 13764 df-hash 14026 df-cj 14791 df-re 14792 df-im 14793 df-sqrt 14927 df-abs 14928 df-clim 15178 df-prod 15597 df-dvds 15945 df-gcd 16183 df-prm 16358 df-fmtno 44932 |
This theorem is referenced by: prmdvdsfmtnof1lem2 44989 |
Copyright terms: Public domain | W3C validator |