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Mirrors > Home > MPE Home > Th. List > Mathboxes > fmtno5nprm | Structured version Visualization version GIF version |
Description: The 5 th Fermat number is a not a prime. (Contributed by AV, 22-Jul-2021.) |
Ref | Expression |
---|---|
fmtno5nprm | ⊢ (FermatNo‘5) ∉ ℙ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 6nn0 11910 | . . . . . . . . 9 ⊢ 6 ∈ ℕ0 | |
2 | 7nn0 11911 | . . . . . . . . 9 ⊢ 7 ∈ ℕ0 | |
3 | 1, 2 | deccl 12105 | . . . . . . . 8 ⊢ ;67 ∈ ℕ0 |
4 | 0nn0 11904 | . . . . . . . 8 ⊢ 0 ∈ ℕ0 | |
5 | 3, 4 | deccl 12105 | . . . . . . 7 ⊢ ;;670 ∈ ℕ0 |
6 | 5, 4 | deccl 12105 | . . . . . 6 ⊢ ;;;6700 ∈ ℕ0 |
7 | 4nn0 11908 | . . . . . 6 ⊢ 4 ∈ ℕ0 | |
8 | 6, 7 | deccl 12105 | . . . . 5 ⊢ ;;;;67004 ∈ ℕ0 |
9 | 1nn0 11905 | . . . . 5 ⊢ 1 ∈ ℕ0 | |
10 | 8, 9 | deccl 12105 | . . . 4 ⊢ ;;;;;670041 ∈ ℕ0 |
11 | 7nn 11721 | . . . 4 ⊢ 7 ∈ ℕ | |
12 | 10, 11 | decnncl 12110 | . . 3 ⊢ ;;;;;;6700417 ∈ ℕ |
13 | 1, 7 | deccl 12105 | . . . 4 ⊢ ;64 ∈ ℕ0 |
14 | 1nn 11641 | . . . 4 ⊢ 1 ∈ ℕ | |
15 | 13, 14 | decnncl 12110 | . . 3 ⊢ ;;641 ∈ ℕ |
16 | 8, 14 | decnncl 12110 | . . . 4 ⊢ ;;;;;670041 ∈ ℕ |
17 | 1lt10 12229 | . . . 4 ⊢ 1 < ;10 | |
18 | 16, 2, 9, 17 | declti 12128 | . . 3 ⊢ 1 < ;;;;;;6700417 |
19 | 4nn 11712 | . . . . 5 ⊢ 4 ∈ ℕ | |
20 | 1, 19 | decnncl 12110 | . . . 4 ⊢ ;64 ∈ ℕ |
21 | 20, 9, 9, 17 | declti 12128 | . . 3 ⊢ 1 < ;;641 |
22 | fmtno5fac 43735 | . . . 4 ⊢ (FermatNo‘5) = (;;;;;;6700417 · ;;641) | |
23 | 22 | eqcomi 2828 | . . 3 ⊢ (;;;;;;6700417 · ;;641) = (FermatNo‘5) |
24 | 12, 15, 18, 21, 23 | nprmi 16025 | . 2 ⊢ ¬ (FermatNo‘5) ∈ ℙ |
25 | 24 | nelir 3124 | 1 ⊢ (FermatNo‘5) ∉ ℙ |
Colors of variables: wff setvar class |
Syntax hints: ∉ wnel 3121 ‘cfv 6348 (class class class)co 7148 0cc0 10529 1c1 10530 · cmul 10534 4c4 11686 5c5 11687 6c6 11688 7c7 11689 ;cdc 12090 ℙcprime 16007 FermatNocfmtno 43680 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-cnex 10585 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-pre-mulgt0 10606 ax-pre-sup 10607 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rmo 3144 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-om 7573 df-2nd 7682 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-1o 8094 df-2o 8095 df-er 8281 df-en 8502 df-dom 8503 df-sdom 8504 df-fin 8505 df-sup 8898 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-div 11290 df-nn 11631 df-2 11692 df-3 11693 df-4 11694 df-5 11695 df-6 11696 df-7 11697 df-8 11698 df-9 11699 df-n0 11890 df-z 11974 df-dec 12091 df-uz 12236 df-rp 12382 df-seq 13362 df-exp 13422 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-dvds 15600 df-prm 16008 df-fmtno 43681 |
This theorem is referenced by: (None) |
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