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Mathbox for Alexander van der Vekens |
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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 12497 | . . . . . . . . 9 ⊢ 6 ∈ ℕ0 | |
2 | 7nn0 12498 | . . . . . . . . 9 ⊢ 7 ∈ ℕ0 | |
3 | 1, 2 | deccl 12696 | . . . . . . . 8 ⊢ ;67 ∈ ℕ0 |
4 | 0nn0 12491 | . . . . . . . 8 ⊢ 0 ∈ ℕ0 | |
5 | 3, 4 | deccl 12696 | . . . . . . 7 ⊢ ;;670 ∈ ℕ0 |
6 | 5, 4 | deccl 12696 | . . . . . 6 ⊢ ;;;6700 ∈ ℕ0 |
7 | 4nn0 12495 | . . . . . 6 ⊢ 4 ∈ ℕ0 | |
8 | 6, 7 | deccl 12696 | . . . . 5 ⊢ ;;;;67004 ∈ ℕ0 |
9 | 1nn0 12492 | . . . . 5 ⊢ 1 ∈ ℕ0 | |
10 | 8, 9 | deccl 12696 | . . . 4 ⊢ ;;;;;670041 ∈ ℕ0 |
11 | 7nn 12308 | . . . 4 ⊢ 7 ∈ ℕ | |
12 | 10, 11 | decnncl 12701 | . . 3 ⊢ ;;;;;;6700417 ∈ ℕ |
13 | 1, 7 | deccl 12696 | . . . 4 ⊢ ;64 ∈ ℕ0 |
14 | 1nn 12227 | . . . 4 ⊢ 1 ∈ ℕ | |
15 | 13, 14 | decnncl 12701 | . . 3 ⊢ ;;641 ∈ ℕ |
16 | 8, 14 | decnncl 12701 | . . . 4 ⊢ ;;;;;670041 ∈ ℕ |
17 | 1lt10 12820 | . . . 4 ⊢ 1 < ;10 | |
18 | 16, 2, 9, 17 | declti 12719 | . . 3 ⊢ 1 < ;;;;;;6700417 |
19 | 4nn 12299 | . . . . 5 ⊢ 4 ∈ ℕ | |
20 | 1, 19 | decnncl 12701 | . . . 4 ⊢ ;64 ∈ ℕ |
21 | 20, 9, 9, 17 | declti 12719 | . . 3 ⊢ 1 < ;;641 |
22 | fmtno5fac 46822 | . . . 4 ⊢ (FermatNo‘5) = (;;;;;;6700417 · ;;641) | |
23 | 22 | eqcomi 2735 | . . 3 ⊢ (;;;;;;6700417 · ;;641) = (FermatNo‘5) |
24 | 12, 15, 18, 21, 23 | nprmi 16633 | . 2 ⊢ ¬ (FermatNo‘5) ∈ ℙ |
25 | 24 | nelir 3043 | 1 ⊢ (FermatNo‘5) ∉ ℙ |
Colors of variables: wff setvar class |
Syntax hints: ∉ wnel 3040 ‘cfv 6537 (class class class)co 7405 0cc0 11112 1c1 11113 · cmul 11117 4c4 12273 5c5 12274 6c6 12275 7c7 12276 ;cdc 12681 ℙcprime 16615 FermatNocfmtno 46767 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 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 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-2o 8468 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-sup 9439 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-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-rp 12981 df-seq 13973 df-exp 14033 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-dvds 16205 df-prm 16616 df-fmtno 46768 |
This theorem is referenced by: (None) |
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