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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > m7prm | Structured version Visualization version GIF version |
Description: The seventh Mersenne number M7 = 127 is a prime number. (Contributed by AV, 18-Aug-2021.) |
Ref | Expression |
---|---|
m7prm | ⊢ ((2↑7) − 1) ∈ ℙ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1nn0 12504 | . . . 4 ⊢ 1 ∈ ℕ0 | |
2 | 2nn0 12505 | . . . 4 ⊢ 2 ∈ ℕ0 | |
3 | 1, 2 | deccl 12708 | . . 3 ⊢ ;12 ∈ ℕ0 |
4 | 8nn0 12511 | . . 3 ⊢ 8 ∈ ℕ0 | |
5 | 2exp7 17042 | . . 3 ⊢ (2↑7) = ;;128 | |
6 | 2p1e3 12370 | . . . 4 ⊢ (2 + 1) = 3 | |
7 | eqid 2727 | . . . 4 ⊢ ;12 = ;12 | |
8 | 1, 2, 6, 7 | decsuc 12724 | . . 3 ⊢ (;12 + 1) = ;13 |
9 | 7p1e8 12377 | . . . 4 ⊢ (7 + 1) = 8 | |
10 | 8cn 12325 | . . . . 5 ⊢ 8 ∈ ℂ | |
11 | ax-1cn 11182 | . . . . 5 ⊢ 1 ∈ ℂ | |
12 | 7cn 12322 | . . . . 5 ⊢ 7 ∈ ℂ | |
13 | 10, 11, 12 | subadd2i 11564 | . . . 4 ⊢ ((8 − 1) = 7 ↔ (7 + 1) = 8) |
14 | 9, 13 | mpbir 230 | . . 3 ⊢ (8 − 1) = 7 |
15 | 3, 4, 1, 5, 8, 14 | decsubi 12756 | . 2 ⊢ ((2↑7) − 1) = ;;127 |
16 | 127prm 46852 | . 2 ⊢ ;;127 ∈ ℙ | |
17 | 15, 16 | eqeltri 2824 | 1 ⊢ ((2↑7) − 1) ∈ ℙ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 ∈ wcel 2099 (class class class)co 7414 1c1 11125 + caddc 11127 − cmin 11460 2c2 12283 3c3 12284 7c7 12288 8c8 12289 ;cdc 12693 ↑cexp 14044 ℙcprime 16627 |
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 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 ax-pre-sup 11202 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 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-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7863 df-1st 7985 df-2nd 7986 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-1o 8478 df-2o 8479 df-er 8716 df-en 8954 df-dom 8955 df-sdom 8956 df-fin 8957 df-sup 9451 df-inf 9452 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 df-sub 11462 df-neg 11463 df-div 11888 df-nn 12229 df-2 12291 df-3 12292 df-4 12293 df-5 12294 df-6 12295 df-7 12296 df-8 12297 df-9 12298 df-n0 12489 df-z 12575 df-dec 12694 df-uz 12839 df-rp 12993 df-fz 13503 df-seq 13985 df-exp 14045 df-cj 15064 df-re 15065 df-im 15066 df-sqrt 15200 df-abs 15201 df-dvds 16217 df-prm 16628 |
This theorem is referenced by: (None) |
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