Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
||
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 11901 | . . . 4 ⊢ 1 ∈ ℕ0 | |
2 | 2nn0 11902 | . . . 4 ⊢ 2 ∈ ℕ0 | |
3 | 1, 2 | deccl 12101 | . . 3 ⊢ ;12 ∈ ℕ0 |
4 | 8nn0 11908 | . . 3 ⊢ 8 ∈ ℕ0 | |
5 | 2exp7 43639 | . . 3 ⊢ (2↑7) = ;;128 | |
6 | 2p1e3 11767 | . . . 4 ⊢ (2 + 1) = 3 | |
7 | eqid 2818 | . . . 4 ⊢ ;12 = ;12 | |
8 | 1, 2, 6, 7 | decsuc 12117 | . . 3 ⊢ (;12 + 1) = ;13 |
9 | 7p1e8 11774 | . . . 4 ⊢ (7 + 1) = 8 | |
10 | 8cn 11722 | . . . . 5 ⊢ 8 ∈ ℂ | |
11 | ax-1cn 10583 | . . . . 5 ⊢ 1 ∈ ℂ | |
12 | 7cn 11719 | . . . . 5 ⊢ 7 ∈ ℂ | |
13 | 10, 11, 12 | subadd2i 10962 | . . . 4 ⊢ ((8 − 1) = 7 ↔ (7 + 1) = 8) |
14 | 9, 13 | mpbir 232 | . . 3 ⊢ (8 − 1) = 7 |
15 | 3, 4, 1, 5, 8, 14 | decsubi 12149 | . 2 ⊢ ((2↑7) − 1) = ;;127 |
16 | 127prm 43640 | . 2 ⊢ ;;127 ∈ ℙ | |
17 | 15, 16 | eqeltri 2906 | 1 ⊢ ((2↑7) − 1) ∈ ℙ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 ∈ wcel 2105 (class class class)co 7145 1c1 10526 + caddc 10528 − cmin 10858 2c2 11680 3c3 11681 7c7 11685 8c8 11686 ;cdc 12086 ↑cexp 13417 ℙcprime 16003 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 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 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-2o 8092 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-sup 8894 df-inf 8895 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-rp 12378 df-fz 12881 df-seq 13358 df-exp 13418 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-dvds 15596 df-prm 16004 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |