Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > m11nprm | Structured version Visualization version GIF version |
Description: The eleventh Mersenne number M11 = 2047 is not a prime number. (Contributed by AV, 18-Aug-2021.) |
Ref | Expression |
---|---|
m11nprm | ⊢ ((2↑;11) − 1) = (;89 · ;23) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2nn0 11902 | . . . . 5 ⊢ 2 ∈ ℕ0 | |
2 | 0nn0 11900 | . . . . 5 ⊢ 0 ∈ ℕ0 | |
3 | 1, 2 | deccl 12101 | . . . 4 ⊢ ;20 ∈ ℕ0 |
4 | 4nn0 11904 | . . . 4 ⊢ 4 ∈ ℕ0 | |
5 | 3, 4 | deccl 12101 | . . 3 ⊢ ;;204 ∈ ℕ0 |
6 | 8nn0 11908 | . . 3 ⊢ 8 ∈ ℕ0 | |
7 | 1nn0 11901 | . . 3 ⊢ 1 ∈ ℕ0 | |
8 | 2exp11 43642 | . . 3 ⊢ (2↑;11) = ;;;2048 | |
9 | 4p1e5 11771 | . . . 4 ⊢ (4 + 1) = 5 | |
10 | eqid 2818 | . . . 4 ⊢ ;;204 = ;;204 | |
11 | 3, 4, 9, 10 | decsuc 12117 | . . 3 ⊢ (;;204 + 1) = ;;205 |
12 | 8m1e7 11758 | . . 3 ⊢ (8 − 1) = 7 | |
13 | 5, 6, 7, 8, 11, 12 | decsubi 12149 | . 2 ⊢ ((2↑;11) − 1) = ;;;2047 |
14 | 3nn0 11903 | . . . 4 ⊢ 3 ∈ ℕ0 | |
15 | 1, 14 | deccl 12101 | . . 3 ⊢ ;23 ∈ ℕ0 |
16 | 9nn0 11909 | . . 3 ⊢ 9 ∈ ℕ0 | |
17 | eqid 2818 | . . 3 ⊢ ;89 = ;89 | |
18 | 7nn0 11907 | . . 3 ⊢ 7 ∈ ℕ0 | |
19 | eqid 2818 | . . . 4 ⊢ ;23 = ;23 | |
20 | eqid 2818 | . . . 4 ⊢ ;20 = ;20 | |
21 | 8t2e16 12201 | . . . . . 6 ⊢ (8 · 2) = ;16 | |
22 | 2p2e4 11760 | . . . . . 6 ⊢ (2 + 2) = 4 | |
23 | 21, 22 | oveq12i 7157 | . . . . 5 ⊢ ((8 · 2) + (2 + 2)) = (;16 + 4) |
24 | 6nn0 11906 | . . . . . 6 ⊢ 6 ∈ ℕ0 | |
25 | eqid 2818 | . . . . . 6 ⊢ ;16 = ;16 | |
26 | 1p1e2 11750 | . . . . . 6 ⊢ (1 + 1) = 2 | |
27 | 6p4e10 12158 | . . . . . 6 ⊢ (6 + 4) = ;10 | |
28 | 7, 24, 4, 25, 26, 27 | decaddci2 12148 | . . . . 5 ⊢ (;16 + 4) = ;20 |
29 | 23, 28 | eqtri 2841 | . . . 4 ⊢ ((8 · 2) + (2 + 2)) = ;20 |
30 | 8t3e24 12202 | . . . . . 6 ⊢ (8 · 3) = ;24 | |
31 | 30 | oveq1i 7155 | . . . . 5 ⊢ ((8 · 3) + 0) = (;24 + 0) |
32 | 1, 4 | deccl 12101 | . . . . . . 7 ⊢ ;24 ∈ ℕ0 |
33 | 32 | nn0cni 11897 | . . . . . 6 ⊢ ;24 ∈ ℂ |
34 | 33 | addid1i 10815 | . . . . 5 ⊢ (;24 + 0) = ;24 |
35 | 31, 34 | eqtri 2841 | . . . 4 ⊢ ((8 · 3) + 0) = ;24 |
36 | 1, 14, 1, 2, 19, 20, 6, 4, 1, 29, 35 | decma2c 12139 | . . 3 ⊢ ((8 · ;23) + ;20) = ;;204 |
37 | 9t2e18 12208 | . . . . 5 ⊢ (9 · 2) = ;18 | |
38 | 8p2e10 12166 | . . . . 5 ⊢ (8 + 2) = ;10 | |
39 | 7, 6, 1, 37, 26, 38 | decaddci2 12148 | . . . 4 ⊢ ((9 · 2) + 2) = ;20 |
40 | 9t3e27 12209 | . . . 4 ⊢ (9 · 3) = ;27 | |
41 | 16, 1, 14, 19, 18, 1, 39, 40 | decmul2c 12152 | . . 3 ⊢ (9 · ;23) = ;;207 |
42 | 15, 6, 16, 17, 18, 3, 36, 41 | decmul1c 12151 | . 2 ⊢ (;89 · ;23) = ;;;2047 |
43 | 13, 42 | eqtr4i 2844 | 1 ⊢ ((2↑;11) − 1) = (;89 · ;23) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 (class class class)co 7145 0cc0 10525 1c1 10526 + caddc 10528 · cmul 10530 − cmin 10858 2c2 11680 3c3 11681 4c4 11682 5c5 11683 6c6 11684 7c7 11685 8c8 11686 9c9 11687 ;cdc 12086 ↑cexp 13417 |
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 |
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-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-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 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-seq 13358 df-exp 13418 |
This theorem is referenced by: (None) |
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