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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 12437 | . . . . 5 ⊢ 2 ∈ ℕ0 | |
2 | 0nn0 12435 | . . . . 5 ⊢ 0 ∈ ℕ0 | |
3 | 1, 2 | deccl 12640 | . . . 4 ⊢ ;20 ∈ ℕ0 |
4 | 4nn0 12439 | . . . 4 ⊢ 4 ∈ ℕ0 | |
5 | 3, 4 | deccl 12640 | . . 3 ⊢ ;;204 ∈ ℕ0 |
6 | 8nn0 12443 | . . 3 ⊢ 8 ∈ ℕ0 | |
7 | 1nn0 12436 | . . 3 ⊢ 1 ∈ ℕ0 | |
8 | 2exp11 16969 | . . 3 ⊢ (2↑;11) = ;;;2048 | |
9 | 4p1e5 12306 | . . . 4 ⊢ (4 + 1) = 5 | |
10 | eqid 2737 | . . . 4 ⊢ ;;204 = ;;204 | |
11 | 3, 4, 9, 10 | decsuc 12656 | . . 3 ⊢ (;;204 + 1) = ;;205 |
12 | 8m1e7 12293 | . . 3 ⊢ (8 − 1) = 7 | |
13 | 5, 6, 7, 8, 11, 12 | decsubi 12688 | . 2 ⊢ ((2↑;11) − 1) = ;;;2047 |
14 | 3nn0 12438 | . . . 4 ⊢ 3 ∈ ℕ0 | |
15 | 1, 14 | deccl 12640 | . . 3 ⊢ ;23 ∈ ℕ0 |
16 | 9nn0 12444 | . . 3 ⊢ 9 ∈ ℕ0 | |
17 | eqid 2737 | . . 3 ⊢ ;89 = ;89 | |
18 | 7nn0 12442 | . . 3 ⊢ 7 ∈ ℕ0 | |
19 | eqid 2737 | . . . 4 ⊢ ;23 = ;23 | |
20 | eqid 2737 | . . . 4 ⊢ ;20 = ;20 | |
21 | 8t2e16 12740 | . . . . . 6 ⊢ (8 · 2) = ;16 | |
22 | 2p2e4 12295 | . . . . . 6 ⊢ (2 + 2) = 4 | |
23 | 21, 22 | oveq12i 7374 | . . . . 5 ⊢ ((8 · 2) + (2 + 2)) = (;16 + 4) |
24 | 6nn0 12441 | . . . . . 6 ⊢ 6 ∈ ℕ0 | |
25 | eqid 2737 | . . . . . 6 ⊢ ;16 = ;16 | |
26 | 1p1e2 12285 | . . . . . 6 ⊢ (1 + 1) = 2 | |
27 | 6p4e10 12697 | . . . . . 6 ⊢ (6 + 4) = ;10 | |
28 | 7, 24, 4, 25, 26, 27 | decaddci2 12687 | . . . . 5 ⊢ (;16 + 4) = ;20 |
29 | 23, 28 | eqtri 2765 | . . . 4 ⊢ ((8 · 2) + (2 + 2)) = ;20 |
30 | 8t3e24 12741 | . . . . . 6 ⊢ (8 · 3) = ;24 | |
31 | 30 | oveq1i 7372 | . . . . 5 ⊢ ((8 · 3) + 0) = (;24 + 0) |
32 | 1, 4 | deccl 12640 | . . . . . . 7 ⊢ ;24 ∈ ℕ0 |
33 | 32 | nn0cni 12432 | . . . . . 6 ⊢ ;24 ∈ ℂ |
34 | 33 | addid1i 11349 | . . . . 5 ⊢ (;24 + 0) = ;24 |
35 | 31, 34 | eqtri 2765 | . . . 4 ⊢ ((8 · 3) + 0) = ;24 |
36 | 1, 14, 1, 2, 19, 20, 6, 4, 1, 29, 35 | decma2c 12678 | . . 3 ⊢ ((8 · ;23) + ;20) = ;;204 |
37 | 9t2e18 12747 | . . . . 5 ⊢ (9 · 2) = ;18 | |
38 | 8p2e10 12705 | . . . . 5 ⊢ (8 + 2) = ;10 | |
39 | 7, 6, 1, 37, 26, 38 | decaddci2 12687 | . . . 4 ⊢ ((9 · 2) + 2) = ;20 |
40 | 9t3e27 12748 | . . . 4 ⊢ (9 · 3) = ;27 | |
41 | 16, 1, 14, 19, 18, 1, 39, 40 | decmul2c 12691 | . . 3 ⊢ (9 · ;23) = ;;207 |
42 | 15, 6, 16, 17, 18, 3, 36, 41 | decmul1c 12690 | . 2 ⊢ (;89 · ;23) = ;;;2047 |
43 | 13, 42 | eqtr4i 2768 | 1 ⊢ ((2↑;11) − 1) = (;89 · ;23) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 (class class class)co 7362 0cc0 11058 1c1 11059 + caddc 11061 · cmul 11063 − cmin 11392 2c2 12215 3c3 12216 4c4 12217 5c5 12218 6c6 12219 7c7 12220 8c8 12221 9c9 12222 ;cdc 12625 ↑cexp 13974 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-nn 12161 df-2 12223 df-3 12224 df-4 12225 df-5 12226 df-6 12227 df-7 12228 df-8 12229 df-9 12230 df-n0 12421 df-z 12507 df-dec 12626 df-uz 12771 df-seq 13914 df-exp 13975 |
This theorem is referenced by: (None) |
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