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 12250 | . . . . 5 ⊢ 2 ∈ ℕ0 | |
2 | 0nn0 12248 | . . . . 5 ⊢ 0 ∈ ℕ0 | |
3 | 1, 2 | deccl 12452 | . . . 4 ⊢ ;20 ∈ ℕ0 |
4 | 4nn0 12252 | . . . 4 ⊢ 4 ∈ ℕ0 | |
5 | 3, 4 | deccl 12452 | . . 3 ⊢ ;;204 ∈ ℕ0 |
6 | 8nn0 12256 | . . 3 ⊢ 8 ∈ ℕ0 | |
7 | 1nn0 12249 | . . 3 ⊢ 1 ∈ ℕ0 | |
8 | 2exp11 16791 | . . 3 ⊢ (2↑;11) = ;;;2048 | |
9 | 4p1e5 12119 | . . . 4 ⊢ (4 + 1) = 5 | |
10 | eqid 2738 | . . . 4 ⊢ ;;204 = ;;204 | |
11 | 3, 4, 9, 10 | decsuc 12468 | . . 3 ⊢ (;;204 + 1) = ;;205 |
12 | 8m1e7 12106 | . . 3 ⊢ (8 − 1) = 7 | |
13 | 5, 6, 7, 8, 11, 12 | decsubi 12500 | . 2 ⊢ ((2↑;11) − 1) = ;;;2047 |
14 | 3nn0 12251 | . . . 4 ⊢ 3 ∈ ℕ0 | |
15 | 1, 14 | deccl 12452 | . . 3 ⊢ ;23 ∈ ℕ0 |
16 | 9nn0 12257 | . . 3 ⊢ 9 ∈ ℕ0 | |
17 | eqid 2738 | . . 3 ⊢ ;89 = ;89 | |
18 | 7nn0 12255 | . . 3 ⊢ 7 ∈ ℕ0 | |
19 | eqid 2738 | . . . 4 ⊢ ;23 = ;23 | |
20 | eqid 2738 | . . . 4 ⊢ ;20 = ;20 | |
21 | 8t2e16 12552 | . . . . . 6 ⊢ (8 · 2) = ;16 | |
22 | 2p2e4 12108 | . . . . . 6 ⊢ (2 + 2) = 4 | |
23 | 21, 22 | oveq12i 7287 | . . . . 5 ⊢ ((8 · 2) + (2 + 2)) = (;16 + 4) |
24 | 6nn0 12254 | . . . . . 6 ⊢ 6 ∈ ℕ0 | |
25 | eqid 2738 | . . . . . 6 ⊢ ;16 = ;16 | |
26 | 1p1e2 12098 | . . . . . 6 ⊢ (1 + 1) = 2 | |
27 | 6p4e10 12509 | . . . . . 6 ⊢ (6 + 4) = ;10 | |
28 | 7, 24, 4, 25, 26, 27 | decaddci2 12499 | . . . . 5 ⊢ (;16 + 4) = ;20 |
29 | 23, 28 | eqtri 2766 | . . . 4 ⊢ ((8 · 2) + (2 + 2)) = ;20 |
30 | 8t3e24 12553 | . . . . . 6 ⊢ (8 · 3) = ;24 | |
31 | 30 | oveq1i 7285 | . . . . 5 ⊢ ((8 · 3) + 0) = (;24 + 0) |
32 | 1, 4 | deccl 12452 | . . . . . . 7 ⊢ ;24 ∈ ℕ0 |
33 | 32 | nn0cni 12245 | . . . . . 6 ⊢ ;24 ∈ ℂ |
34 | 33 | addid1i 11162 | . . . . 5 ⊢ (;24 + 0) = ;24 |
35 | 31, 34 | eqtri 2766 | . . . 4 ⊢ ((8 · 3) + 0) = ;24 |
36 | 1, 14, 1, 2, 19, 20, 6, 4, 1, 29, 35 | decma2c 12490 | . . 3 ⊢ ((8 · ;23) + ;20) = ;;204 |
37 | 9t2e18 12559 | . . . . 5 ⊢ (9 · 2) = ;18 | |
38 | 8p2e10 12517 | . . . . 5 ⊢ (8 + 2) = ;10 | |
39 | 7, 6, 1, 37, 26, 38 | decaddci2 12499 | . . . 4 ⊢ ((9 · 2) + 2) = ;20 |
40 | 9t3e27 12560 | . . . 4 ⊢ (9 · 3) = ;27 | |
41 | 16, 1, 14, 19, 18, 1, 39, 40 | decmul2c 12503 | . . 3 ⊢ (9 · ;23) = ;;207 |
42 | 15, 6, 16, 17, 18, 3, 36, 41 | decmul1c 12502 | . 2 ⊢ (;89 · ;23) = ;;;2047 |
43 | 13, 42 | eqtr4i 2769 | 1 ⊢ ((2↑;11) − 1) = (;89 · ;23) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1539 (class class class)co 7275 0cc0 10871 1c1 10872 + caddc 10874 · cmul 10876 − cmin 11205 2c2 12028 3c3 12029 4c4 12030 5c5 12031 6c6 12032 7c7 12033 8c8 12034 9c9 12035 ;cdc 12437 ↑cexp 13782 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-seq 13722 df-exp 13783 |
This theorem is referenced by: (None) |
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