Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
||
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 11993 | . . . . 5 ⊢ 2 ∈ ℕ0 | |
2 | 0nn0 11991 | . . . . 5 ⊢ 0 ∈ ℕ0 | |
3 | 1, 2 | deccl 12194 | . . . 4 ⊢ ;20 ∈ ℕ0 |
4 | 4nn0 11995 | . . . 4 ⊢ 4 ∈ ℕ0 | |
5 | 3, 4 | deccl 12194 | . . 3 ⊢ ;;204 ∈ ℕ0 |
6 | 8nn0 11999 | . . 3 ⊢ 8 ∈ ℕ0 | |
7 | 1nn0 11992 | . . 3 ⊢ 1 ∈ ℕ0 | |
8 | 2exp11 16526 | . . 3 ⊢ (2↑;11) = ;;;2048 | |
9 | 4p1e5 11862 | . . . 4 ⊢ (4 + 1) = 5 | |
10 | eqid 2738 | . . . 4 ⊢ ;;204 = ;;204 | |
11 | 3, 4, 9, 10 | decsuc 12210 | . . 3 ⊢ (;;204 + 1) = ;;205 |
12 | 8m1e7 11849 | . . 3 ⊢ (8 − 1) = 7 | |
13 | 5, 6, 7, 8, 11, 12 | decsubi 12242 | . 2 ⊢ ((2↑;11) − 1) = ;;;2047 |
14 | 3nn0 11994 | . . . 4 ⊢ 3 ∈ ℕ0 | |
15 | 1, 14 | deccl 12194 | . . 3 ⊢ ;23 ∈ ℕ0 |
16 | 9nn0 12000 | . . 3 ⊢ 9 ∈ ℕ0 | |
17 | eqid 2738 | . . 3 ⊢ ;89 = ;89 | |
18 | 7nn0 11998 | . . 3 ⊢ 7 ∈ ℕ0 | |
19 | eqid 2738 | . . . 4 ⊢ ;23 = ;23 | |
20 | eqid 2738 | . . . 4 ⊢ ;20 = ;20 | |
21 | 8t2e16 12294 | . . . . . 6 ⊢ (8 · 2) = ;16 | |
22 | 2p2e4 11851 | . . . . . 6 ⊢ (2 + 2) = 4 | |
23 | 21, 22 | oveq12i 7182 | . . . . 5 ⊢ ((8 · 2) + (2 + 2)) = (;16 + 4) |
24 | 6nn0 11997 | . . . . . 6 ⊢ 6 ∈ ℕ0 | |
25 | eqid 2738 | . . . . . 6 ⊢ ;16 = ;16 | |
26 | 1p1e2 11841 | . . . . . 6 ⊢ (1 + 1) = 2 | |
27 | 6p4e10 12251 | . . . . . 6 ⊢ (6 + 4) = ;10 | |
28 | 7, 24, 4, 25, 26, 27 | decaddci2 12241 | . . . . 5 ⊢ (;16 + 4) = ;20 |
29 | 23, 28 | eqtri 2761 | . . . 4 ⊢ ((8 · 2) + (2 + 2)) = ;20 |
30 | 8t3e24 12295 | . . . . . 6 ⊢ (8 · 3) = ;24 | |
31 | 30 | oveq1i 7180 | . . . . 5 ⊢ ((8 · 3) + 0) = (;24 + 0) |
32 | 1, 4 | deccl 12194 | . . . . . . 7 ⊢ ;24 ∈ ℕ0 |
33 | 32 | nn0cni 11988 | . . . . . 6 ⊢ ;24 ∈ ℂ |
34 | 33 | addid1i 10905 | . . . . 5 ⊢ (;24 + 0) = ;24 |
35 | 31, 34 | eqtri 2761 | . . . 4 ⊢ ((8 · 3) + 0) = ;24 |
36 | 1, 14, 1, 2, 19, 20, 6, 4, 1, 29, 35 | decma2c 12232 | . . 3 ⊢ ((8 · ;23) + ;20) = ;;204 |
37 | 9t2e18 12301 | . . . . 5 ⊢ (9 · 2) = ;18 | |
38 | 8p2e10 12259 | . . . . 5 ⊢ (8 + 2) = ;10 | |
39 | 7, 6, 1, 37, 26, 38 | decaddci2 12241 | . . . 4 ⊢ ((9 · 2) + 2) = ;20 |
40 | 9t3e27 12302 | . . . 4 ⊢ (9 · 3) = ;27 | |
41 | 16, 1, 14, 19, 18, 1, 39, 40 | decmul2c 12245 | . . 3 ⊢ (9 · ;23) = ;;207 |
42 | 15, 6, 16, 17, 18, 3, 36, 41 | decmul1c 12244 | . 2 ⊢ (;89 · ;23) = ;;;2047 |
43 | 13, 42 | eqtr4i 2764 | 1 ⊢ ((2↑;11) − 1) = (;89 · ;23) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 (class class class)co 7170 0cc0 10615 1c1 10616 + caddc 10618 · cmul 10620 − cmin 10948 2c2 11771 3c3 11772 4c4 11773 5c5 11774 6c6 11775 7c7 11776 8c8 11777 9c9 11778 ;cdc 12179 ↑cexp 13521 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-cnex 10671 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-om 7600 df-2nd 7715 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-er 8320 df-en 8556 df-dom 8557 df-sdom 8558 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-nn 11717 df-2 11779 df-3 11780 df-4 11781 df-5 11782 df-6 11783 df-7 11784 df-8 11785 df-9 11786 df-n0 11977 df-z 12063 df-dec 12180 df-uz 12325 df-seq 13461 df-exp 13522 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |