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Mirrors > Home > MPE Home > Th. List > Mathboxes > 41prothprm | Structured version Visualization version GIF version |
Description: 41 is a Proth prime. (Contributed by AV, 5-Jul-2020.) |
Ref | Expression |
---|---|
41prothprm.p | ⊢ 𝑃 = ;41 |
Ref | Expression |
---|---|
41prothprm | ⊢ (𝑃 = ((5 · (2↑3)) + 1) ∧ 𝑃 ∈ ℙ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 41prothprm.p | . . 3 ⊢ 𝑃 = ;41 | |
2 | 1 | 41prothprmlem2 41535 | . 2 ⊢ ((3↑((𝑃 − 1) / 2)) mod 𝑃) = (-1 mod 𝑃) |
3 | dfdec10 11497 | . . 3 ⊢ ;41 = ((;10 · 4) + 1) | |
4 | 4t2e8 11181 | . . . . . . . 8 ⊢ (4 · 2) = 8 | |
5 | 4cn 11098 | . . . . . . . . 9 ⊢ 4 ∈ ℂ | |
6 | 2cn 11091 | . . . . . . . . 9 ⊢ 2 ∈ ℂ | |
7 | 5, 6 | mulcomi 10046 | . . . . . . . 8 ⊢ (4 · 2) = (2 · 4) |
8 | 4, 7 | eqtr3i 2646 | . . . . . . 7 ⊢ 8 = (2 · 4) |
9 | 8 | oveq2i 6661 | . . . . . 6 ⊢ (5 · 8) = (5 · (2 · 4)) |
10 | 5cn 11100 | . . . . . . 7 ⊢ 5 ∈ ℂ | |
11 | 10, 6, 5 | mulassi 10049 | . . . . . 6 ⊢ ((5 · 2) · 4) = (5 · (2 · 4)) |
12 | 5t2e10 11634 | . . . . . . 7 ⊢ (5 · 2) = ;10 | |
13 | 12 | oveq1i 6660 | . . . . . 6 ⊢ ((5 · 2) · 4) = (;10 · 4) |
14 | 9, 11, 13 | 3eqtr2i 2650 | . . . . 5 ⊢ (5 · 8) = (;10 · 4) |
15 | cu2 12963 | . . . . . . 7 ⊢ (2↑3) = 8 | |
16 | 15 | eqcomi 2631 | . . . . . 6 ⊢ 8 = (2↑3) |
17 | 16 | oveq2i 6661 | . . . . 5 ⊢ (5 · 8) = (5 · (2↑3)) |
18 | 14, 17 | eqtr3i 2646 | . . . 4 ⊢ (;10 · 4) = (5 · (2↑3)) |
19 | 18 | oveq1i 6660 | . . 3 ⊢ ((;10 · 4) + 1) = ((5 · (2↑3)) + 1) |
20 | 1, 3, 19 | 3eqtri 2648 | . 2 ⊢ 𝑃 = ((5 · (2↑3)) + 1) |
21 | simpr 477 | . . 3 ⊢ ((((3↑((𝑃 − 1) / 2)) mod 𝑃) = (-1 mod 𝑃) ∧ 𝑃 = ((5 · (2↑3)) + 1)) → 𝑃 = ((5 · (2↑3)) + 1)) | |
22 | 3nn 11186 | . . . . 5 ⊢ 3 ∈ ℕ | |
23 | 22 | a1i 11 | . . . 4 ⊢ ((((3↑((𝑃 − 1) / 2)) mod 𝑃) = (-1 mod 𝑃) ∧ 𝑃 = ((5 · (2↑3)) + 1)) → 3 ∈ ℕ) |
24 | 5nn 11188 | . . . . 5 ⊢ 5 ∈ ℕ | |
25 | 24 | a1i 11 | . . . 4 ⊢ ((((3↑((𝑃 − 1) / 2)) mod 𝑃) = (-1 mod 𝑃) ∧ 𝑃 = ((5 · (2↑3)) + 1)) → 5 ∈ ℕ) |
26 | 5lt8 11217 | . . . . . 6 ⊢ 5 < 8 | |
27 | 26, 15 | breqtrri 4680 | . . . . 5 ⊢ 5 < (2↑3) |
28 | 27 | a1i 11 | . . . 4 ⊢ ((((3↑((𝑃 − 1) / 2)) mod 𝑃) = (-1 mod 𝑃) ∧ 𝑃 = ((5 · (2↑3)) + 1)) → 5 < (2↑3)) |
29 | 3z 11410 | . . . . . . 7 ⊢ 3 ∈ ℤ | |
30 | 29 | a1i 11 | . . . . . 6 ⊢ (((3↑((𝑃 − 1) / 2)) mod 𝑃) = (-1 mod 𝑃) → 3 ∈ ℤ) |
31 | oveq1 6657 | . . . . . . . . 9 ⊢ (𝑥 = 3 → (𝑥↑((𝑃 − 1) / 2)) = (3↑((𝑃 − 1) / 2))) | |
32 | 31 | oveq1d 6665 | . . . . . . . 8 ⊢ (𝑥 = 3 → ((𝑥↑((𝑃 − 1) / 2)) mod 𝑃) = ((3↑((𝑃 − 1) / 2)) mod 𝑃)) |
33 | 32 | eqeq1d 2624 | . . . . . . 7 ⊢ (𝑥 = 3 → (((𝑥↑((𝑃 − 1) / 2)) mod 𝑃) = (-1 mod 𝑃) ↔ ((3↑((𝑃 − 1) / 2)) mod 𝑃) = (-1 mod 𝑃))) |
34 | 33 | adantl 482 | . . . . . 6 ⊢ ((((3↑((𝑃 − 1) / 2)) mod 𝑃) = (-1 mod 𝑃) ∧ 𝑥 = 3) → (((𝑥↑((𝑃 − 1) / 2)) mod 𝑃) = (-1 mod 𝑃) ↔ ((3↑((𝑃 − 1) / 2)) mod 𝑃) = (-1 mod 𝑃))) |
35 | id 22 | . . . . . 6 ⊢ (((3↑((𝑃 − 1) / 2)) mod 𝑃) = (-1 mod 𝑃) → ((3↑((𝑃 − 1) / 2)) mod 𝑃) = (-1 mod 𝑃)) | |
36 | 30, 34, 35 | rspcedvd 3317 | . . . . 5 ⊢ (((3↑((𝑃 − 1) / 2)) mod 𝑃) = (-1 mod 𝑃) → ∃𝑥 ∈ ℤ ((𝑥↑((𝑃 − 1) / 2)) mod 𝑃) = (-1 mod 𝑃)) |
37 | 36 | adantr 481 | . . . 4 ⊢ ((((3↑((𝑃 − 1) / 2)) mod 𝑃) = (-1 mod 𝑃) ∧ 𝑃 = ((5 · (2↑3)) + 1)) → ∃𝑥 ∈ ℤ ((𝑥↑((𝑃 − 1) / 2)) mod 𝑃) = (-1 mod 𝑃)) |
38 | 23, 25, 21, 28, 37 | proththd 41531 | . . 3 ⊢ ((((3↑((𝑃 − 1) / 2)) mod 𝑃) = (-1 mod 𝑃) ∧ 𝑃 = ((5 · (2↑3)) + 1)) → 𝑃 ∈ ℙ) |
39 | 21, 38 | jca 554 | . 2 ⊢ ((((3↑((𝑃 − 1) / 2)) mod 𝑃) = (-1 mod 𝑃) ∧ 𝑃 = ((5 · (2↑3)) + 1)) → (𝑃 = ((5 · (2↑3)) + 1) ∧ 𝑃 ∈ ℙ)) |
40 | 2, 20, 39 | mp2an 708 | 1 ⊢ (𝑃 = ((5 · (2↑3)) + 1) ∧ 𝑃 ∈ ℙ) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∃wrex 2913 class class class wbr 4653 (class class class)co 6650 0cc0 9936 1c1 9937 + caddc 9939 · cmul 9941 < clt 10074 − cmin 10266 -cneg 10267 / cdiv 10684 ℕcn 11020 2c2 11070 3c3 11071 4c4 11072 5c5 11073 8c8 11076 ℤcz 11377 ;cdc 11493 mod cmo 12668 ↑cexp 12860 ℙcprime 15385 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-inf 8349 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-xnn0 11364 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-fz 12327 df-fzo 12466 df-fl 12593 df-mod 12669 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-dvds 14984 df-gcd 15217 df-prm 15386 df-odz 15470 df-phi 15471 df-pc 15542 |
This theorem is referenced by: (None) |
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