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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 43782 | . 2 ⊢ ((3↑((𝑃 − 1) / 2)) mod 𝑃) = (-1 mod 𝑃) |
3 | dfdec10 12100 | . . 3 ⊢ ;41 = ((;10 · 4) + 1) | |
4 | 4t2e8 11804 | . . . . . . . 8 ⊢ (4 · 2) = 8 | |
5 | 4cn 11721 | . . . . . . . . 9 ⊢ 4 ∈ ℂ | |
6 | 2cn 11711 | . . . . . . . . 9 ⊢ 2 ∈ ℂ | |
7 | 5, 6 | mulcomi 10648 | . . . . . . . 8 ⊢ (4 · 2) = (2 · 4) |
8 | 4, 7 | eqtr3i 2846 | . . . . . . 7 ⊢ 8 = (2 · 4) |
9 | 8 | oveq2i 7166 | . . . . . 6 ⊢ (5 · 8) = (5 · (2 · 4)) |
10 | 5cn 11724 | . . . . . . 7 ⊢ 5 ∈ ℂ | |
11 | 10, 6, 5 | mulassi 10651 | . . . . . 6 ⊢ ((5 · 2) · 4) = (5 · (2 · 4)) |
12 | 5t2e10 12197 | . . . . . . 7 ⊢ (5 · 2) = ;10 | |
13 | 12 | oveq1i 7165 | . . . . . 6 ⊢ ((5 · 2) · 4) = (;10 · 4) |
14 | 9, 11, 13 | 3eqtr2i 2850 | . . . . 5 ⊢ (5 · 8) = (;10 · 4) |
15 | cu2 13562 | . . . . . . 7 ⊢ (2↑3) = 8 | |
16 | 15 | eqcomi 2830 | . . . . . 6 ⊢ 8 = (2↑3) |
17 | 16 | oveq2i 7166 | . . . . 5 ⊢ (5 · 8) = (5 · (2↑3)) |
18 | 14, 17 | eqtr3i 2846 | . . . 4 ⊢ (;10 · 4) = (5 · (2↑3)) |
19 | 18 | oveq1i 7165 | . . 3 ⊢ ((;10 · 4) + 1) = ((5 · (2↑3)) + 1) |
20 | 1, 3, 19 | 3eqtri 2848 | . 2 ⊢ 𝑃 = ((5 · (2↑3)) + 1) |
21 | simpr 487 | . . 3 ⊢ ((((3↑((𝑃 − 1) / 2)) mod 𝑃) = (-1 mod 𝑃) ∧ 𝑃 = ((5 · (2↑3)) + 1)) → 𝑃 = ((5 · (2↑3)) + 1)) | |
22 | 3nn 11715 | . . . . 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 11722 | . . . . 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 11830 | . . . . . 6 ⊢ 5 < 8 | |
27 | 26, 15 | breqtrri 5092 | . . . . 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 12014 | . . . . . . 7 ⊢ 3 ∈ ℤ | |
30 | 29 | a1i 11 | . . . . . 6 ⊢ (((3↑((𝑃 − 1) / 2)) mod 𝑃) = (-1 mod 𝑃) → 3 ∈ ℤ) |
31 | oveq1 7162 | . . . . . . . . 9 ⊢ (𝑥 = 3 → (𝑥↑((𝑃 − 1) / 2)) = (3↑((𝑃 − 1) / 2))) | |
32 | 31 | oveq1d 7170 | . . . . . . . 8 ⊢ (𝑥 = 3 → ((𝑥↑((𝑃 − 1) / 2)) mod 𝑃) = ((3↑((𝑃 − 1) / 2)) mod 𝑃)) |
33 | 32 | eqeq1d 2823 | . . . . . . 7 ⊢ (𝑥 = 3 → (((𝑥↑((𝑃 − 1) / 2)) mod 𝑃) = (-1 mod 𝑃) ↔ ((3↑((𝑃 − 1) / 2)) mod 𝑃) = (-1 mod 𝑃))) |
34 | 33 | adantl 484 | . . . . . 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 3625 | . . . . 5 ⊢ (((3↑((𝑃 − 1) / 2)) mod 𝑃) = (-1 mod 𝑃) → ∃𝑥 ∈ ℤ ((𝑥↑((𝑃 − 1) / 2)) mod 𝑃) = (-1 mod 𝑃)) |
37 | 36 | adantr 483 | . . . 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 43778 | . . 3 ⊢ ((((3↑((𝑃 − 1) / 2)) mod 𝑃) = (-1 mod 𝑃) ∧ 𝑃 = ((5 · (2↑3)) + 1)) → 𝑃 ∈ ℙ) |
39 | 21, 38 | jca 514 | . 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 690 | 1 ⊢ (𝑃 = ((5 · (2↑3)) + 1) ∧ 𝑃 ∈ ℙ) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∃wrex 3139 class class class wbr 5065 (class class class)co 7155 0cc0 10536 1c1 10537 + caddc 10539 · cmul 10541 < clt 10674 − cmin 10869 -cneg 10870 / cdiv 11296 ℕcn 11637 2c2 11691 3c3 11692 4c4 11693 5c5 11694 8c8 11697 ℤcz 11980 ;cdc 12097 mod cmo 13236 ↑cexp 13428 ℙcprime 16014 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 ax-pre-sup 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-1st 7688 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-2o 8102 df-oadd 8105 df-er 8288 df-map 8407 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-sup 8905 df-inf 8906 df-dju 9329 df-card 9367 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-div 11297 df-nn 11638 df-2 11699 df-3 11700 df-4 11701 df-5 11702 df-6 11703 df-7 11704 df-8 11705 df-9 11706 df-n0 11897 df-xnn0 11967 df-z 11981 df-dec 12098 df-uz 12243 df-q 12348 df-rp 12389 df-fz 12892 df-fzo 13033 df-fl 13161 df-mod 13237 df-seq 13369 df-exp 13429 df-hash 13690 df-cj 14457 df-re 14458 df-im 14459 df-sqrt 14593 df-abs 14594 df-dvds 15607 df-gcd 15843 df-prm 16015 df-odz 16101 df-phi 16102 df-pc 16173 |
This theorem is referenced by: (None) |
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