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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 43790 | . 2 ⊢ ((3↑((𝑃 − 1) / 2)) mod 𝑃) = (-1 mod 𝑃) |
3 | dfdec10 12104 | . . 3 ⊢ ;41 = ((;10 · 4) + 1) | |
4 | 4t2e8 11808 | . . . . . . . 8 ⊢ (4 · 2) = 8 | |
5 | 4cn 11725 | . . . . . . . . 9 ⊢ 4 ∈ ℂ | |
6 | 2cn 11715 | . . . . . . . . 9 ⊢ 2 ∈ ℂ | |
7 | 5, 6 | mulcomi 10651 | . . . . . . . 8 ⊢ (4 · 2) = (2 · 4) |
8 | 4, 7 | eqtr3i 2848 | . . . . . . 7 ⊢ 8 = (2 · 4) |
9 | 8 | oveq2i 7169 | . . . . . 6 ⊢ (5 · 8) = (5 · (2 · 4)) |
10 | 5cn 11728 | . . . . . . 7 ⊢ 5 ∈ ℂ | |
11 | 10, 6, 5 | mulassi 10654 | . . . . . 6 ⊢ ((5 · 2) · 4) = (5 · (2 · 4)) |
12 | 5t2e10 12201 | . . . . . . 7 ⊢ (5 · 2) = ;10 | |
13 | 12 | oveq1i 7168 | . . . . . 6 ⊢ ((5 · 2) · 4) = (;10 · 4) |
14 | 9, 11, 13 | 3eqtr2i 2852 | . . . . 5 ⊢ (5 · 8) = (;10 · 4) |
15 | cu2 13566 | . . . . . . 7 ⊢ (2↑3) = 8 | |
16 | 15 | eqcomi 2832 | . . . . . 6 ⊢ 8 = (2↑3) |
17 | 16 | oveq2i 7169 | . . . . 5 ⊢ (5 · 8) = (5 · (2↑3)) |
18 | 14, 17 | eqtr3i 2848 | . . . 4 ⊢ (;10 · 4) = (5 · (2↑3)) |
19 | 18 | oveq1i 7168 | . . 3 ⊢ ((;10 · 4) + 1) = ((5 · (2↑3)) + 1) |
20 | 1, 3, 19 | 3eqtri 2850 | . 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 11719 | . . . . 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 11726 | . . . . 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 11834 | . . . . . 6 ⊢ 5 < 8 | |
27 | 26, 15 | breqtrri 5095 | . . . . 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 12018 | . . . . . . 7 ⊢ 3 ∈ ℤ | |
30 | 29 | a1i 11 | . . . . . 6 ⊢ (((3↑((𝑃 − 1) / 2)) mod 𝑃) = (-1 mod 𝑃) → 3 ∈ ℤ) |
31 | oveq1 7165 | . . . . . . . . 9 ⊢ (𝑥 = 3 → (𝑥↑((𝑃 − 1) / 2)) = (3↑((𝑃 − 1) / 2))) | |
32 | 31 | oveq1d 7173 | . . . . . . . 8 ⊢ (𝑥 = 3 → ((𝑥↑((𝑃 − 1) / 2)) mod 𝑃) = ((3↑((𝑃 − 1) / 2)) mod 𝑃)) |
33 | 32 | eqeq1d 2825 | . . . . . . 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 3628 | . . . . 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 43786 | . . 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 1537 ∈ wcel 2114 ∃wrex 3141 class class class wbr 5068 (class class class)co 7158 0cc0 10539 1c1 10540 + caddc 10542 · cmul 10544 < clt 10677 − cmin 10872 -cneg 10873 / cdiv 11299 ℕcn 11640 2c2 11695 3c3 11696 4c4 11697 5c5 11698 8c8 11701 ℤcz 11984 ;cdc 12101 mod cmo 13240 ↑cexp 13432 ℙcprime 16017 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-2o 8105 df-oadd 8108 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-sup 8908 df-inf 8909 df-dju 9332 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-xnn0 11971 df-z 11985 df-dec 12102 df-uz 12247 df-q 12352 df-rp 12393 df-fz 12896 df-fzo 13037 df-fl 13165 df-mod 13241 df-seq 13373 df-exp 13433 df-hash 13694 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-dvds 15610 df-gcd 15846 df-prm 16018 df-odz 16104 df-phi 16105 df-pc 16176 |
This theorem is referenced by: (None) |
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