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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 40073 | . 2 ⊢ ((3↑((𝑃 − 1) / 2)) mod 𝑃) = (-1 mod 𝑃) |
3 | dfdec10 11373 | . . 3 ⊢ ;41 = ((;10 · 4) + 1) | |
4 | 4t2e8 11058 | . . . . . . . 8 ⊢ (4 · 2) = 8 | |
5 | 4cn 10975 | . . . . . . . . 9 ⊢ 4 ∈ ℂ | |
6 | 2cn 10968 | . . . . . . . . 9 ⊢ 2 ∈ ℂ | |
7 | 5, 6 | mulcomi 9925 | . . . . . . . 8 ⊢ (4 · 2) = (2 · 4) |
8 | 4, 7 | eqtr3i 2634 | . . . . . . 7 ⊢ 8 = (2 · 4) |
9 | 8 | oveq2i 6560 | . . . . . 6 ⊢ (5 · 8) = (5 · (2 · 4)) |
10 | 5cn 10977 | . . . . . . 7 ⊢ 5 ∈ ℂ | |
11 | 10, 6, 5 | mulassi 9928 | . . . . . 6 ⊢ ((5 · 2) · 4) = (5 · (2 · 4)) |
12 | 5t2e10 11510 | . . . . . . 7 ⊢ (5 · 2) = ;10 | |
13 | 12 | oveq1i 6559 | . . . . . 6 ⊢ ((5 · 2) · 4) = (;10 · 4) |
14 | 9, 11, 13 | 3eqtr2i 2638 | . . . . 5 ⊢ (5 · 8) = (;10 · 4) |
15 | cu2 12825 | . . . . . . 7 ⊢ (2↑3) = 8 | |
16 | 15 | eqcomi 2619 | . . . . . 6 ⊢ 8 = (2↑3) |
17 | 16 | oveq2i 6560 | . . . . 5 ⊢ (5 · 8) = (5 · (2↑3)) |
18 | 14, 17 | eqtr3i 2634 | . . . 4 ⊢ (;10 · 4) = (5 · (2↑3)) |
19 | 18 | oveq1i 6559 | . . 3 ⊢ ((;10 · 4) + 1) = ((5 · (2↑3)) + 1) |
20 | 1, 3, 19 | 3eqtri 2636 | . 2 ⊢ 𝑃 = ((5 · (2↑3)) + 1) |
21 | simpr 476 | . . 3 ⊢ ((((3↑((𝑃 − 1) / 2)) mod 𝑃) = (-1 mod 𝑃) ∧ 𝑃 = ((5 · (2↑3)) + 1)) → 𝑃 = ((5 · (2↑3)) + 1)) | |
22 | 3nn 11063 | . . . . 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 11065 | . . . . 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 11094 | . . . . . 6 ⊢ 5 < 8 | |
27 | 26, 15 | breqtrri 4610 | . . . . 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 11287 | . . . . . . 7 ⊢ 3 ∈ ℤ | |
30 | 29 | a1i 11 | . . . . . 6 ⊢ (((3↑((𝑃 − 1) / 2)) mod 𝑃) = (-1 mod 𝑃) → 3 ∈ ℤ) |
31 | oveq1 6556 | . . . . . . . . 9 ⊢ (𝑥 = 3 → (𝑥↑((𝑃 − 1) / 2)) = (3↑((𝑃 − 1) / 2))) | |
32 | 31 | oveq1d 6564 | . . . . . . . 8 ⊢ (𝑥 = 3 → ((𝑥↑((𝑃 − 1) / 2)) mod 𝑃) = ((3↑((𝑃 − 1) / 2)) mod 𝑃)) |
33 | 32 | eqeq1d 2612 | . . . . . . 7 ⊢ (𝑥 = 3 → (((𝑥↑((𝑃 − 1) / 2)) mod 𝑃) = (-1 mod 𝑃) ↔ ((3↑((𝑃 − 1) / 2)) mod 𝑃) = (-1 mod 𝑃))) |
34 | 33 | adantl 481 | . . . . . 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 3289 | . . . . 5 ⊢ (((3↑((𝑃 − 1) / 2)) mod 𝑃) = (-1 mod 𝑃) → ∃𝑥 ∈ ℤ ((𝑥↑((𝑃 − 1) / 2)) mod 𝑃) = (-1 mod 𝑃)) |
37 | 36 | adantr 480 | . . . 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 40069 | . . 3 ⊢ ((((3↑((𝑃 − 1) / 2)) mod 𝑃) = (-1 mod 𝑃) ∧ 𝑃 = ((5 · (2↑3)) + 1)) → 𝑃 ∈ ℙ) |
39 | 21, 38 | jca 553 | . 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 704 | 1 ⊢ (𝑃 = ((5 · (2↑3)) + 1) ∧ 𝑃 ∈ ℙ) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 class class class wbr 4583 (class class class)co 6549 0cc0 9815 1c1 9816 + caddc 9818 · cmul 9820 < clt 9953 − cmin 10145 -cneg 10146 / cdiv 10563 ℕcn 10897 2c2 10947 3c3 10948 4c4 10949 5c5 10950 8c8 10953 ℤcz 11254 ;cdc 11369 mod cmo 12530 ↑cexp 12722 ℙcprime 15223 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-inf 8232 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-xnn0 11241 df-z 11255 df-dec 11370 df-uz 11564 df-q 11665 df-rp 11709 df-fz 12198 df-fzo 12335 df-fl 12455 df-mod 12531 df-seq 12664 df-exp 12723 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-dvds 14822 df-gcd 15055 df-prm 15224 df-odz 15308 df-phi 15309 df-pc 15380 |
This theorem is referenced by: (None) |
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