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Mathbox for Alexander van der Vekens |
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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 42043 | . 2 ⊢ ((3↑((𝑃 − 1) / 2)) mod 𝑃) = (-1 mod 𝑃) |
3 | dfdec10 11687 | . . 3 ⊢ ;41 = ((;10 · 4) + 1) | |
4 | 4t2e8 11371 | . . . . . . . 8 ⊢ (4 · 2) = 8 | |
5 | 4cn 11288 | . . . . . . . . 9 ⊢ 4 ∈ ℂ | |
6 | 2cn 11281 | . . . . . . . . 9 ⊢ 2 ∈ ℂ | |
7 | 5, 6 | mulcomi 10236 | . . . . . . . 8 ⊢ (4 · 2) = (2 · 4) |
8 | 4, 7 | eqtr3i 2782 | . . . . . . 7 ⊢ 8 = (2 · 4) |
9 | 8 | oveq2i 6822 | . . . . . 6 ⊢ (5 · 8) = (5 · (2 · 4)) |
10 | 5cn 11290 | . . . . . . 7 ⊢ 5 ∈ ℂ | |
11 | 10, 6, 5 | mulassi 10239 | . . . . . 6 ⊢ ((5 · 2) · 4) = (5 · (2 · 4)) |
12 | 5t2e10 11824 | . . . . . . 7 ⊢ (5 · 2) = ;10 | |
13 | 12 | oveq1i 6821 | . . . . . 6 ⊢ ((5 · 2) · 4) = (;10 · 4) |
14 | 9, 11, 13 | 3eqtr2i 2786 | . . . . 5 ⊢ (5 · 8) = (;10 · 4) |
15 | cu2 13155 | . . . . . . 7 ⊢ (2↑3) = 8 | |
16 | 15 | eqcomi 2767 | . . . . . 6 ⊢ 8 = (2↑3) |
17 | 16 | oveq2i 6822 | . . . . 5 ⊢ (5 · 8) = (5 · (2↑3)) |
18 | 14, 17 | eqtr3i 2782 | . . . 4 ⊢ (;10 · 4) = (5 · (2↑3)) |
19 | 18 | oveq1i 6821 | . . 3 ⊢ ((;10 · 4) + 1) = ((5 · (2↑3)) + 1) |
20 | 1, 3, 19 | 3eqtri 2784 | . 2 ⊢ 𝑃 = ((5 · (2↑3)) + 1) |
21 | simpr 479 | . . 3 ⊢ ((((3↑((𝑃 − 1) / 2)) mod 𝑃) = (-1 mod 𝑃) ∧ 𝑃 = ((5 · (2↑3)) + 1)) → 𝑃 = ((5 · (2↑3)) + 1)) | |
22 | 3nn 11376 | . . . . 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 11378 | . . . . 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 11407 | . . . . . 6 ⊢ 5 < 8 | |
27 | 26, 15 | breqtrri 4829 | . . . . 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 11600 | . . . . . . 7 ⊢ 3 ∈ ℤ | |
30 | 29 | a1i 11 | . . . . . 6 ⊢ (((3↑((𝑃 − 1) / 2)) mod 𝑃) = (-1 mod 𝑃) → 3 ∈ ℤ) |
31 | oveq1 6818 | . . . . . . . . 9 ⊢ (𝑥 = 3 → (𝑥↑((𝑃 − 1) / 2)) = (3↑((𝑃 − 1) / 2))) | |
32 | 31 | oveq1d 6826 | . . . . . . . 8 ⊢ (𝑥 = 3 → ((𝑥↑((𝑃 − 1) / 2)) mod 𝑃) = ((3↑((𝑃 − 1) / 2)) mod 𝑃)) |
33 | 32 | eqeq1d 2760 | . . . . . . 7 ⊢ (𝑥 = 3 → (((𝑥↑((𝑃 − 1) / 2)) mod 𝑃) = (-1 mod 𝑃) ↔ ((3↑((𝑃 − 1) / 2)) mod 𝑃) = (-1 mod 𝑃))) |
34 | 33 | adantl 473 | . . . . . 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 3454 | . . . . 5 ⊢ (((3↑((𝑃 − 1) / 2)) mod 𝑃) = (-1 mod 𝑃) → ∃𝑥 ∈ ℤ ((𝑥↑((𝑃 − 1) / 2)) mod 𝑃) = (-1 mod 𝑃)) |
37 | 36 | adantr 472 | . . . 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 42039 | . . 3 ⊢ ((((3↑((𝑃 − 1) / 2)) mod 𝑃) = (-1 mod 𝑃) ∧ 𝑃 = ((5 · (2↑3)) + 1)) → 𝑃 ∈ ℙ) |
39 | 21, 38 | jca 555 | . 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 710 | 1 ⊢ (𝑃 = ((5 · (2↑3)) + 1) ∧ 𝑃 ∈ ℙ) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∧ wa 383 = wceq 1630 ∈ wcel 2137 ∃wrex 3049 class class class wbr 4802 (class class class)co 6811 0cc0 10126 1c1 10127 + caddc 10129 · cmul 10131 < clt 10264 − cmin 10456 -cneg 10457 / cdiv 10874 ℕcn 11210 2c2 11260 3c3 11261 4c4 11262 5c5 11263 8c8 11266 ℤcz 11567 ;cdc 11683 mod cmo 12860 ↑cexp 13052 ℙcprime 15585 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1869 ax-4 1884 ax-5 1986 ax-6 2052 ax-7 2088 ax-8 2139 ax-9 2146 ax-10 2166 ax-11 2181 ax-12 2194 ax-13 2389 ax-ext 2738 ax-rep 4921 ax-sep 4931 ax-nul 4939 ax-pow 4990 ax-pr 5053 ax-un 7112 ax-cnex 10182 ax-resscn 10183 ax-1cn 10184 ax-icn 10185 ax-addcl 10186 ax-addrcl 10187 ax-mulcl 10188 ax-mulrcl 10189 ax-mulcom 10190 ax-addass 10191 ax-mulass 10192 ax-distr 10193 ax-i2m1 10194 ax-1ne0 10195 ax-1rid 10196 ax-rnegex 10197 ax-rrecex 10198 ax-cnre 10199 ax-pre-lttri 10200 ax-pre-lttrn 10201 ax-pre-ltadd 10202 ax-pre-mulgt0 10203 ax-pre-sup 10204 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1633 df-ex 1852 df-nf 1857 df-sb 2045 df-eu 2609 df-mo 2610 df-clab 2745 df-cleq 2751 df-clel 2754 df-nfc 2889 df-ne 2931 df-nel 3034 df-ral 3053 df-rex 3054 df-reu 3055 df-rmo 3056 df-rab 3057 df-v 3340 df-sbc 3575 df-csb 3673 df-dif 3716 df-un 3718 df-in 3720 df-ss 3727 df-pss 3729 df-nul 4057 df-if 4229 df-pw 4302 df-sn 4320 df-pr 4322 df-tp 4324 df-op 4326 df-uni 4587 df-int 4626 df-iun 4672 df-br 4803 df-opab 4863 df-mpt 4880 df-tr 4903 df-id 5172 df-eprel 5177 df-po 5185 df-so 5186 df-fr 5223 df-we 5225 df-xp 5270 df-rel 5271 df-cnv 5272 df-co 5273 df-dm 5274 df-rn 5275 df-res 5276 df-ima 5277 df-pred 5839 df-ord 5885 df-on 5886 df-lim 5887 df-suc 5888 df-iota 6010 df-fun 6049 df-fn 6050 df-f 6051 df-f1 6052 df-fo 6053 df-f1o 6054 df-fv 6055 df-riota 6772 df-ov 6814 df-oprab 6815 df-mpt2 6816 df-om 7229 df-1st 7331 df-2nd 7332 df-wrecs 7574 df-recs 7635 df-rdg 7673 df-1o 7727 df-2o 7728 df-oadd 7731 df-er 7909 df-map 8023 df-en 8120 df-dom 8121 df-sdom 8122 df-fin 8123 df-sup 8511 df-inf 8512 df-card 8953 df-cda 9180 df-pnf 10266 df-mnf 10267 df-xr 10268 df-ltxr 10269 df-le 10270 df-sub 10458 df-neg 10459 df-div 10875 df-nn 11211 df-2 11269 df-3 11270 df-4 11271 df-5 11272 df-6 11273 df-7 11274 df-8 11275 df-9 11276 df-n0 11483 df-xnn0 11554 df-z 11568 df-dec 11684 df-uz 11878 df-q 11980 df-rp 12024 df-fz 12518 df-fzo 12658 df-fl 12785 df-mod 12861 df-seq 12994 df-exp 13053 df-hash 13310 df-cj 14036 df-re 14037 df-im 14038 df-sqrt 14172 df-abs 14173 df-dvds 15181 df-gcd 15417 df-prm 15586 df-odz 15670 df-phi 15671 df-pc 15742 |
This theorem is referenced by: (None) |
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