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Mirrors > Home > MPE Home > Th. List > modprminveq | Structured version Visualization version GIF version |
Description: The modular inverse of 𝐴 mod 𝑃 is unique. (Contributed by Alexander van der Vekens, 17-May-2018.) |
Ref | Expression |
---|---|
modprminv.1 | ⊢ 𝑅 = ((𝐴↑(𝑃 − 2)) mod 𝑃) |
Ref | Expression |
---|---|
modprminveq | ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ ¬ 𝑃 ∥ 𝐴) → ((𝑆 ∈ (0...(𝑃 − 1)) ∧ ((𝐴 · 𝑆) mod 𝑃) = 1) ↔ 𝑆 = 𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfzelz 12902 | . . . . . . 7 ⊢ (𝑆 ∈ (0...(𝑃 − 1)) → 𝑆 ∈ ℤ) | |
2 | zmulcl 12019 | . . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑆 ∈ ℤ) → (𝐴 · 𝑆) ∈ ℤ) | |
3 | 1, 2 | sylan2 595 | . . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑆 ∈ (0...(𝑃 − 1))) → (𝐴 · 𝑆) ∈ ℤ) |
4 | modprm1div 16124 | . . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 · 𝑆) ∈ ℤ) → (((𝐴 · 𝑆) mod 𝑃) = 1 ↔ 𝑃 ∥ ((𝐴 · 𝑆) − 1))) | |
5 | 3, 4 | sylan2 595 | . . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝑆 ∈ (0...(𝑃 − 1)))) → (((𝐴 · 𝑆) mod 𝑃) = 1 ↔ 𝑃 ∥ ((𝐴 · 𝑆) − 1))) |
6 | 5 | expr 460 | . . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ) → (𝑆 ∈ (0...(𝑃 − 1)) → (((𝐴 · 𝑆) mod 𝑃) = 1 ↔ 𝑃 ∥ ((𝐴 · 𝑆) − 1)))) |
7 | 6 | 3adant3 1129 | . . 3 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ ¬ 𝑃 ∥ 𝐴) → (𝑆 ∈ (0...(𝑃 − 1)) → (((𝐴 · 𝑆) mod 𝑃) = 1 ↔ 𝑃 ∥ ((𝐴 · 𝑆) − 1)))) |
8 | 7 | pm5.32d 580 | . 2 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ ¬ 𝑃 ∥ 𝐴) → ((𝑆 ∈ (0...(𝑃 − 1)) ∧ ((𝐴 · 𝑆) mod 𝑃) = 1) ↔ (𝑆 ∈ (0...(𝑃 − 1)) ∧ 𝑃 ∥ ((𝐴 · 𝑆) − 1)))) |
9 | modprminv.1 | . . 3 ⊢ 𝑅 = ((𝐴↑(𝑃 − 2)) mod 𝑃) | |
10 | 9 | prmdiveq 16113 | . 2 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ ¬ 𝑃 ∥ 𝐴) → ((𝑆 ∈ (0...(𝑃 − 1)) ∧ 𝑃 ∥ ((𝐴 · 𝑆) − 1)) ↔ 𝑆 = 𝑅)) |
11 | 8, 10 | bitrd 282 | 1 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ ¬ 𝑃 ∥ 𝐴) → ((𝑆 ∈ (0...(𝑃 − 1)) ∧ ((𝐴 · 𝑆) mod 𝑃) = 1) ↔ 𝑆 = 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 class class class wbr 5030 (class class class)co 7135 0cc0 10526 1c1 10527 · cmul 10531 − cmin 10859 2c2 11680 ℤcz 11969 ...cfz 12885 mod cmo 13232 ↑cexp 13425 ∥ cdvds 15599 ℙcprime 16005 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-sup 8890 df-inf 8891 df-dju 9314 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-n0 11886 df-xnn0 11956 df-z 11970 df-uz 12232 df-rp 12378 df-fz 12886 df-fzo 13029 df-fl 13157 df-mod 13233 df-seq 13365 df-exp 13426 df-hash 13687 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-dvds 15600 df-gcd 15834 df-prm 16006 df-phi 16093 |
This theorem is referenced by: reumodprminv 16131 |
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