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Mirrors > Home > MPE Home > Th. List > prmdivdiv | Structured version Visualization version GIF version |
Description: The (modular) inverse of the inverse of a number is itself. (Contributed by Mario Carneiro, 24-Jan-2015.) |
Ref | Expression |
---|---|
prmdiv.1 | ⊢ 𝑅 = ((𝐴↑(𝑃 − 2)) mod 𝑃) |
Ref | Expression |
---|---|
prmdivdiv | ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ (1...(𝑃 − 1))) → 𝐴 = ((𝑅↑(𝑃 − 2)) mod 𝑃)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fz1ssfz0 13659 | . . 3 ⊢ (1...(𝑃 − 1)) ⊆ (0...(𝑃 − 1)) | |
2 | simpr 484 | . . 3 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ (1...(𝑃 − 1))) → 𝐴 ∈ (1...(𝑃 − 1))) | |
3 | 1, 2 | sselid 3992 | . 2 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ (1...(𝑃 − 1))) → 𝐴 ∈ (0...(𝑃 − 1))) |
4 | simpl 482 | . . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ (1...(𝑃 − 1))) → 𝑃 ∈ ℙ) | |
5 | elfznn 13589 | . . . . . . 7 ⊢ (𝐴 ∈ (1...(𝑃 − 1)) → 𝐴 ∈ ℕ) | |
6 | 5 | adantl 481 | . . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ (1...(𝑃 − 1))) → 𝐴 ∈ ℕ) |
7 | 6 | nnzd 12637 | . . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ (1...(𝑃 − 1))) → 𝐴 ∈ ℤ) |
8 | prmnn 16707 | . . . . . 6 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ) | |
9 | fzm1ndvds 16355 | . . . . . 6 ⊢ ((𝑃 ∈ ℕ ∧ 𝐴 ∈ (1...(𝑃 − 1))) → ¬ 𝑃 ∥ 𝐴) | |
10 | 8, 9 | sylan 580 | . . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ (1...(𝑃 − 1))) → ¬ 𝑃 ∥ 𝐴) |
11 | prmdiv.1 | . . . . . 6 ⊢ 𝑅 = ((𝐴↑(𝑃 − 2)) mod 𝑃) | |
12 | 11 | prmdiv 16818 | . . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ ¬ 𝑃 ∥ 𝐴) → (𝑅 ∈ (1...(𝑃 − 1)) ∧ 𝑃 ∥ ((𝐴 · 𝑅) − 1))) |
13 | 4, 7, 10, 12 | syl3anc 1370 | . . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ (1...(𝑃 − 1))) → (𝑅 ∈ (1...(𝑃 − 1)) ∧ 𝑃 ∥ ((𝐴 · 𝑅) − 1))) |
14 | 13 | simprd 495 | . . 3 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ (1...(𝑃 − 1))) → 𝑃 ∥ ((𝐴 · 𝑅) − 1)) |
15 | 6 | nncnd 12279 | . . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ (1...(𝑃 − 1))) → 𝐴 ∈ ℂ) |
16 | 13 | simpld 494 | . . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ (1...(𝑃 − 1))) → 𝑅 ∈ (1...(𝑃 − 1))) |
17 | elfznn 13589 | . . . . . . 7 ⊢ (𝑅 ∈ (1...(𝑃 − 1)) → 𝑅 ∈ ℕ) | |
18 | 16, 17 | syl 17 | . . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ (1...(𝑃 − 1))) → 𝑅 ∈ ℕ) |
19 | 18 | nncnd 12279 | . . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ (1...(𝑃 − 1))) → 𝑅 ∈ ℂ) |
20 | 15, 19 | mulcomd 11279 | . . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ (1...(𝑃 − 1))) → (𝐴 · 𝑅) = (𝑅 · 𝐴)) |
21 | 20 | oveq1d 7445 | . . 3 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ (1...(𝑃 − 1))) → ((𝐴 · 𝑅) − 1) = ((𝑅 · 𝐴) − 1)) |
22 | 14, 21 | breqtrd 5173 | . 2 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ (1...(𝑃 − 1))) → 𝑃 ∥ ((𝑅 · 𝐴) − 1)) |
23 | 16 | elfzelzd 13561 | . . 3 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ (1...(𝑃 − 1))) → 𝑅 ∈ ℤ) |
24 | fzm1ndvds 16355 | . . . 4 ⊢ ((𝑃 ∈ ℕ ∧ 𝑅 ∈ (1...(𝑃 − 1))) → ¬ 𝑃 ∥ 𝑅) | |
25 | 8, 16, 24 | syl2an2r 685 | . . 3 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ (1...(𝑃 − 1))) → ¬ 𝑃 ∥ 𝑅) |
26 | eqid 2734 | . . . 4 ⊢ ((𝑅↑(𝑃 − 2)) mod 𝑃) = ((𝑅↑(𝑃 − 2)) mod 𝑃) | |
27 | 26 | prmdiveq 16819 | . . 3 ⊢ ((𝑃 ∈ ℙ ∧ 𝑅 ∈ ℤ ∧ ¬ 𝑃 ∥ 𝑅) → ((𝐴 ∈ (0...(𝑃 − 1)) ∧ 𝑃 ∥ ((𝑅 · 𝐴) − 1)) ↔ 𝐴 = ((𝑅↑(𝑃 − 2)) mod 𝑃))) |
28 | 4, 23, 25, 27 | syl3anc 1370 | . 2 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ (1...(𝑃 − 1))) → ((𝐴 ∈ (0...(𝑃 − 1)) ∧ 𝑃 ∥ ((𝑅 · 𝐴) − 1)) ↔ 𝐴 = ((𝑅↑(𝑃 − 2)) mod 𝑃))) |
29 | 3, 22, 28 | mpbi2and 712 | 1 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ (1...(𝑃 − 1))) → 𝐴 = ((𝑅↑(𝑃 − 2)) mod 𝑃)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1536 ∈ wcel 2105 class class class wbr 5147 (class class class)co 7430 0cc0 11152 1c1 11153 · cmul 11157 − cmin 11489 ℕcn 12263 2c2 12318 ℤcz 12610 ...cfz 13543 mod cmo 13905 ↑cexp 14098 ∥ cdvds 16286 ℙcprime 16704 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-2o 8505 df-oadd 8508 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-sup 9479 df-inf 9480 df-dju 9938 df-card 9976 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-n0 12524 df-xnn0 12597 df-z 12611 df-uz 12876 df-rp 13032 df-fz 13544 df-fzo 13691 df-fl 13828 df-mod 13906 df-seq 14039 df-exp 14099 df-hash 14366 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-dvds 16287 df-gcd 16528 df-prm 16705 df-phi 16799 |
This theorem is referenced by: wilthlem2 27126 |
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