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Mirrors > Home > MPE Home > Th. List > Mathboxes > modexp2m1d | Structured version Visualization version GIF version |
Description: The square of an integer which is -1 modulo a number greater than 1 is 1 modulo the same modulus. (Contributed by AV, 5-Jul-2020.) |
Ref | Expression |
---|---|
modexp2m1d.a | ⊢ (𝜑 → 𝐴 ∈ ℤ) |
modexp2m1d.e | ⊢ (𝜑 → 𝐸 ∈ ℝ+) |
modexp2m1d.g | ⊢ (𝜑 → 1 < 𝐸) |
modexp2m1d.m | ⊢ (𝜑 → (𝐴 mod 𝐸) = (-1 mod 𝐸)) |
Ref | Expression |
---|---|
modexp2m1d | ⊢ (𝜑 → ((𝐴↑2) mod 𝐸) = 1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | modexp2m1d.a | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℤ) | |
2 | 1 | zcnd 11840 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
3 | 2 | sqvald 13329 | . . . 4 ⊢ (𝜑 → (𝐴↑2) = (𝐴 · 𝐴)) |
4 | 3 | oveq1d 6939 | . . 3 ⊢ (𝜑 → ((𝐴↑2) mod 𝐸) = ((𝐴 · 𝐴) mod 𝐸)) |
5 | neg1z 11770 | . . . . 5 ⊢ -1 ∈ ℤ | |
6 | 5 | a1i 11 | . . . 4 ⊢ (𝜑 → -1 ∈ ℤ) |
7 | modexp2m1d.e | . . . 4 ⊢ (𝜑 → 𝐸 ∈ ℝ+) | |
8 | modexp2m1d.m | . . . 4 ⊢ (𝜑 → (𝐴 mod 𝐸) = (-1 mod 𝐸)) | |
9 | 1, 6, 1, 6, 7, 8, 8 | modmul12d 13048 | . . 3 ⊢ (𝜑 → ((𝐴 · 𝐴) mod 𝐸) = ((-1 · -1) mod 𝐸)) |
10 | 4, 9 | eqtrd 2814 | . 2 ⊢ (𝜑 → ((𝐴↑2) mod 𝐸) = ((-1 · -1) mod 𝐸)) |
11 | neg1mulneg1e1 11600 | . . . . 5 ⊢ (-1 · -1) = 1 | |
12 | 11 | a1i 11 | . . . 4 ⊢ (𝜑 → (-1 · -1) = 1) |
13 | 12 | oveq1d 6939 | . . 3 ⊢ (𝜑 → ((-1 · -1) mod 𝐸) = (1 mod 𝐸)) |
14 | 7 | rpred 12186 | . . . 4 ⊢ (𝜑 → 𝐸 ∈ ℝ) |
15 | modexp2m1d.g | . . . 4 ⊢ (𝜑 → 1 < 𝐸) | |
16 | 1mod 13026 | . . . 4 ⊢ ((𝐸 ∈ ℝ ∧ 1 < 𝐸) → (1 mod 𝐸) = 1) | |
17 | 14, 15, 16 | syl2anc 579 | . . 3 ⊢ (𝜑 → (1 mod 𝐸) = 1) |
18 | 13, 17 | eqtrd 2814 | . 2 ⊢ (𝜑 → ((-1 · -1) mod 𝐸) = 1) |
19 | 10, 18 | eqtrd 2814 | 1 ⊢ (𝜑 → ((𝐴↑2) mod 𝐸) = 1) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ∈ wcel 2107 class class class wbr 4888 (class class class)co 6924 ℝcr 10273 1c1 10275 · cmul 10279 < clt 10413 -cneg 10609 2c2 11435 ℤcz 11733 ℝ+crp 12142 mod cmo 12992 ↑cexp 13183 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 ax-pre-sup 10352 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-om 7346 df-2nd 7448 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-er 8028 df-en 8244 df-dom 8245 df-sdom 8246 df-sup 8638 df-inf 8639 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-div 11036 df-nn 11380 df-2 11443 df-n0 11648 df-z 11734 df-uz 11998 df-rp 12143 df-fl 12917 df-mod 12993 df-seq 13125 df-exp 13184 |
This theorem is referenced by: proththd 42566 |
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