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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 12266 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
3 | 2 | sqvald 13696 | . . . 4 ⊢ (𝜑 → (𝐴↑2) = (𝐴 · 𝐴)) |
4 | 3 | oveq1d 7217 | . . 3 ⊢ (𝜑 → ((𝐴↑2) mod 𝐸) = ((𝐴 · 𝐴) mod 𝐸)) |
5 | neg1z 12196 | . . . . 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 13481 | . . 3 ⊢ (𝜑 → ((𝐴 · 𝐴) mod 𝐸) = ((-1 · -1) mod 𝐸)) |
10 | 4, 9 | eqtrd 2774 | . 2 ⊢ (𝜑 → ((𝐴↑2) mod 𝐸) = ((-1 · -1) mod 𝐸)) |
11 | neg1mulneg1e1 12026 | . . . . 5 ⊢ (-1 · -1) = 1 | |
12 | 11 | a1i 11 | . . . 4 ⊢ (𝜑 → (-1 · -1) = 1) |
13 | 12 | oveq1d 7217 | . . 3 ⊢ (𝜑 → ((-1 · -1) mod 𝐸) = (1 mod 𝐸)) |
14 | 7 | rpred 12611 | . . . 4 ⊢ (𝜑 → 𝐸 ∈ ℝ) |
15 | modexp2m1d.g | . . . 4 ⊢ (𝜑 → 1 < 𝐸) | |
16 | 1mod 13459 | . . . 4 ⊢ ((𝐸 ∈ ℝ ∧ 1 < 𝐸) → (1 mod 𝐸) = 1) | |
17 | 14, 15, 16 | syl2anc 587 | . . 3 ⊢ (𝜑 → (1 mod 𝐸) = 1) |
18 | 13, 17 | eqtrd 2774 | . 2 ⊢ (𝜑 → ((-1 · -1) mod 𝐸) = 1) |
19 | 10, 18 | eqtrd 2774 | 1 ⊢ (𝜑 → ((𝐴↑2) mod 𝐸) = 1) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2110 class class class wbr 5043 (class class class)co 7202 ℝcr 10711 1c1 10713 · cmul 10717 < clt 10850 -cneg 11046 2c2 11868 ℤcz 12159 ℝ+crp 12569 mod cmo 13425 ↑cexp 13618 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 ax-pre-sup 10790 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-om 7634 df-2nd 7751 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-er 8380 df-en 8616 df-dom 8617 df-sdom 8618 df-sup 9047 df-inf 9048 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-div 11473 df-nn 11814 df-2 11876 df-n0 12074 df-z 12160 df-uz 12422 df-rp 12570 df-fl 13350 df-mod 13426 df-seq 13558 df-exp 13619 |
This theorem is referenced by: proththd 44693 |
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