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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 12744 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
3 | 2 | sqvald 14189 | . . . 4 ⊢ (𝜑 → (𝐴↑2) = (𝐴 · 𝐴)) |
4 | 3 | oveq1d 7460 | . . 3 ⊢ (𝜑 → ((𝐴↑2) mod 𝐸) = ((𝐴 · 𝐴) mod 𝐸)) |
5 | neg1z 12675 | . . . . 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 13972 | . . 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 12502 | . . . . 5 ⊢ (-1 · -1) = 1 | |
12 | 11 | a1i 11 | . . . 4 ⊢ (𝜑 → (-1 · -1) = 1) |
13 | 12 | oveq1d 7460 | . . 3 ⊢ (𝜑 → ((-1 · -1) mod 𝐸) = (1 mod 𝐸)) |
14 | 7 | rpred 13095 | . . . 4 ⊢ (𝜑 → 𝐸 ∈ ℝ) |
15 | modexp2m1d.g | . . . 4 ⊢ (𝜑 → 1 < 𝐸) | |
16 | 1mod 13950 | . . . 4 ⊢ ((𝐸 ∈ ℝ ∧ 1 < 𝐸) → (1 mod 𝐸) = 1) | |
17 | 14, 15, 16 | syl2anc 583 | . . 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 1537 ∈ wcel 2103 class class class wbr 5169 (class class class)co 7445 ℝcr 11179 1c1 11181 · cmul 11185 < clt 11320 -cneg 11517 2c2 12344 ℤcz 12635 ℝ+crp 13053 mod cmo 13916 ↑cexp 14108 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-sep 5320 ax-nul 5327 ax-pow 5386 ax-pr 5450 ax-un 7766 ax-cnex 11236 ax-resscn 11237 ax-1cn 11238 ax-icn 11239 ax-addcl 11240 ax-addrcl 11241 ax-mulcl 11242 ax-mulrcl 11243 ax-mulcom 11244 ax-addass 11245 ax-mulass 11246 ax-distr 11247 ax-i2m1 11248 ax-1ne0 11249 ax-1rid 11250 ax-rnegex 11251 ax-rrecex 11252 ax-cnre 11253 ax-pre-lttri 11254 ax-pre-lttrn 11255 ax-pre-ltadd 11256 ax-pre-mulgt0 11257 ax-pre-sup 11258 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3383 df-reu 3384 df-rab 3439 df-v 3484 df-sbc 3799 df-csb 3916 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-pss 3990 df-nul 4348 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5021 df-br 5170 df-opab 5232 df-mpt 5253 df-tr 5287 df-id 5597 df-eprel 5603 df-po 5611 df-so 5612 df-fr 5654 df-we 5656 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-res 5711 df-ima 5712 df-pred 6331 df-ord 6397 df-on 6398 df-lim 6399 df-suc 6400 df-iota 6524 df-fun 6574 df-fn 6575 df-f 6576 df-f1 6577 df-fo 6578 df-f1o 6579 df-fv 6580 df-riota 7401 df-ov 7448 df-oprab 7449 df-mpo 7450 df-om 7900 df-2nd 8027 df-frecs 8318 df-wrecs 8349 df-recs 8423 df-rdg 8462 df-er 8759 df-en 9000 df-dom 9001 df-sdom 9002 df-sup 9507 df-inf 9508 df-pnf 11322 df-mnf 11323 df-xr 11324 df-ltxr 11325 df-le 11326 df-sub 11518 df-neg 11519 df-div 11944 df-nn 12290 df-2 12352 df-n0 12550 df-z 12636 df-uz 12900 df-rp 13054 df-fl 13839 df-mod 13917 df-seq 14049 df-exp 14109 |
This theorem is referenced by: proththd 47421 |
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