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Mirrors > Home > MPE Home > Th. List > mod2xi | Structured version Visualization version GIF version |
Description: Double exponents in a power mod calculation. (Contributed by Mario Carneiro, 21-Feb-2014.) |
Ref | Expression |
---|---|
modxai.1 | ⊢ 𝑁 ∈ ℕ |
modxai.2 | ⊢ 𝐴 ∈ ℕ |
modxai.3 | ⊢ 𝐵 ∈ ℕ0 |
modxai.4 | ⊢ 𝐷 ∈ ℤ |
modxai.5 | ⊢ 𝐾 ∈ ℕ0 |
modxai.6 | ⊢ 𝑀 ∈ ℕ0 |
mod2xi.9 | ⊢ ((𝐴↑𝐵) mod 𝑁) = (𝐾 mod 𝑁) |
mod2xi.7 | ⊢ (2 · 𝐵) = 𝐸 |
mod2xi.8 | ⊢ ((𝐷 · 𝑁) + 𝑀) = (𝐾 · 𝐾) |
Ref | Expression |
---|---|
mod2xi | ⊢ ((𝐴↑𝐸) mod 𝑁) = (𝑀 mod 𝑁) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | modxai.1 | . 2 ⊢ 𝑁 ∈ ℕ | |
2 | modxai.2 | . 2 ⊢ 𝐴 ∈ ℕ | |
3 | modxai.3 | . 2 ⊢ 𝐵 ∈ ℕ0 | |
4 | modxai.4 | . 2 ⊢ 𝐷 ∈ ℤ | |
5 | modxai.5 | . 2 ⊢ 𝐾 ∈ ℕ0 | |
6 | modxai.6 | . 2 ⊢ 𝑀 ∈ ℕ0 | |
7 | mod2xi.9 | . 2 ⊢ ((𝐴↑𝐵) mod 𝑁) = (𝐾 mod 𝑁) | |
8 | 3 | nn0cni 11713 | . . . 4 ⊢ 𝐵 ∈ ℂ |
9 | 8 | 2timesi 11578 | . . 3 ⊢ (2 · 𝐵) = (𝐵 + 𝐵) |
10 | mod2xi.7 | . . 3 ⊢ (2 · 𝐵) = 𝐸 | |
11 | 9, 10 | eqtr3i 2798 | . 2 ⊢ (𝐵 + 𝐵) = 𝐸 |
12 | mod2xi.8 | . 2 ⊢ ((𝐷 · 𝑁) + 𝑀) = (𝐾 · 𝐾) | |
13 | 1, 2, 3, 4, 5, 6, 3, 5, 7, 7, 11, 12 | modxai 16250 | 1 ⊢ ((𝐴↑𝐸) mod 𝑁) = (𝑀 mod 𝑁) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1507 ∈ wcel 2048 (class class class)co 6970 + caddc 10330 · cmul 10332 ℕcn 11431 2c2 11488 ℕ0cn0 11700 ℤcz 11786 mod cmo 13045 ↑cexp 13237 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-cnex 10383 ax-resscn 10384 ax-1cn 10385 ax-icn 10386 ax-addcl 10387 ax-addrcl 10388 ax-mulcl 10389 ax-mulrcl 10390 ax-mulcom 10391 ax-addass 10392 ax-mulass 10393 ax-distr 10394 ax-i2m1 10395 ax-1ne0 10396 ax-1rid 10397 ax-rnegex 10398 ax-rrecex 10399 ax-cnre 10400 ax-pre-lttri 10401 ax-pre-lttrn 10402 ax-pre-ltadd 10403 ax-pre-mulgt0 10404 ax-pre-sup 10405 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-pss 3841 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-iun 4788 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5305 df-eprel 5310 df-po 5319 df-so 5320 df-fr 5359 df-we 5361 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-om 7391 df-2nd 7495 df-wrecs 7743 df-recs 7805 df-rdg 7843 df-er 8081 df-en 8299 df-dom 8300 df-sdom 8301 df-sup 8693 df-inf 8694 df-pnf 10468 df-mnf 10469 df-xr 10470 df-ltxr 10471 df-le 10472 df-sub 10664 df-neg 10665 df-div 11091 df-nn 11432 df-2 11496 df-n0 11701 df-z 11787 df-uz 12052 df-rp 12198 df-fl 12970 df-mod 13046 df-seq 13178 df-exp 13238 |
This theorem is referenced by: mod2xnegi 16253 1259lem1 16310 1259lem2 16311 1259lem3 16312 1259lem4 16313 2503lem1 16316 2503lem2 16317 4001lem1 16320 4001lem2 16321 4001lem3 16322 2exp340mod341 43206 8exp8mod9 43209 |
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