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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 11896 | . . . 4 ⊢ 𝐵 ∈ ℂ |
9 | 8 | 2timesi 11762 | . . 3 ⊢ (2 · 𝐵) = (𝐵 + 𝐵) |
10 | mod2xi.7 | . . 3 ⊢ (2 · 𝐵) = 𝐸 | |
11 | 9, 10 | eqtr3i 2846 | . 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 16387 | 1 ⊢ ((𝐴↑𝐸) mod 𝑁) = (𝑀 mod 𝑁) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2114 (class class class)co 7142 + caddc 10526 · cmul 10528 ℕcn 11624 2c2 11679 ℕ0cn0 11884 ℤcz 11968 mod cmo 13227 ↑cexp 13419 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5189 ax-nul 5196 ax-pow 5252 ax-pr 5316 ax-un 7447 ax-cnex 10579 ax-resscn 10580 ax-1cn 10581 ax-icn 10582 ax-addcl 10583 ax-addrcl 10584 ax-mulcl 10585 ax-mulrcl 10586 ax-mulcom 10587 ax-addass 10588 ax-mulass 10589 ax-distr 10590 ax-i2m1 10591 ax-1ne0 10592 ax-1rid 10593 ax-rnegex 10594 ax-rrecex 10595 ax-cnre 10596 ax-pre-lttri 10597 ax-pre-lttrn 10598 ax-pre-ltadd 10599 ax-pre-mulgt0 10600 ax-pre-sup 10601 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3488 df-sbc 3764 df-csb 3872 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-pss 3942 df-nul 4280 df-if 4454 df-pw 4527 df-sn 4554 df-pr 4556 df-tp 4558 df-op 4560 df-uni 4825 df-iun 4907 df-br 5053 df-opab 5115 df-mpt 5133 df-tr 5159 df-id 5446 df-eprel 5451 df-po 5460 df-so 5461 df-fr 5500 df-we 5502 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-rn 5552 df-res 5553 df-ima 5554 df-pred 6134 df-ord 6180 df-on 6181 df-lim 6182 df-suc 6183 df-iota 6300 df-fun 6343 df-fn 6344 df-f 6345 df-f1 6346 df-fo 6347 df-f1o 6348 df-fv 6349 df-riota 7100 df-ov 7145 df-oprab 7146 df-mpo 7147 df-om 7567 df-2nd 7676 df-wrecs 7933 df-recs 7994 df-rdg 8032 df-er 8275 df-en 8496 df-dom 8497 df-sdom 8498 df-sup 8892 df-inf 8893 df-pnf 10663 df-mnf 10664 df-xr 10665 df-ltxr 10666 df-le 10667 df-sub 10858 df-neg 10859 df-div 11284 df-nn 11625 df-2 11687 df-n0 11885 df-z 11969 df-uz 12231 df-rp 12377 df-fl 13152 df-mod 13228 df-seq 13360 df-exp 13420 |
This theorem is referenced by: mod2xnegi 16390 1259lem1 16447 1259lem2 16448 1259lem3 16449 1259lem4 16450 2503lem1 16453 2503lem2 16454 4001lem1 16457 4001lem2 16458 4001lem3 16459 2exp340mod341 43983 8exp8mod9 43986 |
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