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Mirrors > Home > MPE Home > Th. List > modxai | Structured version Visualization version GIF version |
Description: Add exponents in a power mod calculation. (Contributed by Mario Carneiro, 21-Feb-2014.) (Revised by Mario Carneiro, 5-Feb-2015.) |
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 |
modxai.7 | ⊢ 𝐶 ∈ ℕ0 |
modxai.8 | ⊢ 𝐿 ∈ ℕ0 |
modxai.11 | ⊢ ((𝐴↑𝐵) mod 𝑁) = (𝐾 mod 𝑁) |
modxai.12 | ⊢ ((𝐴↑𝐶) mod 𝑁) = (𝐿 mod 𝑁) |
modxai.9 | ⊢ (𝐵 + 𝐶) = 𝐸 |
modxai.10 | ⊢ ((𝐷 · 𝑁) + 𝑀) = (𝐾 · 𝐿) |
Ref | Expression |
---|---|
modxai | ⊢ ((𝐴↑𝐸) mod 𝑁) = (𝑀 mod 𝑁) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | modxai.9 | . . . . 5 ⊢ (𝐵 + 𝐶) = 𝐸 | |
2 | 1 | oveq2i 6820 | . . . 4 ⊢ (𝐴↑(𝐵 + 𝐶)) = (𝐴↑𝐸) |
3 | modxai.2 | . . . . . 6 ⊢ 𝐴 ∈ ℕ | |
4 | 3 | nncni 11218 | . . . . 5 ⊢ 𝐴 ∈ ℂ |
5 | modxai.3 | . . . . 5 ⊢ 𝐵 ∈ ℕ0 | |
6 | modxai.7 | . . . . 5 ⊢ 𝐶 ∈ ℕ0 | |
7 | expadd 13092 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) → (𝐴↑(𝐵 + 𝐶)) = ((𝐴↑𝐵) · (𝐴↑𝐶))) | |
8 | 4, 5, 6, 7 | mp3an 1569 | . . . 4 ⊢ (𝐴↑(𝐵 + 𝐶)) = ((𝐴↑𝐵) · (𝐴↑𝐶)) |
9 | 2, 8 | eqtr3i 2780 | . . 3 ⊢ (𝐴↑𝐸) = ((𝐴↑𝐵) · (𝐴↑𝐶)) |
10 | 9 | oveq1i 6819 | . 2 ⊢ ((𝐴↑𝐸) mod 𝑁) = (((𝐴↑𝐵) · (𝐴↑𝐶)) mod 𝑁) |
11 | nnexpcl 13063 | . . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ0) → (𝐴↑𝐵) ∈ ℕ) | |
12 | 3, 5, 11 | mp2an 710 | . . . . . . . 8 ⊢ (𝐴↑𝐵) ∈ ℕ |
13 | 12 | nnzi 11589 | . . . . . . 7 ⊢ (𝐴↑𝐵) ∈ ℤ |
14 | 13 | a1i 11 | . . . . . 6 ⊢ (⊤ → (𝐴↑𝐵) ∈ ℤ) |
15 | modxai.5 | . . . . . . . 8 ⊢ 𝐾 ∈ ℕ0 | |
16 | 15 | nn0zi 11590 | . . . . . . 7 ⊢ 𝐾 ∈ ℤ |
17 | 16 | a1i 11 | . . . . . 6 ⊢ (⊤ → 𝐾 ∈ ℤ) |
18 | nnexpcl 13063 | . . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝐶 ∈ ℕ0) → (𝐴↑𝐶) ∈ ℕ) | |
19 | 3, 6, 18 | mp2an 710 | . . . . . . . 8 ⊢ (𝐴↑𝐶) ∈ ℕ |
20 | 19 | nnzi 11589 | . . . . . . 7 ⊢ (𝐴↑𝐶) ∈ ℤ |
21 | 20 | a1i 11 | . . . . . 6 ⊢ (⊤ → (𝐴↑𝐶) ∈ ℤ) |
22 | modxai.8 | . . . . . . . 8 ⊢ 𝐿 ∈ ℕ0 | |
23 | 22 | nn0zi 11590 | . . . . . . 7 ⊢ 𝐿 ∈ ℤ |
24 | 23 | a1i 11 | . . . . . 6 ⊢ (⊤ → 𝐿 ∈ ℤ) |
25 | modxai.1 | . . . . . . . 8 ⊢ 𝑁 ∈ ℕ | |
26 | nnrp 12031 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ+) | |
27 | 25, 26 | ax-mp 5 | . . . . . . 7 ⊢ 𝑁 ∈ ℝ+ |
28 | 27 | a1i 11 | . . . . . 6 ⊢ (⊤ → 𝑁 ∈ ℝ+) |
29 | modxai.11 | . . . . . . 7 ⊢ ((𝐴↑𝐵) mod 𝑁) = (𝐾 mod 𝑁) | |
30 | 29 | a1i 11 | . . . . . 6 ⊢ (⊤ → ((𝐴↑𝐵) mod 𝑁) = (𝐾 mod 𝑁)) |
31 | modxai.12 | . . . . . . 7 ⊢ ((𝐴↑𝐶) mod 𝑁) = (𝐿 mod 𝑁) | |
32 | 31 | a1i 11 | . . . . . 6 ⊢ (⊤ → ((𝐴↑𝐶) mod 𝑁) = (𝐿 mod 𝑁)) |
33 | 14, 17, 21, 24, 28, 30, 32 | modmul12d 12914 | . . . . 5 ⊢ (⊤ → (((𝐴↑𝐵) · (𝐴↑𝐶)) mod 𝑁) = ((𝐾 · 𝐿) mod 𝑁)) |
34 | 33 | trud 1638 | . . . 4 ⊢ (((𝐴↑𝐵) · (𝐴↑𝐶)) mod 𝑁) = ((𝐾 · 𝐿) mod 𝑁) |
35 | modxai.10 | . . . . . 6 ⊢ ((𝐷 · 𝑁) + 𝑀) = (𝐾 · 𝐿) | |
36 | modxai.4 | . . . . . . . . 9 ⊢ 𝐷 ∈ ℤ | |
37 | zcn 11570 | . . . . . . . . 9 ⊢ (𝐷 ∈ ℤ → 𝐷 ∈ ℂ) | |
38 | 36, 37 | ax-mp 5 | . . . . . . . 8 ⊢ 𝐷 ∈ ℂ |
39 | 25 | nncni 11218 | . . . . . . . 8 ⊢ 𝑁 ∈ ℂ |
40 | 38, 39 | mulcli 10233 | . . . . . . 7 ⊢ (𝐷 · 𝑁) ∈ ℂ |
41 | modxai.6 | . . . . . . . 8 ⊢ 𝑀 ∈ ℕ0 | |
42 | 41 | nn0cni 11492 | . . . . . . 7 ⊢ 𝑀 ∈ ℂ |
43 | 40, 42 | addcomi 10415 | . . . . . 6 ⊢ ((𝐷 · 𝑁) + 𝑀) = (𝑀 + (𝐷 · 𝑁)) |
44 | 35, 43 | eqtr3i 2780 | . . . . 5 ⊢ (𝐾 · 𝐿) = (𝑀 + (𝐷 · 𝑁)) |
45 | 44 | oveq1i 6819 | . . . 4 ⊢ ((𝐾 · 𝐿) mod 𝑁) = ((𝑀 + (𝐷 · 𝑁)) mod 𝑁) |
46 | 34, 45 | eqtri 2778 | . . 3 ⊢ (((𝐴↑𝐵) · (𝐴↑𝐶)) mod 𝑁) = ((𝑀 + (𝐷 · 𝑁)) mod 𝑁) |
47 | 41 | nn0rei 11491 | . . . 4 ⊢ 𝑀 ∈ ℝ |
48 | modcyc 12895 | . . . 4 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ+ ∧ 𝐷 ∈ ℤ) → ((𝑀 + (𝐷 · 𝑁)) mod 𝑁) = (𝑀 mod 𝑁)) | |
49 | 47, 27, 36, 48 | mp3an 1569 | . . 3 ⊢ ((𝑀 + (𝐷 · 𝑁)) mod 𝑁) = (𝑀 mod 𝑁) |
50 | 46, 49 | eqtri 2778 | . 2 ⊢ (((𝐴↑𝐵) · (𝐴↑𝐶)) mod 𝑁) = (𝑀 mod 𝑁) |
51 | 10, 50 | eqtri 2778 | 1 ⊢ ((𝐴↑𝐸) mod 𝑁) = (𝑀 mod 𝑁) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1628 ⊤wtru 1629 ∈ wcel 2135 (class class class)co 6809 ℂcc 10122 ℝcr 10123 + caddc 10127 · cmul 10129 ℕcn 11208 ℕ0cn0 11480 ℤcz 11565 ℝ+crp 12021 mod cmo 12858 ↑cexp 13050 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1867 ax-4 1882 ax-5 1984 ax-6 2050 ax-7 2086 ax-8 2137 ax-9 2144 ax-10 2164 ax-11 2179 ax-12 2192 ax-13 2387 ax-ext 2736 ax-sep 4929 ax-nul 4937 ax-pow 4988 ax-pr 5051 ax-un 7110 ax-cnex 10180 ax-resscn 10181 ax-1cn 10182 ax-icn 10183 ax-addcl 10184 ax-addrcl 10185 ax-mulcl 10186 ax-mulrcl 10187 ax-mulcom 10188 ax-addass 10189 ax-mulass 10190 ax-distr 10191 ax-i2m1 10192 ax-1ne0 10193 ax-1rid 10194 ax-rnegex 10195 ax-rrecex 10196 ax-cnre 10197 ax-pre-lttri 10198 ax-pre-lttrn 10199 ax-pre-ltadd 10200 ax-pre-mulgt0 10201 ax-pre-sup 10202 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1631 df-ex 1850 df-nf 1855 df-sb 2043 df-eu 2607 df-mo 2608 df-clab 2743 df-cleq 2749 df-clel 2752 df-nfc 2887 df-ne 2929 df-nel 3032 df-ral 3051 df-rex 3052 df-reu 3053 df-rmo 3054 df-rab 3055 df-v 3338 df-sbc 3573 df-csb 3671 df-dif 3714 df-un 3716 df-in 3718 df-ss 3725 df-pss 3727 df-nul 4055 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-tp 4322 df-op 4324 df-uni 4585 df-iun 4670 df-br 4801 df-opab 4861 df-mpt 4878 df-tr 4901 df-id 5170 df-eprel 5175 df-po 5183 df-so 5184 df-fr 5221 df-we 5223 df-xp 5268 df-rel 5269 df-cnv 5270 df-co 5271 df-dm 5272 df-rn 5273 df-res 5274 df-ima 5275 df-pred 5837 df-ord 5883 df-on 5884 df-lim 5885 df-suc 5886 df-iota 6008 df-fun 6047 df-fn 6048 df-f 6049 df-f1 6050 df-fo 6051 df-f1o 6052 df-fv 6053 df-riota 6770 df-ov 6812 df-oprab 6813 df-mpt2 6814 df-om 7227 df-2nd 7330 df-wrecs 7572 df-recs 7633 df-rdg 7671 df-er 7907 df-en 8118 df-dom 8119 df-sdom 8120 df-sup 8509 df-inf 8510 df-pnf 10264 df-mnf 10265 df-xr 10266 df-ltxr 10267 df-le 10268 df-sub 10456 df-neg 10457 df-div 10873 df-nn 11209 df-n0 11481 df-z 11566 df-uz 11876 df-rp 12022 df-fl 12783 df-mod 12859 df-seq 12992 df-exp 13051 |
This theorem is referenced by: mod2xi 15971 modxp1i 15972 1259lem3 16038 1259lem4 16039 2503lem2 16043 4001lem3 16048 |
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