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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 7169 | . . . 4 ⊢ (𝐴↑(𝐵 + 𝐶)) = (𝐴↑𝐸) |
3 | modxai.2 | . . . . . 6 ⊢ 𝐴 ∈ ℕ | |
4 | 3 | nncni 11650 | . . . . 5 ⊢ 𝐴 ∈ ℂ |
5 | modxai.3 | . . . . 5 ⊢ 𝐵 ∈ ℕ0 | |
6 | modxai.7 | . . . . 5 ⊢ 𝐶 ∈ ℕ0 | |
7 | expadd 13474 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) → (𝐴↑(𝐵 + 𝐶)) = ((𝐴↑𝐵) · (𝐴↑𝐶))) | |
8 | 4, 5, 6, 7 | mp3an 1457 | . . . 4 ⊢ (𝐴↑(𝐵 + 𝐶)) = ((𝐴↑𝐵) · (𝐴↑𝐶)) |
9 | 2, 8 | eqtr3i 2848 | . . 3 ⊢ (𝐴↑𝐸) = ((𝐴↑𝐵) · (𝐴↑𝐶)) |
10 | 9 | oveq1i 7168 | . 2 ⊢ ((𝐴↑𝐸) mod 𝑁) = (((𝐴↑𝐵) · (𝐴↑𝐶)) mod 𝑁) |
11 | nnexpcl 13445 | . . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ0) → (𝐴↑𝐵) ∈ ℕ) | |
12 | 3, 5, 11 | mp2an 690 | . . . . . . . 8 ⊢ (𝐴↑𝐵) ∈ ℕ |
13 | 12 | nnzi 12009 | . . . . . . 7 ⊢ (𝐴↑𝐵) ∈ ℤ |
14 | 13 | a1i 11 | . . . . . 6 ⊢ (⊤ → (𝐴↑𝐵) ∈ ℤ) |
15 | modxai.5 | . . . . . . . 8 ⊢ 𝐾 ∈ ℕ0 | |
16 | 15 | nn0zi 12010 | . . . . . . 7 ⊢ 𝐾 ∈ ℤ |
17 | 16 | a1i 11 | . . . . . 6 ⊢ (⊤ → 𝐾 ∈ ℤ) |
18 | nnexpcl 13445 | . . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝐶 ∈ ℕ0) → (𝐴↑𝐶) ∈ ℕ) | |
19 | 3, 6, 18 | mp2an 690 | . . . . . . . 8 ⊢ (𝐴↑𝐶) ∈ ℕ |
20 | 19 | nnzi 12009 | . . . . . . 7 ⊢ (𝐴↑𝐶) ∈ ℤ |
21 | 20 | a1i 11 | . . . . . 6 ⊢ (⊤ → (𝐴↑𝐶) ∈ ℤ) |
22 | modxai.8 | . . . . . . . 8 ⊢ 𝐿 ∈ ℕ0 | |
23 | 22 | nn0zi 12010 | . . . . . . 7 ⊢ 𝐿 ∈ ℤ |
24 | 23 | a1i 11 | . . . . . 6 ⊢ (⊤ → 𝐿 ∈ ℤ) |
25 | modxai.1 | . . . . . . . 8 ⊢ 𝑁 ∈ ℕ | |
26 | nnrp 12403 | . . . . . . . 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 13296 | . . . . 5 ⊢ (⊤ → (((𝐴↑𝐵) · (𝐴↑𝐶)) mod 𝑁) = ((𝐾 · 𝐿) mod 𝑁)) |
34 | 33 | mptru 1544 | . . . 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 11989 | . . . . . . . . 9 ⊢ (𝐷 ∈ ℤ → 𝐷 ∈ ℂ) | |
38 | 36, 37 | ax-mp 5 | . . . . . . . 8 ⊢ 𝐷 ∈ ℂ |
39 | 25 | nncni 11650 | . . . . . . . 8 ⊢ 𝑁 ∈ ℂ |
40 | 38, 39 | mulcli 10650 | . . . . . . 7 ⊢ (𝐷 · 𝑁) ∈ ℂ |
41 | modxai.6 | . . . . . . . 8 ⊢ 𝑀 ∈ ℕ0 | |
42 | 41 | nn0cni 11912 | . . . . . . 7 ⊢ 𝑀 ∈ ℂ |
43 | 40, 42 | addcomi 10833 | . . . . . 6 ⊢ ((𝐷 · 𝑁) + 𝑀) = (𝑀 + (𝐷 · 𝑁)) |
44 | 35, 43 | eqtr3i 2848 | . . . . 5 ⊢ (𝐾 · 𝐿) = (𝑀 + (𝐷 · 𝑁)) |
45 | 44 | oveq1i 7168 | . . . 4 ⊢ ((𝐾 · 𝐿) mod 𝑁) = ((𝑀 + (𝐷 · 𝑁)) mod 𝑁) |
46 | 34, 45 | eqtri 2846 | . . 3 ⊢ (((𝐴↑𝐵) · (𝐴↑𝐶)) mod 𝑁) = ((𝑀 + (𝐷 · 𝑁)) mod 𝑁) |
47 | 41 | nn0rei 11911 | . . . 4 ⊢ 𝑀 ∈ ℝ |
48 | modcyc 13277 | . . . 4 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ+ ∧ 𝐷 ∈ ℤ) → ((𝑀 + (𝐷 · 𝑁)) mod 𝑁) = (𝑀 mod 𝑁)) | |
49 | 47, 27, 36, 48 | mp3an 1457 | . . 3 ⊢ ((𝑀 + (𝐷 · 𝑁)) mod 𝑁) = (𝑀 mod 𝑁) |
50 | 46, 49 | eqtri 2846 | . 2 ⊢ (((𝐴↑𝐵) · (𝐴↑𝐶)) mod 𝑁) = (𝑀 mod 𝑁) |
51 | 10, 50 | eqtri 2846 | 1 ⊢ ((𝐴↑𝐸) mod 𝑁) = (𝑀 mod 𝑁) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ⊤wtru 1538 ∈ wcel 2114 (class class class)co 7158 ℂcc 10537 ℝcr 10538 + caddc 10542 · cmul 10544 ℕcn 11640 ℕ0cn0 11900 ℤcz 11984 ℝ+crp 12392 mod cmo 13240 ↑cexp 13432 |
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 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-sup 8908 df-inf 8909 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-n0 11901 df-z 11985 df-uz 12247 df-rp 12393 df-fl 13165 df-mod 13241 df-seq 13373 df-exp 13433 |
This theorem is referenced by: mod2xi 16407 modxp1i 16408 1259lem3 16468 1259lem4 16469 2503lem2 16473 4001lem3 16478 |
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