Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 6560 | . . . 4 ⊢ (𝐴↑(𝐵 + 𝐶)) = (𝐴↑𝐸) |
3 | modxai.2 | . . . . . 6 ⊢ 𝐴 ∈ ℕ | |
4 | 3 | nncni 10907 | . . . . 5 ⊢ 𝐴 ∈ ℂ |
5 | modxai.3 | . . . . 5 ⊢ 𝐵 ∈ ℕ0 | |
6 | modxai.7 | . . . . 5 ⊢ 𝐶 ∈ ℕ0 | |
7 | expadd 12764 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) → (𝐴↑(𝐵 + 𝐶)) = ((𝐴↑𝐵) · (𝐴↑𝐶))) | |
8 | 4, 5, 6, 7 | mp3an 1416 | . . . 4 ⊢ (𝐴↑(𝐵 + 𝐶)) = ((𝐴↑𝐵) · (𝐴↑𝐶)) |
9 | 2, 8 | eqtr3i 2634 | . . 3 ⊢ (𝐴↑𝐸) = ((𝐴↑𝐵) · (𝐴↑𝐶)) |
10 | 9 | oveq1i 6559 | . 2 ⊢ ((𝐴↑𝐸) mod 𝑁) = (((𝐴↑𝐵) · (𝐴↑𝐶)) mod 𝑁) |
11 | nnexpcl 12735 | . . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ0) → (𝐴↑𝐵) ∈ ℕ) | |
12 | 3, 5, 11 | mp2an 704 | . . . . . . . 8 ⊢ (𝐴↑𝐵) ∈ ℕ |
13 | 12 | nnzi 11278 | . . . . . . 7 ⊢ (𝐴↑𝐵) ∈ ℤ |
14 | 13 | a1i 11 | . . . . . 6 ⊢ (⊤ → (𝐴↑𝐵) ∈ ℤ) |
15 | modxai.5 | . . . . . . . 8 ⊢ 𝐾 ∈ ℕ0 | |
16 | 15 | nn0zi 11279 | . . . . . . 7 ⊢ 𝐾 ∈ ℤ |
17 | 16 | a1i 11 | . . . . . 6 ⊢ (⊤ → 𝐾 ∈ ℤ) |
18 | nnexpcl 12735 | . . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝐶 ∈ ℕ0) → (𝐴↑𝐶) ∈ ℕ) | |
19 | 3, 6, 18 | mp2an 704 | . . . . . . . 8 ⊢ (𝐴↑𝐶) ∈ ℕ |
20 | 19 | nnzi 11278 | . . . . . . 7 ⊢ (𝐴↑𝐶) ∈ ℤ |
21 | 20 | a1i 11 | . . . . . 6 ⊢ (⊤ → (𝐴↑𝐶) ∈ ℤ) |
22 | modxai.8 | . . . . . . . 8 ⊢ 𝐿 ∈ ℕ0 | |
23 | 22 | nn0zi 11279 | . . . . . . 7 ⊢ 𝐿 ∈ ℤ |
24 | 23 | a1i 11 | . . . . . 6 ⊢ (⊤ → 𝐿 ∈ ℤ) |
25 | modxai.1 | . . . . . . . 8 ⊢ 𝑁 ∈ ℕ | |
26 | nnrp 11718 | . . . . . . . 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 12586 | . . . . 5 ⊢ (⊤ → (((𝐴↑𝐵) · (𝐴↑𝐶)) mod 𝑁) = ((𝐾 · 𝐿) mod 𝑁)) |
34 | 33 | trud 1484 | . . . 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 11259 | . . . . . . . . 9 ⊢ (𝐷 ∈ ℤ → 𝐷 ∈ ℂ) | |
38 | 36, 37 | ax-mp 5 | . . . . . . . 8 ⊢ 𝐷 ∈ ℂ |
39 | 25 | nncni 10907 | . . . . . . . 8 ⊢ 𝑁 ∈ ℂ |
40 | 38, 39 | mulcli 9924 | . . . . . . 7 ⊢ (𝐷 · 𝑁) ∈ ℂ |
41 | modxai.6 | . . . . . . . 8 ⊢ 𝑀 ∈ ℕ0 | |
42 | 41 | nn0cni 11181 | . . . . . . 7 ⊢ 𝑀 ∈ ℂ |
43 | 40, 42 | addcomi 10106 | . . . . . 6 ⊢ ((𝐷 · 𝑁) + 𝑀) = (𝑀 + (𝐷 · 𝑁)) |
44 | 35, 43 | eqtr3i 2634 | . . . . 5 ⊢ (𝐾 · 𝐿) = (𝑀 + (𝐷 · 𝑁)) |
45 | 44 | oveq1i 6559 | . . . 4 ⊢ ((𝐾 · 𝐿) mod 𝑁) = ((𝑀 + (𝐷 · 𝑁)) mod 𝑁) |
46 | 34, 45 | eqtri 2632 | . . 3 ⊢ (((𝐴↑𝐵) · (𝐴↑𝐶)) mod 𝑁) = ((𝑀 + (𝐷 · 𝑁)) mod 𝑁) |
47 | 41 | nn0rei 11180 | . . . 4 ⊢ 𝑀 ∈ ℝ |
48 | modcyc 12567 | . . . 4 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ+ ∧ 𝐷 ∈ ℤ) → ((𝑀 + (𝐷 · 𝑁)) mod 𝑁) = (𝑀 mod 𝑁)) | |
49 | 47, 27, 36, 48 | mp3an 1416 | . . 3 ⊢ ((𝑀 + (𝐷 · 𝑁)) mod 𝑁) = (𝑀 mod 𝑁) |
50 | 46, 49 | eqtri 2632 | . 2 ⊢ (((𝐴↑𝐵) · (𝐴↑𝐶)) mod 𝑁) = (𝑀 mod 𝑁) |
51 | 10, 50 | eqtri 2632 | 1 ⊢ ((𝐴↑𝐸) mod 𝑁) = (𝑀 mod 𝑁) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ⊤wtru 1476 ∈ wcel 1977 (class class class)co 6549 ℂcc 9813 ℝcr 9814 + caddc 9818 · cmul 9820 ℕcn 10897 ℕ0cn0 11169 ℤcz 11254 ℝ+crp 11708 mod cmo 12530 ↑cexp 12722 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-sup 8231 df-inf 8232 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-fl 12455 df-mod 12531 df-seq 12664 df-exp 12723 |
This theorem is referenced by: mod2xi 15611 modxp1i 15612 1259lem3 15678 1259lem4 15679 2503lem2 15683 4001lem3 15688 |
Copyright terms: Public domain | W3C validator |