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Mirrors > Home > ILE Home > Th. List > dvdsmodexp | GIF version |
Description: If a positive integer divides another integer, this other integer is equal to its positive powers modulo the positive integer. (Formerly part of the proof for fermltl 12113). (Contributed by Mario Carneiro, 28-Feb-2014.) (Revised by AV, 19-Mar-2022.) |
Ref | Expression |
---|---|
dvdsmodexp | ⊢ ((𝑁 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∥ 𝐴) → ((𝐴↑𝐵) mod 𝑁) = (𝐴 mod 𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvdszrcl 11692 | . . 3 ⊢ (𝑁 ∥ 𝐴 → (𝑁 ∈ ℤ ∧ 𝐴 ∈ ℤ)) | |
2 | dvdsmod0 11693 | . . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝐴) → (𝐴 mod 𝑁) = 0) | |
3 | 2 | 3ad2antl2 1145 | . . . . . . . 8 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∥ 𝐴) → (𝐴 mod 𝑁) = 0) |
4 | 3 | ex 114 | . . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝑁 ∥ 𝐴 → (𝐴 mod 𝑁) = 0)) |
5 | simpl3 987 | . . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (𝐴 mod 𝑁) = 0) → 𝐵 ∈ ℕ) | |
6 | 5 | 0expd 10571 | . . . . . . . . . 10 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (𝐴 mod 𝑁) = 0) → (0↑𝐵) = 0) |
7 | 6 | oveq1d 5840 | . . . . . . . . 9 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (𝐴 mod 𝑁) = 0) → ((0↑𝐵) mod 𝑁) = (0 mod 𝑁)) |
8 | simpl1 985 | . . . . . . . . . 10 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (𝐴 mod 𝑁) = 0) → 𝐴 ∈ ℤ) | |
9 | 0zd 9180 | . . . . . . . . . 10 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (𝐴 mod 𝑁) = 0) → 0 ∈ ℤ) | |
10 | nnnn0 9098 | . . . . . . . . . . . 12 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℕ0) | |
11 | 10 | 3ad2ant3 1005 | . . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℕ) → 𝐵 ∈ ℕ0) |
12 | 11 | adantr 274 | . . . . . . . . . 10 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (𝐴 mod 𝑁) = 0) → 𝐵 ∈ ℕ0) |
13 | simpl2 986 | . . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (𝐴 mod 𝑁) = 0) → 𝑁 ∈ ℕ) | |
14 | nnq 9543 | . . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℚ) | |
15 | 13, 14 | syl 14 | . . . . . . . . . 10 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (𝐴 mod 𝑁) = 0) → 𝑁 ∈ ℚ) |
16 | nnrp 9571 | . . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ+) | |
17 | 16 | 3ad2ant2 1004 | . . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℕ) → 𝑁 ∈ ℝ+) |
18 | 17 | adantr 274 | . . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (𝐴 mod 𝑁) = 0) → 𝑁 ∈ ℝ+) |
19 | 18 | rpgt0d 9607 | . . . . . . . . . 10 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (𝐴 mod 𝑁) = 0) → 0 < 𝑁) |
20 | simpr 109 | . . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (𝐴 mod 𝑁) = 0) → (𝐴 mod 𝑁) = 0) | |
21 | q0mod 10258 | . . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℚ ∧ 0 < 𝑁) → (0 mod 𝑁) = 0) | |
22 | 15, 19, 21 | syl2anc 409 | . . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (𝐴 mod 𝑁) = 0) → (0 mod 𝑁) = 0) |
23 | 20, 22 | eqtr4d 2193 | . . . . . . . . . 10 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (𝐴 mod 𝑁) = 0) → (𝐴 mod 𝑁) = (0 mod 𝑁)) |
24 | 8, 9, 12, 15, 19, 23 | modqexp 10548 | . . . . . . . . 9 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (𝐴 mod 𝑁) = 0) → ((𝐴↑𝐵) mod 𝑁) = ((0↑𝐵) mod 𝑁)) |
25 | 7, 24, 23 | 3eqtr4d 2200 | . . . . . . . 8 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (𝐴 mod 𝑁) = 0) → ((𝐴↑𝐵) mod 𝑁) = (𝐴 mod 𝑁)) |
26 | 25 | ex 114 | . . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ((𝐴 mod 𝑁) = 0 → ((𝐴↑𝐵) mod 𝑁) = (𝐴 mod 𝑁))) |
27 | 4, 26 | syld 45 | . . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝑁 ∥ 𝐴 → ((𝐴↑𝐵) mod 𝑁) = (𝐴 mod 𝑁))) |
28 | 27 | 3exp 1184 | . . . . 5 ⊢ (𝐴 ∈ ℤ → (𝑁 ∈ ℕ → (𝐵 ∈ ℕ → (𝑁 ∥ 𝐴 → ((𝐴↑𝐵) mod 𝑁) = (𝐴 mod 𝑁))))) |
29 | 28 | com24 87 | . . . 4 ⊢ (𝐴 ∈ ℤ → (𝑁 ∥ 𝐴 → (𝐵 ∈ ℕ → (𝑁 ∈ ℕ → ((𝐴↑𝐵) mod 𝑁) = (𝐴 mod 𝑁))))) |
30 | 29 | adantl 275 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ 𝐴 ∈ ℤ) → (𝑁 ∥ 𝐴 → (𝐵 ∈ ℕ → (𝑁 ∈ ℕ → ((𝐴↑𝐵) mod 𝑁) = (𝐴 mod 𝑁))))) |
31 | 1, 30 | mpcom 36 | . 2 ⊢ (𝑁 ∥ 𝐴 → (𝐵 ∈ ℕ → (𝑁 ∈ ℕ → ((𝐴↑𝐵) mod 𝑁) = (𝐴 mod 𝑁)))) |
32 | 31 | 3imp31 1178 | 1 ⊢ ((𝑁 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∥ 𝐴) → ((𝐴↑𝐵) mod 𝑁) = (𝐴 mod 𝑁)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∧ w3a 963 = wceq 1335 ∈ wcel 2128 class class class wbr 3966 (class class class)co 5825 0cc0 7733 < clt 7913 ℕcn 8834 ℕ0cn0 9091 ℤcz 9168 ℚcq 9529 ℝ+crp 9561 mod cmo 10225 ↑cexp 10422 ∥ cdvds 11687 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4080 ax-sep 4083 ax-nul 4091 ax-pow 4136 ax-pr 4170 ax-un 4394 ax-setind 4497 ax-iinf 4548 ax-cnex 7824 ax-resscn 7825 ax-1cn 7826 ax-1re 7827 ax-icn 7828 ax-addcl 7829 ax-addrcl 7830 ax-mulcl 7831 ax-mulrcl 7832 ax-addcom 7833 ax-mulcom 7834 ax-addass 7835 ax-mulass 7836 ax-distr 7837 ax-i2m1 7838 ax-0lt1 7839 ax-1rid 7840 ax-0id 7841 ax-rnegex 7842 ax-precex 7843 ax-cnre 7844 ax-pre-ltirr 7845 ax-pre-ltwlin 7846 ax-pre-lttrn 7847 ax-pre-apti 7848 ax-pre-ltadd 7849 ax-pre-mulgt0 7850 ax-pre-mulext 7851 ax-arch 7852 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rmo 2443 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-if 3506 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3774 df-int 3809 df-iun 3852 df-br 3967 df-opab 4027 df-mpt 4028 df-tr 4064 df-id 4254 df-po 4257 df-iso 4258 df-iord 4327 df-on 4329 df-ilim 4330 df-suc 4332 df-iom 4551 df-xp 4593 df-rel 4594 df-cnv 4595 df-co 4596 df-dm 4597 df-rn 4598 df-res 4599 df-ima 4600 df-iota 5136 df-fun 5173 df-fn 5174 df-f 5175 df-f1 5176 df-fo 5177 df-f1o 5178 df-fv 5179 df-riota 5781 df-ov 5828 df-oprab 5829 df-mpo 5830 df-1st 6089 df-2nd 6090 df-recs 6253 df-frec 6339 df-pnf 7915 df-mnf 7916 df-xr 7917 df-ltxr 7918 df-le 7919 df-sub 8049 df-neg 8050 df-reap 8451 df-ap 8458 df-div 8547 df-inn 8835 df-n0 9092 df-z 9169 df-uz 9441 df-q 9530 df-rp 9562 df-fl 10173 df-mod 10226 df-seqfrec 10349 df-exp 10423 df-dvds 11688 |
This theorem is referenced by: fermltl 12113 |
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