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Mirrors > Home > MPE Home > Th. List > modcyc2 | Structured version Visualization version GIF version |
Description: The modulo operation is periodic. (Contributed by NM, 12-Nov-2008.) |
Ref | Expression |
---|---|
modcyc2 | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ∧ 𝑁 ∈ ℤ) → ((𝐴 − (𝐵 · 𝑁)) mod 𝐵) = (𝐴 mod 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | recn 10621 | . . . 4 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
2 | rpcn 12393 | . . . 4 ⊢ (𝐵 ∈ ℝ+ → 𝐵 ∈ ℂ) | |
3 | zcn 11980 | . . . 4 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ) | |
4 | mulneg1 11070 | . . . . . . . . 9 ⊢ ((𝑁 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (-𝑁 · 𝐵) = -(𝑁 · 𝐵)) | |
5 | 4 | ancoms 461 | . . . . . . . 8 ⊢ ((𝐵 ∈ ℂ ∧ 𝑁 ∈ ℂ) → (-𝑁 · 𝐵) = -(𝑁 · 𝐵)) |
6 | mulcom 10617 | . . . . . . . . 9 ⊢ ((𝐵 ∈ ℂ ∧ 𝑁 ∈ ℂ) → (𝐵 · 𝑁) = (𝑁 · 𝐵)) | |
7 | 6 | negeqd 10874 | . . . . . . . 8 ⊢ ((𝐵 ∈ ℂ ∧ 𝑁 ∈ ℂ) → -(𝐵 · 𝑁) = -(𝑁 · 𝐵)) |
8 | 5, 7 | eqtr4d 2859 | . . . . . . 7 ⊢ ((𝐵 ∈ ℂ ∧ 𝑁 ∈ ℂ) → (-𝑁 · 𝐵) = -(𝐵 · 𝑁)) |
9 | 8 | 3adant1 1126 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℂ) → (-𝑁 · 𝐵) = -(𝐵 · 𝑁)) |
10 | 9 | oveq2d 7166 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℂ) → (𝐴 + (-𝑁 · 𝐵)) = (𝐴 + -(𝐵 · 𝑁))) |
11 | mulcl 10615 | . . . . . . 7 ⊢ ((𝐵 ∈ ℂ ∧ 𝑁 ∈ ℂ) → (𝐵 · 𝑁) ∈ ℂ) | |
12 | negsub 10928 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (𝐵 · 𝑁) ∈ ℂ) → (𝐴 + -(𝐵 · 𝑁)) = (𝐴 − (𝐵 · 𝑁))) | |
13 | 11, 12 | sylan2 594 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝑁 ∈ ℂ)) → (𝐴 + -(𝐵 · 𝑁)) = (𝐴 − (𝐵 · 𝑁))) |
14 | 13 | 3impb 1111 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℂ) → (𝐴 + -(𝐵 · 𝑁)) = (𝐴 − (𝐵 · 𝑁))) |
15 | 10, 14 | eqtr2d 2857 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℂ) → (𝐴 − (𝐵 · 𝑁)) = (𝐴 + (-𝑁 · 𝐵))) |
16 | 1, 2, 3, 15 | syl3an 1156 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ∧ 𝑁 ∈ ℤ) → (𝐴 − (𝐵 · 𝑁)) = (𝐴 + (-𝑁 · 𝐵))) |
17 | 16 | oveq1d 7165 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ∧ 𝑁 ∈ ℤ) → ((𝐴 − (𝐵 · 𝑁)) mod 𝐵) = ((𝐴 + (-𝑁 · 𝐵)) mod 𝐵)) |
18 | znegcl 12011 | . . 3 ⊢ (𝑁 ∈ ℤ → -𝑁 ∈ ℤ) | |
19 | modcyc 13268 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ∧ -𝑁 ∈ ℤ) → ((𝐴 + (-𝑁 · 𝐵)) mod 𝐵) = (𝐴 mod 𝐵)) | |
20 | 18, 19 | syl3an3 1161 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ∧ 𝑁 ∈ ℤ) → ((𝐴 + (-𝑁 · 𝐵)) mod 𝐵) = (𝐴 mod 𝐵)) |
21 | 17, 20 | eqtrd 2856 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ∧ 𝑁 ∈ ℤ) → ((𝐴 − (𝐵 · 𝑁)) mod 𝐵) = (𝐴 mod 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 (class class class)co 7150 ℂcc 10529 ℝcr 10530 + caddc 10534 · cmul 10536 − cmin 10864 -cneg 10865 ℤcz 11975 ℝ+crp 12383 mod cmo 13231 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-sup 8900 df-inf 8901 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-n0 11892 df-z 11976 df-uz 12238 df-rp 12384 df-fl 13156 df-mod 13232 |
This theorem is referenced by: modadd1 13270 modmul1 13286 2submod 13294 modsubdir 13302 |
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