modcyc2 - Metamath Proof Explorer

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Theorem modcyc2 13728

Description: The modulo operation is periodic. (Contributed by NM, 12-Nov-2008.)

**Assertion**
Ref	Expression
modcyc2	⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝑁 ∈ ℤ) → ((𝐴 − (𝐵 · 𝑁)) mod 𝐵) = (𝐴 mod 𝐵))

**Proof of Theorem modcyc2**
Step	Hyp	Ref	Expression
1		recn 11062	. . . 4 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ)
2		rpcn 12841	. . . 4 ⊢ (𝐵 ∈ ℝ⁺ → 𝐵 ∈ ℂ)
3		zcn 12425	. . . 4 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ)
4		mulneg1 11512	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (-𝑁 · 𝐵) = -(𝑁 · 𝐵))
5	4	ancoms 459	. . . . . . . 8 ⊢ ((𝐵 ∈ ℂ ∧ 𝑁 ∈ ℂ) → (-𝑁 · 𝐵) = -(𝑁 · 𝐵))
6		mulcom 11058	. . . . . . . . 9 ⊢ ((𝐵 ∈ ℂ ∧ 𝑁 ∈ ℂ) → (𝐵 · 𝑁) = (𝑁 · 𝐵))
7	6	negeqd 11316	. . . . . . . 8 ⊢ ((𝐵 ∈ ℂ ∧ 𝑁 ∈ ℂ) → -(𝐵 · 𝑁) = -(𝑁 · 𝐵))
8	5, 7	eqtr4d 2779	. . . . . . 7 ⊢ ((𝐵 ∈ ℂ ∧ 𝑁 ∈ ℂ) → (-𝑁 · 𝐵) = -(𝐵 · 𝑁))
9	8	3adant1 1129	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℂ) → (-𝑁 · 𝐵) = -(𝐵 · 𝑁))
10	9	oveq2d 7353	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℂ) → (𝐴 + (-𝑁 · 𝐵)) = (𝐴 + -(𝐵 · 𝑁)))
11		mulcl 11056	. . . . . . 7 ⊢ ((𝐵 ∈ ℂ ∧ 𝑁 ∈ ℂ) → (𝐵 · 𝑁) ∈ ℂ)
12		negsub 11370	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (𝐵 · 𝑁) ∈ ℂ) → (𝐴 + -(𝐵 · 𝑁)) = (𝐴 − (𝐵 · 𝑁)))
13	11, 12	sylan2 593	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝑁 ∈ ℂ)) → (𝐴 + -(𝐵 · 𝑁)) = (𝐴 − (𝐵 · 𝑁)))
14	13	3impb 1114	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℂ) → (𝐴 + -(𝐵 · 𝑁)) = (𝐴 − (𝐵 · 𝑁)))
15	10, 14	eqtr2d 2777	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℂ) → (𝐴 − (𝐵 · 𝑁)) = (𝐴 + (-𝑁 · 𝐵)))
16	1, 2, 3, 15	syl3an 1159	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝑁 ∈ ℤ) → (𝐴 − (𝐵 · 𝑁)) = (𝐴 + (-𝑁 · 𝐵)))
17	16	oveq1d 7352	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝑁 ∈ ℤ) → ((𝐴 − (𝐵 · 𝑁)) mod 𝐵) = ((𝐴 + (-𝑁 · 𝐵)) mod 𝐵))
18		znegcl 12456	. . 3 ⊢ (𝑁 ∈ ℤ → -𝑁 ∈ ℤ)
19		modcyc 13727	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ⁺ ∧ -𝑁 ∈ ℤ) → ((𝐴 + (-𝑁 · 𝐵)) mod 𝐵) = (𝐴 mod 𝐵))
20	18, 19	syl3an3 1164	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ⁺ ∧ 𝑁 ∈ ℤ) → ((𝐴 + (-𝑁 · 𝐵)) mod 𝐵) = (𝐴 mod 𝐵))
21	17, 20	eqtrd 2776	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 396 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 (class class class)co 7337 ℂcc 10970 ℝcr 10971 + caddc 10975 · cmul 10977 − cmin 11306 -cneg 11307 ℤcz 12420 ℝ⁺crp 12831 mod cmo 13690

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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-sep 5243 ax-nul 5250 ax-pow 5308 ax-pr 5372 ax-un 7650 ax-cnex 11028 ax-resscn 11029 ax-1cn 11030 ax-icn 11031 ax-addcl 11032 ax-addrcl 11033 ax-mulcl 11034 ax-mulrcl 11035 ax-mulcom 11036 ax-addass 11037 ax-mulass 11038 ax-distr 11039 ax-i2m1 11040 ax-1ne0 11041 ax-1rid 11042 ax-rnegex 11043 ax-rrecex 11044 ax-cnre 11045 ax-pre-lttri 11046 ax-pre-lttrn 11047 ax-pre-ltadd 11048 ax-pre-mulgt0 11049 ax-pre-sup 11050

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3349 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3917 df-nul 4270 df-if 4474 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4853 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5176 df-tr 5210 df-id 5518 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5575 df-we 5577 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-pred 6238 df-ord 6305 df-on 6306 df-lim 6307 df-suc 6308 df-iota 6431 df-fun 6481 df-fn 6482 df-f 6483 df-f1 6484 df-fo 6485 df-f1o 6486 df-fv 6487 df-riota 7293 df-ov 7340 df-oprab 7341 df-mpo 7342 df-om 7781 df-2nd 7900 df-frecs 8167 df-wrecs 8198 df-recs 8272 df-rdg 8311 df-er 8569 df-en 8805 df-dom 8806 df-sdom 8807 df-sup 9299 df-inf 9300 df-pnf 11112 df-mnf 11113 df-xr 11114 df-ltxr 11115 df-le 11116 df-sub 11308 df-neg 11309 df-div 11734 df-nn 12075 df-n0 12335 df-z 12421 df-uz 12684 df-rp 12832 df-fl 13613 df-mod 13691

This theorem is referenced by: modadd1 13729 modmul1 13745 2submod 13753 modsubdir 13761

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