Theorem modval 13912

Description: The value of the modulo operation. The modulo congruence notation of number theory, 𝐽≡𝐾 (modulo 𝑁), can be expressed in our notation as (𝐽 mod 𝑁) = (𝐾 mod 𝑁). Definition 1 in Knuth, The Art of Computer Programming, Vol. I (1972), p. 38. Knuth uses "mod" for the operation and "modulo" for the congruence. Unlike Knuth, we restrict the second argument to positive reals to simplify certain theorems. (This also gives us future flexibility to extend it to any one of several different conventions for a zero or negative second argument, should there be an advantage in doing so.) (Contributed by NM, 10-Nov-2008.) (Revised by Mario Carneiro, 3-Nov-2013.)

Assertion

Ref	Expression
modval	⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (𝐴 mod 𝐵) = (𝐴 − (𝐵 · (⌊‘(𝐴 / 𝐵)))))

Proof of Theorem modval

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvoveq1 7455	. . . 4 ⊢ (𝑥 = 𝐴 → (⌊‘(𝑥 / 𝑦)) = (⌊‘(𝐴 / 𝑦)))
2	1	oveq2d 7448	. . 3 ⊢ (𝑥 = 𝐴 → (𝑦 · (⌊‘(𝑥 / 𝑦))) = (𝑦 · (⌊‘(𝐴 / 𝑦))))
3		oveq12 7441	. . 3 ⊢ ((𝑥 = 𝐴 ∧ (𝑦 · (⌊‘(𝑥 / 𝑦))) = (𝑦 · (⌊‘(𝐴 / 𝑦)))) → (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦)))) = (𝐴 − (𝑦 · (⌊‘(𝐴 / 𝑦)))))
4	2, 3	mpdan 687	. 2 ⊢ (𝑥 = 𝐴 → (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦)))) = (𝐴 − (𝑦 · (⌊‘(𝐴 / 𝑦)))))
5		oveq2 7440	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴 / 𝑦) = (𝐴 / 𝐵))
6	5	fveq2d 6909	. . . 4 ⊢ (𝑦 = 𝐵 → (⌊‘(𝐴 / 𝑦)) = (⌊‘(𝐴 / 𝐵)))
7		oveq12 7441	. . . 4 ⊢ ((𝑦 = 𝐵 ∧ (⌊‘(𝐴 / 𝑦)) = (⌊‘(𝐴 / 𝐵))) → (𝑦 · (⌊‘(𝐴 / 𝑦))) = (𝐵 · (⌊‘(𝐴 / 𝐵))))
8	6, 7	mpdan 687	. . 3 ⊢ (𝑦 = 𝐵 → (𝑦 · (⌊‘(𝐴 / 𝑦))) = (𝐵 · (⌊‘(𝐴 / 𝐵))))
9	8	oveq2d 7448	. 2 ⊢ (𝑦 = 𝐵 → (𝐴 − (𝑦 · (⌊‘(𝐴 / 𝑦)))) = (𝐴 − (𝐵 · (⌊‘(𝐴 / 𝐵)))))
10		df-mod 13911	. 2 ⊢ mod = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ+ ↦ (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦)))))
11		ovex 7465	. 2 ⊢ (𝐴 − (𝐵 · (⌊‘(𝐴 / 𝐵)))) ∈ V
12	4, 9, 10, 11	ovmpo 7594	1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (𝐴 mod 𝐵) = (𝐴 − (𝐵 · (⌊‘(𝐴 / 𝐵)))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2107  ‘cfv 6560  (class class class)co 7432  ℝcr 11155   · cmul 11161   − cmin 11493   / cdiv 11921  ℝ⁺crp 13035  ⌊cfl 13831   mod cmo 13910

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-iota 6513  df-fun 6562  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-mod 13911

This theorem is referenced by:  modvalr  13913  modcl  13914  mod0  13917  modge0  13920  modlt  13921  moddiffl  13923  modfrac  13925  modmulnn  13930  zmodcl  13932  modid  13937  modcyc  13947  modadd1  13949  modmul1  13966  moddi  13981  modsubdir  13982  modirr  13984  iexpcyc  14247  digit2  14276  dvdsmod  16367  divalgmod  16444  modgcd  16570  bezoutlem3  16579  prmdiv  16823  odzdvds  16834  fldivp1  16936  mulgmodid  19132  odmodnn0  19559  odmod  19565  gexdvds  19603  zringlpirlem3  21476  sineq0  26567  efif1olem2  26586  lgseisenlem4  27423  dchrisumlem1  27534  ostth2lem2  27679  sineq0ALT  44962  ltmod  45658  fourierswlem  46250