Theorem modval 13891

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 7419	. . . 4 ⊢ (𝑥 = 𝐴 → (⌊‘(𝑥 / 𝑦)) = (⌊‘(𝐴 / 𝑦)))
2	1	oveq2d 7412	. . 3 ⊢ (𝑥 = 𝐴 → (𝑦 · (⌊‘(𝑥 / 𝑦))) = (𝑦 · (⌊‘(𝐴 / 𝑦))))
3		oveq12 7405	. . 3 ⊢ ((𝑥 = 𝐴 ∧ (𝑦 · (⌊‘(𝑥 / 𝑦))) = (𝑦 · (⌊‘(𝐴 / 𝑦)))) → (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦)))) = (𝐴 − (𝑦 · (⌊‘(𝐴 / 𝑦)))))
4	2, 3	mpdan 697	. 2 ⊢ (𝑥 = 𝐴 → (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦)))) = (𝐴 − (𝑦 · (⌊‘(𝐴 / 𝑦)))))
5		oveq2 7404	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴 / 𝑦) = (𝐴 / 𝐵))
6	5	fveq2d 6871	. . . 4 ⊢ (𝑦 = 𝐵 → (⌊‘(𝐴 / 𝑦)) = (⌊‘(𝐴 / 𝐵)))
7		oveq12 7405	. . . 4 ⊢ ((𝑦 = 𝐵 ∧ (⌊‘(𝐴 / 𝑦)) = (⌊‘(𝐴 / 𝐵))) → (𝑦 · (⌊‘(𝐴 / 𝑦))) = (𝐵 · (⌊‘(𝐴 / 𝐵))))
8	6, 7	mpdan 697	. . 3 ⊢ (𝑦 = 𝐵 → (𝑦 · (⌊‘(𝐴 / 𝑦))) = (𝐵 · (⌊‘(𝐴 / 𝐵))))
9	8	oveq2d 7412	. 2 ⊢ (𝑦 = 𝐵 → (𝐴 − (𝑦 · (⌊‘(𝐴 / 𝑦)))) = (𝐴 − (𝐵 · (⌊‘(𝐴 / 𝐵)))))
10		df-mod 13890	. 2 ⊢ mod = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ+ ↦ (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦)))))
11		ovex 7429	. 2 ⊢ (𝐴 − (𝐵 · (⌊‘(𝐴 / 𝐵)))) ∈ V
12	4, 9, 10, 11	ovmpo 7556	1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (𝐴 mod 𝐵) = (𝐴 − (𝐵 · (⌊‘(𝐴 / 𝐵)))))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1561   ∈ wcel 2143  ‘cfv 6521  (class class class)co 7396  ℝcr 11083   · cmul 11089   − cmin 11425   / cdiv 11855  ℝ⁺crp 13003  ⌊cfl 13810   mod cmo 13889

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5247  ax-nul 5257  ax-pr 5391

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3078  df-rex 3088  df-rab 3416  df-v 3457  df-sbc 3746  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5102  df-opab 5164  df-id 5543  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-iota 6477  df-fun 6523  df-fv 6529  df-ov 7399  df-oprab 7400  df-mpo 7401  df-mod 13890

This theorem is referenced by:  modvalr  13892  modcl  13893  mod0  13896  modge0  13899  modlt  13900  moddiffl  13902  modfrac  13904  modmulnn  13909  zmodcl  13911  modid  13916  modcyc  13926  modadd1  13928  modmul1  13947  moddi  13962  modsubdir  13963  modirr  13965  iexpcyc  14230  digit2  14259  dvdsmod  16373  divalgmod  16450  modgcd  16576  bezoutlem3  16585  prmdiv  16830  odzdvds  16841  fldivp1  16943  mulgmodid  19165  odmodnn0  19590  odmod  19596  gexdvds  19634  zringlpirlem3  21523  sineq0  26596  efif1olem2  26615  lgseisenlem4  27449  dchrisumlem1  27560  ostth2lem2  27705  sineq0ALT  45503  ltmod  46203  fourierswlem  46795