Theorem modval 13877

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 7422	. . . 4 ⊢ (𝑥 = 𝐴 → (⌊‘(𝑥 / 𝑦)) = (⌊‘(𝐴 / 𝑦)))
2	1	oveq2d 7415	. . 3 ⊢ (𝑥 = 𝐴 → (𝑦 · (⌊‘(𝑥 / 𝑦))) = (𝑦 · (⌊‘(𝐴 / 𝑦))))
3		oveq12 7408	. . 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 7407	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴 / 𝑦) = (𝐴 / 𝐵))
6	5	fveq2d 6876	. . . 4 ⊢ (𝑦 = 𝐵 → (⌊‘(𝐴 / 𝑦)) = (⌊‘(𝐴 / 𝐵)))
7		oveq12 7408	. . . 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 7415	. 2 ⊢ (𝑦 = 𝐵 → (𝐴 − (𝑦 · (⌊‘(𝐴 / 𝑦)))) = (𝐴 − (𝐵 · (⌊‘(𝐴 / 𝐵)))))
10		df-mod 13876	. 2 ⊢ mod = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ+ ↦ (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦)))))
11		ovex 7432	. 2 ⊢ (𝐴 − (𝐵 · (⌊‘(𝐴 / 𝐵)))) ∈ V
12	4, 9, 10, 11	ovmpo 7561	1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (𝐴 mod 𝐵) = (𝐴 − (𝐵 · (⌊‘(𝐴 / 𝐵)))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2107  ‘cfv 6527  (class class class)co 7399  ℝcr 11120   · cmul 11126   − cmin 11458   / cdiv 11886  ℝ⁺crp 13000  ⌊cfl 13796   mod cmo 13875

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 2706  ax-sep 5263  ax-nul 5273  ax-pr 5399

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 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rab 3414  df-v 3459  df-sbc 3764  df-dif 3927  df-un 3929  df-ss 3941  df-nul 4307  df-if 4499  df-sn 4600  df-pr 4602  df-op 4606  df-uni 4881  df-br 5117  df-opab 5179  df-id 5545  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-iota 6480  df-fun 6529  df-fv 6535  df-ov 7402  df-oprab 7403  df-mpo 7404  df-mod 13876

This theorem is referenced by:  modvalr  13878  modcl  13879  mod0  13882  modge0  13885  modlt  13886  moddiffl  13888  modfrac  13890  modmulnn  13895  zmodcl  13897  modid  13902  modcyc  13912  modadd1  13914  modmul1  13931  moddi  13946  modsubdir  13947  modirr  13949  iexpcyc  14213  digit2  14242  dvdsmod  16333  divalgmod  16410  modgcd  16536  bezoutlem3  16545  prmdiv  16789  odzdvds  16800  fldivp1  16902  mulgmodid  19081  odmodnn0  19506  odmod  19512  gexdvds  19550  zringlpirlem3  21410  sineq0  26469  efif1olem2  26488  lgseisenlem4  27325  dchrisumlem1  27436  ostth2lem2  27581  sineq0ALT  44888  ltmod  45597  fourierswlem  46189