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Theorem modval 13231
Description: The value of the modulo operation. The modulo congruence notation of number theory, 𝐽𝐾 (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.
StepHypRef Expression
1 fvoveq1 7171 . . . 4 (𝑥 = 𝐴 → (⌊‘(𝑥 / 𝑦)) = (⌊‘(𝐴 / 𝑦)))
21oveq2d 7164 . . 3 (𝑥 = 𝐴 → (𝑦 · (⌊‘(𝑥 / 𝑦))) = (𝑦 · (⌊‘(𝐴 / 𝑦))))
3 oveq12 7157 . . 3 ((𝑥 = 𝐴 ∧ (𝑦 · (⌊‘(𝑥 / 𝑦))) = (𝑦 · (⌊‘(𝐴 / 𝑦)))) → (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦)))) = (𝐴 − (𝑦 · (⌊‘(𝐴 / 𝑦)))))
42, 3mpdan 685 . 2 (𝑥 = 𝐴 → (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦)))) = (𝐴 − (𝑦 · (⌊‘(𝐴 / 𝑦)))))
5 oveq2 7156 . . . . 5 (𝑦 = 𝐵 → (𝐴 / 𝑦) = (𝐴 / 𝐵))
65fveq2d 6667 . . . 4 (𝑦 = 𝐵 → (⌊‘(𝐴 / 𝑦)) = (⌊‘(𝐴 / 𝐵)))
7 oveq12 7157 . . . 4 ((𝑦 = 𝐵 ∧ (⌊‘(𝐴 / 𝑦)) = (⌊‘(𝐴 / 𝐵))) → (𝑦 · (⌊‘(𝐴 / 𝑦))) = (𝐵 · (⌊‘(𝐴 / 𝐵))))
86, 7mpdan 685 . . 3 (𝑦 = 𝐵 → (𝑦 · (⌊‘(𝐴 / 𝑦))) = (𝐵 · (⌊‘(𝐴 / 𝐵))))
98oveq2d 7164 . 2 (𝑦 = 𝐵 → (𝐴 − (𝑦 · (⌊‘(𝐴 / 𝑦)))) = (𝐴 − (𝐵 · (⌊‘(𝐴 / 𝐵)))))
10 df-mod 13230 . 2 mod = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ+ ↦ (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦)))))
11 ovex 7181 . 2 (𝐴 − (𝐵 · (⌊‘(𝐴 / 𝐵)))) ∈ V
124, 9, 10, 11ovmpo 7302 1 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (𝐴 mod 𝐵) = (𝐴 − (𝐵 · (⌊‘(𝐴 / 𝐵)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1530  wcel 2107  cfv 6348  (class class class)co 7148  cr 10528   · cmul 10534  cmin 10862   / cdiv 11289  +crp 12381  cfl 13152   mod cmo 13229
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-ov 7151  df-oprab 7152  df-mpo 7153  df-mod 13230
This theorem is referenced by:  modvalr  13232  modcl  13233  mod0  13236  modge0  13239  modlt  13240  moddiffl  13242  modfrac  13244  modmulnn  13249  zmodcl  13251  modid  13256  modcyc  13266  modadd1  13268  modmul1  13284  moddi  13299  modsubdir  13300  modirr  13302  iexpcyc  13561  digit2  13589  dvdsmod  15670  divalgmod  15749  modgcd  15872  bezoutlem3  15881  prmdiv  16114  odzdvds  16124  fldivp1  16225  mulgmodid  18258  odmodnn0  18660  odmod  18666  gexdvds  18701  zringlpirlem3  20625  sineq0  25101  efif1olem2  25119  lgseisenlem4  25946  dchrisumlem1  26057  ostth2lem2  26202  sineq0ALT  41256  ltmod  41903  fourierswlem  42500
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