Theorem modval 13242

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 7181	. . . 4 ⊢ (𝑥 = 𝐴 → (⌊‘(𝑥 / 𝑦)) = (⌊‘(𝐴 / 𝑦)))
2	1	oveq2d 7174	. . 3 ⊢ (𝑥 = 𝐴 → (𝑦 · (⌊‘(𝑥 / 𝑦))) = (𝑦 · (⌊‘(𝐴 / 𝑦))))
3		oveq12 7167	. . 3 ⊢ ((𝑥 = 𝐴 ∧ (𝑦 · (⌊‘(𝑥 / 𝑦))) = (𝑦 · (⌊‘(𝐴 / 𝑦)))) → (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦)))) = (𝐴 − (𝑦 · (⌊‘(𝐴 / 𝑦)))))
4	2, 3	mpdan 685	. 2 ⊢ (𝑥 = 𝐴 → (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦)))) = (𝐴 − (𝑦 · (⌊‘(𝐴 / 𝑦)))))
5		oveq2 7166	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴 / 𝑦) = (𝐴 / 𝐵))
6	5	fveq2d 6676	. . . 4 ⊢ (𝑦 = 𝐵 → (⌊‘(𝐴 / 𝑦)) = (⌊‘(𝐴 / 𝐵)))
7		oveq12 7167	. . . 4 ⊢ ((𝑦 = 𝐵 ∧ (⌊‘(𝐴 / 𝑦)) = (⌊‘(𝐴 / 𝐵))) → (𝑦 · (⌊‘(𝐴 / 𝑦))) = (𝐵 · (⌊‘(𝐴 / 𝐵))))
8	6, 7	mpdan 685	. . 3 ⊢ (𝑦 = 𝐵 → (𝑦 · (⌊‘(𝐴 / 𝑦))) = (𝐵 · (⌊‘(𝐴 / 𝐵))))
9	8	oveq2d 7174	. 2 ⊢ (𝑦 = 𝐵 → (𝐴 − (𝑦 · (⌊‘(𝐴 / 𝑦)))) = (𝐴 − (𝐵 · (⌊‘(𝐴 / 𝐵)))))
10		df-mod 13241	. 2 ⊢ mod = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ+ ↦ (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦)))))
11		ovex 7191	. 2 ⊢ (𝐴 − (𝐵 · (⌊‘(𝐴 / 𝐵)))) ∈ V
12	4, 9, 10, 11	ovmpo 7312	1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (𝐴 mod 𝐵) = (𝐴 − (𝐵 · (⌊‘(𝐴 / 𝐵)))))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ‘cfv 6357  (class class class)co 7158  ℝcr 10538   · cmul 10544   − cmin 10872   / cdiv 11299  ℝ⁺crp 12392  ⌊cfl 13163   mod cmo 13240

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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-iota 6316  df-fun 6359  df-fv 6365  df-ov 7161  df-oprab 7162  df-mpo 7163  df-mod 13241

This theorem is referenced by:  modvalr  13243  modcl  13244  mod0  13247  modge0  13250  modlt  13251  moddiffl  13253  modfrac  13255  modmulnn  13260  zmodcl  13262  modid  13267  modcyc  13277  modadd1  13279  modmul1  13295  moddi  13310  modsubdir  13311  modirr  13313  iexpcyc  13572  digit2  13600  dvdsmod  15680  divalgmod  15759  modgcd  15882  bezoutlem3  15891  prmdiv  16124  odzdvds  16134  fldivp1  16235  mulgmodid  18268  odmodnn0  18670  odmod  18676  gexdvds  18711  zringlpirlem3  20635  sineq0  25111  efif1olem2  25129  lgseisenlem4  25956  dchrisumlem1  26067  ostth2lem2  26212  sineq0ALT  41278  ltmod  41926  fourierswlem  42522