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Theorem flval 13760
Description: Value of the floor (greatest integer) function. The floor of 𝐴 is the (unique) integer less than or equal to 𝐴 whose successor is strictly greater than 𝐴. (Contributed by NM, 14-Nov-2004.) (Revised by Mario Carneiro, 2-Nov-2013.)
Assertion
Ref Expression
flval (𝐴 ∈ ℝ → (⌊‘𝐴) = (𝑥 ∈ ℤ (𝑥𝐴𝐴 < (𝑥 + 1))))
Distinct variable group:   𝑥,𝐴

Proof of Theorem flval
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 breq2 5143 . . . 4 (𝑦 = 𝐴 → (𝑥𝑦𝑥𝐴))
2 breq1 5142 . . . 4 (𝑦 = 𝐴 → (𝑦 < (𝑥 + 1) ↔ 𝐴 < (𝑥 + 1)))
31, 2anbi12d 630 . . 3 (𝑦 = 𝐴 → ((𝑥𝑦𝑦 < (𝑥 + 1)) ↔ (𝑥𝐴𝐴 < (𝑥 + 1))))
43riotabidv 7360 . 2 (𝑦 = 𝐴 → (𝑥 ∈ ℤ (𝑥𝑦𝑦 < (𝑥 + 1))) = (𝑥 ∈ ℤ (𝑥𝐴𝐴 < (𝑥 + 1))))
5 df-fl 13758 . 2 ⌊ = (𝑦 ∈ ℝ ↦ (𝑥 ∈ ℤ (𝑥𝑦𝑦 < (𝑥 + 1))))
6 riotaex 7362 . 2 (𝑥 ∈ ℤ (𝑥𝐴𝐴 < (𝑥 + 1))) ∈ V
74, 5, 6fvmpt 6989 1 (𝐴 ∈ ℝ → (⌊‘𝐴) = (𝑥 ∈ ℤ (𝑥𝐴𝐴 < (𝑥 + 1))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wcel 2098   class class class wbr 5139  cfv 6534  crio 7357  (class class class)co 7402  cr 11106  1c1 11108   + caddc 11110   < clt 11247  cle 11248  cz 12557  cfl 13756
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-iota 6486  df-fun 6536  df-fv 6542  df-riota 7358  df-fl 13758
This theorem is referenced by:  flcl  13761  fllelt  13763  flflp1  13773  flbi  13782  dfceil2  13805  ltflcei  36979
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