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Theorem flval 12635
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 4689 . . . 4 (𝑦 = 𝐴 → (𝑥𝑦𝑥𝐴))
2 breq1 4688 . . . 4 (𝑦 = 𝐴 → (𝑦 < (𝑥 + 1) ↔ 𝐴 < (𝑥 + 1)))
31, 2anbi12d 747 . . 3 (𝑦 = 𝐴 → ((𝑥𝑦𝑦 < (𝑥 + 1)) ↔ (𝑥𝐴𝐴 < (𝑥 + 1))))
43riotabidv 6653 . 2 (𝑦 = 𝐴 → (𝑥 ∈ ℤ (𝑥𝑦𝑦 < (𝑥 + 1))) = (𝑥 ∈ ℤ (𝑥𝐴𝐴 < (𝑥 + 1))))
5 df-fl 12633 . 2 ⌊ = (𝑦 ∈ ℝ ↦ (𝑥 ∈ ℤ (𝑥𝑦𝑦 < (𝑥 + 1))))
6 riotaex 6655 . 2 (𝑥 ∈ ℤ (𝑥𝐴𝐴 < (𝑥 + 1))) ∈ V
74, 5, 6fvmpt 6321 1 (𝐴 ∈ ℝ → (⌊‘𝐴) = (𝑥 ∈ ℤ (𝑥𝐴𝐴 < (𝑥 + 1))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030   class class class wbr 4685  cfv 5926  crio 6650  (class class class)co 6690  cr 9973  1c1 9975   + caddc 9977   < clt 10112  cle 10113  cz 11415  cfl 12631
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-riota 6651  df-fl 12633
This theorem is referenced by:  flcl  12636  fllelt  12638  flflp1  12648  flbi  12657  dfceil2  12680  ltflcei  33527
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