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Theorem flval 9610
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 3826 . . . 4 (𝑦 = 𝐴 → (𝑥𝑦𝑥𝐴))
2 breq1 3825 . . . 4 (𝑦 = 𝐴 → (𝑦 < (𝑥 + 1) ↔ 𝐴 < (𝑥 + 1)))
31, 2anbi12d 457 . . 3 (𝑦 = 𝐴 → ((𝑥𝑦𝑦 < (𝑥 + 1)) ↔ (𝑥𝐴𝐴 < (𝑥 + 1))))
43riotabidv 5573 . 2 (𝑦 = 𝐴 → (𝑥 ∈ ℤ (𝑥𝑦𝑦 < (𝑥 + 1))) = (𝑥 ∈ ℤ (𝑥𝐴𝐴 < (𝑥 + 1))))
5 df-fl 9608 . 2 ⌊ = (𝑦 ∈ ℝ ↦ (𝑥 ∈ ℤ (𝑥𝑦𝑦 < (𝑥 + 1))))
6 zex 8695 . . 3 ℤ ∈ V
7 riotaexg 5575 . . 3 (ℤ ∈ V → (𝑥 ∈ ℤ (𝑥𝑦𝑦 < (𝑥 + 1))) ∈ V)
86, 7ax-mp 7 . 2 (𝑥 ∈ ℤ (𝑥𝑦𝑦 < (𝑥 + 1))) ∈ V
94, 5, 8fvmpt3i 5349 1 (𝐴 ∈ ℝ → (⌊‘𝐴) = (𝑥 ∈ ℤ (𝑥𝐴𝐴 < (𝑥 + 1))))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1287  wcel 1436  Vcvv 2615   class class class wbr 3822  cfv 4983  crio 5570  (class class class)co 5615  cr 7296  1c1 7298   + caddc 7300   < clt 7469  cle 7470  cz 8686  cfl 9606
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3934  ax-pow 3986  ax-pr 4012  ax-un 4236  ax-cnex 7383  ax-resscn 7384
This theorem depends on definitions:  df-bi 115  df-3or 923  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-rab 2364  df-v 2617  df-sbc 2830  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3639  df-br 3823  df-opab 3877  df-mpt 3878  df-id 4096  df-xp 4419  df-rel 4420  df-cnv 4421  df-co 4422  df-dm 4423  df-iota 4948  df-fun 4985  df-fv 4991  df-riota 5571  df-ov 5618  df-neg 7603  df-z 8687  df-fl 9608
This theorem is referenced by:  flqcl  9611  flapcl  9613  flqlelt  9614  flqbi  9628
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