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 (𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1))))

Distinct variable group:   𝑥,𝐴

Proof of Theorem flval

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq2 3826	. . . 4 ⊢ (𝑦 = 𝐴 → (𝑥 ≤ 𝑦 ↔ 𝑥 ≤ 𝐴))
2		breq1 3825	. . . 4 ⊢ (𝑦 = 𝐴 → (𝑦 < (𝑥 + 1) ↔ 𝐴 < (𝑥 + 1)))
3	1, 2	anbi12d 457	. . 3 ⊢ (𝑦 = 𝐴 → ((𝑥 ≤ 𝑦 ∧ 𝑦 < (𝑥 + 1)) ↔ (𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1))))
4	3	riotabidv 5573	. 2 ⊢ (𝑦 = 𝐴 → (℩𝑥 ∈ ℤ (𝑥 ≤ 𝑦 ∧ 𝑦 < (𝑥 + 1))) = (℩𝑥 ∈ ℤ (𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1))))
5		df-fl 9608	. 2 ⊢ ⌊ = (𝑦 ∈ ℝ ↦ (℩𝑥 ∈ ℤ (𝑥 ≤ 𝑦 ∧ 𝑦 < (𝑥 + 1))))
6		zex 8695	. . 3 ⊢ ℤ ∈ V
7		riotaexg 5575	. . 3 ⊢ (ℤ ∈ V → (℩𝑥 ∈ ℤ (𝑥 ≤ 𝑦 ∧ 𝑦 < (𝑥 + 1))) ∈ V)
8	6, 7	ax-mp 7	. 2 ⊢ (℩𝑥 ∈ ℤ (𝑥 ≤ 𝑦 ∧ 𝑦 < (𝑥 + 1))) ∈ V
9	4, 5, 8	fvmpt3i 5349	1 ⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) = (℩𝑥 ∈ ℤ (𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1))))

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  ℝcr 7296  1c1 7298   + caddc 7300   < clt 7469   ≤ cle 7470  ℤcz 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