Theorem flval 10362

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 4037	. . . 4 ⊢ (𝑦 = 𝐴 → (𝑥 ≤ 𝑦 ↔ 𝑥 ≤ 𝐴))
2		breq1 4036	. . . 4 ⊢ (𝑦 = 𝐴 → (𝑦 < (𝑥 + 1) ↔ 𝐴 < (𝑥 + 1)))
3	1, 2	anbi12d 473	. . 3 ⊢ (𝑦 = 𝐴 → ((𝑥 ≤ 𝑦 ∧ 𝑦 < (𝑥 + 1)) ↔ (𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1))))
4	3	riotabidv 5879	. 2 ⊢ (𝑦 = 𝐴 → (℩𝑥 ∈ ℤ (𝑥 ≤ 𝑦 ∧ 𝑦 < (𝑥 + 1))) = (℩𝑥 ∈ ℤ (𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1))))
5		df-fl 10360	. 2 ⊢ ⌊ = (𝑦 ∈ ℝ ↦ (℩𝑥 ∈ ℤ (𝑥 ≤ 𝑦 ∧ 𝑦 < (𝑥 + 1))))
6		zex 9335	. . 3 ⊢ ℤ ∈ V
7		riotaexg 5881	. . 3 ⊢ (ℤ ∈ V → (℩𝑥 ∈ ℤ (𝑥 ≤ 𝑦 ∧ 𝑦 < (𝑥 + 1))) ∈ V)
8	6, 7	ax-mp 5	. 2 ⊢ (℩𝑥 ∈ ℤ (𝑥 ≤ 𝑦 ∧ 𝑦 < (𝑥 + 1))) ∈ V
9	4, 5, 8	fvmpt3i 5641	1 ⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) = (℩𝑥 ∈ ℤ (𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1))))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2167  Vcvv 2763   class class class wbr 4033  ‘cfv 5258  ℩crio 5876  (class class class)co 5922  ℝcr 7878  1c1 7880   + caddc 7882   < clt 8061   ≤ cle 8062  ℤcz 9326  ⌊cfl 10358

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-cnex 7970  ax-resscn 7971

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-iota 5219  df-fun 5260  df-fv 5266  df-riota 5877  df-ov 5925  df-neg 8200  df-z 9327  df-fl 10360

This theorem is referenced by:  flqcl  10363  flapcl  10365  flqlelt  10366  flqbi  10380