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Mirrors > Home > ILE Home > Th. List > flval | GIF version |
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.) |
Ref | Expression |
---|---|
flval | ⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) = (℩𝑥 ∈ ℤ (𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq2 3815 | . . . 4 ⊢ (𝑦 = 𝐴 → (𝑥 ≤ 𝑦 ↔ 𝑥 ≤ 𝐴)) | |
2 | breq1 3814 | . . . 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 5549 | . 2 ⊢ (𝑦 = 𝐴 → (℩𝑥 ∈ ℤ (𝑥 ≤ 𝑦 ∧ 𝑦 < (𝑥 + 1))) = (℩𝑥 ∈ ℤ (𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)))) |
5 | df-fl 9566 | . 2 ⊢ ⌊ = (𝑦 ∈ ℝ ↦ (℩𝑥 ∈ ℤ (𝑥 ≤ 𝑦 ∧ 𝑦 < (𝑥 + 1)))) | |
6 | zex 8655 | . . 3 ⊢ ℤ ∈ V | |
7 | riotaexg 5551 | . . 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 5329 | 1 ⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) = (℩𝑥 ∈ ℤ (𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 = wceq 1285 ∈ wcel 1434 Vcvv 2612 class class class wbr 3811 ‘cfv 4969 ℩crio 5546 (class class class)co 5591 ℝcr 7252 1c1 7254 + caddc 7256 < clt 7425 ≤ cle 7426 ℤcz 8646 ⌊cfl 9564 |
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 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-sep 3922 ax-pow 3974 ax-pr 4000 ax-un 4224 ax-cnex 7339 ax-resscn 7340 |
This theorem depends on definitions: df-bi 115 df-3or 921 df-3an 922 df-tru 1288 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ral 2358 df-rex 2359 df-rab 2362 df-v 2614 df-sbc 2827 df-un 2988 df-in 2990 df-ss 2997 df-pw 3408 df-sn 3428 df-pr 3429 df-op 3431 df-uni 3628 df-br 3812 df-opab 3866 df-mpt 3867 df-id 4084 df-xp 4407 df-rel 4408 df-cnv 4409 df-co 4410 df-dm 4411 df-iota 4934 df-fun 4971 df-fv 4977 df-riota 5547 df-ov 5594 df-neg 7559 df-z 8647 df-fl 9566 |
This theorem is referenced by: flqcl 9569 flapcl 9571 flqlelt 9572 flqbi 9586 |
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