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Mirrors > Home > ILE Home > Th. List > flval | Unicode version |
Description: Value of the floor (greatest integer) function. The floor of is the (unique) integer less than or equal to whose successor is strictly greater than . (Contributed by NM, 14-Nov-2004.) (Revised by Mario Carneiro, 2-Nov-2013.) |
Ref | Expression |
---|---|
flval |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq2 3768 | . . . 4 | |
2 | breq1 3767 | . . . 4 | |
3 | 1, 2 | anbi12d 442 | . . 3 |
4 | 3 | riotabidv 5470 | . 2 |
5 | df-fl 9114 | . 2 | |
6 | zex 8254 | . . 3 | |
7 | riotaexg 5472 | . . 3 | |
8 | 6, 7 | ax-mp 7 | . 2 |
9 | 4, 5, 8 | fvmpt3i 5252 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wceq 1243 wcel 1393 cvv 2557 class class class wbr 3764 cfv 4902 crio 5467 (class class class)co 5512 cr 6888 c1 6890 caddc 6892 clt 7060 cle 7061 cz 8245 cfl 9112 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-cnex 6975 ax-resscn 6976 |
This theorem depends on definitions: df-bi 110 df-3or 886 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-rab 2315 df-v 2559 df-sbc 2765 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-mpt 3820 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-iota 4867 df-fun 4904 df-fv 4910 df-riota 5468 df-ov 5515 df-neg 7185 df-z 8246 df-fl 9114 |
This theorem is referenced by: flqcl 9117 flqlelt 9118 flqbi 9132 |
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