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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 3933 | . . . 4 | |
2 | breq1 3932 | . . . 4 | |
3 | 1, 2 | anbi12d 464 | . . 3 |
4 | 3 | riotabidv 5732 | . 2 |
5 | df-fl 10043 | . 2 | |
6 | zex 9063 | . . 3 | |
7 | riotaexg 5734 | . . 3 | |
8 | 6, 7 | ax-mp 5 | . 2 |
9 | 4, 5, 8 | fvmpt3i 5501 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wcel 1480 cvv 2686 class class class wbr 3929 cfv 5123 crio 5729 (class class class)co 5774 cr 7619 c1 7621 caddc 7623 clt 7800 cle 7801 cz 9054 cfl 10041 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-cnex 7711 ax-resscn 7712 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-sbc 2910 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fv 5131 df-riota 5730 df-ov 5777 df-neg 7936 df-z 9055 df-fl 10043 |
This theorem is referenced by: flqcl 10046 flapcl 10048 flqlelt 10049 flqbi 10063 |
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