Mathbox for Jim Kingdon |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > Mathboxes > supfz | Unicode version |
Description: The supremum of a finite sequence of integers. (Contributed by Scott Fenton, 8-Aug-2013.) (Revised by Jim Kingdon, 15-Oct-2022.) |
Ref | Expression |
---|---|
supfz |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simprl 520 | . . . 4 | |
2 | 1 | zred 9178 | . . 3 |
3 | simprr 521 | . . . 4 | |
4 | 3 | zred 9178 | . . 3 |
5 | 2, 4 | lttri3d 7883 | . 2 |
6 | eluzelz 9340 | . 2 | |
7 | eluzfz2 9817 | . 2 | |
8 | elfzle2 9813 | . . . 4 | |
9 | 8 | adantl 275 | . . 3 |
10 | elfzelz 9811 | . . . . 5 | |
11 | 10 | zred 9178 | . . . 4 |
12 | 6 | zred 9178 | . . . 4 |
13 | lenlt 7845 | . . . 4 | |
14 | 11, 12, 13 | syl2anr 288 | . . 3 |
15 | 9, 14 | mpbid 146 | . 2 |
16 | 5, 6, 7, 15 | supmaxti 6891 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wceq 1331 wcel 1480 class class class wbr 3929 cfv 5123 (class class class)co 5774 csup 6869 cr 7624 clt 7805 cle 7806 cz 9059 cuz 9331 cfz 9795 |
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-in1 603 ax-in2 604 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-setind 4452 ax-cnex 7716 ax-resscn 7717 ax-pre-ltirr 7737 ax-pre-apti 7740 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 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-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 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-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-sup 6871 df-pnf 7807 df-mnf 7808 df-xr 7809 df-ltxr 7810 df-le 7811 df-neg 7941 df-z 9060 df-uz 9332 df-fz 9796 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |