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Mirrors > Home > ILE Home > Th. List > Mathboxes > inffz | Unicode version |
Description: The infimum of a finite sequence of integers. (Contributed by Scott Fenton, 8-Aug-2013.) (Revised by Jim Kingdon, 15-Oct-2022.) |
Ref | Expression |
---|---|
inffz | inf |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simprl 526 | . . . 4 | |
2 | 1 | zred 9327 | . . 3 |
3 | simprr 527 | . . . 4 | |
4 | 3 | zred 9327 | . . 3 |
5 | 2, 4 | lttri3d 8027 | . 2 |
6 | eluzel2 9485 | . 2 | |
7 | eluzfz1 9980 | . 2 | |
8 | elfzle1 9976 | . . . 4 | |
9 | 8 | adantl 275 | . . 3 |
10 | 6 | zred 9327 | . . . 4 |
11 | elfzelz 9974 | . . . . 5 | |
12 | 11 | zred 9327 | . . . 4 |
13 | lenlt 7988 | . . . 4 | |
14 | 10, 12, 13 | syl2an 287 | . . 3 |
15 | 9, 14 | mpbid 146 | . 2 |
16 | 5, 6, 7, 15 | infminti 7002 | 1 inf |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wceq 1348 wcel 2141 class class class wbr 3987 cfv 5196 (class class class)co 5851 infcinf 6958 cr 7766 clt 7947 cle 7948 cz 9205 cuz 9480 cfz 9958 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-cnex 7858 ax-resscn 7859 ax-pre-ltirr 7879 ax-pre-apti 7882 |
This theorem depends on definitions: df-bi 116 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-br 3988 df-opab 4049 df-mpt 4050 df-id 4276 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-fv 5204 df-riota 5807 df-ov 5854 df-oprab 5855 df-mpo 5856 df-sup 6959 df-inf 6960 df-pnf 7949 df-mnf 7950 df-xr 7951 df-ltxr 7952 df-le 7953 df-neg 8086 df-z 9206 df-uz 9481 df-fz 9959 |
This theorem is referenced by: (None) |
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