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Mathbox for Jim Kingdon |
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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 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simprl 529 |
. . . 4
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2 | 1 | zred 9389 |
. . 3
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3 | simprr 531 |
. . . 4
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4 | 3 | zred 9389 |
. . 3
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5 | 2, 4 | lttri3d 8086 |
. 2
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6 | eluzelz 9551 |
. 2
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7 | eluzfz2 10046 |
. 2
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8 | elfzle2 10042 |
. . . 4
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9 | 8 | adantl 277 |
. . 3
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10 | elfzelz 10039 |
. . . . 5
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11 | 10 | zred 9389 |
. . . 4
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12 | 6 | zred 9389 |
. . . 4
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13 | lenlt 8047 |
. . . 4
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14 | 11, 12, 13 | syl2anr 290 |
. . 3
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15 | 9, 14 | mpbid 147 |
. 2
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16 | 5, 6, 7, 15 | supmaxti 7017 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 ax-cnex 7916 ax-resscn 7917 ax-pre-ltirr 7937 ax-pre-apti 7940 |
This theorem depends on definitions: df-bi 117 df-3or 980 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-nel 2453 df-ral 2470 df-rex 2471 df-reu 2472 df-rmo 2473 df-rab 2474 df-v 2751 df-sbc 2975 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-br 4016 df-opab 4077 df-mpt 4078 df-id 4305 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-ima 4651 df-iota 5190 df-fun 5230 df-fn 5231 df-f 5232 df-fv 5236 df-riota 5844 df-ov 5891 df-oprab 5892 df-mpo 5893 df-sup 6997 df-pnf 8008 df-mnf 8009 df-xr 8010 df-ltxr 8011 df-le 8012 df-neg 8145 df-z 9268 df-uz 9543 df-fz 10023 |
This theorem is referenced by: (None) |
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