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Mirrors > Home > ILE Home > Th. List > infrenegsupex | Unicode version |
Description: The infimum of a set of
reals ![]() |
Ref | Expression |
---|---|
infrenegsupex.ex |
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infrenegsupex.ss |
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Ref | Expression |
---|---|
infrenegsupex |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lttri3 8014 |
. . . . . 6
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2 | 1 | adantl 277 |
. . . . 5
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3 | infrenegsupex.ex |
. . . . 5
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4 | 2, 3 | infclti 7015 |
. . . 4
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5 | 4 | recnd 7963 |
. . 3
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6 | 5 | negnegd 8236 |
. 2
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7 | negeq 8127 |
. . . . . . . . 9
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8 | 7 | cbvmptv 4096 |
. . . . . . . 8
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9 | 8 | mptpreima 5117 |
. . . . . . 7
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10 | eqid 2177 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
11 | 10 | negiso 8888 |
. . . . . . . . 9
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12 | 11 | simpri 113 |
. . . . . . . 8
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13 | 12 | imaeq1i 4962 |
. . . . . . 7
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14 | 9, 13 | eqtr3i 2200 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
15 | 14 | supeq1i 6980 |
. . . . 5
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16 | 11 | simpli 111 |
. . . . . . . . 9
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17 | isocnv 5805 |
. . . . . . . . 9
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18 | 16, 17 | ax-mp 5 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
19 | isoeq1 5795 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
20 | 12, 19 | ax-mp 5 |
. . . . . . . 8
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21 | 18, 20 | mpbi 145 |
. . . . . . 7
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22 | 21 | a1i 9 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
23 | infrenegsupex.ss |
. . . . . 6
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24 | 3 | cnvinfex 7010 |
. . . . . 6
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25 | 2 | cnvti 7011 |
. . . . . 6
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26 | 22, 23, 24, 25 | supisoti 7002 |
. . . . 5
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27 | 15, 26 | eqtrid 2222 |
. . . 4
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28 | df-inf 6977 |
. . . . . . 7
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29 | 28 | eqcomi 2181 |
. . . . . 6
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30 | 29 | fveq2i 5513 |
. . . . 5
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31 | eqidd 2178 |
. . . . . 6
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32 | negeq 8127 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
33 | 32 | adantl 277 |
. . . . . 6
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34 | 5 | negcld 8232 |
. . . . . 6
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35 | 31, 33, 4, 34 | fvmptd 5592 |
. . . . 5
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36 | 30, 35 | eqtrid 2222 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
37 | 27, 36 | eqtr2d 2211 |
. . 3
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38 | 37 | negeqd 8129 |
. 2
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39 | 6, 38 | eqtr3d 2212 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4205 ax-un 4429 ax-setind 4532 ax-cnex 7880 ax-resscn 7881 ax-1cn 7882 ax-1re 7883 ax-icn 7884 ax-addcl 7885 ax-addrcl 7886 ax-mulcl 7887 ax-addcom 7889 ax-addass 7891 ax-distr 7893 ax-i2m1 7894 ax-0id 7897 ax-rnegex 7898 ax-cnre 7900 ax-pre-ltirr 7901 ax-pre-apti 7904 ax-pre-ltadd 7905 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-br 4001 df-opab 4062 df-mpt 4063 df-id 4289 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-rn 4633 df-res 4634 df-ima 4635 df-iota 5173 df-fun 5213 df-fn 5214 df-f 5215 df-f1 5216 df-fo 5217 df-f1o 5218 df-fv 5219 df-isom 5220 df-riota 5824 df-ov 5871 df-oprab 5872 df-mpo 5873 df-sup 6976 df-inf 6977 df-pnf 7971 df-mnf 7972 df-ltxr 7974 df-sub 8107 df-neg 8108 |
This theorem is referenced by: supminfex 9573 minmax 11209 infssuzcldc 11922 |
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