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Mirrors > Home > HSE Home > Th. List > nmbdfnlb | Unicode version |
Description: A lower bound for the norm of a bounded linear functional. (Contributed by NM, 25-Apr-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nmbdfnlb |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq1 5690 |
. . . . . 6
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2 | 1 | fveq2d 5695 |
. . . . 5
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3 | fveq2 5691 |
. . . . . 6
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4 | 3 | oveq1d 6059 |
. . . . 5
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5 | 2, 4 | breq12d 4189 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
6 | 5 | imbi2d 308 |
. . 3
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7 | eleq1 2468 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
8 | 3 | eleq1d 2474 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
9 | 7, 8 | anbi12d 692 |
. . . . 5
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10 | eleq1 2468 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
11 | fveq2 5691 |
. . . . . . 7
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12 | 11 | eleq1d 2474 |
. . . . . 6
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13 | 10, 12 | anbi12d 692 |
. . . . 5
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14 | 0lnfn 23445 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
15 | nmfn0 23447 |
. . . . . . 7
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16 | 0re 9051 |
. . . . . . 7
![]() ![]() ![]() ![]() | |
17 | 15, 16 | eqeltri 2478 |
. . . . . 6
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18 | 14, 17 | pm3.2i 442 |
. . . . 5
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19 | 9, 13, 18 | elimhyp 3751 |
. . . 4
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20 | 19 | nmbdfnlbi 23509 |
. . 3
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21 | 6, 20 | dedth 3744 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
22 | 21 | 3impia 1150 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem is referenced by: lnfncnbd 23517 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1552 ax-5 1563 ax-17 1623 ax-9 1662 ax-8 1683 ax-13 1723 ax-14 1725 ax-6 1740 ax-7 1745 ax-11 1757 ax-12 1946 ax-ext 2389 ax-sep 4294 ax-nul 4302 ax-pow 4341 ax-pr 4367 ax-un 4664 ax-cnex 9006 ax-resscn 9007 ax-1cn 9008 ax-icn 9009 ax-addcl 9010 ax-addrcl 9011 ax-mulcl 9012 ax-mulrcl 9013 ax-mulcom 9014 ax-addass 9015 ax-mulass 9016 ax-distr 9017 ax-i2m1 9018 ax-1ne0 9019 ax-1rid 9020 ax-rnegex 9021 ax-rrecex 9022 ax-cnre 9023 ax-pre-lttri 9024 ax-pre-lttrn 9025 ax-pre-ltadd 9026 ax-pre-mulgt0 9027 ax-pre-sup 9028 ax-hilex 22459 ax-hfvadd 22460 ax-hv0cl 22463 ax-hvaddid 22464 ax-hfvmul 22465 ax-hvmulid 22466 ax-hvmul0 22470 ax-hfi 22538 ax-his1 22541 ax-his3 22543 ax-his4 22544 |
This theorem depends on definitions: df-bi 178 df-or 360 df-an 361 df-3or 937 df-3an 938 df-tru 1325 df-ex 1548 df-nf 1551 df-sb 1656 df-eu 2262 df-mo 2263 df-clab 2395 df-cleq 2401 df-clel 2404 df-nfc 2533 df-ne 2573 df-nel 2574 df-ral 2675 df-rex 2676 df-reu 2677 df-rmo 2678 df-rab 2679 df-v 2922 df-sbc 3126 df-csb 3216 df-dif 3287 df-un 3289 df-in 3291 df-ss 3298 df-pss 3300 df-nul 3593 df-if 3704 df-pw 3765 df-sn 3784 df-pr 3785 df-tp 3786 df-op 3787 df-uni 3980 df-iun 4059 df-br 4177 df-opab 4231 df-mpt 4232 df-tr 4267 df-eprel 4458 df-id 4462 df-po 4467 df-so 4468 df-fr 4505 df-we 4507 df-ord 4548 df-on 4549 df-lim 4550 df-suc 4551 df-om 4809 df-xp 4847 df-rel 4848 df-cnv 4849 df-co 4850 df-dm 4851 df-rn 4852 df-res 4853 df-ima 4854 df-iota 5381 df-fun 5419 df-fn 5420 df-f 5421 df-f1 5422 df-fo 5423 df-f1o 5424 df-fv 5425 df-ov 6047 df-oprab 6048 df-mpt2 6049 df-2nd 6313 df-riota 6512 df-recs 6596 df-rdg 6631 df-er 6868 df-map 6983 df-en 7073 df-dom 7074 df-sdom 7075 df-sup 7408 df-pnf 9082 df-mnf 9083 df-xr 9084 df-ltxr 9085 df-le 9086 df-sub 9253 df-neg 9254 df-div 9638 df-nn 9961 df-2 10018 df-3 10019 df-n0 10182 df-z 10243 df-uz 10449 df-rp 10573 df-seq 11283 df-exp 11342 df-cj 11863 df-re 11864 df-im 11865 df-sqr 11999 df-abs 12000 df-hnorm 22428 df-nmfn 23305 df-lnfn 23308 |
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