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Mirrors > Home > HSE Home > Th. List > hhbloi | Structured version Visualization version GIF version |
Description: A bounded linear operator in Hilbert space. (Contributed by NM, 19-Nov-2007.) (Revised by Mario Carneiro, 19-Nov-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hhnmo.1 | ⊢ 𝑈 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 |
hhblo.2 | ⊢ 𝐵 = (𝑈 BLnOp 𝑈) |
Ref | Expression |
---|---|
hhbloi | ⊢ BndLinOp = 𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-bdop 31769 | . 2 ⊢ BndLinOp = {𝑥 ∈ LinOp ∣ (normop‘𝑥) < +∞} | |
2 | hhnmo.1 | . . . 4 ⊢ 𝑈 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 | |
3 | 2 | hhnv 31092 | . . 3 ⊢ 𝑈 ∈ NrmCVec |
4 | eqid 2726 | . . . . 5 ⊢ (𝑈 normOpOLD 𝑈) = (𝑈 normOpOLD 𝑈) | |
5 | 2, 4 | hhnmoi 31828 | . . . 4 ⊢ normop = (𝑈 normOpOLD 𝑈) |
6 | eqid 2726 | . . . . 5 ⊢ (𝑈 LnOp 𝑈) = (𝑈 LnOp 𝑈) | |
7 | 2, 6 | hhlnoi 31827 | . . . 4 ⊢ LinOp = (𝑈 LnOp 𝑈) |
8 | hhblo.2 | . . . 4 ⊢ 𝐵 = (𝑈 BLnOp 𝑈) | |
9 | 5, 7, 8 | bloval 30708 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑈 ∈ NrmCVec) → 𝐵 = {𝑥 ∈ LinOp ∣ (normop‘𝑥) < +∞}) |
10 | 3, 3, 9 | mp2an 690 | . 2 ⊢ 𝐵 = {𝑥 ∈ LinOp ∣ (normop‘𝑥) < +∞} |
11 | 1, 10 | eqtr4i 2757 | 1 ⊢ BndLinOp = 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 ∈ wcel 2099 {crab 3419 〈cop 4629 class class class wbr 5143 ‘cfv 6543 (class class class)co 7413 +∞cpnf 11283 < clt 11286 NrmCVeccnv 30511 LnOp clno 30667 normOpOLD cnmoo 30668 BLnOp cblo 30669 +ℎ cva 30847 ·ℎ csm 30848 normℎcno 30850 normopcnop 30872 LinOpclo 30874 BndLinOpcbo 30875 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7735 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224 ax-hilex 30926 ax-hfvadd 30927 ax-hvcom 30928 ax-hvass 30929 ax-hv0cl 30930 ax-hvaddid 30931 ax-hfvmul 30932 ax-hvmulid 30933 ax-hvmulass 30934 ax-hvdistr1 30935 ax-hvdistr2 30936 ax-hvmul0 30937 ax-hfi 31006 ax-his1 31009 ax-his2 31010 ax-his3 31011 ax-his4 31012 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-iun 4995 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6302 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7866 df-1st 7992 df-2nd 7993 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-er 8723 df-en 8964 df-dom 8965 df-sdom 8966 df-sup 9475 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12256 df-2 12318 df-3 12319 df-4 12320 df-n0 12516 df-z 12602 df-uz 12866 df-rp 13020 df-seq 14013 df-exp 14073 df-cj 15096 df-re 15097 df-im 15098 df-sqrt 15232 df-abs 15233 df-grpo 30420 df-gid 30421 df-ablo 30472 df-vc 30486 df-nv 30519 df-va 30522 df-ba 30523 df-sm 30524 df-nmcv 30527 df-lno 30671 df-nmoo 30672 df-blo 30673 df-hnorm 30895 df-hvsub 30898 df-nmop 31766 df-lnop 31768 df-bdop 31769 |
This theorem is referenced by: hmopbdoptHIL 31915 |
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