Theorem dfbdop2 11423

Description: Alternate definition of the set of bounded linear Hilbert space operators.

Assertion

Ref	Expression
dfbdop2	|- BndLinOp = {t e. LinOp | (normop` t) < +oo}

Proof of Theorem dfbdop2

Step	Hyp	Ref	Expression
1		incom 2787	. . 3 |- (LinOp i^i {t | (t:~H-->~H /\ (normop` t) < +oo)}) = ({t | (t:~H-->~H /\ (normop` t) < +oo)} i^i LinOp)
2		dfrab2 2869	. . 3 |- {t e. LinOp | (t:~H-->~H /\ (normop` t) < +oo)} = ({t | (t:~H-->~H /\ (normop` t) < +oo)} i^i LinOp)
3	1, 2	eqtr4i 1911	. 2 |- (LinOp i^i {t | (t:~H-->~H /\ (normop` t) < +oo)}) = {t e. LinOp | (t:~H-->~H /\ (normop` t) < +oo)}
4		df-bdop 11405	. 2 |- BndLinOp = (LinOp i^i {t | (t:~H-->~H /\ (normop` t) < +oo)})
5		lnopf 11422	. . . 4 |- (t e. LinOp -> t:~H-->~H)
6	5	biantrurd 796	. . 3 |- (t e. LinOp -> ((normop` t) < +oo <-> (t:~H-->~H /\ (normop` t) < +oo)))
7	6	rabbiia 2285	. 2 |- {t e. LinOp | (normop` t) < +oo} = {t e. LinOp | (t:~H-->~H /\ (normop` t) < +oo)}
8	3, 4, 7	3eqtr4i 1921	1 |- BndLinOp = {t e. LinOp | (normop` t) < +oo}

Syntax hints:   /\ wa 240   = wceq 1298   e. wcel 1300  {cab 1871  {crab 2108   i^i cin 2592   class class class wbr 3338  -->wf 3994  ` cfv 3998   +oocpnf 6650   < clt 6653  ~Hchil 10420  norm_opcnop 10446  LinOpclo 10448  BndLinOpcbo 10449

This theorem is referenced by:  elbdop 11424  hhbloi 11465

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-hilex 10501

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-fv 4014  df-opr 4886  df-lnop 11404  df-bdop 11405

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