Theorem bloval 9781

Description: The class of bounded linear operators between two normed complex vector spaces.

Hypotheses

Ref	Expression
bloval.3	|- N = (UnormOpW)
bloval.4	|- L = (U LnOp W)
bloval.5	|- B = (U BLnOp W)

Assertion

Ref	Expression
bloval	|- ((U e. NrmCVec /\ W e. NrmCVec) -> B = {t e. L | (N` t) < +oo})

Distinct variable groups:   t,L   t,N   t,U   t,W

Proof of Theorem bloval

Step	Hyp	Ref	Expression
1		bloval.4	. . . . 5 |- L = (U LnOp W)
2		oprex 4907	. . . . 5 |- (U LnOp W) e. _V
3	1, 2	eqeltri 1967	. . . 4 |- L e. _V
4	3	rabex 3461	. . 3 |- {t e. L | (N` t) < +oo} e. _V
5		opreq1 4889	. . . . . . 7 |- (u = U -> (unormOpw) = (UnormOpw))
6	5	fveq1d 4683	. . . . . 6 |- (u = U -> ((unormOpw)` t) = ((UnormOpw)` t))
7	6	breq1d 3348	. . . . 5 |- (u = U -> (((unormOpw)` t) < +oo <-> ((UnormOpw)` t) < +oo))
8	7	rabbidv 2287	. . . 4 |- (u = U -> {t e. (u LnOp w) | ((unormOpw)` t) < +oo} = {t e. (u LnOp w) | ((UnormOpw)` t) < +oo})
9		opreq1 4889	. . . . 5 |- (u = U -> (u LnOp w) = (U LnOp w))
10		rabeq 2289	. . . . 5 |- ((u LnOp w) = (U LnOp w) -> {t e. (u LnOp w) | ((UnormOpw)` t) < +oo} = {t e. (U LnOp w) | ((UnormOpw)` t) < +oo})
11	9, 10	syl 12	. . . 4 |- (u = U -> {t e. (u LnOp w) | ((UnormOpw)` t) < +oo} = {t e. (U LnOp w) | ((UnormOpw)` t) < +oo})
12	8, 11	eqtrd 1925	. . 3 |- (u = U -> {t e. (u LnOp w) | ((unormOpw)` t) < +oo} = {t e. (U LnOp w) | ((UnormOpw)` t) < +oo})
13		opreq2 4890	. . . . . 6 |- (w = W -> (U LnOp w) = (U LnOp W))
14	13, 1	syl6eqr 1946	. . . . 5 |- (w = W -> (U LnOp w) = L)
15		rabeq 2289	. . . . 5 |- ((U LnOp w) = L -> {t e. (U LnOp w) | ((UnormOpw)` t) < +oo} = {t e. L | ((UnormOpw)` t) < +oo})
16	14, 15	syl 12	. . . 4 |- (w = W -> {t e. (U LnOp w) | ((UnormOpw)` t) < +oo} = {t e. L | ((UnormOpw)` t) < +oo})
17		opreq2 4890	. . . . . . . 8 |- (w = W -> (UnormOpw) = (UnormOpW))
18		bloval.3	. . . . . . . 8 |- N = (UnormOpW)
19	17, 18	syl6eqr 1946	. . . . . . 7 |- (w = W -> (UnormOpw) = N)
20	19	fveq1d 4683	. . . . . 6 |- (w = W -> ((UnormOpw)` t) = (N` t))
21	20	breq1d 3348	. . . . 5 |- (w = W -> (((UnormOpw)` t) < +oo <-> (N` t) < +oo))
22	21	rabbidv 2287	. . . 4 |- (w = W -> {t e. L | ((UnormOpw)` t) < +oo} = {t e. L | (N` t) < +oo})
23	16, 22	eqtrd 1925	. . 3 |- (w = W -> {t e. (U LnOp w) | ((UnormOpw)` t) < +oo} = {t e. L | (N` t) < +oo})
24		df-blo 9746	. . 3 |- BLnOp = {<.<.u, w>., o>. | ((u e. NrmCVec /\ w e. NrmCVec) /\ o = {t e. (u LnOp w) | ((unormOpw)` t) < +oo})}
25	4, 12, 23, 24	oprabval2 4957	. 2 |- ((U e. NrmCVec /\ W e. NrmCVec) -> (U BLnOp W) = {t e. L | (N` t) < +oo})
26		bloval.5	. 2 |- B = (U BLnOp W)
27	25, 26	syl5eq 1940	1 |- ((U e. NrmCVec /\ W e. NrmCVec) -> B = {t e. L | (N` t) < +oo})

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  {crab 2108  _Vcvv 2292   class class class wbr 3338  ` cfv 3998  (class class class)co 4884   +oocpnf 6650   < clt 6653  NrmCVeccnv 9535   LnOp clno 9740  normOpcnmo 9741   BLnOp cblo 9742

This theorem is referenced by:  isblo 9782  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-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  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-rex 2110  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  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-fv 4014  df-opr 4886  df-oprab 4887  df-blo 9746

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