Theorem bloval 8441

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 3983	. . . . 5 |- (U LnOp W) e. V
3	1, 2	eqeltr 1544	. . . 4 |- L e. V
4	3	rabex 2725	. . 3 |- {t e. L | (N` t) < +oo} e. V
5		opreq1 3968	. . . . . . 7 |- (u = U -> (unormOpw) = (UnormOpw))
6	5	fveq1d 3726	. . . . . 6 |- (u = U -> ((unormOpw)` t) = ((UnormOpw)` t))
7	6	breq1d 2629	. . . . 5 |- (u = U -> (((unormOpw)` t) < +oo <-> ((UnormOpw)` t) < +oo))
8	7	rabbisdv 1807	. . . 4 |- (u = U -> {t e. (u LnOp w) | ((unormOpw)` t) < +oo} = {t e. (u LnOp w) | ((UnormOpw)` t) < +oo})
9		opreq1 3968	. . . . 5 |- (u = U -> (u LnOp w) = (U LnOp w))
10		rabeq 1809	. . . . 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 10	. . . 4 |- (u = U -> {t e. (u LnOp w) | ((UnormOpw)` t) < +oo} = {t e. (U LnOp w) | ((UnormOpw)` t) < +oo})
12	8, 11	eqtrd 1507	. . 3 |- (u = U -> {t e. (u LnOp w) | ((unormOpw)` t) < +oo} = {t e. (U LnOp w) | ((UnormOpw)` t) < +oo})
13		opreq2 3969	. . . . . 6 |- (w = W -> (U LnOp w) = (U LnOp W))
14	13, 1	syl6eqr 1525	. . . . 5 |- (w = W -> (U LnOp w) = L)
15		rabeq 1809	. . . . 5 |- ((U LnOp w) = L -> {t e. (U LnOp w) | ((UnormOpw)` t) < +oo} = {t e. L | ((UnormOpw)` t) < +oo})
16	14, 15	syl 10	. . . 4 |- (w = W -> {t e. (U LnOp w) | ((UnormOpw)` t) < +oo} = {t e. L | ((UnormOpw)` t) < +oo})
17		opreq2 3969	. . . . . . . 8 |- (w = W -> (UnormOpw) = (UnormOpW))
18		bloval.3	. . . . . . . 8 |- N = (UnormOpW)
19	17, 18	syl6eqr 1525	. . . . . . 7 |- (w = W -> (UnormOpw) = N)
20	19	fveq1d 3726	. . . . . 6 |- (w = W -> ((UnormOpw)` t) = (N` t))
21	20	breq1d 2629	. . . . 5 |- (w = W -> (((UnormOpw)` t) < +oo <-> (N` t) < +oo))
22	21	rabbisdv 1807	. . . 4 |- (w = W -> {t e. L | ((UnormOpw)` t) < +oo} = {t e. L | (N` t) < +oo})
23	16, 22	eqtrd 1507	. . 3 |- (w = W -> {t e. (U LnOp w) | ((UnormOpw)` t) < +oo} = {t e. L | (N` t) < +oo})
24		df-blo 8407	. . 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 4028	. 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 1519	1 |- ((U e. NrmCVec /\ W e. NrmCVec) -> B = {t e. L | (N` t) < +oo})

Syntax hints:   -> wi 3   /\ wa 223   = wceq 956   e. wcel 958  {crab 1648  Vcvv 1811   class class class wbr 2619  ` cfv 3182  (class class class)co 3963   +oocpnf 5483   < clt 5486  NrmCVeccnv 8203   LnOp clno 8401  normOpcnmo 8402   BLnOp cblo 8403

This theorem is referenced by:  isblo 8442  hhblo 9828

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-rex 1650  df-rab 1652  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fv 3198  df-opr 3965  df-oprab 3966  df-blo 8407

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