Theorem bloval 27636

Description: The class of bounded linear operators between two normed complex vector spaces. (Contributed by NM, 6-Nov-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bloval.3	|- N = ( U normOpOLD W )
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

Allowed substitution hint:   B( t)

Proof of Theorem bloval

Dummy variables u w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		bloval.5	. 2 |- B = ( U BLnOp W )
2		oveq1 6657	. . . 4 |- ( u = U -> ( u LnOp w ) = ( U LnOp w ) )
3		oveq1 6657	. . . . . 6 |- ( u = U -> ( u normOpOLD w ) = ( U normOpOLD w ) )
4	3	fveq1d 6193	. . . . 5 |- ( u = U -> ( ( u normOpOLD w ) ` t ) = ( ( U normOpOLD w ) ` t ) )
5	4	breq1d 4663	. . . 4 |- ( u = U -> ( ( ( u normOpOLD w ) ` t ) < +oo <-> ( ( U normOpOLD w ) ` t ) < +oo ) )
6	2, 5	rabeqbidv 3195	. . 3 |- ( u = U -> { t e. ( u LnOp w ) | ( ( u normOpOLD w ) ` t ) < +oo } = { t e. ( U LnOp w ) | ( ( U normOpOLD w ) ` t ) < +oo } )
7		oveq2 6658	. . . . 5 |- ( w = W -> ( U LnOp w ) = ( U LnOp W ) )
8		bloval.4	. . . . 5 |- L = ( U LnOp W )
9	7, 8	syl6eqr 2674	. . . 4 |- ( w = W -> ( U LnOp w ) = L )
10		oveq2 6658	. . . . . . 7 |- ( w = W -> ( U normOpOLD w ) = ( U normOpOLD W ) )
11		bloval.3	. . . . . . 7 |- N = ( U normOpOLD W )
12	10, 11	syl6eqr 2674	. . . . . 6 |- ( w = W -> ( U normOpOLD w ) = N )
13	12	fveq1d 6193	. . . . 5 |- ( w = W -> ( ( U normOpOLD w ) ` t ) = ( N ` t ) )
14	13	breq1d 4663	. . . 4 |- ( w = W -> ( ( ( U normOpOLD w ) ` t ) < +oo <-> ( N ` t ) < +oo ) )
15	9, 14	rabeqbidv 3195	. . 3 |- ( w = W -> { t e. ( U LnOp w ) | ( ( U normOpOLD w ) ` t ) < +oo } = { t e. L | ( N ` t ) < +oo } )
16		df-blo 27601	. . 3 |- BLnOp = ( u e. NrmCVec , w e. NrmCVec |-> { t e. ( u LnOp w ) | ( ( u normOpOLD w ) ` t ) < +oo } )
17		ovex 6678	. . . . 5 |- ( U LnOp W ) e. _V
18	8, 17	eqeltri 2697	. . . 4 |- L e. _V
19	18	rabex 4813	. . 3 |- { t e. L | ( N ` t ) < +oo } e. _V
20	6, 15, 16, 19	ovmpt2 6796	. 2 |- ( ( U e. NrmCVec /\ W e. NrmCVec ) -> ( U BLnOp W ) = { t e. L | ( N ` t ) < +oo } )
21	1, 20	syl5eq 2668	1 |- ( ( U e. NrmCVec /\ W e. NrmCVec ) -> B = { t e. L | ( N ` t ) < +oo } )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   _Vcvv 3200   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   +oocpnf 10071   < clt 10074   NrmCVeccnv 27439   LnOp clno 27595   normOpOLDcnmoo 27596   BLnOp cblo 27597

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-blo 27601

This theorem is referenced by:  isblo  27637  hhbloi  28761