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Theorem blof 28571
Description: A bounded operator is an operator. (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
blof.1 𝑋 = (BaseSet‘𝑈)
blof.2 𝑌 = (BaseSet‘𝑊)
blof.5 𝐵 = (𝑈 BLnOp 𝑊)
Assertion
Ref Expression
blof ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇𝐵) → 𝑇:𝑋𝑌)

Proof of Theorem blof
StepHypRef Expression
1 eqid 2801 . . 3 (𝑈 LnOp 𝑊) = (𝑈 LnOp 𝑊)
2 blof.5 . . 3 𝐵 = (𝑈 BLnOp 𝑊)
31, 2bloln 28570 . 2 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇𝐵) → 𝑇 ∈ (𝑈 LnOp 𝑊))
4 blof.1 . . 3 𝑋 = (BaseSet‘𝑈)
5 blof.2 . . 3 𝑌 = (BaseSet‘𝑊)
64, 5, 1lnof 28541 . 2 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ (𝑈 LnOp 𝑊)) → 𝑇:𝑋𝑌)
73, 6syld3an3 1406 1 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇𝐵) → 𝑇:𝑋𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1084   = wceq 1538  wcel 2112  wf 6324  cfv 6328  (class class class)co 7139  NrmCVeccnv 28370  BaseSetcba 28372   LnOp clno 28526   BLnOp cblo 28528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-map 8395  df-lno 28530  df-blo 28532
This theorem is referenced by:  nmblore  28572  nmblolbii  28585  blometi  28589  ubthlem3  28658  htthlem  28703
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