Theorem isblo 28561

Description: The predicate "is a bounded linear operator." (Contributed by NM, 6-Nov-2007.) (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
isblo	⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑇 ∈ 𝐵 ↔ (𝑇 ∈ 𝐿 ∧ (𝑁‘𝑇) < +∞)))

Proof of Theorem isblo

Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bloval.3	. . . 4 ⊢ 𝑁 = (𝑈 normOpOLD 𝑊)
2		bloval.4	. . . 4 ⊢ 𝐿 = (𝑈 LnOp 𝑊)
3		bloval.5	. . . 4 ⊢ 𝐵 = (𝑈 BLnOp 𝑊)
4	1, 2, 3	bloval 28560	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝐵 = {𝑡 ∈ 𝐿 ∣ (𝑁‘𝑡) < +∞})
5	4	eleq2d 2900	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑇 ∈ 𝐵 ↔ 𝑇 ∈ {𝑡 ∈ 𝐿 ∣ (𝑁‘𝑡) < +∞}))
6		fveq2 6672	. . . 4 ⊢ (𝑡 = 𝑇 → (𝑁‘𝑡) = (𝑁‘𝑇))
7	6	breq1d 5078	. . 3 ⊢ (𝑡 = 𝑇 → ((𝑁‘𝑡) < +∞ ↔ (𝑁‘𝑇) < +∞))
8	7	elrab 3682	. 2 ⊢ (𝑇 ∈ {𝑡 ∈ 𝐿 ∣ (𝑁‘𝑡) < +∞} ↔ (𝑇 ∈ 𝐿 ∧ (𝑁‘𝑇) < +∞))
9	5, 8	syl6bb 289	1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑇 ∈ 𝐵 ↔ (𝑇 ∈ 𝐿 ∧ (𝑁‘𝑇) < +∞)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  {crab 3144   class class class wbr 5068  ‘cfv 6357  (class class class)co 7158  +∞cpnf 10674   < clt 10677  NrmCVeccnv 28363   LnOp clno 28519   normOp_OLD cnmoo 28520   BLnOp cblo 28521

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-iota 6316  df-fun 6359  df-fv 6365  df-ov 7161  df-oprab 7162  df-mpo 7163  df-blo 28525

This theorem is referenced by:  isblo2  28562  bloln  28563  nmblore  28565  isblo3i  28580  htthlem  28696