Theorem bloval 28560

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 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝐵 = {𝑡 ∈ 𝐿 ∣ (𝑁‘𝑡) < +∞})

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 7165	. . . 4 ⊢ (𝑢 = 𝑈 → (𝑢 LnOp 𝑤) = (𝑈 LnOp 𝑤))
3		oveq1 7165	. . . . . 6 ⊢ (𝑢 = 𝑈 → (𝑢 normOpOLD 𝑤) = (𝑈 normOpOLD 𝑤))
4	3	fveq1d 6674	. . . . 5 ⊢ (𝑢 = 𝑈 → ((𝑢 normOpOLD 𝑤)‘𝑡) = ((𝑈 normOpOLD 𝑤)‘𝑡))
5	4	breq1d 5078	. . . 4 ⊢ (𝑢 = 𝑈 → (((𝑢 normOpOLD 𝑤)‘𝑡) < +∞ ↔ ((𝑈 normOpOLD 𝑤)‘𝑡) < +∞))
6	2, 5	rabeqbidv 3487	. . 3 ⊢ (𝑢 = 𝑈 → {𝑡 ∈ (𝑢 LnOp 𝑤) ∣ ((𝑢 normOpOLD 𝑤)‘𝑡) < +∞} = {𝑡 ∈ (𝑈 LnOp 𝑤) ∣ ((𝑈 normOpOLD 𝑤)‘𝑡) < +∞})
7		oveq2 7166	. . . . 5 ⊢ (𝑤 = 𝑊 → (𝑈 LnOp 𝑤) = (𝑈 LnOp 𝑊))
8		bloval.4	. . . . 5 ⊢ 𝐿 = (𝑈 LnOp 𝑊)
9	7, 8	syl6eqr 2876	. . . 4 ⊢ (𝑤 = 𝑊 → (𝑈 LnOp 𝑤) = 𝐿)
10		oveq2 7166	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (𝑈 normOpOLD 𝑤) = (𝑈 normOpOLD 𝑊))
11		bloval.3	. . . . . . 7 ⊢ 𝑁 = (𝑈 normOpOLD 𝑊)
12	10, 11	syl6eqr 2876	. . . . . 6 ⊢ (𝑤 = 𝑊 → (𝑈 normOpOLD 𝑤) = 𝑁)
13	12	fveq1d 6674	. . . . 5 ⊢ (𝑤 = 𝑊 → ((𝑈 normOpOLD 𝑤)‘𝑡) = (𝑁‘𝑡))
14	13	breq1d 5078	. . . 4 ⊢ (𝑤 = 𝑊 → (((𝑈 normOpOLD 𝑤)‘𝑡) < +∞ ↔ (𝑁‘𝑡) < +∞))
15	9, 14	rabeqbidv 3487	. . 3 ⊢ (𝑤 = 𝑊 → {𝑡 ∈ (𝑈 LnOp 𝑤) ∣ ((𝑈 normOpOLD 𝑤)‘𝑡) < +∞} = {𝑡 ∈ 𝐿 ∣ (𝑁‘𝑡) < +∞})
16		df-blo 28525	. . 3 ⊢ BLnOp = (𝑢 ∈ NrmCVec, 𝑤 ∈ NrmCVec ↦ {𝑡 ∈ (𝑢 LnOp 𝑤) ∣ ((𝑢 normOpOLD 𝑤)‘𝑡) < +∞})
17	8	ovexi 7192	. . . 4 ⊢ 𝐿 ∈ V
18	17	rabex 5237	. . 3 ⊢ {𝑡 ∈ 𝐿 ∣ (𝑁‘𝑡) < +∞} ∈ V
19	6, 15, 16, 18	ovmpo 7312	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑈 BLnOp 𝑊) = {𝑡 ∈ 𝐿 ∣ (𝑁‘𝑡) < +∞})
20	1, 19	syl5eq 2870	1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝐵 = {𝑡 ∈ 𝐿 ∣ (𝑁‘𝑡) < +∞})

Syntax hints:   → wi 4   ∧ 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:  isblo  28561  hhbloi  29681