Theorem bloval 30761

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 7353	. . . 4 ⊢ (𝑢 = 𝑈 → (𝑢 LnOp 𝑤) = (𝑈 LnOp 𝑤))
3		oveq1 7353	. . . . . 6 ⊢ (𝑢 = 𝑈 → (𝑢 normOpOLD 𝑤) = (𝑈 normOpOLD 𝑤))
4	3	fveq1d 6824	. . . . 5 ⊢ (𝑢 = 𝑈 → ((𝑢 normOpOLD 𝑤)‘𝑡) = ((𝑈 normOpOLD 𝑤)‘𝑡))
5	4	breq1d 5099	. . . 4 ⊢ (𝑢 = 𝑈 → (((𝑢 normOpOLD 𝑤)‘𝑡) < +∞ ↔ ((𝑈 normOpOLD 𝑤)‘𝑡) < +∞))
6	2, 5	rabeqbidv 3413	. . 3 ⊢ (𝑢 = 𝑈 → {𝑡 ∈ (𝑢 LnOp 𝑤) ∣ ((𝑢 normOpOLD 𝑤)‘𝑡) < +∞} = {𝑡 ∈ (𝑈 LnOp 𝑤) ∣ ((𝑈 normOpOLD 𝑤)‘𝑡) < +∞})
7		oveq2 7354	. . . . 5 ⊢ (𝑤 = 𝑊 → (𝑈 LnOp 𝑤) = (𝑈 LnOp 𝑊))
8		bloval.4	. . . . 5 ⊢ 𝐿 = (𝑈 LnOp 𝑊)
9	7, 8	eqtr4di 2784	. . . 4 ⊢ (𝑤 = 𝑊 → (𝑈 LnOp 𝑤) = 𝐿)
10		oveq2 7354	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (𝑈 normOpOLD 𝑤) = (𝑈 normOpOLD 𝑊))
11		bloval.3	. . . . . . 7 ⊢ 𝑁 = (𝑈 normOpOLD 𝑊)
12	10, 11	eqtr4di 2784	. . . . . 6 ⊢ (𝑤 = 𝑊 → (𝑈 normOpOLD 𝑤) = 𝑁)
13	12	fveq1d 6824	. . . . 5 ⊢ (𝑤 = 𝑊 → ((𝑈 normOpOLD 𝑤)‘𝑡) = (𝑁‘𝑡))
14	13	breq1d 5099	. . . 4 ⊢ (𝑤 = 𝑊 → (((𝑈 normOpOLD 𝑤)‘𝑡) < +∞ ↔ (𝑁‘𝑡) < +∞))
15	9, 14	rabeqbidv 3413	. . 3 ⊢ (𝑤 = 𝑊 → {𝑡 ∈ (𝑈 LnOp 𝑤) ∣ ((𝑈 normOpOLD 𝑤)‘𝑡) < +∞} = {𝑡 ∈ 𝐿 ∣ (𝑁‘𝑡) < +∞})
16		df-blo 30726	. . 3 ⊢ BLnOp = (𝑢 ∈ NrmCVec, 𝑤 ∈ NrmCVec ↦ {𝑡 ∈ (𝑢 LnOp 𝑤) ∣ ((𝑢 normOpOLD 𝑤)‘𝑡) < +∞})
17	8	ovexi 7380	. . . 4 ⊢ 𝐿 ∈ V
18	17	rabex 5275	. . 3 ⊢ {𝑡 ∈ 𝐿 ∣ (𝑁‘𝑡) < +∞} ∈ V
19	6, 15, 16, 18	ovmpo 7506	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑈 BLnOp 𝑊) = {𝑡 ∈ 𝐿 ∣ (𝑁‘𝑡) < +∞})
20	1, 19	eqtrid 2778	1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝐵 = {𝑡 ∈ 𝐿 ∣ (𝑁‘𝑡) < +∞})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  {crab 3395   class class class wbr 5089  ‘cfv 6481  (class class class)co 7346  +∞cpnf 11143   < clt 11146  NrmCVeccnv 30564   LnOp clno 30720   normOp_OLD cnmoo 30721   BLnOp cblo 30722

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 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-iota 6437  df-fun 6483  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-blo 30726

This theorem is referenced by:  isblo  30762  hhbloi  31882