Theorem bloval 31133

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 7417	. . . 4 ⊢ (𝑢 = 𝑈 → (𝑢 LnOp 𝑤) = (𝑈 LnOp 𝑤))
3		oveq1 7417	. . . . . 6 ⊢ (𝑢 = 𝑈 → (𝑢 normOpOLD 𝑤) = (𝑈 normOpOLD 𝑤))
4	3	fveq1d 6883	. . . . 5 ⊢ (𝑢 = 𝑈 → ((𝑢 normOpOLD 𝑤)‘𝑡) = ((𝑈 normOpOLD 𝑤)‘𝑡))
5	4	breq1d 5119	. . . 4 ⊢ (𝑢 = 𝑈 → (((𝑢 normOpOLD 𝑤)‘𝑡) < +∞ ↔ ((𝑈 normOpOLD 𝑤)‘𝑡) < +∞))
6	2, 5	rabeqbidv 3434	. . 3 ⊢ (𝑢 = 𝑈 → {𝑡 ∈ (𝑢 LnOp 𝑤) ∣ ((𝑢 normOpOLD 𝑤)‘𝑡) < +∞} = {𝑡 ∈ (𝑈 LnOp 𝑤) ∣ ((𝑈 normOpOLD 𝑤)‘𝑡) < +∞})
7		oveq2 7418	. . . . 5 ⊢ (𝑤 = 𝑊 → (𝑈 LnOp 𝑤) = (𝑈 LnOp 𝑊))
8		bloval.4	. . . . 5 ⊢ 𝐿 = (𝑈 LnOp 𝑊)
9	7, 8	eqtr4di 2816	. . . 4 ⊢ (𝑤 = 𝑊 → (𝑈 LnOp 𝑤) = 𝐿)
10		oveq2 7418	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (𝑈 normOpOLD 𝑤) = (𝑈 normOpOLD 𝑊))
11		bloval.3	. . . . . . 7 ⊢ 𝑁 = (𝑈 normOpOLD 𝑊)
12	10, 11	eqtr4di 2816	. . . . . 6 ⊢ (𝑤 = 𝑊 → (𝑈 normOpOLD 𝑤) = 𝑁)
13	12	fveq1d 6883	. . . . 5 ⊢ (𝑤 = 𝑊 → ((𝑈 normOpOLD 𝑤)‘𝑡) = (𝑁‘𝑡))
14	13	breq1d 5119	. . . 4 ⊢ (𝑤 = 𝑊 → (((𝑈 normOpOLD 𝑤)‘𝑡) < +∞ ↔ (𝑁‘𝑡) < +∞))
15	9, 14	rabeqbidv 3434	. . 3 ⊢ (𝑤 = 𝑊 → {𝑡 ∈ (𝑈 LnOp 𝑤) ∣ ((𝑈 normOpOLD 𝑤)‘𝑡) < +∞} = {𝑡 ∈ 𝐿 ∣ (𝑁‘𝑡) < +∞})
16		df-blo 31098	. . 3 ⊢ BLnOp = (𝑢 ∈ NrmCVec, 𝑤 ∈ NrmCVec ↦ {𝑡 ∈ (𝑢 LnOp 𝑤) ∣ ((𝑢 normOpOLD 𝑤)‘𝑡) < +∞})
17	8	ovexi 7444	. . . 4 ⊢ 𝐿 ∈ V
18	17	rabex 5309	. . 3 ⊢ {𝑡 ∈ 𝐿 ∣ (𝑁‘𝑡) < +∞} ∈ V
19	6, 15, 16, 18	ovmpo 7570	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑈 BLnOp 𝑊) = {𝑡 ∈ 𝐿 ∣ (𝑁‘𝑡) < +∞})
20	1, 19	eqtrid 2810	1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝐵 = {𝑡 ∈ 𝐿 ∣ (𝑁‘𝑡) < +∞})

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143  {crab 3416   class class class wbr 5109  ‘cfv 6536  (class class class)co 7410  +∞cpnf 11235   < clt 11238  NrmCVeccnv 30936   LnOp clno 31092   normOp_OLD cnmoo 31093   BLnOp cblo 31094

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3745  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-iota 6492  df-fun 6538  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-blo 31098

This theorem is referenced by:  isblo  31134  hhbloi  32254