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Theorem isblo 30718

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 normOp_OLD 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 normOp_OLD 𝑊)
2		bloval.4	. . . 4 ⊢ 𝐿 = (𝑈 LnOp 𝑊)
3		bloval.5	. . . 4 ⊢ 𝐵 = (𝑈 BLnOp 𝑊)
4	1, 2, 3	bloval 30717	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝐵 = {𝑡 ∈ 𝐿 ∣ (𝑁‘𝑡) < +∞})
5	4	eleq2d 2815	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑇 ∈ 𝐵 ↔ 𝑇 ∈ {𝑡 ∈ 𝐿 ∣ (𝑁‘𝑡) < +∞}))
6		fveq2 6861	. . . 4 ⊢ (𝑡 = 𝑇 → (𝑁‘𝑡) = (𝑁‘𝑇))
7	6	breq1d 5120	. . 3 ⊢ (𝑡 = 𝑇 → ((𝑁‘𝑡) < +∞ ↔ (𝑁‘𝑇) < +∞))
8	7	elrab 3662	. 2 ⊢ (𝑇 ∈ {𝑡 ∈ 𝐿 ∣ (𝑁‘𝑡) < +∞} ↔ (𝑇 ∈ 𝐿 ∧ (𝑁‘𝑇) < +∞))
9	5, 8	bitrdi 287	1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑇 ∈ 𝐵 ↔ (𝑇 ∈ 𝐿 ∧ (𝑁‘𝑇) < +∞)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2109 {crab 3408 class class class wbr 5110 ‘cfv 6514 (class class class)co 7390 +∞cpnf 11212 < clt 11215 NrmCVeccnv 30520 LnOp clno 30676 normOp_OLD cnmoo 30677 BLnOp cblo 30678

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-sep 5254 ax-nul 5264 ax-pr 5390

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-ral 3046 df-rex 3055 df-rab 3409 df-v 3452 df-sbc 3757 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-br 5111 df-opab 5173 df-id 5536 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-iota 6467 df-fun 6516 df-fv 6522 df-ov 7393 df-oprab 7394 df-mpo 7395 df-blo 30682

This theorem is referenced by: isblo2 30719 bloln 30720 nmblore 30722 isblo3i 30737 htthlem 30853

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