isblo2 - Metamath Proof Explorer

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Theorem isblo2 29046

Description: The predicate "is a bounded linear operator." (Contributed by NM, 8-Dec-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
isblo2	⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑇 ∈ 𝐵 ↔ (𝑇 ∈ 𝐿 ∧ (𝑁‘𝑇) ∈ ℝ)))

**Proof of Theorem isblo2**
Step	Hyp	Ref	Expression
1		bloval.3	. . 3 ⊢ 𝑁 = (𝑈 normOp_OLD 𝑊)
2		bloval.4	. . 3 ⊢ 𝐿 = (𝑈 LnOp 𝑊)
3		bloval.5	. . 3 ⊢ 𝐵 = (𝑈 BLnOp 𝑊)
4	1, 2, 3	isblo 29045	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑇 ∈ 𝐵 ↔ (𝑇 ∈ 𝐿 ∧ (𝑁‘𝑇) < +∞)))
5		eqid 2738	. . . . . 6 ⊢ (BaseSet‘𝑈) = (BaseSet‘𝑈)
6		eqid 2738	. . . . . 6 ⊢ (BaseSet‘𝑊) = (BaseSet‘𝑊)
7	5, 6, 2	lnof 29018	. . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → 𝑇:(BaseSet‘𝑈)⟶(BaseSet‘𝑊))
8	5, 6, 1	nmoreltpnf 29032	. . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:(BaseSet‘𝑈)⟶(BaseSet‘𝑊)) → ((𝑁‘𝑇) ∈ ℝ ↔ (𝑁‘𝑇) < +∞))
9	7, 8	syld3an3 1407	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → ((𝑁‘𝑇) ∈ ℝ ↔ (𝑁‘𝑇) < +∞))
10	9	3expa 1116	. . 3 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) ∧ 𝑇 ∈ 𝐿) → ((𝑁‘𝑇) ∈ ℝ ↔ (𝑁‘𝑇) < +∞))
11	10	pm5.32da 578	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → ((𝑇 ∈ 𝐿 ∧ (𝑁‘𝑇) ∈ ℝ) ↔ (𝑇 ∈ 𝐿 ∧ (𝑁‘𝑇) < +∞)))
12	4, 11	bitr4d 281	1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑇 ∈ 𝐵 ↔ (𝑇 ∈ 𝐿 ∧ (𝑁‘𝑇) ∈ ℝ)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1539 ∈ wcel 2108 class class class wbr 5070 ⟶wf 6414 ‘cfv 6418 (class class class)co 7255 ℝcr 10801 +∞cpnf 10937 < clt 10940 NrmCVeccnv 28847 BaseSetcba 28849 LnOp clno 29003 normOp_OLD cnmoo 29004 BLnOp cblo 29005

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-er 8456 df-map 8575 df-en 8692 df-dom 8693 df-sdom 8694 df-sup 9131 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-n0 12164 df-z 12250 df-uz 12512 df-rp 12660 df-seq 13650 df-exp 13711 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-grpo 28756 df-gid 28757 df-ginv 28758 df-ablo 28808 df-vc 28822 df-nv 28855 df-va 28858 df-ba 28859 df-sm 28860 df-0v 28861 df-nmcv 28863 df-lno 29007 df-nmoo 29008 df-blo 29009

This theorem is referenced by: 0blo 29055

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