isblo2 - Metamath Proof Explorer

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Theorem isblo2 30806

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 30805	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑇 ∈ 𝐵 ↔ (𝑇 ∈ 𝐿 ∧ (𝑁‘𝑇) < +∞)))
5		eqid 2734	. . . . . 6 ⊢ (BaseSet‘𝑈) = (BaseSet‘𝑈)
6		eqid 2734	. . . . . 6 ⊢ (BaseSet‘𝑊) = (BaseSet‘𝑊)
7	5, 6, 2	lnof 30778	. . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → 𝑇:(BaseSet‘𝑈)⟶(BaseSet‘𝑊))
8	5, 6, 1	nmoreltpnf 30792	. . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:(BaseSet‘𝑈)⟶(BaseSet‘𝑊)) → ((𝑁‘𝑇) ∈ ℝ ↔ (𝑁‘𝑇) < +∞))
9	7, 8	syld3an3 1409	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → ((𝑁‘𝑇) ∈ ℝ ↔ (𝑁‘𝑇) < +∞))
10	9	3expa 1118	. . 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 282	1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑇 ∈ 𝐵 ↔ (𝑇 ∈ 𝐿 ∧ (𝑁‘𝑇) ∈ ℝ)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2103 class class class wbr 5169 ⟶wf 6568 ‘cfv 6572 (class class class)co 7445 ℝcr 11179 +∞cpnf 11317 < clt 11320 NrmCVeccnv 30607 BaseSetcba 30609 LnOp clno 30763 normOp_OLD cnmoo 30764 BLnOp cblo 30765

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-rep 5306 ax-sep 5320 ax-nul 5327 ax-pow 5386 ax-pr 5450 ax-un 7766 ax-cnex 11236 ax-resscn 11237 ax-1cn 11238 ax-icn 11239 ax-addcl 11240 ax-addrcl 11241 ax-mulcl 11242 ax-mulrcl 11243 ax-mulcom 11244 ax-addass 11245 ax-mulass 11246 ax-distr 11247 ax-i2m1 11248 ax-1ne0 11249 ax-1rid 11250 ax-rnegex 11251 ax-rrecex 11252 ax-cnre 11253 ax-pre-lttri 11254 ax-pre-lttrn 11255 ax-pre-ltadd 11256 ax-pre-mulgt0 11257 ax-pre-sup 11258

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3383 df-reu 3384 df-rab 3439 df-v 3484 df-sbc 3799 df-csb 3916 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-pss 3990 df-nul 4348 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5021 df-br 5170 df-opab 5232 df-mpt 5253 df-tr 5287 df-id 5597 df-eprel 5603 df-po 5611 df-so 5612 df-fr 5654 df-we 5656 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-res 5711 df-ima 5712 df-pred 6331 df-ord 6397 df-on 6398 df-lim 6399 df-suc 6400 df-iota 6524 df-fun 6574 df-fn 6575 df-f 6576 df-f1 6577 df-fo 6578 df-f1o 6579 df-fv 6580 df-riota 7401 df-ov 7448 df-oprab 7449 df-mpo 7450 df-om 7900 df-1st 8026 df-2nd 8027 df-frecs 8318 df-wrecs 8349 df-recs 8423 df-rdg 8462 df-er 8759 df-map 8882 df-en 9000 df-dom 9001 df-sdom 9002 df-sup 9507 df-pnf 11322 df-mnf 11323 df-xr 11324 df-ltxr 11325 df-le 11326 df-sub 11518 df-neg 11519 df-div 11944 df-nn 12290 df-2 12352 df-3 12353 df-n0 12550 df-z 12636 df-uz 12900 df-rp 13054 df-seq 14049 df-exp 14109 df-cj 15144 df-re 15145 df-im 15146 df-sqrt 15280 df-abs 15281 df-grpo 30516 df-gid 30517 df-ginv 30518 df-ablo 30568 df-vc 30582 df-nv 30615 df-va 30618 df-ba 30619 df-sm 30620 df-0v 30621 df-nmcv 30623 df-lno 30767 df-nmoo 30768 df-blo 30769

This theorem is referenced by: 0blo 30815

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