isblo2 - Metamath Proof Explorer

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Theorem isblo2 30718

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 30717	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑇 ∈ 𝐵 ↔ (𝑇 ∈ 𝐿 ∧ (𝑁‘𝑇) < +∞)))
5		eqid 2730	. . . . . 6 ⊢ (BaseSet‘𝑈) = (BaseSet‘𝑈)
6		eqid 2730	. . . . . 6 ⊢ (BaseSet‘𝑊) = (BaseSet‘𝑊)
7	5, 6, 2	lnof 30690	. . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → 𝑇:(BaseSet‘𝑈)⟶(BaseSet‘𝑊))
8	5, 6, 1	nmoreltpnf 30704	. . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:(BaseSet‘𝑈)⟶(BaseSet‘𝑊)) → ((𝑁‘𝑇) ∈ ℝ ↔ (𝑁‘𝑇) < +∞))
9	7, 8	syld3an3 1411	. . . 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 579	. 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 1540 ∈ wcel 2109 class class class wbr 5109 ⟶wf 6509 ‘cfv 6513 (class class class)co 7389 ℝcr 11073 +∞cpnf 11211 < clt 11214 NrmCVeccnv 30519 BaseSetcba 30521 LnOp clno 30675 normOp_OLD cnmoo 30676 BLnOp cblo 30677

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-rep 5236 ax-sep 5253 ax-nul 5263 ax-pow 5322 ax-pr 5389 ax-un 7713 ax-cnex 11130 ax-resscn 11131 ax-1cn 11132 ax-icn 11133 ax-addcl 11134 ax-addrcl 11135 ax-mulcl 11136 ax-mulrcl 11137 ax-mulcom 11138 ax-addass 11139 ax-mulass 11140 ax-distr 11141 ax-i2m1 11142 ax-1ne0 11143 ax-1rid 11144 ax-rnegex 11145 ax-rrecex 11146 ax-cnre 11147 ax-pre-lttri 11148 ax-pre-lttrn 11149 ax-pre-ltadd 11150 ax-pre-mulgt0 11151 ax-pre-sup 11152

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 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-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3756 df-csb 3865 df-dif 3919 df-un 3921 df-in 3923 df-ss 3933 df-pss 3936 df-nul 4299 df-if 4491 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4874 df-iun 4959 df-br 5110 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5535 df-eprel 5540 df-po 5548 df-so 5549 df-fr 5593 df-we 5595 df-xp 5646 df-rel 5647 df-cnv 5648 df-co 5649 df-dm 5650 df-rn 5651 df-res 5652 df-ima 5653 df-pred 6276 df-ord 6337 df-on 6338 df-lim 6339 df-suc 6340 df-iota 6466 df-fun 6515 df-fn 6516 df-f 6517 df-f1 6518 df-fo 6519 df-f1o 6520 df-fv 6521 df-riota 7346 df-ov 7392 df-oprab 7393 df-mpo 7394 df-om 7845 df-1st 7970 df-2nd 7971 df-frecs 8262 df-wrecs 8293 df-recs 8342 df-rdg 8380 df-er 8673 df-map 8803 df-en 8921 df-dom 8922 df-sdom 8923 df-sup 9399 df-pnf 11216 df-mnf 11217 df-xr 11218 df-ltxr 11219 df-le 11220 df-sub 11413 df-neg 11414 df-div 11842 df-nn 12188 df-2 12250 df-3 12251 df-n0 12449 df-z 12536 df-uz 12800 df-rp 12958 df-seq 13973 df-exp 14033 df-cj 15071 df-re 15072 df-im 15073 df-sqrt 15207 df-abs 15208 df-grpo 30428 df-gid 30429 df-ginv 30430 df-ablo 30480 df-vc 30494 df-nv 30527 df-va 30530 df-ba 30531 df-sm 30532 df-0v 30533 df-nmcv 30535 df-lno 30679 df-nmoo 30680 df-blo 30681

This theorem is referenced by: 0blo 30727

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