bdopcoi - Hilbert Space Explorer

	Hilbert Space Explorer		< Previous Next > Nearby theorems
Mirrors > Home > HSE Home > Th. List > bdopcoi		Structured version Visualization version GIF version

Theorem bdopcoi 31606

Description: The composition of two bounded linear operators is bounded. (Contributed by NM, 9-Mar-2006.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
nmoptri.1	⊢ 𝑆 ∈ BndLinOp
nmoptri.2	⊢ 𝑇 ∈ BndLinOp

**Assertion**
Ref	Expression
bdopcoi	⊢ (𝑆 ∘ 𝑇) ∈ BndLinOp

**Proof of Theorem bdopcoi**
Step	Hyp	Ref	Expression
1		nmoptri.1	. . . 4 ⊢ 𝑆 ∈ BndLinOp
2		bdopln 31369	. . . 4 ⊢ (𝑆 ∈ BndLinOp → 𝑆 ∈ LinOp)
3	1, 2	ax-mp 5	. . 3 ⊢ 𝑆 ∈ LinOp
4		nmoptri.2	. . . 4 ⊢ 𝑇 ∈ BndLinOp
5		bdopln 31369	. . . 4 ⊢ (𝑇 ∈ BndLinOp → 𝑇 ∈ LinOp)
6	4, 5	ax-mp 5	. . 3 ⊢ 𝑇 ∈ LinOp
7	3, 6	lnopcoi 31511	. 2 ⊢ (𝑆 ∘ 𝑇) ∈ LinOp
8	3	lnopfi 31477	. . . . 5 ⊢ 𝑆: ℋ⟶ ℋ
9	6	lnopfi 31477	. . . . 5 ⊢ 𝑇: ℋ⟶ ℋ
10	8, 9	hocofi 31274	. . . 4 ⊢ (𝑆 ∘ 𝑇): ℋ⟶ ℋ
11		nmopxr 31374	. . . 4 ⊢ ((𝑆 ∘ 𝑇): ℋ⟶ ℋ → (norm_op‘(𝑆 ∘ 𝑇)) ∈ ℝ^*)
12	10, 11	ax-mp 5	. . 3 ⊢ (norm_op‘(𝑆 ∘ 𝑇)) ∈ ℝ^*
13		nmopre 31378	. . . . 5 ⊢ (𝑆 ∈ BndLinOp → (norm_op‘𝑆) ∈ ℝ)
14	1, 13	ax-mp 5	. . . 4 ⊢ (norm_op‘𝑆) ∈ ℝ
15		nmopre 31378	. . . . 5 ⊢ (𝑇 ∈ BndLinOp → (norm_op‘𝑇) ∈ ℝ)
16	4, 15	ax-mp 5	. . . 4 ⊢ (norm_op‘𝑇) ∈ ℝ
17	14, 16	remulcli 11234	. . 3 ⊢ ((norm_op‘𝑆) · (norm_op‘𝑇)) ∈ ℝ
18		nmopgtmnf 31376	. . . 4 ⊢ ((𝑆 ∘ 𝑇): ℋ⟶ ℋ → -∞ < (norm_op‘(𝑆 ∘ 𝑇)))
19	10, 18	ax-mp 5	. . 3 ⊢ -∞ < (norm_op‘(𝑆 ∘ 𝑇))
20	1, 4	nmopcoi 31603	. . 3 ⊢ (norm_op‘(𝑆 ∘ 𝑇)) ≤ ((norm_op‘𝑆) · (norm_op‘𝑇))
21		xrre 13152	. . 3 ⊢ ((((norm_op‘(𝑆 ∘ 𝑇)) ∈ ℝ^* ∧ ((norm_op‘𝑆) · (norm_op‘𝑇)) ∈ ℝ) ∧ (-∞ < (norm_op‘(𝑆 ∘ 𝑇)) ∧ (norm_op‘(𝑆 ∘ 𝑇)) ≤ ((norm_op‘𝑆) · (norm_op‘𝑇)))) → (norm_op‘(𝑆 ∘ 𝑇)) ∈ ℝ)
22	12, 17, 19, 20, 21	mp4an 691	. 2 ⊢ (norm_op‘(𝑆 ∘ 𝑇)) ∈ ℝ
23		elbdop2 31379	. 2 ⊢ ((𝑆 ∘ 𝑇) ∈ BndLinOp ↔ ((𝑆 ∘ 𝑇) ∈ LinOp ∧ (norm_op‘(𝑆 ∘ 𝑇)) ∈ ℝ))
24	7, 22, 23	mpbir2an 709	1 ⊢ (𝑆 ∘ 𝑇) ∈ BndLinOp

Colors of variables: wff setvar class

Syntax hints: ∈ wcel 2106 class class class wbr 5148 ∘ ccom 5680 ⟶wf 6539 ‘cfv 6543 (class class class)co 7411 ℝcr 11111 · cmul 11117 -∞cmnf 11250 ℝ^*cxr 11251 < clt 11252 ≤ cle 11253 ℋchba 30427 norm_opcnop 30453 LinOpclo 30455 BndLinOpcbo 30456

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-inf2 9638 ax-cc 10432 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 ax-addf 11191 ax-mulf 11192 ax-hilex 30507 ax-hfvadd 30508 ax-hvcom 30509 ax-hvass 30510 ax-hv0cl 30511 ax-hvaddid 30512 ax-hfvmul 30513 ax-hvmulid 30514 ax-hvmulass 30515 ax-hvdistr1 30516 ax-hvdistr2 30517 ax-hvmul0 30518 ax-hfi 30587 ax-his1 30590 ax-his2 30591 ax-his3 30592 ax-his4 30593 ax-hcompl 30710

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7672 df-om 7858 df-1st 7977 df-2nd 7978 df-supp 8149 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-2o 8469 df-oadd 8472 df-omul 8473 df-er 8705 df-map 8824 df-pm 8825 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-fi 9408 df-sup 9439 df-inf 9440 df-oi 9507 df-card 9936 df-acn 9939 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-q 12937 df-rp 12979 df-xneg 13096 df-xadd 13097 df-xmul 13098 df-ioo 13332 df-ico 13334 df-icc 13335 df-fz 13489 df-fzo 13632 df-fl 13761 df-seq 13971 df-exp 14032 df-hash 14295 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-clim 15436 df-rlim 15437 df-sum 15637 df-struct 17084 df-sets 17101 df-slot 17119 df-ndx 17131 df-base 17149 df-ress 17178 df-plusg 17214 df-mulr 17215 df-starv 17216 df-sca 17217 df-vsca 17218 df-ip 17219 df-tset 17220 df-ple 17221 df-ds 17223 df-unif 17224 df-hom 17225 df-cco 17226 df-rest 17372 df-topn 17373 df-0g 17391 df-gsum 17392 df-topgen 17393 df-pt 17394 df-prds 17397 df-xrs 17452 df-qtop 17457 df-imas 17458 df-xps 17460 df-mre 17534 df-mrc 17535 df-acs 17537 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-submnd 18706 df-mulg 18987 df-cntz 19222 df-cmn 19691 df-psmet 21136 df-xmet 21137 df-met 21138 df-bl 21139 df-mopn 21140 df-fbas 21141 df-fg 21142 df-cnfld 21145 df-top 22616 df-topon 22633 df-topsp 22655 df-bases 22669 df-cld 22743 df-ntr 22744 df-cls 22745 df-nei 22822 df-cn 22951 df-cnp 22952 df-lm 22953 df-haus 23039 df-tx 23286 df-hmeo 23479 df-fil 23570 df-fm 23662 df-flim 23663 df-flf 23664 df-xms 24046 df-ms 24047 df-tms 24048 df-cfil 24996 df-cau 24997 df-cmet 24998 df-grpo 30001 df-gid 30002 df-ginv 30003 df-gdiv 30004 df-ablo 30053 df-vc 30067 df-nv 30100 df-va 30103 df-ba 30104 df-sm 30105 df-0v 30106 df-vs 30107 df-nmcv 30108 df-ims 30109 df-dip 30209 df-ssp 30230 df-lno 30252 df-nmoo 30253 df-0o 30255 df-ph 30321 df-cbn 30371 df-hnorm 30476 df-hba 30477 df-hvsub 30479 df-hlim 30480 df-hcau 30481 df-sh 30715 df-ch 30729 df-oc 30760 df-ch0 30761 df-shs 30816 df-pjh 30903 df-h0op 31256 df-nmop 31347 df-lnop 31349 df-bdop 31350 df-hmop 31352

This theorem is referenced by: adjcoi 31608 nmopcoadji 31609 unierri 31612

W3C validator