bdopcoi - Hilbert Space Explorer

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Theorem bdopcoi 31382

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 31145	. . . 4 ⊢ (𝑆 ∈ BndLinOp → 𝑆 ∈ LinOp)
3	1, 2	ax-mp 5	. . 3 ⊢ 𝑆 ∈ LinOp
4		nmoptri.2	. . . 4 ⊢ 𝑇 ∈ BndLinOp
5		bdopln 31145	. . . 4 ⊢ (𝑇 ∈ BndLinOp → 𝑇 ∈ LinOp)
6	4, 5	ax-mp 5	. . 3 ⊢ 𝑇 ∈ LinOp
7	3, 6	lnopcoi 31287	. 2 ⊢ (𝑆 ∘ 𝑇) ∈ LinOp
8	3	lnopfi 31253	. . . . 5 ⊢ 𝑆: ℋ⟶ ℋ
9	6	lnopfi 31253	. . . . 5 ⊢ 𝑇: ℋ⟶ ℋ
10	8, 9	hocofi 31050	. . . 4 ⊢ (𝑆 ∘ 𝑇): ℋ⟶ ℋ
11		nmopxr 31150	. . . 4 ⊢ ((𝑆 ∘ 𝑇): ℋ⟶ ℋ → (norm_op‘(𝑆 ∘ 𝑇)) ∈ ℝ^*)
12	10, 11	ax-mp 5	. . 3 ⊢ (norm_op‘(𝑆 ∘ 𝑇)) ∈ ℝ^*
13		nmopre 31154	. . . . 5 ⊢ (𝑆 ∈ BndLinOp → (norm_op‘𝑆) ∈ ℝ)
14	1, 13	ax-mp 5	. . . 4 ⊢ (norm_op‘𝑆) ∈ ℝ
15		nmopre 31154	. . . . 5 ⊢ (𝑇 ∈ BndLinOp → (norm_op‘𝑇) ∈ ℝ)
16	4, 15	ax-mp 5	. . . 4 ⊢ (norm_op‘𝑇) ∈ ℝ
17	14, 16	remulcli 11230	. . 3 ⊢ ((norm_op‘𝑆) · (norm_op‘𝑇)) ∈ ℝ
18		nmopgtmnf 31152	. . . 4 ⊢ ((𝑆 ∘ 𝑇): ℋ⟶ ℋ → -∞ < (norm_op‘(𝑆 ∘ 𝑇)))
19	10, 18	ax-mp 5	. . 3 ⊢ -∞ < (norm_op‘(𝑆 ∘ 𝑇))
20	1, 4	nmopcoi 31379	. . 3 ⊢ (norm_op‘(𝑆 ∘ 𝑇)) ≤ ((norm_op‘𝑆) · (norm_op‘𝑇))
21		xrre 13148	. . 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 692	. 2 ⊢ (norm_op‘(𝑆 ∘ 𝑇)) ∈ ℝ
23		elbdop2 31155	. 2 ⊢ ((𝑆 ∘ 𝑇) ∈ BndLinOp ↔ ((𝑆 ∘ 𝑇) ∈ LinOp ∧ (norm_op‘(𝑆 ∘ 𝑇)) ∈ ℝ))
24	7, 22, 23	mpbir2an 710	1 ⊢ (𝑆 ∘ 𝑇) ∈ BndLinOp

Colors of variables: wff setvar class

Syntax hints: ∈ wcel 2107 class class class wbr 5149 ∘ ccom 5681 ⟶wf 6540 ‘cfv 6544 (class class class)co 7409 ℝcr 11109 · cmul 11115 -∞cmnf 11246 ℝ^*cxr 11247 < clt 11248 ≤ cle 11249 ℋchba 30203 norm_opcnop 30229 LinOpclo 30231 BndLinOpcbo 30232

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-inf2 9636 ax-cc 10430 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188 ax-addf 11189 ax-mulf 11190 ax-hilex 30283 ax-hfvadd 30284 ax-hvcom 30285 ax-hvass 30286 ax-hv0cl 30287 ax-hvaddid 30288 ax-hfvmul 30289 ax-hvmulid 30290 ax-hvmulass 30291 ax-hvdistr1 30292 ax-hvdistr2 30293 ax-hvmul0 30294 ax-hfi 30363 ax-his1 30366 ax-his2 30367 ax-his3 30368 ax-his4 30369 ax-hcompl 30486

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-iin 5001 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-of 7670 df-om 7856 df-1st 7975 df-2nd 7976 df-supp 8147 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-2o 8467 df-oadd 8470 df-omul 8471 df-er 8703 df-map 8822 df-pm 8823 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fsupp 9362 df-fi 9406 df-sup 9437 df-inf 9438 df-oi 9505 df-card 9934 df-acn 9937 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-4 12277 df-5 12278 df-6 12279 df-7 12280 df-8 12281 df-9 12282 df-n0 12473 df-z 12559 df-dec 12678 df-uz 12823 df-q 12933 df-rp 12975 df-xneg 13092 df-xadd 13093 df-xmul 13094 df-ioo 13328 df-ico 13330 df-icc 13331 df-fz 13485 df-fzo 13628 df-fl 13757 df-seq 13967 df-exp 14028 df-hash 14291 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-clim 15432 df-rlim 15433 df-sum 15633 df-struct 17080 df-sets 17097 df-slot 17115 df-ndx 17127 df-base 17145 df-ress 17174 df-plusg 17210 df-mulr 17211 df-starv 17212 df-sca 17213 df-vsca 17214 df-ip 17215 df-tset 17216 df-ple 17217 df-ds 17219 df-unif 17220 df-hom 17221 df-cco 17222 df-rest 17368 df-topn 17369 df-0g 17387 df-gsum 17388 df-topgen 17389 df-pt 17390 df-prds 17393 df-xrs 17448 df-qtop 17453 df-imas 17454 df-xps 17456 df-mre 17530 df-mrc 17531 df-acs 17533 df-mgm 18561 df-sgrp 18610 df-mnd 18626 df-submnd 18672 df-mulg 18951 df-cntz 19181 df-cmn 19650 df-psmet 20936 df-xmet 20937 df-met 20938 df-bl 20939 df-mopn 20940 df-fbas 20941 df-fg 20942 df-cnfld 20945 df-top 22396 df-topon 22413 df-topsp 22435 df-bases 22449 df-cld 22523 df-ntr 22524 df-cls 22525 df-nei 22602 df-cn 22731 df-cnp 22732 df-lm 22733 df-haus 22819 df-tx 23066 df-hmeo 23259 df-fil 23350 df-fm 23442 df-flim 23443 df-flf 23444 df-xms 23826 df-ms 23827 df-tms 23828 df-cfil 24772 df-cau 24773 df-cmet 24774 df-grpo 29777 df-gid 29778 df-ginv 29779 df-gdiv 29780 df-ablo 29829 df-vc 29843 df-nv 29876 df-va 29879 df-ba 29880 df-sm 29881 df-0v 29882 df-vs 29883 df-nmcv 29884 df-ims 29885 df-dip 29985 df-ssp 30006 df-lno 30028 df-nmoo 30029 df-0o 30031 df-ph 30097 df-cbn 30147 df-hnorm 30252 df-hba 30253 df-hvsub 30255 df-hlim 30256 df-hcau 30257 df-sh 30491 df-ch 30505 df-oc 30536 df-ch0 30537 df-shs 30592 df-pjh 30679 df-h0op 31032 df-nmop 31123 df-lnop 31125 df-bdop 31126 df-hmop 31128

This theorem is referenced by: adjcoi 31384 nmopcoadji 31385 unierri 31388

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