adjbdln - Hilbert Space Explorer

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

Theorem adjbdln 32115

Description: The adjoint of a bounded linear operator is a bounded linear operator. (Contributed by NM, 19-Feb-2006.) (New usage is discouraged.)

**Assertion**
Ref	Expression
adjbdln	⊢ (𝑇 ∈ BndLinOp → (adj_ℎ‘𝑇) ∈ BndLinOp)

Proof of Theorem adjbdln Dummy variables 𝑡 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		bdopadj 32114	. . . 4 ⊢ (𝑇 ∈ BndLinOp → 𝑇 ∈ dom adj_ℎ)
2		adjval 31922	. . . 4 ⊢ (𝑇 ∈ dom adj_ℎ → (adj_ℎ‘𝑇) = (℩𝑡 ∈ ( ℋ ↑_m ℋ)∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑡‘𝑥) ·_ih 𝑦)))
3	1, 2	syl 17	. . 3 ⊢ (𝑇 ∈ BndLinOp → (adj_ℎ‘𝑇) = (℩𝑡 ∈ ( ℋ ↑_m ℋ)∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑡‘𝑥) ·_ih 𝑦)))
4		cnlnadj 32111	. . . . . 6 ⊢ (𝑇 ∈ (LinOp ∩ ContOp) → ∃𝑡 ∈ (LinOp ∩ ContOp)∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇‘𝑥) ·_ih 𝑦) = (𝑥 ·_ih (𝑡‘𝑦)))
5		lncnopbd 32069	. . . . . 6 ⊢ (𝑇 ∈ (LinOp ∩ ContOp) ↔ 𝑇 ∈ BndLinOp)
6		lncnbd 32070	. . . . . . 7 ⊢ (LinOp ∩ ContOp) = BndLinOp
7	6	rexeqi 3333	. . . . . 6 ⊢ (∃𝑡 ∈ (LinOp ∩ ContOp)∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇‘𝑥) ·_ih 𝑦) = (𝑥 ·_ih (𝑡‘𝑦)) ↔ ∃𝑡 ∈ BndLinOp ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇‘𝑥) ·_ih 𝑦) = (𝑥 ·_ih (𝑡‘𝑦)))
8	4, 5, 7	3imtr3i 291	. . . . 5 ⊢ (𝑇 ∈ BndLinOp → ∃𝑡 ∈ BndLinOp ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇‘𝑥) ·_ih 𝑦) = (𝑥 ·_ih (𝑡‘𝑦)))
9		bdopf 31894	. . . . . . . 8 ⊢ (𝑇 ∈ BndLinOp → 𝑇: ℋ⟶ ℋ)
10		bdopf 31894	. . . . . . . 8 ⊢ (𝑡 ∈ BndLinOp → 𝑡: ℋ⟶ ℋ)
11		adjsym 31865	. . . . . . . 8 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑡: ℋ⟶ ℋ) → (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑡‘𝑥) ·_ih 𝑦) ↔ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑡‘𝑦)) = ((𝑇‘𝑥) ·_ih 𝑦)))
12	9, 10, 11	syl2an 595	. . . . . . 7 ⊢ ((𝑇 ∈ BndLinOp ∧ 𝑡 ∈ BndLinOp) → (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑡‘𝑥) ·_ih 𝑦) ↔ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑡‘𝑦)) = ((𝑇‘𝑥) ·_ih 𝑦)))
13		eqcom 2747	. . . . . . . 8 ⊢ (((𝑇‘𝑥) ·_ih 𝑦) = (𝑥 ·_ih (𝑡‘𝑦)) ↔ (𝑥 ·_ih (𝑡‘𝑦)) = ((𝑇‘𝑥) ·_ih 𝑦))
14	13	2ralbii 3134	. . . . . . 7 ⊢ (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇‘𝑥) ·_ih 𝑦) = (𝑥 ·_ih (𝑡‘𝑦)) ↔ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑡‘𝑦)) = ((𝑇‘𝑥) ·_ih 𝑦))
15	12, 14	bitr4di 289	. . . . . 6 ⊢ ((𝑇 ∈ BndLinOp ∧ 𝑡 ∈ BndLinOp) → (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑡‘𝑥) ·_ih 𝑦) ↔ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇‘𝑥) ·_ih 𝑦) = (𝑥 ·_ih (𝑡‘𝑦))))
16	15	rexbidva 3183	. . . . 5 ⊢ (𝑇 ∈ BndLinOp → (∃𝑡 ∈ BndLinOp ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑡‘𝑥) ·_ih 𝑦) ↔ ∃𝑡 ∈ BndLinOp ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇‘𝑥) ·_ih 𝑦) = (𝑥 ·_ih (𝑡‘𝑦))))
17	8, 16	mpbird 257	. . . 4 ⊢ (𝑇 ∈ BndLinOp → ∃𝑡 ∈ BndLinOp ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑡‘𝑥) ·_ih 𝑦))
18		adjeu 31921	. . . . . 6 ⊢ (𝑇: ℋ⟶ ℋ → (𝑇 ∈ dom adj_ℎ ↔ ∃!𝑡 ∈ ( ℋ ↑_m ℋ)∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑡‘𝑥) ·_ih 𝑦)))
19	9, 18	syl 17	. . . . 5 ⊢ (𝑇 ∈ BndLinOp → (𝑇 ∈ dom adj_ℎ ↔ ∃!𝑡 ∈ ( ℋ ↑_m ℋ)∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑡‘𝑥) ·_ih 𝑦)))
20	1, 19	mpbid 232	. . . 4 ⊢ (𝑇 ∈ BndLinOp → ∃!𝑡 ∈ ( ℋ ↑_m ℋ)∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑡‘𝑥) ·_ih 𝑦))
21		ax-hilex 31031	. . . . . . . 8 ⊢ ℋ ∈ V
22	21, 21	elmap 8929	. . . . . . 7 ⊢ (𝑡 ∈ ( ℋ ↑_m ℋ) ↔ 𝑡: ℋ⟶ ℋ)
23	10, 22	sylibr 234	. . . . . 6 ⊢ (𝑡 ∈ BndLinOp → 𝑡 ∈ ( ℋ ↑_m ℋ))
24	23	ssriv 4012	. . . . 5 ⊢ BndLinOp ⊆ ( ℋ ↑_m ℋ)
25		id 22	. . . . . 6 ⊢ (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑡‘𝑥) ·_ih 𝑦) → ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑡‘𝑥) ·_ih 𝑦))
26	25	rgenw 3071	. . . . 5 ⊢ ∀𝑡 ∈ BndLinOp (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑡‘𝑥) ·_ih 𝑦) → ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑡‘𝑥) ·_ih 𝑦))
27		riotass2 7435	. . . . 5 ⊢ (((BndLinOp ⊆ ( ℋ ↑_m ℋ) ∧ ∀𝑡 ∈ BndLinOp (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑡‘𝑥) ·_ih 𝑦) → ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑡‘𝑥) ·_ih 𝑦))) ∧ (∃𝑡 ∈ BndLinOp ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑡‘𝑥) ·_ih 𝑦) ∧ ∃!𝑡 ∈ ( ℋ ↑_m ℋ)∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑡‘𝑥) ·_ih 𝑦))) → (℩𝑡 ∈ BndLinOp ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑡‘𝑥) ·_ih 𝑦)) = (℩𝑡 ∈ ( ℋ ↑_m ℋ)∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑡‘𝑥) ·_ih 𝑦)))
28	24, 26, 27	mpanl12 701	. . . 4 ⊢ ((∃𝑡 ∈ BndLinOp ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑡‘𝑥) ·_ih 𝑦) ∧ ∃!𝑡 ∈ ( ℋ ↑_m ℋ)∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑡‘𝑥) ·_ih 𝑦)) → (℩𝑡 ∈ BndLinOp ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑡‘𝑥) ·_ih 𝑦)) = (℩𝑡 ∈ ( ℋ ↑_m ℋ)∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑡‘𝑥) ·_ih 𝑦)))
29	17, 20, 28	syl2anc 583	. . 3 ⊢ (𝑇 ∈ BndLinOp → (℩𝑡 ∈ BndLinOp ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑡‘𝑥) ·_ih 𝑦)) = (℩𝑡 ∈ ( ℋ ↑_m ℋ)∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑡‘𝑥) ·_ih 𝑦)))
30	3, 29	eqtr4d 2783	. 2 ⊢ (𝑇 ∈ BndLinOp → (adj_ℎ‘𝑇) = (℩𝑡 ∈ BndLinOp ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑡‘𝑥) ·_ih 𝑦)))
31	24	a1i 11	. . . 4 ⊢ (𝑇 ∈ BndLinOp → BndLinOp ⊆ ( ℋ ↑_m ℋ))
32		reuss 4346	. . . 4 ⊢ ((BndLinOp ⊆ ( ℋ ↑_m ℋ) ∧ ∃𝑡 ∈ BndLinOp ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑡‘𝑥) ·_ih 𝑦) ∧ ∃!𝑡 ∈ ( ℋ ↑_m ℋ)∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑡‘𝑥) ·_ih 𝑦)) → ∃!𝑡 ∈ BndLinOp ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑡‘𝑥) ·_ih 𝑦))
33	31, 17, 20, 32	syl3anc 1371	. . 3 ⊢ (𝑇 ∈ BndLinOp → ∃!𝑡 ∈ BndLinOp ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑡‘𝑥) ·_ih 𝑦))
34		riotacl 7422	. . 3 ⊢ (∃!𝑡 ∈ BndLinOp ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑡‘𝑥) ·_ih 𝑦) → (℩𝑡 ∈ BndLinOp ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑡‘𝑥) ·_ih 𝑦)) ∈ BndLinOp)
35	33, 34	syl 17	. 2 ⊢ (𝑇 ∈ BndLinOp → (℩𝑡 ∈ BndLinOp ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑡‘𝑥) ·_ih 𝑦)) ∈ BndLinOp)
36	30, 35	eqeltrd 2844	1 ⊢ (𝑇 ∈ BndLinOp → (adj_ℎ‘𝑇) ∈ BndLinOp)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2108 ∀wral 3067 ∃wrex 3076 ∃!wreu 3386 ∩ cin 3975 ⊆ wss 3976 dom cdm 5700 ⟶wf 6569 ‘cfv 6573 ℩crio 7403 (class class class)co 7448 ↑_m cmap 8884 ℋchba 30951 ·_ih csp 30954 ContOpccop 30978 LinOpclo 30979 BndLinOpcbo 30980 adj_ℎcado 30987

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 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-inf2 9710 ax-cc 10504 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262 ax-addf 11263 ax-mulf 11264 ax-hilex 31031 ax-hfvadd 31032 ax-hvcom 31033 ax-hvass 31034 ax-hv0cl 31035 ax-hvaddid 31036 ax-hfvmul 31037 ax-hvmulid 31038 ax-hvmulass 31039 ax-hvdistr1 31040 ax-hvdistr2 31041 ax-hvmul0 31042 ax-hfi 31111 ax-his1 31114 ax-his2 31115 ax-his3 31116 ax-his4 31117 ax-hcompl 31234

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 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-of 7714 df-om 7904 df-1st 8030 df-2nd 8031 df-supp 8202 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-oadd 8526 df-omul 8527 df-er 8763 df-map 8886 df-pm 8887 df-ixp 8956 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-fsupp 9432 df-fi 9480 df-sup 9511 df-inf 9512 df-oi 9579 df-card 10008 df-acn 10011 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-z 12640 df-dec 12759 df-uz 12904 df-q 13014 df-rp 13058 df-xneg 13175 df-xadd 13176 df-xmul 13177 df-ioo 13411 df-ico 13413 df-icc 13414 df-fz 13568 df-fzo 13712 df-fl 13843 df-seq 14053 df-exp 14113 df-hash 14380 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-clim 15534 df-rlim 15535 df-sum 15735 df-struct 17194 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-mulr 17325 df-starv 17326 df-sca 17327 df-vsca 17328 df-ip 17329 df-tset 17330 df-ple 17331 df-ds 17333 df-unif 17334 df-hom 17335 df-cco 17336 df-rest 17482 df-topn 17483 df-0g 17501 df-gsum 17502 df-topgen 17503 df-pt 17504 df-prds 17507 df-xrs 17562 df-qtop 17567 df-imas 17568 df-xps 17570 df-mre 17644 df-mrc 17645 df-acs 17647 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-submnd 18819 df-mulg 19108 df-cntz 19357 df-cmn 19824 df-psmet 21379 df-xmet 21380 df-met 21381 df-bl 21382 df-mopn 21383 df-fbas 21384 df-fg 21385 df-cnfld 21388 df-top 22921 df-topon 22938 df-topsp 22960 df-bases 22974 df-cld 23048 df-ntr 23049 df-cls 23050 df-nei 23127 df-cn 23256 df-cnp 23257 df-lm 23258 df-t1 23343 df-haus 23344 df-tx 23591 df-hmeo 23784 df-fil 23875 df-fm 23967 df-flim 23968 df-flf 23969 df-xms 24351 df-ms 24352 df-tms 24353 df-cfil 25308 df-cau 25309 df-cmet 25310 df-grpo 30525 df-gid 30526 df-ginv 30527 df-gdiv 30528 df-ablo 30577 df-vc 30591 df-nv 30624 df-va 30627 df-ba 30628 df-sm 30629 df-0v 30630 df-vs 30631 df-nmcv 30632 df-ims 30633 df-dip 30733 df-ssp 30754 df-ph 30845 df-cbn 30895 df-hnorm 31000 df-hba 31001 df-hvsub 31003 df-hlim 31004 df-hcau 31005 df-sh 31239 df-ch 31253 df-oc 31284 df-ch0 31285 df-shs 31340 df-pjh 31427 df-h0op 31780 df-nmop 31871 df-cnop 31872 df-lnop 31873 df-bdop 31874 df-unop 31875 df-hmop 31876 df-nmfn 31877 df-nlfn 31878 df-cnfn 31879 df-lnfn 31880 df-adjh 31881

This theorem is referenced by: adjbdlnb 32116 adjbd1o 32117 nmopadjlem 32121 nmopadji 32122 adjcoi 32132 nmopcoadj2i 32134 nmopcoadj0i 32135

W3C validator