adjbdln - Hilbert Space Explorer

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Theorem adjbdln 32055

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 32054	. . . 4 ⊢ (𝑇 ∈ BndLinOp → 𝑇 ∈ dom adj_ℎ)
2		adjval 31862	. . . 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 32051	. . . . . 6 ⊢ (𝑇 ∈ (LinOp ∩ ContOp) → ∃𝑡 ∈ (LinOp ∩ ContOp)∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇‘𝑥) ·_ih 𝑦) = (𝑥 ·_ih (𝑡‘𝑦)))
5		lncnopbd 32009	. . . . . 6 ⊢ (𝑇 ∈ (LinOp ∩ ContOp) ↔ 𝑇 ∈ BndLinOp)
6		lncnbd 32010	. . . . . . 7 ⊢ (LinOp ∩ ContOp) = BndLinOp
7	6	rexeqi 3291	. . . . . 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 31834	. . . . . . . 8 ⊢ (𝑇 ∈ BndLinOp → 𝑇: ℋ⟶ ℋ)
10		bdopf 31834	. . . . . . . 8 ⊢ (𝑡 ∈ BndLinOp → 𝑡: ℋ⟶ ℋ)
11		adjsym 31805	. . . . . . . 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 596	. . . . . . 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 2738	. . . . . . . 8 ⊢ (((𝑇‘𝑥) ·_ih 𝑦) = (𝑥 ·_ih (𝑡‘𝑦)) ↔ (𝑥 ·_ih (𝑡‘𝑦)) = ((𝑇‘𝑥) ·_ih 𝑦))
14	13	2ralbii 3107	. . . . . . 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 3154	. . . . 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 31861	. . . . . 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 30971	. . . . . . . 8 ⊢ ℋ ∈ V
22	21, 21	elmap 8790	. . . . . . 7 ⊢ (𝑡 ∈ ( ℋ ↑_m ℋ) ↔ 𝑡: ℋ⟶ ℋ)
23	10, 22	sylibr 234	. . . . . 6 ⊢ (𝑡 ∈ BndLinOp → 𝑡 ∈ ( ℋ ↑_m ℋ))
24	23	ssriv 3933	. . . . 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 3051	. . . . 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 7328	. . . . 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 702	. . . 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 584	. . 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 2769	. 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 4272	. . . 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 1373	. . 3 ⊢ (𝑇 ∈ BndLinOp → ∃!𝑡 ∈ BndLinOp ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑡‘𝑥) ·_ih 𝑦))
34		riotacl 7315	. . 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 2831	1 ⊢ (𝑇 ∈ BndLinOp → (adj_ℎ‘𝑇) ∈ BndLinOp)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1541 ∈ wcel 2111 ∀wral 3047 ∃wrex 3056 ∃!wreu 3344 ∩ cin 3896 ⊆ wss 3897 dom cdm 5611 ⟶wf 6472 ‘cfv 6476 ℩crio 7297 (class class class)co 7341 ↑_m cmap 8745 ℋchba 30891 ·_ih csp 30894 ContOpccop 30918 LinOpclo 30919 BndLinOpcbo 30920 adj_ℎcado 30927

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5212 ax-sep 5229 ax-nul 5239 ax-pow 5298 ax-pr 5365 ax-un 7663 ax-inf2 9526 ax-cc 10321 ax-cnex 11057 ax-resscn 11058 ax-1cn 11059 ax-icn 11060 ax-addcl 11061 ax-addrcl 11062 ax-mulcl 11063 ax-mulrcl 11064 ax-mulcom 11065 ax-addass 11066 ax-mulass 11067 ax-distr 11068 ax-i2m1 11069 ax-1ne0 11070 ax-1rid 11071 ax-rnegex 11072 ax-rrecex 11073 ax-cnre 11074 ax-pre-lttri 11075 ax-pre-lttrn 11076 ax-pre-ltadd 11077 ax-pre-mulgt0 11078 ax-pre-sup 11079 ax-addf 11080 ax-mulf 11081 ax-hilex 30971 ax-hfvadd 30972 ax-hvcom 30973 ax-hvass 30974 ax-hv0cl 30975 ax-hvaddid 30976 ax-hfvmul 30977 ax-hvmulid 30978 ax-hvmulass 30979 ax-hvdistr1 30980 ax-hvdistr2 30981 ax-hvmul0 30982 ax-hfi 31051 ax-his1 31054 ax-his2 31055 ax-his3 31056 ax-his4 31057 ax-hcompl 31174

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4279 df-if 4471 df-pw 4547 df-sn 4572 df-pr 4574 df-tp 4576 df-op 4578 df-uni 4855 df-int 4893 df-iun 4938 df-iin 4939 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5506 df-eprel 5511 df-po 5519 df-so 5520 df-fr 5564 df-se 5565 df-we 5566 df-xp 5617 df-rel 5618 df-cnv 5619 df-co 5620 df-dm 5621 df-rn 5622 df-res 5623 df-ima 5624 df-pred 6243 df-ord 6304 df-on 6305 df-lim 6306 df-suc 6307 df-iota 6432 df-fun 6478 df-fn 6479 df-f 6480 df-f1 6481 df-fo 6482 df-f1o 6483 df-fv 6484 df-isom 6485 df-riota 7298 df-ov 7344 df-oprab 7345 df-mpo 7346 df-of 7605 df-om 7792 df-1st 7916 df-2nd 7917 df-supp 8086 df-frecs 8206 df-wrecs 8237 df-recs 8286 df-rdg 8324 df-1o 8380 df-2o 8381 df-oadd 8384 df-omul 8385 df-er 8617 df-map 8747 df-pm 8748 df-ixp 8817 df-en 8865 df-dom 8866 df-sdom 8867 df-fin 8868 df-fsupp 9241 df-fi 9290 df-sup 9321 df-inf 9322 df-oi 9391 df-card 9827 df-acn 9830 df-pnf 11143 df-mnf 11144 df-xr 11145 df-ltxr 11146 df-le 11147 df-sub 11341 df-neg 11342 df-div 11770 df-nn 12121 df-2 12183 df-3 12184 df-4 12185 df-5 12186 df-6 12187 df-7 12188 df-8 12189 df-9 12190 df-n0 12377 df-z 12464 df-dec 12584 df-uz 12728 df-q 12842 df-rp 12886 df-xneg 13006 df-xadd 13007 df-xmul 13008 df-ioo 13244 df-ico 13246 df-icc 13247 df-fz 13403 df-fzo 13550 df-fl 13691 df-seq 13904 df-exp 13964 df-hash 14233 df-cj 15001 df-re 15002 df-im 15003 df-sqrt 15137 df-abs 15138 df-clim 15390 df-rlim 15391 df-sum 15589 df-struct 17053 df-sets 17070 df-slot 17088 df-ndx 17100 df-base 17116 df-ress 17137 df-plusg 17169 df-mulr 17170 df-starv 17171 df-sca 17172 df-vsca 17173 df-ip 17174 df-tset 17175 df-ple 17176 df-ds 17178 df-unif 17179 df-hom 17180 df-cco 17181 df-rest 17321 df-topn 17322 df-0g 17340 df-gsum 17341 df-topgen 17342 df-pt 17343 df-prds 17346 df-xrs 17401 df-qtop 17406 df-imas 17407 df-xps 17409 df-mre 17483 df-mrc 17484 df-acs 17486 df-mgm 18543 df-sgrp 18622 df-mnd 18638 df-submnd 18687 df-mulg 18976 df-cntz 19224 df-cmn 19689 df-psmet 21278 df-xmet 21279 df-met 21280 df-bl 21281 df-mopn 21282 df-fbas 21283 df-fg 21284 df-cnfld 21287 df-top 22804 df-topon 22821 df-topsp 22843 df-bases 22856 df-cld 22929 df-ntr 22930 df-cls 22931 df-nei 23008 df-cn 23137 df-cnp 23138 df-lm 23139 df-t1 23224 df-haus 23225 df-tx 23472 df-hmeo 23665 df-fil 23756 df-fm 23848 df-flim 23849 df-flf 23850 df-xms 24230 df-ms 24231 df-tms 24232 df-cfil 25177 df-cau 25178 df-cmet 25179 df-grpo 30465 df-gid 30466 df-ginv 30467 df-gdiv 30468 df-ablo 30517 df-vc 30531 df-nv 30564 df-va 30567 df-ba 30568 df-sm 30569 df-0v 30570 df-vs 30571 df-nmcv 30572 df-ims 30573 df-dip 30673 df-ssp 30694 df-ph 30785 df-cbn 30835 df-hnorm 30940 df-hba 30941 df-hvsub 30943 df-hlim 30944 df-hcau 30945 df-sh 31179 df-ch 31193 df-oc 31224 df-ch0 31225 df-shs 31280 df-pjh 31367 df-h0op 31720 df-nmop 31811 df-cnop 31812 df-lnop 31813 df-bdop 31814 df-unop 31815 df-hmop 31816 df-nmfn 31817 df-nlfn 31818 df-cnfn 31819 df-lnfn 31820 df-adjh 31821

This theorem is referenced by: adjbdlnb 32056 adjbd1o 32057 nmopadjlem 32061 nmopadji 32062 adjcoi 32072 nmopcoadj2i 32074 nmopcoadj0i 32075

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