adjbdln - Hilbert Space Explorer

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Theorem adjbdln 31025

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 31024	. . . 4 ⊢ (𝑇 ∈ BndLinOp → 𝑇 ∈ dom adj_ℎ)
2		adjval 30832	. . . 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 31021	. . . . . 6 ⊢ (𝑇 ∈ (LinOp ∩ ContOp) → ∃𝑡 ∈ (LinOp ∩ ContOp)∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇‘𝑥) ·_ih 𝑦) = (𝑥 ·_ih (𝑡‘𝑦)))
5		lncnopbd 30979	. . . . . 6 ⊢ (𝑇 ∈ (LinOp ∩ ContOp) ↔ 𝑇 ∈ BndLinOp)
6		lncnbd 30980	. . . . . . 7 ⊢ (LinOp ∩ ContOp) = BndLinOp
7	6	rexeqi 3312	. . . . . 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 290	. . . . 5 ⊢ (𝑇 ∈ BndLinOp → ∃𝑡 ∈ BndLinOp ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇‘𝑥) ·_ih 𝑦) = (𝑥 ·_ih (𝑡‘𝑦)))
9		bdopf 30804	. . . . . . . 8 ⊢ (𝑇 ∈ BndLinOp → 𝑇: ℋ⟶ ℋ)
10		bdopf 30804	. . . . . . . 8 ⊢ (𝑡 ∈ BndLinOp → 𝑡: ℋ⟶ ℋ)
11		adjsym 30775	. . . . . . . 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 2743	. . . . . . . 8 ⊢ (((𝑇‘𝑥) ·_ih 𝑦) = (𝑥 ·_ih (𝑡‘𝑦)) ↔ (𝑥 ·_ih (𝑡‘𝑦)) = ((𝑇‘𝑥) ·_ih 𝑦))
14	13	2ralbii 3127	. . . . . . 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 288	. . . . . 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 3173	. . . . 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 256	. . . 4 ⊢ (𝑇 ∈ BndLinOp → ∃𝑡 ∈ BndLinOp ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑡‘𝑥) ·_ih 𝑦))
18		adjeu 30831	. . . . . 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 231	. . . 4 ⊢ (𝑇 ∈ BndLinOp → ∃!𝑡 ∈ ( ℋ ↑_m ℋ)∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑡‘𝑥) ·_ih 𝑦))
21		ax-hilex 29941	. . . . . . . 8 ⊢ ℋ ∈ V
22	21, 21	elmap 8809	. . . . . . 7 ⊢ (𝑡 ∈ ( ℋ ↑_m ℋ) ↔ 𝑡: ℋ⟶ ℋ)
23	10, 22	sylibr 233	. . . . . 6 ⊢ (𝑡 ∈ BndLinOp → 𝑡 ∈ ( ℋ ↑_m ℋ))
24	23	ssriv 3948	. . . . 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 3068	. . . . 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 7344	. . . . 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 700	. . . 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 2779	. 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 4276	. . . 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 7331	. . 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 2838	1 ⊢ (𝑇 ∈ BndLinOp → (adj_ℎ‘𝑇) ∈ BndLinOp)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3064 ∃wrex 3073 ∃!wreu 3351 ∩ cin 3909 ⊆ wss 3910 dom cdm 5633 ⟶wf 6492 ‘cfv 6496 ℩crio 7312 (class class class)co 7357 ↑_m cmap 8765 ℋchba 29861 ·_ih csp 29864 ContOpccop 29888 LinOpclo 29889 BndLinOpcbo 29890 adj_ℎcado 29897

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 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-inf2 9577 ax-cc 10371 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129 ax-addf 11130 ax-mulf 11131 ax-hilex 29941 ax-hfvadd 29942 ax-hvcom 29943 ax-hvass 29944 ax-hv0cl 29945 ax-hvaddid 29946 ax-hfvmul 29947 ax-hvmulid 29948 ax-hvmulass 29949 ax-hvdistr1 29950 ax-hvdistr2 29951 ax-hvmul0 29952 ax-hfi 30021 ax-his1 30024 ax-his2 30025 ax-his3 30026 ax-his4 30027 ax-hcompl 30144

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 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-iin 4957 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7617 df-om 7803 df-1st 7921 df-2nd 7922 df-supp 8093 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-2o 8413 df-oadd 8416 df-omul 8417 df-er 8648 df-map 8767 df-pm 8768 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9306 df-fi 9347 df-sup 9378 df-inf 9379 df-oi 9446 df-card 9875 df-acn 9878 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-z 12500 df-dec 12619 df-uz 12764 df-q 12874 df-rp 12916 df-xneg 13033 df-xadd 13034 df-xmul 13035 df-ioo 13268 df-ico 13270 df-icc 13271 df-fz 13425 df-fzo 13568 df-fl 13697 df-seq 13907 df-exp 13968 df-hash 14231 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-clim 15370 df-rlim 15371 df-sum 15571 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-starv 17148 df-sca 17149 df-vsca 17150 df-ip 17151 df-tset 17152 df-ple 17153 df-ds 17155 df-unif 17156 df-hom 17157 df-cco 17158 df-rest 17304 df-topn 17305 df-0g 17323 df-gsum 17324 df-topgen 17325 df-pt 17326 df-prds 17329 df-xrs 17384 df-qtop 17389 df-imas 17390 df-xps 17392 df-mre 17466 df-mrc 17467 df-acs 17469 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-submnd 18602 df-mulg 18873 df-cntz 19097 df-cmn 19564 df-psmet 20788 df-xmet 20789 df-met 20790 df-bl 20791 df-mopn 20792 df-fbas 20793 df-fg 20794 df-cnfld 20797 df-top 22243 df-topon 22260 df-topsp 22282 df-bases 22296 df-cld 22370 df-ntr 22371 df-cls 22372 df-nei 22449 df-cn 22578 df-cnp 22579 df-lm 22580 df-t1 22665 df-haus 22666 df-tx 22913 df-hmeo 23106 df-fil 23197 df-fm 23289 df-flim 23290 df-flf 23291 df-xms 23673 df-ms 23674 df-tms 23675 df-cfil 24619 df-cau 24620 df-cmet 24621 df-grpo 29435 df-gid 29436 df-ginv 29437 df-gdiv 29438 df-ablo 29487 df-vc 29501 df-nv 29534 df-va 29537 df-ba 29538 df-sm 29539 df-0v 29540 df-vs 29541 df-nmcv 29542 df-ims 29543 df-dip 29643 df-ssp 29664 df-ph 29755 df-cbn 29805 df-hnorm 29910 df-hba 29911 df-hvsub 29913 df-hlim 29914 df-hcau 29915 df-sh 30149 df-ch 30163 df-oc 30194 df-ch0 30195 df-shs 30250 df-pjh 30337 df-h0op 30690 df-nmop 30781 df-cnop 30782 df-lnop 30783 df-bdop 30784 df-unop 30785 df-hmop 30786 df-nmfn 30787 df-nlfn 30788 df-cnfn 30789 df-lnfn 30790 df-adjh 30791

This theorem is referenced by: adjbdlnb 31026 adjbd1o 31027 nmopadjlem 31031 nmopadji 31032 adjcoi 31042 nmopcoadj2i 31044 nmopcoadj0i 31045

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