adjbdln - Hilbert Space Explorer

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Theorem adjbdln 32154

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 32153	. . . 4 ⊢ (𝑇 ∈ BndLinOp → 𝑇 ∈ dom adj_ℎ)
2		adjval 31961	. . . 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 32150	. . . . . 6 ⊢ (𝑇 ∈ (LinOp ∩ ContOp) → ∃𝑡 ∈ (LinOp ∩ ContOp)∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇‘𝑥) ·_ih 𝑦) = (𝑥 ·_ih (𝑡‘𝑦)))
5		lncnopbd 32108	. . . . . 6 ⊢ (𝑇 ∈ (LinOp ∩ ContOp) ↔ 𝑇 ∈ BndLinOp)
6		lncnbd 32109	. . . . . . 7 ⊢ (LinOp ∩ ContOp) = BndLinOp
7	6	rexeqi 3294	. . . . . 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 31933	. . . . . . . 8 ⊢ (𝑇 ∈ BndLinOp → 𝑇: ℋ⟶ ℋ)
10		bdopf 31933	. . . . . . . 8 ⊢ (𝑡 ∈ BndLinOp → 𝑡: ℋ⟶ ℋ)
11		adjsym 31904	. . . . . . . 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 597	. . . . . . 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 3112	. . . . . . 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 3159	. . . . 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 31960	. . . . . 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 31070	. . . . . . . 8 ⊢ ℋ ∈ V
22	21, 21	elmap 8819	. . . . . . 7 ⊢ (𝑡 ∈ ( ℋ ↑_m ℋ) ↔ 𝑡: ℋ⟶ ℋ)
23	10, 22	sylibr 234	. . . . . 6 ⊢ (𝑡 ∈ BndLinOp → 𝑡 ∈ ( ℋ ↑_m ℋ))
24	23	ssriv 3925	. . . . 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 3055	. . . . 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 7354	. . . . 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 703	. . . 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 585	. . 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 2774	. 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 4267	. . . 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 1374	. . 3 ⊢ (𝑇 ∈ BndLinOp → ∃!𝑡 ∈ BndLinOp ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑡‘𝑥) ·_ih 𝑦))
34		riotacl 7341	. . 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 2836	1 ⊢ (𝑇 ∈ BndLinOp → (adj_ℎ‘𝑇) ∈ BndLinOp)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ∀wral 3051 ∃wrex 3061 ∃!wreu 3340 ∩ cin 3888 ⊆ wss 3889 dom cdm 5631 ⟶wf 6494 ‘cfv 6498 ℩crio 7323 (class class class)co 7367 ↑_m cmap 8773 ℋchba 30990 ·_ih csp 30993 ContOpccop 31017 LinOpclo 31018 BndLinOpcbo 31019 adj_ℎcado 31026

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-inf2 9562 ax-cc 10357 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116 ax-addf 11117 ax-mulf 11118 ax-hilex 31070 ax-hfvadd 31071 ax-hvcom 31072 ax-hvass 31073 ax-hv0cl 31074 ax-hvaddid 31075 ax-hfvmul 31076 ax-hvmulid 31077 ax-hvmulass 31078 ax-hvdistr1 31079 ax-hvdistr2 31080 ax-hvmul0 31081 ax-hfi 31150 ax-his1 31153 ax-his2 31154 ax-his3 31155 ax-his4 31156 ax-hcompl 31273

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-iin 4936 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-isom 6507 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-of 7631 df-om 7818 df-1st 7942 df-2nd 7943 df-supp 8111 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-2o 8406 df-oadd 8409 df-omul 8410 df-er 8643 df-map 8775 df-pm 8776 df-ixp 8846 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-fsupp 9275 df-fi 9324 df-sup 9355 df-inf 9356 df-oi 9425 df-card 9863 df-acn 9866 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-4 12246 df-5 12247 df-6 12248 df-7 12249 df-8 12250 df-9 12251 df-n0 12438 df-z 12525 df-dec 12645 df-uz 12789 df-q 12899 df-rp 12943 df-xneg 13063 df-xadd 13064 df-xmul 13065 df-ioo 13302 df-ico 13304 df-icc 13305 df-fz 13462 df-fzo 13609 df-fl 13751 df-seq 13964 df-exp 14024 df-hash 14293 df-cj 15061 df-re 15062 df-im 15063 df-sqrt 15197 df-abs 15198 df-clim 15450 df-rlim 15451 df-sum 15649 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-hom 17244 df-cco 17245 df-rest 17385 df-topn 17386 df-0g 17404 df-gsum 17405 df-topgen 17406 df-pt 17407 df-prds 17410 df-xrs 17466 df-qtop 17471 df-imas 17472 df-xps 17474 df-mre 17548 df-mrc 17549 df-acs 17551 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-submnd 18752 df-mulg 19044 df-cntz 19292 df-cmn 19757 df-psmet 21344 df-xmet 21345 df-met 21346 df-bl 21347 df-mopn 21348 df-fbas 21349 df-fg 21350 df-cnfld 21353 df-top 22859 df-topon 22876 df-topsp 22898 df-bases 22911 df-cld 22984 df-ntr 22985 df-cls 22986 df-nei 23063 df-cn 23192 df-cnp 23193 df-lm 23194 df-t1 23279 df-haus 23280 df-tx 23527 df-hmeo 23720 df-fil 23811 df-fm 23903 df-flim 23904 df-flf 23905 df-xms 24285 df-ms 24286 df-tms 24287 df-cfil 25222 df-cau 25223 df-cmet 25224 df-grpo 30564 df-gid 30565 df-ginv 30566 df-gdiv 30567 df-ablo 30616 df-vc 30630 df-nv 30663 df-va 30666 df-ba 30667 df-sm 30668 df-0v 30669 df-vs 30670 df-nmcv 30671 df-ims 30672 df-dip 30772 df-ssp 30793 df-ph 30884 df-cbn 30934 df-hnorm 31039 df-hba 31040 df-hvsub 31042 df-hlim 31043 df-hcau 31044 df-sh 31278 df-ch 31292 df-oc 31323 df-ch0 31324 df-shs 31379 df-pjh 31466 df-h0op 31819 df-nmop 31910 df-cnop 31911 df-lnop 31912 df-bdop 31913 df-unop 31914 df-hmop 31915 df-nmfn 31916 df-nlfn 31917 df-cnfn 31918 df-lnfn 31919 df-adjh 31920

This theorem is referenced by: adjbdlnb 32155 adjbd1o 32156 nmopadjlem 32160 nmopadji 32161 adjcoi 32171 nmopcoadj2i 32173 nmopcoadj0i 32174

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