adjbdln - Hilbert Space Explorer

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Theorem adjbdln 32435

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 32434	. . . 4 ⊢ (𝑇 ∈ BndLinOp → 𝑇 ∈ dom adj_ℎ)
2		adjval 32242	. . . 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 18	. . 3 ⊢ (𝑇 ∈ BndLinOp → (adj_ℎ‘𝑇) = (℩𝑡 ∈ ( ℋ ↑_m ℋ)∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑡‘𝑥) ·_ih 𝑦)))
4		cnlnadj 32431	. . . . . 6 ⊢ (𝑇 ∈ (LinOp ∩ ContOp) → ∃𝑡 ∈ (LinOp ∩ ContOp)∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇‘𝑥) ·_ih 𝑦) = (𝑥 ·_ih (𝑡‘𝑦)))
5		lncnopbd 32389	. . . . . 6 ⊢ (𝑇 ∈ (LinOp ∩ ContOp) ↔ 𝑇 ∈ BndLinOp)
6		lncnbd 32390	. . . . . . 7 ⊢ (LinOp ∩ ContOp) = BndLinOp
7	6	rexeqi 3322	. . . . . 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 294	. . . . 5 ⊢ (𝑇 ∈ BndLinOp → ∃𝑡 ∈ BndLinOp ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇‘𝑥) ·_ih 𝑦) = (𝑥 ·_ih (𝑡‘𝑦)))
9		bdopf 32214	. . . . . . . 8 ⊢ (𝑇 ∈ BndLinOp → 𝑇: ℋ⟶ ℋ)
10		bdopf 32214	. . . . . . . 8 ⊢ (𝑡 ∈ BndLinOp → 𝑡: ℋ⟶ ℋ)
11		adjsym 32185	. . . . . . . 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 607	. . . . . . 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 2770	. . . . . . . 8 ⊢ (((𝑇‘𝑥) ·_ih 𝑦) = (𝑥 ·_ih (𝑡‘𝑦)) ↔ (𝑥 ·_ih (𝑡‘𝑦)) = ((𝑇‘𝑥) ·_ih 𝑦))
14	13	2ralbii 3140	. . . . . . 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 292	. . . . . 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 3187	. . . . 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 260	. . . 4 ⊢ (𝑇 ∈ BndLinOp → ∃𝑡 ∈ BndLinOp ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑡‘𝑥) ·_ih 𝑦))
18		adjeu 32241	. . . . . 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 18	. . . . 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 235	. . . 4 ⊢ (𝑇 ∈ BndLinOp → ∃!𝑡 ∈ ( ℋ ↑_m ℋ)∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑡‘𝑥) ·_ih 𝑦))
21		ax-hilex 31351	. . . . . . . 8 ⊢ ℋ ∈ V
22	21, 21	elmap 8865	. . . . . . 7 ⊢ (𝑡 ∈ ( ℋ ↑_m ℋ) ↔ 𝑡: ℋ⟶ ℋ)
23	10, 22	sylibr 237	. . . . . 6 ⊢ (𝑡 ∈ BndLinOp → 𝑡 ∈ ( ℋ ↑_m ℋ))
24	23	ssriv 3941	. . . . 5 ⊢ BndLinOp ⊆ ( ℋ ↑_m ℋ)
25		id 23	. . . . . 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 3083	. . . . 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 7397	. . . . 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 714	. . . 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 595	. . 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 2801	. 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 4280	. . . 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 1398	. . 3 ⊢ (𝑇 ∈ BndLinOp → ∃!𝑡 ∈ BndLinOp ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑡‘𝑥) ·_ih 𝑦))
34		riotacl 7384	. . 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 18	. 2 ⊢ (𝑇 ∈ BndLinOp → (℩𝑡 ∈ BndLinOp ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑡‘𝑥) ·_ih 𝑦)) ∈ BndLinOp)
36	30, 35	eqeltrd 2863	1 ⊢ (𝑇 ∈ BndLinOp → (adj_ℎ‘𝑇) ∈ BndLinOp)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 = wceq 1570 ∈ wcel 2143 ∀wral 3079 ∃wrex 3089 ∃!wreu 3367 ∩ cin 3904 ⊆ wss 3905 dom cdm 5661 ⟶wf 6532 ‘cfv 6536 ℩crio 7366 (class class class)co 7410 ↑_m cmap 8820 ℋchba 31271 ·_ih csp 31274 ContOpccop 31298 LinOpclo 31299 BndLinOpcbo 31300 adj_ℎcado 31307

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5238 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-inf2 9606 ax-cc 10414 ax-cnex 11151 ax-resscn 11152 ax-1cn 11153 ax-icn 11154 ax-addcl 11155 ax-addrcl 11156 ax-mulcl 11157 ax-mulrcl 11158 ax-mulcom 11159 ax-addass 11160 ax-mulass 11161 ax-distr 11162 ax-i2m1 11163 ax-1ne0 11164 ax-1rid 11165 ax-rnegex 11166 ax-rrecex 11167 ax-cnre 11168 ax-pre-lttri 11169 ax-pre-lttrn 11170 ax-pre-ltadd 11171 ax-pre-mulgt0 11172 ax-pre-sup 11173 ax-addf 11174 ax-mulf 11175 ax-hilex 31351 ax-hfvadd 31352 ax-hvcom 31353 ax-hvass 31354 ax-hv0cl 31355 ax-hvaddid 31356 ax-hfvmul 31357 ax-hvmulid 31358 ax-hvmulass 31359 ax-hvdistr1 31360 ax-hvdistr2 31361 ax-hvmul0 31362 ax-hfi 31431 ax-his1 31434 ax-his2 31435 ax-his3 31436 ax-his4 31437 ax-hcompl 31554

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4873 df-int 4913 df-iun 4958 df-iin 4959 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-se 5615 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-of 7674 df-om 7859 df-1st 7982 df-2nd 7983 df-supp 8153 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-2o 8450 df-oadd 8453 df-omul 8454 df-er 8690 df-map 8822 df-pm 8823 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fsupp 9318 df-fi 9367 df-sup 9398 df-inf 9399 df-oi 9468 df-card 9921 df-acn 9924 df-pnf 11240 df-mnf 11241 df-xr 11242 df-ltxr 11243 df-le 11244 df-sub 11438 df-neg 11439 df-div 11867 df-nn 12229 df-2 12298 df-3 12299 df-4 12300 df-5 12301 df-6 12302 df-7 12303 df-8 12304 df-9 12305 df-n0 12500 df-z 12587 df-dec 12707 df-uz 12858 df-q 12968 df-rp 13012 df-xneg 13132 df-xadd 13133 df-xmul 13134 df-ioo 13371 df-ico 13373 df-icc 13374 df-fz 13531 df-fzo 13679 df-fl 13821 df-seq 14034 df-exp 14094 df-hash 14363 df-cj 15146 df-re 15147 df-im 15148 df-sqrt 15282 df-abs 15283 df-clim 15535 df-rlim 15536 df-sum 15734 df-struct 17202 df-sets 17219 df-slot 17237 df-ndx 17249 df-base 17265 df-ress 17286 df-plusg 17318 df-mulr 17319 df-starv 17320 df-sca 17321 df-vsca 17322 df-ip 17323 df-tset 17324 df-ple 17325 df-ds 17327 df-unif 17328 df-hom 17329 df-cco 17330 df-rest 17470 df-topn 17471 df-0g 17489 df-gsum 17490 df-topgen 17491 df-pt 17492 df-prds 17495 df-xrs 17551 df-qtop 17556 df-imas 17557 df-xps 17559 df-mre 17633 df-mrc 17634 df-acs 17636 df-mgm 18693 df-sgrp 18772 df-mnd 18788 df-submnd 18837 df-mulg 19129 df-cntz 19382 df-cmn 19847 df-psmet 21514 df-xmet 21515 df-met 21516 df-bl 21517 df-mopn 21518 df-fbas 21519 df-fg 21520 df-cnfld 21523 df-top 23051 df-topon 23068 df-topsp 23090 df-bases 23103 df-cld 23176 df-ntr 23177 df-cls 23178 df-nei 23255 df-cn 23384 df-cnp 23385 df-lm 23386 df-t1 23471 df-haus 23472 df-tx 23719 df-hmeo 23912 df-fil 24003 df-fm 24095 df-flim 24096 df-flf 24097 df-xms 24477 df-ms 24478 df-tms 24479 df-cfil 25414 df-cau 25415 df-cmet 25416 df-grpo 30845 df-gid 30846 df-ginv 30847 df-gdiv 30848 df-ablo 30897 df-vc 30911 df-nv 30944 df-va 30947 df-ba 30948 df-sm 30949 df-0v 30950 df-vs 30951 df-nmcv 30952 df-ims 30953 df-dip 31053 df-ssp 31074 df-ph 31165 df-cbn 31215 df-hnorm 31320 df-hba 31321 df-hvsub 31323 df-hlim 31324 df-hcau 31325 df-sh 31559 df-ch 31573 df-oc 31604 df-ch0 31605 df-shs 31660 df-pjh 31747 df-h0op 32100 df-nmop 32191 df-cnop 32192 df-lnop 32193 df-bdop 32194 df-unop 32195 df-hmop 32196 df-nmfn 32197 df-nlfn 32198 df-cnfn 32199 df-lnfn 32200 df-adjh 32201

This theorem is referenced by: adjbdlnb 32436 adjbd1o 32437 nmopadjlem 32441 nmopadji 32442 adjcoi 32452 nmopcoadj2i 32454 nmopcoadj0i 32455

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