cnlnadjlem5 - Hilbert Space Explorer

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Theorem cnlnadjlem5 32050

Description: Lemma for cnlnadji 32055. 𝐹 is an adjoint of 𝑇 (later, we will show it is unique). (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
cnlnadjlem.1	⊢ 𝑇 ∈ LinOp
cnlnadjlem.2	⊢ 𝑇 ∈ ContOp
cnlnadjlem.3	⊢ 𝐺 = (𝑔 ∈ ℋ ↦ ((𝑇‘𝑔) ·_ih 𝑦))
cnlnadjlem.4	⊢ 𝐵 = (℩𝑤 ∈ ℋ ∀𝑣 ∈ ℋ ((𝑇‘𝑣) ·_ih 𝑦) = (𝑣 ·_ih 𝑤))
cnlnadjlem.5	⊢ 𝐹 = (𝑦 ∈ ℋ ↦ 𝐵)

**Assertion**
Ref	Expression
cnlnadjlem5	⊢ ((𝐴 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝑇‘𝐶) ·_ih 𝐴) = (𝐶 ·_ih (𝐹‘𝐴)))

Distinct variable groups: 𝑣,𝑔,𝑤,𝑦,𝐴 𝑤,𝐹 𝑇,𝑔,𝑣,𝑤,𝑦 𝑣,𝐺,𝑤 Allowed substitution hints: 𝐵(𝑦,𝑤,𝑣,𝑔) 𝐶(𝑦,𝑤,𝑣,𝑔) 𝐹(𝑦,𝑣,𝑔) 𝐺(𝑦,𝑔)

Proof of Theorem cnlnadjlem5 Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfcv 2891	. . 3 ⊢ Ⅎ𝑦𝐴
2		nfcv 2891	. . . 4 ⊢ Ⅎ𝑦 ℋ
3		nfcv 2891	. . . . . 6 ⊢ Ⅎ𝑦𝑓
4		nfcv 2891	. . . . . 6 ⊢ Ⅎ𝑦 ·_ih
5		cnlnadjlem.5	. . . . . . . 8 ⊢ 𝐹 = (𝑦 ∈ ℋ ↦ 𝐵)
6		nfmpt1 5201	. . . . . . . 8 ⊢ Ⅎ𝑦(𝑦 ∈ ℋ ↦ 𝐵)
7	5, 6	nfcxfr 2889	. . . . . . 7 ⊢ Ⅎ𝑦𝐹
8	7, 1	nffv 6850	. . . . . 6 ⊢ Ⅎ𝑦(𝐹‘𝐴)
9	3, 4, 8	nfov 7399	. . . . 5 ⊢ Ⅎ𝑦(𝑓 ·_ih (𝐹‘𝐴))
10	9	nfeq2 2909	. . . 4 ⊢ Ⅎ𝑦((𝑇‘𝑓) ·_ih 𝐴) = (𝑓 ·_ih (𝐹‘𝐴))
11	2, 10	nfralw 3283	. . 3 ⊢ Ⅎ𝑦∀𝑓 ∈ ℋ ((𝑇‘𝑓) ·_ih 𝐴) = (𝑓 ·_ih (𝐹‘𝐴))
12		oveq2 7377	. . . . 5 ⊢ (𝑦 = 𝐴 → ((𝑇‘𝑓) ·_ih 𝑦) = ((𝑇‘𝑓) ·_ih 𝐴))
13		fveq2 6840	. . . . . 6 ⊢ (𝑦 = 𝐴 → (𝐹‘𝑦) = (𝐹‘𝐴))
14	13	oveq2d 7385	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝑓 ·_ih (𝐹‘𝑦)) = (𝑓 ·_ih (𝐹‘𝐴)))
15	12, 14	eqeq12d 2745	. . . 4 ⊢ (𝑦 = 𝐴 → (((𝑇‘𝑓) ·_ih 𝑦) = (𝑓 ·_ih (𝐹‘𝑦)) ↔ ((𝑇‘𝑓) ·_ih 𝐴) = (𝑓 ·_ih (𝐹‘𝐴))))
16	15	ralbidv 3156	. . 3 ⊢ (𝑦 = 𝐴 → (∀𝑓 ∈ ℋ ((𝑇‘𝑓) ·_ih 𝑦) = (𝑓 ·_ih (𝐹‘𝑦)) ↔ ∀𝑓 ∈ ℋ ((𝑇‘𝑓) ·_ih 𝐴) = (𝑓 ·_ih (𝐹‘𝐴))))
17		cnlnadjlem.4	. . . . . . 7 ⊢ 𝐵 = (℩𝑤 ∈ ℋ ∀𝑣 ∈ ℋ ((𝑇‘𝑣) ·_ih 𝑦) = (𝑣 ·_ih 𝑤))
18		riotaex 7330	. . . . . . 7 ⊢ (℩𝑤 ∈ ℋ ∀𝑣 ∈ ℋ ((𝑇‘𝑣) ·_ih 𝑦) = (𝑣 ·_ih 𝑤)) ∈ V
19	17, 18	eqeltri 2824	. . . . . 6 ⊢ 𝐵 ∈ V
20	5	fvmpt2 6961	. . . . . 6 ⊢ ((𝑦 ∈ ℋ ∧ 𝐵 ∈ V) → (𝐹‘𝑦) = 𝐵)
21	19, 20	mpan2 691	. . . . 5 ⊢ (𝑦 ∈ ℋ → (𝐹‘𝑦) = 𝐵)
22		fveq2 6840	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑓 → (𝑇‘𝑣) = (𝑇‘𝑓))
23	22	oveq1d 7384	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑓 → ((𝑇‘𝑣) ·_ih 𝑦) = ((𝑇‘𝑓) ·_ih 𝑦))
24		oveq1 7376	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑓 → (𝑣 ·_ih 𝑤) = (𝑓 ·_ih 𝑤))
25	23, 24	eqeq12d 2745	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑓 → (((𝑇‘𝑣) ·_ih 𝑦) = (𝑣 ·_ih 𝑤) ↔ ((𝑇‘𝑓) ·_ih 𝑦) = (𝑓 ·_ih 𝑤)))
26	25	cbvralvw 3213	. . . . . . . . . 10 ⊢ (∀𝑣 ∈ ℋ ((𝑇‘𝑣) ·_ih 𝑦) = (𝑣 ·_ih 𝑤) ↔ ∀𝑓 ∈ ℋ ((𝑇‘𝑓) ·_ih 𝑦) = (𝑓 ·_ih 𝑤))
27	26	a1i 11	. . . . . . . . 9 ⊢ (𝑤 ∈ ℋ → (∀𝑣 ∈ ℋ ((𝑇‘𝑣) ·_ih 𝑦) = (𝑣 ·_ih 𝑤) ↔ ∀𝑓 ∈ ℋ ((𝑇‘𝑓) ·_ih 𝑦) = (𝑓 ·_ih 𝑤)))
28		cnlnadjlem.1	. . . . . . . . . . . 12 ⊢ 𝑇 ∈ LinOp
29		cnlnadjlem.2	. . . . . . . . . . . 12 ⊢ 𝑇 ∈ ContOp
30		cnlnadjlem.3	. . . . . . . . . . . 12 ⊢ 𝐺 = (𝑔 ∈ ℋ ↦ ((𝑇‘𝑔) ·_ih 𝑦))
31	28, 29, 30	cnlnadjlem1 32046	. . . . . . . . . . 11 ⊢ (𝑓 ∈ ℋ → (𝐺‘𝑓) = ((𝑇‘𝑓) ·_ih 𝑦))
32	31	eqeq1d 2731	. . . . . . . . . 10 ⊢ (𝑓 ∈ ℋ → ((𝐺‘𝑓) = (𝑓 ·_ih 𝑤) ↔ ((𝑇‘𝑓) ·_ih 𝑦) = (𝑓 ·_ih 𝑤)))
33	32	ralbiia 3073	. . . . . . . . 9 ⊢ (∀𝑓 ∈ ℋ (𝐺‘𝑓) = (𝑓 ·_ih 𝑤) ↔ ∀𝑓 ∈ ℋ ((𝑇‘𝑓) ·_ih 𝑦) = (𝑓 ·_ih 𝑤))
34	27, 33	bitr4di 289	. . . . . . . 8 ⊢ (𝑤 ∈ ℋ → (∀𝑣 ∈ ℋ ((𝑇‘𝑣) ·_ih 𝑦) = (𝑣 ·_ih 𝑤) ↔ ∀𝑓 ∈ ℋ (𝐺‘𝑓) = (𝑓 ·_ih 𝑤)))
35	34	riotabiia 7346	. . . . . . 7 ⊢ (℩𝑤 ∈ ℋ ∀𝑣 ∈ ℋ ((𝑇‘𝑣) ·_ih 𝑦) = (𝑣 ·_ih 𝑤)) = (℩𝑤 ∈ ℋ ∀𝑓 ∈ ℋ (𝐺‘𝑓) = (𝑓 ·_ih 𝑤))
36	17, 35	eqtri 2752	. . . . . 6 ⊢ 𝐵 = (℩𝑤 ∈ ℋ ∀𝑓 ∈ ℋ (𝐺‘𝑓) = (𝑓 ·_ih 𝑤))
37	28, 29, 30	cnlnadjlem2 32047	. . . . . . . 8 ⊢ (𝑦 ∈ ℋ → (𝐺 ∈ LinFn ∧ 𝐺 ∈ ContFn))
38		elin 3927	. . . . . . . 8 ⊢ (𝐺 ∈ (LinFn ∩ ContFn) ↔ (𝐺 ∈ LinFn ∧ 𝐺 ∈ ContFn))
39	37, 38	sylibr 234	. . . . . . 7 ⊢ (𝑦 ∈ ℋ → 𝐺 ∈ (LinFn ∩ ContFn))
40		riesz4 32043	. . . . . . 7 ⊢ (𝐺 ∈ (LinFn ∩ ContFn) → ∃!𝑤 ∈ ℋ ∀𝑓 ∈ ℋ (𝐺‘𝑓) = (𝑓 ·_ih 𝑤))
41		riotacl2 7342	. . . . . . 7 ⊢ (∃!𝑤 ∈ ℋ ∀𝑓 ∈ ℋ (𝐺‘𝑓) = (𝑓 ·_ih 𝑤) → (℩𝑤 ∈ ℋ ∀𝑓 ∈ ℋ (𝐺‘𝑓) = (𝑓 ·_ih 𝑤)) ∈ {𝑤 ∈ ℋ ∣ ∀𝑓 ∈ ℋ (𝐺‘𝑓) = (𝑓 ·_ih 𝑤)})
42	39, 40, 41	3syl 18	. . . . . 6 ⊢ (𝑦 ∈ ℋ → (℩𝑤 ∈ ℋ ∀𝑓 ∈ ℋ (𝐺‘𝑓) = (𝑓 ·_ih 𝑤)) ∈ {𝑤 ∈ ℋ ∣ ∀𝑓 ∈ ℋ (𝐺‘𝑓) = (𝑓 ·_ih 𝑤)})
43	36, 42	eqeltrid 2832	. . . . 5 ⊢ (𝑦 ∈ ℋ → 𝐵 ∈ {𝑤 ∈ ℋ ∣ ∀𝑓 ∈ ℋ (𝐺‘𝑓) = (𝑓 ·_ih 𝑤)})
44	21, 43	eqeltrd 2828	. . . 4 ⊢ (𝑦 ∈ ℋ → (𝐹‘𝑦) ∈ {𝑤 ∈ ℋ ∣ ∀𝑓 ∈ ℋ (𝐺‘𝑓) = (𝑓 ·_ih 𝑤)})
45		oveq2 7377	. . . . . . . . 9 ⊢ (𝑤 = (𝐹‘𝑦) → (𝑓 ·_ih 𝑤) = (𝑓 ·_ih (𝐹‘𝑦)))
46	45	eqeq2d 2740	. . . . . . . 8 ⊢ (𝑤 = (𝐹‘𝑦) → (((𝑇‘𝑓) ·_ih 𝑦) = (𝑓 ·_ih 𝑤) ↔ ((𝑇‘𝑓) ·_ih 𝑦) = (𝑓 ·_ih (𝐹‘𝑦))))
47	46	ralbidv 3156	. . . . . . 7 ⊢ (𝑤 = (𝐹‘𝑦) → (∀𝑓 ∈ ℋ ((𝑇‘𝑓) ·_ih 𝑦) = (𝑓 ·_ih 𝑤) ↔ ∀𝑓 ∈ ℋ ((𝑇‘𝑓) ·_ih 𝑦) = (𝑓 ·_ih (𝐹‘𝑦))))
48	33, 47	bitrid 283	. . . . . 6 ⊢ (𝑤 = (𝐹‘𝑦) → (∀𝑓 ∈ ℋ (𝐺‘𝑓) = (𝑓 ·_ih 𝑤) ↔ ∀𝑓 ∈ ℋ ((𝑇‘𝑓) ·_ih 𝑦) = (𝑓 ·_ih (𝐹‘𝑦))))
49	48	elrab 3656	. . . . 5 ⊢ ((𝐹‘𝑦) ∈ {𝑤 ∈ ℋ ∣ ∀𝑓 ∈ ℋ (𝐺‘𝑓) = (𝑓 ·_ih 𝑤)} ↔ ((𝐹‘𝑦) ∈ ℋ ∧ ∀𝑓 ∈ ℋ ((𝑇‘𝑓) ·_ih 𝑦) = (𝑓 ·_ih (𝐹‘𝑦))))
50	49	simprbi 496	. . . 4 ⊢ ((𝐹‘𝑦) ∈ {𝑤 ∈ ℋ ∣ ∀𝑓 ∈ ℋ (𝐺‘𝑓) = (𝑓 ·_ih 𝑤)} → ∀𝑓 ∈ ℋ ((𝑇‘𝑓) ·_ih 𝑦) = (𝑓 ·_ih (𝐹‘𝑦)))
51	44, 50	syl 17	. . 3 ⊢ (𝑦 ∈ ℋ → ∀𝑓 ∈ ℋ ((𝑇‘𝑓) ·_ih 𝑦) = (𝑓 ·_ih (𝐹‘𝑦)))
52	1, 11, 16, 51	vtoclgaf 3539	. 2 ⊢ (𝐴 ∈ ℋ → ∀𝑓 ∈ ℋ ((𝑇‘𝑓) ·_ih 𝐴) = (𝑓 ·_ih (𝐹‘𝐴)))
53		fveq2 6840	. . . . 5 ⊢ (𝑓 = 𝐶 → (𝑇‘𝑓) = (𝑇‘𝐶))
54	53	oveq1d 7384	. . . 4 ⊢ (𝑓 = 𝐶 → ((𝑇‘𝑓) ·_ih 𝐴) = ((𝑇‘𝐶) ·_ih 𝐴))
55		oveq1 7376	. . . 4 ⊢ (𝑓 = 𝐶 → (𝑓 ·_ih (𝐹‘𝐴)) = (𝐶 ·_ih (𝐹‘𝐴)))
56	54, 55	eqeq12d 2745	. . 3 ⊢ (𝑓 = 𝐶 → (((𝑇‘𝑓) ·_ih 𝐴) = (𝑓 ·_ih (𝐹‘𝐴)) ↔ ((𝑇‘𝐶) ·_ih 𝐴) = (𝐶 ·_ih (𝐹‘𝐴))))
57	56	rspccva 3584	. 2 ⊢ ((∀𝑓 ∈ ℋ ((𝑇‘𝑓) ·_ih 𝐴) = (𝑓 ·_ih (𝐹‘𝐴)) ∧ 𝐶 ∈ ℋ) → ((𝑇‘𝐶) ·_ih 𝐴) = (𝐶 ·_ih (𝐹‘𝐴)))
58	52, 57	sylan 580	1 ⊢ ((𝐴 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝑇‘𝐶) ·_ih 𝐴) = (𝐶 ·_ih (𝐹‘𝐴)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ∀wral 3044 ∃!wreu 3349 {crab 3402 Vcvv 3444 ∩ cin 3910 ↦ cmpt 5183 ‘cfv 6499 ℩crio 7325 (class class class)co 7369 ℋchba 30898 ·_ih csp 30901 ContOpccop 30925 LinOpclo 30926 ContFnccnfn 30932 LinFnclf 30933

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-inf2 9570 ax-cc 10364 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121 ax-pre-sup 11122 ax-addf 11123 ax-mulf 11124 ax-hilex 30978 ax-hfvadd 30979 ax-hvcom 30980 ax-hvass 30981 ax-hv0cl 30982 ax-hvaddid 30983 ax-hfvmul 30984 ax-hvmulid 30985 ax-hvmulass 30986 ax-hvdistr1 30987 ax-hvdistr2 30988 ax-hvmul0 30989 ax-hfi 31058 ax-his1 31061 ax-his2 31062 ax-his3 31063 ax-his4 31064 ax-hcompl 31181

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4868 df-int 4907 df-iun 4953 df-iin 4954 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 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 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-isom 6508 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-of 7633 df-om 7823 df-1st 7947 df-2nd 7948 df-supp 8117 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-1o 8411 df-2o 8412 df-oadd 8415 df-omul 8416 df-er 8648 df-map 8778 df-pm 8779 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9289 df-fi 9338 df-sup 9369 df-inf 9370 df-oi 9439 df-card 9868 df-acn 9871 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-div 11812 df-nn 12163 df-2 12225 df-3 12226 df-4 12227 df-5 12228 df-6 12229 df-7 12230 df-8 12231 df-9 12232 df-n0 12419 df-z 12506 df-dec 12626 df-uz 12770 df-q 12884 df-rp 12928 df-xneg 13048 df-xadd 13049 df-xmul 13050 df-ioo 13286 df-ico 13288 df-icc 13289 df-fz 13445 df-fzo 13592 df-fl 13730 df-seq 13943 df-exp 14003 df-hash 14272 df-cj 15041 df-re 15042 df-im 15043 df-sqrt 15177 df-abs 15178 df-clim 15430 df-rlim 15431 df-sum 15629 df-struct 17093 df-sets 17110 df-slot 17128 df-ndx 17140 df-base 17156 df-ress 17177 df-plusg 17209 df-mulr 17210 df-starv 17211 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-hom 17220 df-cco 17221 df-rest 17361 df-topn 17362 df-0g 17380 df-gsum 17381 df-topgen 17382 df-pt 17383 df-prds 17386 df-xrs 17441 df-qtop 17446 df-imas 17447 df-xps 17449 df-mre 17523 df-mrc 17524 df-acs 17526 df-mgm 18549 df-sgrp 18628 df-mnd 18644 df-submnd 18693 df-mulg 18982 df-cntz 19231 df-cmn 19696 df-psmet 21288 df-xmet 21289 df-met 21290 df-bl 21291 df-mopn 21292 df-fbas 21293 df-fg 21294 df-cnfld 21297 df-top 22814 df-topon 22831 df-topsp 22853 df-bases 22866 df-cld 22939 df-ntr 22940 df-cls 22941 df-nei 23018 df-cn 23147 df-cnp 23148 df-lm 23149 df-t1 23234 df-haus 23235 df-tx 23482 df-hmeo 23675 df-fil 23766 df-fm 23858 df-flim 23859 df-flf 23860 df-xms 24241 df-ms 24242 df-tms 24243 df-cfil 25188 df-cau 25189 df-cmet 25190 df-grpo 30472 df-gid 30473 df-ginv 30474 df-gdiv 30475 df-ablo 30524 df-vc 30538 df-nv 30571 df-va 30574 df-ba 30575 df-sm 30576 df-0v 30577 df-vs 30578 df-nmcv 30579 df-ims 30580 df-dip 30680 df-ssp 30701 df-ph 30792 df-cbn 30842 df-hnorm 30947 df-hba 30948 df-hvsub 30950 df-hlim 30951 df-hcau 30952 df-sh 31186 df-ch 31200 df-oc 31231 df-ch0 31232 df-nmop 31818 df-cnop 31819 df-lnop 31820 df-nmfn 31824 df-nlfn 31825 df-cnfn 31826 df-lnfn 31827

This theorem is referenced by: cnlnadjlem6 32051 cnlnadjlem7 32052 cnlnadjlem9 32054

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