cnlnadjlem5 - Hilbert Space Explorer

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Theorem cnlnadjlem5 29854

Description: Lemma for cnlnadji 29859. 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 2955	. . 3 ⊢ Ⅎ𝑦𝐴
2		nfcv 2955	. . . 4 ⊢ Ⅎ𝑦 ℋ
3		nfcv 2955	. . . . . 6 ⊢ Ⅎ𝑦𝑓
4		nfcv 2955	. . . . . 6 ⊢ Ⅎ𝑦 ·_ih
5		cnlnadjlem.5	. . . . . . . 8 ⊢ 𝐹 = (𝑦 ∈ ℋ ↦ 𝐵)
6		nfmpt1 5128	. . . . . . . 8 ⊢ Ⅎ𝑦(𝑦 ∈ ℋ ↦ 𝐵)
7	5, 6	nfcxfr 2953	. . . . . . 7 ⊢ Ⅎ𝑦𝐹
8	7, 1	nffv 6655	. . . . . 6 ⊢ Ⅎ𝑦(𝐹‘𝐴)
9	3, 4, 8	nfov 7165	. . . . 5 ⊢ Ⅎ𝑦(𝑓 ·_ih (𝐹‘𝐴))
10	9	nfeq2 2972	. . . 4 ⊢ Ⅎ𝑦((𝑇‘𝑓) ·_ih 𝐴) = (𝑓 ·_ih (𝐹‘𝐴))
11	2, 10	nfralw 3189	. . 3 ⊢ Ⅎ𝑦∀𝑓 ∈ ℋ ((𝑇‘𝑓) ·_ih 𝐴) = (𝑓 ·_ih (𝐹‘𝐴))
12		oveq2 7143	. . . . 5 ⊢ (𝑦 = 𝐴 → ((𝑇‘𝑓) ·_ih 𝑦) = ((𝑇‘𝑓) ·_ih 𝐴))
13		fveq2 6645	. . . . . 6 ⊢ (𝑦 = 𝐴 → (𝐹‘𝑦) = (𝐹‘𝐴))
14	13	oveq2d 7151	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝑓 ·_ih (𝐹‘𝑦)) = (𝑓 ·_ih (𝐹‘𝐴)))
15	12, 14	eqeq12d 2814	. . . 4 ⊢ (𝑦 = 𝐴 → (((𝑇‘𝑓) ·_ih 𝑦) = (𝑓 ·_ih (𝐹‘𝑦)) ↔ ((𝑇‘𝑓) ·_ih 𝐴) = (𝑓 ·_ih (𝐹‘𝐴))))
16	15	ralbidv 3162	. . 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 7097	. . . . . . 7 ⊢ (℩𝑤 ∈ ℋ ∀𝑣 ∈ ℋ ((𝑇‘𝑣) ·_ih 𝑦) = (𝑣 ·_ih 𝑤)) ∈ V
19	17, 18	eqeltri 2886	. . . . . 6 ⊢ 𝐵 ∈ V
20	5	fvmpt2 6756	. . . . . 6 ⊢ ((𝑦 ∈ ℋ ∧ 𝐵 ∈ V) → (𝐹‘𝑦) = 𝐵)
21	19, 20	mpan2 690	. . . . 5 ⊢ (𝑦 ∈ ℋ → (𝐹‘𝑦) = 𝐵)
22		fveq2 6645	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑓 → (𝑇‘𝑣) = (𝑇‘𝑓))
23	22	oveq1d 7150	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑓 → ((𝑇‘𝑣) ·_ih 𝑦) = ((𝑇‘𝑓) ·_ih 𝑦))
24		oveq1 7142	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑓 → (𝑣 ·_ih 𝑤) = (𝑓 ·_ih 𝑤))
25	23, 24	eqeq12d 2814	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑓 → (((𝑇‘𝑣) ·_ih 𝑦) = (𝑣 ·_ih 𝑤) ↔ ((𝑇‘𝑓) ·_ih 𝑦) = (𝑓 ·_ih 𝑤)))
26	25	cbvralvw 3396	. . . . . . . . . 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 29850	. . . . . . . . . . 11 ⊢ (𝑓 ∈ ℋ → (𝐺‘𝑓) = ((𝑇‘𝑓) ·_ih 𝑦))
32	31	eqeq1d 2800	. . . . . . . . . 10 ⊢ (𝑓 ∈ ℋ → ((𝐺‘𝑓) = (𝑓 ·_ih 𝑤) ↔ ((𝑇‘𝑓) ·_ih 𝑦) = (𝑓 ·_ih 𝑤)))
33	32	ralbiia 3132	. . . . . . . . 9 ⊢ (∀𝑓 ∈ ℋ (𝐺‘𝑓) = (𝑓 ·_ih 𝑤) ↔ ∀𝑓 ∈ ℋ ((𝑇‘𝑓) ·_ih 𝑦) = (𝑓 ·_ih 𝑤))
34	27, 33	syl6bbr 292	. . . . . . . 8 ⊢ (𝑤 ∈ ℋ → (∀𝑣 ∈ ℋ ((𝑇‘𝑣) ·_ih 𝑦) = (𝑣 ·_ih 𝑤) ↔ ∀𝑓 ∈ ℋ (𝐺‘𝑓) = (𝑓 ·_ih 𝑤)))
35	34	riotabiia 7113	. . . . . . 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 2821	. . . . . 6 ⊢ 𝐵 = (℩𝑤 ∈ ℋ ∀𝑓 ∈ ℋ (𝐺‘𝑓) = (𝑓 ·_ih 𝑤))
37	28, 29, 30	cnlnadjlem2 29851	. . . . . . . 8 ⊢ (𝑦 ∈ ℋ → (𝐺 ∈ LinFn ∧ 𝐺 ∈ ContFn))
38		elin 3897	. . . . . . . 8 ⊢ (𝐺 ∈ (LinFn ∩ ContFn) ↔ (𝐺 ∈ LinFn ∧ 𝐺 ∈ ContFn))
39	37, 38	sylibr 237	. . . . . . 7 ⊢ (𝑦 ∈ ℋ → 𝐺 ∈ (LinFn ∩ ContFn))
40		riesz4 29847	. . . . . . 7 ⊢ (𝐺 ∈ (LinFn ∩ ContFn) → ∃!𝑤 ∈ ℋ ∀𝑓 ∈ ℋ (𝐺‘𝑓) = (𝑓 ·_ih 𝑤))
41		riotacl2 7109	. . . . . . 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 2894	. . . . 5 ⊢ (𝑦 ∈ ℋ → 𝐵 ∈ {𝑤 ∈ ℋ ∣ ∀𝑓 ∈ ℋ (𝐺‘𝑓) = (𝑓 ·_ih 𝑤)})
44	21, 43	eqeltrd 2890	. . . 4 ⊢ (𝑦 ∈ ℋ → (𝐹‘𝑦) ∈ {𝑤 ∈ ℋ ∣ ∀𝑓 ∈ ℋ (𝐺‘𝑓) = (𝑓 ·_ih 𝑤)})
45		oveq2 7143	. . . . . . . . 9 ⊢ (𝑤 = (𝐹‘𝑦) → (𝑓 ·_ih 𝑤) = (𝑓 ·_ih (𝐹‘𝑦)))
46	45	eqeq2d 2809	. . . . . . . 8 ⊢ (𝑤 = (𝐹‘𝑦) → (((𝑇‘𝑓) ·_ih 𝑦) = (𝑓 ·_ih 𝑤) ↔ ((𝑇‘𝑓) ·_ih 𝑦) = (𝑓 ·_ih (𝐹‘𝑦))))
47	46	ralbidv 3162	. . . . . . 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	syl5bb 286	. . . . . 6 ⊢ (𝑤 = (𝐹‘𝑦) → (∀𝑓 ∈ ℋ (𝐺‘𝑓) = (𝑓 ·_ih 𝑤) ↔ ∀𝑓 ∈ ℋ ((𝑇‘𝑓) ·_ih 𝑦) = (𝑓 ·_ih (𝐹‘𝑦))))
49	48	elrab 3628	. . . . 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 500	. . . 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 3521	. 2 ⊢ (𝐴 ∈ ℋ → ∀𝑓 ∈ ℋ ((𝑇‘𝑓) ·_ih 𝐴) = (𝑓 ·_ih (𝐹‘𝐴)))
53		fveq2 6645	. . . . 5 ⊢ (𝑓 = 𝐶 → (𝑇‘𝑓) = (𝑇‘𝐶))
54	53	oveq1d 7150	. . . 4 ⊢ (𝑓 = 𝐶 → ((𝑇‘𝑓) ·_ih 𝐴) = ((𝑇‘𝐶) ·_ih 𝐴))
55		oveq1 7142	. . . 4 ⊢ (𝑓 = 𝐶 → (𝑓 ·_ih (𝐹‘𝐴)) = (𝐶 ·_ih (𝐹‘𝐴)))
56	54, 55	eqeq12d 2814	. . 3 ⊢ (𝑓 = 𝐶 → (((𝑇‘𝑓) ·_ih 𝐴) = (𝑓 ·_ih (𝐹‘𝐴)) ↔ ((𝑇‘𝐶) ·_ih 𝐴) = (𝐶 ·_ih (𝐹‘𝐴))))
57	56	rspccva 3570	. 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 583	1 ⊢ ((𝐴 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝑇‘𝐶) ·_ih 𝐴) = (𝐶 ·_ih (𝐹‘𝐴)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∀wral 3106 ∃!wreu 3108 {crab 3110 Vcvv 3441 ∩ cin 3880 ↦ cmpt 5110 ‘cfv 6324 ℩crio 7092 (class class class)co 7135 ℋchba 28702 ·_ih csp 28705 ContOpccop 28729 LinOpclo 28730 ContFnccnfn 28736 LinFnclf 28737

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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cc 9846 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-addf 10605 ax-mulf 10606 ax-hilex 28782 ax-hfvadd 28783 ax-hvcom 28784 ax-hvass 28785 ax-hv0cl 28786 ax-hvaddid 28787 ax-hfvmul 28788 ax-hvmulid 28789 ax-hvmulass 28790 ax-hvdistr1 28791 ax-hvdistr2 28792 ax-hvmul0 28793 ax-hfi 28862 ax-his1 28865 ax-his2 28866 ax-his3 28867 ax-his4 28868 ax-hcompl 28985

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-omul 8090 df-er 8272 df-map 8391 df-pm 8392 df-ixp 8445 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-fi 8859 df-sup 8890 df-inf 8891 df-oi 8958 df-card 9352 df-acn 9355 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-ioo 12730 df-ico 12732 df-icc 12733 df-fz 12886 df-fzo 13029 df-fl 13157 df-seq 13365 df-exp 13426 df-hash 13687 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-clim 14837 df-rlim 14838 df-sum 15035 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-hom 16581 df-cco 16582 df-rest 16688 df-topn 16689 df-0g 16707 df-gsum 16708 df-topgen 16709 df-pt 16710 df-prds 16713 df-xrs 16767 df-qtop 16772 df-imas 16773 df-xps 16775 df-mre 16849 df-mrc 16850 df-acs 16852 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-mulg 18217 df-cntz 18439 df-cmn 18900 df-psmet 20083 df-xmet 20084 df-met 20085 df-bl 20086 df-mopn 20087 df-fbas 20088 df-fg 20089 df-cnfld 20092 df-top 21499 df-topon 21516 df-topsp 21538 df-bases 21551 df-cld 21624 df-ntr 21625 df-cls 21626 df-nei 21703 df-cn 21832 df-cnp 21833 df-lm 21834 df-t1 21919 df-haus 21920 df-tx 22167 df-hmeo 22360 df-fil 22451 df-fm 22543 df-flim 22544 df-flf 22545 df-xms 22927 df-ms 22928 df-tms 22929 df-cfil 23859 df-cau 23860 df-cmet 23861 df-grpo 28276 df-gid 28277 df-ginv 28278 df-gdiv 28279 df-ablo 28328 df-vc 28342 df-nv 28375 df-va 28378 df-ba 28379 df-sm 28380 df-0v 28381 df-vs 28382 df-nmcv 28383 df-ims 28384 df-dip 28484 df-ssp 28505 df-ph 28596 df-cbn 28646 df-hnorm 28751 df-hba 28752 df-hvsub 28754 df-hlim 28755 df-hcau 28756 df-sh 28990 df-ch 29004 df-oc 29035 df-ch0 29036 df-nmop 29622 df-cnop 29623 df-lnop 29624 df-nmfn 29628 df-nlfn 29629 df-cnfn 29630 df-lnfn 29631

This theorem is referenced by: cnlnadjlem6 29855 cnlnadjlem7 29856 cnlnadjlem9 29858

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