cnlnadjlem5 - Hilbert Space Explorer

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Theorem cnlnadjlem5 32007

Description: Lemma for cnlnadji 32012. 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 2892	. . 3 ⊢ Ⅎ𝑦𝐴
2		nfcv 2892	. . . 4 ⊢ Ⅎ𝑦 ℋ
3		nfcv 2892	. . . . . 6 ⊢ Ⅎ𝑦𝑓
4		nfcv 2892	. . . . . 6 ⊢ Ⅎ𝑦 ·_ih
5		cnlnadjlem.5	. . . . . . . 8 ⊢ 𝐹 = (𝑦 ∈ ℋ ↦ 𝐵)
6		nfmpt1 5209	. . . . . . . 8 ⊢ Ⅎ𝑦(𝑦 ∈ ℋ ↦ 𝐵)
7	5, 6	nfcxfr 2890	. . . . . . 7 ⊢ Ⅎ𝑦𝐹
8	7, 1	nffv 6871	. . . . . 6 ⊢ Ⅎ𝑦(𝐹‘𝐴)
9	3, 4, 8	nfov 7420	. . . . 5 ⊢ Ⅎ𝑦(𝑓 ·_ih (𝐹‘𝐴))
10	9	nfeq2 2910	. . . 4 ⊢ Ⅎ𝑦((𝑇‘𝑓) ·_ih 𝐴) = (𝑓 ·_ih (𝐹‘𝐴))
11	2, 10	nfralw 3287	. . 3 ⊢ Ⅎ𝑦∀𝑓 ∈ ℋ ((𝑇‘𝑓) ·_ih 𝐴) = (𝑓 ·_ih (𝐹‘𝐴))
12		oveq2 7398	. . . . 5 ⊢ (𝑦 = 𝐴 → ((𝑇‘𝑓) ·_ih 𝑦) = ((𝑇‘𝑓) ·_ih 𝐴))
13		fveq2 6861	. . . . . 6 ⊢ (𝑦 = 𝐴 → (𝐹‘𝑦) = (𝐹‘𝐴))
14	13	oveq2d 7406	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝑓 ·_ih (𝐹‘𝑦)) = (𝑓 ·_ih (𝐹‘𝐴)))
15	12, 14	eqeq12d 2746	. . . 4 ⊢ (𝑦 = 𝐴 → (((𝑇‘𝑓) ·_ih 𝑦) = (𝑓 ·_ih (𝐹‘𝑦)) ↔ ((𝑇‘𝑓) ·_ih 𝐴) = (𝑓 ·_ih (𝐹‘𝐴))))
16	15	ralbidv 3157	. . 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 7351	. . . . . . 7 ⊢ (℩𝑤 ∈ ℋ ∀𝑣 ∈ ℋ ((𝑇‘𝑣) ·_ih 𝑦) = (𝑣 ·_ih 𝑤)) ∈ V
19	17, 18	eqeltri 2825	. . . . . 6 ⊢ 𝐵 ∈ V
20	5	fvmpt2 6982	. . . . . 6 ⊢ ((𝑦 ∈ ℋ ∧ 𝐵 ∈ V) → (𝐹‘𝑦) = 𝐵)
21	19, 20	mpan2 691	. . . . 5 ⊢ (𝑦 ∈ ℋ → (𝐹‘𝑦) = 𝐵)
22		fveq2 6861	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑓 → (𝑇‘𝑣) = (𝑇‘𝑓))
23	22	oveq1d 7405	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑓 → ((𝑇‘𝑣) ·_ih 𝑦) = ((𝑇‘𝑓) ·_ih 𝑦))
24		oveq1 7397	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑓 → (𝑣 ·_ih 𝑤) = (𝑓 ·_ih 𝑤))
25	23, 24	eqeq12d 2746	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑓 → (((𝑇‘𝑣) ·_ih 𝑦) = (𝑣 ·_ih 𝑤) ↔ ((𝑇‘𝑓) ·_ih 𝑦) = (𝑓 ·_ih 𝑤)))
26	25	cbvralvw 3216	. . . . . . . . . 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 32003	. . . . . . . . . . 11 ⊢ (𝑓 ∈ ℋ → (𝐺‘𝑓) = ((𝑇‘𝑓) ·_ih 𝑦))
32	31	eqeq1d 2732	. . . . . . . . . 10 ⊢ (𝑓 ∈ ℋ → ((𝐺‘𝑓) = (𝑓 ·_ih 𝑤) ↔ ((𝑇‘𝑓) ·_ih 𝑦) = (𝑓 ·_ih 𝑤)))
33	32	ralbiia 3074	. . . . . . . . 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 7367	. . . . . . 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 2753	. . . . . 6 ⊢ 𝐵 = (℩𝑤 ∈ ℋ ∀𝑓 ∈ ℋ (𝐺‘𝑓) = (𝑓 ·_ih 𝑤))
37	28, 29, 30	cnlnadjlem2 32004	. . . . . . . 8 ⊢ (𝑦 ∈ ℋ → (𝐺 ∈ LinFn ∧ 𝐺 ∈ ContFn))
38		elin 3933	. . . . . . . 8 ⊢ (𝐺 ∈ (LinFn ∩ ContFn) ↔ (𝐺 ∈ LinFn ∧ 𝐺 ∈ ContFn))
39	37, 38	sylibr 234	. . . . . . 7 ⊢ (𝑦 ∈ ℋ → 𝐺 ∈ (LinFn ∩ ContFn))
40		riesz4 32000	. . . . . . 7 ⊢ (𝐺 ∈ (LinFn ∩ ContFn) → ∃!𝑤 ∈ ℋ ∀𝑓 ∈ ℋ (𝐺‘𝑓) = (𝑓 ·_ih 𝑤))
41		riotacl2 7363	. . . . . . 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 2833	. . . . 5 ⊢ (𝑦 ∈ ℋ → 𝐵 ∈ {𝑤 ∈ ℋ ∣ ∀𝑓 ∈ ℋ (𝐺‘𝑓) = (𝑓 ·_ih 𝑤)})
44	21, 43	eqeltrd 2829	. . . 4 ⊢ (𝑦 ∈ ℋ → (𝐹‘𝑦) ∈ {𝑤 ∈ ℋ ∣ ∀𝑓 ∈ ℋ (𝐺‘𝑓) = (𝑓 ·_ih 𝑤)})
45		oveq2 7398	. . . . . . . . 9 ⊢ (𝑤 = (𝐹‘𝑦) → (𝑓 ·_ih 𝑤) = (𝑓 ·_ih (𝐹‘𝑦)))
46	45	eqeq2d 2741	. . . . . . . 8 ⊢ (𝑤 = (𝐹‘𝑦) → (((𝑇‘𝑓) ·_ih 𝑦) = (𝑓 ·_ih 𝑤) ↔ ((𝑇‘𝑓) ·_ih 𝑦) = (𝑓 ·_ih (𝐹‘𝑦))))
47	46	ralbidv 3157	. . . . . . 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 3662	. . . . 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 3545	. 2 ⊢ (𝐴 ∈ ℋ → ∀𝑓 ∈ ℋ ((𝑇‘𝑓) ·_ih 𝐴) = (𝑓 ·_ih (𝐹‘𝐴)))
53		fveq2 6861	. . . . 5 ⊢ (𝑓 = 𝐶 → (𝑇‘𝑓) = (𝑇‘𝐶))
54	53	oveq1d 7405	. . . 4 ⊢ (𝑓 = 𝐶 → ((𝑇‘𝑓) ·_ih 𝐴) = ((𝑇‘𝐶) ·_ih 𝐴))
55		oveq1 7397	. . . 4 ⊢ (𝑓 = 𝐶 → (𝑓 ·_ih (𝐹‘𝐴)) = (𝐶 ·_ih (𝐹‘𝐴)))
56	54, 55	eqeq12d 2746	. . 3 ⊢ (𝑓 = 𝐶 → (((𝑇‘𝑓) ·_ih 𝐴) = (𝑓 ·_ih (𝐹‘𝐴)) ↔ ((𝑇‘𝐶) ·_ih 𝐴) = (𝐶 ·_ih (𝐹‘𝐴))))
57	56	rspccva 3590	. 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 3045 ∃!wreu 3354 {crab 3408 Vcvv 3450 ∩ cin 3916 ↦ cmpt 5191 ‘cfv 6514 ℩crio 7346 (class class class)co 7390 ℋchba 30855 ·_ih csp 30858 ContOpccop 30882 LinOpclo 30883 ContFnccnfn 30889 LinFnclf 30890

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 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-inf2 9601 ax-cc 10395 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-pre-sup 11153 ax-addf 11154 ax-mulf 11155 ax-hilex 30935 ax-hfvadd 30936 ax-hvcom 30937 ax-hvass 30938 ax-hv0cl 30939 ax-hvaddid 30940 ax-hfvmul 30941 ax-hvmulid 30942 ax-hvmulass 30943 ax-hvdistr1 30944 ax-hvdistr2 30945 ax-hvmul0 30946 ax-hfi 31015 ax-his1 31018 ax-his2 31019 ax-his3 31020 ax-his4 31021 ax-hcompl 31138

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 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-tp 4597 df-op 4599 df-uni 4875 df-int 4914 df-iun 4960 df-iin 4961 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-se 5595 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-isom 6523 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-of 7656 df-om 7846 df-1st 7971 df-2nd 7972 df-supp 8143 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-2o 8438 df-oadd 8441 df-omul 8442 df-er 8674 df-map 8804 df-pm 8805 df-ixp 8874 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-fsupp 9320 df-fi 9369 df-sup 9400 df-inf 9401 df-oi 9470 df-card 9899 df-acn 9902 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-div 11843 df-nn 12194 df-2 12256 df-3 12257 df-4 12258 df-5 12259 df-6 12260 df-7 12261 df-8 12262 df-9 12263 df-n0 12450 df-z 12537 df-dec 12657 df-uz 12801 df-q 12915 df-rp 12959 df-xneg 13079 df-xadd 13080 df-xmul 13081 df-ioo 13317 df-ico 13319 df-icc 13320 df-fz 13476 df-fzo 13623 df-fl 13761 df-seq 13974 df-exp 14034 df-hash 14303 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-clim 15461 df-rlim 15462 df-sum 15660 df-struct 17124 df-sets 17141 df-slot 17159 df-ndx 17171 df-base 17187 df-ress 17208 df-plusg 17240 df-mulr 17241 df-starv 17242 df-sca 17243 df-vsca 17244 df-ip 17245 df-tset 17246 df-ple 17247 df-ds 17249 df-unif 17250 df-hom 17251 df-cco 17252 df-rest 17392 df-topn 17393 df-0g 17411 df-gsum 17412 df-topgen 17413 df-pt 17414 df-prds 17417 df-xrs 17472 df-qtop 17477 df-imas 17478 df-xps 17480 df-mre 17554 df-mrc 17555 df-acs 17557 df-mgm 18574 df-sgrp 18653 df-mnd 18669 df-submnd 18718 df-mulg 19007 df-cntz 19256 df-cmn 19719 df-psmet 21263 df-xmet 21264 df-met 21265 df-bl 21266 df-mopn 21267 df-fbas 21268 df-fg 21269 df-cnfld 21272 df-top 22788 df-topon 22805 df-topsp 22827 df-bases 22840 df-cld 22913 df-ntr 22914 df-cls 22915 df-nei 22992 df-cn 23121 df-cnp 23122 df-lm 23123 df-t1 23208 df-haus 23209 df-tx 23456 df-hmeo 23649 df-fil 23740 df-fm 23832 df-flim 23833 df-flf 23834 df-xms 24215 df-ms 24216 df-tms 24217 df-cfil 25162 df-cau 25163 df-cmet 25164 df-grpo 30429 df-gid 30430 df-ginv 30431 df-gdiv 30432 df-ablo 30481 df-vc 30495 df-nv 30528 df-va 30531 df-ba 30532 df-sm 30533 df-0v 30534 df-vs 30535 df-nmcv 30536 df-ims 30537 df-dip 30637 df-ssp 30658 df-ph 30749 df-cbn 30799 df-hnorm 30904 df-hba 30905 df-hvsub 30907 df-hlim 30908 df-hcau 30909 df-sh 31143 df-ch 31157 df-oc 31188 df-ch0 31189 df-nmop 31775 df-cnop 31776 df-lnop 31777 df-nmfn 31781 df-nlfn 31782 df-cnfn 31783 df-lnfn 31784

This theorem is referenced by: cnlnadjlem6 32008 cnlnadjlem7 32009 cnlnadjlem9 32011

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