cnlnadjlem5 - Hilbert Space Explorer

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Theorem cnlnadjlem5 32150

Description: Lemma for cnlnadji 32155. 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 2899	. . 3 ⊢ Ⅎ𝑦𝐴
2		nfcv 2899	. . . 4 ⊢ Ⅎ𝑦 ℋ
3		nfcv 2899	. . . . . 6 ⊢ Ⅎ𝑦𝑓
4		nfcv 2899	. . . . . 6 ⊢ Ⅎ𝑦 ·_ih
5		cnlnadjlem.5	. . . . . . . 8 ⊢ 𝐹 = (𝑦 ∈ ℋ ↦ 𝐵)
6		nfmpt1 5198	. . . . . . . 8 ⊢ Ⅎ𝑦(𝑦 ∈ ℋ ↦ 𝐵)
7	5, 6	nfcxfr 2897	. . . . . . 7 ⊢ Ⅎ𝑦𝐹
8	7, 1	nffv 6845	. . . . . 6 ⊢ Ⅎ𝑦(𝐹‘𝐴)
9	3, 4, 8	nfov 7390	. . . . 5 ⊢ Ⅎ𝑦(𝑓 ·_ih (𝐹‘𝐴))
10	9	nfeq2 2917	. . . 4 ⊢ Ⅎ𝑦((𝑇‘𝑓) ·_ih 𝐴) = (𝑓 ·_ih (𝐹‘𝐴))
11	2, 10	nfralw 3284	. . 3 ⊢ Ⅎ𝑦∀𝑓 ∈ ℋ ((𝑇‘𝑓) ·_ih 𝐴) = (𝑓 ·_ih (𝐹‘𝐴))
12		oveq2 7368	. . . . 5 ⊢ (𝑦 = 𝐴 → ((𝑇‘𝑓) ·_ih 𝑦) = ((𝑇‘𝑓) ·_ih 𝐴))
13		fveq2 6835	. . . . . 6 ⊢ (𝑦 = 𝐴 → (𝐹‘𝑦) = (𝐹‘𝐴))
14	13	oveq2d 7376	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝑓 ·_ih (𝐹‘𝑦)) = (𝑓 ·_ih (𝐹‘𝐴)))
15	12, 14	eqeq12d 2753	. . . 4 ⊢ (𝑦 = 𝐴 → (((𝑇‘𝑓) ·_ih 𝑦) = (𝑓 ·_ih (𝐹‘𝑦)) ↔ ((𝑇‘𝑓) ·_ih 𝐴) = (𝑓 ·_ih (𝐹‘𝐴))))
16	15	ralbidv 3160	. . 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 7321	. . . . . . 7 ⊢ (℩𝑤 ∈ ℋ ∀𝑣 ∈ ℋ ((𝑇‘𝑣) ·_ih 𝑦) = (𝑣 ·_ih 𝑤)) ∈ V
19	17, 18	eqeltri 2833	. . . . . 6 ⊢ 𝐵 ∈ V
20	5	fvmpt2 6954	. . . . . 6 ⊢ ((𝑦 ∈ ℋ ∧ 𝐵 ∈ V) → (𝐹‘𝑦) = 𝐵)
21	19, 20	mpan2 692	. . . . 5 ⊢ (𝑦 ∈ ℋ → (𝐹‘𝑦) = 𝐵)
22		fveq2 6835	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑓 → (𝑇‘𝑣) = (𝑇‘𝑓))
23	22	oveq1d 7375	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑓 → ((𝑇‘𝑣) ·_ih 𝑦) = ((𝑇‘𝑓) ·_ih 𝑦))
24		oveq1 7367	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑓 → (𝑣 ·_ih 𝑤) = (𝑓 ·_ih 𝑤))
25	23, 24	eqeq12d 2753	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑓 → (((𝑇‘𝑣) ·_ih 𝑦) = (𝑣 ·_ih 𝑤) ↔ ((𝑇‘𝑓) ·_ih 𝑦) = (𝑓 ·_ih 𝑤)))
26	25	cbvralvw 3215	. . . . . . . . . 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 32146	. . . . . . . . . . 11 ⊢ (𝑓 ∈ ℋ → (𝐺‘𝑓) = ((𝑇‘𝑓) ·_ih 𝑦))
32	31	eqeq1d 2739	. . . . . . . . . 10 ⊢ (𝑓 ∈ ℋ → ((𝐺‘𝑓) = (𝑓 ·_ih 𝑤) ↔ ((𝑇‘𝑓) ·_ih 𝑦) = (𝑓 ·_ih 𝑤)))
33	32	ralbiia 3081	. . . . . . . . 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 7337	. . . . . . 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 2760	. . . . . 6 ⊢ 𝐵 = (℩𝑤 ∈ ℋ ∀𝑓 ∈ ℋ (𝐺‘𝑓) = (𝑓 ·_ih 𝑤))
37	28, 29, 30	cnlnadjlem2 32147	. . . . . . . 8 ⊢ (𝑦 ∈ ℋ → (𝐺 ∈ LinFn ∧ 𝐺 ∈ ContFn))
38		elin 3918	. . . . . . . 8 ⊢ (𝐺 ∈ (LinFn ∩ ContFn) ↔ (𝐺 ∈ LinFn ∧ 𝐺 ∈ ContFn))
39	37, 38	sylibr 234	. . . . . . 7 ⊢ (𝑦 ∈ ℋ → 𝐺 ∈ (LinFn ∩ ContFn))
40		riesz4 32143	. . . . . . 7 ⊢ (𝐺 ∈ (LinFn ∩ ContFn) → ∃!𝑤 ∈ ℋ ∀𝑓 ∈ ℋ (𝐺‘𝑓) = (𝑓 ·_ih 𝑤))
41		riotacl2 7333	. . . . . . 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 2841	. . . . 5 ⊢ (𝑦 ∈ ℋ → 𝐵 ∈ {𝑤 ∈ ℋ ∣ ∀𝑓 ∈ ℋ (𝐺‘𝑓) = (𝑓 ·_ih 𝑤)})
44	21, 43	eqeltrd 2837	. . . 4 ⊢ (𝑦 ∈ ℋ → (𝐹‘𝑦) ∈ {𝑤 ∈ ℋ ∣ ∀𝑓 ∈ ℋ (𝐺‘𝑓) = (𝑓 ·_ih 𝑤)})
45		oveq2 7368	. . . . . . . . 9 ⊢ (𝑤 = (𝐹‘𝑦) → (𝑓 ·_ih 𝑤) = (𝑓 ·_ih (𝐹‘𝑦)))
46	45	eqeq2d 2748	. . . . . . . 8 ⊢ (𝑤 = (𝐹‘𝑦) → (((𝑇‘𝑓) ·_ih 𝑦) = (𝑓 ·_ih 𝑤) ↔ ((𝑇‘𝑓) ·_ih 𝑦) = (𝑓 ·_ih (𝐹‘𝑦))))
47	46	ralbidv 3160	. . . . . . 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 3647	. . . . 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 3532	. 2 ⊢ (𝐴 ∈ ℋ → ∀𝑓 ∈ ℋ ((𝑇‘𝑓) ·_ih 𝐴) = (𝑓 ·_ih (𝐹‘𝐴)))
53		fveq2 6835	. . . . 5 ⊢ (𝑓 = 𝐶 → (𝑇‘𝑓) = (𝑇‘𝐶))
54	53	oveq1d 7375	. . . 4 ⊢ (𝑓 = 𝐶 → ((𝑇‘𝑓) ·_ih 𝐴) = ((𝑇‘𝐶) ·_ih 𝐴))
55		oveq1 7367	. . . 4 ⊢ (𝑓 = 𝐶 → (𝑓 ·_ih (𝐹‘𝐴)) = (𝐶 ·_ih (𝐹‘𝐴)))
56	54, 55	eqeq12d 2753	. . 3 ⊢ (𝑓 = 𝐶 → (((𝑇‘𝑓) ·_ih 𝐴) = (𝑓 ·_ih (𝐹‘𝐴)) ↔ ((𝑇‘𝐶) ·_ih 𝐴) = (𝐶 ·_ih (𝐹‘𝐴))))
57	56	rspccva 3576	. 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 581	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 1542 ∈ wcel 2114 ∀wral 3052 ∃!wreu 3349 {crab 3400 Vcvv 3441 ∩ cin 3901 ↦ cmpt 5180 ‘cfv 6493 ℩crio 7316 (class class class)co 7360 ℋchba 30998 ·_ih csp 31001 ContOpccop 31025 LinOpclo 31026 ContFnccnfn 31032 LinFnclf 31033

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 2709 ax-rep 5225 ax-sep 5242 ax-nul 5252 ax-pow 5311 ax-pr 5378 ax-un 7682 ax-inf2 9554 ax-cc 10349 ax-cnex 11086 ax-resscn 11087 ax-1cn 11088 ax-icn 11089 ax-addcl 11090 ax-addrcl 11091 ax-mulcl 11092 ax-mulrcl 11093 ax-mulcom 11094 ax-addass 11095 ax-mulass 11096 ax-distr 11097 ax-i2m1 11098 ax-1ne0 11099 ax-1rid 11100 ax-rnegex 11101 ax-rrecex 11102 ax-cnre 11103 ax-pre-lttri 11104 ax-pre-lttrn 11105 ax-pre-ltadd 11106 ax-pre-mulgt0 11107 ax-pre-sup 11108 ax-addf 11109 ax-mulf 11110 ax-hilex 31078 ax-hfvadd 31079 ax-hvcom 31080 ax-hvass 31081 ax-hv0cl 31082 ax-hvaddid 31083 ax-hfvmul 31084 ax-hvmulid 31085 ax-hvmulass 31086 ax-hvdistr1 31087 ax-hvdistr2 31088 ax-hvmul0 31089 ax-hfi 31158 ax-his1 31161 ax-his2 31162 ax-his3 31163 ax-his4 31164 ax-hcompl 31281

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 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3351 df-reu 3352 df-rab 3401 df-v 3443 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4287 df-if 4481 df-pw 4557 df-sn 4582 df-pr 4584 df-tp 4586 df-op 4588 df-uni 4865 df-int 4904 df-iun 4949 df-iin 4950 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-se 5579 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-of 7624 df-om 7811 df-1st 7935 df-2nd 7936 df-supp 8105 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-2o 8400 df-oadd 8403 df-omul 8404 df-er 8637 df-map 8769 df-pm 8770 df-ixp 8840 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-fsupp 9269 df-fi 9318 df-sup 9349 df-inf 9350 df-oi 9419 df-card 9855 df-acn 9858 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12150 df-2 12212 df-3 12213 df-4 12214 df-5 12215 df-6 12216 df-7 12217 df-8 12218 df-9 12219 df-n0 12406 df-z 12493 df-dec 12612 df-uz 12756 df-q 12866 df-rp 12910 df-xneg 13030 df-xadd 13031 df-xmul 13032 df-ioo 13269 df-ico 13271 df-icc 13272 df-fz 13428 df-fzo 13575 df-fl 13716 df-seq 13929 df-exp 13989 df-hash 14258 df-cj 15026 df-re 15027 df-im 15028 df-sqrt 15162 df-abs 15163 df-clim 15415 df-rlim 15416 df-sum 15614 df-struct 17078 df-sets 17095 df-slot 17113 df-ndx 17125 df-base 17141 df-ress 17162 df-plusg 17194 df-mulr 17195 df-starv 17196 df-sca 17197 df-vsca 17198 df-ip 17199 df-tset 17200 df-ple 17201 df-ds 17203 df-unif 17204 df-hom 17205 df-cco 17206 df-rest 17346 df-topn 17347 df-0g 17365 df-gsum 17366 df-topgen 17367 df-pt 17368 df-prds 17371 df-xrs 17427 df-qtop 17432 df-imas 17433 df-xps 17435 df-mre 17509 df-mrc 17510 df-acs 17512 df-mgm 18569 df-sgrp 18648 df-mnd 18664 df-submnd 18713 df-mulg 19002 df-cntz 19250 df-cmn 19715 df-psmet 21305 df-xmet 21306 df-met 21307 df-bl 21308 df-mopn 21309 df-fbas 21310 df-fg 21311 df-cnfld 21314 df-top 22842 df-topon 22859 df-topsp 22881 df-bases 22894 df-cld 22967 df-ntr 22968 df-cls 22969 df-nei 23046 df-cn 23175 df-cnp 23176 df-lm 23177 df-t1 23262 df-haus 23263 df-tx 23510 df-hmeo 23703 df-fil 23794 df-fm 23886 df-flim 23887 df-flf 23888 df-xms 24268 df-ms 24269 df-tms 24270 df-cfil 25215 df-cau 25216 df-cmet 25217 df-grpo 30572 df-gid 30573 df-ginv 30574 df-gdiv 30575 df-ablo 30624 df-vc 30638 df-nv 30671 df-va 30674 df-ba 30675 df-sm 30676 df-0v 30677 df-vs 30678 df-nmcv 30679 df-ims 30680 df-dip 30780 df-ssp 30801 df-ph 30892 df-cbn 30942 df-hnorm 31047 df-hba 31048 df-hvsub 31050 df-hlim 31051 df-hcau 31052 df-sh 31286 df-ch 31300 df-oc 31331 df-ch0 31332 df-nmop 31918 df-cnop 31919 df-lnop 31920 df-nmfn 31924 df-nlfn 31925 df-cnfn 31926 df-lnfn 31927

This theorem is referenced by: cnlnadjlem6 32151 cnlnadjlem7 32152 cnlnadjlem9 32154

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