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| Mirrors > Home > HSE Home > Th. List > cnlnadjlem5 | Structured version Visualization version GIF version | ||
| Description: Lemma for cnlnadji 32095. 𝐹 is an adjoint of 𝑇 (later, we will show it is unique). (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.) |
| 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) |
| Ref | Expression |
|---|---|
| cnlnadjlem5 | ⊢ ((𝐴 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝑇‘𝐶) ·ih 𝐴) = (𝐶 ·ih (𝐹‘𝐴))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | nfcv 2905 | . . 3 ⊢ Ⅎ𝑦𝐴 | |
| 2 | nfcv 2905 | . . . 4 ⊢ Ⅎ𝑦 ℋ | |
| 3 | nfcv 2905 | . . . . . 6 ⊢ Ⅎ𝑦𝑓 | |
| 4 | nfcv 2905 | . . . . . 6 ⊢ Ⅎ𝑦 ·ih | |
| 5 | cnlnadjlem.5 | . . . . . . . 8 ⊢ 𝐹 = (𝑦 ∈ ℋ ↦ 𝐵) | |
| 6 | nfmpt1 5250 | . . . . . . . 8 ⊢ Ⅎ𝑦(𝑦 ∈ ℋ ↦ 𝐵) | |
| 7 | 5, 6 | nfcxfr 2903 | . . . . . . 7 ⊢ Ⅎ𝑦𝐹 |
| 8 | 7, 1 | nffv 6916 | . . . . . 6 ⊢ Ⅎ𝑦(𝐹‘𝐴) |
| 9 | 3, 4, 8 | nfov 7461 | . . . . 5 ⊢ Ⅎ𝑦(𝑓 ·ih (𝐹‘𝐴)) |
| 10 | 9 | nfeq2 2923 | . . . 4 ⊢ Ⅎ𝑦((𝑇‘𝑓) ·ih 𝐴) = (𝑓 ·ih (𝐹‘𝐴)) |
| 11 | 2, 10 | nfralw 3311 | . . 3 ⊢ Ⅎ𝑦∀𝑓 ∈ ℋ ((𝑇‘𝑓) ·ih 𝐴) = (𝑓 ·ih (𝐹‘𝐴)) |
| 12 | oveq2 7439 | . . . . 5 ⊢ (𝑦 = 𝐴 → ((𝑇‘𝑓) ·ih 𝑦) = ((𝑇‘𝑓) ·ih 𝐴)) | |
| 13 | fveq2 6906 | . . . . . 6 ⊢ (𝑦 = 𝐴 → (𝐹‘𝑦) = (𝐹‘𝐴)) | |
| 14 | 13 | oveq2d 7447 | . . . . 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 3178 | . . 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 7392 | . . . . . . 7 ⊢ (℩𝑤 ∈ ℋ ∀𝑣 ∈ ℋ ((𝑇‘𝑣) ·ih 𝑦) = (𝑣 ·ih 𝑤)) ∈ V | |
| 19 | 17, 18 | eqeltri 2837 | . . . . . 6 ⊢ 𝐵 ∈ V |
| 20 | 5 | fvmpt2 7027 | . . . . . 6 ⊢ ((𝑦 ∈ ℋ ∧ 𝐵 ∈ V) → (𝐹‘𝑦) = 𝐵) |
| 21 | 19, 20 | mpan2 691 | . . . . 5 ⊢ (𝑦 ∈ ℋ → (𝐹‘𝑦) = 𝐵) |
| 22 | fveq2 6906 | . . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑓 → (𝑇‘𝑣) = (𝑇‘𝑓)) | |
| 23 | 22 | oveq1d 7446 | . . . . . . . . . . . 12 ⊢ (𝑣 = 𝑓 → ((𝑇‘𝑣) ·ih 𝑦) = ((𝑇‘𝑓) ·ih 𝑦)) |
| 24 | oveq1 7438 | . . . . . . . . . . . 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 3237 | . . . . . . . . . 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 32086 | . . . . . . . . . . 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 3091 | . . . . . . . . 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 7408 | . . . . . . 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 2765 | . . . . . 6 ⊢ 𝐵 = (℩𝑤 ∈ ℋ ∀𝑓 ∈ ℋ (𝐺‘𝑓) = (𝑓 ·ih 𝑤)) |
| 37 | 28, 29, 30 | cnlnadjlem2 32087 | . . . . . . . 8 ⊢ (𝑦 ∈ ℋ → (𝐺 ∈ LinFn ∧ 𝐺 ∈ ContFn)) |
| 38 | elin 3967 | . . . . . . . 8 ⊢ (𝐺 ∈ (LinFn ∩ ContFn) ↔ (𝐺 ∈ LinFn ∧ 𝐺 ∈ ContFn)) | |
| 39 | 37, 38 | sylibr 234 | . . . . . . 7 ⊢ (𝑦 ∈ ℋ → 𝐺 ∈ (LinFn ∩ ContFn)) |
| 40 | riesz4 32083 | . . . . . . 7 ⊢ (𝐺 ∈ (LinFn ∩ ContFn) → ∃!𝑤 ∈ ℋ ∀𝑓 ∈ ℋ (𝐺‘𝑓) = (𝑓 ·ih 𝑤)) | |
| 41 | riotacl2 7404 | . . . . . . 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 2845 | . . . . 5 ⊢ (𝑦 ∈ ℋ → 𝐵 ∈ {𝑤 ∈ ℋ ∣ ∀𝑓 ∈ ℋ (𝐺‘𝑓) = (𝑓 ·ih 𝑤)}) |
| 44 | 21, 43 | eqeltrd 2841 | . . . 4 ⊢ (𝑦 ∈ ℋ → (𝐹‘𝑦) ∈ {𝑤 ∈ ℋ ∣ ∀𝑓 ∈ ℋ (𝐺‘𝑓) = (𝑓 ·ih 𝑤)}) |
| 45 | oveq2 7439 | . . . . . . . . 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 3178 | . . . . . . 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 3692 | . . . . 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 3576 | . 2 ⊢ (𝐴 ∈ ℋ → ∀𝑓 ∈ ℋ ((𝑇‘𝑓) ·ih 𝐴) = (𝑓 ·ih (𝐹‘𝐴))) |
| 53 | fveq2 6906 | . . . . 5 ⊢ (𝑓 = 𝐶 → (𝑇‘𝑓) = (𝑇‘𝐶)) | |
| 54 | 53 | oveq1d 7446 | . . . 4 ⊢ (𝑓 = 𝐶 → ((𝑇‘𝑓) ·ih 𝐴) = ((𝑇‘𝐶) ·ih 𝐴)) |
| 55 | oveq1 7438 | . . . 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 3621 | . 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 2108 ∀wral 3061 ∃!wreu 3378 {crab 3436 Vcvv 3480 ∩ cin 3950 ↦ cmpt 5225 ‘cfv 6561 ℩crio 7387 (class class class)co 7431 ℋchba 30938 ·ih csp 30941 ContOpccop 30965 LinOpclo 30966 ContFnccnfn 30972 LinFnclf 30973 |
| 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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cc 10475 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234 ax-mulf 11235 ax-hilex 31018 ax-hfvadd 31019 ax-hvcom 31020 ax-hvass 31021 ax-hv0cl 31022 ax-hvaddid 31023 ax-hfvmul 31024 ax-hvmulid 31025 ax-hvmulass 31026 ax-hvdistr1 31027 ax-hvdistr2 31028 ax-hvmul0 31029 ax-hfi 31098 ax-his1 31101 ax-his2 31102 ax-his3 31103 ax-his4 31104 ax-hcompl 31221 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-oadd 8510 df-omul 8511 df-er 8745 df-map 8868 df-pm 8869 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-fi 9451 df-sup 9482 df-inf 9483 df-oi 9550 df-card 9979 df-acn 9982 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-ioo 13391 df-ico 13393 df-icc 13394 df-fz 13548 df-fzo 13695 df-fl 13832 df-seq 14043 df-exp 14103 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-clim 15524 df-rlim 15525 df-sum 15723 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-rest 17467 df-topn 17468 df-0g 17486 df-gsum 17487 df-topgen 17488 df-pt 17489 df-prds 17492 df-xrs 17547 df-qtop 17552 df-imas 17553 df-xps 17555 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-mulg 19086 df-cntz 19335 df-cmn 19800 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-fbas 21361 df-fg 21362 df-cnfld 21365 df-top 22900 df-topon 22917 df-topsp 22939 df-bases 22953 df-cld 23027 df-ntr 23028 df-cls 23029 df-nei 23106 df-cn 23235 df-cnp 23236 df-lm 23237 df-t1 23322 df-haus 23323 df-tx 23570 df-hmeo 23763 df-fil 23854 df-fm 23946 df-flim 23947 df-flf 23948 df-xms 24330 df-ms 24331 df-tms 24332 df-cfil 25289 df-cau 25290 df-cmet 25291 df-grpo 30512 df-gid 30513 df-ginv 30514 df-gdiv 30515 df-ablo 30564 df-vc 30578 df-nv 30611 df-va 30614 df-ba 30615 df-sm 30616 df-0v 30617 df-vs 30618 df-nmcv 30619 df-ims 30620 df-dip 30720 df-ssp 30741 df-ph 30832 df-cbn 30882 df-hnorm 30987 df-hba 30988 df-hvsub 30990 df-hlim 30991 df-hcau 30992 df-sh 31226 df-ch 31240 df-oc 31271 df-ch0 31272 df-nmop 31858 df-cnop 31859 df-lnop 31860 df-nmfn 31864 df-nlfn 31865 df-cnfn 31866 df-lnfn 31867 |
| This theorem is referenced by: cnlnadjlem6 32091 cnlnadjlem7 32092 cnlnadjlem9 32094 |
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