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Mirrors > Home > HSE Home > Th. List > cnlnadjlem5 | Structured version Visualization version GIF version |
Description: Lemma for cnlnadji 29855. 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 2979 | . . 3 ⊢ Ⅎ𝑦𝐴 | |
2 | nfcv 2979 | . . . 4 ⊢ Ⅎ𝑦 ℋ | |
3 | nfcv 2979 | . . . . . 6 ⊢ Ⅎ𝑦𝑓 | |
4 | nfcv 2979 | . . . . . 6 ⊢ Ⅎ𝑦 ·ih | |
5 | cnlnadjlem.5 | . . . . . . . 8 ⊢ 𝐹 = (𝑦 ∈ ℋ ↦ 𝐵) | |
6 | nfmpt1 5166 | . . . . . . . 8 ⊢ Ⅎ𝑦(𝑦 ∈ ℋ ↦ 𝐵) | |
7 | 5, 6 | nfcxfr 2977 | . . . . . . 7 ⊢ Ⅎ𝑦𝐹 |
8 | 7, 1 | nffv 6682 | . . . . . 6 ⊢ Ⅎ𝑦(𝐹‘𝐴) |
9 | 3, 4, 8 | nfov 7188 | . . . . 5 ⊢ Ⅎ𝑦(𝑓 ·ih (𝐹‘𝐴)) |
10 | 9 | nfeq2 2997 | . . . 4 ⊢ Ⅎ𝑦((𝑇‘𝑓) ·ih 𝐴) = (𝑓 ·ih (𝐹‘𝐴)) |
11 | 2, 10 | nfralw 3227 | . . 3 ⊢ Ⅎ𝑦∀𝑓 ∈ ℋ ((𝑇‘𝑓) ·ih 𝐴) = (𝑓 ·ih (𝐹‘𝐴)) |
12 | oveq2 7166 | . . . . 5 ⊢ (𝑦 = 𝐴 → ((𝑇‘𝑓) ·ih 𝑦) = ((𝑇‘𝑓) ·ih 𝐴)) | |
13 | fveq2 6672 | . . . . . 6 ⊢ (𝑦 = 𝐴 → (𝐹‘𝑦) = (𝐹‘𝐴)) | |
14 | 13 | oveq2d 7174 | . . . . 5 ⊢ (𝑦 = 𝐴 → (𝑓 ·ih (𝐹‘𝑦)) = (𝑓 ·ih (𝐹‘𝐴))) |
15 | 12, 14 | eqeq12d 2839 | . . . 4 ⊢ (𝑦 = 𝐴 → (((𝑇‘𝑓) ·ih 𝑦) = (𝑓 ·ih (𝐹‘𝑦)) ↔ ((𝑇‘𝑓) ·ih 𝐴) = (𝑓 ·ih (𝐹‘𝐴)))) |
16 | 15 | ralbidv 3199 | . . 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 7120 | . . . . . . 7 ⊢ (℩𝑤 ∈ ℋ ∀𝑣 ∈ ℋ ((𝑇‘𝑣) ·ih 𝑦) = (𝑣 ·ih 𝑤)) ∈ V | |
19 | 17, 18 | eqeltri 2911 | . . . . . 6 ⊢ 𝐵 ∈ V |
20 | 5 | fvmpt2 6781 | . . . . . 6 ⊢ ((𝑦 ∈ ℋ ∧ 𝐵 ∈ V) → (𝐹‘𝑦) = 𝐵) |
21 | 19, 20 | mpan2 689 | . . . . 5 ⊢ (𝑦 ∈ ℋ → (𝐹‘𝑦) = 𝐵) |
22 | fveq2 6672 | . . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑓 → (𝑇‘𝑣) = (𝑇‘𝑓)) | |
23 | 22 | oveq1d 7173 | . . . . . . . . . . . 12 ⊢ (𝑣 = 𝑓 → ((𝑇‘𝑣) ·ih 𝑦) = ((𝑇‘𝑓) ·ih 𝑦)) |
24 | oveq1 7165 | . . . . . . . . . . . 12 ⊢ (𝑣 = 𝑓 → (𝑣 ·ih 𝑤) = (𝑓 ·ih 𝑤)) | |
25 | 23, 24 | eqeq12d 2839 | . . . . . . . . . . 11 ⊢ (𝑣 = 𝑓 → (((𝑇‘𝑣) ·ih 𝑦) = (𝑣 ·ih 𝑤) ↔ ((𝑇‘𝑓) ·ih 𝑦) = (𝑓 ·ih 𝑤))) |
26 | 25 | cbvralvw 3451 | . . . . . . . . . 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 29846 | . . . . . . . . . . 11 ⊢ (𝑓 ∈ ℋ → (𝐺‘𝑓) = ((𝑇‘𝑓) ·ih 𝑦)) |
32 | 31 | eqeq1d 2825 | . . . . . . . . . 10 ⊢ (𝑓 ∈ ℋ → ((𝐺‘𝑓) = (𝑓 ·ih 𝑤) ↔ ((𝑇‘𝑓) ·ih 𝑦) = (𝑓 ·ih 𝑤))) |
33 | 32 | ralbiia 3166 | . . . . . . . . 9 ⊢ (∀𝑓 ∈ ℋ (𝐺‘𝑓) = (𝑓 ·ih 𝑤) ↔ ∀𝑓 ∈ ℋ ((𝑇‘𝑓) ·ih 𝑦) = (𝑓 ·ih 𝑤)) |
34 | 27, 33 | syl6bbr 291 | . . . . . . . 8 ⊢ (𝑤 ∈ ℋ → (∀𝑣 ∈ ℋ ((𝑇‘𝑣) ·ih 𝑦) = (𝑣 ·ih 𝑤) ↔ ∀𝑓 ∈ ℋ (𝐺‘𝑓) = (𝑓 ·ih 𝑤))) |
35 | 34 | riotabiia 7136 | . . . . . . 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 2846 | . . . . . 6 ⊢ 𝐵 = (℩𝑤 ∈ ℋ ∀𝑓 ∈ ℋ (𝐺‘𝑓) = (𝑓 ·ih 𝑤)) |
37 | 28, 29, 30 | cnlnadjlem2 29847 | . . . . . . . 8 ⊢ (𝑦 ∈ ℋ → (𝐺 ∈ LinFn ∧ 𝐺 ∈ ContFn)) |
38 | elin 4171 | . . . . . . . 8 ⊢ (𝐺 ∈ (LinFn ∩ ContFn) ↔ (𝐺 ∈ LinFn ∧ 𝐺 ∈ ContFn)) | |
39 | 37, 38 | sylibr 236 | . . . . . . 7 ⊢ (𝑦 ∈ ℋ → 𝐺 ∈ (LinFn ∩ ContFn)) |
40 | riesz4 29843 | . . . . . . 7 ⊢ (𝐺 ∈ (LinFn ∩ ContFn) → ∃!𝑤 ∈ ℋ ∀𝑓 ∈ ℋ (𝐺‘𝑓) = (𝑓 ·ih 𝑤)) | |
41 | riotacl2 7132 | . . . . . . 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 2919 | . . . . 5 ⊢ (𝑦 ∈ ℋ → 𝐵 ∈ {𝑤 ∈ ℋ ∣ ∀𝑓 ∈ ℋ (𝐺‘𝑓) = (𝑓 ·ih 𝑤)}) |
44 | 21, 43 | eqeltrd 2915 | . . . 4 ⊢ (𝑦 ∈ ℋ → (𝐹‘𝑦) ∈ {𝑤 ∈ ℋ ∣ ∀𝑓 ∈ ℋ (𝐺‘𝑓) = (𝑓 ·ih 𝑤)}) |
45 | oveq2 7166 | . . . . . . . . 9 ⊢ (𝑤 = (𝐹‘𝑦) → (𝑓 ·ih 𝑤) = (𝑓 ·ih (𝐹‘𝑦))) | |
46 | 45 | eqeq2d 2834 | . . . . . . . 8 ⊢ (𝑤 = (𝐹‘𝑦) → (((𝑇‘𝑓) ·ih 𝑦) = (𝑓 ·ih 𝑤) ↔ ((𝑇‘𝑓) ·ih 𝑦) = (𝑓 ·ih (𝐹‘𝑦)))) |
47 | 46 | ralbidv 3199 | . . . . . . 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 285 | . . . . . 6 ⊢ (𝑤 = (𝐹‘𝑦) → (∀𝑓 ∈ ℋ (𝐺‘𝑓) = (𝑓 ·ih 𝑤) ↔ ∀𝑓 ∈ ℋ ((𝑇‘𝑓) ·ih 𝑦) = (𝑓 ·ih (𝐹‘𝑦)))) |
49 | 48 | elrab 3682 | . . . . 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 499 | . . . 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 3575 | . 2 ⊢ (𝐴 ∈ ℋ → ∀𝑓 ∈ ℋ ((𝑇‘𝑓) ·ih 𝐴) = (𝑓 ·ih (𝐹‘𝐴))) |
53 | fveq2 6672 | . . . . 5 ⊢ (𝑓 = 𝐶 → (𝑇‘𝑓) = (𝑇‘𝐶)) | |
54 | 53 | oveq1d 7173 | . . . 4 ⊢ (𝑓 = 𝐶 → ((𝑇‘𝑓) ·ih 𝐴) = ((𝑇‘𝐶) ·ih 𝐴)) |
55 | oveq1 7165 | . . . 4 ⊢ (𝑓 = 𝐶 → (𝑓 ·ih (𝐹‘𝐴)) = (𝐶 ·ih (𝐹‘𝐴))) | |
56 | 54, 55 | eqeq12d 2839 | . . 3 ⊢ (𝑓 = 𝐶 → (((𝑇‘𝑓) ·ih 𝐴) = (𝑓 ·ih (𝐹‘𝐴)) ↔ ((𝑇‘𝐶) ·ih 𝐴) = (𝐶 ·ih (𝐹‘𝐴)))) |
57 | 56 | rspccva 3624 | . 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 582 | 1 ⊢ ((𝐴 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝑇‘𝐶) ·ih 𝐴) = (𝐶 ·ih (𝐹‘𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3140 ∃!wreu 3142 {crab 3144 Vcvv 3496 ∩ cin 3937 ↦ cmpt 5148 ‘cfv 6357 ℩crio 7115 (class class class)co 7158 ℋchba 28698 ·ih csp 28701 ContOpccop 28725 LinOpclo 28726 ContFnccnfn 28732 LinFnclf 28733 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 ax-cc 9859 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 ax-addf 10618 ax-mulf 10619 ax-hilex 28778 ax-hfvadd 28779 ax-hvcom 28780 ax-hvass 28781 ax-hv0cl 28782 ax-hvaddid 28783 ax-hfvmul 28784 ax-hvmulid 28785 ax-hvmulass 28786 ax-hvdistr1 28787 ax-hvdistr2 28788 ax-hvmul0 28789 ax-hfi 28858 ax-his1 28861 ax-his2 28862 ax-his3 28863 ax-his4 28864 ax-hcompl 28981 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-om 7583 df-1st 7691 df-2nd 7692 df-supp 7833 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-2o 8105 df-oadd 8108 df-omul 8109 df-er 8291 df-map 8410 df-pm 8411 df-ixp 8464 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fsupp 8836 df-fi 8877 df-sup 8908 df-inf 8909 df-oi 8976 df-card 9370 df-acn 9373 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-q 12352 df-rp 12393 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-ioo 12745 df-ico 12747 df-icc 12748 df-fz 12896 df-fzo 13037 df-fl 13165 df-seq 13373 df-exp 13433 df-hash 13694 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-clim 14847 df-rlim 14848 df-sum 15045 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-starv 16582 df-sca 16583 df-vsca 16584 df-ip 16585 df-tset 16586 df-ple 16587 df-ds 16589 df-unif 16590 df-hom 16591 df-cco 16592 df-rest 16698 df-topn 16699 df-0g 16717 df-gsum 16718 df-topgen 16719 df-pt 16720 df-prds 16723 df-xrs 16777 df-qtop 16782 df-imas 16783 df-xps 16785 df-mre 16859 df-mrc 16860 df-acs 16862 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-submnd 17959 df-mulg 18227 df-cntz 18449 df-cmn 18910 df-psmet 20539 df-xmet 20540 df-met 20541 df-bl 20542 df-mopn 20543 df-fbas 20544 df-fg 20545 df-cnfld 20548 df-top 21504 df-topon 21521 df-topsp 21543 df-bases 21556 df-cld 21629 df-ntr 21630 df-cls 21631 df-nei 21708 df-cn 21837 df-cnp 21838 df-lm 21839 df-t1 21924 df-haus 21925 df-tx 22172 df-hmeo 22365 df-fil 22456 df-fm 22548 df-flim 22549 df-flf 22550 df-xms 22932 df-ms 22933 df-tms 22934 df-cfil 23860 df-cau 23861 df-cmet 23862 df-grpo 28272 df-gid 28273 df-ginv 28274 df-gdiv 28275 df-ablo 28324 df-vc 28338 df-nv 28371 df-va 28374 df-ba 28375 df-sm 28376 df-0v 28377 df-vs 28378 df-nmcv 28379 df-ims 28380 df-dip 28480 df-ssp 28501 df-ph 28592 df-cbn 28642 df-hnorm 28747 df-hba 28748 df-hvsub 28750 df-hlim 28751 df-hcau 28752 df-sh 28986 df-ch 29000 df-oc 29031 df-ch0 29032 df-nmop 29618 df-cnop 29619 df-lnop 29620 df-nmfn 29624 df-nlfn 29625 df-cnfn 29626 df-lnfn 29627 |
This theorem is referenced by: cnlnadjlem6 29851 cnlnadjlem7 29852 cnlnadjlem9 29854 |
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