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Mirrors > Home > HSE Home > Th. List > cnlnadjlem5 | Structured version Visualization version GIF version |
Description: Lemma for cnlnadji 32105. 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 2903 | . . 3 ⊢ Ⅎ𝑦𝐴 | |
2 | nfcv 2903 | . . . 4 ⊢ Ⅎ𝑦 ℋ | |
3 | nfcv 2903 | . . . . . 6 ⊢ Ⅎ𝑦𝑓 | |
4 | nfcv 2903 | . . . . . 6 ⊢ Ⅎ𝑦 ·ih | |
5 | cnlnadjlem.5 | . . . . . . . 8 ⊢ 𝐹 = (𝑦 ∈ ℋ ↦ 𝐵) | |
6 | nfmpt1 5256 | . . . . . . . 8 ⊢ Ⅎ𝑦(𝑦 ∈ ℋ ↦ 𝐵) | |
7 | 5, 6 | nfcxfr 2901 | . . . . . . 7 ⊢ Ⅎ𝑦𝐹 |
8 | 7, 1 | nffv 6917 | . . . . . 6 ⊢ Ⅎ𝑦(𝐹‘𝐴) |
9 | 3, 4, 8 | nfov 7461 | . . . . 5 ⊢ Ⅎ𝑦(𝑓 ·ih (𝐹‘𝐴)) |
10 | 9 | nfeq2 2921 | . . . 4 ⊢ Ⅎ𝑦((𝑇‘𝑓) ·ih 𝐴) = (𝑓 ·ih (𝐹‘𝐴)) |
11 | 2, 10 | nfralw 3309 | . . 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 6907 | . . . . . 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 2751 | . . . 4 ⊢ (𝑦 = 𝐴 → (((𝑇‘𝑓) ·ih 𝑦) = (𝑓 ·ih (𝐹‘𝑦)) ↔ ((𝑇‘𝑓) ·ih 𝐴) = (𝑓 ·ih (𝐹‘𝐴)))) |
16 | 15 | ralbidv 3176 | . . 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 2835 | . . . . . 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 6907 | . . . . . . . . . . . . 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 2751 | . . . . . . . . . . 11 ⊢ (𝑣 = 𝑓 → (((𝑇‘𝑣) ·ih 𝑦) = (𝑣 ·ih 𝑤) ↔ ((𝑇‘𝑓) ·ih 𝑦) = (𝑓 ·ih 𝑤))) |
26 | 25 | cbvralvw 3235 | . . . . . . . . . 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 32096 | . . . . . . . . . . 11 ⊢ (𝑓 ∈ ℋ → (𝐺‘𝑓) = ((𝑇‘𝑓) ·ih 𝑦)) |
32 | 31 | eqeq1d 2737 | . . . . . . . . . 10 ⊢ (𝑓 ∈ ℋ → ((𝐺‘𝑓) = (𝑓 ·ih 𝑤) ↔ ((𝑇‘𝑓) ·ih 𝑦) = (𝑓 ·ih 𝑤))) |
33 | 32 | ralbiia 3089 | . . . . . . . . 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 2763 | . . . . . 6 ⊢ 𝐵 = (℩𝑤 ∈ ℋ ∀𝑓 ∈ ℋ (𝐺‘𝑓) = (𝑓 ·ih 𝑤)) |
37 | 28, 29, 30 | cnlnadjlem2 32097 | . . . . . . . 8 ⊢ (𝑦 ∈ ℋ → (𝐺 ∈ LinFn ∧ 𝐺 ∈ ContFn)) |
38 | elin 3979 | . . . . . . . 8 ⊢ (𝐺 ∈ (LinFn ∩ ContFn) ↔ (𝐺 ∈ LinFn ∧ 𝐺 ∈ ContFn)) | |
39 | 37, 38 | sylibr 234 | . . . . . . 7 ⊢ (𝑦 ∈ ℋ → 𝐺 ∈ (LinFn ∩ ContFn)) |
40 | riesz4 32093 | . . . . . . 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 2843 | . . . . 5 ⊢ (𝑦 ∈ ℋ → 𝐵 ∈ {𝑤 ∈ ℋ ∣ ∀𝑓 ∈ ℋ (𝐺‘𝑓) = (𝑓 ·ih 𝑤)}) |
44 | 21, 43 | eqeltrd 2839 | . . . 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 2746 | . . . . . . . 8 ⊢ (𝑤 = (𝐹‘𝑦) → (((𝑇‘𝑓) ·ih 𝑦) = (𝑓 ·ih 𝑤) ↔ ((𝑇‘𝑓) ·ih 𝑦) = (𝑓 ·ih (𝐹‘𝑦)))) |
47 | 46 | ralbidv 3176 | . . . . . . 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 3695 | . . . . 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 6907 | . . . . 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 2751 | . . 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 1537 ∈ wcel 2106 ∀wral 3059 ∃!wreu 3376 {crab 3433 Vcvv 3478 ∩ cin 3962 ↦ cmpt 5231 ‘cfv 6563 ℩crio 7387 (class class class)co 7431 ℋchba 30948 ·ih csp 30951 ContOpccop 30975 LinOpclo 30976 ContFnccnfn 30982 LinFnclf 30983 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cc 10473 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 ax-addf 11232 ax-mulf 11233 ax-hilex 31028 ax-hfvadd 31029 ax-hvcom 31030 ax-hvass 31031 ax-hv0cl 31032 ax-hvaddid 31033 ax-hfvmul 31034 ax-hvmulid 31035 ax-hvmulass 31036 ax-hvdistr1 31037 ax-hvdistr2 31038 ax-hvmul0 31039 ax-hfi 31108 ax-his1 31111 ax-his2 31112 ax-his3 31113 ax-his4 31114 ax-hcompl 31231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-oadd 8509 df-omul 8510 df-er 8744 df-map 8867 df-pm 8868 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-fi 9449 df-sup 9480 df-inf 9481 df-oi 9548 df-card 9977 df-acn 9980 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-q 12989 df-rp 13033 df-xneg 13152 df-xadd 13153 df-xmul 13154 df-ioo 13388 df-ico 13390 df-icc 13391 df-fz 13545 df-fzo 13692 df-fl 13829 df-seq 14040 df-exp 14100 df-hash 14367 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-clim 15521 df-rlim 15522 df-sum 15720 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-rest 17469 df-topn 17470 df-0g 17488 df-gsum 17489 df-topgen 17490 df-pt 17491 df-prds 17494 df-xrs 17549 df-qtop 17554 df-imas 17555 df-xps 17557 df-mre 17631 df-mrc 17632 df-acs 17634 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-submnd 18810 df-mulg 19099 df-cntz 19348 df-cmn 19815 df-psmet 21374 df-xmet 21375 df-met 21376 df-bl 21377 df-mopn 21378 df-fbas 21379 df-fg 21380 df-cnfld 21383 df-top 22916 df-topon 22933 df-topsp 22955 df-bases 22969 df-cld 23043 df-ntr 23044 df-cls 23045 df-nei 23122 df-cn 23251 df-cnp 23252 df-lm 23253 df-t1 23338 df-haus 23339 df-tx 23586 df-hmeo 23779 df-fil 23870 df-fm 23962 df-flim 23963 df-flf 23964 df-xms 24346 df-ms 24347 df-tms 24348 df-cfil 25303 df-cau 25304 df-cmet 25305 df-grpo 30522 df-gid 30523 df-ginv 30524 df-gdiv 30525 df-ablo 30574 df-vc 30588 df-nv 30621 df-va 30624 df-ba 30625 df-sm 30626 df-0v 30627 df-vs 30628 df-nmcv 30629 df-ims 30630 df-dip 30730 df-ssp 30751 df-ph 30842 df-cbn 30892 df-hnorm 30997 df-hba 30998 df-hvsub 31000 df-hlim 31001 df-hcau 31002 df-sh 31236 df-ch 31250 df-oc 31281 df-ch0 31282 df-nmop 31868 df-cnop 31869 df-lnop 31870 df-nmfn 31874 df-nlfn 31875 df-cnfn 31876 df-lnfn 31877 |
This theorem is referenced by: cnlnadjlem6 32101 cnlnadjlem7 32102 cnlnadjlem9 32104 |
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