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Mirrors > Home > HSE Home > Th. List > cnlnadjlem5 | Structured version Visualization version GIF version |
Description: Lemma for cnlnadji 28319. 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 ∈ ConOp |
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 2751 | . . 3 ⊢ Ⅎ𝑦𝐴 | |
2 | nfcv 2751 | . . . 4 ⊢ Ⅎ𝑦 ℋ | |
3 | nfcv 2751 | . . . . . 6 ⊢ Ⅎ𝑦𝑓 | |
4 | nfcv 2751 | . . . . . 6 ⊢ Ⅎ𝑦 ·ih | |
5 | cnlnadjlem.5 | . . . . . . . 8 ⊢ 𝐹 = (𝑦 ∈ ℋ ↦ 𝐵) | |
6 | nfmpt1 4675 | . . . . . . . 8 ⊢ Ⅎ𝑦(𝑦 ∈ ℋ ↦ 𝐵) | |
7 | 5, 6 | nfcxfr 2749 | . . . . . . 7 ⊢ Ⅎ𝑦𝐹 |
8 | 7, 1 | nffv 6110 | . . . . . 6 ⊢ Ⅎ𝑦(𝐹‘𝐴) |
9 | 3, 4, 8 | nfov 6575 | . . . . 5 ⊢ Ⅎ𝑦(𝑓 ·ih (𝐹‘𝐴)) |
10 | 9 | nfeq2 2766 | . . . 4 ⊢ Ⅎ𝑦((𝑇‘𝑓) ·ih 𝐴) = (𝑓 ·ih (𝐹‘𝐴)) |
11 | 2, 10 | nfral 2929 | . . 3 ⊢ Ⅎ𝑦∀𝑓 ∈ ℋ ((𝑇‘𝑓) ·ih 𝐴) = (𝑓 ·ih (𝐹‘𝐴)) |
12 | oveq2 6557 | . . . . 5 ⊢ (𝑦 = 𝐴 → ((𝑇‘𝑓) ·ih 𝑦) = ((𝑇‘𝑓) ·ih 𝐴)) | |
13 | fveq2 6103 | . . . . . 6 ⊢ (𝑦 = 𝐴 → (𝐹‘𝑦) = (𝐹‘𝐴)) | |
14 | 13 | oveq2d 6565 | . . . . 5 ⊢ (𝑦 = 𝐴 → (𝑓 ·ih (𝐹‘𝑦)) = (𝑓 ·ih (𝐹‘𝐴))) |
15 | 12, 14 | eqeq12d 2625 | . . . 4 ⊢ (𝑦 = 𝐴 → (((𝑇‘𝑓) ·ih 𝑦) = (𝑓 ·ih (𝐹‘𝑦)) ↔ ((𝑇‘𝑓) ·ih 𝐴) = (𝑓 ·ih (𝐹‘𝐴)))) |
16 | 15 | ralbidv 2969 | . . 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 6515 | . . . . . . 7 ⊢ (℩𝑤 ∈ ℋ ∀𝑣 ∈ ℋ ((𝑇‘𝑣) ·ih 𝑦) = (𝑣 ·ih 𝑤)) ∈ V | |
19 | 17, 18 | eqeltri 2684 | . . . . . 6 ⊢ 𝐵 ∈ V |
20 | 5 | fvmpt2 6200 | . . . . . 6 ⊢ ((𝑦 ∈ ℋ ∧ 𝐵 ∈ V) → (𝐹‘𝑦) = 𝐵) |
21 | 19, 20 | mpan2 703 | . . . . 5 ⊢ (𝑦 ∈ ℋ → (𝐹‘𝑦) = 𝐵) |
22 | fveq2 6103 | . . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑓 → (𝑇‘𝑣) = (𝑇‘𝑓)) | |
23 | 22 | oveq1d 6564 | . . . . . . . . . . . 12 ⊢ (𝑣 = 𝑓 → ((𝑇‘𝑣) ·ih 𝑦) = ((𝑇‘𝑓) ·ih 𝑦)) |
24 | oveq1 6556 | . . . . . . . . . . . 12 ⊢ (𝑣 = 𝑓 → (𝑣 ·ih 𝑤) = (𝑓 ·ih 𝑤)) | |
25 | 23, 24 | eqeq12d 2625 | . . . . . . . . . . 11 ⊢ (𝑣 = 𝑓 → (((𝑇‘𝑣) ·ih 𝑦) = (𝑣 ·ih 𝑤) ↔ ((𝑇‘𝑓) ·ih 𝑦) = (𝑓 ·ih 𝑤))) |
26 | 25 | cbvralv 3147 | . . . . . . . . . 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 ∈ ConOp | |
30 | cnlnadjlem.3 | . . . . . . . . . . . 12 ⊢ 𝐺 = (𝑔 ∈ ℋ ↦ ((𝑇‘𝑔) ·ih 𝑦)) | |
31 | 28, 29, 30 | cnlnadjlem1 28310 | . . . . . . . . . . 11 ⊢ (𝑓 ∈ ℋ → (𝐺‘𝑓) = ((𝑇‘𝑓) ·ih 𝑦)) |
32 | 31 | eqeq1d 2612 | . . . . . . . . . 10 ⊢ (𝑓 ∈ ℋ → ((𝐺‘𝑓) = (𝑓 ·ih 𝑤) ↔ ((𝑇‘𝑓) ·ih 𝑦) = (𝑓 ·ih 𝑤))) |
33 | 32 | ralbiia 2962 | . . . . . . . . 9 ⊢ (∀𝑓 ∈ ℋ (𝐺‘𝑓) = (𝑓 ·ih 𝑤) ↔ ∀𝑓 ∈ ℋ ((𝑇‘𝑓) ·ih 𝑦) = (𝑓 ·ih 𝑤)) |
34 | 27, 33 | syl6bbr 277 | . . . . . . . 8 ⊢ (𝑤 ∈ ℋ → (∀𝑣 ∈ ℋ ((𝑇‘𝑣) ·ih 𝑦) = (𝑣 ·ih 𝑤) ↔ ∀𝑓 ∈ ℋ (𝐺‘𝑓) = (𝑓 ·ih 𝑤))) |
35 | 34 | riotabiia 6528 | . . . . . . 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 2632 | . . . . . 6 ⊢ 𝐵 = (℩𝑤 ∈ ℋ ∀𝑓 ∈ ℋ (𝐺‘𝑓) = (𝑓 ·ih 𝑤)) |
37 | 28, 29, 30 | cnlnadjlem2 28311 | . . . . . . . 8 ⊢ (𝑦 ∈ ℋ → (𝐺 ∈ LinFn ∧ 𝐺 ∈ ConFn)) |
38 | elin 3758 | . . . . . . . 8 ⊢ (𝐺 ∈ (LinFn ∩ ConFn) ↔ (𝐺 ∈ LinFn ∧ 𝐺 ∈ ConFn)) | |
39 | 37, 38 | sylibr 223 | . . . . . . 7 ⊢ (𝑦 ∈ ℋ → 𝐺 ∈ (LinFn ∩ ConFn)) |
40 | riesz4 28307 | . . . . . . 7 ⊢ (𝐺 ∈ (LinFn ∩ ConFn) → ∃!𝑤 ∈ ℋ ∀𝑓 ∈ ℋ (𝐺‘𝑓) = (𝑓 ·ih 𝑤)) | |
41 | riotacl2 6524 | . . . . . . 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 | syl5eqel 2692 | . . . . 5 ⊢ (𝑦 ∈ ℋ → 𝐵 ∈ {𝑤 ∈ ℋ ∣ ∀𝑓 ∈ ℋ (𝐺‘𝑓) = (𝑓 ·ih 𝑤)}) |
44 | 21, 43 | eqeltrd 2688 | . . . 4 ⊢ (𝑦 ∈ ℋ → (𝐹‘𝑦) ∈ {𝑤 ∈ ℋ ∣ ∀𝑓 ∈ ℋ (𝐺‘𝑓) = (𝑓 ·ih 𝑤)}) |
45 | oveq2 6557 | . . . . . . . . 9 ⊢ (𝑤 = (𝐹‘𝑦) → (𝑓 ·ih 𝑤) = (𝑓 ·ih (𝐹‘𝑦))) | |
46 | 45 | eqeq2d 2620 | . . . . . . . 8 ⊢ (𝑤 = (𝐹‘𝑦) → (((𝑇‘𝑓) ·ih 𝑦) = (𝑓 ·ih 𝑤) ↔ ((𝑇‘𝑓) ·ih 𝑦) = (𝑓 ·ih (𝐹‘𝑦)))) |
47 | 46 | ralbidv 2969 | . . . . . . 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 271 | . . . . . 6 ⊢ (𝑤 = (𝐹‘𝑦) → (∀𝑓 ∈ ℋ (𝐺‘𝑓) = (𝑓 ·ih 𝑤) ↔ ∀𝑓 ∈ ℋ ((𝑇‘𝑓) ·ih 𝑦) = (𝑓 ·ih (𝐹‘𝑦)))) |
49 | 48 | elrab 3331 | . . . . 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 479 | . . . 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 3244 | . 2 ⊢ (𝐴 ∈ ℋ → ∀𝑓 ∈ ℋ ((𝑇‘𝑓) ·ih 𝐴) = (𝑓 ·ih (𝐹‘𝐴))) |
53 | fveq2 6103 | . . . . 5 ⊢ (𝑓 = 𝐶 → (𝑇‘𝑓) = (𝑇‘𝐶)) | |
54 | 53 | oveq1d 6564 | . . . 4 ⊢ (𝑓 = 𝐶 → ((𝑇‘𝑓) ·ih 𝐴) = ((𝑇‘𝐶) ·ih 𝐴)) |
55 | oveq1 6556 | . . . 4 ⊢ (𝑓 = 𝐶 → (𝑓 ·ih (𝐹‘𝐴)) = (𝐶 ·ih (𝐹‘𝐴))) | |
56 | 54, 55 | eqeq12d 2625 | . . 3 ⊢ (𝑓 = 𝐶 → (((𝑇‘𝑓) ·ih 𝐴) = (𝑓 ·ih (𝐹‘𝐴)) ↔ ((𝑇‘𝐶) ·ih 𝐴) = (𝐶 ·ih (𝐹‘𝐴)))) |
57 | 56 | rspccva 3281 | . 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 487 | 1 ⊢ ((𝐴 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝑇‘𝐶) ·ih 𝐴) = (𝐶 ·ih (𝐹‘𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ∃!wreu 2898 {crab 2900 Vcvv 3173 ∩ cin 3539 ↦ cmpt 4643 ‘cfv 5804 ℩crio 6510 (class class class)co 6549 ℋchil 27160 ·ih csp 27163 ConOpccop 27187 LinOpclo 27188 ConFnccnfn 27194 LinFnclf 27195 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cc 9140 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 ax-mulf 9895 ax-hilex 27240 ax-hfvadd 27241 ax-hvcom 27242 ax-hvass 27243 ax-hv0cl 27244 ax-hvaddid 27245 ax-hfvmul 27246 ax-hvmulid 27247 ax-hvmulass 27248 ax-hvdistr1 27249 ax-hvdistr2 27250 ax-hvmul0 27251 ax-hfi 27320 ax-his1 27323 ax-his2 27324 ax-his3 27325 ax-his4 27326 ax-hcompl 27443 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-omul 7452 df-er 7629 df-map 7746 df-pm 7747 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-fi 8200 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-acn 8651 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-ioo 12050 df-ico 12052 df-icc 12053 df-fz 12198 df-fzo 12335 df-fl 12455 df-seq 12664 df-exp 12723 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-clim 14067 df-rlim 14068 df-sum 14265 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-hom 15793 df-cco 15794 df-rest 15906 df-topn 15907 df-0g 15925 df-gsum 15926 df-topgen 15927 df-pt 15928 df-prds 15931 df-xrs 15985 df-qtop 15990 df-imas 15991 df-xps 15993 df-mre 16069 df-mrc 16070 df-acs 16072 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-submnd 17159 df-mulg 17364 df-cntz 17573 df-cmn 18018 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-fbas 19564 df-fg 19565 df-cnfld 19568 df-top 20521 df-bases 20522 df-topon 20523 df-topsp 20524 df-cld 20633 df-ntr 20634 df-cls 20635 df-nei 20712 df-cn 20841 df-cnp 20842 df-lm 20843 df-t1 20928 df-haus 20929 df-tx 21175 df-hmeo 21368 df-fil 21460 df-fm 21552 df-flim 21553 df-flf 21554 df-xms 21935 df-ms 21936 df-tms 21937 df-cfil 22861 df-cau 22862 df-cmet 22863 df-grpo 26731 df-gid 26732 df-ginv 26733 df-gdiv 26734 df-ablo 26783 df-vc 26798 df-nv 26831 df-va 26834 df-ba 26835 df-sm 26836 df-0v 26837 df-vs 26838 df-nmcv 26839 df-ims 26840 df-dip 26940 df-ssp 26961 df-ph 27052 df-cbn 27103 df-hnorm 27209 df-hba 27210 df-hvsub 27212 df-hlim 27213 df-hcau 27214 df-sh 27448 df-ch 27462 df-oc 27493 df-ch0 27494 df-nmop 28082 df-cnop 28083 df-lnop 28084 df-nmfn 28088 df-nlfn 28089 df-cnfn 28090 df-lnfn 28091 |
This theorem is referenced by: cnlnadjlem6 28315 cnlnadjlem7 28316 cnlnadjlem9 28318 |
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