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Mirrors > Home > HSE Home > Th. List > cnlnadjlem1 | Structured version Visualization version GIF version |
Description: Lemma for cnlnadji 28935 (Theorem 3.10 of [Beran] p. 104: every continuous linear operator has an adjoint). The value of the auxiliary functional 𝐺. (Contributed by NM, 16-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)) |
Ref | Expression |
---|---|
cnlnadjlem1 | ⊢ (𝐴 ∈ ℋ → (𝐺‘𝐴) = ((𝑇‘𝐴) ·ih 𝑦)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6191 | . . 3 ⊢ (𝑔 = 𝐴 → (𝑇‘𝑔) = (𝑇‘𝐴)) | |
2 | 1 | oveq1d 6665 | . 2 ⊢ (𝑔 = 𝐴 → ((𝑇‘𝑔) ·ih 𝑦) = ((𝑇‘𝐴) ·ih 𝑦)) |
3 | cnlnadjlem.3 | . 2 ⊢ 𝐺 = (𝑔 ∈ ℋ ↦ ((𝑇‘𝑔) ·ih 𝑦)) | |
4 | ovex 6678 | . 2 ⊢ ((𝑇‘𝐴) ·ih 𝑦) ∈ V | |
5 | 2, 3, 4 | fvmpt 6282 | 1 ⊢ (𝐴 ∈ ℋ → (𝐺‘𝐴) = ((𝑇‘𝐴) ·ih 𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 ↦ cmpt 4729 ‘cfv 5888 (class class class)co 6650 ℋchil 27776 ·ih csp 27779 ContOpccop 27803 LinOpclo 27804 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 df-ov 6653 |
This theorem is referenced by: cnlnadjlem2 28927 cnlnadjlem3 28928 cnlnadjlem5 28930 |
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