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Mirrors > Home > HSE Home > Th. List > cnlnadjlem1 | Structured version Visualization version GIF version |
Description: Lemma for cnlnadji 31067 (Theorem 3.10 of [Beran] p. 104: every continuous linear operator has an adjoint). The value of the auxiliary functional ๐บ. (Contributed by NM, 16-Feb-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
cnlnadjlem.1 | โข ๐ โ LinOp |
cnlnadjlem.2 | โข ๐ โ ContOp |
cnlnadjlem.3 | โข ๐บ = (๐ โ โ โฆ ((๐โ๐) ยทih ๐ฆ)) |
Ref | Expression |
---|---|
cnlnadjlem1 | โข (๐ด โ โ โ (๐บโ๐ด) = ((๐โ๐ด) ยทih ๐ฆ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6846 | . . 3 โข (๐ = ๐ด โ (๐โ๐) = (๐โ๐ด)) | |
2 | 1 | oveq1d 7376 | . 2 โข (๐ = ๐ด โ ((๐โ๐) ยทih ๐ฆ) = ((๐โ๐ด) ยทih ๐ฆ)) |
3 | cnlnadjlem.3 | . 2 โข ๐บ = (๐ โ โ โฆ ((๐โ๐) ยทih ๐ฆ)) | |
4 | ovex 7394 | . 2 โข ((๐โ๐ด) ยทih ๐ฆ) โ V | |
5 | 2, 3, 4 | fvmpt 6952 | 1 โข (๐ด โ โ โ (๐บโ๐ด) = ((๐โ๐ด) ยทih ๐ฆ)) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 = wceq 1542 โ wcel 2107 โฆ cmpt 5192 โcfv 6500 (class class class)co 7361 โchba 29910 ยทih csp 29913 ContOpccop 29937 LinOpclo 29938 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pr 5388 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-iota 6452 df-fun 6502 df-fv 6508 df-ov 7364 |
This theorem is referenced by: cnlnadjlem2 31059 cnlnadjlem3 31060 cnlnadjlem5 31062 |
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