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Mirrors > Home > HSE Home > Th. List > shelii | Structured version Visualization version GIF version |
Description: A member of a subspace of a Hilbert space is a vector. (Contributed by NM, 6-Oct-1999.) (New usage is discouraged.) |
Ref | Expression |
---|---|
shssi.1 | ⊢ 𝐻 ∈ Sℋ |
sheli.1 | ⊢ 𝐴 ∈ 𝐻 |
Ref | Expression |
---|---|
shelii | ⊢ 𝐴 ∈ ℋ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | shssi.1 | . . 3 ⊢ 𝐻 ∈ Sℋ | |
2 | 1 | shssii 28070 | . 2 ⊢ 𝐻 ⊆ ℋ |
3 | sheli.1 | . 2 ⊢ 𝐴 ∈ 𝐻 | |
4 | 2, 3 | sselii 3600 | 1 ⊢ 𝐴 ∈ ℋ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 1990 ℋchil 27776 Sℋ csh 27785 |
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-hilex 27856 |
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-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-xp 5120 df-cnv 5122 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-sh 28064 |
This theorem is referenced by: omlsilem 28261 |
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