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Mirrors > Home > HSE Home > Th. List > sheli | 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 |
Ref | Expression |
---|---|
sheli | ⊢ (𝐴 ∈ 𝐻 → 𝐴 ∈ ℋ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | shssi.1 | . . 3 ⊢ 𝐻 ∈ Sℋ | |
2 | 1 | shssii 28990 | . 2 ⊢ 𝐻 ⊆ ℋ |
3 | 2 | sseli 3963 | 1 ⊢ (𝐴 ∈ 𝐻 → 𝐴 ∈ ℋ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 ℋchba 28696 Sℋ csh 28705 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-hilex 28776 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-xp 5561 df-cnv 5563 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-sh 28984 |
This theorem is referenced by: norm1exi 29027 hhssabloi 29039 hhssnv 29041 shscli 29094 shunssi 29145 shmodsi 29166 omlsii 29180 5oalem1 29431 5oalem2 29432 5oalem3 29433 5oalem5 29435 imaelshi 29835 pjimai 29953 shatomici 30135 shatomistici 30138 cdjreui 30209 cdj1i 30210 cdj3lem1 30211 cdj3lem2b 30214 cdj3lem3 30215 cdj3lem3b 30217 cdj3i 30218 |
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