Theorem shss 29473

Description: A subspace is a subset of Hilbert space. (Contributed by NM, 9-Oct-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
shss	⊢ (𝐻 ∈ Sℋ → 𝐻 ⊆ ℋ)

Proof of Theorem shss

Step	Hyp	Ref	Expression
1		issh 29471	. . 3 ⊢ (𝐻 ∈ Sℋ ↔ ((𝐻 ⊆ ℋ ∧ 0ℎ ∈ 𝐻) ∧ (( +ℎ “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( ·ℎ “ (ℂ × 𝐻)) ⊆ 𝐻)))
2	1	simplbi 497	. 2 ⊢ (𝐻 ∈ Sℋ → (𝐻 ⊆ ℋ ∧ 0ℎ ∈ 𝐻))
3	2	simpld 494	1 ⊢ (𝐻 ∈ Sℋ → 𝐻 ⊆ ℋ)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2108   ⊆ wss 3883   × cxp 5578   “ cima 5583  ℂcc 10800   ℋchba 29182   +_ℎ cva 29183   ·_ℎ csm 29184  0_ℎc0v 29187   S_ℋ csh 29191

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709  ax-sep 5218  ax-hilex 29262

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-xp 5586  df-cnv 5588  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-sh 29470

This theorem is referenced by:  shel  29474  shex  29475  shssii  29476  shsubcl  29483  chss  29492  shsspwh  29509  hhsssh  29532  shocel  29545  shocsh  29547  ocss  29548  shocss  29549  shocorth  29555  shococss  29557  shorth  29558  shoccl  29568  shsel  29577  shintcli  29592  spanid  29610  shjval  29614  shjcl  29619  shlej1  29623  shlub  29677  chscllem2  29901  chscllem4  29903