Theorem hlcph 23160

Description: Every subcomplex Hilbert space is a subcomplex pre-Hilbert space. (Contributed by Mario Carneiro, 15-Oct-2015.)

Assertion

Ref	Expression
hlcph	|- ( W e. CHil -> W e. CPreHil )

Proof of Theorem hlcph

Step	Hyp	Ref	Expression
1		ishl 23158	. 2 |- ( W e. CHil <-> ( W e. Ban /\ W e. CPreHil ) )
2	1	simprbi 480	1 |- ( W e. CHil -> W e. CPreHil )

Syntax hints:   -> wi 4   e. wcel 1990   CPreHilccph 22966  Bancbn 23130   CHilchl 23131

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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-v 3202  df-in 3581  df-hl 23134

This theorem is referenced by:  hlphl  23161  hlprlem  23163  pjthlem1  23208  pjthlem2  23209  cldcss  23212