Theorem cphphl 22971

Description: A subcomplex pre-Hilbert space is a pre-Hilbert space. (Contributed by Mario Carneiro, 7-Oct-2015.)

Assertion

Ref	Expression
cphphl	⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ PreHil)

Proof of Theorem cphphl

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊)
2		eqid 2622	. . . 4 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝑊)
3		eqid 2622	. . . 4 ⊢ (norm‘𝑊) = (norm‘𝑊)
4		eqid 2622	. . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
5		eqid 2622	. . . 4 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
6	1, 2, 3, 4, 5	iscph 22970	. . 3 ⊢ (𝑊 ∈ ℂPreHil ↔ ((𝑊 ∈ PreHil ∧ 𝑊 ∈ NrmMod ∧ (Scalar‘𝑊) = (ℂfld ↾s (Base‘(Scalar‘𝑊)))) ∧ (√ “ ((Base‘(Scalar‘𝑊)) ∩ (0[,)+∞))) ⊆ (Base‘(Scalar‘𝑊)) ∧ (norm‘𝑊) = (𝑥 ∈ (Base‘𝑊) ↦ (√‘(𝑥(·𝑖‘𝑊)𝑥)))))
7	6	simp1bi 1076	. 2 ⊢ (𝑊 ∈ ℂPreHil → (𝑊 ∈ PreHil ∧ 𝑊 ∈ NrmMod ∧ (Scalar‘𝑊) = (ℂfld ↾s (Base‘(Scalar‘𝑊)))))
8	7	simp1d 1073	1 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ PreHil)

Syntax hints:   → wi 4   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ∩ cin 3573   ⊆ wss 3574   ↦ cmpt 4729   “ cima 5117  ‘cfv 5888  (class class class)co 6650  0cc0 9936  +∞cpnf 10071  [,)cico 12177  √csqrt 13973  Basecbs 15857   ↾_s cress 15858  Scalarcsca 15944  ·_𝑖cip 15946  ℂ_fldccnfld 19746  PreHilcphl 19969  normcnm 22381  NrmModcnlm 22385  ℂPreHilccph 22966

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-nul 4789

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-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fv 5896  df-ov 6653  df-cph 22968

This theorem is referenced by:  cphlvec  22975  cphcjcl  22983  cphipcl  22991  cphnmf  22995  cphipcj  22999  cphorthcom  23001  cphip0l  23002  cphip0r  23003  cphipeq0  23004  cphdir  23005  cphdi  23006  cph2di  23007  cphsubdir  23008  cphsubdi  23009  cph2subdi  23010  cphass  23011  cphassr  23012  ipcau  23037  nmparlem  23038  ipcn  23045  hlphl  23161  pjthlem2  23209