Theorem cphsca 22979

Description: A subcomplex pre-Hilbert space is a vector space over a subfield of ℂ_fld. (Contributed by Mario Carneiro, 8-Oct-2015.)

Hypotheses

Ref	Expression
cphsca.f	|- F = (Scalar ` W )
cphsca.k	|- K = ( Base ` F )

Assertion

Ref	Expression
cphsca	|- ( W e. CPreHil -> F = (ℂfld ↾s K ) )

Proof of Theorem cphsca

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		cphsca.f	. . . 4 |- F = (Scalar ` W )
5		cphsca.k	. . . 4 |- K = ( Base ` F )
6	1, 2, 3, 4, 5	iscph 22970	. . 3 |- ( W e. CPreHil <-> ( ( W e. PreHil /\ W e. NrmMod /\ F = (ℂfld ↾s K ) ) /\ ( sqr " ( K i^i ( 0 [,) +oo ) ) ) C_ K /\ ( norm ` W ) = ( x e. ( Base ` W ) |-> ( sqr ` ( x ( .i ` W ) x ) ) ) ) )
7	6	simp1bi 1076	. 2 |- ( W e. CPreHil -> ( W e. PreHil /\ W e. NrmMod /\ F = (ℂfld ↾s K ) ) )
8	7	simp3d 1075	1 |- ( W e. CPreHil -> F = (ℂfld ↾s K ) )

Syntax hints:   -> wi 4   /\ w3a 1037   = wceq 1483   e. wcel 1990   i^i cin 3573   C_ wss 3574   |-> cmpt 4729   "cima 5117   ` cfv 5888  (class class class)co 6650   0cc0 9936   +oocpnf 10071   [,)cico 12177   sqrcsqrt 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:  cphsubrg  22980  cphreccl  22981  cphcjcl  22983  cphqss  22988  cphclm  22989  ipcau  23037  hlprlem  23163  ishl2  23166