Theorem cphnmfval 22992

Description: The value of the norm in a subcomplex pre-Hilbert space is the square root of the inner product of a vector with itself. (Contributed by Mario Carneiro, 7-Oct-2015.)

Hypotheses

Ref	Expression
nmsq.v	|- V = ( Base ` W )
nmsq.h	|- ., = ( .i ` W )
nmsq.n	|- N = ( norm ` W )

Assertion

Ref	Expression
cphnmfval	|- ( W e. CPreHil -> N = ( x e. V |-> ( sqr ` ( x ., x ) ) ) )

Distinct variable groups:   x, .,   x, V   x, W

Allowed substitution hint:   N( x)

Proof of Theorem cphnmfval

Step	Hyp	Ref	Expression
1		nmsq.v	. . 3 |- V = ( Base ` W )
2		nmsq.h	. . 3 |- ., = ( .i ` W )
3		nmsq.n	. . 3 |- N = ( norm ` W )
4		eqid 2622	. . 3 |- (Scalar ` W ) = (Scalar ` W )
5		eqid 2622	. . 3 |- ( Base ` (Scalar ` W ) ) = ( Base ` (Scalar ` W ) )
6	1, 2, 3, 4, 5	iscph 22970	. 2 |- ( W e. CPreHil <-> ( ( W e. PreHil /\ W e. NrmMod /\ (Scalar ` W ) = (ℂfld ↾s ( Base ` (Scalar ` W ) ) ) ) /\ ( sqr " ( ( Base ` (Scalar ` W ) ) i^i ( 0 [,) +oo ) ) ) C_ ( Base ` (Scalar ` W ) ) /\ N = ( x e. V |-> ( sqr ` ( x ., x ) ) ) ) )
7	6	simp3bi 1078	1 |- ( W e. CPreHil -> N = ( x e. V |-> ( sqr ` ( x ., x ) ) ) )

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:  cphnm  22993  cphnmf  22995  cphtchnm  23029