Theorem obsip 20065

Description: The inner product of two elements of an orthonormal basis. (Contributed by Mario Carneiro, 23-Oct-2015.)

Hypotheses

Ref	Expression
isobs.v	|- V = ( Base ` W )
isobs.h	|- ., = ( .i ` W )
isobs.f	|- F = (Scalar ` W )
isobs.u	|- .1. = ( 1r ` F )
isobs.z	|- .0. = ( 0g ` F )

Assertion

Ref	Expression
obsip	|- ( ( B e. (OBasis ` W ) /\ P e. B /\ Q e. B ) -> ( P ., Q ) = if ( P = Q , .1. , .0. ) )

Proof of Theorem obsip

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isobs.v	. . . . . 6 |- V = ( Base ` W )
2		isobs.h	. . . . . 6 |- ., = ( .i ` W )
3		isobs.f	. . . . . 6 |- F = (Scalar ` W )
4		isobs.u	. . . . . 6 |- .1. = ( 1r ` F )
5		isobs.z	. . . . . 6 |- .0. = ( 0g ` F )
6		eqid 2622	. . . . . 6 |- ( ocv ` W ) = ( ocv ` W )
7		eqid 2622	. . . . . 6 |- ( 0g ` W ) = ( 0g ` W )
8	1, 2, 3, 4, 5, 6, 7	isobs 20064	. . . . 5 |- ( B e. (OBasis ` W ) <-> ( W e. PreHil /\ B C_ V /\ ( A. x e. B A. y e. B ( x ., y ) = if ( x = y , .1. , .0. ) /\ ( ( ocv ` W ) ` B ) = { ( 0g ` W ) } ) ) )
9	8	simp3bi 1078	. . . 4 |- ( B e. (OBasis ` W ) -> ( A. x e. B A. y e. B ( x ., y ) = if ( x = y , .1. , .0. ) /\ ( ( ocv ` W ) ` B ) = { ( 0g ` W ) } ) )
10	9	simpld 475	. . 3 |- ( B e. (OBasis ` W ) -> A. x e. B A. y e. B ( x ., y ) = if ( x = y , .1. , .0. ) )
11		oveq1 6657	. . . . 5 |- ( x = P -> ( x ., y ) = ( P ., y ) )
12		eqeq1 2626	. . . . . 6 |- ( x = P -> ( x = y <-> P = y ) )
13	12	ifbid 4108	. . . . 5 |- ( x = P -> if ( x = y , .1. , .0. ) = if ( P = y , .1. , .0. ) )
14	11, 13	eqeq12d 2637	. . . 4 |- ( x = P -> ( ( x ., y ) = if ( x = y , .1. , .0. ) <-> ( P ., y ) = if ( P = y , .1. , .0. ) ) )
15		oveq2 6658	. . . . 5 |- ( y = Q -> ( P ., y ) = ( P ., Q ) )
16		eqeq2 2633	. . . . . 6 |- ( y = Q -> ( P = y <-> P = Q ) )
17	16	ifbid 4108	. . . . 5 |- ( y = Q -> if ( P = y , .1. , .0. ) = if ( P = Q , .1. , .0. ) )
18	15, 17	eqeq12d 2637	. . . 4 |- ( y = Q -> ( ( P ., y ) = if ( P = y , .1. , .0. ) <-> ( P ., Q ) = if ( P = Q , .1. , .0. ) ) )
19	14, 18	rspc2v 3322	. . 3 |- ( ( P e. B /\ Q e. B ) -> ( A. x e. B A. y e. B ( x ., y ) = if ( x = y , .1. , .0. ) -> ( P ., Q ) = if ( P = Q , .1. , .0. ) ) )
20	10, 19	syl5com 31	. 2 |- ( B e. (OBasis ` W ) -> ( ( P e. B /\ Q e. B ) -> ( P ., Q ) = if ( P = Q , .1. , .0. ) ) )
21	20	3impib 1262	1 |- ( ( B e. (OBasis ` W ) /\ P e. B /\ Q e. B ) -> ( P ., Q ) = if ( P = Q , .1. , .0. ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   C_ wss 3574   ifcif 4086   {csn 4177   ` cfv 5888  (class class class)co 6650   Basecbs 15857  Scalarcsca 15944   .icip 15946   0gc0g 16100   1rcur 18501   PreHilcphl 19969   ocvcocv 20004  OBasiscobs 20046

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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

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-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  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-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-obs 20049

This theorem is referenced by:  obsipid  20066  obselocv  20072