Theorem obsipid 20066

Description: A basis element has unit length. (Contributed by Mario Carneiro, 23-Oct-2015.)

Hypotheses

Ref	Expression
obsipid.h	|- ., = ( .i ` W )
obsipid.f	|- F = (Scalar ` W )
obsipid.u	|- .1. = ( 1r ` F )

Assertion

Ref	Expression
obsipid	|- ( ( B e. (OBasis ` W ) /\ A e. B ) -> ( A ., A ) = .1. )

Proof of Theorem obsipid

Step	Hyp	Ref	Expression
1		eqid 2622	. . . 4 |- ( Base ` W ) = ( Base ` W )
2		obsipid.h	. . . 4 |- ., = ( .i ` W )
3		obsipid.f	. . . 4 |- F = (Scalar ` W )
4		obsipid.u	. . . 4 |- .1. = ( 1r ` F )
5		eqid 2622	. . . 4 |- ( 0g ` F ) = ( 0g ` F )
6	1, 2, 3, 4, 5	obsip 20065	. . 3 |- ( ( B e. (OBasis ` W ) /\ A e. B /\ A e. B ) -> ( A ., A ) = if ( A = A , .1. , ( 0g ` F ) ) )
7	6	3anidm23 1385	. 2 |- ( ( B e. (OBasis ` W ) /\ A e. B ) -> ( A ., A ) = if ( A = A , .1. , ( 0g ` F ) ) )
8		eqid 2622	. . 3 |- A = A
9	8	iftruei 4093	. 2 |- if ( A = A , .1. , ( 0g ` F ) ) = .1.
10	7, 9	syl6eq 2672	1 |- ( ( B e. (OBasis ` W ) /\ A e. B ) -> ( A ., A ) = .1. )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   ifcif 4086   ` cfv 5888  (class class class)co 6650   Basecbs 15857  Scalarcsca 15944   .icip 15946   0gc0g 16100   1rcur 18501  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:  obsne0  20069