Theorem ocvi 20013

Description: Property of a member of the orthocomplement of a subset. (Contributed by Mario Carneiro, 13-Oct-2015.)

Hypotheses

Ref	Expression
ocvfval.v	|- V = ( Base ` W )
ocvfval.i	|- ., = ( .i ` W )
ocvfval.f	|- F = (Scalar ` W )
ocvfval.z	|- .0. = ( 0g ` F )
ocvfval.o	|- ._|_ = ( ocv ` W )

Assertion

Ref	Expression
ocvi	|- ( ( A e. ( ._|_ ` S ) /\ B e. S ) -> ( A ., B ) = .0. )

Proof of Theorem ocvi

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ocvfval.v	. . . 4 |- V = ( Base ` W )
2		ocvfval.i	. . . 4 |- ., = ( .i ` W )
3		ocvfval.f	. . . 4 |- F = (Scalar ` W )
4		ocvfval.z	. . . 4 |- .0. = ( 0g ` F )
5		ocvfval.o	. . . 4 |- ._|_ = ( ocv ` W )
6	1, 2, 3, 4, 5	elocv 20012	. . 3 |- ( A e. ( ._|_ ` S ) <-> ( S C_ V /\ A e. V /\ A. x e. S ( A ., x ) = .0. ) )
7	6	simp3bi 1078	. 2 |- ( A e. ( ._|_ ` S ) -> A. x e. S ( A ., x ) = .0. )
8		oveq2 6658	. . . 4 |- ( x = B -> ( A ., x ) = ( A ., B ) )
9	8	eqeq1d 2624	. . 3 |- ( x = B -> ( ( A ., x ) = .0. <-> ( A ., B ) = .0. ) )
10	9	rspccva 3308	. 2 |- ( ( A. x e. S ( A ., x ) = .0. /\ B e. S ) -> ( A ., B ) = .0. )
11	7, 10	sylan 488	1 |- ( ( A e. ( ._|_ ` S ) /\ B e. S ) -> ( A ., B ) = .0. )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   C_ wss 3574   ` cfv 5888  (class class class)co 6650   Basecbs 15857  Scalarcsca 15944   .icip 15946   0gc0g 16100   ocvcocv 20004

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  ax-un 6949

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-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-ocv 20007

This theorem is referenced by:  ocvocv  20015  ocvlss  20016  ocvin  20018  lsmcss  20036  clsocv  23049