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Theorem ocv2ss 16589
Description: Orthocomplements reverse subset inclusion. (Contributed by Mario Carneiro, 13-Oct-2015.)
Hypothesis
Ref Expression
ocv2ss.o  |-  ._|_  =  ( ocv `  W )
Assertion
Ref Expression
ocv2ss  |-  ( T 
C_  S  ->  (  ._|_  `  S )  C_  (  ._|_  `  T )
)

Proof of Theorem ocv2ss
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sstr2 3199 . . . 4  |-  ( T 
C_  S  ->  ( S  C_  ( Base `  W
)  ->  T  C_  ( Base `  W ) ) )
2 idd 21 . . . 4  |-  ( T 
C_  S  ->  (
x  e.  ( Base `  W )  ->  x  e.  ( Base `  W
) ) )
3 ssralv 3250 . . . 4  |-  ( T 
C_  S  ->  ( A. y  e.  S  ( x ( .i
`  W ) y )  =  ( 0g
`  (Scalar `  W )
)  ->  A. y  e.  T  ( x
( .i `  W
) y )  =  ( 0g `  (Scalar `  W ) ) ) )
41, 2, 33anim123d 1259 . . 3  |-  ( T 
C_  S  ->  (
( S  C_  ( Base `  W )  /\  x  e.  ( Base `  W )  /\  A. y  e.  S  (
x ( .i `  W ) y )  =  ( 0g `  (Scalar `  W ) ) )  ->  ( T  C_  ( Base `  W
)  /\  x  e.  ( Base `  W )  /\  A. y  e.  T  ( x ( .i
`  W ) y )  =  ( 0g
`  (Scalar `  W )
) ) ) )
5 eqid 2296 . . . 4  |-  ( Base `  W )  =  (
Base `  W )
6 eqid 2296 . . . 4  |-  ( .i
`  W )  =  ( .i `  W
)
7 eqid 2296 . . . 4  |-  (Scalar `  W )  =  (Scalar `  W )
8 eqid 2296 . . . 4  |-  ( 0g
`  (Scalar `  W )
)  =  ( 0g
`  (Scalar `  W )
)
9 ocv2ss.o . . . 4  |-  ._|_  =  ( ocv `  W )
105, 6, 7, 8, 9elocv 16584 . . 3  |-  ( x  e.  (  ._|_  `  S
)  <->  ( S  C_  ( Base `  W )  /\  x  e.  ( Base `  W )  /\  A. y  e.  S  ( x ( .i `  W ) y )  =  ( 0g `  (Scalar `  W ) ) ) )
115, 6, 7, 8, 9elocv 16584 . . 3  |-  ( x  e.  (  ._|_  `  T
)  <->  ( T  C_  ( Base `  W )  /\  x  e.  ( Base `  W )  /\  A. y  e.  T  ( x ( .i `  W ) y )  =  ( 0g `  (Scalar `  W ) ) ) )
124, 10, 113imtr4g 261 . 2  |-  ( T 
C_  S  ->  (
x  e.  (  ._|_  `  S )  ->  x  e.  (  ._|_  `  T
) ) )
1312ssrdv 3198 1  |-  ( T 
C_  S  ->  (  ._|_  `  S )  C_  (  ._|_  `  T )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556    C_ wss 3165   ` cfv 5271  (class class class)co 5874   Basecbs 13164  Scalarcsca 13227   .icip 13229   0gc0g 13416   ocvcocv 16576
This theorem is referenced by:  ocvsscon  16591  ocvlsp  16592  ocvcss  16603  cssmre  16609  mrccss  16610  clsocv  18693
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5877  df-ocv 16579
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