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Theorem ocvlss 18467
Description: The orthocomplement of a subset is a linear subspace of the pre-Hilbert space. (Contributed by Mario Carneiro, 13-Oct-2015.)
Hypotheses
Ref Expression
ocvss.v  |-  V  =  ( Base `  W
)
ocvss.o  |-  ._|_  =  ( ocv `  W )
ocvlss.l  |-  L  =  ( LSubSp `  W )
Assertion
Ref Expression
ocvlss  |-  ( ( W  e.  PreHil  /\  S  C_  V )  ->  (  ._|_  `  S )  e.  L )

Proof of Theorem ocvlss
Dummy variables  x  r  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ocvss.v . . . 4  |-  V  =  ( Base `  W
)
2 ocvss.o . . . 4  |-  ._|_  =  ( ocv `  W )
31, 2ocvss 18465 . . 3  |-  (  ._|_  `  S )  C_  V
43a1i 11 . 2  |-  ( ( W  e.  PreHil  /\  S  C_  V )  ->  (  ._|_  `  S )  C_  V )
5 simpr 461 . . . 4  |-  ( ( W  e.  PreHil  /\  S  C_  V )  ->  S  C_  V )
6 phllmod 18429 . . . . . 6  |-  ( W  e.  PreHil  ->  W  e.  LMod )
76adantr 465 . . . . 5  |-  ( ( W  e.  PreHil  /\  S  C_  V )  ->  W  e.  LMod )
8 eqid 2467 . . . . . 6  |-  ( 0g
`  W )  =  ( 0g `  W
)
91, 8lmod0vcl 17321 . . . . 5  |-  ( W  e.  LMod  ->  ( 0g
`  W )  e.  V )
107, 9syl 16 . . . 4  |-  ( ( W  e.  PreHil  /\  S  C_  V )  ->  ( 0g `  W )  e.  V )
11 simpll 753 . . . . . 6  |-  ( ( ( W  e.  PreHil  /\  S  C_  V )  /\  x  e.  S
)  ->  W  e.  PreHil )
125sselda 3504 . . . . . 6  |-  ( ( ( W  e.  PreHil  /\  S  C_  V )  /\  x  e.  S
)  ->  x  e.  V )
13 eqid 2467 . . . . . . 7  |-  (Scalar `  W )  =  (Scalar `  W )
14 eqid 2467 . . . . . . 7  |-  ( .i
`  W )  =  ( .i `  W
)
15 eqid 2467 . . . . . . 7  |-  ( 0g
`  (Scalar `  W )
)  =  ( 0g
`  (Scalar `  W )
)
1613, 14, 1, 15, 8ip0l 18435 . . . . . 6  |-  ( ( W  e.  PreHil  /\  x  e.  V )  ->  (
( 0g `  W
) ( .i `  W ) x )  =  ( 0g `  (Scalar `  W ) ) )
1711, 12, 16syl2anc 661 . . . . 5  |-  ( ( ( W  e.  PreHil  /\  S  C_  V )  /\  x  e.  S
)  ->  ( ( 0g `  W ) ( .i `  W ) x )  =  ( 0g `  (Scalar `  W ) ) )
1817ralrimiva 2878 . . . 4  |-  ( ( W  e.  PreHil  /\  S  C_  V )  ->  A. x  e.  S  ( ( 0g `  W ) ( .i `  W ) x )  =  ( 0g `  (Scalar `  W ) ) )
191, 14, 13, 15, 2elocv 18463 . . . 4  |-  ( ( 0g `  W )  e.  (  ._|_  `  S
)  <->  ( S  C_  V  /\  ( 0g `  W )  e.  V  /\  A. x  e.  S  ( ( 0g `  W ) ( .i
`  W ) x )  =  ( 0g
`  (Scalar `  W )
) ) )
205, 10, 18, 19syl3anbrc 1180 . . 3  |-  ( ( W  e.  PreHil  /\  S  C_  V )  ->  ( 0g `  W )  e.  (  ._|_  `  S ) )
21 ne0i 3791 . . 3  |-  ( ( 0g `  W )  e.  (  ._|_  `  S
)  ->  (  ._|_  `  S )  =/=  (/) )
2220, 21syl 16 . 2  |-  ( ( W  e.  PreHil  /\  S  C_  V )  ->  (  ._|_  `  S )  =/=  (/) )
235adantr 465 . . . 4  |-  ( ( ( W  e.  PreHil  /\  S  C_  V )  /\  ( r  e.  (
Base `  (Scalar `  W
) )  /\  y  e.  (  ._|_  `  S
)  /\  z  e.  (  ._|_  `  S )
) )  ->  S  C_  V )
247adantr 465 . . . . 5  |-  ( ( ( W  e.  PreHil  /\  S  C_  V )  /\  ( r  e.  (
Base `  (Scalar `  W
) )  /\  y  e.  (  ._|_  `  S
)  /\  z  e.  (  ._|_  `  S )
) )  ->  W  e.  LMod )
25 simpr1 1002 . . . . . 6  |-  ( ( ( W  e.  PreHil  /\  S  C_  V )  /\  ( r  e.  (
Base `  (Scalar `  W
) )  /\  y  e.  (  ._|_  `  S
)  /\  z  e.  (  ._|_  `  S )
) )  ->  r  e.  ( Base `  (Scalar `  W ) ) )
26 simpr2 1003 . . . . . . 7  |-  ( ( ( W  e.  PreHil  /\  S  C_  V )  /\  ( r  e.  (
Base `  (Scalar `  W
) )  /\  y  e.  (  ._|_  `  S
)  /\  z  e.  (  ._|_  `  S )
) )  ->  y  e.  (  ._|_  `  S
) )
273, 26sseldi 3502 . . . . . 6  |-  ( ( ( W  e.  PreHil  /\  S  C_  V )  /\  ( r  e.  (
Base `  (Scalar `  W
) )  /\  y  e.  (  ._|_  `  S
)  /\  z  e.  (  ._|_  `  S )
) )  ->  y  e.  V )
28 eqid 2467 . . . . . . 7  |-  ( .s
`  W )  =  ( .s `  W
)
29 eqid 2467 . . . . . . 7  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
301, 13, 28, 29lmodvscl 17309 . . . . . 6  |-  ( ( W  e.  LMod  /\  r  e.  ( Base `  (Scalar `  W ) )  /\  y  e.  V )  ->  ( r ( .s
`  W ) y )  e.  V )
3124, 25, 27, 30syl3anc 1228 . . . . 5  |-  ( ( ( W  e.  PreHil  /\  S  C_  V )  /\  ( r  e.  (
Base `  (Scalar `  W
) )  /\  y  e.  (  ._|_  `  S
)  /\  z  e.  (  ._|_  `  S )
) )  ->  (
r ( .s `  W ) y )  e.  V )
32 simpr3 1004 . . . . . 6  |-  ( ( ( W  e.  PreHil  /\  S  C_  V )  /\  ( r  e.  (
Base `  (Scalar `  W
) )  /\  y  e.  (  ._|_  `  S
)  /\  z  e.  (  ._|_  `  S )
) )  ->  z  e.  (  ._|_  `  S
) )
333, 32sseldi 3502 . . . . 5  |-  ( ( ( W  e.  PreHil  /\  S  C_  V )  /\  ( r  e.  (
Base `  (Scalar `  W
) )  /\  y  e.  (  ._|_  `  S
)  /\  z  e.  (  ._|_  `  S )
) )  ->  z  e.  V )
34 eqid 2467 . . . . . 6  |-  ( +g  `  W )  =  ( +g  `  W )
351, 34lmodvacl 17306 . . . . 5  |-  ( ( W  e.  LMod  /\  (
r ( .s `  W ) y )  e.  V  /\  z  e.  V )  ->  (
( r ( .s
`  W ) y ) ( +g  `  W
) z )  e.  V )
3624, 31, 33, 35syl3anc 1228 . . . 4  |-  ( ( ( W  e.  PreHil  /\  S  C_  V )  /\  ( r  e.  (
Base `  (Scalar `  W
) )  /\  y  e.  (  ._|_  `  S
)  /\  z  e.  (  ._|_  `  S )
) )  ->  (
( r ( .s
`  W ) y ) ( +g  `  W
) z )  e.  V )
3711adantlr 714 . . . . . . 7  |-  ( ( ( ( W  e. 
PreHil  /\  S  C_  V
)  /\  ( r  e.  ( Base `  (Scalar `  W ) )  /\  y  e.  (  ._|_  `  S )  /\  z  e.  (  ._|_  `  S
) ) )  /\  x  e.  S )  ->  W  e.  PreHil )
3831adantr 465 . . . . . . 7  |-  ( ( ( ( W  e. 
PreHil  /\  S  C_  V
)  /\  ( r  e.  ( Base `  (Scalar `  W ) )  /\  y  e.  (  ._|_  `  S )  /\  z  e.  (  ._|_  `  S
) ) )  /\  x  e.  S )  ->  ( r ( .s
`  W ) y )  e.  V )
3933adantr 465 . . . . . . 7  |-  ( ( ( ( W  e. 
PreHil  /\  S  C_  V
)  /\  ( r  e.  ( Base `  (Scalar `  W ) )  /\  y  e.  (  ._|_  `  S )  /\  z  e.  (  ._|_  `  S
) ) )  /\  x  e.  S )  ->  z  e.  V )
4012adantlr 714 . . . . . . 7  |-  ( ( ( ( W  e. 
PreHil  /\  S  C_  V
)  /\  ( r  e.  ( Base `  (Scalar `  W ) )  /\  y  e.  (  ._|_  `  S )  /\  z  e.  (  ._|_  `  S
) ) )  /\  x  e.  S )  ->  x  e.  V )
41 eqid 2467 . . . . . . . 8  |-  ( +g  `  (Scalar `  W )
)  =  ( +g  `  (Scalar `  W )
)
4213, 14, 1, 34, 41ipdir 18438 . . . . . . 7  |-  ( ( W  e.  PreHil  /\  (
( r ( .s
`  W ) y )  e.  V  /\  z  e.  V  /\  x  e.  V )
)  ->  ( (
( r ( .s
`  W ) y ) ( +g  `  W
) z ) ( .i `  W ) x )  =  ( ( ( r ( .s `  W ) y ) ( .i
`  W ) x ) ( +g  `  (Scalar `  W ) ) ( z ( .i `  W ) x ) ) )
4337, 38, 39, 40, 42syl13anc 1230 . . . . . 6  |-  ( ( ( ( W  e. 
PreHil  /\  S  C_  V
)  /\  ( r  e.  ( Base `  (Scalar `  W ) )  /\  y  e.  (  ._|_  `  S )  /\  z  e.  (  ._|_  `  S
) ) )  /\  x  e.  S )  ->  ( ( ( r ( .s `  W
) y ) ( +g  `  W ) z ) ( .i
`  W ) x )  =  ( ( ( r ( .s
`  W ) y ) ( .i `  W ) x ) ( +g  `  (Scalar `  W ) ) ( z ( .i `  W ) x ) ) )
4425adantr 465 . . . . . . . . 9  |-  ( ( ( ( W  e. 
PreHil  /\  S  C_  V
)  /\  ( r  e.  ( Base `  (Scalar `  W ) )  /\  y  e.  (  ._|_  `  S )  /\  z  e.  (  ._|_  `  S
) ) )  /\  x  e.  S )  ->  r  e.  ( Base `  (Scalar `  W )
) )
4527adantr 465 . . . . . . . . 9  |-  ( ( ( ( W  e. 
PreHil  /\  S  C_  V
)  /\  ( r  e.  ( Base `  (Scalar `  W ) )  /\  y  e.  (  ._|_  `  S )  /\  z  e.  (  ._|_  `  S
) ) )  /\  x  e.  S )  ->  y  e.  V )
46 eqid 2467 . . . . . . . . . 10  |-  ( .r
`  (Scalar `  W )
)  =  ( .r
`  (Scalar `  W )
)
4713, 14, 1, 29, 28, 46ipass 18444 . . . . . . . . 9  |-  ( ( W  e.  PreHil  /\  (
r  e.  ( Base `  (Scalar `  W )
)  /\  y  e.  V  /\  x  e.  V
) )  ->  (
( r ( .s
`  W ) y ) ( .i `  W ) x )  =  ( r ( .r `  (Scalar `  W ) ) ( y ( .i `  W ) x ) ) )
4837, 44, 45, 40, 47syl13anc 1230 . . . . . . . 8  |-  ( ( ( ( W  e. 
PreHil  /\  S  C_  V
)  /\  ( r  e.  ( Base `  (Scalar `  W ) )  /\  y  e.  (  ._|_  `  S )  /\  z  e.  (  ._|_  `  S
) ) )  /\  x  e.  S )  ->  ( ( r ( .s `  W ) y ) ( .i
`  W ) x )  =  ( r ( .r `  (Scalar `  W ) ) ( y ( .i `  W ) x ) ) )
491, 14, 13, 15, 2ocvi 18464 . . . . . . . . . 10  |-  ( ( y  e.  (  ._|_  `  S )  /\  x  e.  S )  ->  (
y ( .i `  W ) x )  =  ( 0g `  (Scalar `  W ) ) )
5026, 49sylan 471 . . . . . . . . 9  |-  ( ( ( ( W  e. 
PreHil  /\  S  C_  V
)  /\  ( r  e.  ( Base `  (Scalar `  W ) )  /\  y  e.  (  ._|_  `  S )  /\  z  e.  (  ._|_  `  S
) ) )  /\  x  e.  S )  ->  ( y ( .i
`  W ) x )  =  ( 0g
`  (Scalar `  W )
) )
5150oveq2d 6298 . . . . . . . 8  |-  ( ( ( ( W  e. 
PreHil  /\  S  C_  V
)  /\  ( r  e.  ( Base `  (Scalar `  W ) )  /\  y  e.  (  ._|_  `  S )  /\  z  e.  (  ._|_  `  S
) ) )  /\  x  e.  S )  ->  ( r ( .r
`  (Scalar `  W )
) ( y ( .i `  W ) x ) )  =  ( r ( .r
`  (Scalar `  W )
) ( 0g `  (Scalar `  W ) ) ) )
5224adantr 465 . . . . . . . . . 10  |-  ( ( ( ( W  e. 
PreHil  /\  S  C_  V
)  /\  ( r  e.  ( Base `  (Scalar `  W ) )  /\  y  e.  (  ._|_  `  S )  /\  z  e.  (  ._|_  `  S
) ) )  /\  x  e.  S )  ->  W  e.  LMod )
5313lmodrng 17300 . . . . . . . . . 10  |-  ( W  e.  LMod  ->  (Scalar `  W )  e.  Ring )
5452, 53syl 16 . . . . . . . . 9  |-  ( ( ( ( W  e. 
PreHil  /\  S  C_  V
)  /\  ( r  e.  ( Base `  (Scalar `  W ) )  /\  y  e.  (  ._|_  `  S )  /\  z  e.  (  ._|_  `  S
) ) )  /\  x  e.  S )  ->  (Scalar `  W )  e.  Ring )
5529, 46, 15rngrz 17020 . . . . . . . . 9  |-  ( ( (Scalar `  W )  e.  Ring  /\  r  e.  ( Base `  (Scalar `  W
) ) )  -> 
( r ( .r
`  (Scalar `  W )
) ( 0g `  (Scalar `  W ) ) )  =  ( 0g
`  (Scalar `  W )
) )
5654, 44, 55syl2anc 661 . . . . . . . 8  |-  ( ( ( ( W  e. 
PreHil  /\  S  C_  V
)  /\  ( r  e.  ( Base `  (Scalar `  W ) )  /\  y  e.  (  ._|_  `  S )  /\  z  e.  (  ._|_  `  S
) ) )  /\  x  e.  S )  ->  ( r ( .r
`  (Scalar `  W )
) ( 0g `  (Scalar `  W ) ) )  =  ( 0g
`  (Scalar `  W )
) )
5748, 51, 563eqtrd 2512 . . . . . . 7  |-  ( ( ( ( W  e. 
PreHil  /\  S  C_  V
)  /\  ( r  e.  ( Base `  (Scalar `  W ) )  /\  y  e.  (  ._|_  `  S )  /\  z  e.  (  ._|_  `  S
) ) )  /\  x  e.  S )  ->  ( ( r ( .s `  W ) y ) ( .i
`  W ) x )  =  ( 0g
`  (Scalar `  W )
) )
581, 14, 13, 15, 2ocvi 18464 . . . . . . . 8  |-  ( ( z  e.  (  ._|_  `  S )  /\  x  e.  S )  ->  (
z ( .i `  W ) x )  =  ( 0g `  (Scalar `  W ) ) )
5932, 58sylan 471 . . . . . . 7  |-  ( ( ( ( W  e. 
PreHil  /\  S  C_  V
)  /\  ( r  e.  ( Base `  (Scalar `  W ) )  /\  y  e.  (  ._|_  `  S )  /\  z  e.  (  ._|_  `  S
) ) )  /\  x  e.  S )  ->  ( z ( .i
`  W ) x )  =  ( 0g
`  (Scalar `  W )
) )
6057, 59oveq12d 6300 . . . . . 6  |-  ( ( ( ( W  e. 
PreHil  /\  S  C_  V
)  /\  ( r  e.  ( Base `  (Scalar `  W ) )  /\  y  e.  (  ._|_  `  S )  /\  z  e.  (  ._|_  `  S
) ) )  /\  x  e.  S )  ->  ( ( ( r ( .s `  W
) y ) ( .i `  W ) x ) ( +g  `  (Scalar `  W )
) ( z ( .i `  W ) x ) )  =  ( ( 0g `  (Scalar `  W ) ) ( +g  `  (Scalar `  W ) ) ( 0g `  (Scalar `  W ) ) ) )
6113lmodfgrp 17301 . . . . . . 7  |-  ( W  e.  LMod  ->  (Scalar `  W )  e.  Grp )
6229, 15grpidcl 15876 . . . . . . . 8  |-  ( (Scalar `  W )  e.  Grp  ->  ( 0g `  (Scalar `  W ) )  e.  ( Base `  (Scalar `  W ) ) )
6329, 41, 15grplid 15878 . . . . . . . 8  |-  ( ( (Scalar `  W )  e.  Grp  /\  ( 0g
`  (Scalar `  W )
)  e.  ( Base `  (Scalar `  W )
) )  ->  (
( 0g `  (Scalar `  W ) ) ( +g  `  (Scalar `  W ) ) ( 0g `  (Scalar `  W ) ) )  =  ( 0g `  (Scalar `  W ) ) )
6462, 63mpdan 668 . . . . . . 7  |-  ( (Scalar `  W )  e.  Grp  ->  ( ( 0g `  (Scalar `  W ) ) ( +g  `  (Scalar `  W ) ) ( 0g `  (Scalar `  W ) ) )  =  ( 0g `  (Scalar `  W ) ) )
6552, 61, 643syl 20 . . . . . 6  |-  ( ( ( ( W  e. 
PreHil  /\  S  C_  V
)  /\  ( r  e.  ( Base `  (Scalar `  W ) )  /\  y  e.  (  ._|_  `  S )  /\  z  e.  (  ._|_  `  S
) ) )  /\  x  e.  S )  ->  ( ( 0g `  (Scalar `  W ) ) ( +g  `  (Scalar `  W ) ) ( 0g `  (Scalar `  W ) ) )  =  ( 0g `  (Scalar `  W ) ) )
6643, 60, 653eqtrd 2512 . . . . 5  |-  ( ( ( ( W  e. 
PreHil  /\  S  C_  V
)  /\  ( r  e.  ( Base `  (Scalar `  W ) )  /\  y  e.  (  ._|_  `  S )  /\  z  e.  (  ._|_  `  S
) ) )  /\  x  e.  S )  ->  ( ( ( r ( .s `  W
) y ) ( +g  `  W ) z ) ( .i
`  W ) x )  =  ( 0g
`  (Scalar `  W )
) )
6766ralrimiva 2878 . . . 4  |-  ( ( ( W  e.  PreHil  /\  S  C_  V )  /\  ( r  e.  (
Base `  (Scalar `  W
) )  /\  y  e.  (  ._|_  `  S
)  /\  z  e.  (  ._|_  `  S )
) )  ->  A. x  e.  S  ( (
( r ( .s
`  W ) y ) ( +g  `  W
) z ) ( .i `  W ) x )  =  ( 0g `  (Scalar `  W ) ) )
681, 14, 13, 15, 2elocv 18463 . . . 4  |-  ( ( ( r ( .s
`  W ) y ) ( +g  `  W
) z )  e.  (  ._|_  `  S )  <-> 
( S  C_  V  /\  ( ( r ( .s `  W ) y ) ( +g  `  W ) z )  e.  V  /\  A. x  e.  S  (
( ( r ( .s `  W ) y ) ( +g  `  W ) z ) ( .i `  W
) x )  =  ( 0g `  (Scalar `  W ) ) ) )
6923, 36, 67, 68syl3anbrc 1180 . . 3  |-  ( ( ( W  e.  PreHil  /\  S  C_  V )  /\  ( r  e.  (
Base `  (Scalar `  W
) )  /\  y  e.  (  ._|_  `  S
)  /\  z  e.  (  ._|_  `  S )
) )  ->  (
( r ( .s
`  W ) y ) ( +g  `  W
) z )  e.  (  ._|_  `  S ) )
7069ralrimivvva 2886 . 2  |-  ( ( W  e.  PreHil  /\  S  C_  V )  ->  A. r  e.  ( Base `  (Scalar `  W ) ) A. y  e.  (  ._|_  `  S ) A. z  e.  (  ._|_  `  S
) ( ( r ( .s `  W
) y ) ( +g  `  W ) z )  e.  ( 
._|_  `  S ) )
71 ocvlss.l . . 3  |-  L  =  ( LSubSp `  W )
7213, 29, 1, 34, 28, 71islss 17361 . 2  |-  ( ( 
._|_  `  S )  e.  L  <->  ( (  ._|_  `  S )  C_  V  /\  (  ._|_  `  S
)  =/=  (/)  /\  A. r  e.  ( Base `  (Scalar `  W )
) A. y  e.  (  ._|_  `  S ) A. z  e.  ( 
._|_  `  S ) ( ( r ( .s
`  W ) y ) ( +g  `  W
) z )  e.  (  ._|_  `  S ) ) )
734, 22, 70, 72syl3anbrc 1180 1  |-  ( ( W  e.  PreHil  /\  S  C_  V )  ->  (  ._|_  `  S )  e.  L )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814    C_ wss 3476   (/)c0 3785   ` cfv 5586  (class class class)co 6282   Basecbs 14483   +g cplusg 14548   .rcmulr 14549  Scalarcsca 14551   .scvsca 14552   .icip 14553   0gc0g 14688   Grpcgrp 15720   Ringcrg 16983   LModclmod 17292   LSubSpclss 17358   PreHilcphl 18423   ocvcocv 18455
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-recs 7039  df-rdg 7073  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-ndx 14486  df-slot 14487  df-base 14488  df-sets 14489  df-plusg 14561  df-sca 14564  df-vsca 14565  df-ip 14566  df-0g 14690  df-mnd 15725  df-grp 15855  df-ghm 16057  df-mgp 16929  df-rng 16985  df-lmod 17294  df-lss 17359  df-lmhm 17448  df-lvec 17529  df-sra 17598  df-rgmod 17599  df-phl 18425  df-ocv 18458
This theorem is referenced by:  ocvin  18469  ocvlsp  18471  csslss  18486  pjdm2  18506  pjff  18507  pjf2  18509  pjfo  18510  ocvpj  18512  pjthlem2  21585  pjth  21586
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