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Theorem ocvlss 16899
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 16897 . . 3  |-  (  ._|_  `  S )  C_  V
43a1i 11 . 2  |-  ( ( W  e.  PreHil  /\  S  C_  V )  ->  (  ._|_  `  S )  C_  V )
5 simpr 448 . . . 4  |-  ( ( W  e.  PreHil  /\  S  C_  V )  ->  S  C_  V )
6 phllmod 16861 . . . . . 6  |-  ( W  e.  PreHil  ->  W  e.  LMod )
76adantr 452 . . . . 5  |-  ( ( W  e.  PreHil  /\  S  C_  V )  ->  W  e.  LMod )
8 eqid 2436 . . . . . 6  |-  ( 0g
`  W )  =  ( 0g `  W
)
91, 8lmod0vcl 15979 . . . . 5  |-  ( W  e.  LMod  ->  ( 0g
`  W )  e.  V )
107, 9syl 16 . . . 4  |-  ( ( W  e.  PreHil  /\  S  C_  V )  ->  ( 0g `  W )  e.  V )
11 simpll 731 . . . . . 6  |-  ( ( ( W  e.  PreHil  /\  S  C_  V )  /\  x  e.  S
)  ->  W  e.  PreHil )
125sselda 3348 . . . . . 6  |-  ( ( ( W  e.  PreHil  /\  S  C_  V )  /\  x  e.  S
)  ->  x  e.  V )
13 eqid 2436 . . . . . . 7  |-  (Scalar `  W )  =  (Scalar `  W )
14 eqid 2436 . . . . . . 7  |-  ( .i
`  W )  =  ( .i `  W
)
15 eqid 2436 . . . . . . 7  |-  ( 0g
`  (Scalar `  W )
)  =  ( 0g
`  (Scalar `  W )
)
1613, 14, 1, 15, 8ip0l 16867 . . . . . 6  |-  ( ( W  e.  PreHil  /\  x  e.  V )  ->  (
( 0g `  W
) ( .i `  W ) x )  =  ( 0g `  (Scalar `  W ) ) )
1711, 12, 16syl2anc 643 . . . . 5  |-  ( ( ( W  e.  PreHil  /\  S  C_  V )  /\  x  e.  S
)  ->  ( ( 0g `  W ) ( .i `  W ) x )  =  ( 0g `  (Scalar `  W ) ) )
1817ralrimiva 2789 . . . 4  |-  ( ( W  e.  PreHil  /\  S  C_  V )  ->  A. x  e.  S  ( ( 0g `  W ) ( .i `  W ) x )  =  ( 0g `  (Scalar `  W ) ) )
191, 14, 13, 15, 2elocv 16895 . . . 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 1138 . . 3  |-  ( ( W  e.  PreHil  /\  S  C_  V )  ->  ( 0g `  W )  e.  (  ._|_  `  S ) )
21 ne0i 3634 . . 3  |-  ( ( 0g `  W )  e.  (  ._|_  `  S
)  ->  (  ._|_  `  S )  =/=  (/) )
2220, 21syl 16 . 2  |-  ( ( W  e.  PreHil  /\  S  C_  V )  ->  (  ._|_  `  S )  =/=  (/) )
235adantr 452 . . . 4  |-  ( ( ( W  e.  PreHil  /\  S  C_  V )  /\  ( r  e.  (
Base `  (Scalar `  W
) )  /\  y  e.  (  ._|_  `  S
)  /\  z  e.  (  ._|_  `  S )
) )  ->  S  C_  V )
247adantr 452 . . . . 5  |-  ( ( ( W  e.  PreHil  /\  S  C_  V )  /\  ( r  e.  (
Base `  (Scalar `  W
) )  /\  y  e.  (  ._|_  `  S
)  /\  z  e.  (  ._|_  `  S )
) )  ->  W  e.  LMod )
25 simpr1 963 . . . . . 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 964 . . . . . . 7  |-  ( ( ( W  e.  PreHil  /\  S  C_  V )  /\  ( r  e.  (
Base `  (Scalar `  W
) )  /\  y  e.  (  ._|_  `  S
)  /\  z  e.  (  ._|_  `  S )
) )  ->  y  e.  (  ._|_  `  S
) )
273, 26sseldi 3346 . . . . . 6  |-  ( ( ( W  e.  PreHil  /\  S  C_  V )  /\  ( r  e.  (
Base `  (Scalar `  W
) )  /\  y  e.  (  ._|_  `  S
)  /\  z  e.  (  ._|_  `  S )
) )  ->  y  e.  V )
28 eqid 2436 . . . . . . 7  |-  ( .s
`  W )  =  ( .s `  W
)
29 eqid 2436 . . . . . . 7  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
301, 13, 28, 29lmodvscl 15967 . . . . . 6  |-  ( ( W  e.  LMod  /\  r  e.  ( Base `  (Scalar `  W ) )  /\  y  e.  V )  ->  ( r ( .s
`  W ) y )  e.  V )
3124, 25, 27, 30syl3anc 1184 . . . . 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 965 . . . . . 6  |-  ( ( ( W  e.  PreHil  /\  S  C_  V )  /\  ( r  e.  (
Base `  (Scalar `  W
) )  /\  y  e.  (  ._|_  `  S
)  /\  z  e.  (  ._|_  `  S )
) )  ->  z  e.  (  ._|_  `  S
) )
333, 32sseldi 3346 . . . . 5  |-  ( ( ( W  e.  PreHil  /\  S  C_  V )  /\  ( r  e.  (
Base `  (Scalar `  W
) )  /\  y  e.  (  ._|_  `  S
)  /\  z  e.  (  ._|_  `  S )
) )  ->  z  e.  V )
34 eqid 2436 . . . . . 6  |-  ( +g  `  W )  =  ( +g  `  W )
351, 34lmodvacl 15964 . . . . 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 1184 . . . 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 696 . . . . . . 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 452 . . . . . . 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 452 . . . . . . 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 696 . . . . . . 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 2436 . . . . . . . 8  |-  ( +g  `  (Scalar `  W )
)  =  ( +g  `  (Scalar `  W )
)
4213, 14, 1, 34, 41ipdir 16870 . . . . . . 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 1186 . . . . . 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 452 . . . . . . . . 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 452 . . . . . . . . 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 2436 . . . . . . . . . 10  |-  ( .r
`  (Scalar `  W )
)  =  ( .r
`  (Scalar `  W )
)
4713, 14, 1, 29, 28, 46ipass 16876 . . . . . . . . 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 1186 . . . . . . . 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 16896 . . . . . . . . . 10  |-  ( ( y  e.  (  ._|_  `  S )  /\  x  e.  S )  ->  (
y ( .i `  W ) x )  =  ( 0g `  (Scalar `  W ) ) )
5026, 49sylan 458 . . . . . . . . 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 6097 . . . . . . . 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 452 . . . . . . . . . 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 15958 . . . . . . . . . 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 15701 . . . . . . . . 9  |-  ( ( (Scalar `  W )  e.  Ring  /\  r  e.  ( Base `  (Scalar `  W
) ) )  -> 
( r ( .r
`  (Scalar `  W )
) ( 0g `  (Scalar `  W ) ) )  =  ( 0g
`  (Scalar `  W )
) )
5654, 44, 55syl2anc 643 . . . . . . . 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 2472 . . . . . . 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 16896 . . . . . . . 8  |-  ( ( z  e.  (  ._|_  `  S )  /\  x  e.  S )  ->  (
z ( .i `  W ) x )  =  ( 0g `  (Scalar `  W ) ) )
5932, 58sylan 458 . . . . . . 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 6099 . . . . . 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 15959 . . . . . . 7  |-  ( W  e.  LMod  ->  (Scalar `  W )  e.  Grp )
6229, 15grpidcl 14833 . . . . . . . 8  |-  ( (Scalar `  W )  e.  Grp  ->  ( 0g `  (Scalar `  W ) )  e.  ( Base `  (Scalar `  W ) ) )
6329, 41, 15grplid 14835 . . . . . . . 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 650 . . . . . . 7  |-  ( (Scalar `  W )  e.  Grp  ->  ( ( 0g `  (Scalar `  W ) ) ( +g  `  (Scalar `  W ) ) ( 0g `  (Scalar `  W ) ) )  =  ( 0g `  (Scalar `  W ) ) )
6552, 61, 643syl 19 . . . . . 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 2472 . . . . 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 2789 . . . 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 16895 . . . 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 1138 . . 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 2799 . 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 16011 . 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 1138 1  |-  ( ( W  e.  PreHil  /\  S  C_  V )  ->  (  ._|_  `  S )  e.  L )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2599   A.wral 2705    C_ wss 3320   (/)c0 3628   ` cfv 5454  (class class class)co 6081   Basecbs 13469   +g cplusg 13529   .rcmulr 13530  Scalarcsca 13532   .scvsca 13533   .icip 13534   0gc0g 13723   Grpcgrp 14685   Ringcrg 15660   LModclmod 15950   LSubSpclss 16008   PreHilcphl 16855   ocvcocv 16887
This theorem is referenced by:  ocvin  16901  ocvlsp  16903  csslss  16918  pjdm2  16938  pjff  16939  pjf2  16941  pjfo  16942  ocvpj  16944  pjthlem2  19339  pjth  19340
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-3 10059  df-4 10060  df-5 10061  df-6 10062  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-plusg 13542  df-sca 13545  df-vsca 13546  df-0g 13727  df-mnd 14690  df-grp 14812  df-ghm 15004  df-mgp 15649  df-rng 15663  df-lmod 15952  df-lss 16009  df-lmhm 16098  df-lvec 16175  df-sra 16244  df-rgmod 16245  df-phl 16857  df-ocv 16890
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