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Theorem ocvlss 18221
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 18219 . . 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 18183 . . . . . 6  |-  ( W  e.  PreHil  ->  W  e.  LMod )
76adantr 465 . . . . 5  |-  ( ( W  e.  PreHil  /\  S  C_  V )  ->  W  e.  LMod )
8 eqid 2454 . . . . . 6  |-  ( 0g
`  W )  =  ( 0g `  W
)
91, 8lmod0vcl 17099 . . . . 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 3463 . . . . . 6  |-  ( ( ( W  e.  PreHil  /\  S  C_  V )  /\  x  e.  S
)  ->  x  e.  V )
13 eqid 2454 . . . . . . 7  |-  (Scalar `  W )  =  (Scalar `  W )
14 eqid 2454 . . . . . . 7  |-  ( .i
`  W )  =  ( .i `  W
)
15 eqid 2454 . . . . . . 7  |-  ( 0g
`  (Scalar `  W )
)  =  ( 0g
`  (Scalar `  W )
)
1613, 14, 1, 15, 8ip0l 18189 . . . . . 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 2829 . . . 4  |-  ( ( W  e.  PreHil  /\  S  C_  V )  ->  A. x  e.  S  ( ( 0g `  W ) ( .i `  W ) x )  =  ( 0g `  (Scalar `  W ) ) )
191, 14, 13, 15, 2elocv 18217 . . . 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 1172 . . 3  |-  ( ( W  e.  PreHil  /\  S  C_  V )  ->  ( 0g `  W )  e.  (  ._|_  `  S ) )
21 ne0i 3750 . . 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 994 . . . . . 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 995 . . . . . . 7  |-  ( ( ( W  e.  PreHil  /\  S  C_  V )  /\  ( r  e.  (
Base `  (Scalar `  W
) )  /\  y  e.  (  ._|_  `  S
)  /\  z  e.  (  ._|_  `  S )
) )  ->  y  e.  (  ._|_  `  S
) )
273, 26sseldi 3461 . . . . . 6  |-  ( ( ( W  e.  PreHil  /\  S  C_  V )  /\  ( r  e.  (
Base `  (Scalar `  W
) )  /\  y  e.  (  ._|_  `  S
)  /\  z  e.  (  ._|_  `  S )
) )  ->  y  e.  V )
28 eqid 2454 . . . . . . 7  |-  ( .s
`  W )  =  ( .s `  W
)
29 eqid 2454 . . . . . . 7  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
301, 13, 28, 29lmodvscl 17087 . . . . . 6  |-  ( ( W  e.  LMod  /\  r  e.  ( Base `  (Scalar `  W ) )  /\  y  e.  V )  ->  ( r ( .s
`  W ) y )  e.  V )
3124, 25, 27, 30syl3anc 1219 . . . . 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 996 . . . . . 6  |-  ( ( ( W  e.  PreHil  /\  S  C_  V )  /\  ( r  e.  (
Base `  (Scalar `  W
) )  /\  y  e.  (  ._|_  `  S
)  /\  z  e.  (  ._|_  `  S )
) )  ->  z  e.  (  ._|_  `  S
) )
333, 32sseldi 3461 . . . . 5  |-  ( ( ( W  e.  PreHil  /\  S  C_  V )  /\  ( r  e.  (
Base `  (Scalar `  W
) )  /\  y  e.  (  ._|_  `  S
)  /\  z  e.  (  ._|_  `  S )
) )  ->  z  e.  V )
34 eqid 2454 . . . . . 6  |-  ( +g  `  W )  =  ( +g  `  W )
351, 34lmodvacl 17084 . . . . 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 1219 . . . 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 2454 . . . . . . . 8  |-  ( +g  `  (Scalar `  W )
)  =  ( +g  `  (Scalar `  W )
)
4213, 14, 1, 34, 41ipdir 18192 . . . . . . 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 1221 . . . . . 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 2454 . . . . . . . . . 10  |-  ( .r
`  (Scalar `  W )
)  =  ( .r
`  (Scalar `  W )
)
4713, 14, 1, 29, 28, 46ipass 18198 . . . . . . . . 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 1221 . . . . . . . 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 18218 . . . . . . . . . 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 6215 . . . . . . . 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 17078 . . . . . . . . . 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 16804 . . . . . . . . 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 2499 . . . . . . 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 18218 . . . . . . . 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 6217 . . . . . 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 17079 . . . . . . 7  |-  ( W  e.  LMod  ->  (Scalar `  W )  e.  Grp )
6229, 15grpidcl 15684 . . . . . . . 8  |-  ( (Scalar `  W )  e.  Grp  ->  ( 0g `  (Scalar `  W ) )  e.  ( Base `  (Scalar `  W ) ) )
6329, 41, 15grplid 15686 . . . . . . . 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 2499 . . . . 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 2829 . . . 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 18217 . . . 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 1172 . . 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 2913 . 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 17138 . 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 1172 1  |-  ( ( W  e.  PreHil  /\  S  C_  V )  ->  (  ._|_  `  S )  e.  L )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2647   A.wral 2798    C_ wss 3435   (/)c0 3744   ` cfv 5525  (class class class)co 6199   Basecbs 14291   +g cplusg 14356   .rcmulr 14357  Scalarcsca 14359   .scvsca 14360   .icip 14361   0gc0g 14496   Grpcgrp 15528   Ringcrg 16767   LModclmod 17070   LSubSpclss 17135   PreHilcphl 18177   ocvcocv 18209
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4510  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481  ax-cnex 9448  ax-resscn 9449  ax-1cn 9450  ax-icn 9451  ax-addcl 9452  ax-addrcl 9453  ax-mulcl 9454  ax-mulrcl 9455  ax-mulcom 9456  ax-addass 9457  ax-mulass 9458  ax-distr 9459  ax-i2m1 9460  ax-1ne0 9461  ax-1rid 9462  ax-rnegex 9463  ax-rrecex 9464  ax-cnre 9465  ax-pre-lttri 9466  ax-pre-lttrn 9467  ax-pre-ltadd 9468  ax-pre-mulgt0 9469
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-nel 2650  df-ral 2803  df-rex 2804  df-reu 2805  df-rmo 2806  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-pss 3451  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-tp 3989  df-op 3991  df-uni 4199  df-iun 4280  df-br 4400  df-opab 4458  df-mpt 4459  df-tr 4493  df-eprel 4739  df-id 4743  df-po 4748  df-so 4749  df-fr 4786  df-we 4788  df-ord 4829  df-on 4830  df-lim 4831  df-suc 4832  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6160  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-om 6586  df-recs 6941  df-rdg 6975  df-er 7210  df-en 7420  df-dom 7421  df-sdom 7422  df-pnf 9530  df-mnf 9531  df-xr 9532  df-ltxr 9533  df-le 9534  df-sub 9707  df-neg 9708  df-nn 10433  df-2 10490  df-3 10491  df-4 10492  df-5 10493  df-6 10494  df-7 10495  df-8 10496  df-ndx 14294  df-slot 14295  df-base 14296  df-sets 14297  df-plusg 14369  df-sca 14372  df-vsca 14373  df-ip 14374  df-0g 14498  df-mnd 15533  df-grp 15663  df-ghm 15863  df-mgp 16713  df-rng 16769  df-lmod 17072  df-lss 17136  df-lmhm 17225  df-lvec 17306  df-sra 17375  df-rgmod 17376  df-phl 18179  df-ocv 18212
This theorem is referenced by:  ocvin  18223  ocvlsp  18225  csslss  18240  pjdm2  18260  pjff  18261  pjf2  18263  pjfo  18264  ocvpj  18266  pjthlem2  21056  pjth  21057
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