MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ocvlss Structured version   Unicode version

Theorem ocvlss 18891
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 18889 . . 3  |-  (  ._|_  `  S )  C_  V
43a1i 11 . 2  |-  ( ( W  e.  PreHil  /\  S  C_  V )  ->  (  ._|_  `  S )  C_  V )
5 simpr 459 . . . 4  |-  ( ( W  e.  PreHil  /\  S  C_  V )  ->  S  C_  V )
6 phllmod 18853 . . . . . 6  |-  ( W  e.  PreHil  ->  W  e.  LMod )
76adantr 463 . . . . 5  |-  ( ( W  e.  PreHil  /\  S  C_  V )  ->  W  e.  LMod )
8 eqid 2400 . . . . . 6  |-  ( 0g
`  W )  =  ( 0g `  W
)
91, 8lmod0vcl 17751 . . . . 5  |-  ( W  e.  LMod  ->  ( 0g
`  W )  e.  V )
107, 9syl 17 . . . 4  |-  ( ( W  e.  PreHil  /\  S  C_  V )  ->  ( 0g `  W )  e.  V )
11 simpll 752 . . . . . 6  |-  ( ( ( W  e.  PreHil  /\  S  C_  V )  /\  x  e.  S
)  ->  W  e.  PreHil )
125sselda 3439 . . . . . 6  |-  ( ( ( W  e.  PreHil  /\  S  C_  V )  /\  x  e.  S
)  ->  x  e.  V )
13 eqid 2400 . . . . . . 7  |-  (Scalar `  W )  =  (Scalar `  W )
14 eqid 2400 . . . . . . 7  |-  ( .i
`  W )  =  ( .i `  W
)
15 eqid 2400 . . . . . . 7  |-  ( 0g
`  (Scalar `  W )
)  =  ( 0g
`  (Scalar `  W )
)
1613, 14, 1, 15, 8ip0l 18859 . . . . . 6  |-  ( ( W  e.  PreHil  /\  x  e.  V )  ->  (
( 0g `  W
) ( .i `  W ) x )  =  ( 0g `  (Scalar `  W ) ) )
1711, 12, 16syl2anc 659 . . . . 5  |-  ( ( ( W  e.  PreHil  /\  S  C_  V )  /\  x  e.  S
)  ->  ( ( 0g `  W ) ( .i `  W ) x )  =  ( 0g `  (Scalar `  W ) ) )
1817ralrimiva 2815 . . . 4  |-  ( ( W  e.  PreHil  /\  S  C_  V )  ->  A. x  e.  S  ( ( 0g `  W ) ( .i `  W ) x )  =  ( 0g `  (Scalar `  W ) ) )
191, 14, 13, 15, 2elocv 18887 . . . 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 1179 . . 3  |-  ( ( W  e.  PreHil  /\  S  C_  V )  ->  ( 0g `  W )  e.  (  ._|_  `  S ) )
21 ne0i 3741 . . 3  |-  ( ( 0g `  W )  e.  (  ._|_  `  S
)  ->  (  ._|_  `  S )  =/=  (/) )
2220, 21syl 17 . 2  |-  ( ( W  e.  PreHil  /\  S  C_  V )  ->  (  ._|_  `  S )  =/=  (/) )
235adantr 463 . . . 4  |-  ( ( ( W  e.  PreHil  /\  S  C_  V )  /\  ( r  e.  (
Base `  (Scalar `  W
) )  /\  y  e.  (  ._|_  `  S
)  /\  z  e.  (  ._|_  `  S )
) )  ->  S  C_  V )
247adantr 463 . . . . 5  |-  ( ( ( W  e.  PreHil  /\  S  C_  V )  /\  ( r  e.  (
Base `  (Scalar `  W
) )  /\  y  e.  (  ._|_  `  S
)  /\  z  e.  (  ._|_  `  S )
) )  ->  W  e.  LMod )
25 simpr1 1001 . . . . . 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 1002 . . . . . . 7  |-  ( ( ( W  e.  PreHil  /\  S  C_  V )  /\  ( r  e.  (
Base `  (Scalar `  W
) )  /\  y  e.  (  ._|_  `  S
)  /\  z  e.  (  ._|_  `  S )
) )  ->  y  e.  (  ._|_  `  S
) )
273, 26sseldi 3437 . . . . . 6  |-  ( ( ( W  e.  PreHil  /\  S  C_  V )  /\  ( r  e.  (
Base `  (Scalar `  W
) )  /\  y  e.  (  ._|_  `  S
)  /\  z  e.  (  ._|_  `  S )
) )  ->  y  e.  V )
28 eqid 2400 . . . . . . 7  |-  ( .s
`  W )  =  ( .s `  W
)
29 eqid 2400 . . . . . . 7  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
301, 13, 28, 29lmodvscl 17739 . . . . . 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 1003 . . . . . 6  |-  ( ( ( W  e.  PreHil  /\  S  C_  V )  /\  ( r  e.  (
Base `  (Scalar `  W
) )  /\  y  e.  (  ._|_  `  S
)  /\  z  e.  (  ._|_  `  S )
) )  ->  z  e.  (  ._|_  `  S
) )
333, 32sseldi 3437 . . . . 5  |-  ( ( ( W  e.  PreHil  /\  S  C_  V )  /\  ( r  e.  (
Base `  (Scalar `  W
) )  /\  y  e.  (  ._|_  `  S
)  /\  z  e.  (  ._|_  `  S )
) )  ->  z  e.  V )
34 eqid 2400 . . . . . 6  |-  ( +g  `  W )  =  ( +g  `  W )
351, 34lmodvacl 17736 . . . . 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 713 . . . . . . 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 463 . . . . . . 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 463 . . . . . . 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 713 . . . . . . 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 2400 . . . . . . . 8  |-  ( +g  `  (Scalar `  W )
)  =  ( +g  `  (Scalar `  W )
)
4213, 14, 1, 34, 41ipdir 18862 . . . . . . 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 463 . . . . . . . . 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 463 . . . . . . . . 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 2400 . . . . . . . . . 10  |-  ( .r
`  (Scalar `  W )
)  =  ( .r
`  (Scalar `  W )
)
4713, 14, 1, 29, 28, 46ipass 18868 . . . . . . . . 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 18888 . . . . . . . . . 10  |-  ( ( y  e.  (  ._|_  `  S )  /\  x  e.  S )  ->  (
y ( .i `  W ) x )  =  ( 0g `  (Scalar `  W ) ) )
5026, 49sylan 469 . . . . . . . . 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 6248 . . . . . . . 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 463 . . . . . . . . . 10  |-  ( ( ( ( W  e. 
PreHil  /\  S  C_  V
)  /\  ( r  e.  ( Base `  (Scalar `  W ) )  /\  y  e.  (  ._|_  `  S )  /\  z  e.  (  ._|_  `  S
) ) )  /\  x  e.  S )  ->  W  e.  LMod )
5313lmodring 17730 . . . . . . . . . 10  |-  ( W  e.  LMod  ->  (Scalar `  W )  e.  Ring )
5452, 53syl 17 . . . . . . . . 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, 15ringrz 17446 . . . . . . . . 9  |-  ( ( (Scalar `  W )  e.  Ring  /\  r  e.  ( Base `  (Scalar `  W
) ) )  -> 
( r ( .r
`  (Scalar `  W )
) ( 0g `  (Scalar `  W ) ) )  =  ( 0g
`  (Scalar `  W )
) )
5654, 44, 55syl2anc 659 . . . . . . . 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 2445 . . . . . . 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 18888 . . . . . . . 8  |-  ( ( z  e.  (  ._|_  `  S )  /\  x  e.  S )  ->  (
z ( .i `  W ) x )  =  ( 0g `  (Scalar `  W ) ) )
5932, 58sylan 469 . . . . . . 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 6250 . . . . . 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 17731 . . . . . . 7  |-  ( W  e.  LMod  ->  (Scalar `  W )  e.  Grp )
6229, 15grpidcl 16292 . . . . . . . 8  |-  ( (Scalar `  W )  e.  Grp  ->  ( 0g `  (Scalar `  W ) )  e.  ( Base `  (Scalar `  W ) ) )
6329, 41, 15grplid 16294 . . . . . . . 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 666 . . . . . . 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 2445 . . . . 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 2815 . . . 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 18887 . . . 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 1179 . . 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 2823 . 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 17791 . 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 1179 1  |-  ( ( W  e.  PreHil  /\  S  C_  V )  ->  (  ._|_  `  S )  e.  L )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 972    = wceq 1403    e. wcel 1840    =/= wne 2596   A.wral 2751    C_ wss 3411   (/)c0 3735   ` cfv 5523  (class class class)co 6232   Basecbs 14731   +g cplusg 14799   .rcmulr 14800  Scalarcsca 14802   .scvsca 14803   .icip 14804   0gc0g 14944   Grpcgrp 16267   Ringcrg 17408   LModclmod 17722   LSubSpclss 17788   PreHilcphl 18847   ocvcocv 18879
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528  ax-cnex 9496  ax-resscn 9497  ax-1cn 9498  ax-icn 9499  ax-addcl 9500  ax-addrcl 9501  ax-mulcl 9502  ax-mulrcl 9503  ax-mulcom 9504  ax-addass 9505  ax-mulass 9506  ax-distr 9507  ax-i2m1 9508  ax-1ne0 9509  ax-1rid 9510  ax-rnegex 9511  ax-rrecex 9512  ax-cnre 9513  ax-pre-lttri 9514  ax-pre-lttrn 9515  ax-pre-ltadd 9516  ax-pre-mulgt0 9517
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 973  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-nel 2599  df-ral 2756  df-rex 2757  df-reu 2758  df-rmo 2759  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-pss 3427  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-tp 3974  df-op 3976  df-uni 4189  df-iun 4270  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4487  df-eprel 4731  df-id 4735  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-riota 6194  df-ov 6235  df-oprab 6236  df-mpt2 6237  df-om 6637  df-recs 6997  df-rdg 7031  df-er 7266  df-en 7473  df-dom 7474  df-sdom 7475  df-pnf 9578  df-mnf 9579  df-xr 9580  df-ltxr 9581  df-le 9582  df-sub 9761  df-neg 9762  df-nn 10495  df-2 10553  df-3 10554  df-4 10555  df-5 10556  df-6 10557  df-7 10558  df-8 10559  df-ndx 14734  df-slot 14735  df-base 14736  df-sets 14737  df-plusg 14812  df-sca 14815  df-vsca 14816  df-ip 14817  df-0g 14946  df-mgm 16086  df-sgrp 16125  df-mnd 16135  df-grp 16271  df-ghm 16479  df-mgp 17352  df-ring 17410  df-lmod 17724  df-lss 17789  df-lmhm 17878  df-lvec 17959  df-sra 18028  df-rgmod 18029  df-phl 18849  df-ocv 18882
This theorem is referenced by:  ocvin  18893  ocvlsp  18895  csslss  18910  pjdm2  18930  pjff  18931  pjf2  18933  pjfo  18934  ocvpj  18936  pjthlem2  22035  pjth  22036
  Copyright terms: Public domain W3C validator