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

Theorem ocvlss 16572
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 16570 . . 3  |-  (  ._|_  `  S )  C_  V
43a1i 10 . 2  |-  ( ( W  e.  PreHil  /\  S  C_  V )  ->  (  ._|_  `  S )  C_  V )
5 simpr 447 . . . 4  |-  ( ( W  e.  PreHil  /\  S  C_  V )  ->  S  C_  V )
6 phllmod 16534 . . . . . 6  |-  ( W  e.  PreHil  ->  W  e.  LMod )
76adantr 451 . . . . 5  |-  ( ( W  e.  PreHil  /\  S  C_  V )  ->  W  e.  LMod )
8 eqid 2283 . . . . . 6  |-  ( 0g
`  W )  =  ( 0g `  W
)
91, 8lmod0vcl 15659 . . . . 5  |-  ( W  e.  LMod  ->  ( 0g
`  W )  e.  V )
107, 9syl 15 . . . 4  |-  ( ( W  e.  PreHil  /\  S  C_  V )  ->  ( 0g `  W )  e.  V )
11 simpll 730 . . . . . 6  |-  ( ( ( W  e.  PreHil  /\  S  C_  V )  /\  x  e.  S
)  ->  W  e.  PreHil )
125sselda 3180 . . . . . 6  |-  ( ( ( W  e.  PreHil  /\  S  C_  V )  /\  x  e.  S
)  ->  x  e.  V )
13 eqid 2283 . . . . . . 7  |-  (Scalar `  W )  =  (Scalar `  W )
14 eqid 2283 . . . . . . 7  |-  ( .i
`  W )  =  ( .i `  W
)
15 eqid 2283 . . . . . . 7  |-  ( 0g
`  (Scalar `  W )
)  =  ( 0g
`  (Scalar `  W )
)
1613, 14, 1, 15, 8ip0l 16540 . . . . . 6  |-  ( ( W  e.  PreHil  /\  x  e.  V )  ->  (
( 0g `  W
) ( .i `  W ) x )  =  ( 0g `  (Scalar `  W ) ) )
1711, 12, 16syl2anc 642 . . . . 5  |-  ( ( ( W  e.  PreHil  /\  S  C_  V )  /\  x  e.  S
)  ->  ( ( 0g `  W ) ( .i `  W ) x )  =  ( 0g `  (Scalar `  W ) ) )
1817ralrimiva 2626 . . . 4  |-  ( ( W  e.  PreHil  /\  S  C_  V )  ->  A. x  e.  S  ( ( 0g `  W ) ( .i `  W ) x )  =  ( 0g `  (Scalar `  W ) ) )
191, 14, 13, 15, 2elocv 16568 . . . 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 1136 . . 3  |-  ( ( W  e.  PreHil  /\  S  C_  V )  ->  ( 0g `  W )  e.  (  ._|_  `  S ) )
21 ne0i 3461 . . 3  |-  ( ( 0g `  W )  e.  (  ._|_  `  S
)  ->  (  ._|_  `  S )  =/=  (/) )
2220, 21syl 15 . 2  |-  ( ( W  e.  PreHil  /\  S  C_  V )  ->  (  ._|_  `  S )  =/=  (/) )
235adantr 451 . . . 4  |-  ( ( ( W  e.  PreHil  /\  S  C_  V )  /\  ( r  e.  (
Base `  (Scalar `  W
) )  /\  y  e.  (  ._|_  `  S
)  /\  z  e.  (  ._|_  `  S )
) )  ->  S  C_  V )
247adantr 451 . . . . 5  |-  ( ( ( W  e.  PreHil  /\  S  C_  V )  /\  ( r  e.  (
Base `  (Scalar `  W
) )  /\  y  e.  (  ._|_  `  S
)  /\  z  e.  (  ._|_  `  S )
) )  ->  W  e.  LMod )
25 simpr1 961 . . . . . 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 962 . . . . . . 7  |-  ( ( ( W  e.  PreHil  /\  S  C_  V )  /\  ( r  e.  (
Base `  (Scalar `  W
) )  /\  y  e.  (  ._|_  `  S
)  /\  z  e.  (  ._|_  `  S )
) )  ->  y  e.  (  ._|_  `  S
) )
273, 26sseldi 3178 . . . . . 6  |-  ( ( ( W  e.  PreHil  /\  S  C_  V )  /\  ( r  e.  (
Base `  (Scalar `  W
) )  /\  y  e.  (  ._|_  `  S
)  /\  z  e.  (  ._|_  `  S )
) )  ->  y  e.  V )
28 eqid 2283 . . . . . . 7  |-  ( .s
`  W )  =  ( .s `  W
)
29 eqid 2283 . . . . . . 7  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
301, 13, 28, 29lmodvscl 15644 . . . . . 6  |-  ( ( W  e.  LMod  /\  r  e.  ( Base `  (Scalar `  W ) )  /\  y  e.  V )  ->  ( r ( .s
`  W ) y )  e.  V )
3124, 25, 27, 30syl3anc 1182 . . . . 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 963 . . . . . 6  |-  ( ( ( W  e.  PreHil  /\  S  C_  V )  /\  ( r  e.  (
Base `  (Scalar `  W
) )  /\  y  e.  (  ._|_  `  S
)  /\  z  e.  (  ._|_  `  S )
) )  ->  z  e.  (  ._|_  `  S
) )
333, 32sseldi 3178 . . . . 5  |-  ( ( ( W  e.  PreHil  /\  S  C_  V )  /\  ( r  e.  (
Base `  (Scalar `  W
) )  /\  y  e.  (  ._|_  `  S
)  /\  z  e.  (  ._|_  `  S )
) )  ->  z  e.  V )
34 eqid 2283 . . . . . 6  |-  ( +g  `  W )  =  ( +g  `  W )
351, 34lmodvacl 15641 . . . . 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 1182 . . . 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 695 . . . . . . 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 451 . . . . . . 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 451 . . . . . . 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 695 . . . . . . 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 2283 . . . . . . . 8  |-  ( +g  `  (Scalar `  W )
)  =  ( +g  `  (Scalar `  W )
)
4213, 14, 1, 34, 41ipdir 16543 . . . . . . 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 1184 . . . . . 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 451 . . . . . . . . 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 451 . . . . . . . . 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 2283 . . . . . . . . . 10  |-  ( .r
`  (Scalar `  W )
)  =  ( .r
`  (Scalar `  W )
)
4713, 14, 1, 29, 28, 46ipass 16549 . . . . . . . . 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 1184 . . . . . . . 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 16569 . . . . . . . . . 10  |-  ( ( y  e.  (  ._|_  `  S )  /\  x  e.  S )  ->  (
y ( .i `  W ) x )  =  ( 0g `  (Scalar `  W ) ) )
5026, 49sylan 457 . . . . . . . . 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 5874 . . . . . . . 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 451 . . . . . . . . . 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 15635 . . . . . . . . . 10  |-  ( W  e.  LMod  ->  (Scalar `  W )  e.  Ring )
5452, 53syl 15 . . . . . . . . 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 15378 . . . . . . . . 9  |-  ( ( (Scalar `  W )  e.  Ring  /\  r  e.  ( Base `  (Scalar `  W
) ) )  -> 
( r ( .r
`  (Scalar `  W )
) ( 0g `  (Scalar `  W ) ) )  =  ( 0g
`  (Scalar `  W )
) )
5654, 44, 55syl2anc 642 . . . . . . . 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 2319 . . . . . . 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 16569 . . . . . . . 8  |-  ( ( z  e.  (  ._|_  `  S )  /\  x  e.  S )  ->  (
z ( .i `  W ) x )  =  ( 0g `  (Scalar `  W ) ) )
5932, 58sylan 457 . . . . . . 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 5876 . . . . . 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 15636 . . . . . . 7  |-  ( W  e.  LMod  ->  (Scalar `  W )  e.  Grp )
6229, 15grpidcl 14510 . . . . . . . 8  |-  ( (Scalar `  W )  e.  Grp  ->  ( 0g `  (Scalar `  W ) )  e.  ( Base `  (Scalar `  W ) ) )
6329, 41, 15grplid 14512 . . . . . . . 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 649 . . . . . . 7  |-  ( (Scalar `  W )  e.  Grp  ->  ( ( 0g `  (Scalar `  W ) ) ( +g  `  (Scalar `  W ) ) ( 0g `  (Scalar `  W ) ) )  =  ( 0g `  (Scalar `  W ) ) )
6552, 61, 643syl 18 . . . . . 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 2319 . . . . 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 2626 . . . 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 16568 . . . 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 1136 . . 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 2636 . 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 15692 . 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 1136 1  |-  ( ( W  e.  PreHil  /\  S  C_  V )  ->  (  ._|_  `  S )  e.  L )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543    C_ wss 3152   (/)c0 3455   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208   .rcmulr 13209  Scalarcsca 13211   .scvsca 13212   .icip 13213   0gc0g 13400   Grpcgrp 14362   Ringcrg 15337   LModclmod 15627   LSubSpclss 15689   PreHilcphl 16528   ocvcocv 16560
This theorem is referenced by:  ocvin  16574  ocvlsp  16576  csslss  16591  pjdm2  16611  pjff  16612  pjf2  16614  pjfo  16615  ocvpj  16617  pjthlem2  18802  pjth  18803
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-plusg 13221  df-sca 13224  df-vsca 13225  df-0g 13404  df-mnd 14367  df-grp 14489  df-ghm 14681  df-mgp 15326  df-rng 15340  df-lmod 15629  df-lss 15690  df-lmhm 15779  df-lvec 15856  df-sra 15925  df-rgmod 15926  df-phl 16530  df-ocv 16563
  Copyright terms: Public domain W3C validator