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Theorem erngdvlem4 34023
Description: Lemma for erngdv 34025. (Contributed by NM, 11-Aug-2013.)
Hypotheses
Ref Expression
ernggrp.h  |-  H  =  ( LHyp `  K
)
ernggrp.d  |-  D  =  ( ( EDRing `  K
) `  W )
erngdv.b  |-  B  =  ( Base `  K
)
erngdv.t  |-  T  =  ( ( LTrn `  K
) `  W )
erngdv.e  |-  E  =  ( ( TEndo `  K
) `  W )
erngdv.p  |-  P  =  ( a  e.  E ,  b  e.  E  |->  ( f  e.  T  |->  ( ( a `  f )  o.  (
b `  f )
) ) )
erngdv.o  |-  .0.  =  ( f  e.  T  |->  (  _I  |`  B ) )
erngdv.i  |-  I  =  ( a  e.  E  |->  ( f  e.  T  |->  `' ( a `  f ) ) )
erngrnglem.m  |-  .+  =  ( a  e.  E ,  b  e.  E  |->  ( a  o.  b
) )
edlemk6.j  |-  .\/  =  ( join `  K )
edlemk6.m  |-  ./\  =  ( meet `  K )
edlemk6.r  |-  R  =  ( ( trL `  K
) `  W )
edlemk6.p  |-  Q  =  ( ( oc `  K ) `  W
)
edlemk6.z  |-  Z  =  ( ( Q  .\/  ( R `  b ) )  ./\  ( (
h `  Q )  .\/  ( R `  (
b  o.  `' ( s `  h ) ) ) ) )
edlemk6.y  |-  Y  =  ( ( Q  .\/  ( R `  g ) )  ./\  ( Z  .\/  ( R `  (
g  o.  `' b ) ) ) )
edlemk6.x  |-  X  =  ( iota_ z  e.  T  A. b  e.  T  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  ( s `  h ) )  /\  ( R `  b )  =/=  ( R `  g ) )  -> 
( z `  Q
)  =  Y ) )
edlemk6.u  |-  U  =  ( g  e.  T  |->  if ( ( s `
 h )  =  h ,  g ,  X ) )
Assertion
Ref Expression
erngdvlem4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( h  e.  T  /\  h  =/=  (  _I  |`  B ) ) )  ->  D  e.  DivRing )
Distinct variable groups:    B, f    D, s    a, b, s, E    f, a, K, b, s    f, H, s    .0. , s    T, a, b, f, s    W, a, b, f, s    P, s    g, b, z,  ./\    .\/ , b, g, z    B, b   
g, s, B, z    H, b, g, z    g, K, z    .+ , s    P, g, z    Q, b, g, z    R, b, g, z    T, g, z    g, W, z    z, Y    g, Z    f, g, z    h, b, g, s, z
Allowed substitution hints:    B( h, a)    D( z, f, g, h, a, b)    P( f, h, a, b)    .+ ( z,
f, g, h, a, b)    Q( f, h, s, a)    R( f, h, s, a)    T( h)    U( z,
f, g, h, s, a, b)    E( z, f, g, h)    H( h, a)    I( z, f, g, h, s, a, b)    .\/ ( f, h, s, a)    K( h)    ./\ ( f, h, s, a)    W( h)    X( z, f, g, h, s, a, b)    Y( f, g, h, s, a, b)    .0. ( z, f, g, h, a, b)    Z( z, f, h, s, a, b)

Proof of Theorem erngdvlem4
Dummy variable  t is distinct from all other variables.
StepHypRef Expression
1 ernggrp.h . . . . 5  |-  H  =  ( LHyp `  K
)
2 erngdv.t . . . . 5  |-  T  =  ( ( LTrn `  K
) `  W )
3 erngdv.e . . . . 5  |-  E  =  ( ( TEndo `  K
) `  W )
4 ernggrp.d . . . . 5  |-  D  =  ( ( EDRing `  K
) `  W )
5 eqid 2404 . . . . 5  |-  ( Base `  D )  =  (
Base `  D )
61, 2, 3, 4, 5erngbase 33833 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( Base `  D
)  =  E )
76eqcomd 2412 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E  =  ( Base `  D ) )
87adantr 465 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( h  e.  T  /\  h  =/=  (  _I  |`  B ) ) )  ->  E  =  ( Base `  D
) )
9 eqid 2404 . . . . 5  |-  ( .r
`  D )  =  ( .r `  D
)
101, 2, 3, 4, 9erngfmul 33837 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( .r `  D
)  =  ( a  e.  E ,  b  e.  E  |->  ( a  o.  b ) ) )
11 erngrnglem.m . . . 4  |-  .+  =  ( a  e.  E ,  b  e.  E  |->  ( a  o.  b
) )
1210, 11syl6reqr 2464 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  .+  =  ( .r
`  D ) )
1312adantr 465 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( h  e.  T  /\  h  =/=  (  _I  |`  B ) ) )  ->  .+  =  ( .r `  D ) )
14 erngdv.b . . . . . . 7  |-  B  =  ( Base `  K
)
15 erngdv.o . . . . . . 7  |-  .0.  =  ( f  e.  T  |->  (  _I  |`  B ) )
1614, 1, 2, 3, 15tendo0cl 33822 . . . . . 6  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  .0.  e.  E )
1716, 6eleqtrrd 2495 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  .0.  e.  ( Base `  D ) )
18 eqid 2404 . . . . . . . . 9  |-  ( +g  `  D )  =  ( +g  `  D )
191, 2, 3, 4, 18erngfplus 33834 . . . . . . . 8  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( +g  `  D
)  =  ( a  e.  E ,  b  e.  E  |->  ( f  e.  T  |->  ( ( a `  f )  o.  ( b `  f ) ) ) ) )
20 erngdv.p . . . . . . . 8  |-  P  =  ( a  e.  E ,  b  e.  E  |->  ( f  e.  T  |->  ( ( a `  f )  o.  (
b `  f )
) ) )
2119, 20syl6reqr 2464 . . . . . . 7  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  P  =  ( +g  `  D ) )
2221oveqd 6297 . . . . . 6  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  .0.  P  .0.  )  =  (  .0.  ( +g  `  D
)  .0.  ) )
2314, 1, 2, 3, 15, 20tendo0pl 33823 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  .0.  e.  E
)  ->  (  .0.  P  .0.  )  =  .0.  )
2416, 23mpdan 668 . . . . . 6  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  .0.  P  .0.  )  =  .0.  )
2522, 24eqtr3d 2447 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  .0.  ( +g  `  D )  .0.  )  =  .0.  )
26 erngdv.i . . . . . . 7  |-  I  =  ( a  e.  E  |->  ( f  e.  T  |->  `' ( a `  f ) ) )
271, 4, 14, 2, 3, 20, 15, 26erngdvlem1 34020 . . . . . 6  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  D  e.  Grp )
28 eqid 2404 . . . . . . 7  |-  ( 0g
`  D )  =  ( 0g `  D
)
295, 18, 28isgrpid2 16412 . . . . . 6  |-  ( D  e.  Grp  ->  (
(  .0.  e.  (
Base `  D )  /\  (  .0.  ( +g  `  D )  .0.  )  =  .0.  )  <->  ( 0g `  D )  =  .0.  ) )
3027, 29syl 17 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( (  .0.  e.  ( Base `  D )  /\  (  .0.  ( +g  `  D )  .0.  )  =  .0.  )  <->  ( 0g `  D )  =  .0.  ) )
3117, 25, 30mpbi2and 924 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( 0g `  D
)  =  .0.  )
3231eqcomd 2412 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  .0.  =  ( 0g
`  D ) )
3332adantr 465 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( h  e.  T  /\  h  =/=  (  _I  |`  B ) ) )  ->  .0.  =  ( 0g `  D ) )
341, 4, 14, 2, 3, 20, 15, 26, 11erngdvlem3 34022 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  D  e.  Ring )
351, 2, 3, 4, 34erng1lem 34019 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( 1r `  D
)  =  (  _I  |`  T ) )
3635eqcomd 2412 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  _I  |`  T )  =  ( 1r `  D ) )
3736adantr 465 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( h  e.  T  /\  h  =/=  (  _I  |`  B ) ) )  ->  (  _I  |`  T )  =  ( 1r `  D
) )
3834adantr 465 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( h  e.  T  /\  h  =/=  (  _I  |`  B ) ) )  ->  D  e.  Ring )
39 simp1l 1023 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
h  e.  T  /\  h  =/=  (  _I  |`  B ) ) )  /\  (
s  e.  E  /\  s  =/=  .0.  )  /\  ( t  e.  E  /\  t  =/=  .0.  ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
4012oveqd 6297 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( s  .+  t
)  =  ( s ( .r `  D
) t ) )
4139, 40syl 17 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
h  e.  T  /\  h  =/=  (  _I  |`  B ) ) )  /\  (
s  e.  E  /\  s  =/=  .0.  )  /\  ( t  e.  E  /\  t  =/=  .0.  ) )  ->  (
s  .+  t )  =  ( s ( .r `  D ) t ) )
42 simp2l 1025 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
h  e.  T  /\  h  =/=  (  _I  |`  B ) ) )  /\  (
s  e.  E  /\  s  =/=  .0.  )  /\  ( t  e.  E  /\  t  =/=  .0.  ) )  ->  s  e.  E )
43 simp3l 1027 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
h  e.  T  /\  h  =/=  (  _I  |`  B ) ) )  /\  (
s  e.  E  /\  s  =/=  .0.  )  /\  ( t  e.  E  /\  t  =/=  .0.  ) )  ->  t  e.  E )
441, 2, 3, 4, 9erngmul 33838 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  E ) )  -> 
( s ( .r
`  D ) t )  =  ( s  o.  t ) )
4539, 42, 43, 44syl12anc 1230 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
h  e.  T  /\  h  =/=  (  _I  |`  B ) ) )  /\  (
s  e.  E  /\  s  =/=  .0.  )  /\  ( t  e.  E  /\  t  =/=  .0.  ) )  ->  (
s ( .r `  D ) t )  =  ( s  o.  t ) )
4641, 45eqtrd 2445 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
h  e.  T  /\  h  =/=  (  _I  |`  B ) ) )  /\  (
s  e.  E  /\  s  =/=  .0.  )  /\  ( t  e.  E  /\  t  =/=  .0.  ) )  ->  (
s  .+  t )  =  ( s  o.  t ) )
4714, 1, 2, 3, 15tendoconid 33861 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  s  =/= 
.0.  )  /\  (
t  e.  E  /\  t  =/=  .0.  ) )  ->  ( s  o.  t )  =/=  .0.  )
48473adant1r 1225 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
h  e.  T  /\  h  =/=  (  _I  |`  B ) ) )  /\  (
s  e.  E  /\  s  =/=  .0.  )  /\  ( t  e.  E  /\  t  =/=  .0.  ) )  ->  (
s  o.  t )  =/=  .0.  )
4946, 48eqnetrd 2698 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
h  e.  T  /\  h  =/=  (  _I  |`  B ) ) )  /\  (
s  e.  E  /\  s  =/=  .0.  )  /\  ( t  e.  E  /\  t  =/=  .0.  ) )  ->  (
s  .+  t )  =/=  .0.  )
5014, 1, 2, 3, 15tendo1ne0 33860 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  _I  |`  T )  =/=  .0.  )
5150adantr 465 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( h  e.  T  /\  h  =/=  (  _I  |`  B ) ) )  ->  (  _I  |`  T )  =/= 
.0.  )
52 simpll 754 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
h  e.  T  /\  h  =/=  (  _I  |`  B ) ) )  /\  (
s  e.  E  /\  s  =/=  .0.  ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
53 simplrl 764 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
h  e.  T  /\  h  =/=  (  _I  |`  B ) ) )  /\  (
s  e.  E  /\  s  =/=  .0.  ) )  ->  h  e.  T
)
54 simpr 461 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
h  e.  T  /\  h  =/=  (  _I  |`  B ) ) )  /\  (
s  e.  E  /\  s  =/=  .0.  ) )  ->  ( s  e.  E  /\  s  =/= 
.0.  ) )
55 edlemk6.j . . . . 5  |-  .\/  =  ( join `  K )
56 edlemk6.m . . . . 5  |-  ./\  =  ( meet `  K )
57 edlemk6.r . . . . 5  |-  R  =  ( ( trL `  K
) `  W )
58 edlemk6.p . . . . 5  |-  Q  =  ( ( oc `  K ) `  W
)
59 edlemk6.z . . . . 5  |-  Z  =  ( ( Q  .\/  ( R `  b ) )  ./\  ( (
h `  Q )  .\/  ( R `  (
b  o.  `' ( s `  h ) ) ) ) )
60 edlemk6.y . . . . 5  |-  Y  =  ( ( Q  .\/  ( R `  g ) )  ./\  ( Z  .\/  ( R `  (
g  o.  `' b ) ) ) )
61 edlemk6.x . . . . 5  |-  X  =  ( iota_ z  e.  T  A. b  e.  T  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  ( s `  h ) )  /\  ( R `  b )  =/=  ( R `  g ) )  -> 
( z `  Q
)  =  Y ) )
62 edlemk6.u . . . . 5  |-  U  =  ( g  e.  T  |->  if ( ( s `
 h )  =  h ,  g ,  X ) )
6314, 55, 56, 1, 2, 57, 58, 59, 60, 61, 62, 3, 15cdleml6 34013 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  h  e.  T  /\  ( s  e.  E  /\  s  =/=  .0.  ) )  ->  ( U  e.  E  /\  ( U `  ( s `
 h ) )  =  h ) )
6463simpld 459 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  h  e.  T  /\  ( s  e.  E  /\  s  =/=  .0.  ) )  ->  U  e.  E )
6552, 53, 54, 64syl3anc 1232 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
h  e.  T  /\  h  =/=  (  _I  |`  B ) ) )  /\  (
s  e.  E  /\  s  =/=  .0.  ) )  ->  U  e.  E
)
6614, 55, 56, 1, 2, 57, 58, 59, 60, 61, 62, 3, 15cdleml9 34016 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( h  e.  T  /\  h  =/=  (  _I  |`  B ) )  /\  ( s  e.  E  /\  s  =/=  .0.  ) )  ->  U  =/=  .0.  )
67663expa 1199 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
h  e.  T  /\  h  =/=  (  _I  |`  B ) ) )  /\  (
s  e.  E  /\  s  =/=  .0.  ) )  ->  U  =/=  .0.  )
6812oveqd 6297 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( U  .+  s
)  =  ( U ( .r `  D
) s ) )
6968ad2antrr 726 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
h  e.  T  /\  h  =/=  (  _I  |`  B ) ) )  /\  (
s  e.  E  /\  s  =/=  .0.  ) )  ->  ( U  .+  s )  =  ( U ( .r `  D ) s ) )
70 simprl 758 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
h  e.  T  /\  h  =/=  (  _I  |`  B ) ) )  /\  (
s  e.  E  /\  s  =/=  .0.  ) )  ->  s  e.  E
)
711, 2, 3, 4, 9erngmul 33838 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  s  e.  E ) )  -> 
( U ( .r
`  D ) s )  =  ( U  o.  s ) )
7252, 65, 70, 71syl12anc 1230 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
h  e.  T  /\  h  =/=  (  _I  |`  B ) ) )  /\  (
s  e.  E  /\  s  =/=  .0.  ) )  ->  ( U ( .r `  D ) s )  =  ( U  o.  s ) )
7314, 55, 56, 1, 2, 57, 58, 59, 60, 61, 62, 3, 15cdleml8 34015 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( h  e.  T  /\  h  =/=  (  _I  |`  B ) )  /\  ( s  e.  E  /\  s  =/=  .0.  ) )  -> 
( U  o.  s
)  =  (  _I  |`  T ) )
74733expa 1199 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
h  e.  T  /\  h  =/=  (  _I  |`  B ) ) )  /\  (
s  e.  E  /\  s  =/=  .0.  ) )  ->  ( U  o.  s )  =  (  _I  |`  T )
)
7569, 72, 743eqtrd 2449 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
h  e.  T  /\  h  =/=  (  _I  |`  B ) ) )  /\  (
s  e.  E  /\  s  =/=  .0.  ) )  ->  ( U  .+  s )  =  (  _I  |`  T )
)
768, 13, 33, 37, 38, 49, 51, 65, 67, 75isdrngd 17743 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( h  e.  T  /\  h  =/=  (  _I  |`  B ) ) )  ->  D  e.  DivRing )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 186    /\ wa 369    /\ w3a 976    = wceq 1407    e. wcel 1844    =/= wne 2600   A.wral 2756   ifcif 3887    |-> cmpt 4455    _I cid 4735   `'ccnv 4824    |` cres 4827    o. ccom 4829   ` cfv 5571   iota_crio 6241  (class class class)co 6280    |-> cmpt2 6282   Basecbs 14843   +g cplusg 14911   .rcmulr 14912   occoc 14919   0gc0g 15056   joincjn 15899   meetcmee 15900   Grpcgrp 16379   1rcur 17475   Ringcrg 17520   DivRingcdr 17718   HLchlt 32381   LHypclh 33014   LTrncltrn 33131   trLctrl 33189   TEndoctendo 33784   EDRingcedring 33785
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576  ax-cnex 9580  ax-resscn 9581  ax-1cn 9582  ax-icn 9583  ax-addcl 9584  ax-addrcl 9585  ax-mulcl 9586  ax-mulrcl 9587  ax-mulcom 9588  ax-addass 9589  ax-mulass 9590  ax-distr 9591  ax-i2m1 9592  ax-1ne0 9593  ax-1rid 9594  ax-rnegex 9595  ax-rrecex 9596  ax-cnre 9597  ax-pre-lttri 9598  ax-pre-lttrn 9599  ax-pre-ltadd 9600  ax-pre-mulgt0 9601  ax-riotaBAD 31990
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-3an 978  df-tru 1410  df-fal 1413  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-nel 2603  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-tp 3979  df-op 3981  df-uni 4194  df-int 4230  df-iun 4275  df-iin 4276  df-br 4398  df-opab 4456  df-mpt 4457  df-tr 4492  df-eprel 4736  df-id 4740  df-po 4746  df-so 4747  df-fr 4784  df-we 4786  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-pred 5369  df-ord 5415  df-on 5416  df-lim 5417  df-suc 5418  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-riota 6242  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-om 6686  df-1st 6786  df-2nd 6787  df-tpos 6960  df-undef 7007  df-wrecs 7015  df-recs 7077  df-rdg 7115  df-1o 7169  df-oadd 7173  df-er 7350  df-map 7461  df-en 7557  df-dom 7558  df-sdom 7559  df-fin 7560  df-pnf 9662  df-mnf 9663  df-xr 9664  df-ltxr 9665  df-le 9666  df-sub 9845  df-neg 9846  df-nn 10579  df-2 10637  df-3 10638  df-n0 10839  df-z 10908  df-uz 11130  df-fz 11729  df-struct 14845  df-ndx 14846  df-slot 14847  df-base 14848  df-sets 14849  df-ress 14850  df-plusg 14924  df-mulr 14925  df-0g 15058  df-preset 15883  df-poset 15901  df-plt 15914  df-lub 15930  df-glb 15931  df-join 15932  df-meet 15933  df-p0 15995  df-p1 15996  df-lat 16002  df-clat 16064  df-mgm 16198  df-sgrp 16237  df-mnd 16247  df-grp 16383  df-minusg 16384  df-mgp 17464  df-ur 17476  df-ring 17522  df-oppr 17594  df-dvdsr 17612  df-unit 17613  df-invr 17643  df-dvr 17654  df-drng 17720  df-oposet 32207  df-ol 32209  df-oml 32210  df-covers 32297  df-ats 32298  df-atl 32329  df-cvlat 32353  df-hlat 32382  df-llines 32528  df-lplanes 32529  df-lvols 32530  df-lines 32531  df-psubsp 32533  df-pmap 32534  df-padd 32826  df-lhyp 33018  df-laut 33019  df-ldil 33134  df-ltrn 33135  df-trl 33190  df-tendo 33787  df-edring 33789
This theorem is referenced by:  erngdv  34025
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