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Theorem erngdvlem4 34943
Description: Lemma for erngdv 34945. (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 2451 . . . . 5  |-  ( Base `  D )  =  (
Base `  D )
61, 2, 3, 4, 5erngbase 34753 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( Base `  D
)  =  E )
76eqcomd 2459 . . 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 2451 . . . . 5  |-  ( .r
`  D )  =  ( .r `  D
)
101, 2, 3, 4, 9erngfmul 34757 . . . 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 2511 . . 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 34742 . . . . . 6  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  .0.  e.  E )
1716, 6eleqtrrd 2542 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  .0.  e.  ( Base `  D ) )
18 eqid 2451 . . . . . . . . 9  |-  ( +g  `  D )  =  ( +g  `  D )
191, 2, 3, 4, 18erngfplus 34754 . . . . . . . 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 2511 . . . . . . 7  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  P  =  ( +g  `  D ) )
2221oveqd 6209 . . . . . 6  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  .0.  P  .0.  )  =  (  .0.  ( +g  `  D
)  .0.  ) )
2314, 1, 2, 3, 15, 20tendo0pl 34743 . . . . . . 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 2494 . . . . 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 34940 . . . . . 6  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  D  e.  Grp )
28 eqid 2451 . . . . . . 7  |-  ( 0g
`  D )  =  ( 0g `  D
)
295, 18, 28isgrpid2 15678 . . . . . 6  |-  ( D  e.  Grp  ->  (
(  .0.  e.  (
Base `  D )  /\  (  .0.  ( +g  `  D )  .0.  )  =  .0.  )  <->  ( 0g `  D )  =  .0.  ) )
3027, 29syl 16 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( (  .0.  e.  ( Base `  D )  /\  (  .0.  ( +g  `  D )  .0.  )  =  .0.  )  <->  ( 0g `  D )  =  .0.  ) )
3117, 25, 30mpbi2and 912 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( 0g `  D
)  =  .0.  )
3231eqcomd 2459 . . 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 34942 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  D  e.  Ring )
351, 2, 3, 4, 34erng1lem 34939 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( 1r `  D
)  =  (  _I  |`  T ) )
3635eqcomd 2459 . . 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 1012 . . . . 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 6209 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( s  .+  t
)  =  ( s ( .r `  D
) t ) )
4139, 40syl 16 . . . 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 1014 . . . . 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 1016 . . . . 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 34758 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  E ) )  -> 
( s ( .r
`  D ) t )  =  ( s  o.  t ) )
4539, 42, 43, 44syl12anc 1217 . . . 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 2492 . . 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 34781 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  s  =/= 
.0.  )  /\  (
t  e.  E  /\  t  =/=  .0.  ) )  ->  ( s  o.  t )  =/=  .0.  )
48473adant1r 1212 . . 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 2741 . 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 34780 . . 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 753 . . 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 759 . . 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 34933 . . . 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 1219 . 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 34936 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( h  e.  T  /\  h  =/=  (  _I  |`  B ) )  /\  ( s  e.  E  /\  s  =/=  .0.  ) )  ->  U  =/=  .0.  )
67663expa 1188 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
h  e.  T  /\  h  =/=  (  _I  |`  B ) ) )  /\  (
s  e.  E  /\  s  =/=  .0.  ) )  ->  U  =/=  .0.  )
6812oveqd 6209 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( U  .+  s
)  =  ( U ( .r `  D
) s ) )
6968ad2antrr 725 . . 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 755 . . . 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 34758 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  s  e.  E ) )  -> 
( U ( .r
`  D ) s )  =  ( U  o.  s ) )
7252, 65, 70, 71syl12anc 1217 . . 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 34935 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( h  e.  T  /\  h  =/=  (  _I  |`  B ) )  /\  ( s  e.  E  /\  s  =/=  .0.  ) )  -> 
( U  o.  s
)  =  (  _I  |`  T ) )
74733expa 1188 . . 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 2496 . 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 16965 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 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2644   A.wral 2795   ifcif 3891    |-> cmpt 4450    _I cid 4731   `'ccnv 4939    |` cres 4942    o. ccom 4944   ` cfv 5518   iota_crio 6152  (class class class)co 6192    |-> cmpt2 6194   Basecbs 14278   +g cplusg 14342   .rcmulr 14343   occoc 14350   0gc0g 14482   joincjn 15218   meetcmee 15219   Grpcgrp 15514   1rcur 16710   Ringcrg 16753   DivRingcdr 16940   HLchlt 33303   LHypclh 33936   LTrncltrn 34053   trLctrl 34110   TEndoctendo 34704   EDRingcedring 34705
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 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-cnex 9441  ax-resscn 9442  ax-1cn 9443  ax-icn 9444  ax-addcl 9445  ax-addrcl 9446  ax-mulcl 9447  ax-mulrcl 9448  ax-mulcom 9449  ax-addass 9450  ax-mulass 9451  ax-distr 9452  ax-i2m1 9453  ax-1ne0 9454  ax-1rid 9455  ax-rnegex 9456  ax-rrecex 9457  ax-cnre 9458  ax-pre-lttri 9459  ax-pre-lttrn 9460  ax-pre-ltadd 9461  ax-pre-mulgt0 9462  ax-riotaBAD 32912
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-int 4229  df-iun 4273  df-iin 4274  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-eprel 4732  df-id 4736  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 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-om 6579  df-1st 6679  df-2nd 6680  df-tpos 6847  df-undef 6894  df-recs 6934  df-rdg 6968  df-1o 7022  df-oadd 7026  df-er 7203  df-map 7318  df-en 7413  df-dom 7414  df-sdom 7415  df-fin 7416  df-pnf 9523  df-mnf 9524  df-xr 9525  df-ltxr 9526  df-le 9527  df-sub 9700  df-neg 9701  df-nn 10426  df-2 10483  df-3 10484  df-n0 10683  df-z 10750  df-uz 10965  df-fz 11541  df-struct 14280  df-ndx 14281  df-slot 14282  df-base 14283  df-sets 14284  df-ress 14285  df-plusg 14355  df-mulr 14356  df-0g 14484  df-poset 15220  df-plt 15232  df-lub 15248  df-glb 15249  df-join 15250  df-meet 15251  df-p0 15313  df-p1 15314  df-lat 15320  df-clat 15382  df-mnd 15519  df-grp 15649  df-minusg 15650  df-mgp 16699  df-ur 16711  df-rng 16755  df-oppr 16823  df-dvdsr 16841  df-unit 16842  df-invr 16872  df-dvr 16883  df-drng 16942  df-oposet 33129  df-ol 33131  df-oml 33132  df-covers 33219  df-ats 33220  df-atl 33251  df-cvlat 33275  df-hlat 33304  df-llines 33450  df-lplanes 33451  df-lvols 33452  df-lines 33453  df-psubsp 33455  df-pmap 33456  df-padd 33748  df-lhyp 33940  df-laut 33941  df-ldil 34056  df-ltrn 34057  df-trl 34111  df-tendo 34707  df-edring 34709
This theorem is referenced by:  erngdv  34945
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