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

Theorem lmhmpropd 16183
Description: Module homomorphism depends only on the module attributes of structures. (Contributed by Mario Carneiro, 8-Oct-2015.)
Hypotheses
Ref Expression
lmhmpropd.a  |-  ( ph  ->  B  =  ( Base `  J ) )
lmhmpropd.b  |-  ( ph  ->  C  =  ( Base `  K ) )
lmhmpropd.c  |-  ( ph  ->  B  =  ( Base `  L ) )
lmhmpropd.d  |-  ( ph  ->  C  =  ( Base `  M ) )
lmhmpropd.1  |-  ( ph  ->  F  =  (Scalar `  J ) )
lmhmpropd.2  |-  ( ph  ->  G  =  (Scalar `  K ) )
lmhmpropd.3  |-  ( ph  ->  F  =  (Scalar `  L ) )
lmhmpropd.4  |-  ( ph  ->  G  =  (Scalar `  M ) )
lmhmpropd.p  |-  P  =  ( Base `  F
)
lmhmpropd.q  |-  Q  =  ( Base `  G
)
lmhmpropd.e  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x ( +g  `  J ) y )  =  ( x ( +g  `  L ) y ) )
lmhmpropd.f  |-  ( (
ph  /\  ( x  e.  C  /\  y  e.  C ) )  -> 
( x ( +g  `  K ) y )  =  ( x ( +g  `  M ) y ) )
lmhmpropd.g  |-  ( (
ph  /\  ( x  e.  P  /\  y  e.  B ) )  -> 
( x ( .s
`  J ) y )  =  ( x ( .s `  L
) y ) )
lmhmpropd.h  |-  ( (
ph  /\  ( x  e.  Q  /\  y  e.  C ) )  -> 
( x ( .s
`  K ) y )  =  ( x ( .s `  M
) y ) )
Assertion
Ref Expression
lmhmpropd  |-  ( ph  ->  ( J LMHom  K )  =  ( L LMHom  M
) )
Distinct variable groups:    x, y, C    x, J, y    x, K, y    x, L, y   
x, M, y    x, P, y    ph, x, y   
x, B, y    x, Q, y
Allowed substitution hints:    F( x, y)    G( x, y)

Proof of Theorem lmhmpropd
Dummy variables  z  w  f are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lmhmpropd.a . . . . . 6  |-  ( ph  ->  B  =  ( Base `  J ) )
2 lmhmpropd.c . . . . . 6  |-  ( ph  ->  B  =  ( Base `  L ) )
3 lmhmpropd.e . . . . . 6  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x ( +g  `  J ) y )  =  ( x ( +g  `  L ) y ) )
4 lmhmpropd.1 . . . . . 6  |-  ( ph  ->  F  =  (Scalar `  J ) )
5 lmhmpropd.3 . . . . . 6  |-  ( ph  ->  F  =  (Scalar `  L ) )
6 lmhmpropd.p . . . . . 6  |-  P  =  ( Base `  F
)
7 lmhmpropd.g . . . . . 6  |-  ( (
ph  /\  ( x  e.  P  /\  y  e.  B ) )  -> 
( x ( .s
`  J ) y )  =  ( x ( .s `  L
) y ) )
81, 2, 3, 4, 5, 6, 7lmodpropd 16045 . . . . 5  |-  ( ph  ->  ( J  e.  LMod  <->  L  e.  LMod ) )
9 lmhmpropd.b . . . . . 6  |-  ( ph  ->  C  =  ( Base `  K ) )
10 lmhmpropd.d . . . . . 6  |-  ( ph  ->  C  =  ( Base `  M ) )
11 lmhmpropd.f . . . . . 6  |-  ( (
ph  /\  ( x  e.  C  /\  y  e.  C ) )  -> 
( x ( +g  `  K ) y )  =  ( x ( +g  `  M ) y ) )
12 lmhmpropd.2 . . . . . 6  |-  ( ph  ->  G  =  (Scalar `  K ) )
13 lmhmpropd.4 . . . . . 6  |-  ( ph  ->  G  =  (Scalar `  M ) )
14 lmhmpropd.q . . . . . 6  |-  Q  =  ( Base `  G
)
15 lmhmpropd.h . . . . . 6  |-  ( (
ph  /\  ( x  e.  Q  /\  y  e.  C ) )  -> 
( x ( .s
`  K ) y )  =  ( x ( .s `  M
) y ) )
169, 10, 11, 12, 13, 14, 15lmodpropd 16045 . . . . 5  |-  ( ph  ->  ( K  e.  LMod  <->  M  e.  LMod ) )
178, 16anbi12d 693 . . . 4  |-  ( ph  ->  ( ( J  e. 
LMod  /\  K  e.  LMod ) 
<->  ( L  e.  LMod  /\  M  e.  LMod )
) )
187proplem 13953 . . . . . . . . . . 11  |-  ( (
ph  /\  ( z  e.  P  /\  w  e.  B ) )  -> 
( z ( .s
`  J ) w )  =  ( z ( .s `  L
) w ) )
1918adantlr 697 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
f  e.  ( J 
GrpHom  K )  /\  G  =  F ) )  /\  ( z  e.  P  /\  w  e.  B
) )  ->  (
z ( .s `  J ) w )  =  ( z ( .s `  L ) w ) )
2019fveq2d 5767 . . . . . . . . 9  |-  ( ( ( ph  /\  (
f  e.  ( J 
GrpHom  K )  /\  G  =  F ) )  /\  ( z  e.  P  /\  w  e.  B
) )  ->  (
f `  ( z
( .s `  J
) w ) )  =  ( f `  ( z ( .s
`  L ) w ) ) )
21 simpll 732 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
f  e.  ( J 
GrpHom  K )  /\  G  =  F ) )  /\  ( z  e.  P  /\  w  e.  B
) )  ->  ph )
22 simprl 734 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
f  e.  ( J 
GrpHom  K )  /\  G  =  F ) )  /\  ( z  e.  P  /\  w  e.  B
) )  ->  z  e.  P )
23 simplrr 739 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
f  e.  ( J 
GrpHom  K )  /\  G  =  F ) )  /\  ( z  e.  P  /\  w  e.  B
) )  ->  G  =  F )
2423fveq2d 5767 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
f  e.  ( J 
GrpHom  K )  /\  G  =  F ) )  /\  ( z  e.  P  /\  w  e.  B
) )  ->  ( Base `  G )  =  ( Base `  F
) )
2524, 14, 63eqtr4g 2500 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
f  e.  ( J 
GrpHom  K )  /\  G  =  F ) )  /\  ( z  e.  P  /\  w  e.  B
) )  ->  Q  =  P )
2622, 25eleqtrrd 2520 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
f  e.  ( J 
GrpHom  K )  /\  G  =  F ) )  /\  ( z  e.  P  /\  w  e.  B
) )  ->  z  e.  Q )
27 simplrl 738 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
f  e.  ( J 
GrpHom  K )  /\  G  =  F ) )  /\  ( z  e.  P  /\  w  e.  B
) )  ->  f  e.  ( J  GrpHom  K ) )
28 eqid 2443 . . . . . . . . . . . . . 14  |-  ( Base `  J )  =  (
Base `  J )
29 eqid 2443 . . . . . . . . . . . . . 14  |-  ( Base `  K )  =  (
Base `  K )
3028, 29ghmf 15048 . . . . . . . . . . . . 13  |-  ( f  e.  ( J  GrpHom  K )  ->  f :
( Base `  J ) --> ( Base `  K )
)
3127, 30syl 16 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
f  e.  ( J 
GrpHom  K )  /\  G  =  F ) )  /\  ( z  e.  P  /\  w  e.  B
) )  ->  f : ( Base `  J
) --> ( Base `  K
) )
32 simprr 735 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
f  e.  ( J 
GrpHom  K )  /\  G  =  F ) )  /\  ( z  e.  P  /\  w  e.  B
) )  ->  w  e.  B )
3321, 1syl 16 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
f  e.  ( J 
GrpHom  K )  /\  G  =  F ) )  /\  ( z  e.  P  /\  w  e.  B
) )  ->  B  =  ( Base `  J
) )
3432, 33eleqtrd 2519 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
f  e.  ( J 
GrpHom  K )  /\  G  =  F ) )  /\  ( z  e.  P  /\  w  e.  B
) )  ->  w  e.  ( Base `  J
) )
3531, 34ffvelrnd 5907 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
f  e.  ( J 
GrpHom  K )  /\  G  =  F ) )  /\  ( z  e.  P  /\  w  e.  B
) )  ->  (
f `  w )  e.  ( Base `  K
) )
3621, 9syl 16 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
f  e.  ( J 
GrpHom  K )  /\  G  =  F ) )  /\  ( z  e.  P  /\  w  e.  B
) )  ->  C  =  ( Base `  K
) )
3735, 36eleqtrrd 2520 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
f  e.  ( J 
GrpHom  K )  /\  G  =  F ) )  /\  ( z  e.  P  /\  w  e.  B
) )  ->  (
f `  w )  e.  C )
3815proplem 13953 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  Q  /\  (
f `  w )  e.  C ) )  -> 
( z ( .s
`  K ) ( f `  w ) )  =  ( z ( .s `  M
) ( f `  w ) ) )
3921, 26, 37, 38syl12anc 1183 . . . . . . . . 9  |-  ( ( ( ph  /\  (
f  e.  ( J 
GrpHom  K )  /\  G  =  F ) )  /\  ( z  e.  P  /\  w  e.  B
) )  ->  (
z ( .s `  K ) ( f `
 w ) )  =  ( z ( .s `  M ) ( f `  w
) ) )
4020, 39eqeq12d 2457 . . . . . . . 8  |-  ( ( ( ph  /\  (
f  e.  ( J 
GrpHom  K )  /\  G  =  F ) )  /\  ( z  e.  P  /\  w  e.  B
) )  ->  (
( f `  (
z ( .s `  J ) w ) )  =  ( z ( .s `  K
) ( f `  w ) )  <->  ( f `  ( z ( .s
`  L ) w ) )  =  ( z ( .s `  M ) ( f `
 w ) ) ) )
41402ralbidva 2752 . . . . . . 7  |-  ( (
ph  /\  ( f  e.  ( J  GrpHom  K )  /\  G  =  F ) )  ->  ( A. z  e.  P  A. w  e.  B  ( f `  (
z ( .s `  J ) w ) )  =  ( z ( .s `  K
) ( f `  w ) )  <->  A. z  e.  P  A. w  e.  B  ( f `  ( z ( .s
`  L ) w ) )  =  ( z ( .s `  M ) ( f `
 w ) ) ) )
4241pm5.32da 624 . . . . . 6  |-  ( ph  ->  ( ( ( f  e.  ( J  GrpHom  K )  /\  G  =  F )  /\  A. z  e.  P  A. w  e.  B  (
f `  ( z
( .s `  J
) w ) )  =  ( z ( .s `  K ) ( f `  w
) ) )  <->  ( (
f  e.  ( J 
GrpHom  K )  /\  G  =  F )  /\  A. z  e.  P  A. w  e.  B  (
f `  ( z
( .s `  L
) w ) )  =  ( z ( .s `  M ) ( f `  w
) ) ) ) )
43 df-3an 939 . . . . . 6  |-  ( ( f  e.  ( J 
GrpHom  K )  /\  G  =  F  /\  A. z  e.  P  A. w  e.  B  ( f `  ( z ( .s
`  J ) w ) )  =  ( z ( .s `  K ) ( f `
 w ) ) )  <->  ( ( f  e.  ( J  GrpHom  K )  /\  G  =  F )  /\  A. z  e.  P  A. w  e.  B  (
f `  ( z
( .s `  J
) w ) )  =  ( z ( .s `  K ) ( f `  w
) ) ) )
44 df-3an 939 . . . . . 6  |-  ( ( f  e.  ( J 
GrpHom  K )  /\  G  =  F  /\  A. z  e.  P  A. w  e.  B  ( f `  ( z ( .s
`  L ) w ) )  =  ( z ( .s `  M ) ( f `
 w ) ) )  <->  ( ( f  e.  ( J  GrpHom  K )  /\  G  =  F )  /\  A. z  e.  P  A. w  e.  B  (
f `  ( z
( .s `  L
) w ) )  =  ( z ( .s `  M ) ( f `  w
) ) ) )
4542, 43, 443bitr4g 281 . . . . 5  |-  ( ph  ->  ( ( f  e.  ( J  GrpHom  K )  /\  G  =  F  /\  A. z  e.  P  A. w  e.  B  ( f `  ( z ( .s
`  J ) w ) )  =  ( z ( .s `  K ) ( f `
 w ) ) )  <->  ( f  e.  ( J  GrpHom  K )  /\  G  =  F  /\  A. z  e.  P  A. w  e.  B  ( f `  ( z ( .s
`  L ) w ) )  =  ( z ( .s `  M ) ( f `
 w ) ) ) ) )
4612, 4eqeq12d 2457 . . . . . 6  |-  ( ph  ->  ( G  =  F  <-> 
(Scalar `  K )  =  (Scalar `  J )
) )
474fveq2d 5767 . . . . . . . 8  |-  ( ph  ->  ( Base `  F
)  =  ( Base `  (Scalar `  J )
) )
486, 47syl5eq 2487 . . . . . . 7  |-  ( ph  ->  P  =  ( Base `  (Scalar `  J )
) )
491raleqdv 2917 . . . . . . 7  |-  ( ph  ->  ( A. w  e.  B  ( f `  ( z ( .s
`  J ) w ) )  =  ( z ( .s `  K ) ( f `
 w ) )  <->  A. w  e.  ( Base `  J ) ( f `  ( z ( .s `  J
) w ) )  =  ( z ( .s `  K ) ( f `  w
) ) ) )
5048, 49raleqbidv 2925 . . . . . 6  |-  ( ph  ->  ( A. z  e.  P  A. w  e.  B  ( f `  ( z ( .s
`  J ) w ) )  =  ( z ( .s `  K ) ( f `
 w ) )  <->  A. z  e.  ( Base `  (Scalar `  J
) ) A. w  e.  ( Base `  J
) ( f `  ( z ( .s
`  J ) w ) )  =  ( z ( .s `  K ) ( f `
 w ) ) ) )
5146, 503anbi23d 1258 . . . . 5  |-  ( ph  ->  ( ( f  e.  ( J  GrpHom  K )  /\  G  =  F  /\  A. z  e.  P  A. w  e.  B  ( f `  ( z ( .s
`  J ) w ) )  =  ( z ( .s `  K ) ( f `
 w ) ) )  <->  ( f  e.  ( J  GrpHom  K )  /\  (Scalar `  K
)  =  (Scalar `  J )  /\  A. z  e.  ( Base `  (Scalar `  J )
) A. w  e.  ( Base `  J
) ( f `  ( z ( .s
`  J ) w ) )  =  ( z ( .s `  K ) ( f `
 w ) ) ) ) )
521, 9, 2, 10, 3, 11ghmpropd 15081 . . . . . . 7  |-  ( ph  ->  ( J  GrpHom  K )  =  ( L  GrpHom  M ) )
5352eleq2d 2510 . . . . . 6  |-  ( ph  ->  ( f  e.  ( J  GrpHom  K )  <->  f  e.  ( L  GrpHom  M ) ) )
5413, 5eqeq12d 2457 . . . . . 6  |-  ( ph  ->  ( G  =  F  <-> 
(Scalar `  M )  =  (Scalar `  L )
) )
555fveq2d 5767 . . . . . . . 8  |-  ( ph  ->  ( Base `  F
)  =  ( Base `  (Scalar `  L )
) )
566, 55syl5eq 2487 . . . . . . 7  |-  ( ph  ->  P  =  ( Base `  (Scalar `  L )
) )
572raleqdv 2917 . . . . . . 7  |-  ( ph  ->  ( A. w  e.  B  ( f `  ( z ( .s
`  L ) w ) )  =  ( z ( .s `  M ) ( f `
 w ) )  <->  A. w  e.  ( Base `  L ) ( f `  ( z ( .s `  L
) w ) )  =  ( z ( .s `  M ) ( f `  w
) ) ) )
5856, 57raleqbidv 2925 . . . . . 6  |-  ( ph  ->  ( A. z  e.  P  A. w  e.  B  ( f `  ( z ( .s
`  L ) w ) )  =  ( z ( .s `  M ) ( f `
 w ) )  <->  A. z  e.  ( Base `  (Scalar `  L
) ) A. w  e.  ( Base `  L
) ( f `  ( z ( .s
`  L ) w ) )  =  ( z ( .s `  M ) ( f `
 w ) ) ) )
5953, 54, 583anbi123d 1255 . . . . 5  |-  ( ph  ->  ( ( f  e.  ( J  GrpHom  K )  /\  G  =  F  /\  A. z  e.  P  A. w  e.  B  ( f `  ( z ( .s
`  L ) w ) )  =  ( z ( .s `  M ) ( f `
 w ) ) )  <->  ( f  e.  ( L  GrpHom  M )  /\  (Scalar `  M
)  =  (Scalar `  L )  /\  A. z  e.  ( Base `  (Scalar `  L )
) A. w  e.  ( Base `  L
) ( f `  ( z ( .s
`  L ) w ) )  =  ( z ( .s `  M ) ( f `
 w ) ) ) ) )
6045, 51, 593bitr3d 276 . . . 4  |-  ( ph  ->  ( ( f  e.  ( J  GrpHom  K )  /\  (Scalar `  K
)  =  (Scalar `  J )  /\  A. z  e.  ( Base `  (Scalar `  J )
) A. w  e.  ( Base `  J
) ( f `  ( z ( .s
`  J ) w ) )  =  ( z ( .s `  K ) ( f `
 w ) ) )  <->  ( f  e.  ( L  GrpHom  M )  /\  (Scalar `  M
)  =  (Scalar `  L )  /\  A. z  e.  ( Base `  (Scalar `  L )
) A. w  e.  ( Base `  L
) ( f `  ( z ( .s
`  L ) w ) )  =  ( z ( .s `  M ) ( f `
 w ) ) ) ) )
6117, 60anbi12d 693 . . 3  |-  ( ph  ->  ( ( ( J  e.  LMod  /\  K  e. 
LMod )  /\  (
f  e.  ( J 
GrpHom  K )  /\  (Scalar `  K )  =  (Scalar `  J )  /\  A. z  e.  ( Base `  (Scalar `  J )
) A. w  e.  ( Base `  J
) ( f `  ( z ( .s
`  J ) w ) )  =  ( z ( .s `  K ) ( f `
 w ) ) ) )  <->  ( ( L  e.  LMod  /\  M  e.  LMod )  /\  (
f  e.  ( L 
GrpHom  M )  /\  (Scalar `  M )  =  (Scalar `  L )  /\  A. z  e.  ( Base `  (Scalar `  L )
) A. w  e.  ( Base `  L
) ( f `  ( z ( .s
`  L ) w ) )  =  ( z ( .s `  M ) ( f `
 w ) ) ) ) ) )
62 eqid 2443 . . . 4  |-  (Scalar `  J )  =  (Scalar `  J )
63 eqid 2443 . . . 4  |-  (Scalar `  K )  =  (Scalar `  K )
64 eqid 2443 . . . 4  |-  ( Base `  (Scalar `  J )
)  =  ( Base `  (Scalar `  J )
)
65 eqid 2443 . . . 4  |-  ( .s
`  J )  =  ( .s `  J
)
66 eqid 2443 . . . 4  |-  ( .s
`  K )  =  ( .s `  K
)
6762, 63, 64, 28, 65, 66islmhm 16141 . . 3  |-  ( f  e.  ( J LMHom  K
)  <->  ( ( J  e.  LMod  /\  K  e. 
LMod )  /\  (
f  e.  ( J 
GrpHom  K )  /\  (Scalar `  K )  =  (Scalar `  J )  /\  A. z  e.  ( Base `  (Scalar `  J )
) A. w  e.  ( Base `  J
) ( f `  ( z ( .s
`  J ) w ) )  =  ( z ( .s `  K ) ( f `
 w ) ) ) ) )
68 eqid 2443 . . . 4  |-  (Scalar `  L )  =  (Scalar `  L )
69 eqid 2443 . . . 4  |-  (Scalar `  M )  =  (Scalar `  M )
70 eqid 2443 . . . 4  |-  ( Base `  (Scalar `  L )
)  =  ( Base `  (Scalar `  L )
)
71 eqid 2443 . . . 4  |-  ( Base `  L )  =  (
Base `  L )
72 eqid 2443 . . . 4  |-  ( .s
`  L )  =  ( .s `  L
)
73 eqid 2443 . . . 4  |-  ( .s
`  M )  =  ( .s `  M
)
7468, 69, 70, 71, 72, 73islmhm 16141 . . 3  |-  ( f  e.  ( L LMHom  M
)  <->  ( ( L  e.  LMod  /\  M  e. 
LMod )  /\  (
f  e.  ( L 
GrpHom  M )  /\  (Scalar `  M )  =  (Scalar `  L )  /\  A. z  e.  ( Base `  (Scalar `  L )
) A. w  e.  ( Base `  L
) ( f `  ( z ( .s
`  L ) w ) )  =  ( z ( .s `  M ) ( f `
 w ) ) ) ) )
7561, 67, 743bitr4g 281 . 2  |-  ( ph  ->  ( f  e.  ( J LMHom  K )  <->  f  e.  ( L LMHom  M ) ) )
7675eqrdv 2441 1  |-  ( ph  ->  ( J LMHom  K )  =  ( L LMHom  M
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1654    e. wcel 1728   A.wral 2712   -->wf 5485   ` cfv 5489  (class class class)co 6117   Basecbs 13507   +g cplusg 13567  Scalarcsca 13570   .scvsca 13571    GrpHom cghm 15041   LModclmod 15988   LMHom clmhm 16133
This theorem is referenced by:  phlpropd  16924
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-13 1730  ax-14 1732  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424  ax-rep 4351  ax-sep 4361  ax-nul 4369  ax-pow 4412  ax-pr 4438  ax-un 4736  ax-cnex 9084  ax-resscn 9085  ax-1cn 9086  ax-icn 9087  ax-addcl 9088  ax-addrcl 9089  ax-mulcl 9090  ax-mulrcl 9091  ax-mulcom 9092  ax-addass 9093  ax-mulass 9094  ax-distr 9095  ax-i2m1 9096  ax-1ne0 9097  ax-1rid 9098  ax-rnegex 9099  ax-rrecex 9100  ax-cnre 9101  ax-pre-lttri 9102  ax-pre-lttrn 9103  ax-pre-ltadd 9104  ax-pre-mulgt0 9105
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1661  df-eu 2292  df-mo 2293  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2717  df-rex 2718  df-reu 2719  df-rmo 2720  df-rab 2721  df-v 2967  df-sbc 3171  df-csb 3271  df-dif 3312  df-un 3314  df-in 3316  df-ss 3323  df-pss 3325  df-nul 3617  df-if 3768  df-pw 3830  df-sn 3849  df-pr 3850  df-tp 3851  df-op 3852  df-uni 4045  df-iun 4124  df-br 4244  df-opab 4298  df-mpt 4299  df-tr 4334  df-eprel 4529  df-id 4533  df-po 4538  df-so 4539  df-fr 4576  df-we 4578  df-ord 4619  df-on 4620  df-lim 4621  df-suc 4622  df-om 4881  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5453  df-fun 5491  df-fn 5492  df-f 5493  df-f1 5494  df-fo 5495  df-f1o 5496  df-fv 5497  df-ov 6120  df-oprab 6121  df-mpt2 6122  df-riota 6585  df-recs 6669  df-rdg 6704  df-er 6941  df-map 7056  df-en 7146  df-dom 7147  df-sdom 7148  df-pnf 9160  df-mnf 9161  df-xr 9162  df-ltxr 9163  df-le 9164  df-sub 9331  df-neg 9332  df-nn 10039  df-2 10096  df-ndx 13510  df-slot 13511  df-base 13512  df-sets 13513  df-plusg 13580  df-0g 13765  df-mnd 14728  df-mhm 14776  df-grp 14850  df-ghm 15042  df-mgp 15687  df-rng 15701  df-ur 15703  df-lmod 15990  df-lmhm 16136
  Copyright terms: Public domain W3C validator