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

Theorem lmhmpropd 15826
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 15688 . . . . 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 15688 . . . . 5  |-  ( ph  ->  ( K  e.  LMod  <->  M  e.  LMod ) )
178, 16anbi12d 691 . . . 4  |-  ( ph  ->  ( ( J  e. 
LMod  /\  K  e.  LMod ) 
<->  ( L  e.  LMod  /\  M  e.  LMod )
) )
187proplem 13592 . . . . . . . . . . 11  |-  ( (
ph  /\  ( z  e.  P  /\  w  e.  B ) )  -> 
( z ( .s
`  J ) w )  =  ( z ( .s `  L
) w ) )
1918adantlr 695 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
f  e.  ( J 
GrpHom  K )  /\  G  =  F ) )  /\  ( z  e.  P  /\  w  e.  B
) )  ->  (
z ( .s `  J ) w )  =  ( z ( .s `  L ) w ) )
2019fveq2d 5529 . . . . . . . . 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 730 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
f  e.  ( J 
GrpHom  K )  /\  G  =  F ) )  /\  ( z  e.  P  /\  w  e.  B
) )  ->  ph )
22 simprl 732 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
f  e.  ( J 
GrpHom  K )  /\  G  =  F ) )  /\  ( z  e.  P  /\  w  e.  B
) )  ->  z  e.  P )
23 simplrr 737 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
f  e.  ( J 
GrpHom  K )  /\  G  =  F ) )  /\  ( z  e.  P  /\  w  e.  B
) )  ->  G  =  F )
2423fveq2d 5529 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
f  e.  ( J 
GrpHom  K )  /\  G  =  F ) )  /\  ( z  e.  P  /\  w  e.  B
) )  ->  ( Base `  G )  =  ( Base `  F
) )
2524, 14, 63eqtr4g 2340 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
f  e.  ( J 
GrpHom  K )  /\  G  =  F ) )  /\  ( z  e.  P  /\  w  e.  B
) )  ->  Q  =  P )
2622, 25eleqtrrd 2360 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
f  e.  ( J 
GrpHom  K )  /\  G  =  F ) )  /\  ( z  e.  P  /\  w  e.  B
) )  ->  z  e.  Q )
27 simplrl 736 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
f  e.  ( J 
GrpHom  K )  /\  G  =  F ) )  /\  ( z  e.  P  /\  w  e.  B
) )  ->  f  e.  ( J  GrpHom  K ) )
28 eqid 2283 . . . . . . . . . . . . . 14  |-  ( Base `  J )  =  (
Base `  J )
29 eqid 2283 . . . . . . . . . . . . . 14  |-  ( Base `  K )  =  (
Base `  K )
3028, 29ghmf 14687 . . . . . . . . . . . . 13  |-  ( f  e.  ( J  GrpHom  K )  ->  f :
( Base `  J ) --> ( Base `  K )
)
3127, 30syl 15 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
f  e.  ( J 
GrpHom  K )  /\  G  =  F ) )  /\  ( z  e.  P  /\  w  e.  B
) )  ->  f : ( Base `  J
) --> ( Base `  K
) )
32 simprr 733 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
f  e.  ( J 
GrpHom  K )  /\  G  =  F ) )  /\  ( z  e.  P  /\  w  e.  B
) )  ->  w  e.  B )
3321, 1syl 15 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
f  e.  ( J 
GrpHom  K )  /\  G  =  F ) )  /\  ( z  e.  P  /\  w  e.  B
) )  ->  B  =  ( Base `  J
) )
3432, 33eleqtrd 2359 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
f  e.  ( J 
GrpHom  K )  /\  G  =  F ) )  /\  ( z  e.  P  /\  w  e.  B
) )  ->  w  e.  ( Base `  J
) )
35 ffvelrn 5663 . . . . . . . . . . . 12  |-  ( ( f : ( Base `  J ) --> ( Base `  K )  /\  w  e.  ( Base `  J
) )  ->  (
f `  w )  e.  ( Base `  K
) )
3631, 34, 35syl2anc 642 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
f  e.  ( J 
GrpHom  K )  /\  G  =  F ) )  /\  ( z  e.  P  /\  w  e.  B
) )  ->  (
f `  w )  e.  ( Base `  K
) )
3721, 9syl 15 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
f  e.  ( J 
GrpHom  K )  /\  G  =  F ) )  /\  ( z  e.  P  /\  w  e.  B
) )  ->  C  =  ( Base `  K
) )
3836, 37eleqtrrd 2360 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
f  e.  ( J 
GrpHom  K )  /\  G  =  F ) )  /\  ( z  e.  P  /\  w  e.  B
) )  ->  (
f `  w )  e.  C )
3915proplem 13592 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  Q  /\  (
f `  w )  e.  C ) )  -> 
( z ( .s
`  K ) ( f `  w ) )  =  ( z ( .s `  M
) ( f `  w ) ) )
4021, 26, 38, 39syl12anc 1180 . . . . . . . . 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
) ) )
4120, 40eqeq12d 2297 . . . . . . . 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 ) ) ) )
42412ralbidva 2583 . . . . . . 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 ) ) ) )
4342pm5.32da 622 . . . . . 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
) ) ) ) )
44 df-3an 936 . . . . . 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
) ) ) )
45 df-3an 936 . . . . . 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
) ) ) )
4643, 44, 453bitr4g 279 . . . . 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 ) ) ) ) )
4712, 4eqeq12d 2297 . . . . . 6  |-  ( ph  ->  ( G  =  F  <-> 
(Scalar `  K )  =  (Scalar `  J )
) )
484fveq2d 5529 . . . . . . . 8  |-  ( ph  ->  ( Base `  F
)  =  ( Base `  (Scalar `  J )
) )
496, 48syl5eq 2327 . . . . . . 7  |-  ( ph  ->  P  =  ( Base `  (Scalar `  J )
) )
501raleqdv 2742 . . . . . . 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
) ) ) )
5149, 50raleqbidv 2748 . . . . . 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 ) ) ) )
5247, 513anbi23d 1255 . . . . 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 ) ) ) ) )
531, 9, 2, 10, 3, 11ghmpropd 14720 . . . . . . 7  |-  ( ph  ->  ( J  GrpHom  K )  =  ( L  GrpHom  M ) )
5453eleq2d 2350 . . . . . 6  |-  ( ph  ->  ( f  e.  ( J  GrpHom  K )  <->  f  e.  ( L  GrpHom  M ) ) )
5513, 5eqeq12d 2297 . . . . . 6  |-  ( ph  ->  ( G  =  F  <-> 
(Scalar `  M )  =  (Scalar `  L )
) )
565fveq2d 5529 . . . . . . . 8  |-  ( ph  ->  ( Base `  F
)  =  ( Base `  (Scalar `  L )
) )
576, 56syl5eq 2327 . . . . . . 7  |-  ( ph  ->  P  =  ( Base `  (Scalar `  L )
) )
582raleqdv 2742 . . . . . . 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
) ) ) )
5957, 58raleqbidv 2748 . . . . . 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 ) ) ) )
6054, 55, 593anbi123d 1252 . . . . 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 ) ) ) ) )
6146, 52, 603bitr3d 274 . . . 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 ) ) ) ) )
6217, 61anbi12d 691 . . 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 ) ) ) ) ) )
63 eqid 2283 . . . 4  |-  (Scalar `  J )  =  (Scalar `  J )
64 eqid 2283 . . . 4  |-  (Scalar `  K )  =  (Scalar `  K )
65 eqid 2283 . . . 4  |-  ( Base `  (Scalar `  J )
)  =  ( Base `  (Scalar `  J )
)
66 eqid 2283 . . . 4  |-  ( .s
`  J )  =  ( .s `  J
)
67 eqid 2283 . . . 4  |-  ( .s
`  K )  =  ( .s `  K
)
6863, 64, 65, 28, 66, 67islmhm 15784 . . 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 ) ) ) ) )
69 eqid 2283 . . . 4  |-  (Scalar `  L )  =  (Scalar `  L )
70 eqid 2283 . . . 4  |-  (Scalar `  M )  =  (Scalar `  M )
71 eqid 2283 . . . 4  |-  ( Base `  (Scalar `  L )
)  =  ( Base `  (Scalar `  L )
)
72 eqid 2283 . . . 4  |-  ( Base `  L )  =  (
Base `  L )
73 eqid 2283 . . . 4  |-  ( .s
`  L )  =  ( .s `  L
)
74 eqid 2283 . . . 4  |-  ( .s
`  M )  =  ( .s `  M
)
7569, 70, 71, 72, 73, 74islmhm 15784 . . 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 ) ) ) ) )
7662, 68, 753bitr4g 279 . 2  |-  ( ph  ->  ( f  e.  ( J LMHom  K )  <->  f  e.  ( L LMHom  M ) ) )
7776eqrdv 2281 1  |-  ( ph  ->  ( J LMHom  K )  =  ( L LMHom  M
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   -->wf 5251   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208  Scalarcsca 13211   .scvsca 13212    GrpHom cghm 14680   LModclmod 15627   LMHom clmhm 15776
This theorem is referenced by:  phlpropd  16559
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-map 6774  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-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-plusg 13221  df-0g 13404  df-mnd 14367  df-mhm 14415  df-grp 14489  df-ghm 14681  df-mgp 15326  df-rng 15340  df-ur 15342  df-lmod 15629  df-lmhm 15779
  Copyright terms: Public domain W3C validator