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Theorem lmhmpropd 16108
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 15970 . . . . 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 15970 . . . . 5  |-  ( ph  ->  ( K  e.  LMod  <->  M  e.  LMod ) )
178, 16anbi12d 692 . . . 4  |-  ( ph  ->  ( ( J  e. 
LMod  /\  K  e.  LMod ) 
<->  ( L  e.  LMod  /\  M  e.  LMod )
) )
187proplem 13878 . . . . . . . . . . 11  |-  ( (
ph  /\  ( z  e.  P  /\  w  e.  B ) )  -> 
( z ( .s
`  J ) w )  =  ( z ( .s `  L
) w ) )
1918adantlr 696 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
f  e.  ( J 
GrpHom  K )  /\  G  =  F ) )  /\  ( z  e.  P  /\  w  e.  B
) )  ->  (
z ( .s `  J ) w )  =  ( z ( .s `  L ) w ) )
2019fveq2d 5699 . . . . . . . . 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 731 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
f  e.  ( J 
GrpHom  K )  /\  G  =  F ) )  /\  ( z  e.  P  /\  w  e.  B
) )  ->  ph )
22 simprl 733 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
f  e.  ( J 
GrpHom  K )  /\  G  =  F ) )  /\  ( z  e.  P  /\  w  e.  B
) )  ->  z  e.  P )
23 simplrr 738 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
f  e.  ( J 
GrpHom  K )  /\  G  =  F ) )  /\  ( z  e.  P  /\  w  e.  B
) )  ->  G  =  F )
2423fveq2d 5699 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
f  e.  ( J 
GrpHom  K )  /\  G  =  F ) )  /\  ( z  e.  P  /\  w  e.  B
) )  ->  ( Base `  G )  =  ( Base `  F
) )
2524, 14, 63eqtr4g 2469 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
f  e.  ( J 
GrpHom  K )  /\  G  =  F ) )  /\  ( z  e.  P  /\  w  e.  B
) )  ->  Q  =  P )
2622, 25eleqtrrd 2489 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
f  e.  ( J 
GrpHom  K )  /\  G  =  F ) )  /\  ( z  e.  P  /\  w  e.  B
) )  ->  z  e.  Q )
27 simplrl 737 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
f  e.  ( J 
GrpHom  K )  /\  G  =  F ) )  /\  ( z  e.  P  /\  w  e.  B
) )  ->  f  e.  ( J  GrpHom  K ) )
28 eqid 2412 . . . . . . . . . . . . . 14  |-  ( Base `  J )  =  (
Base `  J )
29 eqid 2412 . . . . . . . . . . . . . 14  |-  ( Base `  K )  =  (
Base `  K )
3028, 29ghmf 14973 . . . . . . . . . . . . 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 734 . . . . . . . . . . . . 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 2488 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
f  e.  ( J 
GrpHom  K )  /\  G  =  F ) )  /\  ( z  e.  P  /\  w  e.  B
) )  ->  w  e.  ( Base `  J
) )
3531, 34ffvelrnd 5838 . . . . . . . . . . 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 2489 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
f  e.  ( J 
GrpHom  K )  /\  G  =  F ) )  /\  ( z  e.  P  /\  w  e.  B
) )  ->  (
f `  w )  e.  C )
3815proplem 13878 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  Q  /\  (
f `  w )  e.  C ) )  -> 
( z ( .s
`  K ) ( f `  w ) )  =  ( z ( .s `  M
) ( f `  w ) ) )
3921, 26, 37, 38syl12anc 1182 . . . . . . . . 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 2426 . . . . . . . 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 2714 . . . . . . 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 623 . . . . . 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 938 . . . . . 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 938 . . . . . 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 280 . . . . 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 2426 . . . . . 6  |-  ( ph  ->  ( G  =  F  <-> 
(Scalar `  K )  =  (Scalar `  J )
) )
474fveq2d 5699 . . . . . . . 8  |-  ( ph  ->  ( Base `  F
)  =  ( Base `  (Scalar `  J )
) )
486, 47syl5eq 2456 . . . . . . 7  |-  ( ph  ->  P  =  ( Base `  (Scalar `  J )
) )
491raleqdv 2878 . . . . . . 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 2884 . . . . . 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 1257 . . . . 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 15006 . . . . . . 7  |-  ( ph  ->  ( J  GrpHom  K )  =  ( L  GrpHom  M ) )
5352eleq2d 2479 . . . . . 6  |-  ( ph  ->  ( f  e.  ( J  GrpHom  K )  <->  f  e.  ( L  GrpHom  M ) ) )
5413, 5eqeq12d 2426 . . . . . 6  |-  ( ph  ->  ( G  =  F  <-> 
(Scalar `  M )  =  (Scalar `  L )
) )
555fveq2d 5699 . . . . . . . 8  |-  ( ph  ->  ( Base `  F
)  =  ( Base `  (Scalar `  L )
) )
566, 55syl5eq 2456 . . . . . . 7  |-  ( ph  ->  P  =  ( Base `  (Scalar `  L )
) )
572raleqdv 2878 . . . . . . 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 2884 . . . . . 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 1254 . . . . 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 275 . . . 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 692 . . 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 2412 . . . 4  |-  (Scalar `  J )  =  (Scalar `  J )
63 eqid 2412 . . . 4  |-  (Scalar `  K )  =  (Scalar `  K )
64 eqid 2412 . . . 4  |-  ( Base `  (Scalar `  J )
)  =  ( Base `  (Scalar `  J )
)
65 eqid 2412 . . . 4  |-  ( .s
`  J )  =  ( .s `  J
)
66 eqid 2412 . . . 4  |-  ( .s
`  K )  =  ( .s `  K
)
6762, 63, 64, 28, 65, 66islmhm 16066 . . 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 2412 . . . 4  |-  (Scalar `  L )  =  (Scalar `  L )
69 eqid 2412 . . . 4  |-  (Scalar `  M )  =  (Scalar `  M )
70 eqid 2412 . . . 4  |-  ( Base `  (Scalar `  L )
)  =  ( Base `  (Scalar `  L )
)
71 eqid 2412 . . . 4  |-  ( Base `  L )  =  (
Base `  L )
72 eqid 2412 . . . 4  |-  ( .s
`  L )  =  ( .s `  L
)
73 eqid 2412 . . . 4  |-  ( .s
`  M )  =  ( .s `  M
)
7468, 69, 70, 71, 72, 73islmhm 16066 . . 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 280 . 2  |-  ( ph  ->  ( f  e.  ( J LMHom  K )  <->  f  e.  ( L LMHom  M ) ) )
7675eqrdv 2410 1  |-  ( ph  ->  ( J LMHom  K )  =  ( L LMHom  M
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   A.wral 2674   -->wf 5417   ` cfv 5421  (class class class)co 6048   Basecbs 13432   +g cplusg 13492  Scalarcsca 13495   .scvsca 13496    GrpHom cghm 14966   LModclmod 15913   LMHom clmhm 16058
This theorem is referenced by:  phlpropd  16849
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-riota 6516  df-recs 6600  df-rdg 6635  df-er 6872  df-map 6987  df-en 7077  df-dom 7078  df-sdom 7079  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-nn 9965  df-2 10022  df-ndx 13435  df-slot 13436  df-base 13437  df-sets 13438  df-plusg 13505  df-0g 13690  df-mnd 14653  df-mhm 14701  df-grp 14775  df-ghm 14967  df-mgp 15612  df-rng 15626  df-ur 15628  df-lmod 15915  df-lmhm 16061
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