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

Theorem lmhmpropd 17514
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 17368 . . . . 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 17368 . . . . 5  |-  ( ph  ->  ( K  e.  LMod  <->  M  e.  LMod ) )
178, 16anbi12d 710 . . . 4  |-  ( ph  ->  ( ( J  e. 
LMod  /\  K  e.  LMod ) 
<->  ( L  e.  LMod  /\  M  e.  LMod )
) )
187proplem 14944 . . . . . . . . . . 11  |-  ( (
ph  /\  ( z  e.  P  /\  w  e.  B ) )  -> 
( z ( .s
`  J ) w )  =  ( z ( .s `  L
) w ) )
1918adantlr 714 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
f  e.  ( J 
GrpHom  K )  /\  G  =  F ) )  /\  ( z  e.  P  /\  w  e.  B
) )  ->  (
z ( .s `  J ) w )  =  ( z ( .s `  L ) w ) )
2019fveq2d 5869 . . . . . . . . 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 753 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
f  e.  ( J 
GrpHom  K )  /\  G  =  F ) )  /\  ( z  e.  P  /\  w  e.  B
) )  ->  ph )
22 simprl 755 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
f  e.  ( J 
GrpHom  K )  /\  G  =  F ) )  /\  ( z  e.  P  /\  w  e.  B
) )  ->  z  e.  P )
23 simplrr 760 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
f  e.  ( J 
GrpHom  K )  /\  G  =  F ) )  /\  ( z  e.  P  /\  w  e.  B
) )  ->  G  =  F )
2423fveq2d 5869 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
f  e.  ( J 
GrpHom  K )  /\  G  =  F ) )  /\  ( z  e.  P  /\  w  e.  B
) )  ->  ( Base `  G )  =  ( Base `  F
) )
2524, 14, 63eqtr4g 2533 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
f  e.  ( J 
GrpHom  K )  /\  G  =  F ) )  /\  ( z  e.  P  /\  w  e.  B
) )  ->  Q  =  P )
2622, 25eleqtrrd 2558 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
f  e.  ( J 
GrpHom  K )  /\  G  =  F ) )  /\  ( z  e.  P  /\  w  e.  B
) )  ->  z  e.  Q )
27 simplrl 759 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
f  e.  ( J 
GrpHom  K )  /\  G  =  F ) )  /\  ( z  e.  P  /\  w  e.  B
) )  ->  f  e.  ( J  GrpHom  K ) )
28 eqid 2467 . . . . . . . . . . . . . 14  |-  ( Base `  J )  =  (
Base `  J )
29 eqid 2467 . . . . . . . . . . . . . 14  |-  ( Base `  K )  =  (
Base `  K )
3028, 29ghmf 16073 . . . . . . . . . . . . 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 756 . . . . . . . . . . . . 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 2557 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
f  e.  ( J 
GrpHom  K )  /\  G  =  F ) )  /\  ( z  e.  P  /\  w  e.  B
) )  ->  w  e.  ( Base `  J
) )
3531, 34ffvelrnd 6021 . . . . . . . . . . 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 2558 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
f  e.  ( J 
GrpHom  K )  /\  G  =  F ) )  /\  ( z  e.  P  /\  w  e.  B
) )  ->  (
f `  w )  e.  C )
3815proplem 14944 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  Q  /\  (
f `  w )  e.  C ) )  -> 
( z ( .s
`  K ) ( f `  w ) )  =  ( z ( .s `  M
) ( f `  w ) ) )
3921, 26, 37, 38syl12anc 1226 . . . . . . . . 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 2489 . . . . . . . 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 2906 . . . . . . 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 641 . . . . . 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 975 . . . . . 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 975 . . . . . 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 288 . . . . 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 2489 . . . . . 6  |-  ( ph  ->  ( G  =  F  <-> 
(Scalar `  K )  =  (Scalar `  J )
) )
474fveq2d 5869 . . . . . . . 8  |-  ( ph  ->  ( Base `  F
)  =  ( Base `  (Scalar `  J )
) )
486, 47syl5eq 2520 . . . . . . 7  |-  ( ph  ->  P  =  ( Base `  (Scalar `  J )
) )
491raleqdv 3064 . . . . . . 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 3072 . . . . . 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 1302 . . . . 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 16106 . . . . . . 7  |-  ( ph  ->  ( J  GrpHom  K )  =  ( L  GrpHom  M ) )
5352eleq2d 2537 . . . . . 6  |-  ( ph  ->  ( f  e.  ( J  GrpHom  K )  <->  f  e.  ( L  GrpHom  M ) ) )
5413, 5eqeq12d 2489 . . . . . 6  |-  ( ph  ->  ( G  =  F  <-> 
(Scalar `  M )  =  (Scalar `  L )
) )
555fveq2d 5869 . . . . . . . 8  |-  ( ph  ->  ( Base `  F
)  =  ( Base `  (Scalar `  L )
) )
566, 55syl5eq 2520 . . . . . . 7  |-  ( ph  ->  P  =  ( Base `  (Scalar `  L )
) )
572raleqdv 3064 . . . . . . 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 3072 . . . . . 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 1299 . . . . 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 283 . . . 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 710 . . 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 2467 . . . 4  |-  (Scalar `  J )  =  (Scalar `  J )
63 eqid 2467 . . . 4  |-  (Scalar `  K )  =  (Scalar `  K )
64 eqid 2467 . . . 4  |-  ( Base `  (Scalar `  J )
)  =  ( Base `  (Scalar `  J )
)
65 eqid 2467 . . . 4  |-  ( .s
`  J )  =  ( .s `  J
)
66 eqid 2467 . . . 4  |-  ( .s
`  K )  =  ( .s `  K
)
6762, 63, 64, 28, 65, 66islmhm 17468 . . 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 2467 . . . 4  |-  (Scalar `  L )  =  (Scalar `  L )
69 eqid 2467 . . . 4  |-  (Scalar `  M )  =  (Scalar `  M )
70 eqid 2467 . . . 4  |-  ( Base `  (Scalar `  L )
)  =  ( Base `  (Scalar `  L )
)
71 eqid 2467 . . . 4  |-  ( Base `  L )  =  (
Base `  L )
72 eqid 2467 . . . 4  |-  ( .s
`  L )  =  ( .s `  L
)
73 eqid 2467 . . . 4  |-  ( .s
`  M )  =  ( .s `  M
)
7468, 69, 70, 71, 72, 73islmhm 17468 . . 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 288 . 2  |-  ( ph  ->  ( f  e.  ( J LMHom  K )  <->  f  e.  ( L LMHom  M ) ) )
7675eqrdv 2464 1  |-  ( ph  ->  ( J LMHom  K )  =  ( L LMHom  M
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2814   -->wf 5583   ` cfv 5587  (class class class)co 6283   Basecbs 14489   +g cplusg 14554  Scalarcsca 14557   .scvsca 14558    GrpHom cghm 16066   LModclmod 17307   LMHom clmhm 17460
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575  ax-cnex 9547  ax-resscn 9548  ax-1cn 9549  ax-icn 9550  ax-addcl 9551  ax-addrcl 9552  ax-mulcl 9553  ax-mulrcl 9554  ax-mulcom 9555  ax-addass 9556  ax-mulass 9557  ax-distr 9558  ax-i2m1 9559  ax-1ne0 9560  ax-1rid 9561  ax-rnegex 9562  ax-rrecex 9563  ax-cnre 9564  ax-pre-lttri 9565  ax-pre-lttrn 9566  ax-pre-ltadd 9567  ax-pre-mulgt0 9568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-riota 6244  df-ov 6286  df-oprab 6287  df-mpt2 6288  df-om 6680  df-recs 7042  df-rdg 7076  df-er 7311  df-map 7422  df-en 7517  df-dom 7518  df-sdom 7519  df-pnf 9629  df-mnf 9630  df-xr 9631  df-ltxr 9632  df-le 9633  df-sub 9806  df-neg 9807  df-nn 10536  df-2 10593  df-ndx 14492  df-slot 14493  df-base 14494  df-sets 14495  df-plusg 14567  df-0g 14696  df-mnd 15731  df-mhm 15783  df-grp 15864  df-ghm 16067  df-mgp 16941  df-ur 16953  df-rng 16997  df-lmod 17309  df-lmhm 17463
This theorem is referenced by:  phlpropd  18473
  Copyright terms: Public domain W3C validator