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Theorem funcrngcsetc 40508
Description: The "natural forgetful functor" from the category of non-unital rings into the category of sets which sends each non-unital ring to its underlying set (base set) and the morphisms (non-unital ring homomorphisms) to mappings of the corresponding base sets. An alternate proof is provided in funcrngcsetcALT 40509, using cofuval2 15870 to construct the "natural forgetful functor" from the category of non-unital rings into the category of sets by composing the "inclusion functor" from the category of non-unital rings into the category of extensible structures, see rngcifuestrc 40507, and the "natural forgetful functor" from the category of extensible structures into the category of sets, see funcestrcsetc 16112. (Contributed by AV, 26-Mar-2020.)
Hypotheses
Ref Expression
funcrngcsetc.r  |-  R  =  (RngCat `  U )
funcrngcsetc.s  |-  S  =  ( SetCat `  U )
funcrngcsetc.b  |-  B  =  ( Base `  R
)
funcrngcsetc.u  |-  ( ph  ->  U  e. WUni )
funcrngcsetc.f  |-  ( ph  ->  F  =  ( x  e.  B  |->  ( Base `  x ) ) )
funcrngcsetc.g  |-  ( ph  ->  G  =  ( x  e.  B ,  y  e.  B  |->  (  _I  |`  ( x RngHomo  y ) ) ) )
Assertion
Ref Expression
funcrngcsetc  |-  ( ph  ->  F ( R  Func  S ) G )
Distinct variable groups:    x, B, y    x, R, y    x, S    x, U, y    ph, x, y
Allowed substitution hints:    S( y)    F( x, y)    G( x, y)

Proof of Theorem funcrngcsetc
Dummy variables  a 
b  f are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2471 . . . . . 6  |-  (ExtStrCat `  U
)  =  (ExtStrCat `  U
)
2 funcrngcsetc.s . . . . . 6  |-  S  =  ( SetCat `  U )
3 eqid 2471 . . . . . 6  |-  ( Base `  (ExtStrCat `  U )
)  =  ( Base `  (ExtStrCat `  U )
)
4 eqid 2471 . . . . . 6  |-  ( Base `  S )  =  (
Base `  S )
5 funcrngcsetc.u . . . . . 6  |-  ( ph  ->  U  e. WUni )
61, 5estrcbas 16088 . . . . . . 7  |-  ( ph  ->  U  =  ( Base `  (ExtStrCat `  U )
) )
76mpteq1d 4477 . . . . . 6  |-  ( ph  ->  ( x  e.  U  |->  ( Base `  x
) )  =  ( x  e.  ( Base `  (ExtStrCat `  U )
)  |->  ( Base `  x
) ) )
8 mpt2eq12 6370 . . . . . . 7  |-  ( ( U  =  ( Base `  (ExtStrCat `  U )
)  /\  U  =  ( Base `  (ExtStrCat `  U
) ) )  -> 
( x  e.  U ,  y  e.  U  |->  (  _I  |`  (
( Base `  y )  ^m  ( Base `  x
) ) ) )  =  ( x  e.  ( Base `  (ExtStrCat `  U ) ) ,  y  e.  ( Base `  (ExtStrCat `  U )
)  |->  (  _I  |`  (
( Base `  y )  ^m  ( Base `  x
) ) ) ) )
96, 6, 8syl2anc 673 . . . . . 6  |-  ( ph  ->  ( x  e.  U ,  y  e.  U  |->  (  _I  |`  (
( Base `  y )  ^m  ( Base `  x
) ) ) )  =  ( x  e.  ( Base `  (ExtStrCat `  U ) ) ,  y  e.  ( Base `  (ExtStrCat `  U )
)  |->  (  _I  |`  (
( Base `  y )  ^m  ( Base `  x
) ) ) ) )
101, 2, 3, 4, 5, 7, 9funcestrcsetc 16112 . . . . 5  |-  ( ph  ->  ( x  e.  U  |->  ( Base `  x
) ) ( (ExtStrCat `  U )  Func  S
) ( x  e.  U ,  y  e.  U  |->  (  _I  |`  (
( Base `  y )  ^m  ( Base `  x
) ) ) ) )
11 df-br 4396 . . . . 5  |-  ( ( x  e.  U  |->  (
Base `  x )
) ( (ExtStrCat `  U
)  Func  S )
( x  e.  U ,  y  e.  U  |->  (  _I  |`  (
( Base `  y )  ^m  ( Base `  x
) ) ) )  <->  <. ( x  e.  U  |->  ( Base `  x
) ) ,  ( x  e.  U , 
y  e.  U  |->  (  _I  |`  ( ( Base `  y )  ^m  ( Base `  x )
) ) ) >.  e.  ( (ExtStrCat `  U
)  Func  S )
)
1210, 11sylib 201 . . . 4  |-  ( ph  -> 
<. ( x  e.  U  |->  ( Base `  x
) ) ,  ( x  e.  U , 
y  e.  U  |->  (  _I  |`  ( ( Base `  y )  ^m  ( Base `  x )
) ) ) >.  e.  ( (ExtStrCat `  U
)  Func  S )
)
13 funcrngcsetc.r . . . . . . 7  |-  R  =  (RngCat `  U )
14 eqid 2471 . . . . . . 7  |-  ( Base `  R )  =  (
Base `  R )
1513, 14, 5rngcbas 40475 . . . . . 6  |-  ( ph  ->  ( Base `  R
)  =  ( U  i^i Rng ) )
16 incom 3616 . . . . . 6  |-  ( U  i^i Rng )  =  (Rng 
i^i  U )
1715, 16syl6eq 2521 . . . . 5  |-  ( ph  ->  ( Base `  R
)  =  (Rng  i^i  U ) )
18 eqid 2471 . . . . . 6  |-  ( Hom  `  R )  =  ( Hom  `  R )
1913, 14, 5, 18rngchomfval 40476 . . . . 5  |-  ( ph  ->  ( Hom  `  R
)  =  ( RngHomo  |`  (
( Base `  R )  X.  ( Base `  R
) ) ) )
201, 5, 17, 19rnghmsubcsetc 40487 . . . 4  |-  ( ph  ->  ( Hom  `  R
)  e.  (Subcat `  (ExtStrCat `  U ) ) )
2112, 20funcres 15879 . . 3  |-  ( ph  ->  ( <. ( x  e.  U  |->  ( Base `  x
) ) ,  ( x  e.  U , 
y  e.  U  |->  (  _I  |`  ( ( Base `  y )  ^m  ( Base `  x )
) ) ) >.  |`f  ( Hom  `  R )
)  e.  ( ( (ExtStrCat `  U )  |`cat  ( Hom  `  R )
)  Func  S )
)
22 mptexg 6151 . . . . . 6  |-  ( U  e. WUni  ->  ( x  e.  U  |->  ( Base `  x
) )  e.  _V )
235, 22syl 17 . . . . 5  |-  ( ph  ->  ( x  e.  U  |->  ( Base `  x
) )  e.  _V )
24 fvex 5889 . . . . . 6  |-  ( Hom  `  R )  e.  _V
2524a1i 11 . . . . 5  |-  ( ph  ->  ( Hom  `  R
)  e.  _V )
26 mpt2exga 6888 . . . . . 6  |-  ( ( U  e. WUni  /\  U  e. WUni )  ->  ( x  e.  U ,  y  e.  U  |->  (  _I  |`  (
( Base `  y )  ^m  ( Base `  x
) ) ) )  e.  _V )
275, 5, 26syl2anc 673 . . . . 5  |-  ( ph  ->  ( x  e.  U ,  y  e.  U  |->  (  _I  |`  (
( Base `  y )  ^m  ( Base `  x
) ) ) )  e.  _V )
2815, 19rnghmresfn 40473 . . . . 5  |-  ( ph  ->  ( Hom  `  R
)  Fn  ( (
Base `  R )  X.  ( Base `  R
) ) )
2923, 25, 27, 28resfval2 15876 . . . 4  |-  ( ph  ->  ( <. ( x  e.  U  |->  ( Base `  x
) ) ,  ( x  e.  U , 
y  e.  U  |->  (  _I  |`  ( ( Base `  y )  ^m  ( Base `  x )
) ) ) >.  |`f  ( Hom  `  R )
)  =  <. (
( x  e.  U  |->  ( Base `  x
) )  |`  ( Base `  R ) ) ,  ( a  e.  ( Base `  R
) ,  b  e.  ( Base `  R
)  |->  ( ( a ( x  e.  U ,  y  e.  U  |->  (  _I  |`  (
( Base `  y )  ^m  ( Base `  x
) ) ) ) b )  |`  (
a ( Hom  `  R
) b ) ) ) >. )
30 inss1 3643 . . . . . . . 8  |-  ( U  i^i Rng )  C_  U
3115, 30syl6eqss 3468 . . . . . . 7  |-  ( ph  ->  ( Base `  R
)  C_  U )
3231resmptd 5162 . . . . . 6  |-  ( ph  ->  ( ( x  e.  U  |->  ( Base `  x
) )  |`  ( Base `  R ) )  =  ( x  e.  ( Base `  R
)  |->  ( Base `  x
) ) )
33 funcrngcsetc.f . . . . . . 7  |-  ( ph  ->  F  =  ( x  e.  B  |->  ( Base `  x ) ) )
34 funcrngcsetc.b . . . . . . . . 9  |-  B  =  ( Base `  R
)
3534a1i 11 . . . . . . . 8  |-  ( ph  ->  B  =  ( Base `  R ) )
3635mpteq1d 4477 . . . . . . 7  |-  ( ph  ->  ( x  e.  B  |->  ( Base `  x
) )  =  ( x  e.  ( Base `  R )  |->  ( Base `  x ) ) )
3733, 36eqtr2d 2506 . . . . . 6  |-  ( ph  ->  ( x  e.  (
Base `  R )  |->  ( Base `  x
) )  =  F )
3832, 37eqtrd 2505 . . . . 5  |-  ( ph  ->  ( ( x  e.  U  |->  ( Base `  x
) )  |`  ( Base `  R ) )  =  F )
39 funcrngcsetc.g . . . . . 6  |-  ( ph  ->  G  =  ( x  e.  B ,  y  e.  B  |->  (  _I  |`  ( x RngHomo  y ) ) ) )
40 oveq1 6315 . . . . . . . . 9  |-  ( x  =  a  ->  (
x RngHomo  y )  =  ( a RngHomo  y ) )
4140reseq2d 5111 . . . . . . . 8  |-  ( x  =  a  ->  (  _I  |`  ( x RngHomo  y
) )  =  (  _I  |`  ( a RngHomo  y ) ) )
42 oveq2 6316 . . . . . . . . 9  |-  ( y  =  b  ->  (
a RngHomo  y )  =  ( a RngHomo  b ) )
4342reseq2d 5111 . . . . . . . 8  |-  ( y  =  b  ->  (  _I  |`  ( a RngHomo  y
) )  =  (  _I  |`  ( a RngHomo  b ) ) )
4441, 43cbvmpt2v 6390 . . . . . . 7  |-  ( x  e.  B ,  y  e.  B  |->  (  _I  |`  ( x RngHomo  y ) ) )  =  ( a  e.  B , 
b  e.  B  |->  (  _I  |`  ( a RngHomo  b ) ) )
4544a1i 11 . . . . . 6  |-  ( ph  ->  ( x  e.  B ,  y  e.  B  |->  (  _I  |`  (
x RngHomo  y ) ) )  =  ( a  e.  B ,  b  e.  B  |->  (  _I  |`  (
a RngHomo  b ) ) ) )
4634a1i 11 . . . . . . 7  |-  ( (
ph  /\  a  e.  B )  ->  B  =  ( Base `  R
) )
47 eqidd 2472 . . . . . . . . . 10  |-  ( (
ph  /\  ( a  e.  B  /\  b  e.  B ) )  -> 
( x  e.  U ,  y  e.  U  |->  (  _I  |`  (
( Base `  y )  ^m  ( Base `  x
) ) ) )  =  ( x  e.  U ,  y  e.  U  |->  (  _I  |`  (
( Base `  y )  ^m  ( Base `  x
) ) ) ) )
48 fveq2 5879 . . . . . . . . . . . . 13  |-  ( y  =  b  ->  ( Base `  y )  =  ( Base `  b
) )
49 fveq2 5879 . . . . . . . . . . . . 13  |-  ( x  =  a  ->  ( Base `  x )  =  ( Base `  a
) )
5048, 49oveqan12rd 6328 . . . . . . . . . . . 12  |-  ( ( x  =  a  /\  y  =  b )  ->  ( ( Base `  y
)  ^m  ( Base `  x ) )  =  ( ( Base `  b
)  ^m  ( Base `  a ) ) )
5150reseq2d 5111 . . . . . . . . . . 11  |-  ( ( x  =  a  /\  y  =  b )  ->  (  _I  |`  (
( Base `  y )  ^m  ( Base `  x
) ) )  =  (  _I  |`  (
( Base `  b )  ^m  ( Base `  a
) ) ) )
5251adantl 473 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
a  e.  B  /\  b  e.  B )
)  /\  ( x  =  a  /\  y  =  b ) )  ->  (  _I  |`  (
( Base `  y )  ^m  ( Base `  x
) ) )  =  (  _I  |`  (
( Base `  b )  ^m  ( Base `  a
) ) ) )
5334, 31syl5eqss 3462 . . . . . . . . . . . . . 14  |-  ( ph  ->  B  C_  U )
5453sseld 3417 . . . . . . . . . . . . 13  |-  ( ph  ->  ( a  e.  B  ->  a  e.  U ) )
5554com12 31 . . . . . . . . . . . 12  |-  ( a  e.  B  ->  ( ph  ->  a  e.  U
) )
5655adantr 472 . . . . . . . . . . 11  |-  ( ( a  e.  B  /\  b  e.  B )  ->  ( ph  ->  a  e.  U ) )
5756impcom 437 . . . . . . . . . 10  |-  ( (
ph  /\  ( a  e.  B  /\  b  e.  B ) )  -> 
a  e.  U )
5853sseld 3417 . . . . . . . . . . . 12  |-  ( ph  ->  ( b  e.  B  ->  b  e.  U ) )
5958adantld 474 . . . . . . . . . . 11  |-  ( ph  ->  ( ( a  e.  B  /\  b  e.  B )  ->  b  e.  U ) )
6059imp 436 . . . . . . . . . 10  |-  ( (
ph  /\  ( a  e.  B  /\  b  e.  B ) )  -> 
b  e.  U )
61 ovex 6336 . . . . . . . . . . . 12  |-  ( (
Base `  b )  ^m  ( Base `  a
) )  e.  _V
6261a1i 11 . . . . . . . . . . 11  |-  ( (
ph  /\  ( a  e.  B  /\  b  e.  B ) )  -> 
( ( Base `  b
)  ^m  ( Base `  a ) )  e. 
_V )
6362resiexd 6147 . . . . . . . . . 10  |-  ( (
ph  /\  ( a  e.  B  /\  b  e.  B ) )  -> 
(  _I  |`  (
( Base `  b )  ^m  ( Base `  a
) ) )  e. 
_V )
6447, 52, 57, 60, 63ovmpt2d 6443 . . . . . . . . 9  |-  ( (
ph  /\  ( a  e.  B  /\  b  e.  B ) )  -> 
( a ( x  e.  U ,  y  e.  U  |->  (  _I  |`  ( ( Base `  y
)  ^m  ( Base `  x ) ) ) ) b )  =  (  _I  |`  (
( Base `  b )  ^m  ( Base `  a
) ) ) )
6564reseq1d 5110 . . . . . . . 8  |-  ( (
ph  /\  ( a  e.  B  /\  b  e.  B ) )  -> 
( ( a ( x  e.  U , 
y  e.  U  |->  (  _I  |`  ( ( Base `  y )  ^m  ( Base `  x )
) ) ) b )  |`  ( a
( Hom  `  R ) b ) )  =  ( (  _I  |`  (
( Base `  b )  ^m  ( Base `  a
) ) )  |`  ( a ( Hom  `  R ) b ) ) )
665adantr 472 . . . . . . . . . 10  |-  ( (
ph  /\  ( a  e.  B  /\  b  e.  B ) )  ->  U  e. WUni )
67 simprl 772 . . . . . . . . . 10  |-  ( (
ph  /\  ( a  e.  B  /\  b  e.  B ) )  -> 
a  e.  B )
68 simprr 774 . . . . . . . . . 10  |-  ( (
ph  /\  ( a  e.  B  /\  b  e.  B ) )  -> 
b  e.  B )
6913, 34, 66, 18, 67, 68rngchom 40477 . . . . . . . . 9  |-  ( (
ph  /\  ( a  e.  B  /\  b  e.  B ) )  -> 
( a ( Hom  `  R ) b )  =  ( a RngHomo  b
) )
7069reseq2d 5111 . . . . . . . 8  |-  ( (
ph  /\  ( a  e.  B  /\  b  e.  B ) )  -> 
( (  _I  |`  (
( Base `  b )  ^m  ( Base `  a
) ) )  |`  ( a ( Hom  `  R ) b ) )  =  ( (  _I  |`  ( ( Base `  b )  ^m  ( Base `  a )
) )  |`  (
a RngHomo  b ) ) )
71 eqid 2471 . . . . . . . . . . . 12  |-  ( Base `  a )  =  (
Base `  a )
72 eqid 2471 . . . . . . . . . . . 12  |-  ( Base `  b )  =  (
Base `  b )
7371, 72rnghmf 40407 . . . . . . . . . . 11  |-  ( f  e.  ( a RngHomo  b
)  ->  f :
( Base `  a ) --> ( Base `  b )
)
74 fvex 5889 . . . . . . . . . . . . . 14  |-  ( Base `  b )  e.  _V
75 fvex 5889 . . . . . . . . . . . . . 14  |-  ( Base `  a )  e.  _V
7674, 75pm3.2i 462 . . . . . . . . . . . . 13  |-  ( (
Base `  b )  e.  _V  /\  ( Base `  a )  e.  _V )
7776a1i 11 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( a  e.  B  /\  b  e.  B ) )  -> 
( ( Base `  b
)  e.  _V  /\  ( Base `  a )  e.  _V ) )
78 elmapg 7503 . . . . . . . . . . . 12  |-  ( ( ( Base `  b
)  e.  _V  /\  ( Base `  a )  e.  _V )  ->  (
f  e.  ( (
Base `  b )  ^m  ( Base `  a
) )  <->  f :
( Base `  a ) --> ( Base `  b )
) )
7977, 78syl 17 . . . . . . . . . . 11  |-  ( (
ph  /\  ( a  e.  B  /\  b  e.  B ) )  -> 
( f  e.  ( ( Base `  b
)  ^m  ( Base `  a ) )  <->  f :
( Base `  a ) --> ( Base `  b )
) )
8073, 79syl5ibr 229 . . . . . . . . . 10  |-  ( (
ph  /\  ( a  e.  B  /\  b  e.  B ) )  -> 
( f  e.  ( a RngHomo  b )  -> 
f  e.  ( (
Base `  b )  ^m  ( Base `  a
) ) ) )
8180ssrdv 3424 . . . . . . . . 9  |-  ( (
ph  /\  ( a  e.  B  /\  b  e.  B ) )  -> 
( a RngHomo  b )  C_  ( ( Base `  b
)  ^m  ( Base `  a ) ) )
8281resabs1d 5140 . . . . . . . 8  |-  ( (
ph  /\  ( a  e.  B  /\  b  e.  B ) )  -> 
( (  _I  |`  (
( Base `  b )  ^m  ( Base `  a
) ) )  |`  ( a RngHomo  b ) )  =  (  _I  |`  (
a RngHomo  b ) ) )
8365, 70, 823eqtrrd 2510 . . . . . . 7  |-  ( (
ph  /\  ( a  e.  B  /\  b  e.  B ) )  -> 
(  _I  |`  (
a RngHomo  b ) )  =  ( ( a ( x  e.  U , 
y  e.  U  |->  (  _I  |`  ( ( Base `  y )  ^m  ( Base `  x )
) ) ) b )  |`  ( a
( Hom  `  R ) b ) ) )
8435, 46, 83mpt2eq123dva 6371 . . . . . 6  |-  ( ph  ->  ( a  e.  B ,  b  e.  B  |->  (  _I  |`  (
a RngHomo  b ) ) )  =  ( a  e.  ( Base `  R
) ,  b  e.  ( Base `  R
)  |->  ( ( a ( x  e.  U ,  y  e.  U  |->  (  _I  |`  (
( Base `  y )  ^m  ( Base `  x
) ) ) ) b )  |`  (
a ( Hom  `  R
) b ) ) ) )
8539, 45, 843eqtrrd 2510 . . . . 5  |-  ( ph  ->  ( a  e.  (
Base `  R ) ,  b  e.  ( Base `  R )  |->  ( ( a ( x  e.  U ,  y  e.  U  |->  (  _I  |`  ( ( Base `  y
)  ^m  ( Base `  x ) ) ) ) b )  |`  ( a ( Hom  `  R ) b ) ) )  =  G )
8638, 85opeq12d 4166 . . . 4  |-  ( ph  -> 
<. ( ( x  e.  U  |->  ( Base `  x
) )  |`  ( Base `  R ) ) ,  ( a  e.  ( Base `  R
) ,  b  e.  ( Base `  R
)  |->  ( ( a ( x  e.  U ,  y  e.  U  |->  (  _I  |`  (
( Base `  y )  ^m  ( Base `  x
) ) ) ) b )  |`  (
a ( Hom  `  R
) b ) ) ) >.  =  <. F ,  G >. )
8729, 86eqtr2d 2506 . . 3  |-  ( ph  -> 
<. F ,  G >.  =  ( <. ( x  e.  U  |->  ( Base `  x
) ) ,  ( x  e.  U , 
y  e.  U  |->  (  _I  |`  ( ( Base `  y )  ^m  ( Base `  x )
) ) ) >.  |`f  ( Hom  `  R )
) )
8813, 5, 15, 19rngcval 40472 . . . 4  |-  ( ph  ->  R  =  ( (ExtStrCat `  U )  |`cat  ( Hom  `  R ) ) )
8988oveq1d 6323 . . 3  |-  ( ph  ->  ( R  Func  S
)  =  ( ( (ExtStrCat `  U )  |`cat  ( Hom  `  R )
)  Func  S )
)
9021, 87, 893eltr4d 2564 . 2  |-  ( ph  -> 
<. F ,  G >.  e.  ( R  Func  S
) )
91 df-br 4396 . 2  |-  ( F ( R  Func  S
) G  <->  <. F ,  G >.  e.  ( R 
Func  S ) )
9290, 91sylibr 217 1  |-  ( ph  ->  F ( R  Func  S ) G )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    = wceq 1452    e. wcel 1904   _Vcvv 3031    i^i cin 3389   <.cop 3965   class class class wbr 4395    |-> cmpt 4454    _I cid 4749    |` cres 4841   -->wf 5585   ` cfv 5589  (class class class)co 6308    |-> cmpt2 6310    ^m cmap 7490  WUnicwun 9143   Basecbs 15199   Hom chom 15279    |`cat cresc 15791    Func cfunc 15837    |`f cresf 15840   SetCatcsetc 16048  ExtStrCatcestrc 16085  Rngcrng 40382   RngHomo crngh 40393  RngCatcrngc 40467
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-ixp 7541  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-wun 9145  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-fz 11811  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-hom 15292  df-cco 15293  df-0g 15418  df-cat 15652  df-cid 15653  df-homf 15654  df-ssc 15793  df-resc 15794  df-subc 15795  df-func 15841  df-resf 15844  df-setc 16049  df-estrc 16086  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-mhm 16660  df-grp 16751  df-ghm 16959  df-abl 17511  df-mgp 17802  df-mgmhm 40287  df-rng0 40383  df-rnghomo 40395  df-rngc 40469
This theorem is referenced by: (None)
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