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Theorem funcringcsetc 32987
Description: The "natural forgetful functor" from the category of unital rings into the category of sets which sends each ring to its underlying set (base set) and the morphisms (ring homomorphisms) to mappings of the corresponding base sets. (Contributed by AV, 26-Mar-2020.)
Hypotheses
Ref Expression
funcringcsetc.r  |-  R  =  (RingCat `  U )
funcringcsetc.s  |-  S  =  ( SetCat `  U )
funcringcsetc.b  |-  B  =  ( Base `  R
)
funcringcsetc.u  |-  ( ph  ->  U  e. WUni )
funcringcsetc.f  |-  ( ph  ->  F  =  ( x  e.  B  |->  ( Base `  x ) ) )
funcringcsetc.g  |-  ( ph  ->  G  =  ( x  e.  B ,  y  e.  B  |->  (  _I  |`  ( x RingHom  y ) ) ) )
Assertion
Ref Expression
funcringcsetc  |-  ( ph  ->  F ( R  Func  S ) G )
Distinct variable groups:    x, B, y    x, R    x, S    x, U, y    ph, x, y
Allowed substitution hints:    R( y)    S( y)    F( x, y)    G( x, y)

Proof of Theorem funcringcsetc
Dummy variables  a 
b  f are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2457 . . . . . 6  |-  (ExtStrCat `  U
)  =  (ExtStrCat `  U
)
2 funcringcsetc.s . . . . . 6  |-  S  =  ( SetCat `  U )
3 eqid 2457 . . . . . 6  |-  ( Base `  (ExtStrCat `  U )
)  =  ( Base `  (ExtStrCat `  U )
)
4 eqid 2457 . . . . . 6  |-  ( Base `  S )  =  (
Base `  S )
5 funcringcsetc.u . . . . . 6  |-  ( ph  ->  U  e. WUni )
61, 5estrcbas 15521 . . . . . . 7  |-  ( ph  ->  U  =  ( Base `  (ExtStrCat `  U )
) )
76mpteq1d 4538 . . . . . 6  |-  ( ph  ->  ( x  e.  U  |->  ( Base `  x
) )  =  ( x  e.  ( Base `  (ExtStrCat `  U )
)  |->  ( Base `  x
) ) )
8 mpt2eq12 6356 . . . . . . 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 661 . . . . . 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 15545 . . . . 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 4457 . . . . 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 196 . . . 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 funcringcsetc.r . . . . . . 7  |-  R  =  (RingCat `  U )
14 eqid 2457 . . . . . . 7  |-  ( Base `  R )  =  (
Base `  R )
1513, 14, 5ringcbas 32963 . . . . . 6  |-  ( ph  ->  ( Base `  R
)  =  ( U  i^i  Ring ) )
16 incom 3687 . . . . . 6  |-  ( U  i^i  Ring )  =  (
Ring  i^i  U )
1715, 16syl6eq 2514 . . . . 5  |-  ( ph  ->  ( Base `  R
)  =  ( Ring 
i^i  U ) )
18 eqid 2457 . . . . . 6  |-  ( Hom  `  R )  =  ( Hom  `  R )
1913, 14, 5, 18ringchomfval 32964 . . . . 5  |-  ( ph  ->  ( Hom  `  R
)  =  ( RingHom  |`  (
( Base `  R )  X.  ( Base `  R
) ) ) )
201, 5, 17, 19rhmsubcsetc 32975 . . . 4  |-  ( ph  ->  ( Hom  `  R
)  e.  (Subcat `  (ExtStrCat `  U ) ) )
2112, 20funcres 15312 . . 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 6143 . . . . . 6  |-  ( U  e. WUni  ->  ( x  e.  U  |->  ( Base `  x
) )  e.  _V )
235, 22syl 16 . . . . 5  |-  ( ph  ->  ( x  e.  U  |->  ( Base `  x
) )  e.  _V )
24 fvex 5882 . . . . . 6  |-  ( Hom  `  R )  e.  _V
2524a1i 11 . . . . 5  |-  ( ph  ->  ( Hom  `  R
)  e.  _V )
26 mpt2exga 6875 . . . . . 6  |-  ( ( U  e. WUni  /\  U  e. WUni )  ->  ( x  e.  U ,  y  e.  U  |->  (  _I  |`  (
( Base `  y )  ^m  ( Base `  x
) ) ) )  e.  _V )
275, 5, 26syl2anc 661 . . . . 5  |-  ( ph  ->  ( x  e.  U ,  y  e.  U  |->  (  _I  |`  (
( Base `  y )  ^m  ( Base `  x
) ) ) )  e.  _V )
2815, 19rhmresfn 32961 . . . . 5  |-  ( ph  ->  ( Hom  `  R
)  Fn  ( (
Base `  R )  X.  ( Base `  R
) ) )
2923, 25, 27, 28resfval2 15309 . . . 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 3714 . . . . . . . 8  |-  ( U  i^i  Ring )  C_  U
3115, 30syl6eqss 3549 . . . . . . 7  |-  ( ph  ->  ( Base `  R
)  C_  U )
3231resmptd 5335 . . . . . 6  |-  ( ph  ->  ( ( x  e.  U  |->  ( Base `  x
) )  |`  ( Base `  R ) )  =  ( x  e.  ( Base `  R
)  |->  ( Base `  x
) ) )
33 funcringcsetc.f . . . . . . 7  |-  ( ph  ->  F  =  ( x  e.  B  |->  ( Base `  x ) ) )
34 funcringcsetc.b . . . . . . . . 9  |-  B  =  ( Base `  R
)
3534a1i 11 . . . . . . . 8  |-  ( ph  ->  B  =  ( Base `  R ) )
3635mpteq1d 4538 . . . . . . 7  |-  ( ph  ->  ( x  e.  B  |->  ( Base `  x
) )  =  ( x  e.  ( Base `  R )  |->  ( Base `  x ) ) )
3733, 36eqtr2d 2499 . . . . . 6  |-  ( ph  ->  ( x  e.  (
Base `  R )  |->  ( Base `  x
) )  =  F )
3832, 37eqtrd 2498 . . . . 5  |-  ( ph  ->  ( ( x  e.  U  |->  ( Base `  x
) )  |`  ( Base `  R ) )  =  F )
39 funcringcsetc.g . . . . . 6  |-  ( ph  ->  G  =  ( x  e.  B ,  y  e.  B  |->  (  _I  |`  ( x RingHom  y ) ) ) )
40 oveq1 6303 . . . . . . . . 9  |-  ( x  =  a  ->  (
x RingHom  y )  =  ( a RingHom  y ) )
4140reseq2d 5283 . . . . . . . 8  |-  ( x  =  a  ->  (  _I  |`  ( x RingHom  y
) )  =  (  _I  |`  ( a RingHom  y ) ) )
42 oveq2 6304 . . . . . . . . 9  |-  ( y  =  b  ->  (
a RingHom  y )  =  ( a RingHom  b ) )
4342reseq2d 5283 . . . . . . . 8  |-  ( y  =  b  ->  (  _I  |`  ( a RingHom  y
) )  =  (  _I  |`  ( a RingHom  b ) ) )
4441, 43cbvmpt2v 6376 . . . . . . 7  |-  ( x  e.  B ,  y  e.  B  |->  (  _I  |`  ( x RingHom  y ) ) )  =  ( a  e.  B , 
b  e.  B  |->  (  _I  |`  ( a RingHom  b ) ) )
4544a1i 11 . . . . . 6  |-  ( ph  ->  ( x  e.  B ,  y  e.  B  |->  (  _I  |`  (
x RingHom  y ) ) )  =  ( a  e.  B ,  b  e.  B  |->  (  _I  |`  (
a RingHom  b ) ) ) )
4634a1i 11 . . . . . . 7  |-  ( (
ph  /\  a  e.  B )  ->  B  =  ( Base `  R
) )
47 eqidd 2458 . . . . . . . . . 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 5872 . . . . . . . . . . . . 13  |-  ( y  =  b  ->  ( Base `  y )  =  ( Base `  b
) )
49 fveq2 5872 . . . . . . . . . . . . 13  |-  ( x  =  a  ->  ( Base `  x )  =  ( Base `  a
) )
5048, 49oveqan12rd 6316 . . . . . . . . . . . 12  |-  ( ( x  =  a  /\  y  =  b )  ->  ( ( Base `  y
)  ^m  ( Base `  x ) )  =  ( ( Base `  b
)  ^m  ( Base `  a ) ) )
5150reseq2d 5283 . . . . . . . . . . 11  |-  ( ( x  =  a  /\  y  =  b )  ->  (  _I  |`  (
( Base `  y )  ^m  ( Base `  x
) ) )  =  (  _I  |`  (
( Base `  b )  ^m  ( Base `  a
) ) ) )
5251adantl 466 . . . . . . . . . 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 3543 . . . . . . . . . . . . . 14  |-  ( ph  ->  B  C_  U )
5453sseld 3498 . . . . . . . . . . . . 13  |-  ( ph  ->  ( a  e.  B  ->  a  e.  U ) )
5554com12 31 . . . . . . . . . . . 12  |-  ( a  e.  B  ->  ( ph  ->  a  e.  U
) )
5655adantr 465 . . . . . . . . . . 11  |-  ( ( a  e.  B  /\  b  e.  B )  ->  ( ph  ->  a  e.  U ) )
5756impcom 430 . . . . . . . . . 10  |-  ( (
ph  /\  ( a  e.  B  /\  b  e.  B ) )  -> 
a  e.  U )
5853sseld 3498 . . . . . . . . . . . 12  |-  ( ph  ->  ( b  e.  B  ->  b  e.  U ) )
5958adantld 467 . . . . . . . . . . 11  |-  ( ph  ->  ( ( a  e.  B  /\  b  e.  B )  ->  b  e.  U ) )
6059imp 429 . . . . . . . . . 10  |-  ( (
ph  /\  ( a  e.  B  /\  b  e.  B ) )  -> 
b  e.  U )
61 ovex 6324 . . . . . . . . . . . 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 6139 . . . . . . . . . 10  |-  ( (
ph  /\  ( a  e.  B  /\  b  e.  B ) )  -> 
(  _I  |`  (
( Base `  b )  ^m  ( Base `  a
) ) )  e. 
_V )
6447, 52, 57, 60, 63ovmpt2d 6429 . . . . . . . . 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 5282 . . . . . . . 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 465 . . . . . . . . . 10  |-  ( (
ph  /\  ( a  e.  B  /\  b  e.  B ) )  ->  U  e. WUni )
67 simprl 756 . . . . . . . . . 10  |-  ( (
ph  /\  ( a  e.  B  /\  b  e.  B ) )  -> 
a  e.  B )
68 simprr 757 . . . . . . . . . 10  |-  ( (
ph  /\  ( a  e.  B  /\  b  e.  B ) )  -> 
b  e.  B )
6913, 34, 66, 18, 67, 68ringchom 32965 . . . . . . . . 9  |-  ( (
ph  /\  ( a  e.  B  /\  b  e.  B ) )  -> 
( a ( Hom  `  R ) b )  =  ( a RingHom  b
) )
7069reseq2d 5283 . . . . . . . 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 RingHom  b ) ) )
71 eqid 2457 . . . . . . . . . . . 12  |-  ( Base `  a )  =  (
Base `  a )
72 eqid 2457 . . . . . . . . . . . 12  |-  ( Base `  b )  =  (
Base `  b )
7371, 72rhmf 17502 . . . . . . . . . . 11  |-  ( f  e.  ( a RingHom  b
)  ->  f :
( Base `  a ) --> ( Base `  b )
)
74 fvex 5882 . . . . . . . . . . . . . 14  |-  ( Base `  b )  e.  _V
75 fvex 5882 . . . . . . . . . . . . . 14  |-  ( Base `  a )  e.  _V
7674, 75pm3.2i 455 . . . . . . . . . . . . 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 7451 . . . . . . . . . . . 12  |-  ( ( ( Base `  b
)  e.  _V  /\  ( Base `  a )  e.  _V )  ->  (
f  e.  ( (
Base `  b )  ^m  ( Base `  a
) )  <->  f :
( Base `  a ) --> ( Base `  b )
) )
7977, 78syl 16 . . . . . . . . . . 11  |-  ( (
ph  /\  ( a  e.  B  /\  b  e.  B ) )  -> 
( f  e.  ( ( Base `  b
)  ^m  ( Base `  a ) )  <->  f :
( Base `  a ) --> ( Base `  b )
) )
8073, 79syl5ibr 221 . . . . . . . . . 10  |-  ( (
ph  /\  ( a  e.  B  /\  b  e.  B ) )  -> 
( f  e.  ( a RingHom  b )  -> 
f  e.  ( (
Base `  b )  ^m  ( Base `  a
) ) ) )
8180ssrdv 3505 . . . . . . . . 9  |-  ( (
ph  /\  ( a  e.  B  /\  b  e.  B ) )  -> 
( a RingHom  b )  C_  ( ( Base `  b
)  ^m  ( Base `  a ) ) )
8281resabs1d 5313 . . . . . . . 8  |-  ( (
ph  /\  ( a  e.  B  /\  b  e.  B ) )  -> 
( (  _I  |`  (
( Base `  b )  ^m  ( Base `  a
) ) )  |`  ( a RingHom  b ) )  =  (  _I  |`  (
a RingHom  b ) ) )
8365, 70, 823eqtrrd 2503 . . . . . . 7  |-  ( (
ph  /\  ( a  e.  B  /\  b  e.  B ) )  -> 
(  _I  |`  (
a RingHom  b ) )  =  ( ( a ( x  e.  U , 
y  e.  U  |->  (  _I  |`  ( ( Base `  y )  ^m  ( Base `  x )
) ) ) b )  |`  ( a
( Hom  `  R ) b ) ) )
8435, 46, 83mpt2eq123dva 6357 . . . . . 6  |-  ( ph  ->  ( a  e.  B ,  b  e.  B  |->  (  _I  |`  (
a RingHom  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 2503 . . . . 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 4227 . . . 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 2499 . . 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, 19ringcval 32960 . . . 4  |-  ( ph  ->  R  =  ( (ExtStrCat `  U )  |`cat  ( Hom  `  R ) ) )
8988oveq1d 6311 . . 3  |-  ( ph  ->  ( R  Func  S
)  =  ( ( (ExtStrCat `  U )  |`cat  ( Hom  `  R )
)  Func  S )
)
9021, 87, 893eltr4d 2560 . 2  |-  ( ph  -> 
<. F ,  G >.  e.  ( R  Func  S
) )
91 df-br 4457 . 2  |-  ( F ( R  Func  S
) G  <->  <. F ,  G >.  e.  ( R 
Func  S ) )
9290, 91sylibr 212 1  |-  ( ph  ->  F ( R  Func  S ) G )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819   _Vcvv 3109    i^i cin 3470   <.cop 4038   class class class wbr 4456    |-> cmpt 4515    _I cid 4799    |` cres 5010   -->wf 5590   ` cfv 5594  (class class class)co 6296    |-> cmpt2 6298    ^m cmap 7438  WUnicwun 9095   Basecbs 14644   Hom chom 14723    |`cat cresc 15224    Func cfunc 15270    |`f cresf 15273   SetCatcsetc 15481  ExtStrCatcestrc 15518   Ringcrg 17325   RingHom crh 17488  RingCatcringc 32955
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-pm 7441  df-ixp 7489  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-wun 9097  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-10 10623  df-n0 10817  df-z 10886  df-dec 11001  df-uz 11107  df-fz 11698  df-struct 14646  df-ndx 14647  df-slot 14648  df-base 14649  df-sets 14650  df-ress 14651  df-plusg 14725  df-hom 14736  df-cco 14737  df-0g 14859  df-cat 15085  df-cid 15086  df-homf 15087  df-ssc 15226  df-resc 15227  df-subc 15228  df-func 15274  df-resf 15277  df-setc 15482  df-estrc 15519  df-mgm 15999  df-sgrp 16038  df-mnd 16048  df-mhm 16093  df-grp 16184  df-ghm 16392  df-mgp 17269  df-ur 17281  df-ring 17327  df-rnghom 17491  df-ringc 32957
This theorem is referenced by: (None)
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