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Theorem msubco 30217
Description: The composition of two substitutions is a substitution. (Contributed by Mario Carneiro, 18-Jul-2016.)
Hypothesis
Ref Expression
msubco.s  |-  S  =  (mSubst `  T )
Assertion
Ref Expression
msubco  |-  ( ( F  e.  ran  S  /\  G  e.  ran  S )  ->  ( F  o.  G )  e.  ran  S )

Proof of Theorem msubco
Dummy variables  f 
g  h  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2461 . . . . 5  |-  (mEx `  T )  =  (mEx
`  T )
2 eqid 2461 . . . . 5  |-  (mRSubst `  T
)  =  (mRSubst `  T
)
3 msubco.s . . . . 5  |-  S  =  (mSubst `  T )
41, 2, 3elmsubrn 30214 . . . 4  |-  ran  S  =  ran  ( f  e. 
ran  (mRSubst `  T )  |->  ( x  e.  (mEx
`  T )  |->  <.
( 1st `  x
) ,  ( f `
 ( 2nd `  x
) ) >. )
)
54eleq2i 2531 . . 3  |-  ( F  e.  ran  S  <->  F  e.  ran  ( f  e.  ran  (mRSubst `  T )  |->  ( x  e.  (mEx `  T )  |->  <. ( 1st `  x ) ,  ( f `  ( 2nd `  x ) )
>. ) ) )
6 eqid 2461 . . . 4  |-  ( f  e.  ran  (mRSubst `  T
)  |->  ( x  e.  (mEx `  T )  |-> 
<. ( 1st `  x
) ,  ( f `
 ( 2nd `  x
) ) >. )
)  =  ( f  e.  ran  (mRSubst `  T
)  |->  ( x  e.  (mEx `  T )  |-> 
<. ( 1st `  x
) ,  ( f `
 ( 2nd `  x
) ) >. )
)
7 fvex 5897 . . . . 5  |-  (mEx `  T )  e.  _V
87mptex 6160 . . . 4  |-  ( x  e.  (mEx `  T
)  |->  <. ( 1st `  x
) ,  ( f `
 ( 2nd `  x
) ) >. )  e.  _V
96, 8elrnmpti 5103 . . 3  |-  ( F  e.  ran  ( f  e.  ran  (mRSubst `  T
)  |->  ( x  e.  (mEx `  T )  |-> 
<. ( 1st `  x
) ,  ( f `
 ( 2nd `  x
) ) >. )
)  <->  E. f  e.  ran  (mRSubst `  T ) F  =  ( x  e.  (mEx `  T )  |-> 
<. ( 1st `  x
) ,  ( f `
 ( 2nd `  x
) ) >. )
)
105, 9bitri 257 . 2  |-  ( F  e.  ran  S  <->  E. f  e.  ran  (mRSubst `  T
) F  =  ( x  e.  (mEx `  T )  |->  <. ( 1st `  x ) ,  ( f `  ( 2nd `  x ) )
>. ) )
111, 2, 3elmsubrn 30214 . . . 4  |-  ran  S  =  ran  ( g  e. 
ran  (mRSubst `  T )  |->  ( y  e.  (mEx
`  T )  |->  <.
( 1st `  y
) ,  ( g `
 ( 2nd `  y
) ) >. )
)
1211eleq2i 2531 . . 3  |-  ( G  e.  ran  S  <->  G  e.  ran  ( g  e.  ran  (mRSubst `  T )  |->  ( y  e.  (mEx `  T )  |->  <. ( 1st `  y ) ,  ( g `  ( 2nd `  y ) )
>. ) ) )
13 eqid 2461 . . . 4  |-  ( g  e.  ran  (mRSubst `  T
)  |->  ( y  e.  (mEx `  T )  |-> 
<. ( 1st `  y
) ,  ( g `
 ( 2nd `  y
) ) >. )
)  =  ( g  e.  ran  (mRSubst `  T
)  |->  ( y  e.  (mEx `  T )  |-> 
<. ( 1st `  y
) ,  ( g `
 ( 2nd `  y
) ) >. )
)
147mptex 6160 . . . 4  |-  ( y  e.  (mEx `  T
)  |->  <. ( 1st `  y
) ,  ( g `
 ( 2nd `  y
) ) >. )  e.  _V
1513, 14elrnmpti 5103 . . 3  |-  ( G  e.  ran  ( g  e.  ran  (mRSubst `  T
)  |->  ( y  e.  (mEx `  T )  |-> 
<. ( 1st `  y
) ,  ( g `
 ( 2nd `  y
) ) >. )
)  <->  E. g  e.  ran  (mRSubst `  T ) G  =  ( y  e.  (mEx `  T )  |-> 
<. ( 1st `  y
) ,  ( g `
 ( 2nd `  y
) ) >. )
)
1612, 15bitri 257 . 2  |-  ( G  e.  ran  S  <->  E. g  e.  ran  (mRSubst `  T
) G  =  ( y  e.  (mEx `  T )  |->  <. ( 1st `  y ) ,  ( g `  ( 2nd `  y ) )
>. ) )
17 reeanv 2969 . . 3  |-  ( E. f  e.  ran  (mRSubst `  T ) E. g  e.  ran  (mRSubst `  T
) ( F  =  ( x  e.  (mEx
`  T )  |->  <.
( 1st `  x
) ,  ( f `
 ( 2nd `  x
) ) >. )  /\  G  =  (
y  e.  (mEx `  T )  |->  <. ( 1st `  y ) ,  ( g `  ( 2nd `  y ) )
>. ) )  <->  ( E. f  e.  ran  (mRSubst `  T
) F  =  ( x  e.  (mEx `  T )  |->  <. ( 1st `  x ) ,  ( f `  ( 2nd `  x ) )
>. )  /\  E. g  e.  ran  (mRSubst `  T
) G  =  ( y  e.  (mEx `  T )  |->  <. ( 1st `  y ) ,  ( g `  ( 2nd `  y ) )
>. ) ) )
18 simpr 467 . . . . . . . . . . . 12  |-  ( ( ( f  e.  ran  (mRSubst `  T )  /\  g  e.  ran  (mRSubst `  T
) )  /\  y  e.  (mEx `  T )
)  ->  y  e.  (mEx `  T ) )
19 eqid 2461 . . . . . . . . . . . . 13  |-  (mTC `  T )  =  (mTC
`  T )
20 eqid 2461 . . . . . . . . . . . . 13  |-  (mREx `  T )  =  (mREx `  T )
2119, 1, 20mexval 30188 . . . . . . . . . . . 12  |-  (mEx `  T )  =  ( (mTC `  T )  X.  (mREx `  T )
)
2218, 21syl6eleq 2549 . . . . . . . . . . 11  |-  ( ( ( f  e.  ran  (mRSubst `  T )  /\  g  e.  ran  (mRSubst `  T
) )  /\  y  e.  (mEx `  T )
)  ->  y  e.  ( (mTC `  T )  X.  (mREx `  T )
) )
23 xp1st 6849 . . . . . . . . . . 11  |-  ( y  e.  ( (mTC `  T )  X.  (mREx `  T ) )  -> 
( 1st `  y
)  e.  (mTC `  T ) )
2422, 23syl 17 . . . . . . . . . 10  |-  ( ( ( f  e.  ran  (mRSubst `  T )  /\  g  e.  ran  (mRSubst `  T
) )  /\  y  e.  (mEx `  T )
)  ->  ( 1st `  y )  e.  (mTC
`  T ) )
252, 20mrsubf 30203 . . . . . . . . . . . 12  |-  ( g  e.  ran  (mRSubst `  T
)  ->  g :
(mREx `  T ) --> (mREx `  T ) )
2625ad2antlr 738 . . . . . . . . . . 11  |-  ( ( ( f  e.  ran  (mRSubst `  T )  /\  g  e.  ran  (mRSubst `  T
) )  /\  y  e.  (mEx `  T )
)  ->  g :
(mREx `  T ) --> (mREx `  T ) )
27 xp2nd 6850 . . . . . . . . . . . 12  |-  ( y  e.  ( (mTC `  T )  X.  (mREx `  T ) )  -> 
( 2nd `  y
)  e.  (mREx `  T ) )
2822, 27syl 17 . . . . . . . . . . 11  |-  ( ( ( f  e.  ran  (mRSubst `  T )  /\  g  e.  ran  (mRSubst `  T
) )  /\  y  e.  (mEx `  T )
)  ->  ( 2nd `  y )  e.  (mREx `  T ) )
2926, 28ffvelrnd 6045 . . . . . . . . . 10  |-  ( ( ( f  e.  ran  (mRSubst `  T )  /\  g  e.  ran  (mRSubst `  T
) )  /\  y  e.  (mEx `  T )
)  ->  ( g `  ( 2nd `  y
) )  e.  (mREx `  T ) )
30 opelxpi 4884 . . . . . . . . . 10  |-  ( ( ( 1st `  y
)  e.  (mTC `  T )  /\  (
g `  ( 2nd `  y ) )  e.  (mREx `  T )
)  ->  <. ( 1st `  y ) ,  ( g `  ( 2nd `  y ) ) >.  e.  ( (mTC `  T
)  X.  (mREx `  T ) ) )
3124, 29, 30syl2anc 671 . . . . . . . . 9  |-  ( ( ( f  e.  ran  (mRSubst `  T )  /\  g  e.  ran  (mRSubst `  T
) )  /\  y  e.  (mEx `  T )
)  ->  <. ( 1st `  y ) ,  ( g `  ( 2nd `  y ) ) >.  e.  ( (mTC `  T
)  X.  (mREx `  T ) ) )
3231, 21syl6eleqr 2550 . . . . . . . 8  |-  ( ( ( f  e.  ran  (mRSubst `  T )  /\  g  e.  ran  (mRSubst `  T
) )  /\  y  e.  (mEx `  T )
)  ->  <. ( 1st `  y ) ,  ( g `  ( 2nd `  y ) ) >.  e.  (mEx `  T )
)
33 eqidd 2462 . . . . . . . 8  |-  ( ( f  e.  ran  (mRSubst `  T )  /\  g  e.  ran  (mRSubst `  T
) )  ->  (
y  e.  (mEx `  T )  |->  <. ( 1st `  y ) ,  ( g `  ( 2nd `  y ) )
>. )  =  (
y  e.  (mEx `  T )  |->  <. ( 1st `  y ) ,  ( g `  ( 2nd `  y ) )
>. ) )
34 eqidd 2462 . . . . . . . 8  |-  ( ( f  e.  ran  (mRSubst `  T )  /\  g  e.  ran  (mRSubst `  T
) )  ->  (
x  e.  (mEx `  T )  |->  <. ( 1st `  x ) ,  ( f `  ( 2nd `  x ) )
>. )  =  (
x  e.  (mEx `  T )  |->  <. ( 1st `  x ) ,  ( f `  ( 2nd `  x ) )
>. ) )
35 fvex 5897 . . . . . . . . . 10  |-  ( 1st `  y )  e.  _V
36 fvex 5897 . . . . . . . . . 10  |-  ( g `
 ( 2nd `  y
) )  e.  _V
3735, 36op1std 6829 . . . . . . . . 9  |-  ( x  =  <. ( 1st `  y
) ,  ( g `
 ( 2nd `  y
) ) >.  ->  ( 1st `  x )  =  ( 1st `  y
) )
3835, 36op2ndd 6830 . . . . . . . . . 10  |-  ( x  =  <. ( 1st `  y
) ,  ( g `
 ( 2nd `  y
) ) >.  ->  ( 2nd `  x )  =  ( g `  ( 2nd `  y ) ) )
3938fveq2d 5891 . . . . . . . . 9  |-  ( x  =  <. ( 1st `  y
) ,  ( g `
 ( 2nd `  y
) ) >.  ->  (
f `  ( 2nd `  x ) )  =  ( f `  (
g `  ( 2nd `  y ) ) ) )
4037, 39opeq12d 4187 . . . . . . . 8  |-  ( x  =  <. ( 1st `  y
) ,  ( g `
 ( 2nd `  y
) ) >.  ->  <. ( 1st `  x ) ,  ( f `  ( 2nd `  x ) )
>.  =  <. ( 1st `  y ) ,  ( f `  ( g `
 ( 2nd `  y
) ) ) >.
)
4132, 33, 34, 40fmptco 6079 . . . . . . 7  |-  ( ( f  e.  ran  (mRSubst `  T )  /\  g  e.  ran  (mRSubst `  T
) )  ->  (
( x  e.  (mEx
`  T )  |->  <.
( 1st `  x
) ,  ( f `
 ( 2nd `  x
) ) >. )  o.  ( y  e.  (mEx
`  T )  |->  <.
( 1st `  y
) ,  ( g `
 ( 2nd `  y
) ) >. )
)  =  ( y  e.  (mEx `  T
)  |->  <. ( 1st `  y
) ,  ( f `
 ( g `  ( 2nd `  y ) ) ) >. )
)
42 fvco3 5964 . . . . . . . . . 10  |-  ( ( g : (mREx `  T ) --> (mREx `  T )  /\  ( 2nd `  y )  e.  (mREx `  T )
)  ->  ( (
f  o.  g ) `
 ( 2nd `  y
) )  =  ( f `  ( g `
 ( 2nd `  y
) ) ) )
4326, 28, 42syl2anc 671 . . . . . . . . 9  |-  ( ( ( f  e.  ran  (mRSubst `  T )  /\  g  e.  ran  (mRSubst `  T
) )  /\  y  e.  (mEx `  T )
)  ->  ( (
f  o.  g ) `
 ( 2nd `  y
) )  =  ( f `  ( g `
 ( 2nd `  y
) ) ) )
4443opeq2d 4186 . . . . . . . 8  |-  ( ( ( f  e.  ran  (mRSubst `  T )  /\  g  e.  ran  (mRSubst `  T
) )  /\  y  e.  (mEx `  T )
)  ->  <. ( 1st `  y ) ,  ( ( f  o.  g
) `  ( 2nd `  y ) ) >.  =  <. ( 1st `  y
) ,  ( f `
 ( g `  ( 2nd `  y ) ) ) >. )
4544mpteq2dva 4502 . . . . . . 7  |-  ( ( f  e.  ran  (mRSubst `  T )  /\  g  e.  ran  (mRSubst `  T
) )  ->  (
y  e.  (mEx `  T )  |->  <. ( 1st `  y ) ,  ( ( f  o.  g ) `  ( 2nd `  y ) )
>. )  =  (
y  e.  (mEx `  T )  |->  <. ( 1st `  y ) ,  ( f `  (
g `  ( 2nd `  y ) ) )
>. ) )
4641, 45eqtr4d 2498 . . . . . 6  |-  ( ( f  e.  ran  (mRSubst `  T )  /\  g  e.  ran  (mRSubst `  T
) )  ->  (
( x  e.  (mEx
`  T )  |->  <.
( 1st `  x
) ,  ( f `
 ( 2nd `  x
) ) >. )  o.  ( y  e.  (mEx
`  T )  |->  <.
( 1st `  y
) ,  ( g `
 ( 2nd `  y
) ) >. )
)  =  ( y  e.  (mEx `  T
)  |->  <. ( 1st `  y
) ,  ( ( f  o.  g ) `
 ( 2nd `  y
) ) >. )
)
472mrsubco 30207 . . . . . . . 8  |-  ( ( f  e.  ran  (mRSubst `  T )  /\  g  e.  ran  (mRSubst `  T
) )  ->  (
f  o.  g )  e.  ran  (mRSubst `  T
) )
487mptex 6160 . . . . . . . 8  |-  ( y  e.  (mEx `  T
)  |->  <. ( 1st `  y
) ,  ( ( f  o.  g ) `
 ( 2nd `  y
) ) >. )  e.  _V
49 eqid 2461 . . . . . . . . 9  |-  ( h  e.  ran  (mRSubst `  T
)  |->  ( y  e.  (mEx `  T )  |-> 
<. ( 1st `  y
) ,  ( h `
 ( 2nd `  y
) ) >. )
)  =  ( h  e.  ran  (mRSubst `  T
)  |->  ( y  e.  (mEx `  T )  |-> 
<. ( 1st `  y
) ,  ( h `
 ( 2nd `  y
) ) >. )
)
50 fveq1 5886 . . . . . . . . . . 11  |-  ( h  =  ( f  o.  g )  ->  (
h `  ( 2nd `  y ) )  =  ( ( f  o.  g ) `  ( 2nd `  y ) ) )
5150opeq2d 4186 . . . . . . . . . 10  |-  ( h  =  ( f  o.  g )  ->  <. ( 1st `  y ) ,  ( h `  ( 2nd `  y ) )
>.  =  <. ( 1st `  y ) ,  ( ( f  o.  g
) `  ( 2nd `  y ) ) >.
)
5251mpteq2dv 4503 . . . . . . . . 9  |-  ( h  =  ( f  o.  g )  ->  (
y  e.  (mEx `  T )  |->  <. ( 1st `  y ) ,  ( h `  ( 2nd `  y ) )
>. )  =  (
y  e.  (mEx `  T )  |->  <. ( 1st `  y ) ,  ( ( f  o.  g ) `  ( 2nd `  y ) )
>. ) )
5349, 52elrnmpt1s 5100 . . . . . . . 8  |-  ( ( ( f  o.  g
)  e.  ran  (mRSubst `  T )  /\  (
y  e.  (mEx `  T )  |->  <. ( 1st `  y ) ,  ( ( f  o.  g ) `  ( 2nd `  y ) )
>. )  e.  _V )  ->  ( y  e.  (mEx `  T )  |-> 
<. ( 1st `  y
) ,  ( ( f  o.  g ) `
 ( 2nd `  y
) ) >. )  e.  ran  ( h  e. 
ran  (mRSubst `  T )  |->  ( y  e.  (mEx
`  T )  |->  <.
( 1st `  y
) ,  ( h `
 ( 2nd `  y
) ) >. )
) )
5447, 48, 53sylancl 673 . . . . . . 7  |-  ( ( f  e.  ran  (mRSubst `  T )  /\  g  e.  ran  (mRSubst `  T
) )  ->  (
y  e.  (mEx `  T )  |->  <. ( 1st `  y ) ,  ( ( f  o.  g ) `  ( 2nd `  y ) )
>. )  e.  ran  ( h  e.  ran  (mRSubst `  T )  |->  ( y  e.  (mEx `  T )  |->  <. ( 1st `  y ) ,  ( h `  ( 2nd `  y ) )
>. ) ) )
551, 2, 3elmsubrn 30214 . . . . . . 7  |-  ran  S  =  ran  ( h  e. 
ran  (mRSubst `  T )  |->  ( y  e.  (mEx
`  T )  |->  <.
( 1st `  y
) ,  ( h `
 ( 2nd `  y
) ) >. )
)
5654, 55syl6eleqr 2550 . . . . . 6  |-  ( ( f  e.  ran  (mRSubst `  T )  /\  g  e.  ran  (mRSubst `  T
) )  ->  (
y  e.  (mEx `  T )  |->  <. ( 1st `  y ) ,  ( ( f  o.  g ) `  ( 2nd `  y ) )
>. )  e.  ran  S )
5746, 56eqeltrd 2539 . . . . 5  |-  ( ( f  e.  ran  (mRSubst `  T )  /\  g  e.  ran  (mRSubst `  T
) )  ->  (
( x  e.  (mEx
`  T )  |->  <.
( 1st `  x
) ,  ( f `
 ( 2nd `  x
) ) >. )  o.  ( y  e.  (mEx
`  T )  |->  <.
( 1st `  y
) ,  ( g `
 ( 2nd `  y
) ) >. )
)  e.  ran  S
)
58 coeq1 5010 . . . . . . 7  |-  ( F  =  ( x  e.  (mEx `  T )  |-> 
<. ( 1st `  x
) ,  ( f `
 ( 2nd `  x
) ) >. )  ->  ( F  o.  G
)  =  ( ( x  e.  (mEx `  T )  |->  <. ( 1st `  x ) ,  ( f `  ( 2nd `  x ) )
>. )  o.  G
) )
59 coeq2 5011 . . . . . . 7  |-  ( G  =  ( y  e.  (mEx `  T )  |-> 
<. ( 1st `  y
) ,  ( g `
 ( 2nd `  y
) ) >. )  ->  ( ( x  e.  (mEx `  T )  |-> 
<. ( 1st `  x
) ,  ( f `
 ( 2nd `  x
) ) >. )  o.  G )  =  ( ( x  e.  (mEx
`  T )  |->  <.
( 1st `  x
) ,  ( f `
 ( 2nd `  x
) ) >. )  o.  ( y  e.  (mEx
`  T )  |->  <.
( 1st `  y
) ,  ( g `
 ( 2nd `  y
) ) >. )
) )
6058, 59sylan9eq 2515 . . . . . 6  |-  ( ( F  =  ( x  e.  (mEx `  T
)  |->  <. ( 1st `  x
) ,  ( f `
 ( 2nd `  x
) ) >. )  /\  G  =  (
y  e.  (mEx `  T )  |->  <. ( 1st `  y ) ,  ( g `  ( 2nd `  y ) )
>. ) )  ->  ( F  o.  G )  =  ( ( x  e.  (mEx `  T
)  |->  <. ( 1st `  x
) ,  ( f `
 ( 2nd `  x
) ) >. )  o.  ( y  e.  (mEx
`  T )  |->  <.
( 1st `  y
) ,  ( g `
 ( 2nd `  y
) ) >. )
) )
6160eleq1d 2523 . . . . 5  |-  ( ( F  =  ( x  e.  (mEx `  T
)  |->  <. ( 1st `  x
) ,  ( f `
 ( 2nd `  x
) ) >. )  /\  G  =  (
y  e.  (mEx `  T )  |->  <. ( 1st `  y ) ,  ( g `  ( 2nd `  y ) )
>. ) )  ->  (
( F  o.  G
)  e.  ran  S  <->  ( ( x  e.  (mEx
`  T )  |->  <.
( 1st `  x
) ,  ( f `
 ( 2nd `  x
) ) >. )  o.  ( y  e.  (mEx
`  T )  |->  <.
( 1st `  y
) ,  ( g `
 ( 2nd `  y
) ) >. )
)  e.  ran  S
) )
6257, 61syl5ibrcom 230 . . . 4  |-  ( ( f  e.  ran  (mRSubst `  T )  /\  g  e.  ran  (mRSubst `  T
) )  ->  (
( F  =  ( x  e.  (mEx `  T )  |->  <. ( 1st `  x ) ,  ( f `  ( 2nd `  x ) )
>. )  /\  G  =  ( y  e.  (mEx
`  T )  |->  <.
( 1st `  y
) ,  ( g `
 ( 2nd `  y
) ) >. )
)  ->  ( F  o.  G )  e.  ran  S ) )
6362rexlimivv 2895 . . 3  |-  ( E. f  e.  ran  (mRSubst `  T ) E. g  e.  ran  (mRSubst `  T
) ( F  =  ( x  e.  (mEx
`  T )  |->  <.
( 1st `  x
) ,  ( f `
 ( 2nd `  x
) ) >. )  /\  G  =  (
y  e.  (mEx `  T )  |->  <. ( 1st `  y ) ,  ( g `  ( 2nd `  y ) )
>. ) )  ->  ( F  o.  G )  e.  ran  S )
6417, 63sylbir 218 . 2  |-  ( ( E. f  e.  ran  (mRSubst `  T ) F  =  ( x  e.  (mEx `  T )  |-> 
<. ( 1st `  x
) ,  ( f `
 ( 2nd `  x
) ) >. )  /\  E. g  e.  ran  (mRSubst `  T ) G  =  ( y  e.  (mEx `  T )  |-> 
<. ( 1st `  y
) ,  ( g `
 ( 2nd `  y
) ) >. )
)  ->  ( F  o.  G )  e.  ran  S )
6510, 16, 64syl2anb 486 1  |-  ( ( F  e.  ran  S  /\  G  e.  ran  S )  ->  ( F  o.  G )  e.  ran  S )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 375    = wceq 1454    e. wcel 1897   E.wrex 2749   _Vcvv 3056   <.cop 3985    |-> cmpt 4474    X. cxp 4850   ran crn 4853    o. ccom 4856   -->wf 5596   ` cfv 5600   1stc1st 6817   2ndc2nd 6818  mTCcmtc 30150  mRExcmrex 30152  mExcmex 30153  mRSubstcmrsub 30156  mSubstcmsub 30157
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-rep 4528  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6609  ax-cnex 9620  ax-resscn 9621  ax-1cn 9622  ax-icn 9623  ax-addcl 9624  ax-addrcl 9625  ax-mulcl 9626  ax-mulrcl 9627  ax-mulcom 9628  ax-addass 9629  ax-mulass 9630  ax-distr 9631  ax-i2m1 9632  ax-1ne0 9633  ax-1rid 9634  ax-rnegex 9635  ax-rrecex 9636  ax-cnre 9637  ax-pre-lttri 9638  ax-pre-lttrn 9639  ax-pre-ltadd 9640  ax-pre-mulgt0 9641
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-nel 2635  df-ral 2753  df-rex 2754  df-reu 2755  df-rmo 2756  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-pss 3431  df-nul 3743  df-if 3893  df-pw 3964  df-sn 3980  df-pr 3982  df-tp 3984  df-op 3986  df-uni 4212  df-int 4248  df-iun 4293  df-br 4416  df-opab 4475  df-mpt 4476  df-tr 4511  df-eprel 4763  df-id 4767  df-po 4773  df-so 4774  df-fr 4811  df-we 4813  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-pred 5398  df-ord 5444  df-on 5445  df-lim 5446  df-suc 5447  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-fv 5608  df-riota 6276  df-ov 6317  df-oprab 6318  df-mpt2 6319  df-om 6719  df-1st 6819  df-2nd 6820  df-wrecs 7053  df-recs 7115  df-rdg 7153  df-1o 7207  df-oadd 7211  df-er 7388  df-map 7499  df-pm 7500  df-en 7595  df-dom 7596  df-sdom 7597  df-fin 7598  df-card 8398  df-cda 8623  df-pnf 9702  df-mnf 9703  df-xr 9704  df-ltxr 9705  df-le 9706  df-sub 9887  df-neg 9888  df-nn 10637  df-2 10695  df-n0 10898  df-z 10966  df-uz 11188  df-fz 11813  df-fzo 11946  df-seq 12245  df-hash 12547  df-word 12696  df-lsw 12697  df-concat 12698  df-s1 12699  df-substr 12700  df-struct 15171  df-ndx 15172  df-slot 15173  df-base 15174  df-sets 15175  df-ress 15176  df-plusg 15251  df-0g 15388  df-gsum 15389  df-mgm 16536  df-sgrp 16575  df-mnd 16585  df-mhm 16630  df-submnd 16631  df-frmd 16681  df-vrmd 16682  df-mrex 30172  df-mex 30173  df-mrsub 30176  df-msub 30177
This theorem is referenced by:  mclsppslem  30269
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