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Theorem msubco 29158
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 2454 . . . . 5  |-  (mEx `  T )  =  (mEx
`  T )
2 eqid 2454 . . . . 5  |-  (mRSubst `  T
)  =  (mRSubst `  T
)
3 msubco.s . . . . 5  |-  S  =  (mSubst `  T )
41, 2, 3elmsubrn 29155 . . . 4  |-  ran  S  =  ran  ( f  e. 
ran  (mRSubst `  T )  |->  ( x  e.  (mEx
`  T )  |->  <.
( 1st `  x
) ,  ( f `
 ( 2nd `  x
) ) >. )
)
54eleq2i 2532 . . 3  |-  ( F  e.  ran  S  <->  F  e.  ran  ( f  e.  ran  (mRSubst `  T )  |->  ( x  e.  (mEx `  T )  |->  <. ( 1st `  x ) ,  ( f `  ( 2nd `  x ) )
>. ) ) )
6 eqid 2454 . . . 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 5858 . . . . 5  |-  (mEx `  T )  e.  _V
87mptex 6118 . . . 4  |-  ( x  e.  (mEx `  T
)  |->  <. ( 1st `  x
) ,  ( f `
 ( 2nd `  x
) ) >. )  e.  _V
96, 8elrnmpti 5242 . . 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 249 . 2  |-  ( F  e.  ran  S  <->  E. f  e.  ran  (mRSubst `  T
) F  =  ( x  e.  (mEx `  T )  |->  <. ( 1st `  x ) ,  ( f `  ( 2nd `  x ) )
>. ) )
111, 2, 3elmsubrn 29155 . . . 4  |-  ran  S  =  ran  ( g  e. 
ran  (mRSubst `  T )  |->  ( y  e.  (mEx
`  T )  |->  <.
( 1st `  y
) ,  ( g `
 ( 2nd `  y
) ) >. )
)
1211eleq2i 2532 . . 3  |-  ( G  e.  ran  S  <->  G  e.  ran  ( g  e.  ran  (mRSubst `  T )  |->  ( y  e.  (mEx `  T )  |->  <. ( 1st `  y ) ,  ( g `  ( 2nd `  y ) )
>. ) ) )
13 eqid 2454 . . . 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 6118 . . . 4  |-  ( y  e.  (mEx `  T
)  |->  <. ( 1st `  y
) ,  ( g `
 ( 2nd `  y
) ) >. )  e.  _V
1513, 14elrnmpti 5242 . . 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 249 . 2  |-  ( G  e.  ran  S  <->  E. g  e.  ran  (mRSubst `  T
) G  =  ( y  e.  (mEx `  T )  |->  <. ( 1st `  y ) ,  ( g `  ( 2nd `  y ) )
>. ) )
17 reeanv 3022 . . 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 459 . . . . . . . . . . . 12  |-  ( ( ( f  e.  ran  (mRSubst `  T )  /\  g  e.  ran  (mRSubst `  T
) )  /\  y  e.  (mEx `  T )
)  ->  y  e.  (mEx `  T ) )
19 eqid 2454 . . . . . . . . . . . . 13  |-  (mTC `  T )  =  (mTC
`  T )
20 eqid 2454 . . . . . . . . . . . . 13  |-  (mREx `  T )  =  (mREx `  T )
2119, 1, 20mexval 29129 . . . . . . . . . . . 12  |-  (mEx `  T )  =  ( (mTC `  T )  X.  (mREx `  T )
)
2218, 21syl6eleq 2552 . . . . . . . . . . 11  |-  ( ( ( f  e.  ran  (mRSubst `  T )  /\  g  e.  ran  (mRSubst `  T
) )  /\  y  e.  (mEx `  T )
)  ->  y  e.  ( (mTC `  T )  X.  (mREx `  T )
) )
23 xp1st 6803 . . . . . . . . . . 11  |-  ( y  e.  ( (mTC `  T )  X.  (mREx `  T ) )  -> 
( 1st `  y
)  e.  (mTC `  T ) )
2422, 23syl 16 . . . . . . . . . 10  |-  ( ( ( f  e.  ran  (mRSubst `  T )  /\  g  e.  ran  (mRSubst `  T
) )  /\  y  e.  (mEx `  T )
)  ->  ( 1st `  y )  e.  (mTC
`  T ) )
252, 20mrsubf 29144 . . . . . . . . . . . 12  |-  ( g  e.  ran  (mRSubst `  T
)  ->  g :
(mREx `  T ) --> (mREx `  T ) )
2625ad2antlr 724 . . . . . . . . . . 11  |-  ( ( ( f  e.  ran  (mRSubst `  T )  /\  g  e.  ran  (mRSubst `  T
) )  /\  y  e.  (mEx `  T )
)  ->  g :
(mREx `  T ) --> (mREx `  T ) )
27 xp2nd 6804 . . . . . . . . . . . 12  |-  ( y  e.  ( (mTC `  T )  X.  (mREx `  T ) )  -> 
( 2nd `  y
)  e.  (mREx `  T ) )
2822, 27syl 16 . . . . . . . . . . 11  |-  ( ( ( f  e.  ran  (mRSubst `  T )  /\  g  e.  ran  (mRSubst `  T
) )  /\  y  e.  (mEx `  T )
)  ->  ( 2nd `  y )  e.  (mREx `  T ) )
2926, 28ffvelrnd 6008 . . . . . . . . . 10  |-  ( ( ( f  e.  ran  (mRSubst `  T )  /\  g  e.  ran  (mRSubst `  T
) )  /\  y  e.  (mEx `  T )
)  ->  ( g `  ( 2nd `  y
) )  e.  (mREx `  T ) )
30 opelxpi 5020 . . . . . . . . . 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 659 . . . . . . . . 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 2553 . . . . . . . 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 2455 . . . . . . . 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 2455 . . . . . . . 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 5858 . . . . . . . . . 10  |-  ( 1st `  y )  e.  _V
36 fvex 5858 . . . . . . . . . 10  |-  ( g `
 ( 2nd `  y
) )  e.  _V
3735, 36op1std 6783 . . . . . . . . 9  |-  ( x  =  <. ( 1st `  y
) ,  ( g `
 ( 2nd `  y
) ) >.  ->  ( 1st `  x )  =  ( 1st `  y
) )
3835, 36op2ndd 6784 . . . . . . . . . 10  |-  ( x  =  <. ( 1st `  y
) ,  ( g `
 ( 2nd `  y
) ) >.  ->  ( 2nd `  x )  =  ( g `  ( 2nd `  y ) ) )
3938fveq2d 5852 . . . . . . . . 9  |-  ( x  =  <. ( 1st `  y
) ,  ( g `
 ( 2nd `  y
) ) >.  ->  (
f `  ( 2nd `  x ) )  =  ( f `  (
g `  ( 2nd `  y ) ) ) )
4037, 39opeq12d 4211 . . . . . . . 8  |-  ( x  =  <. ( 1st `  y
) ,  ( g `
 ( 2nd `  y
) ) >.  ->  <. ( 1st `  x ) ,  ( f `  ( 2nd `  x ) )
>.  =  <. ( 1st `  y ) ,  ( f `  ( g `
 ( 2nd `  y
) ) ) >.
)
4132, 33, 34, 40fmptco 6040 . . . . . . 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 5925 . . . . . . . . . 10  |-  ( ( g : (mREx `  T ) --> (mREx `  T )  /\  ( 2nd `  y )  e.  (mREx `  T )
)  ->  ( (
f  o.  g ) `
 ( 2nd `  y
) )  =  ( f `  ( g `
 ( 2nd `  y
) ) ) )
4326, 28, 42syl2anc 659 . . . . . . . . 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 4210 . . . . . . . 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 4525 . . . . . . 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 29148 . . . . . . . 8  |-  ( ( f  e.  ran  (mRSubst `  T )  /\  g  e.  ran  (mRSubst `  T
) )  ->  (
f  o.  g )  e.  ran  (mRSubst `  T
) )
487mptex 6118 . . . . . . . 8  |-  ( y  e.  (mEx `  T
)  |->  <. ( 1st `  y
) ,  ( ( f  o.  g ) `
 ( 2nd `  y
) ) >. )  e.  _V
49 eqid 2454 . . . . . . . . 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 5847 . . . . . . . . . . 11  |-  ( h  =  ( f  o.  g )  ->  (
h `  ( 2nd `  y ) )  =  ( ( f  o.  g ) `  ( 2nd `  y ) ) )
5150opeq2d 4210 . . . . . . . . . 10  |-  ( h  =  ( f  o.  g )  ->  <. ( 1st `  y ) ,  ( h `  ( 2nd `  y ) )
>.  =  <. ( 1st `  y ) ,  ( ( f  o.  g
) `  ( 2nd `  y ) ) >.
)
5251mpteq2dv 4526 . . . . . . . . 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 5239 . . . . . . . 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 660 . . . . . . 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 29155 . . . . . . 7  |-  ran  S  =  ran  ( h  e. 
ran  (mRSubst `  T )  |->  ( y  e.  (mEx
`  T )  |->  <.
( 1st `  y
) ,  ( h `
 ( 2nd `  y
) ) >. )
)
5654, 55syl6eleqr 2553 . . . . . 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 2542 . . . . 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 5149 . . . . . . 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 5150 . . . . . . 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 222 . . . 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 2951 . . 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 213 . 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 477 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 367    = wceq 1398    e. wcel 1823   E.wrex 2805   _Vcvv 3106   <.cop 4022    |-> cmpt 4497    X. cxp 4986   ran crn 4989    o. ccom 4992   -->wf 5566   ` cfv 5570   1stc1st 6771   2ndc2nd 6772  mTCcmtc 29091  mRExcmrex 29093  mExcmex 29094  mRSubstcmrsub 29097  mSubstcmsub 29098
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676  df-fzo 11800  df-seq 12093  df-hash 12391  df-word 12529  df-lsw 12530  df-concat 12531  df-s1 12532  df-substr 12533  df-struct 14721  df-ndx 14722  df-slot 14723  df-base 14724  df-sets 14725  df-ress 14726  df-plusg 14800  df-0g 14934  df-gsum 14935  df-mgm 16074  df-sgrp 16113  df-mnd 16123  df-mhm 16168  df-submnd 16169  df-frmd 16219  df-vrmd 16220  df-mrex 29113  df-mex 29114  df-mrsub 29117  df-msub 29118
This theorem is referenced by:  mclsppslem  29210
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