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Theorem lmhmfgsplit 27184
Description: If the kernel and range of a homomorphism of left modules are finitely generated, then so is the domain. (Contributed by Stefan O'Rear, 1-Jan-2015.) (Revised by Stefan O'Rear, 6-May-2015.)
Hypotheses
Ref Expression
lmhmfgsplit.z  |-  .0.  =  ( 0g `  T )
lmhmfgsplit.k  |-  K  =  ( `' F " {  .0.  } )
lmhmfgsplit.u  |-  U  =  ( Ss  K )
lmhmfgsplit.v  |-  V  =  ( Ts  ran  F )
Assertion
Ref Expression
lmhmfgsplit  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  ->  S  e. LFinGen )

Proof of Theorem lmhmfgsplit
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp3 957 . . 3  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  ->  V  e. LFinGen )
2 lmhmlmod2 15789 . . . . 5  |-  ( F  e.  ( S LMHom  T
)  ->  T  e.  LMod )
323ad2ant1 976 . . . 4  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  ->  T  e.  LMod )
4 lmhmrnlss 15807 . . . . 5  |-  ( F  e.  ( S LMHom  T
)  ->  ran  F  e.  ( LSubSp `  T )
)
543ad2ant1 976 . . . 4  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  ->  ran  F  e.  (
LSubSp `  T ) )
6 lmhmfgsplit.v . . . . 5  |-  V  =  ( Ts  ran  F )
7 eqid 2283 . . . . 5  |-  ( LSubSp `  T )  =  (
LSubSp `  T )
8 eqid 2283 . . . . 5  |-  ( LSpan `  T )  =  (
LSpan `  T )
96, 7, 8islssfg 27168 . . . 4  |-  ( ( T  e.  LMod  /\  ran  F  e.  ( LSubSp `  T
) )  ->  ( V  e. LFinGen  <->  E. a  e.  ~P  ran  F ( a  e. 
Fin  /\  ( ( LSpan `  T ) `  a )  =  ran  F ) ) )
103, 5, 9syl2anc 642 . . 3  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  ->  ( V  e. LFinGen  <->  E. a  e.  ~P  ran  F ( a  e.  Fin  /\  ( ( LSpan `  T
) `  a )  =  ran  F ) ) )
111, 10mpbid 201 . 2  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  ->  E. a  e.  ~P  ran  F ( a  e. 
Fin  /\  ( ( LSpan `  T ) `  a )  =  ran  F ) )
12 simpl1 958 . . . . . . 7  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  /\  ( a  e. 
~P ran  F  /\  ( a  e.  Fin  /\  ( ( LSpan `  T
) `  a )  =  ran  F ) ) )  ->  F  e.  ( S LMHom  T ) )
13 eqid 2283 . . . . . . . 8  |-  ( Base `  S )  =  (
Base `  S )
14 eqid 2283 . . . . . . . 8  |-  ( Base `  T )  =  (
Base `  T )
1513, 14lmhmf 15791 . . . . . . 7  |-  ( F  e.  ( S LMHom  T
)  ->  F :
( Base `  S ) --> ( Base `  T )
)
16 ffn 5389 . . . . . . 7  |-  ( F : ( Base `  S
) --> ( Base `  T
)  ->  F  Fn  ( Base `  S )
)
1712, 15, 163syl 18 . . . . . 6  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  /\  ( a  e. 
~P ran  F  /\  ( a  e.  Fin  /\  ( ( LSpan `  T
) `  a )  =  ran  F ) ) )  ->  F  Fn  ( Base `  S )
)
18 elpwi 3633 . . . . . . 7  |-  ( a  e.  ~P ran  F  ->  a  C_  ran  F )
1918ad2antrl 708 . . . . . 6  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  /\  ( a  e. 
~P ran  F  /\  ( a  e.  Fin  /\  ( ( LSpan `  T
) `  a )  =  ran  F ) ) )  ->  a  C_  ran  F )
20 simprrl 740 . . . . . 6  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  /\  ( a  e. 
~P ran  F  /\  ( a  e.  Fin  /\  ( ( LSpan `  T
) `  a )  =  ran  F ) ) )  ->  a  e.  Fin )
21 fipreima 7161 . . . . . 6  |-  ( ( F  Fn  ( Base `  S )  /\  a  C_ 
ran  F  /\  a  e.  Fin )  ->  E. b  e.  ( ~P ( Base `  S )  i^i  Fin ) ( F "
b )  =  a )
2217, 19, 20, 21syl3anc 1182 . . . . 5  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  /\  ( a  e. 
~P ran  F  /\  ( a  e.  Fin  /\  ( ( LSpan `  T
) `  a )  =  ran  F ) ) )  ->  E. b  e.  ( ~P ( Base `  S )  i^i  Fin ) ( F "
b )  =  a )
23 eqid 2283 . . . . . . . . . . 11  |-  ( LSubSp `  S )  =  (
LSubSp `  S )
24 eqid 2283 . . . . . . . . . . 11  |-  ( LSSum `  S )  =  (
LSSum `  S )
25 lmhmfgsplit.z . . . . . . . . . . 11  |-  .0.  =  ( 0g `  T )
26 lmhmfgsplit.k . . . . . . . . . . 11  |-  K  =  ( `' F " {  .0.  } )
27 simpll1 994 . . . . . . . . . . 11  |-  ( ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  /\  ( a  e.  ~P ran  F  /\  ( a  e.  Fin  /\  ( ( LSpan `  T
) `  a )  =  ran  F ) ) )  /\  ( b  e.  ( ~P ( Base `  S )  i^i 
Fin )  /\  ( F " b )  =  a ) )  ->  F  e.  ( S LMHom  T ) )
28 lmhmlmod1 15790 . . . . . . . . . . . . . 14  |-  ( F  e.  ( S LMHom  T
)  ->  S  e.  LMod )
29283ad2ant1 976 . . . . . . . . . . . . 13  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  ->  S  e.  LMod )
3029ad2antrr 706 . . . . . . . . . . . 12  |-  ( ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  /\  ( a  e.  ~P ran  F  /\  ( a  e.  Fin  /\  ( ( LSpan `  T
) `  a )  =  ran  F ) ) )  /\  ( b  e.  ( ~P ( Base `  S )  i^i 
Fin )  /\  ( F " b )  =  a ) )  ->  S  e.  LMod )
31 inss1 3389 . . . . . . . . . . . . . . 15  |-  ( ~P ( Base `  S
)  i^i  Fin )  C_ 
~P ( Base `  S
)
3231sseli 3176 . . . . . . . . . . . . . 14  |-  ( b  e.  ( ~P ( Base `  S )  i^i 
Fin )  ->  b  e.  ~P ( Base `  S
) )
33 elpwi 3633 . . . . . . . . . . . . . 14  |-  ( b  e.  ~P ( Base `  S )  ->  b  C_  ( Base `  S
) )
3432, 33syl 15 . . . . . . . . . . . . 13  |-  ( b  e.  ( ~P ( Base `  S )  i^i 
Fin )  ->  b  C_  ( Base `  S
) )
3534ad2antrl 708 . . . . . . . . . . . 12  |-  ( ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  /\  ( a  e.  ~P ran  F  /\  ( a  e.  Fin  /\  ( ( LSpan `  T
) `  a )  =  ran  F ) ) )  /\  ( b  e.  ( ~P ( Base `  S )  i^i 
Fin )  /\  ( F " b )  =  a ) )  -> 
b  C_  ( Base `  S ) )
36 eqid 2283 . . . . . . . . . . . . 13  |-  ( LSpan `  S )  =  (
LSpan `  S )
3713, 23, 36lspcl 15733 . . . . . . . . . . . 12  |-  ( ( S  e.  LMod  /\  b  C_  ( Base `  S
) )  ->  (
( LSpan `  S ) `  b )  e.  (
LSubSp `  S ) )
3830, 35, 37syl2anc 642 . . . . . . . . . . 11  |-  ( ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  /\  ( a  e.  ~P ran  F  /\  ( a  e.  Fin  /\  ( ( LSpan `  T
) `  a )  =  ran  F ) ) )  /\  ( b  e.  ( ~P ( Base `  S )  i^i 
Fin )  /\  ( F " b )  =  a ) )  -> 
( ( LSpan `  S
) `  b )  e.  ( LSubSp `  S )
)
3913, 36, 8lmhmlsp 15806 . . . . . . . . . . . . 13  |-  ( ( F  e.  ( S LMHom 
T )  /\  b  C_  ( Base `  S
) )  ->  ( F " ( ( LSpan `  S ) `  b
) )  =  ( ( LSpan `  T ) `  ( F " b
) ) )
4027, 35, 39syl2anc 642 . . . . . . . . . . . 12  |-  ( ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  /\  ( a  e.  ~P ran  F  /\  ( a  e.  Fin  /\  ( ( LSpan `  T
) `  a )  =  ran  F ) ) )  /\  ( b  e.  ( ~P ( Base `  S )  i^i 
Fin )  /\  ( F " b )  =  a ) )  -> 
( F " (
( LSpan `  S ) `  b ) )  =  ( ( LSpan `  T
) `  ( F " b ) ) )
41 fveq2 5525 . . . . . . . . . . . . 13  |-  ( ( F " b )  =  a  ->  (
( LSpan `  T ) `  ( F " b
) )  =  ( ( LSpan `  T ) `  a ) )
4241ad2antll 709 . . . . . . . . . . . 12  |-  ( ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  /\  ( a  e.  ~P ran  F  /\  ( a  e.  Fin  /\  ( ( LSpan `  T
) `  a )  =  ran  F ) ) )  /\  ( b  e.  ( ~P ( Base `  S )  i^i 
Fin )  /\  ( F " b )  =  a ) )  -> 
( ( LSpan `  T
) `  ( F " b ) )  =  ( ( LSpan `  T
) `  a )
)
43 simp2rr 1025 . . . . . . . . . . . . 13  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  /\  ( a  e. 
~P ran  F  /\  ( a  e.  Fin  /\  ( ( LSpan `  T
) `  a )  =  ran  F ) )  /\  ( b  e.  ( ~P ( Base `  S )  i^i  Fin )  /\  ( F "
b )  =  a ) )  ->  (
( LSpan `  T ) `  a )  =  ran  F )
44433expa 1151 . . . . . . . . . . . 12  |-  ( ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  /\  ( a  e.  ~P ran  F  /\  ( a  e.  Fin  /\  ( ( LSpan `  T
) `  a )  =  ran  F ) ) )  /\  ( b  e.  ( ~P ( Base `  S )  i^i 
Fin )  /\  ( F " b )  =  a ) )  -> 
( ( LSpan `  T
) `  a )  =  ran  F )
4540, 42, 443eqtrd 2319 . . . . . . . . . . 11  |-  ( ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  /\  ( a  e.  ~P ran  F  /\  ( a  e.  Fin  /\  ( ( LSpan `  T
) `  a )  =  ran  F ) ) )  /\  ( b  e.  ( ~P ( Base `  S )  i^i 
Fin )  /\  ( F " b )  =  a ) )  -> 
( F " (
( LSpan `  S ) `  b ) )  =  ran  F )
4623, 24, 25, 26, 13, 27, 38, 45kercvrlsm 27181 . . . . . . . . . 10  |-  ( ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  /\  ( a  e.  ~P ran  F  /\  ( a  e.  Fin  /\  ( ( LSpan `  T
) `  a )  =  ran  F ) ) )  /\  ( b  e.  ( ~P ( Base `  S )  i^i 
Fin )  /\  ( F " b )  =  a ) )  -> 
( K ( LSSum `  S ) ( (
LSpan `  S ) `  b ) )  =  ( Base `  S
) )
4746oveq2d 5874 . . . . . . . . 9  |-  ( ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  /\  ( a  e.  ~P ran  F  /\  ( a  e.  Fin  /\  ( ( LSpan `  T
) `  a )  =  ran  F ) ) )  /\  ( b  e.  ( ~P ( Base `  S )  i^i 
Fin )  /\  ( F " b )  =  a ) )  -> 
( Ss  ( K (
LSSum `  S ) ( ( LSpan `  S ) `  b ) ) )  =  ( Ss  ( Base `  S ) ) )
4813ressid 13203 . . . . . . . . . . 11  |-  ( S  e.  LMod  ->  ( Ss  (
Base `  S )
)  =  S )
4929, 48syl 15 . . . . . . . . . 10  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  ->  ( Ss  ( Base `  S
) )  =  S )
5049ad2antrr 706 . . . . . . . . 9  |-  ( ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  /\  ( a  e.  ~P ran  F  /\  ( a  e.  Fin  /\  ( ( LSpan `  T
) `  a )  =  ran  F ) ) )  /\  ( b  e.  ( ~P ( Base `  S )  i^i 
Fin )  /\  ( F " b )  =  a ) )  -> 
( Ss  ( Base `  S
) )  =  S )
5147, 50eqtr2d 2316 . . . . . . . 8  |-  ( ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  /\  ( a  e.  ~P ran  F  /\  ( a  e.  Fin  /\  ( ( LSpan `  T
) `  a )  =  ran  F ) ) )  /\  ( b  e.  ( ~P ( Base `  S )  i^i 
Fin )  /\  ( F " b )  =  a ) )  ->  S  =  ( Ss  ( K ( LSSum `  S
) ( ( LSpan `  S ) `  b
) ) ) )
52 lmhmfgsplit.u . . . . . . . . 9  |-  U  =  ( Ss  K )
53 eqid 2283 . . . . . . . . 9  |-  ( Ss  ( ( LSpan `  S ) `  b ) )  =  ( Ss  ( ( LSpan `  S ) `  b
) )
54 eqid 2283 . . . . . . . . 9  |-  ( Ss  ( K ( LSSum `  S
) ( ( LSpan `  S ) `  b
) ) )  =  ( Ss  ( K (
LSSum `  S ) ( ( LSpan `  S ) `  b ) ) )
5526, 25, 23lmhmkerlss 15808 . . . . . . . . . . 11  |-  ( F  e.  ( S LMHom  T
)  ->  K  e.  ( LSubSp `  S )
)
56553ad2ant1 976 . . . . . . . . . 10  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  ->  K  e.  ( LSubSp `  S ) )
5756ad2antrr 706 . . . . . . . . 9  |-  ( ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  /\  ( a  e.  ~P ran  F  /\  ( a  e.  Fin  /\  ( ( LSpan `  T
) `  a )  =  ran  F ) ) )  /\  ( b  e.  ( ~P ( Base `  S )  i^i 
Fin )  /\  ( F " b )  =  a ) )  ->  K  e.  ( LSubSp `  S ) )
58 simpll2 995 . . . . . . . . 9  |-  ( ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  /\  ( a  e.  ~P ran  F  /\  ( a  e.  Fin  /\  ( ( LSpan `  T
) `  a )  =  ran  F ) ) )  /\  ( b  e.  ( ~P ( Base `  S )  i^i 
Fin )  /\  ( F " b )  =  a ) )  ->  U  e. LFinGen )
59 inss2 3390 . . . . . . . . . . . 12  |-  ( ~P ( Base `  S
)  i^i  Fin )  C_ 
Fin
6059sseli 3176 . . . . . . . . . . 11  |-  ( b  e.  ( ~P ( Base `  S )  i^i 
Fin )  ->  b  e.  Fin )
6160ad2antrl 708 . . . . . . . . . 10  |-  ( ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  /\  ( a  e.  ~P ran  F  /\  ( a  e.  Fin  /\  ( ( LSpan `  T
) `  a )  =  ran  F ) ) )  /\  ( b  e.  ( ~P ( Base `  S )  i^i 
Fin )  /\  ( F " b )  =  a ) )  -> 
b  e.  Fin )
6236, 13, 53islssfgi 27170 . . . . . . . . . 10  |-  ( ( S  e.  LMod  /\  b  C_  ( Base `  S
)  /\  b  e.  Fin )  ->  ( Ss  ( ( LSpan `  S ) `  b ) )  e. LFinGen )
6330, 35, 61, 62syl3anc 1182 . . . . . . . . 9  |-  ( ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  /\  ( a  e.  ~P ran  F  /\  ( a  e.  Fin  /\  ( ( LSpan `  T
) `  a )  =  ran  F ) ) )  /\  ( b  e.  ( ~P ( Base `  S )  i^i 
Fin )  /\  ( F " b )  =  a ) )  -> 
( Ss  ( ( LSpan `  S ) `  b
) )  e. LFinGen )
6423, 24, 52, 53, 54, 30, 57, 38, 58, 63lsmfgcl 27172 . . . . . . . 8  |-  ( ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  /\  ( a  e.  ~P ran  F  /\  ( a  e.  Fin  /\  ( ( LSpan `  T
) `  a )  =  ran  F ) ) )  /\  ( b  e.  ( ~P ( Base `  S )  i^i 
Fin )  /\  ( F " b )  =  a ) )  -> 
( Ss  ( K (
LSSum `  S ) ( ( LSpan `  S ) `  b ) ) )  e. LFinGen )
6551, 64eqeltrd 2357 . . . . . . 7  |-  ( ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  /\  ( a  e.  ~P ran  F  /\  ( a  e.  Fin  /\  ( ( LSpan `  T
) `  a )  =  ran  F ) ) )  /\  ( b  e.  ( ~P ( Base `  S )  i^i 
Fin )  /\  ( F " b )  =  a ) )  ->  S  e. LFinGen )
6665expr 598 . . . . . 6  |-  ( ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  /\  ( a  e.  ~P ran  F  /\  ( a  e.  Fin  /\  ( ( LSpan `  T
) `  a )  =  ran  F ) ) )  /\  b  e.  ( ~P ( Base `  S )  i^i  Fin ) )  ->  (
( F " b
)  =  a  ->  S  e. LFinGen ) )
6766rexlimdva 2667 . . . . 5  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  /\  ( a  e. 
~P ran  F  /\  ( a  e.  Fin  /\  ( ( LSpan `  T
) `  a )  =  ran  F ) ) )  ->  ( E. b  e.  ( ~P ( Base `  S )  i^i  Fin ) ( F
" b )  =  a  ->  S  e. LFinGen ) )
6822, 67mpd 14 . . . 4  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  /\  ( a  e. 
~P ran  F  /\  ( a  e.  Fin  /\  ( ( LSpan `  T
) `  a )  =  ran  F ) ) )  ->  S  e. LFinGen )
6968expr 598 . . 3  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  /\  a  e.  ~P ran  F )  ->  (
( a  e.  Fin  /\  ( ( LSpan `  T
) `  a )  =  ran  F )  ->  S  e. LFinGen ) )
7069rexlimdva 2667 . 2  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  ->  ( E. a  e. 
~P  ran  F (
a  e.  Fin  /\  ( ( LSpan `  T
) `  a )  =  ran  F )  ->  S  e. LFinGen ) )
7111, 70mpd 14 1  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  ->  S  e. LFinGen )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   E.wrex 2544    i^i cin 3151    C_ wss 3152   ~Pcpw 3625   {csn 3640   `'ccnv 4688   ran crn 4690   "cima 4692    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   Fincfn 6863   Basecbs 13148   ↾s cress 13149   0gc0g 13400   LSSumclsm 14945   LModclmod 15627   LSubSpclss 15689   LSpanclspn 15728   LMHom clmhm 15776  LFinGenclfig 27165
This theorem is referenced by:  lmhmlnmsplit  27185
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-sca 13224  df-vsca 13225  df-0g 13404  df-mnd 14367  df-submnd 14416  df-grp 14489  df-minusg 14490  df-sbg 14491  df-subg 14618  df-ghm 14681  df-cntz 14793  df-lsm 14947  df-cmn 15091  df-abl 15092  df-mgp 15326  df-rng 15340  df-ur 15342  df-lmod 15629  df-lss 15690  df-lsp 15729  df-lmhm 15779  df-lfig 27166
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