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Theorem idomsubgmo 29516
Description: The units of an integral domain have at most one subgroup of any single finite cardinality. (Contributed by Stefan O'Rear, 12-Sep-2015.) (Revised by NM, 17-Jun-2017.)
Hypothesis
Ref Expression
idomsubgmo.g  |-  G  =  ( (mulGrp `  R
)s  (Unit `  R )
)
Assertion
Ref Expression
idomsubgmo  |-  ( ( R  e. IDomn  /\  N  e.  NN )  ->  E* y  e.  (SubGrp `  G
) ( # `  y
)  =  N )
Distinct variable groups:    y, G    y, N    y, R

Proof of Theorem idomsubgmo
Dummy variables  x  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fvex 5696 . . . . . . . . 9  |-  ( Base `  G )  e.  _V
21rabex 4438 . . . . . . . 8  |-  { z  e.  ( Base `  G
)  |  ( ( od `  G ) `
 z )  ||  N }  e.  _V
3 simp2l 1014 . . . . . . . . . . 11  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  (
y  e.  (SubGrp `  G )  /\  x  e.  (SubGrp `  G )
)  /\  ( ( # `
 y )  =  N  /\  ( # `  x )  =  N ) )  ->  y  e.  (SubGrp `  G )
)
4 eqid 2438 . . . . . . . . . . . 12  |-  ( Base `  G )  =  (
Base `  G )
54subgss 15673 . . . . . . . . . . 11  |-  ( y  e.  (SubGrp `  G
)  ->  y  C_  ( Base `  G )
)
63, 5syl 16 . . . . . . . . . 10  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  (
y  e.  (SubGrp `  G )  /\  x  e.  (SubGrp `  G )
)  /\  ( ( # `
 y )  =  N  /\  ( # `  x )  =  N ) )  ->  y  C_  ( Base `  G
) )
7 simpl2l 1041 . . . . . . . . . . . 12  |-  ( ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  ( y  e.  (SubGrp `  G )  /\  x  e.  (SubGrp `  G ) )  /\  ( ( # `  y
)  =  N  /\  ( # `  x )  =  N ) )  /\  z  e.  y )  ->  y  e.  (SubGrp `  G ) )
8 simp3l 1016 . . . . . . . . . . . . . . 15  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  (
y  e.  (SubGrp `  G )  /\  x  e.  (SubGrp `  G )
)  /\  ( ( # `
 y )  =  N  /\  ( # `  x )  =  N ) )  ->  ( # `
 y )  =  N )
9 simp1r 1013 . . . . . . . . . . . . . . . 16  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  (
y  e.  (SubGrp `  G )  /\  x  e.  (SubGrp `  G )
)  /\  ( ( # `
 y )  =  N  /\  ( # `  x )  =  N ) )  ->  N  e.  NN )
109nnnn0d 10628 . . . . . . . . . . . . . . 15  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  (
y  e.  (SubGrp `  G )  /\  x  e.  (SubGrp `  G )
)  /\  ( ( # `
 y )  =  N  /\  ( # `  x )  =  N ) )  ->  N  e.  NN0 )
118, 10eqeltrd 2512 . . . . . . . . . . . . . 14  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  (
y  e.  (SubGrp `  G )  /\  x  e.  (SubGrp `  G )
)  /\  ( ( # `
 y )  =  N  /\  ( # `  x )  =  N ) )  ->  ( # `
 y )  e. 
NN0 )
12 vex 2970 . . . . . . . . . . . . . . 15  |-  y  e. 
_V
13 hashclb 12120 . . . . . . . . . . . . . . 15  |-  ( y  e.  _V  ->  (
y  e.  Fin  <->  ( # `  y
)  e.  NN0 )
)
1412, 13ax-mp 5 . . . . . . . . . . . . . 14  |-  ( y  e.  Fin  <->  ( # `  y
)  e.  NN0 )
1511, 14sylibr 212 . . . . . . . . . . . . 13  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  (
y  e.  (SubGrp `  G )  /\  x  e.  (SubGrp `  G )
)  /\  ( ( # `
 y )  =  N  /\  ( # `  x )  =  N ) )  ->  y  e.  Fin )
1615adantr 465 . . . . . . . . . . . 12  |-  ( ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  ( y  e.  (SubGrp `  G )  /\  x  e.  (SubGrp `  G ) )  /\  ( ( # `  y
)  =  N  /\  ( # `  x )  =  N ) )  /\  z  e.  y )  ->  y  e.  Fin )
17 simpr 461 . . . . . . . . . . . 12  |-  ( ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  ( y  e.  (SubGrp `  G )  /\  x  e.  (SubGrp `  G ) )  /\  ( ( # `  y
)  =  N  /\  ( # `  x )  =  N ) )  /\  z  e.  y )  ->  z  e.  y )
18 eqid 2438 . . . . . . . . . . . . 13  |-  ( od
`  G )  =  ( od `  G
)
1918odsubdvds 16061 . . . . . . . . . . . 12  |-  ( ( y  e.  (SubGrp `  G )  /\  y  e.  Fin  /\  z  e.  y )  ->  (
( od `  G
) `  z )  ||  ( # `  y
) )
207, 16, 17, 19syl3anc 1218 . . . . . . . . . . 11  |-  ( ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  ( y  e.  (SubGrp `  G )  /\  x  e.  (SubGrp `  G ) )  /\  ( ( # `  y
)  =  N  /\  ( # `  x )  =  N ) )  /\  z  e.  y )  ->  ( ( od `  G ) `  z )  ||  ( # `
 y ) )
218adantr 465 . . . . . . . . . . 11  |-  ( ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  ( y  e.  (SubGrp `  G )  /\  x  e.  (SubGrp `  G ) )  /\  ( ( # `  y
)  =  N  /\  ( # `  x )  =  N ) )  /\  z  e.  y )  ->  ( # `  y
)  =  N )
2220, 21breqtrd 4311 . . . . . . . . . 10  |-  ( ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  ( y  e.  (SubGrp `  G )  /\  x  e.  (SubGrp `  G ) )  /\  ( ( # `  y
)  =  N  /\  ( # `  x )  =  N ) )  /\  z  e.  y )  ->  ( ( od `  G ) `  z )  ||  N
)
236, 22ssrabdv 3426 . . . . . . . . 9  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  (
y  e.  (SubGrp `  G )  /\  x  e.  (SubGrp `  G )
)  /\  ( ( # `
 y )  =  N  /\  ( # `  x )  =  N ) )  ->  y  C_ 
{ z  e.  (
Base `  G )  |  ( ( od
`  G ) `  z )  ||  N } )
24 simp2r 1015 . . . . . . . . . . 11  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  (
y  e.  (SubGrp `  G )  /\  x  e.  (SubGrp `  G )
)  /\  ( ( # `
 y )  =  N  /\  ( # `  x )  =  N ) )  ->  x  e.  (SubGrp `  G )
)
254subgss 15673 . . . . . . . . . . 11  |-  ( x  e.  (SubGrp `  G
)  ->  x  C_  ( Base `  G ) )
2624, 25syl 16 . . . . . . . . . 10  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  (
y  e.  (SubGrp `  G )  /\  x  e.  (SubGrp `  G )
)  /\  ( ( # `
 y )  =  N  /\  ( # `  x )  =  N ) )  ->  x  C_  ( Base `  G
) )
27 simpl2r 1042 . . . . . . . . . . . 12  |-  ( ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  ( y  e.  (SubGrp `  G )  /\  x  e.  (SubGrp `  G ) )  /\  ( ( # `  y
)  =  N  /\  ( # `  x )  =  N ) )  /\  z  e.  x
)  ->  x  e.  (SubGrp `  G ) )
28 simp3r 1017 . . . . . . . . . . . . . . 15  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  (
y  e.  (SubGrp `  G )  /\  x  e.  (SubGrp `  G )
)  /\  ( ( # `
 y )  =  N  /\  ( # `  x )  =  N ) )  ->  ( # `
 x )  =  N )
2928, 10eqeltrd 2512 . . . . . . . . . . . . . 14  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  (
y  e.  (SubGrp `  G )  /\  x  e.  (SubGrp `  G )
)  /\  ( ( # `
 y )  =  N  /\  ( # `  x )  =  N ) )  ->  ( # `
 x )  e. 
NN0 )
30 vex 2970 . . . . . . . . . . . . . . 15  |-  x  e. 
_V
31 hashclb 12120 . . . . . . . . . . . . . . 15  |-  ( x  e.  _V  ->  (
x  e.  Fin  <->  ( # `  x
)  e.  NN0 )
)
3230, 31ax-mp 5 . . . . . . . . . . . . . 14  |-  ( x  e.  Fin  <->  ( # `  x
)  e.  NN0 )
3329, 32sylibr 212 . . . . . . . . . . . . 13  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  (
y  e.  (SubGrp `  G )  /\  x  e.  (SubGrp `  G )
)  /\  ( ( # `
 y )  =  N  /\  ( # `  x )  =  N ) )  ->  x  e.  Fin )
3433adantr 465 . . . . . . . . . . . 12  |-  ( ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  ( y  e.  (SubGrp `  G )  /\  x  e.  (SubGrp `  G ) )  /\  ( ( # `  y
)  =  N  /\  ( # `  x )  =  N ) )  /\  z  e.  x
)  ->  x  e.  Fin )
35 simpr 461 . . . . . . . . . . . 12  |-  ( ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  ( y  e.  (SubGrp `  G )  /\  x  e.  (SubGrp `  G ) )  /\  ( ( # `  y
)  =  N  /\  ( # `  x )  =  N ) )  /\  z  e.  x
)  ->  z  e.  x )
3618odsubdvds 16061 . . . . . . . . . . . 12  |-  ( ( x  e.  (SubGrp `  G )  /\  x  e.  Fin  /\  z  e.  x )  ->  (
( od `  G
) `  z )  ||  ( # `  x
) )
3727, 34, 35, 36syl3anc 1218 . . . . . . . . . . 11  |-  ( ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  ( y  e.  (SubGrp `  G )  /\  x  e.  (SubGrp `  G ) )  /\  ( ( # `  y
)  =  N  /\  ( # `  x )  =  N ) )  /\  z  e.  x
)  ->  ( ( od `  G ) `  z )  ||  ( # `
 x ) )
3828adantr 465 . . . . . . . . . . 11  |-  ( ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  ( y  e.  (SubGrp `  G )  /\  x  e.  (SubGrp `  G ) )  /\  ( ( # `  y
)  =  N  /\  ( # `  x )  =  N ) )  /\  z  e.  x
)  ->  ( # `  x
)  =  N )
3937, 38breqtrd 4311 . . . . . . . . . 10  |-  ( ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  ( y  e.  (SubGrp `  G )  /\  x  e.  (SubGrp `  G ) )  /\  ( ( # `  y
)  =  N  /\  ( # `  x )  =  N ) )  /\  z  e.  x
)  ->  ( ( od `  G ) `  z )  ||  N
)
4026, 39ssrabdv 3426 . . . . . . . . 9  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  (
y  e.  (SubGrp `  G )  /\  x  e.  (SubGrp `  G )
)  /\  ( ( # `
 y )  =  N  /\  ( # `  x )  =  N ) )  ->  x  C_ 
{ z  e.  (
Base `  G )  |  ( ( od
`  G ) `  z )  ||  N } )
4123, 40unssd 3527 . . . . . . . 8  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  (
y  e.  (SubGrp `  G )  /\  x  e.  (SubGrp `  G )
)  /\  ( ( # `
 y )  =  N  /\  ( # `  x )  =  N ) )  ->  (
y  u.  x ) 
C_  { z  e.  ( Base `  G
)  |  ( ( od `  G ) `
 z )  ||  N } )
42 ssdomg 7347 . . . . . . . 8  |-  ( { z  e.  ( Base `  G )  |  ( ( od `  G
) `  z )  ||  N }  e.  _V  ->  ( ( y  u.  x )  C_  { z  e.  ( Base `  G
)  |  ( ( od `  G ) `
 z )  ||  N }  ->  ( y  u.  x )  ~<_  { z  e.  ( Base `  G )  |  ( ( od `  G
) `  z )  ||  N } ) )
432, 41, 42mpsyl 63 . . . . . . 7  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  (
y  e.  (SubGrp `  G )  /\  x  e.  (SubGrp `  G )
)  /\  ( ( # `
 y )  =  N  /\  ( # `  x )  =  N ) )  ->  (
y  u.  x )  ~<_  { z  e.  (
Base `  G )  |  ( ( od
`  G ) `  z )  ||  N } )
44 idomsubgmo.g . . . . . . . . . . 11  |-  G  =  ( (mulGrp `  R
)s  (Unit `  R )
)
4544, 4, 18idomodle 29514 . . . . . . . . . 10  |-  ( ( R  e. IDomn  /\  N  e.  NN )  ->  ( # `
 { z  e.  ( Base `  G
)  |  ( ( od `  G ) `
 z )  ||  N } )  <_  N
)
46453ad2ant1 1009 . . . . . . . . 9  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  (
y  e.  (SubGrp `  G )  /\  x  e.  (SubGrp `  G )
)  /\  ( ( # `
 y )  =  N  /\  ( # `  x )  =  N ) )  ->  ( # `
 { z  e.  ( Base `  G
)  |  ( ( od `  G ) `
 z )  ||  N } )  <_  N
)
4746, 8breqtrrd 4313 . . . . . . . 8  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  (
y  e.  (SubGrp `  G )  /\  x  e.  (SubGrp `  G )
)  /\  ( ( # `
 y )  =  N  /\  ( # `  x )  =  N ) )  ->  ( # `
 { z  e.  ( Base `  G
)  |  ( ( od `  G ) `
 z )  ||  N } )  <_  ( # `
 y ) )
482a1i 11 . . . . . . . . . 10  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  (
y  e.  (SubGrp `  G )  /\  x  e.  (SubGrp `  G )
)  /\  ( ( # `
 y )  =  N  /\  ( # `  x )  =  N ) )  ->  { z  e.  ( Base `  G
)  |  ( ( od `  G ) `
 z )  ||  N }  e.  _V )
49 hashbnd 12101 . . . . . . . . . 10  |-  ( ( { z  e.  (
Base `  G )  |  ( ( od
`  G ) `  z )  ||  N }  e.  _V  /\  ( # `
 y )  e. 
NN0  /\  ( # `  {
z  e.  ( Base `  G )  |  ( ( od `  G
) `  z )  ||  N } )  <_ 
( # `  y ) )  ->  { z  e.  ( Base `  G
)  |  ( ( od `  G ) `
 z )  ||  N }  e.  Fin )
5048, 11, 47, 49syl3anc 1218 . . . . . . . . 9  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  (
y  e.  (SubGrp `  G )  /\  x  e.  (SubGrp `  G )
)  /\  ( ( # `
 y )  =  N  /\  ( # `  x )  =  N ) )  ->  { z  e.  ( Base `  G
)  |  ( ( od `  G ) `
 z )  ||  N }  e.  Fin )
51 hashdom 12134 . . . . . . . . 9  |-  ( ( { z  e.  (
Base `  G )  |  ( ( od
`  G ) `  z )  ||  N }  e.  Fin  /\  y  e.  _V )  ->  (
( # `  { z  e.  ( Base `  G
)  |  ( ( od `  G ) `
 z )  ||  N } )  <_  ( # `
 y )  <->  { z  e.  ( Base `  G
)  |  ( ( od `  G ) `
 z )  ||  N }  ~<_  y )
)
5250, 12, 51sylancl 662 . . . . . . . 8  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  (
y  e.  (SubGrp `  G )  /\  x  e.  (SubGrp `  G )
)  /\  ( ( # `
 y )  =  N  /\  ( # `  x )  =  N ) )  ->  (
( # `  { z  e.  ( Base `  G
)  |  ( ( od `  G ) `
 z )  ||  N } )  <_  ( # `
 y )  <->  { z  e.  ( Base `  G
)  |  ( ( od `  G ) `
 z )  ||  N }  ~<_  y )
)
5347, 52mpbid 210 . . . . . . 7  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  (
y  e.  (SubGrp `  G )  /\  x  e.  (SubGrp `  G )
)  /\  ( ( # `
 y )  =  N  /\  ( # `  x )  =  N ) )  ->  { z  e.  ( Base `  G
)  |  ( ( od `  G ) `
 z )  ||  N }  ~<_  y )
54 domtr 7354 . . . . . . 7  |-  ( ( ( y  u.  x
)  ~<_  { z  e.  ( Base `  G
)  |  ( ( od `  G ) `
 z )  ||  N }  /\  { z  e.  ( Base `  G
)  |  ( ( od `  G ) `
 z )  ||  N }  ~<_  y )  ->  ( y  u.  x
)  ~<_  y )
5543, 53, 54syl2anc 661 . . . . . 6  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  (
y  e.  (SubGrp `  G )  /\  x  e.  (SubGrp `  G )
)  /\  ( ( # `
 y )  =  N  /\  ( # `  x )  =  N ) )  ->  (
y  u.  x )  ~<_  y )
5612, 30unex 6373 . . . . . . 7  |-  ( y  u.  x )  e. 
_V
57 ssun1 3514 . . . . . . 7  |-  y  C_  ( y  u.  x
)
58 ssdomg 7347 . . . . . . 7  |-  ( ( y  u.  x )  e.  _V  ->  (
y  C_  ( y  u.  x )  ->  y  ~<_  ( y  u.  x
) ) )
5956, 57, 58mp2 9 . . . . . 6  |-  y  ~<_  ( y  u.  x )
60 sbth 7423 . . . . . 6  |-  ( ( ( y  u.  x
)  ~<_  y  /\  y  ~<_  ( y  u.  x
) )  ->  (
y  u.  x ) 
~~  y )
6155, 59, 60sylancl 662 . . . . 5  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  (
y  e.  (SubGrp `  G )  /\  x  e.  (SubGrp `  G )
)  /\  ( ( # `
 y )  =  N  /\  ( # `  x )  =  N ) )  ->  (
y  u.  x ) 
~~  y )
628, 28eqtr4d 2473 . . . . . . 7  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  (
y  e.  (SubGrp `  G )  /\  x  e.  (SubGrp `  G )
)  /\  ( ( # `
 y )  =  N  /\  ( # `  x )  =  N ) )  ->  ( # `
 y )  =  ( # `  x
) )
63 hashen 12110 . . . . . . . 8  |-  ( ( y  e.  Fin  /\  x  e.  Fin )  ->  ( ( # `  y
)  =  ( # `  x )  <->  y  ~~  x ) )
6415, 33, 63syl2anc 661 . . . . . . 7  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  (
y  e.  (SubGrp `  G )  /\  x  e.  (SubGrp `  G )
)  /\  ( ( # `
 y )  =  N  /\  ( # `  x )  =  N ) )  ->  (
( # `  y )  =  ( # `  x
)  <->  y  ~~  x
) )
6562, 64mpbid 210 . . . . . 6  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  (
y  e.  (SubGrp `  G )  /\  x  e.  (SubGrp `  G )
)  /\  ( ( # `
 y )  =  N  /\  ( # `  x )  =  N ) )  ->  y  ~~  x )
66 fiuneneq 29515 . . . . . 6  |-  ( ( y  ~~  x  /\  y  e.  Fin )  ->  ( ( y  u.  x )  ~~  y  <->  y  =  x ) )
6765, 15, 66syl2anc 661 . . . . 5  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  (
y  e.  (SubGrp `  G )  /\  x  e.  (SubGrp `  G )
)  /\  ( ( # `
 y )  =  N  /\  ( # `  x )  =  N ) )  ->  (
( y  u.  x
)  ~~  y  <->  y  =  x ) )
6861, 67mpbid 210 . . . 4  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  (
y  e.  (SubGrp `  G )  /\  x  e.  (SubGrp `  G )
)  /\  ( ( # `
 y )  =  N  /\  ( # `  x )  =  N ) )  ->  y  =  x )
69683expia 1189 . . 3  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  (
y  e.  (SubGrp `  G )  /\  x  e.  (SubGrp `  G )
) )  ->  (
( ( # `  y
)  =  N  /\  ( # `  x )  =  N )  -> 
y  =  x ) )
7069ralrimivva 2803 . 2  |-  ( ( R  e. IDomn  /\  N  e.  NN )  ->  A. y  e.  (SubGrp `  G ) A. x  e.  (SubGrp `  G ) ( ( ( # `  y
)  =  N  /\  ( # `  x )  =  N )  -> 
y  =  x ) )
71 fveq2 5686 . . . 4  |-  ( y  =  x  ->  ( # `
 y )  =  ( # `  x
) )
7271eqeq1d 2446 . . 3  |-  ( y  =  x  ->  (
( # `  y )  =  N  <->  ( # `  x
)  =  N ) )
7372rmo4 3147 . 2  |-  ( E* y  e.  (SubGrp `  G ) ( # `  y )  =  N  <->  A. y  e.  (SubGrp `  G ) A. x  e.  (SubGrp `  G )
( ( ( # `  y )  =  N  /\  ( # `  x
)  =  N )  ->  y  =  x ) )
7470, 73sylibr 212 1  |-  ( ( R  e. IDomn  /\  N  e.  NN )  ->  E* y  e.  (SubGrp `  G
) ( # `  y
)  =  N )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2710   E*wrmo 2713   {crab 2714   _Vcvv 2967    u. cun 3321    C_ wss 3323   class class class wbr 4287   ` cfv 5413  (class class class)co 6086    ~~ cen 7299    ~<_ cdom 7300   Fincfn 7302    <_ cle 9411   NNcn 10314   NN0cn0 10571   #chash 12095    || cdivides 13527   Basecbs 14166   ↾s cress 14167  SubGrpcsubg 15666   odcod 16019  mulGrpcmgp 16579  Unitcui 16719  IDomncidom 17329
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-inf2 7839  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352  ax-addf 9353  ax-mulf 9354
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-iin 4169  df-disj 4258  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-se 4675  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-of 6315  df-ofr 6316  df-om 6472  df-1st 6572  df-2nd 6573  df-supp 6686  df-tpos 6740  df-recs 6824  df-rdg 6858  df-1o 6912  df-2o 6913  df-oadd 6916  df-omul 6917  df-er 7093  df-ec 7095  df-qs 7099  df-map 7208  df-pm 7209  df-ixp 7256  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-fsupp 7613  df-sup 7683  df-oi 7716  df-card 8101  df-acn 8104  df-cda 8329  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-4 10374  df-5 10375  df-6 10376  df-7 10377  df-8 10378  df-9 10379  df-10 10380  df-n0 10572  df-z 10639  df-dec 10748  df-uz 10854  df-rp 10984  df-fz 11430  df-fzo 11541  df-fl 11634  df-mod 11701  df-seq 11799  df-exp 11858  df-hash 12096  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-clim 12958  df-sum 13156  df-dvds 13528  df-struct 14168  df-ndx 14169  df-slot 14170  df-base 14171  df-sets 14172  df-ress 14173  df-plusg 14243  df-mulr 14244  df-starv 14245  df-sca 14246  df-vsca 14247  df-ip 14248  df-tset 14249  df-ple 14250  df-ds 14252  df-unif 14253  df-hom 14254  df-cco 14255  df-0g 14372  df-gsum 14373  df-prds 14378  df-pws 14380  df-mre 14516  df-mrc 14517  df-acs 14519  df-mnd 15407  df-mhm 15456  df-submnd 15457  df-grp 15536  df-minusg 15537  df-sbg 15538  df-mulg 15539  df-subg 15669  df-eqg 15671  df-ghm 15736  df-cntz 15826  df-od 16023  df-cmn 16270  df-abl 16271  df-mgp 16580  df-ur 16592  df-rng 16635  df-cring 16636  df-oppr 16703  df-dvdsr 16721  df-unit 16722  df-invr 16752  df-rnghom 16794  df-subrg 16841  df-lmod 16928  df-lss 16991  df-lsp 17030  df-nzr 17317  df-rlreg 17331  df-domn 17332  df-idom 17333  df-assa 17361  df-asp 17362  df-ascl 17363  df-psr 17400  df-mvr 17401  df-mpl 17402  df-opsr 17404  df-evls 17563  df-evl 17564  df-psr1 17611  df-vr1 17612  df-ply1 17613  df-coe1 17614  df-evl1 17726  df-cnfld 17794  df-mdeg 21499  df-deg1 21500  df-mon1 21577  df-uc1p 21578  df-q1p 21579  df-r1p 21580
This theorem is referenced by:  proot1mul  29517
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