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Theorem idomsubgmo 27386
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 5705 . . . . . . . . 9  |-  ( Base `  G )  e.  _V
21rabex 4318 . . . . . . . 8  |-  { z  e.  ( Base `  G
)  |  ( ( od `  G ) `
 z )  ||  N }  e.  _V
3 simp2l 983 . . . . . . . . . . 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 2408 . . . . . . . . . . . 12  |-  ( Base `  G )  =  (
Base `  G )
54subgss 14904 . . . . . . . . . . 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 1010 . . . . . . . . . . . 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 985 . . . . . . . . . . . . . . 15  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  (
y  e.  (SubGrp `  G )  /\  x  e.  (SubGrp `  G )
)  /\  ( ( # `
 y )  =  N  /\  ( # `  x )  =  N ) )  ->  ( # `
 y )  =  N )
9 simp1r 982 . . . . . . . . . . . . . . . 16  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  (
y  e.  (SubGrp `  G )  /\  x  e.  (SubGrp `  G )
)  /\  ( ( # `
 y )  =  N  /\  ( # `  x )  =  N ) )  ->  N  e.  NN )
109nnnn0d 10234 . . . . . . . . . . . . . . 15  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  (
y  e.  (SubGrp `  G )  /\  x  e.  (SubGrp `  G )
)  /\  ( ( # `
 y )  =  N  /\  ( # `  x )  =  N ) )  ->  N  e.  NN0 )
118, 10eqeltrd 2482 . . . . . . . . . . . . . 14  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  (
y  e.  (SubGrp `  G )  /\  x  e.  (SubGrp `  G )
)  /\  ( ( # `
 y )  =  N  /\  ( # `  x )  =  N ) )  ->  ( # `
 y )  e. 
NN0 )
12 vex 2923 . . . . . . . . . . . . . . 15  |-  y  e. 
_V
13 hashclb 11600 . . . . . . . . . . . . . . 15  |-  ( y  e.  _V  ->  (
y  e.  Fin  <->  ( # `  y
)  e.  NN0 )
)
1412, 13ax-mp 8 . . . . . . . . . . . . . 14  |-  ( y  e.  Fin  <->  ( # `  y
)  e.  NN0 )
1511, 14sylibr 204 . . . . . . . . . . . . 13  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  (
y  e.  (SubGrp `  G )  /\  x  e.  (SubGrp `  G )
)  /\  ( ( # `
 y )  =  N  /\  ( # `  x )  =  N ) )  ->  y  e.  Fin )
1615adantr 452 . . . . . . . . . . . 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 448 . . . . . . . . . . . 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 2408 . . . . . . . . . . . . 13  |-  ( od
`  G )  =  ( od `  G
)
1918odsubdvds 15164 . . . . . . . . . . . 12  |-  ( ( y  e.  (SubGrp `  G )  /\  y  e.  Fin  /\  z  e.  y )  ->  (
( od `  G
) `  z )  ||  ( # `  y
) )
207, 16, 17, 19syl3anc 1184 . . . . . . . . . . 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 452 . . . . . . . . . . 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 4200 . . . . . . . . . 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 3386 . . . . . . . . 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 984 . . . . . . . . . . 11  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  (
y  e.  (SubGrp `  G )  /\  x  e.  (SubGrp `  G )
)  /\  ( ( # `
 y )  =  N  /\  ( # `  x )  =  N ) )  ->  x  e.  (SubGrp `  G )
)
254subgss 14904 . . . . . . . . . . 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 1011 . . . . . . . . . . . 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 986 . . . . . . . . . . . . . . 15  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  (
y  e.  (SubGrp `  G )  /\  x  e.  (SubGrp `  G )
)  /\  ( ( # `
 y )  =  N  /\  ( # `  x )  =  N ) )  ->  ( # `
 x )  =  N )
2928, 10eqeltrd 2482 . . . . . . . . . . . . . 14  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  (
y  e.  (SubGrp `  G )  /\  x  e.  (SubGrp `  G )
)  /\  ( ( # `
 y )  =  N  /\  ( # `  x )  =  N ) )  ->  ( # `
 x )  e. 
NN0 )
30 vex 2923 . . . . . . . . . . . . . . 15  |-  x  e. 
_V
31 hashclb 11600 . . . . . . . . . . . . . . 15  |-  ( x  e.  _V  ->  (
x  e.  Fin  <->  ( # `  x
)  e.  NN0 )
)
3230, 31ax-mp 8 . . . . . . . . . . . . . 14  |-  ( x  e.  Fin  <->  ( # `  x
)  e.  NN0 )
3329, 32sylibr 204 . . . . . . . . . . . . 13  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  (
y  e.  (SubGrp `  G )  /\  x  e.  (SubGrp `  G )
)  /\  ( ( # `
 y )  =  N  /\  ( # `  x )  =  N ) )  ->  x  e.  Fin )
3433adantr 452 . . . . . . . . . . . 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 448 . . . . . . . . . . . 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 15164 . . . . . . . . . . . 12  |-  ( ( x  e.  (SubGrp `  G )  /\  x  e.  Fin  /\  z  e.  x )  ->  (
( od `  G
) `  z )  ||  ( # `  x
) )
3727, 34, 35, 36syl3anc 1184 . . . . . . . . . . 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 452 . . . . . . . . . . 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 4200 . . . . . . . . . 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 3386 . . . . . . . . 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 3487 . . . . . . . 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 7116 . . . . . . . 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 61 . . . . . . 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 27384 . . . . . . . . . 10  |-  ( ( R  e. IDomn  /\  N  e.  NN )  ->  ( # `
 { z  e.  ( Base `  G
)  |  ( ( od `  G ) `
 z )  ||  N } )  <_  N
)
46453ad2ant1 978 . . . . . . . . 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 4202 . . . . . . . 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 11583 . . . . . . . . . 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 1184 . . . . . . . . 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 11612 . . . . . . . . 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 644 . . . . . . . 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 202 . . . . . . 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 7123 . . . . . . 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 643 . . . . . 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 4670 . . . . . . 7  |-  ( y  u.  x )  e. 
_V
57 ssun1 3474 . . . . . . 7  |-  y  C_  ( y  u.  x
)
58 ssdomg 7116 . . . . . . 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 7190 . . . . . 6  |-  ( ( ( y  u.  x
)  ~<_  y  /\  y  ~<_  ( y  u.  x
) )  ->  (
y  u.  x ) 
~~  y )
6155, 59, 60sylancl 644 . . . . 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 2443 . . . . . . 7  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  (
y  e.  (SubGrp `  G )  /\  x  e.  (SubGrp `  G )
)  /\  ( ( # `
 y )  =  N  /\  ( # `  x )  =  N ) )  ->  ( # `
 y )  =  ( # `  x
) )
63 hashen 11590 . . . . . . . 8  |-  ( ( y  e.  Fin  /\  x  e.  Fin )  ->  ( ( # `  y
)  =  ( # `  x )  <->  y  ~~  x ) )
6415, 33, 63syl2anc 643 . . . . . . 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 202 . . . . . 6  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  (
y  e.  (SubGrp `  G )  /\  x  e.  (SubGrp `  G )
)  /\  ( ( # `
 y )  =  N  /\  ( # `  x )  =  N ) )  ->  y  ~~  x )
66 fiuneneq 27385 . . . . . 6  |-  ( ( y  ~~  x  /\  y  e.  Fin )  ->  ( ( y  u.  x )  ~~  y  <->  y  =  x ) )
6765, 15, 66syl2anc 643 . . . . 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 202 . . . 4  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  (
y  e.  (SubGrp `  G )  /\  x  e.  (SubGrp `  G )
)  /\  ( ( # `
 y )  =  N  /\  ( # `  x )  =  N ) )  ->  y  =  x )
69683expia 1155 . . 3  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  (
y  e.  (SubGrp `  G )  /\  x  e.  (SubGrp `  G )
) )  ->  (
( ( # `  y
)  =  N  /\  ( # `  x )  =  N )  -> 
y  =  x ) )
7069ralrimivva 2762 . 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 5691 . . . 4  |-  ( y  =  x  ->  ( # `
 y )  =  ( # `  x
) )
7271eqeq1d 2416 . . 3  |-  ( y  =  x  ->  (
( # `  y )  =  N  <->  ( # `  x
)  =  N ) )
7372rmo4 3091 . 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 204 1  |-  ( ( R  e. IDomn  /\  N  e.  NN )  ->  E* y  e.  (SubGrp `  G
) ( # `  y
)  =  N )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   A.wral 2670   E*wrmo 2673   {crab 2674   _Vcvv 2920    u. cun 3282    C_ wss 3284   class class class wbr 4176   ` cfv 5417  (class class class)co 6044    ~~ cen 7069    ~<_ cdom 7070   Fincfn 7072    <_ cle 9081   NNcn 9960   NN0cn0 10181   #chash 11577    || cdivides 12811   Basecbs 13428   ↾s cress 13429  SubGrpcsubg 14897   odcod 15122  mulGrpcmgp 15607  Unitcui 15703  IDomncidom 16300
This theorem is referenced by:  proot1mul  27387
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-inf2 7556  ax-cnex 9006  ax-resscn 9007  ax-1cn 9008  ax-icn 9009  ax-addcl 9010  ax-addrcl 9011  ax-mulcl 9012  ax-mulrcl 9013  ax-mulcom 9014  ax-addass 9015  ax-mulass 9016  ax-distr 9017  ax-i2m1 9018  ax-1ne0 9019  ax-1rid 9020  ax-rnegex 9021  ax-rrecex 9022  ax-cnre 9023  ax-pre-lttri 9024  ax-pre-lttrn 9025  ax-pre-ltadd 9026  ax-pre-mulgt0 9027  ax-pre-sup 9028  ax-addf 9029  ax-mulf 9030
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-int 4015  df-iun 4059  df-iin 4060  df-disj 4147  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-se 4506  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-isom 5426  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-of 6268  df-ofr 6269  df-1st 6312  df-2nd 6313  df-tpos 6442  df-riota 6512  df-recs 6596  df-rdg 6631  df-1o 6687  df-2o 6688  df-oadd 6691  df-omul 6692  df-er 6868  df-ec 6870  df-qs 6874  df-map 6983  df-pm 6984  df-ixp 7027  df-en 7073  df-dom 7074  df-sdom 7075  df-fin 7076  df-sup 7408  df-oi 7439  df-card 7786  df-acn 7789  df-cda 8008  df-pnf 9082  df-mnf 9083  df-xr 9084  df-ltxr 9085  df-le 9086  df-sub 9253  df-neg 9254  df-div 9638  df-nn 9961  df-2 10018  df-3 10019  df-4 10020  df-5 10021  df-6 10022  df-7 10023  df-8 10024  df-9 10025  df-10 10026  df-n0 10182  df-z 10243  df-dec 10343  df-uz 10449  df-rp 10573  df-fz 11004  df-fzo 11095  df-fl 11161  df-mod 11210  df-seq 11283  df-exp 11342  df-hash 11578  df-cj 11863  df-re 11864  df-im 11865  df-sqr 11999  df-abs 12000  df-clim 12241  df-sum 12439  df-dvds 12812  df-struct 13430  df-ndx 13431  df-slot 13432  df-base 13433  df-sets 13434  df-ress 13435  df-plusg 13501  df-mulr 13502  df-starv 13503  df-sca 13504  df-vsca 13505  df-tset 13507  df-ple 13508  df-ds 13510  df-unif 13511  df-hom 13512  df-cco 13513  df-prds 13630  df-pws 13632  df-0g 13686  df-gsum 13687  df-mre 13770  df-mrc 13771  df-acs 13773  df-mnd 14649  df-mhm 14697  df-submnd 14698  df-grp 14771  df-minusg 14772  df-sbg 14773  df-mulg 14774  df-subg 14900  df-eqg 14902  df-ghm 14963  df-cntz 15075  df-od 15126  df-cmn 15373  df-abl 15374  df-mgp 15608  df-rng 15622  df-cring 15623  df-ur 15624  df-oppr 15687  df-dvdsr 15705  df-unit 15706  df-invr 15736  df-rnghom 15778  df-subrg 15825  df-lmod 15911  df-lss 15968  df-lsp 16007  df-nzr 16288  df-rlreg 16302  df-domn 16303  df-idom 16304  df-assa 16331  df-asp 16332  df-ascl 16333  df-psr 16376  df-mvr 16377  df-mpl 16378  df-evls 16379  df-evl 16380  df-opsr 16384  df-psr1 16535  df-vr1 16536  df-ply1 16537  df-evl1 16539  df-coe1 16540  df-cnfld 16663  df-mdeg 19935  df-deg1 19936  df-mon1 20010  df-uc1p 20011  df-q1p 20012  df-r1p 20013
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