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Theorem idomsubgmo 27514
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 5539 . . . . . . . . 9  |-  ( Base `  G )  e.  _V
21rabex 4165 . . . . . . . 8  |-  { z  e.  ( Base `  G
)  |  ( ( od `  G ) `
 z )  ||  N }  e.  _V
3 simp2l 981 . . . . . . . . . . 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 2283 . . . . . . . . . . . 12  |-  ( Base `  G )  =  (
Base `  G )
54subgss 14622 . . . . . . . . . . 11  |-  ( y  e.  (SubGrp `  G
)  ->  y  C_  ( Base `  G )
)
63, 5syl 15 . . . . . . . . . 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 1008 . . . . . . . . . . . 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 983 . . . . . . . . . . . . . . 15  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  (
y  e.  (SubGrp `  G )  /\  x  e.  (SubGrp `  G )
)  /\  ( ( # `
 y )  =  N  /\  ( # `  x )  =  N ) )  ->  ( # `
 y )  =  N )
9 simp1r 980 . . . . . . . . . . . . . . . 16  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  (
y  e.  (SubGrp `  G )  /\  x  e.  (SubGrp `  G )
)  /\  ( ( # `
 y )  =  N  /\  ( # `  x )  =  N ) )  ->  N  e.  NN )
109nnnn0d 10018 . . . . . . . . . . . . . . 15  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  (
y  e.  (SubGrp `  G )  /\  x  e.  (SubGrp `  G )
)  /\  ( ( # `
 y )  =  N  /\  ( # `  x )  =  N ) )  ->  N  e.  NN0 )
118, 10eqeltrd 2357 . . . . . . . . . . . . . 14  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  (
y  e.  (SubGrp `  G )  /\  x  e.  (SubGrp `  G )
)  /\  ( ( # `
 y )  =  N  /\  ( # `  x )  =  N ) )  ->  ( # `
 y )  e. 
NN0 )
12 vex 2791 . . . . . . . . . . . . . . 15  |-  y  e. 
_V
13 hashclb 11352 . . . . . . . . . . . . . . 15  |-  ( y  e.  _V  ->  (
y  e.  Fin  <->  ( # `  y
)  e.  NN0 )
)
1412, 13ax-mp 8 . . . . . . . . . . . . . 14  |-  ( y  e.  Fin  <->  ( # `  y
)  e.  NN0 )
1511, 14sylibr 203 . . . . . . . . . . . . 13  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  (
y  e.  (SubGrp `  G )  /\  x  e.  (SubGrp `  G )
)  /\  ( ( # `
 y )  =  N  /\  ( # `  x )  =  N ) )  ->  y  e.  Fin )
1615adantr 451 . . . . . . . . . . . 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 447 . . . . . . . . . . . 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 2283 . . . . . . . . . . . . 13  |-  ( od
`  G )  =  ( od `  G
)
1918odsubdvds 14882 . . . . . . . . . . . 12  |-  ( ( y  e.  (SubGrp `  G )  /\  y  e.  Fin  /\  z  e.  y )  ->  (
( od `  G
) `  z )  ||  ( # `  y
) )
207, 16, 17, 19syl3anc 1182 . . . . . . . . . . 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 451 . . . . . . . . . . 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 4047 . . . . . . . . . 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 3252 . . . . . . . . 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 982 . . . . . . . . . . 11  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  (
y  e.  (SubGrp `  G )  /\  x  e.  (SubGrp `  G )
)  /\  ( ( # `
 y )  =  N  /\  ( # `  x )  =  N ) )  ->  x  e.  (SubGrp `  G )
)
254subgss 14622 . . . . . . . . . . 11  |-  ( x  e.  (SubGrp `  G
)  ->  x  C_  ( Base `  G ) )
2624, 25syl 15 . . . . . . . . . 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 1009 . . . . . . . . . . . 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 984 . . . . . . . . . . . . . . 15  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  (
y  e.  (SubGrp `  G )  /\  x  e.  (SubGrp `  G )
)  /\  ( ( # `
 y )  =  N  /\  ( # `  x )  =  N ) )  ->  ( # `
 x )  =  N )
2928, 10eqeltrd 2357 . . . . . . . . . . . . . 14  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  (
y  e.  (SubGrp `  G )  /\  x  e.  (SubGrp `  G )
)  /\  ( ( # `
 y )  =  N  /\  ( # `  x )  =  N ) )  ->  ( # `
 x )  e. 
NN0 )
30 vex 2791 . . . . . . . . . . . . . . 15  |-  x  e. 
_V
31 hashclb 11352 . . . . . . . . . . . . . . 15  |-  ( x  e.  _V  ->  (
x  e.  Fin  <->  ( # `  x
)  e.  NN0 )
)
3230, 31ax-mp 8 . . . . . . . . . . . . . 14  |-  ( x  e.  Fin  <->  ( # `  x
)  e.  NN0 )
3329, 32sylibr 203 . . . . . . . . . . . . 13  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  (
y  e.  (SubGrp `  G )  /\  x  e.  (SubGrp `  G )
)  /\  ( ( # `
 y )  =  N  /\  ( # `  x )  =  N ) )  ->  x  e.  Fin )
3433adantr 451 . . . . . . . . . . . 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 447 . . . . . . . . . . . 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 14882 . . . . . . . . . . . 12  |-  ( ( x  e.  (SubGrp `  G )  /\  x  e.  Fin  /\  z  e.  x )  ->  (
( od `  G
) `  z )  ||  ( # `  x
) )
3727, 34, 35, 36syl3anc 1182 . . . . . . . . . . 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 451 . . . . . . . . . . 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 4047 . . . . . . . . . 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 3252 . . . . . . . . 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 3351 . . . . . . . 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 6907 . . . . . . . 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 59 . . . . . . 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 27512 . . . . . . . . . 10  |-  ( ( R  e. IDomn  /\  N  e.  NN )  ->  ( # `
 { z  e.  ( Base `  G
)  |  ( ( od `  G ) `
 z )  ||  N } )  <_  N
)
46453ad2ant1 976 . . . . . . . . 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 4049 . . . . . . . 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 10 . . . . . . . . . 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 11343 . . . . . . . . . 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 1182 . . . . . . . . 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 11361 . . . . . . . . 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 643 . . . . . . . 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 201 . . . . . . 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 6914 . . . . . . 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 642 . . . . . 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 4518 . . . . . . 7  |-  ( y  u.  x )  e. 
_V
57 ssun1 3338 . . . . . . 7  |-  y  C_  ( y  u.  x
)
58 ssdomg 6907 . . . . . . 7  |-  ( ( y  u.  x )  e.  _V  ->  (
y  C_  ( y  u.  x )  ->  y  ~<_  ( y  u.  x
) ) )
5956, 57, 58mp2 17 . . . . . 6  |-  y  ~<_  ( y  u.  x )
60 sbth 6981 . . . . . 6  |-  ( ( ( y  u.  x
)  ~<_  y  /\  y  ~<_  ( y  u.  x
) )  ->  (
y  u.  x ) 
~~  y )
6155, 59, 60sylancl 643 . . . . 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 2318 . . . . . . 7  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  (
y  e.  (SubGrp `  G )  /\  x  e.  (SubGrp `  G )
)  /\  ( ( # `
 y )  =  N  /\  ( # `  x )  =  N ) )  ->  ( # `
 y )  =  ( # `  x
) )
63 hashen 11346 . . . . . . . 8  |-  ( ( y  e.  Fin  /\  x  e.  Fin )  ->  ( ( # `  y
)  =  ( # `  x )  <->  y  ~~  x ) )
6415, 33, 63syl2anc 642 . . . . . . 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 201 . . . . . 6  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  (
y  e.  (SubGrp `  G )  /\  x  e.  (SubGrp `  G )
)  /\  ( ( # `
 y )  =  N  /\  ( # `  x )  =  N ) )  ->  y  ~~  x )
66 fiuneneq 27513 . . . . . 6  |-  ( ( y  ~~  x  /\  y  e.  Fin )  ->  ( ( y  u.  x )  ~~  y  <->  y  =  x ) )
6765, 15, 66syl2anc 642 . . . . 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 201 . . . 4  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  (
y  e.  (SubGrp `  G )  /\  x  e.  (SubGrp `  G )
)  /\  ( ( # `
 y )  =  N  /\  ( # `  x )  =  N ) )  ->  y  =  x )
69683expia 1153 . . 3  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  (
y  e.  (SubGrp `  G )  /\  x  e.  (SubGrp `  G )
) )  ->  (
( ( # `  y
)  =  N  /\  ( # `  x )  =  N )  -> 
y  =  x ) )
7069ralrimivva 2635 . 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 5525 . . . 4  |-  ( y  =  x  ->  ( # `
 y )  =  ( # `  x
) )
7271eqeq1d 2291 . . 3  |-  ( y  =  x  ->  (
( # `  y )  =  N  <->  ( # `  x
)  =  N ) )
7372rmo4 2958 . 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 203 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   E*wrmo 2546   {crab 2547   _Vcvv 2788    u. cun 3150    C_ wss 3152   class class class wbr 4023   ` cfv 5255  (class class class)co 5858    ~~ cen 6860    ~<_ cdom 6861   Fincfn 6863    <_ cle 8868   NNcn 9746   NN0cn0 9965   #chash 11337    || cdivides 12531   Basecbs 13148   ↾s cress 13149  SubGrpcsubg 14615   odcod 14840  mulGrpcmgp 15325  Unitcui 15421  IDomncidom 16022
This theorem is referenced by:  proot1mul  27515
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-inf2 7342  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  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
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-iin 3908  df-disj 3994  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-se 4353  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-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-ofr 6079  df-1st 6122  df-2nd 6123  df-tpos 6234  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-omul 6484  df-er 6660  df-ec 6662  df-qs 6666  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-acn 7575  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-rp 10355  df-fz 10783  df-fzo 10871  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-sum 12159  df-dvds 12532  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-prds 13348  df-pws 13350  df-0g 13404  df-gsum 13405  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-mhm 14415  df-submnd 14416  df-grp 14489  df-minusg 14490  df-sbg 14491  df-mulg 14492  df-subg 14618  df-eqg 14620  df-ghm 14681  df-cntz 14793  df-od 14844  df-cmn 15091  df-abl 15092  df-mgp 15326  df-rng 15340  df-cring 15341  df-ur 15342  df-oppr 15405  df-dvdsr 15423  df-unit 15424  df-invr 15454  df-rnghom 15496  df-subrg 15543  df-lmod 15629  df-lss 15690  df-lsp 15729  df-nzr 16010  df-rlreg 16024  df-domn 16025  df-idom 16026  df-assa 16053  df-asp 16054  df-ascl 16055  df-psr 16098  df-mvr 16099  df-mpl 16100  df-evls 16101  df-evl 16102  df-opsr 16106  df-psr1 16257  df-vr1 16258  df-ply1 16259  df-evl1 16261  df-coe1 16262  df-cnfld 16378  df-mdeg 19441  df-deg1 19442  df-mon1 19516  df-uc1p 19517  df-q1p 19518  df-r1p 19519
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