Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  idomsubgmo Structured version   Unicode version

Theorem idomsubgmo 35499
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 5858 . . . . . . . . 9  |-  ( Base `  G )  e.  _V
21rabex 4544 . . . . . . . 8  |-  { z  e.  ( Base `  G
)  |  ( ( od `  G ) `
 z )  ||  N }  e.  _V
3 simp2l 1023 . . . . . . . . . . 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 2402 . . . . . . . . . . . 12  |-  ( Base `  G )  =  (
Base `  G )
54subgss 16524 . . . . . . . . . . 11  |-  ( y  e.  (SubGrp `  G
)  ->  y  C_  ( Base `  G )
)
63, 5syl 17 . . . . . . . . . 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 1050 . . . . . . . . . . . 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 1025 . . . . . . . . . . . . . . 15  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  (
y  e.  (SubGrp `  G )  /\  x  e.  (SubGrp `  G )
)  /\  ( ( # `
 y )  =  N  /\  ( # `  x )  =  N ) )  ->  ( # `
 y )  =  N )
9 simp1r 1022 . . . . . . . . . . . . . . . 16  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  (
y  e.  (SubGrp `  G )  /\  x  e.  (SubGrp `  G )
)  /\  ( ( # `
 y )  =  N  /\  ( # `  x )  =  N ) )  ->  N  e.  NN )
109nnnn0d 10892 . . . . . . . . . . . . . . 15  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  (
y  e.  (SubGrp `  G )  /\  x  e.  (SubGrp `  G )
)  /\  ( ( # `
 y )  =  N  /\  ( # `  x )  =  N ) )  ->  N  e.  NN0 )
118, 10eqeltrd 2490 . . . . . . . . . . . . . 14  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  (
y  e.  (SubGrp `  G )  /\  x  e.  (SubGrp `  G )
)  /\  ( ( # `
 y )  =  N  /\  ( # `  x )  =  N ) )  ->  ( # `
 y )  e. 
NN0 )
12 vex 3061 . . . . . . . . . . . . . . 15  |-  y  e. 
_V
13 hashclb 12475 . . . . . . . . . . . . . . 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 463 . . . . . . . . . . . 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 459 . . . . . . . . . . . 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 2402 . . . . . . . . . . . . 13  |-  ( od
`  G )  =  ( od `  G
)
1918odsubdvds 16913 . . . . . . . . . . . 12  |-  ( ( y  e.  (SubGrp `  G )  /\  y  e.  Fin  /\  z  e.  y )  ->  (
( od `  G
) `  z )  ||  ( # `  y
) )
207, 16, 17, 19syl3anc 1230 . . . . . . . . . . 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 463 . . . . . . . . . . 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 4418 . . . . . . . . . 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 3517 . . . . . . . . 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 1024 . . . . . . . . . . 11  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  (
y  e.  (SubGrp `  G )  /\  x  e.  (SubGrp `  G )
)  /\  ( ( # `
 y )  =  N  /\  ( # `  x )  =  N ) )  ->  x  e.  (SubGrp `  G )
)
254subgss 16524 . . . . . . . . . . 11  |-  ( x  e.  (SubGrp `  G
)  ->  x  C_  ( Base `  G ) )
2624, 25syl 17 . . . . . . . . . 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 1051 . . . . . . . . . . . 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 1026 . . . . . . . . . . . . . . 15  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  (
y  e.  (SubGrp `  G )  /\  x  e.  (SubGrp `  G )
)  /\  ( ( # `
 y )  =  N  /\  ( # `  x )  =  N ) )  ->  ( # `
 x )  =  N )
2928, 10eqeltrd 2490 . . . . . . . . . . . . . 14  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  (
y  e.  (SubGrp `  G )  /\  x  e.  (SubGrp `  G )
)  /\  ( ( # `
 y )  =  N  /\  ( # `  x )  =  N ) )  ->  ( # `
 x )  e. 
NN0 )
30 vex 3061 . . . . . . . . . . . . . . 15  |-  x  e. 
_V
31 hashclb 12475 . . . . . . . . . . . . . . 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 463 . . . . . . . . . . . 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 459 . . . . . . . . . . . 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 16913 . . . . . . . . . . . 12  |-  ( ( x  e.  (SubGrp `  G )  /\  x  e.  Fin  /\  z  e.  x )  ->  (
( od `  G
) `  z )  ||  ( # `  x
) )
3727, 34, 35, 36syl3anc 1230 . . . . . . . . . . 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 463 . . . . . . . . . . 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 4418 . . . . . . . . . 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 3517 . . . . . . . . 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 3618 . . . . . . . 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 7598 . . . . . . . 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 62 . . . . . . 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 35497 . . . . . . . . . 10  |-  ( ( R  e. IDomn  /\  N  e.  NN )  ->  ( # `
 { z  e.  ( Base `  G
)  |  ( ( od `  G ) `
 z )  ||  N } )  <_  N
)
46453ad2ant1 1018 . . . . . . . . 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 4420 . . . . . . . 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 12456 . . . . . . . . . 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 1230 . . . . . . . . 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 12493 . . . . . . . . 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 660 . . . . . . . 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 7605 . . . . . . 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 659 . . . . . 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 6579 . . . . . . 7  |-  ( y  u.  x )  e. 
_V
57 ssun1 3605 . . . . . . 7  |-  y  C_  ( y  u.  x
)
58 ssdomg 7598 . . . . . . 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 7674 . . . . . 6  |-  ( ( ( y  u.  x
)  ~<_  y  /\  y  ~<_  ( y  u.  x
) )  ->  (
y  u.  x ) 
~~  y )
6155, 59, 60sylancl 660 . . . . 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 2446 . . . . . . 7  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  (
y  e.  (SubGrp `  G )  /\  x  e.  (SubGrp `  G )
)  /\  ( ( # `
 y )  =  N  /\  ( # `  x )  =  N ) )  ->  ( # `
 y )  =  ( # `  x
) )
63 hashen 12465 . . . . . . . 8  |-  ( ( y  e.  Fin  /\  x  e.  Fin )  ->  ( ( # `  y
)  =  ( # `  x )  <->  y  ~~  x ) )
6415, 33, 63syl2anc 659 . . . . . . 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 35498 . . . . . 6  |-  ( ( y  ~~  x  /\  y  e.  Fin )  ->  ( ( y  u.  x )  ~~  y  <->  y  =  x ) )
6765, 15, 66syl2anc 659 . . . . 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 1199 . . 3  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  (
y  e.  (SubGrp `  G )  /\  x  e.  (SubGrp `  G )
) )  ->  (
( ( # `  y
)  =  N  /\  ( # `  x )  =  N )  -> 
y  =  x ) )
7069ralrimivva 2824 . 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 5848 . . . 4  |-  ( y  =  x  ->  ( # `
 y )  =  ( # `  x
) )
7271eqeq1d 2404 . . 3  |-  ( y  =  x  ->  (
( # `  y )  =  N  <->  ( # `  x
)  =  N ) )
7372rmo4 3241 . 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 367    /\ w3a 974    = wceq 1405    e. wcel 1842   A.wral 2753   E*wrmo 2756   {crab 2757   _Vcvv 3058    u. cun 3411    C_ wss 3413   class class class wbr 4394   ` cfv 5568  (class class class)co 6277    ~~ cen 7550    ~<_ cdom 7551   Fincfn 7553    <_ cle 9658   NNcn 10575   NN0cn0 10835   #chash 12450    || cdvds 14193   Basecbs 14839   ↾s cress 14840  SubGrpcsubg 16517   odcod 16871  mulGrpcmgp 17459  Unitcui 17606  IDomncidom 18247
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-inf2 8090  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597  ax-pre-mulgt0 9598  ax-pre-sup 9599  ax-addf 9600  ax-mulf 9601
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-iin 4273  df-disj 4366  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-se 4782  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-isom 5577  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-of 6520  df-ofr 6521  df-om 6683  df-1st 6783  df-2nd 6784  df-supp 6902  df-tpos 6957  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-1o 7166  df-2o 7167  df-oadd 7170  df-omul 7171  df-er 7347  df-ec 7349  df-qs 7353  df-map 7458  df-pm 7459  df-ixp 7507  df-en 7554  df-dom 7555  df-sdom 7556  df-fin 7557  df-fsupp 7863  df-sup 7934  df-oi 7968  df-card 8351  df-acn 8354  df-cda 8579  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-le 9663  df-sub 9842  df-neg 9843  df-div 10247  df-nn 10576  df-2 10634  df-3 10635  df-4 10636  df-5 10637  df-6 10638  df-7 10639  df-8 10640  df-9 10641  df-10 10642  df-n0 10836  df-z 10905  df-dec 11019  df-uz 11127  df-rp 11265  df-fz 11725  df-fzo 11853  df-fl 11964  df-mod 12033  df-seq 12150  df-exp 12209  df-hash 12451  df-cj 13079  df-re 13080  df-im 13081  df-sqrt 13215  df-abs 13216  df-clim 13458  df-sum 13656  df-dvds 14194  df-struct 14841  df-ndx 14842  df-slot 14843  df-base 14844  df-sets 14845  df-ress 14846  df-plusg 14920  df-mulr 14921  df-starv 14922  df-sca 14923  df-vsca 14924  df-ip 14925  df-tset 14926  df-ple 14927  df-ds 14929  df-unif 14930  df-hom 14931  df-cco 14932  df-0g 15054  df-gsum 15055  df-prds 15060  df-pws 15062  df-mre 15198  df-mrc 15199  df-acs 15201  df-mgm 16194  df-sgrp 16233  df-mnd 16243  df-mhm 16288  df-submnd 16289  df-grp 16379  df-minusg 16380  df-sbg 16381  df-mulg 16382  df-subg 16520  df-eqg 16522  df-ghm 16587  df-cntz 16677  df-od 16875  df-cmn 17122  df-abl 17123  df-mgp 17460  df-ur 17472  df-srg 17476  df-ring 17518  df-cring 17519  df-oppr 17590  df-dvdsr 17608  df-unit 17609  df-invr 17639  df-rnghom 17682  df-subrg 17745  df-lmod 17832  df-lss 17897  df-lsp 17936  df-nzr 18224  df-rlreg 18249  df-domn 18250  df-idom 18251  df-assa 18279  df-asp 18280  df-ascl 18281  df-psr 18323  df-mvr 18324  df-mpl 18325  df-opsr 18327  df-evls 18489  df-evl 18490  df-psr1 18537  df-vr1 18538  df-ply1 18539  df-coe1 18540  df-evl1 18671  df-cnfld 18739  df-mdeg 22743  df-deg1 22744  df-mon1 22821  df-uc1p 22822  df-q1p 22823  df-r1p 22824
This theorem is referenced by:  proot1mul  35500
  Copyright terms: Public domain W3C validator