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Theorem idomsubgmo 29704
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 5802 . . . . . . . . 9  |-  ( Base `  G )  e.  _V
21rabex 4544 . . . . . . . 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 2451 . . . . . . . . . . . 12  |-  ( Base `  G )  =  (
Base `  G )
54subgss 15793 . . . . . . . . . . 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 10740 . . . . . . . . . . . . . . 15  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  (
y  e.  (SubGrp `  G )  /\  x  e.  (SubGrp `  G )
)  /\  ( ( # `
 y )  =  N  /\  ( # `  x )  =  N ) )  ->  N  e.  NN0 )
118, 10eqeltrd 2539 . . . . . . . . . . . . . 14  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  (
y  e.  (SubGrp `  G )  /\  x  e.  (SubGrp `  G )
)  /\  ( ( # `
 y )  =  N  /\  ( # `  x )  =  N ) )  ->  ( # `
 y )  e. 
NN0 )
12 vex 3074 . . . . . . . . . . . . . . 15  |-  y  e. 
_V
13 hashclb 12238 . . . . . . . . . . . . . . 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 2451 . . . . . . . . . . . . 13  |-  ( od
`  G )  =  ( od `  G
)
1918odsubdvds 16183 . . . . . . . . . . . 12  |-  ( ( y  e.  (SubGrp `  G )  /\  y  e.  Fin  /\  z  e.  y )  ->  (
( od `  G
) `  z )  ||  ( # `  y
) )
207, 16, 17, 19syl3anc 1219 . . . . . . . . . . 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 4417 . . . . . . . . . 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 3532 . . . . . . . . 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 15793 . . . . . . . . . . 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 2539 . . . . . . . . . . . . . 14  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  (
y  e.  (SubGrp `  G )  /\  x  e.  (SubGrp `  G )
)  /\  ( ( # `
 y )  =  N  /\  ( # `  x )  =  N ) )  ->  ( # `
 x )  e. 
NN0 )
30 vex 3074 . . . . . . . . . . . . . . 15  |-  x  e. 
_V
31 hashclb 12238 . . . . . . . . . . . . . . 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 16183 . . . . . . . . . . . 12  |-  ( ( x  e.  (SubGrp `  G )  /\  x  e.  Fin  /\  z  e.  x )  ->  (
( od `  G
) `  z )  ||  ( # `  x
) )
3727, 34, 35, 36syl3anc 1219 . . . . . . . . . . 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 4417 . . . . . . . . . 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 3532 . . . . . . . . 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 3633 . . . . . . . 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 7458 . . . . . . . 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 29702 . . . . . . . . . 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 4419 . . . . . . . 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 12219 . . . . . . . . . 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 1219 . . . . . . . . 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 12253 . . . . . . . . 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 7465 . . . . . . 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 6481 . . . . . . 7  |-  ( y  u.  x )  e. 
_V
57 ssun1 3620 . . . . . . 7  |-  y  C_  ( y  u.  x
)
58 ssdomg 7458 . . . . . . 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 7534 . . . . . 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 2495 . . . . . . 7  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  (
y  e.  (SubGrp `  G )  /\  x  e.  (SubGrp `  G )
)  /\  ( ( # `
 y )  =  N  /\  ( # `  x )  =  N ) )  ->  ( # `
 y )  =  ( # `  x
) )
63 hashen 12228 . . . . . . . 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 29703 . . . . . 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 1190 . . 3  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  (
y  e.  (SubGrp `  G )  /\  x  e.  (SubGrp `  G )
) )  ->  (
( ( # `  y
)  =  N  /\  ( # `  x )  =  N )  -> 
y  =  x ) )
7069ralrimivva 2907 . 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 5792 . . . 4  |-  ( y  =  x  ->  ( # `
 y )  =  ( # `  x
) )
7271eqeq1d 2453 . . 3  |-  ( y  =  x  ->  (
( # `  y )  =  N  <->  ( # `  x
)  =  N ) )
7372rmo4 3252 . 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 1370    e. wcel 1758   A.wral 2795   E*wrmo 2798   {crab 2799   _Vcvv 3071    u. cun 3427    C_ wss 3429   class class class wbr 4393   ` cfv 5519  (class class class)co 6193    ~~ cen 7410    ~<_ cdom 7411   Fincfn 7413    <_ cle 9523   NNcn 10426   NN0cn0 10683   #chash 12213    || cdivides 13646   Basecbs 14285   ↾s cress 14286  SubGrpcsubg 15786   odcod 16141  mulGrpcmgp 16705  Unitcui 16846  IDomncidom 17467
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-inf2 7951  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463  ax-pre-sup 9464  ax-addf 9465  ax-mulf 9466
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-int 4230  df-iun 4274  df-iin 4275  df-disj 4364  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-se 4781  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-isom 5528  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-of 6423  df-ofr 6424  df-om 6580  df-1st 6680  df-2nd 6681  df-supp 6794  df-tpos 6848  df-recs 6935  df-rdg 6969  df-1o 7023  df-2o 7024  df-oadd 7027  df-omul 7028  df-er 7204  df-ec 7206  df-qs 7210  df-map 7319  df-pm 7320  df-ixp 7367  df-en 7414  df-dom 7415  df-sdom 7416  df-fin 7417  df-fsupp 7725  df-sup 7795  df-oi 7828  df-card 8213  df-acn 8216  df-cda 8441  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-div 10098  df-nn 10427  df-2 10484  df-3 10485  df-4 10486  df-5 10487  df-6 10488  df-7 10489  df-8 10490  df-9 10491  df-10 10492  df-n0 10684  df-z 10751  df-dec 10860  df-uz 10966  df-rp 11096  df-fz 11548  df-fzo 11659  df-fl 11752  df-mod 11819  df-seq 11917  df-exp 11976  df-hash 12214  df-cj 12699  df-re 12700  df-im 12701  df-sqr 12835  df-abs 12836  df-clim 13077  df-sum 13275  df-dvds 13647  df-struct 14287  df-ndx 14288  df-slot 14289  df-base 14290  df-sets 14291  df-ress 14292  df-plusg 14362  df-mulr 14363  df-starv 14364  df-sca 14365  df-vsca 14366  df-ip 14367  df-tset 14368  df-ple 14369  df-ds 14371  df-unif 14372  df-hom 14373  df-cco 14374  df-0g 14491  df-gsum 14492  df-prds 14497  df-pws 14499  df-mre 14635  df-mrc 14636  df-acs 14638  df-mnd 15526  df-mhm 15575  df-submnd 15576  df-grp 15656  df-minusg 15657  df-sbg 15658  df-mulg 15659  df-subg 15789  df-eqg 15791  df-ghm 15856  df-cntz 15946  df-od 16145  df-cmn 16392  df-abl 16393  df-mgp 16706  df-ur 16718  df-srg 16722  df-rng 16762  df-cring 16763  df-oppr 16830  df-dvdsr 16848  df-unit 16849  df-invr 16879  df-rnghom 16921  df-subrg 16978  df-lmod 17065  df-lss 17129  df-lsp 17168  df-nzr 17455  df-rlreg 17469  df-domn 17470  df-idom 17471  df-assa 17499  df-asp 17500  df-ascl 17501  df-psr 17538  df-mvr 17539  df-mpl 17540  df-opsr 17542  df-evls 17704  df-evl 17705  df-psr1 17752  df-vr1 17753  df-ply1 17754  df-coe1 17755  df-evl1 17869  df-cnfld 17937  df-mdeg 21650  df-deg1 21651  df-mon1 21728  df-uc1p 21729  df-q1p 21730  df-r1p 21731
This theorem is referenced by:  proot1mul  29705
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