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Theorem unitgrp 17100
Description: The group of units is a group under multiplication. (Contributed by Mario Carneiro, 2-Dec-2014.)
Hypotheses
Ref Expression
unitmulcl.1  |-  U  =  (Unit `  R )
unitgrp.2  |-  G  =  ( (mulGrp `  R
)s 
U )
Assertion
Ref Expression
unitgrp  |-  ( R  e.  Ring  ->  G  e. 
Grp )

Proof of Theorem unitgrp
Dummy variables  x  y  z  m are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 unitmulcl.1 . . . 4  |-  U  =  (Unit `  R )
2 unitgrp.2 . . . 4  |-  G  =  ( (mulGrp `  R
)s 
U )
31, 2unitgrpbas 17099 . . 3  |-  U  =  ( Base `  G
)
43a1i 11 . 2  |-  ( R  e.  Ring  ->  U  =  ( Base `  G
) )
5 fvex 5874 . . . 4  |-  ( Base `  G )  e.  _V
63, 5eqeltri 2551 . . 3  |-  U  e. 
_V
7 eqid 2467 . . . . 5  |-  (mulGrp `  R )  =  (mulGrp `  R )
8 eqid 2467 . . . . 5  |-  ( .r
`  R )  =  ( .r `  R
)
97, 8mgpplusg 16935 . . . 4  |-  ( .r
`  R )  =  ( +g  `  (mulGrp `  R ) )
102, 9ressplusg 14593 . . 3  |-  ( U  e.  _V  ->  ( .r `  R )  =  ( +g  `  G
) )
116, 10mp1i 12 . 2  |-  ( R  e.  Ring  ->  ( .r
`  R )  =  ( +g  `  G
) )
121, 8unitmulcl 17097 . 2  |-  ( ( R  e.  Ring  /\  x  e.  U  /\  y  e.  U )  ->  (
x ( .r `  R ) y )  e.  U )
13 eqid 2467 . . . . 5  |-  ( Base `  R )  =  (
Base `  R )
1413, 1unitcl 17092 . . . 4  |-  ( x  e.  U  ->  x  e.  ( Base `  R
) )
1513, 1unitcl 17092 . . . 4  |-  ( y  e.  U  ->  y  e.  ( Base `  R
) )
1613, 1unitcl 17092 . . . 4  |-  ( z  e.  U  ->  z  e.  ( Base `  R
) )
1714, 15, 163anim123i 1181 . . 3  |-  ( ( x  e.  U  /\  y  e.  U  /\  z  e.  U )  ->  ( x  e.  (
Base `  R )  /\  y  e.  ( Base `  R )  /\  z  e.  ( Base `  R ) ) )
1813, 8rngass 17002 . . 3  |-  ( ( R  e.  Ring  /\  (
x  e.  ( Base `  R )  /\  y  e.  ( Base `  R
)  /\  z  e.  ( Base `  R )
) )  ->  (
( x ( .r
`  R ) y ) ( .r `  R ) z )  =  ( x ( .r `  R ) ( y ( .r
`  R ) z ) ) )
1917, 18sylan2 474 . 2  |-  ( ( R  e.  Ring  /\  (
x  e.  U  /\  y  e.  U  /\  z  e.  U )
)  ->  ( (
x ( .r `  R ) y ) ( .r `  R
) z )  =  ( x ( .r
`  R ) ( y ( .r `  R ) z ) ) )
20 eqid 2467 . . 3  |-  ( 1r
`  R )  =  ( 1r `  R
)
211, 201unit 17091 . 2  |-  ( R  e.  Ring  ->  ( 1r
`  R )  e.  U )
2213, 8, 20rnglidm 17009 . . 3  |-  ( ( R  e.  Ring  /\  x  e.  ( Base `  R
) )  ->  (
( 1r `  R
) ( .r `  R ) x )  =  x )
2314, 22sylan2 474 . 2  |-  ( ( R  e.  Ring  /\  x  e.  U )  ->  (
( 1r `  R
) ( .r `  R ) x )  =  x )
24 simpr 461 . . . 4  |-  ( ( R  e.  Ring  /\  x  e.  U )  ->  x  e.  U )
25 eqid 2467 . . . . 5  |-  ( ||r `  R
)  =  ( ||r `  R
)
26 eqid 2467 . . . . 5  |-  (oppr `  R
)  =  (oppr `  R
)
27 eqid 2467 . . . . 5  |-  ( ||r `  (oppr `  R
) )  =  (
||r `  (oppr
`  R ) )
281, 20, 25, 26, 27isunit 17090 . . . 4  |-  ( x  e.  U  <->  ( x
( ||r `
 R ) ( 1r `  R )  /\  x ( ||r `  (oppr `  R
) ) ( 1r
`  R ) ) )
2924, 28sylib 196 . . 3  |-  ( ( R  e.  Ring  /\  x  e.  U )  ->  (
x ( ||r `
 R ) ( 1r `  R )  /\  x ( ||r `  (oppr `  R
) ) ( 1r
`  R ) ) )
3014adantl 466 . . . . . 6  |-  ( ( R  e.  Ring  /\  x  e.  U )  ->  x  e.  ( Base `  R
) )
3113, 25, 8dvdsr2 17080 . . . . . 6  |-  ( x  e.  ( Base `  R
)  ->  ( x
( ||r `
 R ) ( 1r `  R )  <->  E. y  e.  ( Base `  R ) ( y ( .r `  R ) x )  =  ( 1r `  R ) ) )
3230, 31syl 16 . . . . 5  |-  ( ( R  e.  Ring  /\  x  e.  U )  ->  (
x ( ||r `
 R ) ( 1r `  R )  <->  E. y  e.  ( Base `  R ) ( y ( .r `  R ) x )  =  ( 1r `  R ) ) )
3326, 13opprbas 17062 . . . . . . 7  |-  ( Base `  R )  =  (
Base `  (oppr
`  R ) )
34 eqid 2467 . . . . . . 7  |-  ( .r
`  (oppr
`  R ) )  =  ( .r `  (oppr `  R ) )
3533, 27, 34dvdsr2 17080 . . . . . 6  |-  ( x  e.  ( Base `  R
)  ->  ( x
( ||r `
 (oppr
`  R ) ) ( 1r `  R
)  <->  E. m  e.  (
Base `  R )
( m ( .r
`  (oppr
`  R ) ) x )  =  ( 1r `  R ) ) )
3630, 35syl 16 . . . . 5  |-  ( ( R  e.  Ring  /\  x  e.  U )  ->  (
x ( ||r `
 (oppr
`  R ) ) ( 1r `  R
)  <->  E. m  e.  (
Base `  R )
( m ( .r
`  (oppr
`  R ) ) x )  =  ( 1r `  R ) ) )
3732, 36anbi12d 710 . . . 4  |-  ( ( R  e.  Ring  /\  x  e.  U )  ->  (
( x ( ||r `  R
) ( 1r `  R )  /\  x
( ||r `
 (oppr
`  R ) ) ( 1r `  R
) )  <->  ( E. y  e.  ( Base `  R ) ( y ( .r `  R
) x )  =  ( 1r `  R
)  /\  E. m  e.  ( Base `  R
) ( m ( .r `  (oppr `  R
) ) x )  =  ( 1r `  R ) ) ) )
38 reeanv 3029 . . . . 5  |-  ( E. y  e.  ( Base `  R ) E. m  e.  ( Base `  R
) ( ( y ( .r `  R
) x )  =  ( 1r `  R
)  /\  ( m
( .r `  (oppr `  R
) ) x )  =  ( 1r `  R ) )  <->  ( E. y  e.  ( Base `  R ) ( y ( .r `  R
) x )  =  ( 1r `  R
)  /\  E. m  e.  ( Base `  R
) ( m ( .r `  (oppr `  R
) ) x )  =  ( 1r `  R ) ) )
39 simprl 755 . . . . . . . . . . . 12  |-  ( ( ( ( R  e. 
Ring  /\  x  e.  U
)  /\  y  e.  ( Base `  R )
)  /\  ( m  e.  ( Base `  R
)  /\  ( (
y ( .r `  R ) x )  =  ( 1r `  R )  /\  (
m ( .r `  (oppr `  R ) ) x )  =  ( 1r
`  R ) ) ) )  ->  m  e.  ( Base `  R
) )
4030ad2antrr 725 . . . . . . . . . . . 12  |-  ( ( ( ( R  e. 
Ring  /\  x  e.  U
)  /\  y  e.  ( Base `  R )
)  /\  ( m  e.  ( Base `  R
)  /\  ( (
y ( .r `  R ) x )  =  ( 1r `  R )  /\  (
m ( .r `  (oppr `  R ) ) x )  =  ( 1r
`  R ) ) ) )  ->  x  e.  ( Base `  R
) )
4113, 25, 8dvdsrmul 17081 . . . . . . . . . . . 12  |-  ( ( m  e.  ( Base `  R )  /\  x  e.  ( Base `  R
) )  ->  m
( ||r `
 R ) ( x ( .r `  R ) m ) )
4239, 40, 41syl2anc 661 . . . . . . . . . . 11  |-  ( ( ( ( R  e. 
Ring  /\  x  e.  U
)  /\  y  e.  ( Base `  R )
)  /\  ( m  e.  ( Base `  R
)  /\  ( (
y ( .r `  R ) x )  =  ( 1r `  R )  /\  (
m ( .r `  (oppr `  R ) ) x )  =  ( 1r
`  R ) ) ) )  ->  m
( ||r `
 R ) ( x ( .r `  R ) m ) )
43 simplll 757 . . . . . . . . . . . . . 14  |-  ( ( ( ( R  e. 
Ring  /\  x  e.  U
)  /\  y  e.  ( Base `  R )
)  /\  ( m  e.  ( Base `  R
)  /\  ( (
y ( .r `  R ) x )  =  ( 1r `  R )  /\  (
m ( .r `  (oppr `  R ) ) x )  =  ( 1r
`  R ) ) ) )  ->  R  e.  Ring )
44 simplr 754 . . . . . . . . . . . . . 14  |-  ( ( ( ( R  e. 
Ring  /\  x  e.  U
)  /\  y  e.  ( Base `  R )
)  /\  ( m  e.  ( Base `  R
)  /\  ( (
y ( .r `  R ) x )  =  ( 1r `  R )  /\  (
m ( .r `  (oppr `  R ) ) x )  =  ( 1r
`  R ) ) ) )  ->  y  e.  ( Base `  R
) )
4513, 8rngass 17002 . . . . . . . . . . . . . 14  |-  ( ( R  e.  Ring  /\  (
y  e.  ( Base `  R )  /\  x  e.  ( Base `  R
)  /\  m  e.  ( Base `  R )
) )  ->  (
( y ( .r
`  R ) x ) ( .r `  R ) m )  =  ( y ( .r `  R ) ( x ( .r
`  R ) m ) ) )
4643, 44, 40, 39, 45syl13anc 1230 . . . . . . . . . . . . 13  |-  ( ( ( ( R  e. 
Ring  /\  x  e.  U
)  /\  y  e.  ( Base `  R )
)  /\  ( m  e.  ( Base `  R
)  /\  ( (
y ( .r `  R ) x )  =  ( 1r `  R )  /\  (
m ( .r `  (oppr `  R ) ) x )  =  ( 1r
`  R ) ) ) )  ->  (
( y ( .r
`  R ) x ) ( .r `  R ) m )  =  ( y ( .r `  R ) ( x ( .r
`  R ) m ) ) )
47 simprrl 763 . . . . . . . . . . . . . 14  |-  ( ( ( ( R  e. 
Ring  /\  x  e.  U
)  /\  y  e.  ( Base `  R )
)  /\  ( m  e.  ( Base `  R
)  /\  ( (
y ( .r `  R ) x )  =  ( 1r `  R )  /\  (
m ( .r `  (oppr `  R ) ) x )  =  ( 1r
`  R ) ) ) )  ->  (
y ( .r `  R ) x )  =  ( 1r `  R ) )
4847oveq1d 6297 . . . . . . . . . . . . 13  |-  ( ( ( ( R  e. 
Ring  /\  x  e.  U
)  /\  y  e.  ( Base `  R )
)  /\  ( m  e.  ( Base `  R
)  /\  ( (
y ( .r `  R ) x )  =  ( 1r `  R )  /\  (
m ( .r `  (oppr `  R ) ) x )  =  ( 1r
`  R ) ) ) )  ->  (
( y ( .r
`  R ) x ) ( .r `  R ) m )  =  ( ( 1r
`  R ) ( .r `  R ) m ) )
4913, 8, 26, 34opprmul 17059 . . . . . . . . . . . . . . 15  |-  ( m ( .r `  (oppr `  R
) ) x )  =  ( x ( .r `  R ) m )
50 simprrr 764 . . . . . . . . . . . . . . 15  |-  ( ( ( ( R  e. 
Ring  /\  x  e.  U
)  /\  y  e.  ( Base `  R )
)  /\  ( m  e.  ( Base `  R
)  /\  ( (
y ( .r `  R ) x )  =  ( 1r `  R )  /\  (
m ( .r `  (oppr `  R ) ) x )  =  ( 1r
`  R ) ) ) )  ->  (
m ( .r `  (oppr `  R ) ) x )  =  ( 1r
`  R ) )
5149, 50syl5eqr 2522 . . . . . . . . . . . . . 14  |-  ( ( ( ( R  e. 
Ring  /\  x  e.  U
)  /\  y  e.  ( Base `  R )
)  /\  ( m  e.  ( Base `  R
)  /\  ( (
y ( .r `  R ) x )  =  ( 1r `  R )  /\  (
m ( .r `  (oppr `  R ) ) x )  =  ( 1r
`  R ) ) ) )  ->  (
x ( .r `  R ) m )  =  ( 1r `  R ) )
5251oveq2d 6298 . . . . . . . . . . . . 13  |-  ( ( ( ( R  e. 
Ring  /\  x  e.  U
)  /\  y  e.  ( Base `  R )
)  /\  ( m  e.  ( Base `  R
)  /\  ( (
y ( .r `  R ) x )  =  ( 1r `  R )  /\  (
m ( .r `  (oppr `  R ) ) x )  =  ( 1r
`  R ) ) ) )  ->  (
y ( .r `  R ) ( x ( .r `  R
) m ) )  =  ( y ( .r `  R ) ( 1r `  R
) ) )
5346, 48, 523eqtr3d 2516 . . . . . . . . . . . 12  |-  ( ( ( ( R  e. 
Ring  /\  x  e.  U
)  /\  y  e.  ( Base `  R )
)  /\  ( m  e.  ( Base `  R
)  /\  ( (
y ( .r `  R ) x )  =  ( 1r `  R )  /\  (
m ( .r `  (oppr `  R ) ) x )  =  ( 1r
`  R ) ) ) )  ->  (
( 1r `  R
) ( .r `  R ) m )  =  ( y ( .r `  R ) ( 1r `  R
) ) )
5413, 8, 20rnglidm 17009 . . . . . . . . . . . . 13  |-  ( ( R  e.  Ring  /\  m  e.  ( Base `  R
) )  ->  (
( 1r `  R
) ( .r `  R ) m )  =  m )
5543, 39, 54syl2anc 661 . . . . . . . . . . . 12  |-  ( ( ( ( R  e. 
Ring  /\  x  e.  U
)  /\  y  e.  ( Base `  R )
)  /\  ( m  e.  ( Base `  R
)  /\  ( (
y ( .r `  R ) x )  =  ( 1r `  R )  /\  (
m ( .r `  (oppr `  R ) ) x )  =  ( 1r
`  R ) ) ) )  ->  (
( 1r `  R
) ( .r `  R ) m )  =  m )
5613, 8, 20rngridm 17010 . . . . . . . . . . . . 13  |-  ( ( R  e.  Ring  /\  y  e.  ( Base `  R
) )  ->  (
y ( .r `  R ) ( 1r
`  R ) )  =  y )
5743, 44, 56syl2anc 661 . . . . . . . . . . . 12  |-  ( ( ( ( R  e. 
Ring  /\  x  e.  U
)  /\  y  e.  ( Base `  R )
)  /\  ( m  e.  ( Base `  R
)  /\  ( (
y ( .r `  R ) x )  =  ( 1r `  R )  /\  (
m ( .r `  (oppr `  R ) ) x )  =  ( 1r
`  R ) ) ) )  ->  (
y ( .r `  R ) ( 1r
`  R ) )  =  y )
5853, 55, 573eqtr3d 2516 . . . . . . . . . . 11  |-  ( ( ( ( R  e. 
Ring  /\  x  e.  U
)  /\  y  e.  ( Base `  R )
)  /\  ( m  e.  ( Base `  R
)  /\  ( (
y ( .r `  R ) x )  =  ( 1r `  R )  /\  (
m ( .r `  (oppr `  R ) ) x )  =  ( 1r
`  R ) ) ) )  ->  m  =  y )
5942, 58, 513brtr3d 4476 . . . . . . . . . 10  |-  ( ( ( ( R  e. 
Ring  /\  x  e.  U
)  /\  y  e.  ( Base `  R )
)  /\  ( m  e.  ( Base `  R
)  /\  ( (
y ( .r `  R ) x )  =  ( 1r `  R )  /\  (
m ( .r `  (oppr `  R ) ) x )  =  ( 1r
`  R ) ) ) )  ->  y
( ||r `
 R ) ( 1r `  R ) )
6033, 27, 34dvdsrmul 17081 . . . . . . . . . . . 12  |-  ( ( y  e.  ( Base `  R )  /\  x  e.  ( Base `  R
) )  ->  y
( ||r `
 (oppr
`  R ) ) ( x ( .r
`  (oppr
`  R ) ) y ) )
6144, 40, 60syl2anc 661 . . . . . . . . . . 11  |-  ( ( ( ( R  e. 
Ring  /\  x  e.  U
)  /\  y  e.  ( Base `  R )
)  /\  ( m  e.  ( Base `  R
)  /\  ( (
y ( .r `  R ) x )  =  ( 1r `  R )  /\  (
m ( .r `  (oppr `  R ) ) x )  =  ( 1r
`  R ) ) ) )  ->  y
( ||r `
 (oppr
`  R ) ) ( x ( .r
`  (oppr
`  R ) ) y ) )
6213, 8, 26, 34opprmul 17059 . . . . . . . . . . . 12  |-  ( x ( .r `  (oppr `  R
) ) y )  =  ( y ( .r `  R ) x )
6362, 47syl5eq 2520 . . . . . . . . . . 11  |-  ( ( ( ( R  e. 
Ring  /\  x  e.  U
)  /\  y  e.  ( Base `  R )
)  /\  ( m  e.  ( Base `  R
)  /\  ( (
y ( .r `  R ) x )  =  ( 1r `  R )  /\  (
m ( .r `  (oppr `  R ) ) x )  =  ( 1r
`  R ) ) ) )  ->  (
x ( .r `  (oppr `  R ) ) y )  =  ( 1r
`  R ) )
6461, 63breqtrd 4471 . . . . . . . . . 10  |-  ( ( ( ( R  e. 
Ring  /\  x  e.  U
)  /\  y  e.  ( Base `  R )
)  /\  ( m  e.  ( Base `  R
)  /\  ( (
y ( .r `  R ) x )  =  ( 1r `  R )  /\  (
m ( .r `  (oppr `  R ) ) x )  =  ( 1r
`  R ) ) ) )  ->  y
( ||r `
 (oppr
`  R ) ) ( 1r `  R
) )
651, 20, 25, 26, 27isunit 17090 . . . . . . . . . 10  |-  ( y  e.  U  <->  ( y
( ||r `
 R ) ( 1r `  R )  /\  y ( ||r `  (oppr `  R
) ) ( 1r
`  R ) ) )
6659, 64, 65sylanbrc 664 . . . . . . . . 9  |-  ( ( ( ( R  e. 
Ring  /\  x  e.  U
)  /\  y  e.  ( Base `  R )
)  /\  ( m  e.  ( Base `  R
)  /\  ( (
y ( .r `  R ) x )  =  ( 1r `  R )  /\  (
m ( .r `  (oppr `  R ) ) x )  =  ( 1r
`  R ) ) ) )  ->  y  e.  U )
6766, 47jca 532 . . . . . . . 8  |-  ( ( ( ( R  e. 
Ring  /\  x  e.  U
)  /\  y  e.  ( Base `  R )
)  /\  ( m  e.  ( Base `  R
)  /\  ( (
y ( .r `  R ) x )  =  ( 1r `  R )  /\  (
m ( .r `  (oppr `  R ) ) x )  =  ( 1r
`  R ) ) ) )  ->  (
y  e.  U  /\  ( y ( .r
`  R ) x )  =  ( 1r
`  R ) ) )
6867rexlimdvaa 2956 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  x  e.  U )  /\  y  e.  (
Base `  R )
)  ->  ( E. m  e.  ( Base `  R ) ( ( y ( .r `  R ) x )  =  ( 1r `  R )  /\  (
m ( .r `  (oppr `  R ) ) x )  =  ( 1r
`  R ) )  ->  ( y  e.  U  /\  ( y ( .r `  R
) x )  =  ( 1r `  R
) ) ) )
6968expimpd 603 . . . . . 6  |-  ( ( R  e.  Ring  /\  x  e.  U )  ->  (
( y  e.  (
Base `  R )  /\  E. m  e.  (
Base `  R )
( ( y ( .r `  R ) x )  =  ( 1r `  R )  /\  ( m ( .r `  (oppr `  R
) ) x )  =  ( 1r `  R ) ) )  ->  ( y  e.  U  /\  ( y ( .r `  R
) x )  =  ( 1r `  R
) ) ) )
7069reximdv2 2934 . . . . 5  |-  ( ( R  e.  Ring  /\  x  e.  U )  ->  ( E. y  e.  ( Base `  R ) E. m  e.  ( Base `  R ) ( ( y ( .r `  R ) x )  =  ( 1r `  R )  /\  (
m ( .r `  (oppr `  R ) ) x )  =  ( 1r
`  R ) )  ->  E. y  e.  U  ( y ( .r
`  R ) x )  =  ( 1r
`  R ) ) )
7138, 70syl5bir 218 . . . 4  |-  ( ( R  e.  Ring  /\  x  e.  U )  ->  (
( E. y  e.  ( Base `  R
) ( y ( .r `  R ) x )  =  ( 1r `  R )  /\  E. m  e.  ( Base `  R
) ( m ( .r `  (oppr `  R
) ) x )  =  ( 1r `  R ) )  ->  E. y  e.  U  ( y ( .r
`  R ) x )  =  ( 1r
`  R ) ) )
7237, 71sylbid 215 . . 3  |-  ( ( R  e.  Ring  /\  x  e.  U )  ->  (
( x ( ||r `  R
) ( 1r `  R )  /\  x
( ||r `
 (oppr
`  R ) ) ( 1r `  R
) )  ->  E. y  e.  U  ( y
( .r `  R
) x )  =  ( 1r `  R
) ) )
7329, 72mpd 15 . 2  |-  ( ( R  e.  Ring  /\  x  e.  U )  ->  E. y  e.  U  ( y
( .r `  R
) x )  =  ( 1r `  R
) )
744, 11, 12, 19, 21, 23, 73isgrpde 15875 1  |-  ( R  e.  Ring  ->  G  e. 
Grp )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   E.wrex 2815   _Vcvv 3113   class class class wbr 4447   ` cfv 5586  (class class class)co 6282   Basecbs 14486   ↾s cress 14487   +g cplusg 14551   .rcmulr 14552   Grpcgrp 15723  mulGrpcmgp 16931   1rcur 16943   Ringcrg 16986  opprcoppr 17055   ||rcdsr 17071  Unitcui 17072
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-tpos 6952  df-recs 7039  df-rdg 7073  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-3 10591  df-ndx 14489  df-slot 14490  df-base 14491  df-sets 14492  df-ress 14493  df-plusg 14564  df-mulr 14565  df-0g 14693  df-mnd 15728  df-grp 15858  df-mgp 16932  df-ur 16944  df-rng 16988  df-oppr 17056  df-dvdsr 17074  df-unit 17075
This theorem is referenced by:  unitabl  17101  unitsubm  17103  unitinvcl  17107  unitinvinv  17108  unitlinv  17110  unitrinv  17111  isdrng2  17189  subrgugrp  17231  expghm  18296  expghmOLD  18297  invrvald  18945  nrginvrcn  20935  nrgtdrg  20936  dchrfi  23258  dchrghm  23259  dchrabs  23263  dchrptlem1  23267  dchrptlem2  23268  dchrptlem3  23269  dchrsum2  23271  rdivmuldivd  27444  dvrcan5  27446  rhmunitinv  27475  idomodle  30758  proot1mul  30761  proot1hash  30765  proot1ex  30766
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