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Theorem unitgrp 16749
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 16748 . . 3  |-  U  =  ( Base `  G
)
43a1i 11 . 2  |-  ( R  e.  Ring  ->  U  =  ( Base `  G
) )
5 fvex 5696 . . . 4  |-  ( Base `  G )  e.  _V
63, 5eqeltri 2508 . . 3  |-  U  e. 
_V
7 eqid 2438 . . . . 5  |-  (mulGrp `  R )  =  (mulGrp `  R )
8 eqid 2438 . . . . 5  |-  ( .r
`  R )  =  ( .r `  R
)
97, 8mgpplusg 16585 . . . 4  |-  ( .r
`  R )  =  ( +g  `  (mulGrp `  R ) )
102, 9ressplusg 14272 . . 3  |-  ( U  e.  _V  ->  ( .r `  R )  =  ( +g  `  G
) )
116, 10mp1i 12 . 2  |-  ( R  e.  Ring  ->  ( .r
`  R )  =  ( +g  `  G
) )
121, 8unitmulcl 16746 . 2  |-  ( ( R  e.  Ring  /\  x  e.  U  /\  y  e.  U )  ->  (
x ( .r `  R ) y )  e.  U )
13 eqid 2438 . . . . 5  |-  ( Base `  R )  =  (
Base `  R )
1413, 1unitcl 16741 . . . 4  |-  ( x  e.  U  ->  x  e.  ( Base `  R
) )
1513, 1unitcl 16741 . . . 4  |-  ( y  e.  U  ->  y  e.  ( Base `  R
) )
1613, 1unitcl 16741 . . . 4  |-  ( z  e.  U  ->  z  e.  ( Base `  R
) )
1714, 15, 163anim123i 1173 . . 3  |-  ( ( x  e.  U  /\  y  e.  U  /\  z  e.  U )  ->  ( x  e.  (
Base `  R )  /\  y  e.  ( Base `  R )  /\  z  e.  ( Base `  R ) ) )
1813, 8rngass 16651 . . 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 2438 . . 3  |-  ( 1r
`  R )  =  ( 1r `  R
)
211, 201unit 16740 . 2  |-  ( R  e.  Ring  ->  ( 1r
`  R )  e.  U )
2213, 8, 20rnglidm 16658 . . 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 2438 . . . . 5  |-  ( ||r `  R
)  =  ( ||r `  R
)
26 eqid 2438 . . . . 5  |-  (oppr `  R
)  =  (oppr `  R
)
27 eqid 2438 . . . . 5  |-  ( ||r `  (oppr `  R
) )  =  (
||r `  (oppr
`  R ) )
281, 20, 25, 26, 27isunit 16739 . . . 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 16729 . . . . . 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 16711 . . . . . . 7  |-  ( Base `  R )  =  (
Base `  (oppr
`  R ) )
34 eqid 2438 . . . . . . 7  |-  ( .r
`  (oppr
`  R ) )  =  ( .r `  (oppr `  R ) )
3533, 27, 34dvdsr2 16729 . . . . . 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 2883 . . . . 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 16730 . . . . . . . . . . . 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 16651 . . . . . . . . . . . . . 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 1220 . . . . . . . . . . . . 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 6101 . . . . . . . . . . . . 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 16708 . . . . . . . . . . . . . . 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 2484 . . . . . . . . . . . . . 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 6102 . . . . . . . . . . . . 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 2478 . . . . . . . . . . . 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 16658 . . . . . . . . . . . . 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 16659 . . . . . . . . . . . . 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 2478 . . . . . . . . . . 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 4316 . . . . . . . . . 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 16730 . . . . . . . . . . . 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 16708 . . . . . . . . . . . 12  |-  ( x ( .r `  (oppr `  R
) ) y )  =  ( y ( .r `  R ) x )
6362, 47syl5eq 2482 . . . . . . . . . . 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 4311 . . . . . . . . . 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 16739 . . . . . . . . . 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 2837 . . . . . . 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 2820 . . . . 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 15553 1  |-  ( R  e.  Ring  ->  G  e. 
Grp )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   E.wrex 2711   _Vcvv 2967   class class class wbr 4287   ` cfv 5413  (class class class)co 6086   Basecbs 14166   ↾s cress 14167   +g cplusg 14230   .rcmulr 14231   Grpcgrp 15402  mulGrpcmgp 16581   1rcur 16593   Ringcrg 16635  opprcoppr 16704   ||rcdsr 16720  Unitcui 16721
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-tpos 6740  df-recs 6824  df-rdg 6858  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-nn 10315  df-2 10372  df-3 10373  df-ndx 14169  df-slot 14170  df-base 14171  df-sets 14172  df-ress 14173  df-plusg 14243  df-mulr 14244  df-0g 14372  df-mnd 15407  df-grp 15536  df-mgp 16582  df-ur 16594  df-rng 16637  df-oppr 16705  df-dvdsr 16723  df-unit 16724
This theorem is referenced by:  unitabl  16750  unitsubm  16752  unitinvcl  16756  unitinvinv  16757  unitlinv  16759  unitrinv  16760  isdrng2  16822  subrgugrp  16864  expghm  17903  expghmOLD  17904  invrvald  18462  nrginvrcn  20252  nrgtdrg  20253  dchrfi  22574  dchrghm  22575  dchrabs  22579  dchrptlem1  22583  dchrptlem2  22584  dchrptlem3  22585  dchrsum2  22587  rdivmuldivd  26227  dvrcan5  26229  rhmunitinv  26258  idomodle  29532  proot1mul  29535  proot1hash  29539  proot1ex  29540
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