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Theorem unitgrp 16693
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 16692 . . 3  |-  U  =  ( Base `  G
)
43a1i 11 . 2  |-  ( R  e.  Ring  ->  U  =  ( Base `  G
) )
5 fvex 5689 . . . 4  |-  ( Base `  G )  e.  _V
63, 5eqeltri 2503 . . 3  |-  U  e. 
_V
7 eqid 2433 . . . . 5  |-  (mulGrp `  R )  =  (mulGrp `  R )
8 eqid 2433 . . . . 5  |-  ( .r
`  R )  =  ( .r `  R
)
97, 8mgpplusg 16569 . . . 4  |-  ( .r
`  R )  =  ( +g  `  (mulGrp `  R ) )
102, 9ressplusg 14263 . . 3  |-  ( U  e.  _V  ->  ( .r `  R )  =  ( +g  `  G
) )
116, 10mp1i 12 . 2  |-  ( R  e.  Ring  ->  ( .r
`  R )  =  ( +g  `  G
) )
121, 8unitmulcl 16690 . 2  |-  ( ( R  e.  Ring  /\  x  e.  U  /\  y  e.  U )  ->  (
x ( .r `  R ) y )  e.  U )
13 eqid 2433 . . . . 5  |-  ( Base `  R )  =  (
Base `  R )
1413, 1unitcl 16685 . . . 4  |-  ( x  e.  U  ->  x  e.  ( Base `  R
) )
1513, 1unitcl 16685 . . . 4  |-  ( y  e.  U  ->  y  e.  ( Base `  R
) )
1613, 1unitcl 16685 . . . 4  |-  ( z  e.  U  ->  z  e.  ( Base `  R
) )
1714, 15, 163anim123i 1166 . . 3  |-  ( ( x  e.  U  /\  y  e.  U  /\  z  e.  U )  ->  ( x  e.  (
Base `  R )  /\  y  e.  ( Base `  R )  /\  z  e.  ( Base `  R ) ) )
1813, 8rngass 16597 . . 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 471 . 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 2433 . . 3  |-  ( 1r
`  R )  =  ( 1r `  R
)
211, 201unit 16684 . 2  |-  ( R  e.  Ring  ->  ( 1r
`  R )  e.  U )
2213, 8, 20rnglidm 16604 . . 3  |-  ( ( R  e.  Ring  /\  x  e.  ( Base `  R
) )  ->  (
( 1r `  R
) ( .r `  R ) x )  =  x )
2314, 22sylan2 471 . 2  |-  ( ( R  e.  Ring  /\  x  e.  U )  ->  (
( 1r `  R
) ( .r `  R ) x )  =  x )
24 simpr 458 . . . 4  |-  ( ( R  e.  Ring  /\  x  e.  U )  ->  x  e.  U )
25 eqid 2433 . . . . 5  |-  ( ||r `  R
)  =  ( ||r `  R
)
26 eqid 2433 . . . . 5  |-  (oppr `  R
)  =  (oppr `  R
)
27 eqid 2433 . . . . 5  |-  ( ||r `  (oppr `  R
) )  =  (
||r `  (oppr
`  R ) )
281, 20, 25, 26, 27isunit 16683 . . . 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 463 . . . . . 6  |-  ( ( R  e.  Ring  /\  x  e.  U )  ->  x  e.  ( Base `  R
) )
3113, 25, 8dvdsr2 16673 . . . . . 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 16655 . . . . . . 7  |-  ( Base `  R )  =  (
Base `  (oppr
`  R ) )
34 eqid 2433 . . . . . . 7  |-  ( .r
`  (oppr
`  R ) )  =  ( .r `  (oppr `  R ) )
3533, 27, 34dvdsr2 16673 . . . . . 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 703 . . . 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 2878 . . . . 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 748 . . . . . . . . . . . 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 718 . . . . . . . . . . . 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 16674 . . . . . . . . . . . 12  |-  ( ( m  e.  ( Base `  R )  /\  x  e.  ( Base `  R
) )  ->  m
( ||r `
 R ) ( x ( .r `  R ) m ) )
4239, 40, 41syl2anc 654 . . . . . . . . . . 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 750 . . . . . . . . . . . . . 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 747 . . . . . . . . . . . . . 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 16597 . . . . . . . . . . . . . 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 1213 . . . . . . . . . . . . 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 756 . . . . . . . . . . . . . 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 6095 . . . . . . . . . . . . 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 16652 . . . . . . . . . . . . . . 15  |-  ( m ( .r `  (oppr `  R
) ) x )  =  ( x ( .r `  R ) m )
50 simprrr 757 . . . . . . . . . . . . . . 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 2479 . . . . . . . . . . . . . 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 6096 . . . . . . . . . . . . 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 2473 . . . . . . . . . . . 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 16604 . . . . . . . . . . . . 13  |-  ( ( R  e.  Ring  /\  m  e.  ( Base `  R
) )  ->  (
( 1r `  R
) ( .r `  R ) m )  =  m )
5543, 39, 54syl2anc 654 . . . . . . . . . . . 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 16605 . . . . . . . . . . . . 13  |-  ( ( R  e.  Ring  /\  y  e.  ( Base `  R
) )  ->  (
y ( .r `  R ) ( 1r
`  R ) )  =  y )
5743, 44, 56syl2anc 654 . . . . . . . . . . . 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 2473 . . . . . . . . . . 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 4309 . . . . . . . . . 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 16674 . . . . . . . . . . . 12  |-  ( ( y  e.  ( Base `  R )  /\  x  e.  ( Base `  R
) )  ->  y
( ||r `
 (oppr
`  R ) ) ( x ( .r
`  (oppr
`  R ) ) y ) )
6144, 40, 60syl2anc 654 . . . . . . . . . . 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 16652 . . . . . . . . . . . 12  |-  ( x ( .r `  (oppr `  R
) ) y )  =  ( y ( .r `  R ) x )
6362, 47syl5eq 2477 . . . . . . . . . . 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 4304 . . . . . . . . . 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 16683 . . . . . . . . . 10  |-  ( y  e.  U  <->  ( y
( ||r `
 R ) ( 1r `  R )  /\  y ( ||r `  (oppr `  R
) ) ( 1r
`  R ) ) )
6659, 64, 65sylanbrc 657 . . . . . . . . 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 529 . . . . . . . 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 2832 . . . . . . 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 598 . . . . . 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 2815 . . . . 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 15542 1  |-  ( R  e.  Ring  ->  G  e. 
Grp )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 958    = wceq 1362    e. wcel 1755   E.wrex 2706   _Vcvv 2962   class class class wbr 4280   ` cfv 5406  (class class class)co 6080   Basecbs 14157   ↾s cress 14158   +g cplusg 14221   .rcmulr 14222   Grpcgrp 15393  mulGrpcmgp 16565   Ringcrg 16577   1rcur 16579  opprcoppr 16648   ||rcdsr 16664  Unitcui 16665
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-cnex 9326  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-om 6466  df-tpos 6734  df-recs 6818  df-rdg 6852  df-er 7089  df-en 7299  df-dom 7300  df-sdom 7301  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-nn 10311  df-2 10368  df-3 10369  df-ndx 14160  df-slot 14161  df-base 14162  df-sets 14163  df-ress 14164  df-plusg 14234  df-mulr 14235  df-0g 14363  df-mnd 15398  df-grp 15525  df-mgp 16566  df-rng 16580  df-ur 16582  df-oppr 16649  df-dvdsr 16667  df-unit 16668
This theorem is referenced by:  unitabl  16694  unitsubm  16696  unitinvcl  16700  unitinvinv  16701  unitlinv  16703  unitrinv  16704  isdrng2  16766  subrgugrp  16808  expghm  17765  expghmOLD  17766  invrvald  18324  nrginvrcn  20114  nrgtdrg  20115  dchrfi  22479  dchrghm  22480  dchrabs  22484  dchrptlem1  22488  dchrptlem2  22489  dchrptlem3  22490  dchrsum2  22492  rdivmuldivd  26112  dvrcan5  26114  rhmunitinv  26143  idomodle  29406  proot1mul  29409  proot1hash  29413  proot1ex  29414
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