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Theorem unitgrp 13290
Description: The group of units is a group under multiplication. (Contributed by Mario Carneiro, 2-Dec-2014.)
Hypotheses
Ref Expression
unitgrp.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 unitgrp.1 . . . 4  |-  U  =  (Unit `  R )
21a1i 9 . . 3  |-  ( R  e.  Ring  ->  U  =  (Unit `  R )
)
3 unitgrp.2 . . . 4  |-  G  =  ( (mulGrp `  R
)s 
U )
43a1i 9 . . 3  |-  ( R  e.  Ring  ->  G  =  ( (mulGrp `  R
)s 
U ) )
5 ringsrg 13229 . . 3  |-  ( R  e.  Ring  ->  R  e. SRing
)
62, 4, 5unitgrpbasd 13289 . 2  |-  ( R  e.  Ring  ->  U  =  ( Base `  G
) )
7 eqid 2177 . . . 4  |-  (mulGrp `  R )  =  (mulGrp `  R )
8 eqid 2177 . . . 4  |-  ( .r
`  R )  =  ( .r `  R
)
97, 8mgpplusgg 13139 . . 3  |-  ( R  e.  Ring  ->  ( .r
`  R )  =  ( +g  `  (mulGrp `  R ) ) )
10 basfn 12522 . . . . 5  |-  Base  Fn  _V
11 elex 2750 . . . . 5  |-  ( R  e.  Ring  ->  R  e. 
_V )
12 funfvex 5534 . . . . . 6  |-  ( ( Fun  Base  /\  R  e. 
dom  Base )  ->  ( Base `  R )  e. 
_V )
1312funfni 5318 . . . . 5  |-  ( (
Base  Fn  _V  /\  R  e.  _V )  ->  ( Base `  R )  e. 
_V )
1410, 11, 13sylancr 414 . . . 4  |-  ( R  e.  Ring  ->  ( Base `  R )  e.  _V )
15 eqidd 2178 . . . . 5  |-  ( R  e.  Ring  ->  ( Base `  R )  =  (
Base `  R )
)
1615, 2, 5unitssd 13283 . . . 4  |-  ( R  e.  Ring  ->  U  C_  ( Base `  R )
)
1714, 16ssexd 4145 . . 3  |-  ( R  e.  Ring  ->  U  e. 
_V )
187mgpex 13140 . . 3  |-  ( R  e.  Ring  ->  (mulGrp `  R )  e.  _V )
194, 9, 17, 18ressplusgd 12589 . 2  |-  ( R  e.  Ring  ->  ( .r
`  R )  =  ( +g  `  G
) )
201, 8unitmulcl 13287 . 2  |-  ( ( R  e.  Ring  /\  x  e.  U  /\  y  e.  U )  ->  (
x ( .r `  R ) y )  e.  U )
21 eqidd 2178 . . . . 5  |-  ( ( R  e.  Ring  /\  (
x  e.  U  /\  y  e.  U  /\  z  e.  U )
)  ->  ( Base `  R )  =  (
Base `  R )
)
221a1i 9 . . . . 5  |-  ( ( R  e.  Ring  /\  (
x  e.  U  /\  y  e.  U  /\  z  e.  U )
)  ->  U  =  (Unit `  R ) )
235adantr 276 . . . . 5  |-  ( ( R  e.  Ring  /\  (
x  e.  U  /\  y  e.  U  /\  z  e.  U )
)  ->  R  e. SRing )
24 simpr1 1003 . . . . 5  |-  ( ( R  e.  Ring  /\  (
x  e.  U  /\  y  e.  U  /\  z  e.  U )
)  ->  x  e.  U )
2521, 22, 23, 24unitcld 13282 . . . 4  |-  ( ( R  e.  Ring  /\  (
x  e.  U  /\  y  e.  U  /\  z  e.  U )
)  ->  x  e.  ( Base `  R )
)
26 simpr2 1004 . . . . 5  |-  ( ( R  e.  Ring  /\  (
x  e.  U  /\  y  e.  U  /\  z  e.  U )
)  ->  y  e.  U )
2721, 22, 23, 26unitcld 13282 . . . 4  |-  ( ( R  e.  Ring  /\  (
x  e.  U  /\  y  e.  U  /\  z  e.  U )
)  ->  y  e.  ( Base `  R )
)
28 simpr3 1005 . . . . 5  |-  ( ( R  e.  Ring  /\  (
x  e.  U  /\  y  e.  U  /\  z  e.  U )
)  ->  z  e.  U )
2921, 22, 23, 28unitcld 13282 . . . 4  |-  ( ( R  e.  Ring  /\  (
x  e.  U  /\  y  e.  U  /\  z  e.  U )
)  ->  z  e.  ( Base `  R )
)
3025, 27, 293jca 1177 . . 3  |-  ( ( R  e.  Ring  /\  (
x  e.  U  /\  y  e.  U  /\  z  e.  U )
)  ->  ( x  e.  ( Base `  R
)  /\  y  e.  ( Base `  R )  /\  z  e.  ( Base `  R ) ) )
31 eqid 2177 . . . 4  |-  ( Base `  R )  =  (
Base `  R )
3231, 8ringass 13204 . . 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 ) ) )
3330, 32syldan 282 . 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 ) ) )
34 eqid 2177 . . 3  |-  ( 1r
`  R )  =  ( 1r `  R
)
351, 341unit 13281 . 2  |-  ( R  e.  Ring  ->  ( 1r
`  R )  e.  U )
36 eqidd 2178 . . . 4  |-  ( ( R  e.  Ring  /\  x  e.  U )  ->  ( Base `  R )  =  ( Base `  R
) )
371a1i 9 . . . 4  |-  ( ( R  e.  Ring  /\  x  e.  U )  ->  U  =  (Unit `  R )
)
385adantr 276 . . . 4  |-  ( ( R  e.  Ring  /\  x  e.  U )  ->  R  e. SRing )
39 simpr 110 . . . 4  |-  ( ( R  e.  Ring  /\  x  e.  U )  ->  x  e.  U )
4036, 37, 38, 39unitcld 13282 . . 3  |-  ( ( R  e.  Ring  /\  x  e.  U )  ->  x  e.  ( Base `  R
) )
4131, 8, 34ringlidm 13211 . . 3  |-  ( ( R  e.  Ring  /\  x  e.  ( Base `  R
) )  ->  (
( 1r `  R
) ( .r `  R ) x )  =  x )
4240, 41syldan 282 . 2  |-  ( ( R  e.  Ring  /\  x  e.  U )  ->  (
( 1r `  R
) ( .r `  R ) x )  =  x )
43 eqidd 2178 . . . . 5  |-  ( ( R  e.  Ring  /\  x  e.  U )  ->  ( 1r `  R )  =  ( 1r `  R
) )
44 eqidd 2178 . . . . 5  |-  ( ( R  e.  Ring  /\  x  e.  U )  ->  ( ||r `  R )  =  (
||r `  R ) )
45 eqidd 2178 . . . . 5  |-  ( ( R  e.  Ring  /\  x  e.  U )  ->  (oppr `  R
)  =  (oppr `  R
) )
46 eqidd 2178 . . . . 5  |-  ( ( R  e.  Ring  /\  x  e.  U )  ->  ( ||r `  (oppr
`  R ) )  =  ( ||r `
 (oppr
`  R ) ) )
4737, 43, 44, 45, 46, 38isunitd 13280 . . . 4  |-  ( ( R  e.  Ring  /\  x  e.  U )  ->  (
x  e.  U  <->  ( x
( ||r `
 R ) ( 1r `  R )  /\  x ( ||r `  (oppr `  R
) ) ( 1r
`  R ) ) ) )
4839, 47mpbid 147 . . 3  |-  ( ( R  e.  Ring  /\  x  e.  U )  ->  (
x ( ||r `
 R ) ( 1r `  R )  /\  x ( ||r `  (oppr `  R
) ) ( 1r
`  R ) ) )
49 eqidd 2178 . . . . . 6  |-  ( ( R  e.  Ring  /\  x  e.  U )  ->  ( .r `  R )  =  ( .r `  R
) )
5036, 44, 38, 49, 40dvdsr2d 13269 . . . . 5  |-  ( ( R  e.  Ring  /\  x  e.  U )  ->  (
x ( ||r `
 R ) ( 1r `  R )  <->  E. y  e.  ( Base `  R ) ( y ( .r `  R ) x )  =  ( 1r `  R ) ) )
51 eqid 2177 . . . . . . . 8  |-  (oppr `  R
)  =  (oppr `  R
)
5251, 31opprbasg 13252 . . . . . . 7  |-  ( R  e.  Ring  ->  ( Base `  R )  =  (
Base `  (oppr
`  R ) ) )
5352adantr 276 . . . . . 6  |-  ( ( R  e.  Ring  /\  x  e.  U )  ->  ( Base `  R )  =  ( Base `  (oppr `  R
) ) )
5451opprring 13254 . . . . . . . 8  |-  ( R  e.  Ring  ->  (oppr `  R
)  e.  Ring )
55 ringsrg 13229 . . . . . . . 8  |-  ( (oppr `  R )  e.  Ring  -> 
(oppr `  R )  e. SRing )
5654, 55syl 14 . . . . . . 7  |-  ( R  e.  Ring  ->  (oppr `  R
)  e. SRing )
5756adantr 276 . . . . . 6  |-  ( ( R  e.  Ring  /\  x  e.  U )  ->  (oppr `  R
)  e. SRing )
58 eqidd 2178 . . . . . 6  |-  ( ( R  e.  Ring  /\  x  e.  U )  ->  ( .r `  (oppr
`  R ) )  =  ( .r `  (oppr `  R ) ) )
5953, 46, 57, 58, 40dvdsr2d 13269 . . . . 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 ) ) )
6050, 59anbi12d 473 . . . 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 ) ) ) )
61 reeanv 2647 . . . . 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 ) ) )
62 eqidd 2178 . . . . . . . . . . . 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 ) ) ) )  ->  ( Base `  R )  =  ( Base `  R
) )
63 eqidd 2178 . . . . . . . . . . . 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 ) ) ) )  ->  ( ||r `  R )  =  (
||r `  R ) )
6438ad2antrr 488 . . . . . . . . . . . 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 ) ) ) )  ->  R  e. SRing )
65 eqidd 2178 . . . . . . . . . . . 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 ) ) ) )  ->  ( .r `  R )  =  ( .r `  R
) )
66 simprl 529 . . . . . . . . . . . 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
) )
6740ad2antrr 488 . . . . . . . . . . . 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
) )
6862, 63, 64, 65, 66, 67dvdsrmuld 13270 . . . . . . . . . . 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 ) )
69 simplll 533 . . . . . . . . . . . . . 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 )
70 simplr 528 . . . . . . . . . . . . . 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
) )
7131, 8ringass 13204 . . . . . . . . . . . . . 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 ) ) )
7269, 70, 67, 66, 71syl13anc 1240 . . . . . . . . . . . . 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 ) ) )
73 simprrl 539 . . . . . . . . . . . . . 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 ) )
7473oveq1d 5892 . . . . . . . . . . . . 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 ) )
7539ad2antrr 488 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( 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.  U )
76 eqid 2177 . . . . . . . . . . . . . . . . 17  |-  ( .r
`  (oppr
`  R ) )  =  ( .r `  (oppr `  R ) )
7731, 8, 51, 76opprmulg 13248 . . . . . . . . . . . . . . . 16  |-  ( ( R  e.  Ring  /\  m  e.  ( Base `  R
)  /\  x  e.  U )  ->  (
m ( .r `  (oppr `  R ) ) x )  =  ( x ( .r `  R
) m ) )
7869, 66, 75, 77syl3anc 1238 . . . . . . . . . . . . . . 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 )  =  ( x ( .r `  R
) m ) )
79 simprrr 540 . . . . . . . . . . . . . . 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 ) )
8078, 79eqtr3d 2212 . . . . . . . . . . . . . 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 ) )
8180oveq2d 5893 . . . . . . . . . . . . 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
) ) )
8272, 74, 813eqtr3d 2218 . . . . . . . . . . . 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
) ) )
8331, 8, 34ringlidm 13211 . . . . . . . . . . . . 13  |-  ( ( R  e.  Ring  /\  m  e.  ( Base `  R
) )  ->  (
( 1r `  R
) ( .r `  R ) m )  =  m )
8469, 66, 83syl2anc 411 . . . . . . . . . . . 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 )
8531, 8, 34ringridm 13212 . . . . . . . . . . . . 13  |-  ( ( R  e.  Ring  /\  y  e.  ( Base `  R
) )  ->  (
y ( .r `  R ) ( 1r
`  R ) )  =  y )
8669, 70, 85syl2anc 411 . . . . . . . . . . . 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 )
8782, 84, 863eqtr3d 2218 . . . . . . . . . . 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 )
8868, 87, 803brtr3d 4036 . . . . . . . . . 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 ) )
8969, 52syl 14 . . . . . . . . . . . 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 ) ) ) )  ->  ( Base `  R )  =  ( Base `  (oppr `  R
) ) )
90 eqidd 2178 . . . . . . . . . . . 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 ) ) ) )  ->  ( ||r `  (oppr
`  R ) )  =  ( ||r `
 (oppr
`  R ) ) )
9169, 56syl 14 . . . . . . . . . . . 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 ) ) ) )  ->  (oppr `  R
)  e. SRing )
92 eqidd 2178 . . . . . . . . . . . 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 ) ) ) )  ->  ( .r `  (oppr
`  R ) )  =  ( .r `  (oppr `  R ) ) )
9389, 90, 91, 92, 70, 67dvdsrmuld 13270 . . . . . . . . . . 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 ) )
9431, 8, 51, 76opprmulg 13248 . . . . . . . . . . . . 13  |-  ( ( R  e.  Ring  /\  x  e.  U  /\  y  e.  ( Base `  R
) )  ->  (
x ( .r `  (oppr `  R ) ) y )  =  ( y ( .r `  R
) x ) )
9569, 75, 70, 94syl3anc 1238 . . . . . . . . . . . 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 ( .r `  (oppr `  R ) ) y )  =  ( y ( .r `  R
) x ) )
9695, 73eqtrd 2210 . . . . . . . . . . 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 ) )
9793, 96breqtrd 4031 . . . . . . . . . 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
) )
981a1i 9 . . . . . . . . . . 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 ) ) ) )  ->  U  =  (Unit `  R )
)
99 eqidd 2178 . . . . . . . . . . 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 ) ) ) )  ->  ( 1r `  R )  =  ( 1r `  R
) )
100 eqidd 2178 . . . . . . . . . . 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 ) ) ) )  ->  (oppr `  R
)  =  (oppr `  R
) )
10198, 99, 63, 100, 90, 64isunitd 13280 . . . . . . . . . 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  e.  U  <->  ( y
( ||r `
 R ) ( 1r `  R )  /\  y ( ||r `  (oppr `  R
) ) ( 1r
`  R ) ) ) )
10288, 97, 101mpbir2and 944 . . . . . . . . 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 )
103102, 73jca 306 . . . . . . . 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 ) ) )
104103rexlimdvaa 2595 . . . . . . 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
) ) ) )
105104expimpd 363 . . . . . 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
) ) ) )
106105reximdv2 2576 . . . . 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 ) ) )
10761, 106biimtrrid 153 . . . 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 ) ) )
10860, 107sylbid 150 . . 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
) ) )
10948, 108mpd 13 . 2  |-  ( ( R  e.  Ring  /\  x  e.  U )  ->  E. y  e.  U  ( y
( .r `  R
) x )  =  ( 1r `  R
) )
1106, 19, 20, 33, 35, 42, 109isgrpde 12903 1  |-  ( R  e.  Ring  ->  G  e. 
Grp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    /\ w3a 978    = wceq 1353    e. wcel 2148   E.wrex 2456   _Vcvv 2739   class class class wbr 4005    Fn wfn 5213   ` cfv 5218  (class class class)co 5877   Basecbs 12464   ↾s cress 12465   .rcmulr 12539   Grpcgrp 12882  mulGrpcmgp 13135   1rcur 13147  SRingcsrg 13151   Ringcrg 13184  opprcoppr 13244   ||rcdsr 13260  Unitcui 13261
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-addcom 7913  ax-addass 7915  ax-i2m1 7918  ax-0lt1 7919  ax-0id 7921  ax-rnegex 7922  ax-pre-ltirr 7925  ax-pre-lttrn 7927  ax-pre-ltadd 7929
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-tpos 6248  df-pnf 7996  df-mnf 7997  df-ltxr 7999  df-inn 8922  df-2 8980  df-3 8981  df-ndx 12467  df-slot 12468  df-base 12470  df-sets 12471  df-iress 12472  df-plusg 12551  df-mulr 12552  df-0g 12712  df-mgm 12780  df-sgrp 12813  df-mnd 12823  df-grp 12885  df-minusg 12886  df-cmn 13095  df-abl 13096  df-mgp 13136  df-ur 13148  df-srg 13152  df-ring 13186  df-oppr 13245  df-dvdsr 13263  df-unit 13264
This theorem is referenced by:  unitabl  13291  unitsubm  13293  invrfvald  13296  unitinvcl  13297  unitinvinv  13298  unitlinv  13300  unitrinv  13301  rdivmuldivd  13318  subrgugrp  13366
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