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Theorem unitgrp 15772
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 15771 . . 3  |-  U  =  ( Base `  G
)
43a1i 11 . 2  |-  ( R  e.  Ring  ->  U  =  ( Base `  G
) )
5 fvex 5742 . . . 4  |-  ( Base `  G )  e.  _V
63, 5eqeltri 2506 . . 3  |-  U  e. 
_V
7 eqid 2436 . . . . 5  |-  (mulGrp `  R )  =  (mulGrp `  R )
8 eqid 2436 . . . . 5  |-  ( .r
`  R )  =  ( .r `  R
)
97, 8mgpplusg 15652 . . . 4  |-  ( .r
`  R )  =  ( +g  `  (mulGrp `  R ) )
102, 9ressplusg 13571 . . 3  |-  ( U  e.  _V  ->  ( .r `  R )  =  ( +g  `  G
) )
116, 10mp1i 12 . 2  |-  ( R  e.  Ring  ->  ( .r
`  R )  =  ( +g  `  G
) )
121, 8unitmulcl 15769 . 2  |-  ( ( R  e.  Ring  /\  x  e.  U  /\  y  e.  U )  ->  (
x ( .r `  R ) y )  e.  U )
13 eqid 2436 . . . . 5  |-  ( Base `  R )  =  (
Base `  R )
1413, 1unitcl 15764 . . . 4  |-  ( x  e.  U  ->  x  e.  ( Base `  R
) )
1513, 1unitcl 15764 . . . 4  |-  ( y  e.  U  ->  y  e.  ( Base `  R
) )
1613, 1unitcl 15764 . . . 4  |-  ( z  e.  U  ->  z  e.  ( Base `  R
) )
1714, 15, 163anim123i 1139 . . 3  |-  ( ( x  e.  U  /\  y  e.  U  /\  z  e.  U )  ->  ( x  e.  (
Base `  R )  /\  y  e.  ( Base `  R )  /\  z  e.  ( Base `  R ) ) )
1813, 8rngass 15680 . . 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 461 . 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 2436 . . 3  |-  ( 1r
`  R )  =  ( 1r `  R
)
211, 201unit 15763 . 2  |-  ( R  e.  Ring  ->  ( 1r
`  R )  e.  U )
2213, 8, 20rnglidm 15687 . . 3  |-  ( ( R  e.  Ring  /\  x  e.  ( Base `  R
) )  ->  (
( 1r `  R
) ( .r `  R ) x )  =  x )
2314, 22sylan2 461 . 2  |-  ( ( R  e.  Ring  /\  x  e.  U )  ->  (
( 1r `  R
) ( .r `  R ) x )  =  x )
24 simpr 448 . . . 4  |-  ( ( R  e.  Ring  /\  x  e.  U )  ->  x  e.  U )
25 eqid 2436 . . . . 5  |-  ( ||r `  R
)  =  ( ||r `  R
)
26 eqid 2436 . . . . 5  |-  (oppr `  R
)  =  (oppr `  R
)
27 eqid 2436 . . . . 5  |-  ( ||r `  (oppr `  R
) )  =  (
||r `  (oppr
`  R ) )
281, 20, 25, 26, 27isunit 15762 . . . 4  |-  ( x  e.  U  <->  ( x
( ||r `
 R ) ( 1r `  R )  /\  x ( ||r `  (oppr `  R
) ) ( 1r
`  R ) ) )
2924, 28sylib 189 . . 3  |-  ( ( R  e.  Ring  /\  x  e.  U )  ->  (
x ( ||r `
 R ) ( 1r `  R )  /\  x ( ||r `  (oppr `  R
) ) ( 1r
`  R ) ) )
3014adantl 453 . . . . . 6  |-  ( ( R  e.  Ring  /\  x  e.  U )  ->  x  e.  ( Base `  R
) )
3113, 25, 8dvdsr2 15752 . . . . . 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 15734 . . . . . . 7  |-  ( Base `  R )  =  (
Base `  (oppr
`  R ) )
34 eqid 2436 . . . . . . 7  |-  ( .r
`  (oppr
`  R ) )  =  ( .r `  (oppr `  R ) )
3533, 27, 34dvdsr2 15752 . . . . . 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 692 . . . 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 2875 . . . . 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 733 . . . . . . . . . . . 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 707 . . . . . . . . . . . 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 15753 . . . . . . . . . . . 12  |-  ( ( m  e.  ( Base `  R )  /\  x  e.  ( Base `  R
) )  ->  m
( ||r `
 R ) ( x ( .r `  R ) m ) )
4239, 40, 41syl2anc 643 . . . . . . . . . . 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 735 . . . . . . . . . . . . . 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 732 . . . . . . . . . . . . . 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 15680 . . . . . . . . . . . . . 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 1186 . . . . . . . . . . . . 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 741 . . . . . . . . . . . . . 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 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 )  =  ( ( 1r
`  R ) ( .r `  R ) m ) )
4913, 8, 26, 34opprmul 15731 . . . . . . . . . . . . . . 15  |-  ( m ( .r `  (oppr `  R
) ) x )  =  ( x ( .r `  R ) m )
50 simprrr 742 . . . . . . . . . . . . . . 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 2482 . . . . . . . . . . . . . 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 6097 . . . . . . . . . . . . 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 2476 . . . . . . . . . . . 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 15687 . . . . . . . . . . . . 13  |-  ( ( R  e.  Ring  /\  m  e.  ( Base `  R
) )  ->  (
( 1r `  R
) ( .r `  R ) m )  =  m )
5543, 39, 54syl2anc 643 . . . . . . . . . . . 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 15688 . . . . . . . . . . . . 13  |-  ( ( R  e.  Ring  /\  y  e.  ( Base `  R
) )  ->  (
y ( .r `  R ) ( 1r
`  R ) )  =  y )
5743, 44, 56syl2anc 643 . . . . . . . . . . . 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 2476 . . . . . . . . . . 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 4241 . . . . . . . . . 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 15753 . . . . . . . . . . . 12  |-  ( ( y  e.  ( Base `  R )  /\  x  e.  ( Base `  R
) )  ->  y
( ||r `
 (oppr
`  R ) ) ( x ( .r
`  (oppr
`  R ) ) y ) )
6144, 40, 60syl2anc 643 . . . . . . . . . . 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 15731 . . . . . . . . . . . 12  |-  ( x ( .r `  (oppr `  R
) ) y )  =  ( y ( .r `  R ) x )
6362, 47syl5eq 2480 . . . . . . . . . . 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 4236 . . . . . . . . . 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 15762 . . . . . . . . . 10  |-  ( y  e.  U  <->  ( y
( ||r `
 R ) ( 1r `  R )  /\  y ( ||r `  (oppr `  R
) ) ( 1r
`  R ) ) )
6659, 64, 65sylanbrc 646 . . . . . . . . 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 519 . . . . . . . 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 2831 . . . . . . 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 587 . . . . . 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 210 . . . 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 207 . . 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 14829 1  |-  ( R  e.  Ring  ->  G  e. 
Grp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   E.wrex 2706   _Vcvv 2956   class class class wbr 4212   ` cfv 5454  (class class class)co 6081   Basecbs 13469   ↾s cress 13470   +g cplusg 13529   .rcmulr 13530   Grpcgrp 14685  mulGrpcmgp 15648   Ringcrg 15660   1rcur 15662  opprcoppr 15727   ||rcdsr 15743  Unitcui 15744
This theorem is referenced by:  unitabl  15773  unitsubm  15775  unitinvcl  15779  unitinvinv  15780  unitlinv  15782  unitrinv  15783  isdrng2  15845  subrgugrp  15887  expghm  16777  nrginvrcn  18727  nrgtdrg  18728  dchrfi  21039  dchrghm  21040  dchrabs  21044  dchrptlem1  21048  dchrptlem2  21049  dchrptlem3  21050  dchrsum2  21052  rdivmuldivd  24227  dvrcan5  24229  rhmunitinv  24260  idomodle  27489  proot1mul  27492  proot1hash  27496  proot1ex  27497
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-tpos 6479  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-3 10059  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-plusg 13542  df-mulr 13543  df-0g 13727  df-mnd 14690  df-grp 14812  df-mgp 15649  df-rng 15663  df-ur 15665  df-oppr 15728  df-dvdsr 15746  df-unit 15747
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