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Theorem nrginvrcn 18202
Description: The ring inverse function is continuous in a normed ring. (Note that this is true even in rings which are not division rings.) (Contributed by Mario Carneiro, 6-Oct-2015.)
Hypotheses
Ref Expression
nrginvrcn.x  |-  X  =  ( Base `  R
)
nrginvrcn.u  |-  U  =  (Unit `  R )
nrginvrcn.i  |-  I  =  ( invr `  R
)
nrginvrcn.j  |-  J  =  ( TopOpen `  R )
Assertion
Ref Expression
nrginvrcn  |-  ( R  e. NrmRing  ->  I  e.  ( ( Jt  U )  Cn  ( Jt  U ) ) )

Proof of Theorem nrginvrcn
Dummy variables  s 
r  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nrgrng 18174 . . . 4  |-  ( R  e. NrmRing  ->  R  e.  Ring )
2 nrginvrcn.u . . . . 5  |-  U  =  (Unit `  R )
3 eqid 2283 . . . . 5  |-  ( (mulGrp `  R )s  U )  =  ( (mulGrp `  R )s  U
)
42, 3unitgrp 15449 . . . 4  |-  ( R  e.  Ring  ->  ( (mulGrp `  R )s  U )  e.  Grp )
52, 3unitgrpbas 15448 . . . . 5  |-  U  =  ( Base `  (
(mulGrp `  R )s  U
) )
6 nrginvrcn.i . . . . . 6  |-  I  =  ( invr `  R
)
72, 3, 6invrfval 15455 . . . . 5  |-  I  =  ( inv g `  ( (mulGrp `  R )s  U
) )
85, 7grpinvf 14526 . . . 4  |-  ( ( (mulGrp `  R )s  U
)  e.  Grp  ->  I : U --> U )
91, 4, 83syl 18 . . 3  |-  ( R  e. NrmRing  ->  I : U --> U )
10 1rp 10358 . . . . . . . 8  |-  1  e.  RR+
11 ne0i 3461 . . . . . . . 8  |-  ( 1  e.  RR+  ->  RR+  =/=  (/) )
1210, 11ax-mp 8 . . . . . . 7  |-  RR+  =/=  (/)
131ad2antrr 706 . . . . . . . . . . . . . 14  |-  ( ( ( R  e. NrmRing  /\  (
x  e.  U  /\  r  e.  RR+ ) )  /\  y  e.  U
)  ->  R  e.  Ring )
14 nrginvrcn.x . . . . . . . . . . . . . . . 16  |-  X  =  ( Base `  R
)
1514, 2unitss 15442 . . . . . . . . . . . . . . 15  |-  U  C_  X
16 simplrl 736 . . . . . . . . . . . . . . 15  |-  ( ( ( R  e. NrmRing  /\  (
x  e.  U  /\  r  e.  RR+ ) )  /\  y  e.  U
)  ->  x  e.  U )
1715, 16sseldi 3178 . . . . . . . . . . . . . 14  |-  ( ( ( R  e. NrmRing  /\  (
x  e.  U  /\  r  e.  RR+ ) )  /\  y  e.  U
)  ->  x  e.  X )
18 simpr 447 . . . . . . . . . . . . . . 15  |-  ( ( ( R  e. NrmRing  /\  (
x  e.  U  /\  r  e.  RR+ ) )  /\  y  e.  U
)  ->  y  e.  U )
1915, 18sseldi 3178 . . . . . . . . . . . . . 14  |-  ( ( ( R  e. NrmRing  /\  (
x  e.  U  /\  r  e.  RR+ ) )  /\  y  e.  U
)  ->  y  e.  X )
20 eqid 2283 . . . . . . . . . . . . . . 15  |-  ( 1r
`  R )  =  ( 1r `  R
)
21 eqid 2283 . . . . . . . . . . . . . . 15  |-  ( 0g
`  R )  =  ( 0g `  R
)
2214, 20, 21rng1eq0 15379 . . . . . . . . . . . . . 14  |-  ( ( R  e.  Ring  /\  x  e.  X  /\  y  e.  X )  ->  (
( 1r `  R
)  =  ( 0g
`  R )  ->  x  =  y )
)
2313, 17, 19, 22syl3anc 1182 . . . . . . . . . . . . 13  |-  ( ( ( R  e. NrmRing  /\  (
x  e.  U  /\  r  e.  RR+ ) )  /\  y  e.  U
)  ->  ( ( 1r `  R )  =  ( 0g `  R
)  ->  x  =  y ) )
24 eqid 2283 . . . . . . . . . . . . . . . 16  |-  ( I `
 y )  =  ( I `  y
)
25 nrgngp 18173 . . . . . . . . . . . . . . . . . . 19  |-  ( R  e. NrmRing  ->  R  e. NrmGrp )
26 ngpms 18122 . . . . . . . . . . . . . . . . . . 19  |-  ( R  e. NrmGrp  ->  R  e.  MetSp )
27 msxms 18000 . . . . . . . . . . . . . . . . . . 19  |-  ( R  e.  MetSp  ->  R  e.  *
MetSp )
2825, 26, 273syl 18 . . . . . . . . . . . . . . . . . 18  |-  ( R  e. NrmRing  ->  R  e.  * MetSp )
2928ad2antrr 706 . . . . . . . . . . . . . . . . 17  |-  ( ( ( R  e. NrmRing  /\  (
x  e.  U  /\  r  e.  RR+ ) )  /\  y  e.  U
)  ->  R  e.  *
MetSp )
309adantr 451 . . . . . . . . . . . . . . . . . . 19  |-  ( ( R  e. NrmRing  /\  (
x  e.  U  /\  r  e.  RR+ ) )  ->  I : U --> U )
31 ffvelrn 5663 . . . . . . . . . . . . . . . . . . 19  |-  ( ( I : U --> U  /\  y  e.  U )  ->  ( I `  y
)  e.  U )
3230, 31sylan 457 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( R  e. NrmRing  /\  (
x  e.  U  /\  r  e.  RR+ ) )  /\  y  e.  U
)  ->  ( I `  y )  e.  U
)
3315, 32sseldi 3178 . . . . . . . . . . . . . . . . 17  |-  ( ( ( R  e. NrmRing  /\  (
x  e.  U  /\  r  e.  RR+ ) )  /\  y  e.  U
)  ->  ( I `  y )  e.  X
)
34 eqid 2283 . . . . . . . . . . . . . . . . . 18  |-  ( dist `  R )  =  (
dist `  R )
3514, 34xmseq0 18010 . . . . . . . . . . . . . . . . 17  |-  ( ( R  e.  * MetSp  /\  ( I `  y
)  e.  X  /\  ( I `  y
)  e.  X )  ->  ( ( ( I `  y ) ( dist `  R
) ( I `  y ) )  =  0  <->  ( I `  y )  =  ( I `  y ) ) )
3629, 33, 33, 35syl3anc 1182 . . . . . . . . . . . . . . . 16  |-  ( ( ( R  e. NrmRing  /\  (
x  e.  U  /\  r  e.  RR+ ) )  /\  y  e.  U
)  ->  ( (
( I `  y
) ( dist `  R
) ( I `  y ) )  =  0  <->  ( I `  y )  =  ( I `  y ) ) )
3724, 36mpbiri 224 . . . . . . . . . . . . . . 15  |-  ( ( ( R  e. NrmRing  /\  (
x  e.  U  /\  r  e.  RR+ ) )  /\  y  e.  U
)  ->  ( (
I `  y )
( dist `  R )
( I `  y
) )  =  0 )
38 simplrr 737 . . . . . . . . . . . . . . . 16  |-  ( ( ( R  e. NrmRing  /\  (
x  e.  U  /\  r  e.  RR+ ) )  /\  y  e.  U
)  ->  r  e.  RR+ )
3938rpgt0d 10393 . . . . . . . . . . . . . . 15  |-  ( ( ( R  e. NrmRing  /\  (
x  e.  U  /\  r  e.  RR+ ) )  /\  y  e.  U
)  ->  0  <  r )
4037, 39eqbrtrd 4043 . . . . . . . . . . . . . 14  |-  ( ( ( R  e. NrmRing  /\  (
x  e.  U  /\  r  e.  RR+ ) )  /\  y  e.  U
)  ->  ( (
I `  y )
( dist `  R )
( I `  y
) )  <  r
)
41 fveq2 5525 . . . . . . . . . . . . . . . 16  |-  ( x  =  y  ->  (
I `  x )  =  ( I `  y ) )
4241oveq1d 5873 . . . . . . . . . . . . . . 15  |-  ( x  =  y  ->  (
( I `  x
) ( dist `  R
) ( I `  y ) )  =  ( ( I `  y ) ( dist `  R ) ( I `
 y ) ) )
4342breq1d 4033 . . . . . . . . . . . . . 14  |-  ( x  =  y  ->  (
( ( I `  x ) ( dist `  R ) ( I `
 y ) )  <  r  <->  ( (
I `  y )
( dist `  R )
( I `  y
) )  <  r
) )
4440, 43syl5ibrcom 213 . . . . . . . . . . . . 13  |-  ( ( ( R  e. NrmRing  /\  (
x  e.  U  /\  r  e.  RR+ ) )  /\  y  e.  U
)  ->  ( x  =  y  ->  ( ( I `  x ) ( dist `  R
) ( I `  y ) )  < 
r ) )
4523, 44syld 40 . . . . . . . . . . . 12  |-  ( ( ( R  e. NrmRing  /\  (
x  e.  U  /\  r  e.  RR+ ) )  /\  y  e.  U
)  ->  ( ( 1r `  R )  =  ( 0g `  R
)  ->  ( (
I `  x )
( dist `  R )
( I `  y
) )  <  r
) )
4645imp 418 . . . . . . . . . . 11  |-  ( ( ( ( R  e. NrmRing  /\  ( x  e.  U  /\  r  e.  RR+ )
)  /\  y  e.  U )  /\  ( 1r `  R )  =  ( 0g `  R
) )  ->  (
( I `  x
) ( dist `  R
) ( I `  y ) )  < 
r )
4746an32s 779 . . . . . . . . . 10  |-  ( ( ( ( R  e. NrmRing  /\  ( x  e.  U  /\  r  e.  RR+ )
)  /\  ( 1r `  R )  =  ( 0g `  R ) )  /\  y  e.  U )  ->  (
( I `  x
) ( dist `  R
) ( I `  y ) )  < 
r )
4847a1d 22 . . . . . . . . 9  |-  ( ( ( ( R  e. NrmRing  /\  ( x  e.  U  /\  r  e.  RR+ )
)  /\  ( 1r `  R )  =  ( 0g `  R ) )  /\  y  e.  U )  ->  (
( x ( dist `  R ) y )  <  s  ->  (
( I `  x
) ( dist `  R
) ( I `  y ) )  < 
r ) )
4948ralrimiva 2626 . . . . . . . 8  |-  ( ( ( R  e. NrmRing  /\  (
x  e.  U  /\  r  e.  RR+ ) )  /\  ( 1r `  R )  =  ( 0g `  R ) )  ->  A. y  e.  U  ( (
x ( dist `  R
) y )  < 
s  ->  ( (
I `  x )
( dist `  R )
( I `  y
) )  <  r
) )
5049ralrimivw 2627 . . . . . . 7  |-  ( ( ( R  e. NrmRing  /\  (
x  e.  U  /\  r  e.  RR+ ) )  /\  ( 1r `  R )  =  ( 0g `  R ) )  ->  A. s  e.  RR+  A. y  e.  U  ( ( x ( dist `  R
) y )  < 
s  ->  ( (
I `  x )
( dist `  R )
( I `  y
) )  <  r
) )
51 r19.2z 3543 . . . . . . 7  |-  ( (
RR+  =/=  (/)  /\  A. s  e.  RR+  A. y  e.  U  ( (
x ( dist `  R
) y )  < 
s  ->  ( (
I `  x )
( dist `  R )
( I `  y
) )  <  r
) )  ->  E. s  e.  RR+  A. y  e.  U  ( ( x ( dist `  R
) y )  < 
s  ->  ( (
I `  x )
( dist `  R )
( I `  y
) )  <  r
) )
5212, 50, 51sylancr 644 . . . . . 6  |-  ( ( ( R  e. NrmRing  /\  (
x  e.  U  /\  r  e.  RR+ ) )  /\  ( 1r `  R )  =  ( 0g `  R ) )  ->  E. s  e.  RR+  A. y  e.  U  ( ( x ( dist `  R
) y )  < 
s  ->  ( (
I `  x )
( dist `  R )
( I `  y
) )  <  r
) )
53 eqid 2283 . . . . . . 7  |-  ( norm `  R )  =  (
norm `  R )
54 simpll 730 . . . . . . 7  |-  ( ( ( R  e. NrmRing  /\  (
x  e.  U  /\  r  e.  RR+ ) )  /\  ( 1r `  R )  =/=  ( 0g `  R ) )  ->  R  e. NrmRing )
551ad2antrr 706 . . . . . . . 8  |-  ( ( ( R  e. NrmRing  /\  (
x  e.  U  /\  r  e.  RR+ ) )  /\  ( 1r `  R )  =/=  ( 0g `  R ) )  ->  R  e.  Ring )
56 simpr 447 . . . . . . . 8  |-  ( ( ( R  e. NrmRing  /\  (
x  e.  U  /\  r  e.  RR+ ) )  /\  ( 1r `  R )  =/=  ( 0g `  R ) )  ->  ( 1r `  R )  =/=  ( 0g `  R ) )
5720, 21isnzr 16011 . . . . . . . 8  |-  ( R  e. NzRing 
<->  ( R  e.  Ring  /\  ( 1r `  R
)  =/=  ( 0g
`  R ) ) )
5855, 56, 57sylanbrc 645 . . . . . . 7  |-  ( ( ( R  e. NrmRing  /\  (
x  e.  U  /\  r  e.  RR+ ) )  /\  ( 1r `  R )  =/=  ( 0g `  R ) )  ->  R  e. NzRing )
59 simplrl 736 . . . . . . 7  |-  ( ( ( R  e. NrmRing  /\  (
x  e.  U  /\  r  e.  RR+ ) )  /\  ( 1r `  R )  =/=  ( 0g `  R ) )  ->  x  e.  U
)
60 simplrr 737 . . . . . . 7  |-  ( ( ( R  e. NrmRing  /\  (
x  e.  U  /\  r  e.  RR+ ) )  /\  ( 1r `  R )  =/=  ( 0g `  R ) )  ->  r  e.  RR+ )
61 eqid 2283 . . . . . . 7  |-  ( if ( 1  <_  (
( ( norm `  R
) `  x )  x.  r ) ,  1 ,  ( ( (
norm `  R ) `  x )  x.  r
) )  x.  (
( ( norm `  R
) `  x )  /  2 ) )  =  ( if ( 1  <_  ( (
( norm `  R ) `  x )  x.  r
) ,  1 ,  ( ( ( norm `  R ) `  x
)  x.  r ) )  x.  ( ( ( norm `  R
) `  x )  /  2 ) )
6214, 2, 6, 53, 34, 54, 58, 59, 60, 61nrginvrcnlem 18201 . . . . . 6  |-  ( ( ( R  e. NrmRing  /\  (
x  e.  U  /\  r  e.  RR+ ) )  /\  ( 1r `  R )  =/=  ( 0g `  R ) )  ->  E. s  e.  RR+  A. y  e.  U  ( ( x ( dist `  R ) y )  <  s  ->  (
( I `  x
) ( dist `  R
) ( I `  y ) )  < 
r ) )
6352, 62pm2.61dane 2524 . . . . 5  |-  ( ( R  e. NrmRing  /\  (
x  e.  U  /\  r  e.  RR+ ) )  ->  E. s  e.  RR+  A. y  e.  U  ( ( x ( dist `  R ) y )  <  s  ->  (
( I `  x
) ( dist `  R
) ( I `  y ) )  < 
r ) )
6416, 18ovresd 5988 . . . . . . . . 9  |-  ( ( ( R  e. NrmRing  /\  (
x  e.  U  /\  r  e.  RR+ ) )  /\  y  e.  U
)  ->  ( x
( ( dist `  R
)  |`  ( U  X.  U ) ) y )  =  ( x ( dist `  R
) y ) )
6564breq1d 4033 . . . . . . . 8  |-  ( ( ( R  e. NrmRing  /\  (
x  e.  U  /\  r  e.  RR+ ) )  /\  y  e.  U
)  ->  ( (
x ( ( dist `  R )  |`  ( U  X.  U ) ) y )  <  s  <->  ( x ( dist `  R
) y )  < 
s ) )
66 simpl 443 . . . . . . . . . . . 12  |-  ( ( x  e.  U  /\  r  e.  RR+ )  ->  x  e.  U )
67 ffvelrn 5663 . . . . . . . . . . . 12  |-  ( ( I : U --> U  /\  x  e.  U )  ->  ( I `  x
)  e.  U )
689, 66, 67syl2an 463 . . . . . . . . . . 11  |-  ( ( R  e. NrmRing  /\  (
x  e.  U  /\  r  e.  RR+ ) )  ->  ( I `  x )  e.  U
)
6968adantr 451 . . . . . . . . . 10  |-  ( ( ( R  e. NrmRing  /\  (
x  e.  U  /\  r  e.  RR+ ) )  /\  y  e.  U
)  ->  ( I `  x )  e.  U
)
7069, 32ovresd 5988 . . . . . . . . 9  |-  ( ( ( R  e. NrmRing  /\  (
x  e.  U  /\  r  e.  RR+ ) )  /\  y  e.  U
)  ->  ( (
I `  x )
( ( dist `  R
)  |`  ( U  X.  U ) ) ( I `  y ) )  =  ( ( I `  x ) ( dist `  R
) ( I `  y ) ) )
7170breq1d 4033 . . . . . . . 8  |-  ( ( ( R  e. NrmRing  /\  (
x  e.  U  /\  r  e.  RR+ ) )  /\  y  e.  U
)  ->  ( (
( I `  x
) ( ( dist `  R )  |`  ( U  X.  U ) ) ( I `  y
) )  <  r  <->  ( ( I `  x
) ( dist `  R
) ( I `  y ) )  < 
r ) )
7265, 71imbi12d 311 . . . . . . 7  |-  ( ( ( R  e. NrmRing  /\  (
x  e.  U  /\  r  e.  RR+ ) )  /\  y  e.  U
)  ->  ( (
( x ( (
dist `  R )  |`  ( U  X.  U
) ) y )  <  s  ->  (
( I `  x
) ( ( dist `  R )  |`  ( U  X.  U ) ) ( I `  y
) )  <  r
)  <->  ( ( x ( dist `  R
) y )  < 
s  ->  ( (
I `  x )
( dist `  R )
( I `  y
) )  <  r
) ) )
7372ralbidva 2559 . . . . . 6  |-  ( ( R  e. NrmRing  /\  (
x  e.  U  /\  r  e.  RR+ ) )  ->  ( A. y  e.  U  ( (
x ( ( dist `  R )  |`  ( U  X.  U ) ) y )  <  s  ->  ( ( I `  x ) ( (
dist `  R )  |`  ( U  X.  U
) ) ( I `
 y ) )  <  r )  <->  A. y  e.  U  ( (
x ( dist `  R
) y )  < 
s  ->  ( (
I `  x )
( dist `  R )
( I `  y
) )  <  r
) ) )
7473rexbidv 2564 . . . . 5  |-  ( ( R  e. NrmRing  /\  (
x  e.  U  /\  r  e.  RR+ ) )  ->  ( E. s  e.  RR+  A. y  e.  U  ( ( x ( ( dist `  R
)  |`  ( U  X.  U ) ) y )  <  s  -> 
( ( I `  x ) ( (
dist `  R )  |`  ( U  X.  U
) ) ( I `
 y ) )  <  r )  <->  E. s  e.  RR+  A. y  e.  U  ( ( x ( dist `  R
) y )  < 
s  ->  ( (
I `  x )
( dist `  R )
( I `  y
) )  <  r
) ) )
7563, 74mpbird 223 . . . 4  |-  ( ( R  e. NrmRing  /\  (
x  e.  U  /\  r  e.  RR+ ) )  ->  E. s  e.  RR+  A. y  e.  U  ( ( x ( (
dist `  R )  |`  ( U  X.  U
) ) y )  <  s  ->  (
( I `  x
) ( ( dist `  R )  |`  ( U  X.  U ) ) ( I `  y
) )  <  r
) )
7675ralrimivva 2635 . . 3  |-  ( R  e. NrmRing  ->  A. x  e.  U  A. r  e.  RR+  E. s  e.  RR+  A. y  e.  U  ( ( x ( ( dist `  R
)  |`  ( U  X.  U ) ) y )  <  s  -> 
( ( I `  x ) ( (
dist `  R )  |`  ( U  X.  U
) ) ( I `
 y ) )  <  r ) )
77 xpss12 4792 . . . . . . 7  |-  ( ( U  C_  X  /\  U  C_  X )  -> 
( U  X.  U
)  C_  ( X  X.  X ) )
7815, 15, 77mp2an 653 . . . . . 6  |-  ( U  X.  U )  C_  ( X  X.  X
)
79 resabs1 4984 . . . . . 6  |-  ( ( U  X.  U ) 
C_  ( X  X.  X )  ->  (
( ( dist `  R
)  |`  ( X  X.  X ) )  |`  ( U  X.  U
) )  =  ( ( dist `  R
)  |`  ( U  X.  U ) ) )
8078, 79ax-mp 8 . . . . 5  |-  ( ( ( dist `  R
)  |`  ( X  X.  X ) )  |`  ( U  X.  U
) )  =  ( ( dist `  R
)  |`  ( U  X.  U ) )
8125, 26syl 15 . . . . . . 7  |-  ( R  e. NrmRing  ->  R  e.  MetSp )
82 eqid 2283 . . . . . . . 8  |-  ( (
dist `  R )  |`  ( X  X.  X
) )  =  ( ( dist `  R
)  |`  ( X  X.  X ) )
8314, 82xmsxmet 18002 . . . . . . 7  |-  ( R  e.  * MetSp  ->  (
( dist `  R )  |`  ( X  X.  X
) )  e.  ( * Met `  X
) )
8481, 27, 833syl 18 . . . . . 6  |-  ( R  e. NrmRing  ->  ( ( dist `  R )  |`  ( X  X.  X ) )  e.  ( * Met `  X ) )
85 xmetres2 17925 . . . . . 6  |-  ( ( ( ( dist `  R
)  |`  ( X  X.  X ) )  e.  ( * Met `  X
)  /\  U  C_  X
)  ->  ( (
( dist `  R )  |`  ( X  X.  X
) )  |`  ( U  X.  U ) )  e.  ( * Met `  U ) )
8684, 15, 85sylancl 643 . . . . 5  |-  ( R  e. NrmRing  ->  ( ( (
dist `  R )  |`  ( X  X.  X
) )  |`  ( U  X.  U ) )  e.  ( * Met `  U ) )
8780, 86syl5eqelr 2368 . . . 4  |-  ( R  e. NrmRing  ->  ( ( dist `  R )  |`  ( U  X.  U ) )  e.  ( * Met `  U ) )
88 eqid 2283 . . . . 5  |-  ( MetOpen `  ( ( dist `  R
)  |`  ( U  X.  U ) ) )  =  ( MetOpen `  (
( dist `  R )  |`  ( U  X.  U
) ) )
8988, 88metcn 18089 . . . 4  |-  ( ( ( ( dist `  R
)  |`  ( U  X.  U ) )  e.  ( * Met `  U
)  /\  ( ( dist `  R )  |`  ( U  X.  U
) )  e.  ( * Met `  U
) )  ->  (
I  e.  ( (
MetOpen `  ( ( dist `  R )  |`  ( U  X.  U ) ) )  Cn  ( MetOpen `  ( ( dist `  R
)  |`  ( U  X.  U ) ) ) )  <->  ( I : U --> U  /\  A. x  e.  U  A. r  e.  RR+  E. s  e.  RR+  A. y  e.  U  ( ( x ( ( dist `  R
)  |`  ( U  X.  U ) ) y )  <  s  -> 
( ( I `  x ) ( (
dist `  R )  |`  ( U  X.  U
) ) ( I `
 y ) )  <  r ) ) ) )
9087, 87, 89syl2anc 642 . . 3  |-  ( R  e. NrmRing  ->  ( I  e.  ( ( MetOpen `  (
( dist `  R )  |`  ( U  X.  U
) ) )  Cn  ( MetOpen `  ( ( dist `  R )  |`  ( U  X.  U
) ) ) )  <-> 
( I : U --> U  /\  A. x  e.  U  A. r  e.  RR+  E. s  e.  RR+  A. y  e.  U  ( ( x ( (
dist `  R )  |`  ( U  X.  U
) ) y )  <  s  ->  (
( I `  x
) ( ( dist `  R )  |`  ( U  X.  U ) ) ( I `  y
) )  <  r
) ) ) )
919, 76, 90mpbir2and 888 . 2  |-  ( R  e. NrmRing  ->  I  e.  ( ( MetOpen `  ( ( dist `  R )  |`  ( U  X.  U
) ) )  Cn  ( MetOpen `  ( ( dist `  R )  |`  ( U  X.  U
) ) ) ) )
92 nrginvrcn.j . . . . . . 7  |-  J  =  ( TopOpen `  R )
9392, 14, 82mstopn 17998 . . . . . 6  |-  ( R  e.  MetSp  ->  J  =  ( MetOpen `  ( ( dist `  R )  |`  ( X  X.  X
) ) ) )
9425, 26, 933syl 18 . . . . 5  |-  ( R  e. NrmRing  ->  J  =  (
MetOpen `  ( ( dist `  R )  |`  ( X  X.  X ) ) ) )
9594oveq1d 5873 . . . 4  |-  ( R  e. NrmRing  ->  ( Jt  U )  =  ( ( MetOpen `  ( ( dist `  R
)  |`  ( X  X.  X ) ) )t  U ) )
9680eqcomi 2287 . . . . . 6  |-  ( (
dist `  R )  |`  ( U  X.  U
) )  =  ( ( ( dist `  R
)  |`  ( X  X.  X ) )  |`  ( U  X.  U
) )
97 eqid 2283 . . . . . 6  |-  ( MetOpen `  ( ( dist `  R
)  |`  ( X  X.  X ) ) )  =  ( MetOpen `  (
( dist `  R )  |`  ( X  X.  X
) ) )
9896, 97, 88metrest 18070 . . . . 5  |-  ( ( ( ( dist `  R
)  |`  ( X  X.  X ) )  e.  ( * Met `  X
)  /\  U  C_  X
)  ->  ( ( MetOpen
`  ( ( dist `  R )  |`  ( X  X.  X ) ) )t  U )  =  (
MetOpen `  ( ( dist `  R )  |`  ( U  X.  U ) ) ) )
9984, 15, 98sylancl 643 . . . 4  |-  ( R  e. NrmRing  ->  ( ( MetOpen `  ( ( dist `  R
)  |`  ( X  X.  X ) ) )t  U )  =  ( MetOpen `  ( ( dist `  R
)  |`  ( U  X.  U ) ) ) )
10095, 99eqtrd 2315 . . 3  |-  ( R  e. NrmRing  ->  ( Jt  U )  =  ( MetOpen `  (
( dist `  R )  |`  ( U  X.  U
) ) ) )
101100, 100oveq12d 5876 . 2  |-  ( R  e. NrmRing  ->  ( ( Jt  U )  Cn  ( Jt  U ) )  =  ( ( MetOpen `  ( ( dist `  R )  |`  ( U  X.  U
) ) )  Cn  ( MetOpen `  ( ( dist `  R )  |`  ( U  X.  U
) ) ) ) )
10291, 101eleqtrrd 2360 1  |-  ( R  e. NrmRing  ->  I  e.  ( ( Jt  U )  Cn  ( Jt  U ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544    C_ wss 3152   (/)c0 3455   ifcif 3565   class class class wbr 4023    X. cxp 4687    |` cres 4691   -->wf 5251   ` cfv 5255  (class class class)co 5858   0cc0 8737   1c1 8738    x. cmul 8742    < clt 8867    <_ cle 8868    / cdiv 9423   2c2 9795   RR+crp 10354   Basecbs 13148   ↾s cress 13149   distcds 13217   ↾t crest 13325   TopOpenctopn 13326   0gc0g 13400   Grpcgrp 14362  mulGrpcmgp 15325   Ringcrg 15337   1rcur 15339  Unitcui 15421   invrcinvr 15453  NzRingcnzr 16009   * Metcxmt 16369   MetOpencmopn 16372    Cn ccn 16954   *
MetSpcxme 17882   MetSpcmt 17883   normcnm 18099  NrmGrpcngp 18100  NrmRingcnrg 18102
This theorem is referenced by:  nrgtdrg  18203
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-tpos 6234  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-fz 10783  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-tset 13227  df-ple 13228  df-ds 13230  df-rest 13327  df-topgen 13344  df-xrs 13403  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490  df-sbg 14491  df-mgp 15326  df-rng 15340  df-ur 15342  df-oppr 15405  df-dvdsr 15423  df-unit 15424  df-invr 15454  df-abv 15582  df-nzr 16010  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cn 16957  df-cnp 16958  df-xms 17885  df-ms 17886  df-nm 18105  df-ngp 18106  df-nrg 18108
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