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Theorem nrginvrcn 18719
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 18691 . . . 4  |-  ( R  e. NrmRing  ->  R  e.  Ring )
2 nrginvrcn.u . . . . 5  |-  U  =  (Unit `  R )
3 eqid 2435 . . . . 5  |-  ( (mulGrp `  R )s  U )  =  ( (mulGrp `  R )s  U
)
42, 3unitgrp 15764 . . . 4  |-  ( R  e.  Ring  ->  ( (mulGrp `  R )s  U )  e.  Grp )
52, 3unitgrpbas 15763 . . . . 5  |-  U  =  ( Base `  (
(mulGrp `  R )s  U
) )
6 nrginvrcn.i . . . . . 6  |-  I  =  ( invr `  R
)
72, 3, 6invrfval 15770 . . . . 5  |-  I  =  ( inv g `  ( (mulGrp `  R )s  U
) )
85, 7grpinvf 14841 . . . 4  |-  ( ( (mulGrp `  R )s  U
)  e.  Grp  ->  I : U --> U )
91, 4, 83syl 19 . . 3  |-  ( R  e. NrmRing  ->  I : U --> U )
10 1rp 10608 . . . . . . . 8  |-  1  e.  RR+
11 ne0i 3626 . . . . . . . 8  |-  ( 1  e.  RR+  ->  RR+  =/=  (/) )
1210, 11ax-mp 8 . . . . . . 7  |-  RR+  =/=  (/)
131ad2antrr 707 . . . . . . . . . . . . . 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 15757 . . . . . . . . . . . . . . 15  |-  U  C_  X
16 simplrl 737 . . . . . . . . . . . . . . 15  |-  ( ( ( R  e. NrmRing  /\  (
x  e.  U  /\  r  e.  RR+ ) )  /\  y  e.  U
)  ->  x  e.  U )
1715, 16sseldi 3338 . . . . . . . . . . . . . 14  |-  ( ( ( R  e. NrmRing  /\  (
x  e.  U  /\  r  e.  RR+ ) )  /\  y  e.  U
)  ->  x  e.  X )
18 simpr 448 . . . . . . . . . . . . . . 15  |-  ( ( ( R  e. NrmRing  /\  (
x  e.  U  /\  r  e.  RR+ ) )  /\  y  e.  U
)  ->  y  e.  U )
1915, 18sseldi 3338 . . . . . . . . . . . . . 14  |-  ( ( ( R  e. NrmRing  /\  (
x  e.  U  /\  r  e.  RR+ ) )  /\  y  e.  U
)  ->  y  e.  X )
20 eqid 2435 . . . . . . . . . . . . . . 15  |-  ( 1r
`  R )  =  ( 1r `  R
)
21 eqid 2435 . . . . . . . . . . . . . . 15  |-  ( 0g
`  R )  =  ( 0g `  R
)
2214, 20, 21rng1eq0 15694 . . . . . . . . . . . . . 14  |-  ( ( R  e.  Ring  /\  x  e.  X  /\  y  e.  X )  ->  (
( 1r `  R
)  =  ( 0g
`  R )  ->  x  =  y )
)
2313, 17, 19, 22syl3anc 1184 . . . . . . . . . . . . 13  |-  ( ( ( R  e. NrmRing  /\  (
x  e.  U  /\  r  e.  RR+ ) )  /\  y  e.  U
)  ->  ( ( 1r `  R )  =  ( 0g `  R
)  ->  x  =  y ) )
24 eqid 2435 . . . . . . . . . . . . . . . 16  |-  ( I `
 y )  =  ( I `  y
)
25 nrgngp 18690 . . . . . . . . . . . . . . . . . . 19  |-  ( R  e. NrmRing  ->  R  e. NrmGrp )
26 ngpms 18639 . . . . . . . . . . . . . . . . . . 19  |-  ( R  e. NrmGrp  ->  R  e.  MetSp )
27 msxms 18476 . . . . . . . . . . . . . . . . . . 19  |-  ( R  e.  MetSp  ->  R  e.  *
MetSp )
2825, 26, 273syl 19 . . . . . . . . . . . . . . . . . 18  |-  ( R  e. NrmRing  ->  R  e.  * MetSp )
2928ad2antrr 707 . . . . . . . . . . . . . . . . 17  |-  ( ( ( R  e. NrmRing  /\  (
x  e.  U  /\  r  e.  RR+ ) )  /\  y  e.  U
)  ->  R  e.  *
MetSp )
309adantr 452 . . . . . . . . . . . . . . . . . . 19  |-  ( ( R  e. NrmRing  /\  (
x  e.  U  /\  r  e.  RR+ ) )  ->  I : U --> U )
3130ffvelrnda 5862 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( R  e. NrmRing  /\  (
x  e.  U  /\  r  e.  RR+ ) )  /\  y  e.  U
)  ->  ( I `  y )  e.  U
)
3215, 31sseldi 3338 . . . . . . . . . . . . . . . . 17  |-  ( ( ( R  e. NrmRing  /\  (
x  e.  U  /\  r  e.  RR+ ) )  /\  y  e.  U
)  ->  ( I `  y )  e.  X
)
33 eqid 2435 . . . . . . . . . . . . . . . . . 18  |-  ( dist `  R )  =  (
dist `  R )
3414, 33xmseq0 18486 . . . . . . . . . . . . . . . . 17  |-  ( ( R  e.  * MetSp  /\  ( I `  y
)  e.  X  /\  ( I `  y
)  e.  X )  ->  ( ( ( I `  y ) ( dist `  R
) ( I `  y ) )  =  0  <->  ( I `  y )  =  ( I `  y ) ) )
3529, 32, 32, 34syl3anc 1184 . . . . . . . . . . . . . . . 16  |-  ( ( ( R  e. NrmRing  /\  (
x  e.  U  /\  r  e.  RR+ ) )  /\  y  e.  U
)  ->  ( (
( I `  y
) ( dist `  R
) ( I `  y ) )  =  0  <->  ( I `  y )  =  ( I `  y ) ) )
3624, 35mpbiri 225 . . . . . . . . . . . . . . 15  |-  ( ( ( R  e. NrmRing  /\  (
x  e.  U  /\  r  e.  RR+ ) )  /\  y  e.  U
)  ->  ( (
I `  y )
( dist `  R )
( I `  y
) )  =  0 )
37 simplrr 738 . . . . . . . . . . . . . . . 16  |-  ( ( ( R  e. NrmRing  /\  (
x  e.  U  /\  r  e.  RR+ ) )  /\  y  e.  U
)  ->  r  e.  RR+ )
3837rpgt0d 10643 . . . . . . . . . . . . . . 15  |-  ( ( ( R  e. NrmRing  /\  (
x  e.  U  /\  r  e.  RR+ ) )  /\  y  e.  U
)  ->  0  <  r )
3936, 38eqbrtrd 4224 . . . . . . . . . . . . . 14  |-  ( ( ( R  e. NrmRing  /\  (
x  e.  U  /\  r  e.  RR+ ) )  /\  y  e.  U
)  ->  ( (
I `  y )
( dist `  R )
( I `  y
) )  <  r
)
40 fveq2 5720 . . . . . . . . . . . . . . . 16  |-  ( x  =  y  ->  (
I `  x )  =  ( I `  y ) )
4140oveq1d 6088 . . . . . . . . . . . . . . 15  |-  ( x  =  y  ->  (
( I `  x
) ( dist `  R
) ( I `  y ) )  =  ( ( I `  y ) ( dist `  R ) ( I `
 y ) ) )
4241breq1d 4214 . . . . . . . . . . . . . 14  |-  ( x  =  y  ->  (
( ( I `  x ) ( dist `  R ) ( I `
 y ) )  <  r  <->  ( (
I `  y )
( dist `  R )
( I `  y
) )  <  r
) )
4339, 42syl5ibrcom 214 . . . . . . . . . . . . 13  |-  ( ( ( R  e. NrmRing  /\  (
x  e.  U  /\  r  e.  RR+ ) )  /\  y  e.  U
)  ->  ( x  =  y  ->  ( ( I `  x ) ( dist `  R
) ( I `  y ) )  < 
r ) )
4423, 43syld 42 . . . . . . . . . . . 12  |-  ( ( ( R  e. NrmRing  /\  (
x  e.  U  /\  r  e.  RR+ ) )  /\  y  e.  U
)  ->  ( ( 1r `  R )  =  ( 0g `  R
)  ->  ( (
I `  x )
( dist `  R )
( I `  y
) )  <  r
) )
4544imp 419 . . . . . . . . . . 11  |-  ( ( ( ( R  e. NrmRing  /\  ( x  e.  U  /\  r  e.  RR+ )
)  /\  y  e.  U )  /\  ( 1r `  R )  =  ( 0g `  R
) )  ->  (
( I `  x
) ( dist `  R
) ( I `  y ) )  < 
r )
4645an32s 780 . . . . . . . . . 10  |-  ( ( ( ( R  e. NrmRing  /\  ( x  e.  U  /\  r  e.  RR+ )
)  /\  ( 1r `  R )  =  ( 0g `  R ) )  /\  y  e.  U )  ->  (
( I `  x
) ( dist `  R
) ( I `  y ) )  < 
r )
4746a1d 23 . . . . . . . . 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 ) )
4847ralrimiva 2781 . . . . . . . 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
) )
4948ralrimivw 2782 . . . . . . 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
) )
50 r19.2z 3709 . . . . . . 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
) )
5112, 49, 50sylancr 645 . . . . . 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
) )
52 eqid 2435 . . . . . . 7  |-  ( norm `  R )  =  (
norm `  R )
53 simpll 731 . . . . . . 7  |-  ( ( ( R  e. NrmRing  /\  (
x  e.  U  /\  r  e.  RR+ ) )  /\  ( 1r `  R )  =/=  ( 0g `  R ) )  ->  R  e. NrmRing )
541ad2antrr 707 . . . . . . . 8  |-  ( ( ( R  e. NrmRing  /\  (
x  e.  U  /\  r  e.  RR+ ) )  /\  ( 1r `  R )  =/=  ( 0g `  R ) )  ->  R  e.  Ring )
55 simpr 448 . . . . . . . 8  |-  ( ( ( R  e. NrmRing  /\  (
x  e.  U  /\  r  e.  RR+ ) )  /\  ( 1r `  R )  =/=  ( 0g `  R ) )  ->  ( 1r `  R )  =/=  ( 0g `  R ) )
5620, 21isnzr 16322 . . . . . . . 8  |-  ( R  e. NzRing 
<->  ( R  e.  Ring  /\  ( 1r `  R
)  =/=  ( 0g
`  R ) ) )
5754, 55, 56sylanbrc 646 . . . . . . 7  |-  ( ( ( R  e. NrmRing  /\  (
x  e.  U  /\  r  e.  RR+ ) )  /\  ( 1r `  R )  =/=  ( 0g `  R ) )  ->  R  e. NzRing )
58 simplrl 737 . . . . . . 7  |-  ( ( ( R  e. NrmRing  /\  (
x  e.  U  /\  r  e.  RR+ ) )  /\  ( 1r `  R )  =/=  ( 0g `  R ) )  ->  x  e.  U
)
59 simplrr 738 . . . . . . 7  |-  ( ( ( R  e. NrmRing  /\  (
x  e.  U  /\  r  e.  RR+ ) )  /\  ( 1r `  R )  =/=  ( 0g `  R ) )  ->  r  e.  RR+ )
60 eqid 2435 . . . . . . 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 ) )
6114, 2, 6, 52, 33, 53, 57, 58, 59, 60nrginvrcnlem 18718 . . . . . 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 ) )
6251, 61pm2.61dane 2676 . . . . 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 ) )
6316, 18ovresd 6206 . . . . . . . . 9  |-  ( ( ( R  e. NrmRing  /\  (
x  e.  U  /\  r  e.  RR+ ) )  /\  y  e.  U
)  ->  ( x
( ( dist `  R
)  |`  ( U  X.  U ) ) y )  =  ( x ( dist `  R
) y ) )
6463breq1d 4214 . . . . . . . 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 ) )
65 simpl 444 . . . . . . . . . . . 12  |-  ( ( x  e.  U  /\  r  e.  RR+ )  ->  x  e.  U )
66 ffvelrn 5860 . . . . . . . . . . . 12  |-  ( ( I : U --> U  /\  x  e.  U )  ->  ( I `  x
)  e.  U )
679, 65, 66syl2an 464 . . . . . . . . . . 11  |-  ( ( R  e. NrmRing  /\  (
x  e.  U  /\  r  e.  RR+ ) )  ->  ( I `  x )  e.  U
)
6867adantr 452 . . . . . . . . . 10  |-  ( ( ( R  e. NrmRing  /\  (
x  e.  U  /\  r  e.  RR+ ) )  /\  y  e.  U
)  ->  ( I `  x )  e.  U
)
6968, 31ovresd 6206 . . . . . . . . 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 ) ) )
7069breq1d 4214 . . . . . . . 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 ) )
7164, 70imbi12d 312 . . . . . . 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
) ) )
7271ralbidva 2713 . . . . . 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
) ) )
7372rexbidv 2718 . . . . 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
) ) )
7462, 73mpbird 224 . . . 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
) )
7574ralrimivva 2790 . . 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 ) )
76 xpss12 4973 . . . . . . 7  |-  ( ( U  C_  X  /\  U  C_  X )  -> 
( U  X.  U
)  C_  ( X  X.  X ) )
7715, 15, 76mp2an 654 . . . . . 6  |-  ( U  X.  U )  C_  ( X  X.  X
)
78 resabs1 5167 . . . . . 6  |-  ( ( U  X.  U ) 
C_  ( X  X.  X )  ->  (
( ( dist `  R
)  |`  ( X  X.  X ) )  |`  ( U  X.  U
) )  =  ( ( dist `  R
)  |`  ( U  X.  U ) ) )
7977, 78ax-mp 8 . . . . 5  |-  ( ( ( dist `  R
)  |`  ( X  X.  X ) )  |`  ( U  X.  U
) )  =  ( ( dist `  R
)  |`  ( U  X.  U ) )
8025, 26syl 16 . . . . . . 7  |-  ( R  e. NrmRing  ->  R  e.  MetSp )
81 eqid 2435 . . . . . . . 8  |-  ( (
dist `  R )  |`  ( X  X.  X
) )  =  ( ( dist `  R
)  |`  ( X  X.  X ) )
8214, 81xmsxmet 18478 . . . . . . 7  |-  ( R  e.  * MetSp  ->  (
( dist `  R )  |`  ( X  X.  X
) )  e.  ( * Met `  X
) )
8380, 27, 823syl 19 . . . . . 6  |-  ( R  e. NrmRing  ->  ( ( dist `  R )  |`  ( X  X.  X ) )  e.  ( * Met `  X ) )
84 xmetres2 18383 . . . . . 6  |-  ( ( ( ( dist `  R
)  |`  ( X  X.  X ) )  e.  ( * Met `  X
)  /\  U  C_  X
)  ->  ( (
( dist `  R )  |`  ( X  X.  X
) )  |`  ( U  X.  U ) )  e.  ( * Met `  U ) )
8583, 15, 84sylancl 644 . . . . 5  |-  ( R  e. NrmRing  ->  ( ( (
dist `  R )  |`  ( X  X.  X
) )  |`  ( U  X.  U ) )  e.  ( * Met `  U ) )
8679, 85syl5eqelr 2520 . . . 4  |-  ( R  e. NrmRing  ->  ( ( dist `  R )  |`  ( U  X.  U ) )  e.  ( * Met `  U ) )
87 eqid 2435 . . . . 5  |-  ( MetOpen `  ( ( dist `  R
)  |`  ( U  X.  U ) ) )  =  ( MetOpen `  (
( dist `  R )  |`  ( U  X.  U
) ) )
8887, 87metcn 18565 . . . 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 ) ) ) )
8986, 86, 88syl2anc 643 . . 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
) ) ) )
909, 75, 89mpbir2and 889 . 2  |-  ( R  e. NrmRing  ->  I  e.  ( ( MetOpen `  ( ( dist `  R )  |`  ( U  X.  U
) ) )  Cn  ( MetOpen `  ( ( dist `  R )  |`  ( U  X.  U
) ) ) ) )
91 nrginvrcn.j . . . . . . 7  |-  J  =  ( TopOpen `  R )
9291, 14, 81mstopn 18474 . . . . . 6  |-  ( R  e.  MetSp  ->  J  =  ( MetOpen `  ( ( dist `  R )  |`  ( X  X.  X
) ) ) )
9325, 26, 923syl 19 . . . . 5  |-  ( R  e. NrmRing  ->  J  =  (
MetOpen `  ( ( dist `  R )  |`  ( X  X.  X ) ) ) )
9493oveq1d 6088 . . . 4  |-  ( R  e. NrmRing  ->  ( Jt  U )  =  ( ( MetOpen `  ( ( dist `  R
)  |`  ( X  X.  X ) ) )t  U ) )
9579eqcomi 2439 . . . . . 6  |-  ( (
dist `  R )  |`  ( U  X.  U
) )  =  ( ( ( dist `  R
)  |`  ( X  X.  X ) )  |`  ( U  X.  U
) )
96 eqid 2435 . . . . . 6  |-  ( MetOpen `  ( ( dist `  R
)  |`  ( X  X.  X ) ) )  =  ( MetOpen `  (
( dist `  R )  |`  ( X  X.  X
) ) )
9795, 96, 87metrest 18546 . . . . 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 ) ) ) )
9883, 15, 97sylancl 644 . . . 4  |-  ( R  e. NrmRing  ->  ( ( MetOpen `  ( ( dist `  R
)  |`  ( X  X.  X ) ) )t  U )  =  ( MetOpen `  ( ( dist `  R
)  |`  ( U  X.  U ) ) ) )
9994, 98eqtrd 2467 . . 3  |-  ( R  e. NrmRing  ->  ( Jt  U )  =  ( MetOpen `  (
( dist `  R )  |`  ( U  X.  U
) ) ) )
10099, 99oveq12d 6091 . 2  |-  ( R  e. NrmRing  ->  ( ( Jt  U )  Cn  ( Jt  U ) )  =  ( ( MetOpen `  ( ( dist `  R )  |`  ( U  X.  U
) ) )  Cn  ( MetOpen `  ( ( dist `  R )  |`  ( U  X.  U
) ) ) ) )
10190, 100eleqtrrd 2512 1  |-  ( R  e. NrmRing  ->  I  e.  ( ( Jt  U )  Cn  ( Jt  U ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   E.wrex 2698    C_ wss 3312   (/)c0 3620   ifcif 3731   class class class wbr 4204    X. cxp 4868    |` cres 4872   -->wf 5442   ` cfv 5446  (class class class)co 6073   0cc0 8982   1c1 8983    x. cmul 8987    < clt 9112    <_ cle 9113    / cdiv 9669   2c2 10041   RR+crp 10604   Basecbs 13461   ↾s cress 13462   distcds 13530   ↾t crest 13640   TopOpenctopn 13641   0gc0g 13715   Grpcgrp 14677  mulGrpcmgp 15640   Ringcrg 15652   1rcur 15654  Unitcui 15736   invrcinvr 15768  NzRingcnzr 16320   * Metcxmt 16678   MetOpencmopn 16683    Cn ccn 17280   *
MetSpcxme 18339   MetSpcmt 18340   normcnm 18616  NrmGrpcngp 18617  NrmRingcnrg 18619
This theorem is referenced by:  nrgtdrg  18720
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 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-tpos 6471  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-7 10055  df-8 10056  df-9 10057  df-10 10058  df-n0 10214  df-z 10275  df-dec 10375  df-uz 10481  df-q 10567  df-rp 10605  df-xneg 10702  df-xadd 10703  df-xmul 10704  df-fz 11036  df-seq 11316  df-exp 11375  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-struct 13463  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-mulr 13535  df-tset 13540  df-ple 13541  df-ds 13543  df-rest 13642  df-topgen 13659  df-xrs 13718  df-0g 13719  df-mnd 14682  df-grp 14804  df-minusg 14805  df-sbg 14806  df-mgp 15641  df-rng 15655  df-ur 15657  df-oppr 15720  df-dvdsr 15738  df-unit 15739  df-invr 15769  df-abv 15897  df-nzr 16321  df-psmet 16686  df-xmet 16687  df-met 16688  df-bl 16689  df-mopn 16690  df-top 16955  df-bases 16957  df-topon 16958  df-topsp 16959  df-cn 17283  df-cnp 17284  df-xms 18342  df-ms 18343  df-nm 18622  df-ngp 18623  df-nrg 18625
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