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Theorem nrginvrcn 18598
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 18570 . . . 4  |-  ( R  e. NrmRing  ->  R  e.  Ring )
2 nrginvrcn.u . . . . 5  |-  U  =  (Unit `  R )
3 eqid 2387 . . . . 5  |-  ( (mulGrp `  R )s  U )  =  ( (mulGrp `  R )s  U
)
42, 3unitgrp 15699 . . . 4  |-  ( R  e.  Ring  ->  ( (mulGrp `  R )s  U )  e.  Grp )
52, 3unitgrpbas 15698 . . . . 5  |-  U  =  ( Base `  (
(mulGrp `  R )s  U
) )
6 nrginvrcn.i . . . . . 6  |-  I  =  ( invr `  R
)
72, 3, 6invrfval 15705 . . . . 5  |-  I  =  ( inv g `  ( (mulGrp `  R )s  U
) )
85, 7grpinvf 14776 . . . 4  |-  ( ( (mulGrp `  R )s  U
)  e.  Grp  ->  I : U --> U )
91, 4, 83syl 19 . . 3  |-  ( R  e. NrmRing  ->  I : U --> U )
10 1rp 10548 . . . . . . . 8  |-  1  e.  RR+
11 ne0i 3577 . . . . . . . 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 15692 . . . . . . . . . . . . . . 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 3289 . . . . . . . . . . . . . 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 3289 . . . . . . . . . . . . . 14  |-  ( ( ( R  e. NrmRing  /\  (
x  e.  U  /\  r  e.  RR+ ) )  /\  y  e.  U
)  ->  y  e.  X )
20 eqid 2387 . . . . . . . . . . . . . . 15  |-  ( 1r
`  R )  =  ( 1r `  R
)
21 eqid 2387 . . . . . . . . . . . . . . 15  |-  ( 0g
`  R )  =  ( 0g `  R
)
2214, 20, 21rng1eq0 15629 . . . . . . . . . . . . . 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 2387 . . . . . . . . . . . . . . . 16  |-  ( I `
 y )  =  ( I `  y
)
25 nrgngp 18569 . . . . . . . . . . . . . . . . . . 19  |-  ( R  e. NrmRing  ->  R  e. NrmGrp )
26 ngpms 18518 . . . . . . . . . . . . . . . . . . 19  |-  ( R  e. NrmGrp  ->  R  e.  MetSp )
27 msxms 18374 . . . . . . . . . . . . . . . . . . 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 5809 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( R  e. NrmRing  /\  (
x  e.  U  /\  r  e.  RR+ ) )  /\  y  e.  U
)  ->  ( I `  y )  e.  U
)
3215, 31sseldi 3289 . . . . . . . . . . . . . . . . 17  |-  ( ( ( R  e. NrmRing  /\  (
x  e.  U  /\  r  e.  RR+ ) )  /\  y  e.  U
)  ->  ( I `  y )  e.  X
)
33 eqid 2387 . . . . . . . . . . . . . . . . . 18  |-  ( dist `  R )  =  (
dist `  R )
3414, 33xmseq0 18384 . . . . . . . . . . . . . . . . 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 10583 . . . . . . . . . . . . . . 15  |-  ( ( ( R  e. NrmRing  /\  (
x  e.  U  /\  r  e.  RR+ ) )  /\  y  e.  U
)  ->  0  <  r )
3936, 38eqbrtrd 4173 . . . . . . . . . . . . . 14  |-  ( ( ( R  e. NrmRing  /\  (
x  e.  U  /\  r  e.  RR+ ) )  /\  y  e.  U
)  ->  ( (
I `  y )
( dist `  R )
( I `  y
) )  <  r
)
40 fveq2 5668 . . . . . . . . . . . . . . . 16  |-  ( x  =  y  ->  (
I `  x )  =  ( I `  y ) )
4140oveq1d 6035 . . . . . . . . . . . . . . 15  |-  ( x  =  y  ->  (
( I `  x
) ( dist `  R
) ( I `  y ) )  =  ( ( I `  y ) ( dist `  R ) ( I `
 y ) ) )
4241breq1d 4163 . . . . . . . . . . . . . 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 2732 . . . . . . . 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 2733 . . . . . . 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 3660 . . . . . . 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 2387 . . . . . . 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 16257 . . . . . . . 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 2387 . . . . . . 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 18597 . . . . . 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 2628 . . . . 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 6153 . . . . . . . . 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 4163 . . . . . . . 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 5807 . . . . . . . . . . . 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 6153 . . . . . . . . 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 4163 . . . . . . . 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 2665 . . . . . 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 2670 . . . . 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 2741 . . 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 4921 . . . . . . 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 5115 . . . . . 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 2387 . . . . . . . 8  |-  ( (
dist `  R )  |`  ( X  X.  X
) )  =  ( ( dist `  R
)  |`  ( X  X.  X ) )
8214, 81xmsxmet 18376 . . . . . . 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 18299 . . . . . 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 2472 . . . 4  |-  ( R  e. NrmRing  ->  ( ( dist `  R )  |`  ( U  X.  U ) )  e.  ( * Met `  U ) )
87 eqid 2387 . . . . 5  |-  ( MetOpen `  ( ( dist `  R
)  |`  ( U  X.  U ) ) )  =  ( MetOpen `  (
( dist `  R )  |`  ( U  X.  U
) ) )
8887, 87metcn 18463 . . . 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 18372 . . . . . 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 6035 . . . 4  |-  ( R  e. NrmRing  ->  ( Jt  U )  =  ( ( MetOpen `  ( ( dist `  R
)  |`  ( X  X.  X ) ) )t  U ) )
9579eqcomi 2391 . . . . . 6  |-  ( (
dist `  R )  |`  ( U  X.  U
) )  =  ( ( ( dist `  R
)  |`  ( X  X.  X ) )  |`  ( U  X.  U
) )
96 eqid 2387 . . . . . 6  |-  ( MetOpen `  ( ( dist `  R
)  |`  ( X  X.  X ) ) )  =  ( MetOpen `  (
( dist `  R )  |`  ( X  X.  X
) ) )
9795, 96, 87metrest 18444 . . . . 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 2419 . . 3  |-  ( R  e. NrmRing  ->  ( Jt  U )  =  ( MetOpen `  (
( dist `  R )  |`  ( U  X.  U
) ) ) )
10099, 99oveq12d 6038 . 2  |-  ( R  e. NrmRing  ->  ( ( Jt  U )  Cn  ( Jt  U ) )  =  ( ( MetOpen `  ( ( dist `  R )  |`  ( U  X.  U
) ) )  Cn  ( MetOpen `  ( ( dist `  R )  |`  ( U  X.  U
) ) ) ) )
10190, 100eleqtrrd 2464 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 1649    e. wcel 1717    =/= wne 2550   A.wral 2649   E.wrex 2650    C_ wss 3263   (/)c0 3571   ifcif 3682   class class class wbr 4153    X. cxp 4816    |` cres 4820   -->wf 5390   ` cfv 5394  (class class class)co 6020   0cc0 8923   1c1 8924    x. cmul 8928    < clt 9053    <_ cle 9054    / cdiv 9609   2c2 9981   RR+crp 10544   Basecbs 13396   ↾s cress 13397   distcds 13465   ↾t crest 13575   TopOpenctopn 13576   0gc0g 13650   Grpcgrp 14612  mulGrpcmgp 15575   Ringcrg 15587   1rcur 15589  Unitcui 15671   invrcinvr 15703  NzRingcnzr 16255   * Metcxmt 16612   MetOpencmopn 16617    Cn ccn 17210   *
MetSpcxme 18256   MetSpcmt 18257   normcnm 18495  NrmGrpcngp 18496  NrmRingcnrg 18498
This theorem is referenced by:  nrgtdrg  18599
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000  ax-pre-sup 9001
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-tpos 6415  df-riota 6485  df-recs 6569  df-rdg 6604  df-1o 6660  df-oadd 6664  df-er 6841  df-map 6956  df-en 7046  df-dom 7047  df-sdom 7048  df-fin 7049  df-sup 7381  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-div 9610  df-nn 9933  df-2 9990  df-3 9991  df-4 9992  df-5 9993  df-6 9994  df-7 9995  df-8 9996  df-9 9997  df-10 9998  df-n0 10154  df-z 10215  df-dec 10315  df-uz 10421  df-q 10507  df-rp 10545  df-xneg 10642  df-xadd 10643  df-xmul 10644  df-fz 10976  df-seq 11251  df-exp 11310  df-cj 11831  df-re 11832  df-im 11833  df-sqr 11967  df-abs 11968  df-struct 13398  df-ndx 13399  df-slot 13400  df-base 13401  df-sets 13402  df-ress 13403  df-plusg 13469  df-mulr 13470  df-tset 13475  df-ple 13476  df-ds 13478  df-rest 13577  df-topgen 13594  df-xrs 13653  df-0g 13654  df-mnd 14617  df-grp 14739  df-minusg 14740  df-sbg 14741  df-mgp 15576  df-rng 15590  df-ur 15592  df-oppr 15655  df-dvdsr 15673  df-unit 15674  df-invr 15704  df-abv 15832  df-nzr 16256  df-xmet 16619  df-met 16620  df-bl 16621  df-mopn 16622  df-top 16886  df-bases 16888  df-topon 16889  df-topsp 16890  df-cn 17213  df-cnp 17214  df-xms 18259  df-ms 18260  df-nm 18501  df-ngp 18502  df-nrg 18504
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