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Theorem issrng 15867
Description: The predicate "is a star ring." (Contributed by NM, 22-Sep-2011.) (Revised by Mario Carneiro, 6-Oct-2015.)
Hypotheses
Ref Expression
issrng.o  |-  O  =  (oppr
`  R )
issrng.i  |-  .*  =  ( * r f `
 R )
Assertion
Ref Expression
issrng  |-  ( R  e.  *Ring 
<->  (  .*  e.  ( R RingHom  O )  /\  .*  =  `'  .*  )
)

Proof of Theorem issrng
Dummy variables  i 
r are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-srng 15863 . . 3  |-  *Ring  =  {
r  |  [. (
* r f `  r )  /  i ]. ( i  e.  ( r RingHom  (oppr
`  r ) )  /\  i  =  `' i ) }
21eleq2i 2453 . 2  |-  ( R  e.  *Ring 
<->  R  e.  { r  |  [. ( * r f `  r
)  /  i ]. ( i  e.  ( r RingHom  (oppr
`  r ) )  /\  i  =  `' i ) } )
3 rhmrcl1 15751 . . . . 5  |-  (  .*  e.  ( R RingHom  O
)  ->  R  e.  Ring )
4 elex 2909 . . . . 5  |-  ( R  e.  Ring  ->  R  e. 
_V )
53, 4syl 16 . . . 4  |-  (  .*  e.  ( R RingHom  O
)  ->  R  e.  _V )
65adantr 452 . . 3  |-  ( (  .*  e.  ( R RingHom  O )  /\  .*  =  `'  .*  )  ->  R  e.  _V )
7 fvex 5684 . . . . 5  |-  ( * r f `  r
)  e.  _V
87a1i 11 . . . 4  |-  ( r  =  R  ->  (
* r f `  r )  e.  _V )
9 id 20 . . . . . . 7  |-  ( i  =  ( * r f `  r )  ->  i  =  ( * r f `  r ) )
10 fveq2 5670 . . . . . . . 8  |-  ( r  =  R  ->  (
* r f `  r )  =  ( * r f `  R ) )
11 issrng.i . . . . . . . 8  |-  .*  =  ( * r f `
 R )
1210, 11syl6eqr 2439 . . . . . . 7  |-  ( r  =  R  ->  (
* r f `  r )  =  .*  )
139, 12sylan9eqr 2443 . . . . . 6  |-  ( ( r  =  R  /\  i  =  ( * r f `  r ) )  ->  i  =  .*  )
14 simpl 444 . . . . . . 7  |-  ( ( r  =  R  /\  i  =  ( * r f `  r ) )  ->  r  =  R )
1514fveq2d 5674 . . . . . . . 8  |-  ( ( r  =  R  /\  i  =  ( * r f `  r ) )  ->  (oppr
`  r )  =  (oppr
`  R ) )
16 issrng.o . . . . . . . 8  |-  O  =  (oppr
`  R )
1715, 16syl6eqr 2439 . . . . . . 7  |-  ( ( r  =  R  /\  i  =  ( * r f `  r ) )  ->  (oppr
`  r )  =  O )
1814, 17oveq12d 6040 . . . . . 6  |-  ( ( r  =  R  /\  i  =  ( * r f `  r ) )  ->  ( r RingHom  (oppr `  r ) )  =  ( R RingHom  O )
)
1913, 18eleq12d 2457 . . . . 5  |-  ( ( r  =  R  /\  i  =  ( * r f `  r ) )  ->  ( i  e.  ( r RingHom  (oppr `  r
) )  <->  .*  e.  ( R RingHom  O ) ) )
2013cnveqd 4990 . . . . . 6  |-  ( ( r  =  R  /\  i  =  ( * r f `  r ) )  ->  `' i  =  `'  .*  )
2113, 20eqeq12d 2403 . . . . 5  |-  ( ( r  =  R  /\  i  =  ( * r f `  r ) )  ->  ( i  =  `' i  <->  .*  =  `'  .*  ) )
2219, 21anbi12d 692 . . . 4  |-  ( ( r  =  R  /\  i  =  ( * r f `  r ) )  ->  ( (
i  e.  ( r RingHom 
(oppr `  r ) )  /\  i  =  `' i
)  <->  (  .*  e.  ( R RingHom  O )  /\  .*  =  `'  .*  )
) )
238, 22sbcied 3142 . . 3  |-  ( r  =  R  ->  ( [. ( * r f `
 r )  / 
i ]. ( i  e.  ( r RingHom  (oppr `  r
) )  /\  i  =  `' i )  <->  (  .*  e.  ( R RingHom  O )  /\  .*  =  `'  .*  ) ) )
246, 23elab3 3034 . 2  |-  ( R  e.  { r  | 
[. ( * r f `  r )  /  i ]. (
i  e.  ( r RingHom 
(oppr `  r ) )  /\  i  =  `' i
) }  <->  (  .*  e.  ( R RingHom  O )  /\  .*  =  `'  .*  ) )
252, 24bitri 241 1  |-  ( R  e.  *Ring 
<->  (  .*  e.  ( R RingHom  O )  /\  .*  =  `'  .*  )
)
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   {cab 2375   _Vcvv 2901   [.wsbc 3106   `'ccnv 4819   ` cfv 5396  (class class class)co 6022   Ringcrg 15589  opprcoppr 15656   RingHom crh 15746   * r fcstf 15860   *Ringcsr 15861
This theorem is referenced by:  srngrhm  15868  srngcnv  15870  issrngd  15878
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 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-cnex 8981  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002
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 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-riota 6487  df-recs 6571  df-rdg 6606  df-er 6843  df-map 6958  df-en 7048  df-dom 7049  df-sdom 7050  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-nn 9935  df-2 9992  df-ndx 13401  df-slot 13402  df-base 13403  df-sets 13404  df-plusg 13471  df-0g 13656  df-mhm 14667  df-ghm 14933  df-mgp 15578  df-rng 15592  df-ur 15594  df-rnghom 15748  df-srng 15863
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