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Theorem issrng 15615
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 15611 . . 3  |-  *Ring  =  {
r  |  [. (
* r f `  r )  /  i ]. ( i  e.  ( r RingHom  (oppr
`  r ) )  /\  i  =  `' i ) }
21eleq2i 2347 . 2  |-  ( R  e.  *Ring 
<->  R  e.  { r  |  [. ( * r f `  r
)  /  i ]. ( i  e.  ( r RingHom  (oppr
`  r ) )  /\  i  =  `' i ) } )
3 rhmrcl1 15499 . . . . 5  |-  (  .*  e.  ( R RingHom  O
)  ->  R  e.  Ring )
4 elex 2796 . . . . 5  |-  ( R  e.  Ring  ->  R  e. 
_V )
53, 4syl 15 . . . 4  |-  (  .*  e.  ( R RingHom  O
)  ->  R  e.  _V )
65adantr 451 . . 3  |-  ( (  .*  e.  ( R RingHom  O )  /\  .*  =  `'  .*  )  ->  R  e.  _V )
7 fvex 5539 . . . . 5  |-  ( * r f `  r
)  e.  _V
87a1i 10 . . . 4  |-  ( r  =  R  ->  (
* r f `  r )  e.  _V )
9 id 19 . . . . . . 7  |-  ( i  =  ( * r f `  r )  ->  i  =  ( * r f `  r ) )
10 fveq2 5525 . . . . . . . 8  |-  ( r  =  R  ->  (
* r f `  r )  =  ( * r f `  R ) )
11 issrng.i . . . . . . . 8  |-  .*  =  ( * r f `
 R )
1210, 11syl6eqr 2333 . . . . . . 7  |-  ( r  =  R  ->  (
* r f `  r )  =  .*  )
139, 12sylan9eqr 2337 . . . . . 6  |-  ( ( r  =  R  /\  i  =  ( * r f `  r ) )  ->  i  =  .*  )
14 simpl 443 . . . . . . 7  |-  ( ( r  =  R  /\  i  =  ( * r f `  r ) )  ->  r  =  R )
1514fveq2d 5529 . . . . . . . 8  |-  ( ( r  =  R  /\  i  =  ( * r f `  r ) )  ->  (oppr
`  r )  =  (oppr
`  R ) )
16 issrng.o . . . . . . . 8  |-  O  =  (oppr
`  R )
1715, 16syl6eqr 2333 . . . . . . 7  |-  ( ( r  =  R  /\  i  =  ( * r f `  r ) )  ->  (oppr
`  r )  =  O )
1814, 17oveq12d 5876 . . . . . 6  |-  ( ( r  =  R  /\  i  =  ( * r f `  r ) )  ->  ( r RingHom  (oppr `  r ) )  =  ( R RingHom  O )
)
1913, 18eleq12d 2351 . . . . 5  |-  ( ( r  =  R  /\  i  =  ( * r f `  r ) )  ->  ( i  e.  ( r RingHom  (oppr `  r
) )  <->  .*  e.  ( R RingHom  O ) ) )
2013cnveqd 4857 . . . . . 6  |-  ( ( r  =  R  /\  i  =  ( * r f `  r ) )  ->  `' i  =  `'  .*  )
2113, 20eqeq12d 2297 . . . . 5  |-  ( ( r  =  R  /\  i  =  ( * r f `  r ) )  ->  ( i  =  `' i  <->  .*  =  `'  .*  ) )
2219, 21anbi12d 691 . . . 4  |-  ( ( r  =  R  /\  i  =  ( * r f `  r ) )  ->  ( (
i  e.  ( r RingHom 
(oppr `  r ) )  /\  i  =  `' i
)  <->  (  .*  e.  ( R RingHom  O )  /\  .*  =  `'  .*  )
) )
238, 22sbcied 3027 . . 3  |-  ( r  =  R  ->  ( [. ( * r f `
 r )  / 
i ]. ( i  e.  ( r RingHom  (oppr `  r
) )  /\  i  =  `' i )  <->  (  .*  e.  ( R RingHom  O )  /\  .*  =  `'  .*  ) ) )
246, 23elab3 2921 . 2  |-  ( R  e.  { r  | 
[. ( * r f `  r )  /  i ]. (
i  e.  ( r RingHom 
(oppr `  r ) )  /\  i  =  `' i
) }  <->  (  .*  e.  ( R RingHom  O )  /\  .*  =  `'  .*  ) )
252, 24bitri 240 1  |-  ( R  e.  *Ring 
<->  (  .*  e.  ( R RingHom  O )  /\  .*  =  `'  .*  )
)
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   {cab 2269   _Vcvv 2788   [.wsbc 2991   `'ccnv 4688   ` cfv 5255  (class class class)co 5858   Ringcrg 15337  opprcoppr 15404   RingHom crh 15494   * r fcstf 15608   *Ringcsr 15609
This theorem is referenced by:  srngrhm  15616  srngcnv  15618  issrngd  15626
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
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-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-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-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-plusg 13221  df-0g 13404  df-mhm 14415  df-ghm 14681  df-mgp 15326  df-rng 15340  df-ur 15342  df-rnghom 15496  df-srng 15611
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