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Theorem issrng 17279
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  |-  .*  =  ( *rf `  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 17275 . . 3  |-  *Ring  =  {
r  |  [. (
*rf `  r )  /  i ]. ( i  e.  ( r RingHom  (oppr
`  r ) )  /\  i  =  `' i ) }
21eleq2i 2545 . 2  |-  ( R  e.  *Ring 
<->  R  e.  { r  |  [. ( *rf `  r
)  /  i ]. ( i  e.  ( r RingHom  (oppr
`  r ) )  /\  i  =  `' i ) } )
3 rhmrcl1 17149 . . . . 5  |-  (  .*  e.  ( R RingHom  O
)  ->  R  e.  Ring )
4 elex 3122 . . . . 5  |-  ( R  e.  Ring  ->  R  e. 
_V )
53, 4syl 16 . . . 4  |-  (  .*  e.  ( R RingHom  O
)  ->  R  e.  _V )
65adantr 465 . . 3  |-  ( (  .*  e.  ( R RingHom  O )  /\  .*  =  `'  .*  )  ->  R  e.  _V )
7 fvex 5874 . . . . 5  |-  ( *rf `  r
)  e.  _V
87a1i 11 . . . 4  |-  ( r  =  R  ->  (
*rf `  r )  e.  _V )
9 id 22 . . . . . . 7  |-  ( i  =  ( *rf `  r )  ->  i  =  ( *rf `  r ) )
10 fveq2 5864 . . . . . . . 8  |-  ( r  =  R  ->  (
*rf `  r )  =  ( *rf `  R ) )
11 issrng.i . . . . . . . 8  |-  .*  =  ( *rf `  R )
1210, 11syl6eqr 2526 . . . . . . 7  |-  ( r  =  R  ->  (
*rf `  r )  =  .*  )
139, 12sylan9eqr 2530 . . . . . 6  |-  ( ( r  =  R  /\  i  =  ( *rf `  r
) )  ->  i  =  .*  )
14 simpl 457 . . . . . . 7  |-  ( ( r  =  R  /\  i  =  ( *rf `  r
) )  ->  r  =  R )
1514fveq2d 5868 . . . . . . . 8  |-  ( ( r  =  R  /\  i  =  ( *rf `  r
) )  ->  (oppr `  r
)  =  (oppr `  R
) )
16 issrng.o . . . . . . . 8  |-  O  =  (oppr
`  R )
1715, 16syl6eqr 2526 . . . . . . 7  |-  ( ( r  =  R  /\  i  =  ( *rf `  r
) )  ->  (oppr `  r
)  =  O )
1814, 17oveq12d 6300 . . . . . 6  |-  ( ( r  =  R  /\  i  =  ( *rf `  r
) )  ->  (
r RingHom  (oppr
`  r ) )  =  ( R RingHom  O
) )
1913, 18eleq12d 2549 . . . . 5  |-  ( ( r  =  R  /\  i  =  ( *rf `  r
) )  ->  (
i  e.  ( r RingHom 
(oppr `  r ) )  <->  .*  e.  ( R RingHom  O ) ) )
2013cnveqd 5176 . . . . . 6  |-  ( ( r  =  R  /\  i  =  ( *rf `  r
) )  ->  `' i  =  `'  .*  )
2113, 20eqeq12d 2489 . . . . 5  |-  ( ( r  =  R  /\  i  =  ( *rf `  r
) )  ->  (
i  =  `' i  <->  .*  =  `'  .*  )
)
2219, 21anbi12d 710 . . . 4  |-  ( ( r  =  R  /\  i  =  ( *rf `  r
) )  ->  (
( i  e.  ( r RingHom  (oppr
`  r ) )  /\  i  =  `' i )  <->  (  .*  e.  ( R RingHom  O )  /\  .*  =  `'  .*  ) ) )
238, 22sbcied 3368 . . 3  |-  ( r  =  R  ->  ( [. ( *rf `  r )  / 
i ]. ( i  e.  ( r RingHom  (oppr `  r
) )  /\  i  =  `' i )  <->  (  .*  e.  ( R RingHom  O )  /\  .*  =  `'  .*  ) ) )
246, 23elab3 3257 . 2  |-  ( R  e.  { r  | 
[. ( *rf `  r )  /  i ]. (
i  e.  ( r RingHom 
(oppr `  r ) )  /\  i  =  `' i
) }  <->  (  .*  e.  ( R RingHom  O )  /\  .*  =  `'  .*  ) )
252, 24bitri 249 1  |-  ( R  e.  *Ring 
<->  (  .*  e.  ( R RingHom  O )  /\  .*  =  `'  .*  )
)
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   {cab 2452   _Vcvv 3113   [.wsbc 3331   `'ccnv 4998   ` cfv 5586  (class class class)co 6282   Ringcrg 16983  opprcoppr 17052   RingHom crh 17142   *rfcstf 17272   *Ringcsr 17273
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-recs 7039  df-rdg 7073  df-er 7308  df-map 7419  df-en 7514  df-dom 7515  df-sdom 7516  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-ndx 14486  df-slot 14487  df-base 14488  df-sets 14489  df-plusg 14561  df-0g 14690  df-mhm 15774  df-ghm 16057  df-mgp 16929  df-ur 16941  df-rng 16985  df-rnghom 17145  df-srng 17275
This theorem is referenced by:  srngrhm  17280  srngcnv  17282  issrngd  17290
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