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Theorem issrng 15930
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 15926 . . 3  |-  *Ring  =  {
r  |  [. (
* r f `  r )  /  i ]. ( i  e.  ( r RingHom  (oppr
`  r ) )  /\  i  =  `' i ) }
21eleq2i 2499 . 2  |-  ( R  e.  *Ring 
<->  R  e.  { r  |  [. ( * r f `  r
)  /  i ]. ( i  e.  ( r RingHom  (oppr
`  r ) )  /\  i  =  `' i ) } )
3 rhmrcl1 15814 . . . . 5  |-  (  .*  e.  ( R RingHom  O
)  ->  R  e.  Ring )
4 elex 2956 . . . . 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 5734 . . . . 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 5720 . . . . . . . 8  |-  ( r  =  R  ->  (
* r f `  r )  =  ( * r f `  R ) )
11 issrng.i . . . . . . . 8  |-  .*  =  ( * r f `
 R )
1210, 11syl6eqr 2485 . . . . . . 7  |-  ( r  =  R  ->  (
* r f `  r )  =  .*  )
139, 12sylan9eqr 2489 . . . . . 6  |-  ( ( r  =  R  /\  i  =  ( * r f `  r ) )  ->  i  =  .*  )
14 simpl 444 . . . . . . 7  |-  ( ( r  =  R  /\  i  =  ( * r f `  r ) )  ->  r  =  R )
1514fveq2d 5724 . . . . . . . 8  |-  ( ( r  =  R  /\  i  =  ( * r f `  r ) )  ->  (oppr
`  r )  =  (oppr
`  R ) )
16 issrng.o . . . . . . . 8  |-  O  =  (oppr
`  R )
1715, 16syl6eqr 2485 . . . . . . 7  |-  ( ( r  =  R  /\  i  =  ( * r f `  r ) )  ->  (oppr
`  r )  =  O )
1814, 17oveq12d 6091 . . . . . 6  |-  ( ( r  =  R  /\  i  =  ( * r f `  r ) )  ->  ( r RingHom  (oppr `  r ) )  =  ( R RingHom  O )
)
1913, 18eleq12d 2503 . . . . 5  |-  ( ( r  =  R  /\  i  =  ( * r f `  r ) )  ->  ( i  e.  ( r RingHom  (oppr `  r
) )  <->  .*  e.  ( R RingHom  O ) ) )
2013cnveqd 5040 . . . . . 6  |-  ( ( r  =  R  /\  i  =  ( * r f `  r ) )  ->  `' i  =  `'  .*  )
2113, 20eqeq12d 2449 . . . . 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 3189 . . 3  |-  ( r  =  R  ->  ( [. ( * r f `
 r )  / 
i ]. ( i  e.  ( r RingHom  (oppr `  r
) )  /\  i  =  `' i )  <->  (  .*  e.  ( R RingHom  O )  /\  .*  =  `'  .*  ) ) )
246, 23elab3 3081 . 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 1652    e. wcel 1725   {cab 2421   _Vcvv 2948   [.wsbc 3153   `'ccnv 4869   ` cfv 5446  (class class class)co 6073   Ringcrg 15652  opprcoppr 15719   RingHom crh 15809   * r fcstf 15923   *Ringcsr 15924
This theorem is referenced by:  srngrhm  15931  srngcnv  15933  issrngd  15941
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
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-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-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-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-plusg 13534  df-0g 13719  df-mhm 14730  df-ghm 14996  df-mgp 15641  df-rng 15655  df-ur 15657  df-rnghom 15811  df-srng 15926
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