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Theorem issrng 15631
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 15627 . . 3  |-  *Ring  =  {
r  |  [. (
* r f `  r )  /  i ]. ( i  e.  ( r RingHom  (oppr
`  r ) )  /\  i  =  `' i ) }
21eleq2i 2360 . 2  |-  ( R  e.  *Ring 
<->  R  e.  { r  |  [. ( * r f `  r
)  /  i ]. ( i  e.  ( r RingHom  (oppr
`  r ) )  /\  i  =  `' i ) } )
3 rhmrcl1 15515 . . . . 5  |-  (  .*  e.  ( R RingHom  O
)  ->  R  e.  Ring )
4 elex 2809 . . . . 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 5555 . . . . 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 5541 . . . . . . . 8  |-  ( r  =  R  ->  (
* r f `  r )  =  ( * r f `  R ) )
11 issrng.i . . . . . . . 8  |-  .*  =  ( * r f `
 R )
1210, 11syl6eqr 2346 . . . . . . 7  |-  ( r  =  R  ->  (
* r f `  r )  =  .*  )
139, 12sylan9eqr 2350 . . . . . 6  |-  ( ( r  =  R  /\  i  =  ( * r f `  r ) )  ->  i  =  .*  )
14 simpl 443 . . . . . . 7  |-  ( ( r  =  R  /\  i  =  ( * r f `  r ) )  ->  r  =  R )
1514fveq2d 5545 . . . . . . . 8  |-  ( ( r  =  R  /\  i  =  ( * r f `  r ) )  ->  (oppr
`  r )  =  (oppr
`  R ) )
16 issrng.o . . . . . . . 8  |-  O  =  (oppr
`  R )
1715, 16syl6eqr 2346 . . . . . . 7  |-  ( ( r  =  R  /\  i  =  ( * r f `  r ) )  ->  (oppr
`  r )  =  O )
1814, 17oveq12d 5892 . . . . . 6  |-  ( ( r  =  R  /\  i  =  ( * r f `  r ) )  ->  ( r RingHom  (oppr `  r ) )  =  ( R RingHom  O )
)
1913, 18eleq12d 2364 . . . . 5  |-  ( ( r  =  R  /\  i  =  ( * r f `  r ) )  ->  ( i  e.  ( r RingHom  (oppr `  r
) )  <->  .*  e.  ( R RingHom  O ) ) )
2013cnveqd 4873 . . . . . 6  |-  ( ( r  =  R  /\  i  =  ( * r f `  r ) )  ->  `' i  =  `'  .*  )
2113, 20eqeq12d 2310 . . . . 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 3040 . . 3  |-  ( r  =  R  ->  ( [. ( * r f `
 r )  / 
i ]. ( i  e.  ( r RingHom  (oppr `  r
) )  /\  i  =  `' i )  <->  (  .*  e.  ( R RingHom  O )  /\  .*  =  `'  .*  ) ) )
246, 23elab3 2934 . 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 1632    e. wcel 1696   {cab 2282   _Vcvv 2801   [.wsbc 3004   `'ccnv 4704   ` cfv 5271  (class class class)co 5874   Ringcrg 15353  opprcoppr 15420   RingHom crh 15510   * r fcstf 15624   *Ringcsr 15625
This theorem is referenced by:  srngrhm  15632  srngcnv  15634  issrngd  15642
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-plusg 13237  df-0g 13420  df-mhm 14431  df-ghm 14697  df-mgp 15342  df-rng 15356  df-ur 15358  df-rnghom 15512  df-srng 15627
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