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Theorem issrng 16933
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 16929 . . 3  |-  *Ring  =  {
r  |  [. (
*rf `  r )  /  i ]. ( i  e.  ( r RingHom  (oppr
`  r ) )  /\  i  =  `' i ) }
21eleq2i 2505 . 2  |-  ( R  e.  *Ring 
<->  R  e.  { r  |  [. ( *rf `  r
)  /  i ]. ( i  e.  ( r RingHom  (oppr
`  r ) )  /\  i  =  `' i ) } )
3 rhmrcl1 16807 . . . . 5  |-  (  .*  e.  ( R RingHom  O
)  ->  R  e.  Ring )
4 elex 2979 . . . . 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 5699 . . . . 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 5689 . . . . . . . 8  |-  ( r  =  R  ->  (
*rf `  r )  =  ( *rf `  R ) )
11 issrng.i . . . . . . . 8  |-  .*  =  ( *rf `  R )
1210, 11syl6eqr 2491 . . . . . . 7  |-  ( r  =  R  ->  (
*rf `  r )  =  .*  )
139, 12sylan9eqr 2495 . . . . . 6  |-  ( ( r  =  R  /\  i  =  ( *rf `  r
) )  ->  i  =  .*  )
14 simpl 457 . . . . . . 7  |-  ( ( r  =  R  /\  i  =  ( *rf `  r
) )  ->  r  =  R )
1514fveq2d 5693 . . . . . . . 8  |-  ( ( r  =  R  /\  i  =  ( *rf `  r
) )  ->  (oppr `  r
)  =  (oppr `  R
) )
16 issrng.o . . . . . . . 8  |-  O  =  (oppr
`  R )
1715, 16syl6eqr 2491 . . . . . . 7  |-  ( ( r  =  R  /\  i  =  ( *rf `  r
) )  ->  (oppr `  r
)  =  O )
1814, 17oveq12d 6107 . . . . . 6  |-  ( ( r  =  R  /\  i  =  ( *rf `  r
) )  ->  (
r RingHom  (oppr
`  r ) )  =  ( R RingHom  O
) )
1913, 18eleq12d 2509 . . . . 5  |-  ( ( r  =  R  /\  i  =  ( *rf `  r
) )  ->  (
i  e.  ( r RingHom 
(oppr `  r ) )  <->  .*  e.  ( R RingHom  O ) ) )
2013cnveqd 5013 . . . . . 6  |-  ( ( r  =  R  /\  i  =  ( *rf `  r
) )  ->  `' i  =  `'  .*  )
2113, 20eqeq12d 2455 . . . . 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 3221 . . 3  |-  ( r  =  R  ->  ( [. ( *rf `  r )  / 
i ]. ( i  e.  ( r RingHom  (oppr `  r
) )  /\  i  =  `' i )  <->  (  .*  e.  ( R RingHom  O )  /\  .*  =  `'  .*  ) ) )
246, 23elab3 3111 . 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 1369    e. wcel 1756   {cab 2427   _Vcvv 2970   [.wsbc 3184   `'ccnv 4837   ` cfv 5416  (class class class)co 6089   Ringcrg 16643  opprcoppr 16712   RingHom crh 16802   *rfcstf 16926   *Ringcsr 16927
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-cnex 9336  ax-resscn 9337  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-addrcl 9341  ax-mulcl 9342  ax-mulrcl 9343  ax-mulcom 9344  ax-addass 9345  ax-mulass 9346  ax-distr 9347  ax-i2m1 9348  ax-1ne0 9349  ax-1rid 9350  ax-rnegex 9351  ax-rrecex 9352  ax-cnre 9353  ax-pre-lttri 9354  ax-pre-lttrn 9355  ax-pre-ltadd 9356  ax-pre-mulgt0 9357
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-om 6475  df-recs 6830  df-rdg 6864  df-er 7099  df-map 7214  df-en 7309  df-dom 7310  df-sdom 7311  df-pnf 9418  df-mnf 9419  df-xr 9420  df-ltxr 9421  df-le 9422  df-sub 9595  df-neg 9596  df-nn 10321  df-2 10378  df-ndx 14175  df-slot 14176  df-base 14177  df-sets 14178  df-plusg 14249  df-0g 14378  df-mhm 15462  df-ghm 15743  df-mgp 16590  df-ur 16602  df-rng 16645  df-rnghom 16804  df-srng 16929
This theorem is referenced by:  srngrhm  16934  srngcnv  16936  issrngd  16944
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