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Theorem issrng 18673
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 = (oppr𝑅)
issrng.i = (*rf𝑅)
Assertion
Ref Expression
issrng (𝑅 ∈ *-Ring ↔ ( ∈ (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 18669 . . 3 *-Ring = {𝑟[(*rf𝑟) / 𝑖](𝑖 ∈ (𝑟 RingHom (oppr𝑟)) ∧ 𝑖 = 𝑖)}
21eleq2i 2680 . 2 (𝑅 ∈ *-Ring ↔ 𝑅 ∈ {𝑟[(*rf𝑟) / 𝑖](𝑖 ∈ (𝑟 RingHom (oppr𝑟)) ∧ 𝑖 = 𝑖)})
3 rhmrcl1 18542 . . . . 5 ( ∈ (𝑅 RingHom 𝑂) → 𝑅 ∈ Ring)
4 elex 3185 . . . . 5 (𝑅 ∈ Ring → 𝑅 ∈ V)
53, 4syl 17 . . . 4 ( ∈ (𝑅 RingHom 𝑂) → 𝑅 ∈ V)
65adantr 480 . . 3 (( ∈ (𝑅 RingHom 𝑂) ∧ = ) → 𝑅 ∈ V)
7 fvex 6113 . . . . 5 (*rf𝑟) ∈ V
87a1i 11 . . . 4 (𝑟 = 𝑅 → (*rf𝑟) ∈ V)
9 id 22 . . . . . . 7 (𝑖 = (*rf𝑟) → 𝑖 = (*rf𝑟))
10 fveq2 6103 . . . . . . . 8 (𝑟 = 𝑅 → (*rf𝑟) = (*rf𝑅))
11 issrng.i . . . . . . . 8 = (*rf𝑅)
1210, 11syl6eqr 2662 . . . . . . 7 (𝑟 = 𝑅 → (*rf𝑟) = )
139, 12sylan9eqr 2666 . . . . . 6 ((𝑟 = 𝑅𝑖 = (*rf𝑟)) → 𝑖 = )
14 simpl 472 . . . . . . 7 ((𝑟 = 𝑅𝑖 = (*rf𝑟)) → 𝑟 = 𝑅)
1514fveq2d 6107 . . . . . . . 8 ((𝑟 = 𝑅𝑖 = (*rf𝑟)) → (oppr𝑟) = (oppr𝑅))
16 issrng.o . . . . . . . 8 𝑂 = (oppr𝑅)
1715, 16syl6eqr 2662 . . . . . . 7 ((𝑟 = 𝑅𝑖 = (*rf𝑟)) → (oppr𝑟) = 𝑂)
1814, 17oveq12d 6567 . . . . . 6 ((𝑟 = 𝑅𝑖 = (*rf𝑟)) → (𝑟 RingHom (oppr𝑟)) = (𝑅 RingHom 𝑂))
1913, 18eleq12d 2682 . . . . 5 ((𝑟 = 𝑅𝑖 = (*rf𝑟)) → (𝑖 ∈ (𝑟 RingHom (oppr𝑟)) ↔ ∈ (𝑅 RingHom 𝑂)))
2013cnveqd 5220 . . . . . 6 ((𝑟 = 𝑅𝑖 = (*rf𝑟)) → 𝑖 = )
2113, 20eqeq12d 2625 . . . . 5 ((𝑟 = 𝑅𝑖 = (*rf𝑟)) → (𝑖 = 𝑖 = ))
2219, 21anbi12d 743 . . . 4 ((𝑟 = 𝑅𝑖 = (*rf𝑟)) → ((𝑖 ∈ (𝑟 RingHom (oppr𝑟)) ∧ 𝑖 = 𝑖) ↔ ( ∈ (𝑅 RingHom 𝑂) ∧ = )))
238, 22sbcied 3439 . . 3 (𝑟 = 𝑅 → ([(*rf𝑟) / 𝑖](𝑖 ∈ (𝑟 RingHom (oppr𝑟)) ∧ 𝑖 = 𝑖) ↔ ( ∈ (𝑅 RingHom 𝑂) ∧ = )))
246, 23elab3 3327 . 2 (𝑅 ∈ {𝑟[(*rf𝑟) / 𝑖](𝑖 ∈ (𝑟 RingHom (oppr𝑟)) ∧ 𝑖 = 𝑖)} ↔ ( ∈ (𝑅 RingHom 𝑂) ∧ = ))
252, 24bitri 263 1 (𝑅 ∈ *-Ring ↔ ( ∈ (𝑅 RingHom 𝑂) ∧ = ))
Colors of variables: wff setvar class
Syntax hints:  wb 195  wa 383   = wceq 1475  wcel 1977  {cab 2596  Vcvv 3173  [wsbc 3402  ccnv 5037  cfv 5804  (class class class)co 6549  Ringcrg 18370  opprcoppr 18445   RingHom crh 18535  *rfcstf 18666  *-Ringcsr 18667
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-plusg 15781  df-0g 15925  df-mhm 17158  df-ghm 17481  df-mgp 18313  df-ur 18325  df-ring 18372  df-rnghom 18538  df-srng 18669
This theorem is referenced by:  srngrhm  18674  srngcnv  18676  issrngd  18684
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