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Theorem subrgcrng 13284
Description: A subring of a commutative ring is a commutative ring. (Contributed by Mario Carneiro, 10-Jan-2015.)
Hypothesis
Ref Expression
subrgring.1 𝑆 = (𝑅 β†Ύs 𝐴)
Assertion
Ref Expression
subrgcrng ((𝑅 ∈ CRing ∧ 𝐴 ∈ (SubRingβ€˜π‘…)) β†’ 𝑆 ∈ CRing)

Proof of Theorem subrgcrng
StepHypRef Expression
1 subrgring.1 . . . 4 𝑆 = (𝑅 β†Ύs 𝐴)
21subrgring 13283 . . 3 (𝐴 ∈ (SubRingβ€˜π‘…) β†’ 𝑆 ∈ Ring)
32adantl 277 . 2 ((𝑅 ∈ CRing ∧ 𝐴 ∈ (SubRingβ€˜π‘…)) β†’ 𝑆 ∈ Ring)
4 eqid 2177 . . . 4 (mulGrpβ€˜π‘…) = (mulGrpβ€˜π‘…)
51, 4mgpress 13072 . . 3 ((𝑅 ∈ CRing ∧ 𝐴 ∈ (SubRingβ€˜π‘…)) β†’ ((mulGrpβ€˜π‘…) β†Ύs 𝐴) = (mulGrpβ€˜π‘†))
6 eqidd 2178 . . . 4 ((𝑅 ∈ CRing ∧ 𝐴 ∈ (SubRingβ€˜π‘…)) β†’ ((mulGrpβ€˜π‘…) β†Ύs 𝐴) = ((mulGrpβ€˜π‘…) β†Ύs 𝐴))
74crngmgp 13118 . . . . 5 (𝑅 ∈ CRing β†’ (mulGrpβ€˜π‘…) ∈ CMnd)
87adantr 276 . . . 4 ((𝑅 ∈ CRing ∧ 𝐴 ∈ (SubRingβ€˜π‘…)) β†’ (mulGrpβ€˜π‘…) ∈ CMnd)
9 eqid 2177 . . . . . . 7 (mulGrpβ€˜π‘†) = (mulGrpβ€˜π‘†)
109ringmgp 13116 . . . . . 6 (𝑆 ∈ Ring β†’ (mulGrpβ€˜π‘†) ∈ Mnd)
113, 10syl 14 . . . . 5 ((𝑅 ∈ CRing ∧ 𝐴 ∈ (SubRingβ€˜π‘…)) β†’ (mulGrpβ€˜π‘†) ∈ Mnd)
125, 11eqeltrd 2254 . . . 4 ((𝑅 ∈ CRing ∧ 𝐴 ∈ (SubRingβ€˜π‘…)) β†’ ((mulGrpβ€˜π‘…) β†Ύs 𝐴) ∈ Mnd)
13 simpr 110 . . . 4 ((𝑅 ∈ CRing ∧ 𝐴 ∈ (SubRingβ€˜π‘…)) β†’ 𝐴 ∈ (SubRingβ€˜π‘…))
146, 8, 12, 13subcmnd 13060 . . 3 ((𝑅 ∈ CRing ∧ 𝐴 ∈ (SubRingβ€˜π‘…)) β†’ ((mulGrpβ€˜π‘…) β†Ύs 𝐴) ∈ CMnd)
155, 14eqeltrrd 2255 . 2 ((𝑅 ∈ CRing ∧ 𝐴 ∈ (SubRingβ€˜π‘…)) β†’ (mulGrpβ€˜π‘†) ∈ CMnd)
169iscrng 13117 . 2 (𝑆 ∈ CRing ↔ (𝑆 ∈ Ring ∧ (mulGrpβ€˜π‘†) ∈ CMnd))
173, 15, 16sylanbrc 417 1 ((𝑅 ∈ CRing ∧ 𝐴 ∈ (SubRingβ€˜π‘…)) β†’ 𝑆 ∈ CRing)
Colors of variables: wff set class
Syntax hints:   β†’ wi 4   ∧ wa 104   = wceq 1353   ∈ wcel 2148  β€˜cfv 5215  (class class class)co 5872   β†Ύs cress 12455  Mndcmnd 12749  CMndccmn 13019  mulGrpcmgp 13061  Ringcrg 13110  CRingccrg 13111  SubRingcsubrg 13276
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4120  ax-pow 4173  ax-pr 4208  ax-un 4432  ax-setind 4535  ax-cnex 7899  ax-resscn 7900  ax-1cn 7901  ax-1re 7902  ax-icn 7903  ax-addcl 7904  ax-addrcl 7905  ax-mulcl 7906  ax-addcom 7908  ax-addass 7910  ax-i2m1 7913  ax-0lt1 7914  ax-0id 7916  ax-rnegex 7917  ax-pre-ltirr 7920  ax-pre-lttrn 7922  ax-pre-ltadd 7924
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-br 4003  df-opab 4064  df-mpt 4065  df-id 4292  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-ima 4638  df-iota 5177  df-fun 5217  df-fn 5218  df-fv 5223  df-ov 5875  df-oprab 5876  df-mpo 5877  df-pnf 7990  df-mnf 7991  df-ltxr 7993  df-inn 8916  df-2 8974  df-3 8975  df-ndx 12457  df-slot 12458  df-base 12460  df-sets 12461  df-iress 12462  df-plusg 12541  df-mulr 12542  df-cmn 13021  df-mgp 13062  df-ring 13112  df-cring 13113  df-subrg 13278
This theorem is referenced by:  zringcrng  13351
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