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Theorem issdrg 19585
 Description: Property of a division subring. (Contributed by Stefan O'Rear, 3-Oct-2015.)
Assertion
Ref Expression
issdrg (𝑆 ∈ (SubDRing‘𝑅) ↔ (𝑅 ∈ DivRing ∧ 𝑆 ∈ (SubRing‘𝑅) ∧ (𝑅s 𝑆) ∈ DivRing))

Proof of Theorem issdrg
Dummy variables 𝑤 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-sdrg 19584 . . . 4 SubDRing = (𝑤 ∈ DivRing ↦ {𝑠 ∈ (SubRing‘𝑤) ∣ (𝑤s 𝑠) ∈ DivRing})
21mptrcl 6761 . . 3 (𝑆 ∈ (SubDRing‘𝑅) → 𝑅 ∈ DivRing)
3 fveq2 6652 . . . . . . 7 (𝑤 = 𝑅 → (SubRing‘𝑤) = (SubRing‘𝑅))
4 oveq1 7149 . . . . . . . 8 (𝑤 = 𝑅 → (𝑤s 𝑠) = (𝑅s 𝑠))
54eleq1d 2874 . . . . . . 7 (𝑤 = 𝑅 → ((𝑤s 𝑠) ∈ DivRing ↔ (𝑅s 𝑠) ∈ DivRing))
63, 5rabeqbidv 3433 . . . . . 6 (𝑤 = 𝑅 → {𝑠 ∈ (SubRing‘𝑤) ∣ (𝑤s 𝑠) ∈ DivRing} = {𝑠 ∈ (SubRing‘𝑅) ∣ (𝑅s 𝑠) ∈ DivRing})
7 fvex 6665 . . . . . . 7 (SubRing‘𝑅) ∈ V
87rabex 5202 . . . . . 6 {𝑠 ∈ (SubRing‘𝑅) ∣ (𝑅s 𝑠) ∈ DivRing} ∈ V
96, 1, 8fvmpt 6752 . . . . 5 (𝑅 ∈ DivRing → (SubDRing‘𝑅) = {𝑠 ∈ (SubRing‘𝑅) ∣ (𝑅s 𝑠) ∈ DivRing})
109eleq2d 2875 . . . 4 (𝑅 ∈ DivRing → (𝑆 ∈ (SubDRing‘𝑅) ↔ 𝑆 ∈ {𝑠 ∈ (SubRing‘𝑅) ∣ (𝑅s 𝑠) ∈ DivRing}))
11 oveq2 7150 . . . . . 6 (𝑠 = 𝑆 → (𝑅s 𝑠) = (𝑅s 𝑆))
1211eleq1d 2874 . . . . 5 (𝑠 = 𝑆 → ((𝑅s 𝑠) ∈ DivRing ↔ (𝑅s 𝑆) ∈ DivRing))
1312elrab 3629 . . . 4 (𝑆 ∈ {𝑠 ∈ (SubRing‘𝑅) ∣ (𝑅s 𝑠) ∈ DivRing} ↔ (𝑆 ∈ (SubRing‘𝑅) ∧ (𝑅s 𝑆) ∈ DivRing))
1410, 13syl6bb 290 . . 3 (𝑅 ∈ DivRing → (𝑆 ∈ (SubDRing‘𝑅) ↔ (𝑆 ∈ (SubRing‘𝑅) ∧ (𝑅s 𝑆) ∈ DivRing)))
152, 14biadanii 821 . 2 (𝑆 ∈ (SubDRing‘𝑅) ↔ (𝑅 ∈ DivRing ∧ (𝑆 ∈ (SubRing‘𝑅) ∧ (𝑅s 𝑆) ∈ DivRing)))
16 3anass 1092 . 2 ((𝑅 ∈ DivRing ∧ 𝑆 ∈ (SubRing‘𝑅) ∧ (𝑅s 𝑆) ∈ DivRing) ↔ (𝑅 ∈ DivRing ∧ (𝑆 ∈ (SubRing‘𝑅) ∧ (𝑅s 𝑆) ∈ DivRing)))
1715, 16bitr4i 281 1 (𝑆 ∈ (SubDRing‘𝑅) ↔ (𝑅 ∈ DivRing ∧ 𝑆 ∈ (SubRing‘𝑅) ∧ (𝑅s 𝑆) ∈ DivRing))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  {crab 3110  ‘cfv 6329  (class class class)co 7142   ↾s cress 16493  DivRingcdr 19513  SubRingcsubrg 19542  SubDRingcsdrg 19583 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5170  ax-nul 5177  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3722  df-dif 3885  df-un 3887  df-in 3889  df-ss 3899  df-nul 4246  df-if 4428  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5528  df-rel 5529  df-cnv 5530  df-co 5531  df-dm 5532  df-rn 5533  df-res 5534  df-ima 5535  df-iota 6288  df-fun 6331  df-fv 6337  df-ov 7145  df-sdrg 19584 This theorem is referenced by:  sdrgid  19586  sdrgss  19587  issdrg2  19588  sdrgint  19594  primefld  19595  primefld0cl  19596  primefld1cl  19597  primefldchr  30964
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