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Theorem issdrg 20404
Description: Property of a division subring. (Contributed by Stefan O'Rear, 3-Oct-2015.)
Assertion
Ref Expression
issdrg (𝑆 ∈ (SubDRingβ€˜π‘…) ↔ (𝑅 ∈ DivRing ∧ 𝑆 ∈ (SubRingβ€˜π‘…) ∧ (𝑅 β†Ύs 𝑆) ∈ DivRing))
Proof of Theorem issdrg
Dummy variables 𝑀 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-sdrg 20403 . . . 4 SubDRing = (𝑀 ∈ DivRing ↦ {𝑠 ∈ (SubRingβ€˜π‘€) ∣ (𝑀 β†Ύs 𝑠) ∈ DivRing})
21mptrcl 7008 . . 3 (𝑆 ∈ (SubDRingβ€˜π‘…) β†’ 𝑅 ∈ DivRing)
3 fveq2 6892 . . . . . . 7 (𝑀 = 𝑅 β†’ (SubRingβ€˜π‘€) = (SubRingβ€˜π‘…))
4 oveq1 7416 . . . . . . . 8 (𝑀 = 𝑅 β†’ (𝑀 β†Ύs 𝑠) = (𝑅 β†Ύs 𝑠))
54eleq1d 2819 . . . . . . 7 (𝑀 = 𝑅 β†’ ((𝑀 β†Ύs 𝑠) ∈ DivRing ↔ (𝑅 β†Ύs 𝑠) ∈ DivRing))
63, 5rabeqbidv 3450 . . . . . 6 (𝑀 = 𝑅 β†’ {𝑠 ∈ (SubRingβ€˜π‘€) ∣ (𝑀 β†Ύs 𝑠) ∈ DivRing} = {𝑠 ∈ (SubRingβ€˜π‘…) ∣ (𝑅 β†Ύs 𝑠) ∈ DivRing})
7 fvex 6905 . . . . . . 7 (SubRingβ€˜π‘…) ∈ V
87rabex 5333 . . . . . 6 {𝑠 ∈ (SubRingβ€˜π‘…) ∣ (𝑅 β†Ύs 𝑠) ∈ DivRing} ∈ V
96, 1, 8fvmpt 6999 . . . . 5 (𝑅 ∈ DivRing β†’ (SubDRingβ€˜π‘…) = {𝑠 ∈ (SubRingβ€˜π‘…) ∣ (𝑅 β†Ύs 𝑠) ∈ DivRing})
109eleq2d 2820 . . . 4 (𝑅 ∈ DivRing β†’ (𝑆 ∈ (SubDRingβ€˜π‘…) ↔ 𝑆 ∈ {𝑠 ∈ (SubRingβ€˜π‘…) ∣ (𝑅 β†Ύs 𝑠) ∈ DivRing}))
11 oveq2 7417 . . . . . 6 (𝑠 = 𝑆 β†’ (𝑅 β†Ύs 𝑠) = (𝑅 β†Ύs 𝑆))
1211eleq1d 2819 . . . . 5 (𝑠 = 𝑆 β†’ ((𝑅 β†Ύs 𝑠) ∈ DivRing ↔ (𝑅 β†Ύs 𝑆) ∈ DivRing))
1312elrab 3684 . . . 4 (𝑆 ∈ {𝑠 ∈ (SubRingβ€˜π‘…) ∣ (𝑅 β†Ύs 𝑠) ∈ DivRing} ↔ (𝑆 ∈ (SubRingβ€˜π‘…) ∧ (𝑅 β†Ύs 𝑆) ∈ DivRing))
1410, 13bitrdi 287 . . 3 (𝑅 ∈ DivRing β†’ (𝑆 ∈ (SubDRingβ€˜π‘…) ↔ (𝑆 ∈ (SubRingβ€˜π‘…) ∧ (𝑅 β†Ύs 𝑆) ∈ DivRing)))
152, 14biadanii 821 . 2 (𝑆 ∈ (SubDRingβ€˜π‘…) ↔ (𝑅 ∈ DivRing ∧ (𝑆 ∈ (SubRingβ€˜π‘…) ∧ (𝑅 β†Ύs 𝑆) ∈ DivRing)))
16 3anass 1096 . 2 ((𝑅 ∈ DivRing ∧ 𝑆 ∈ (SubRingβ€˜π‘…) ∧ (𝑅 β†Ύs 𝑆) ∈ DivRing) ↔ (𝑅 ∈ DivRing ∧ (𝑆 ∈ (SubRingβ€˜π‘…) ∧ (𝑅 β†Ύs 𝑆) ∈ DivRing)))
1715, 16bitr4i 278 1 (𝑆 ∈ (SubDRingβ€˜π‘…) ↔ (𝑅 ∈ DivRing ∧ 𝑆 ∈ (SubRingβ€˜π‘…) ∧ (𝑅 β†Ύs 𝑆) ∈ DivRing))
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  {crab 3433  β€˜cfv 6544  (class class class)co 7409   β†Ύs cress 17173  SubRingcsubrg 20315  DivRingcdr 20357  SubDRingcsdrg 20402
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fv 6552  df-ov 7412  df-sdrg 20403
This theorem is referenced by:  sdrgrcl  20405  sdrgdrng  20406  sdrgsubrg  20407  sdrgid  20408  sdrgss  20409  issdrg2  20411  fldsdrgfld  20414  sdrgint  20420  primefld  20421  primefld0cl  20422  primefld1cl  20423  sdrgdvcl  32397  sdrginvcl  32398  primefldchr  32399  fldgensdrg  32404  fldgenssp  32408  primefldgen1  32411  1fldgenq  32412  irngnzply1lem  32754  irngnzply1  32755  ply1annig1p  32765  minplycl  32767  ply1annprmidl  32768  algextdeglem1  32772
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