MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  issdrg Structured version   Visualization version   GIF version

Theorem issdrg 20637
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 20636 . . . 4 SubDRing = (𝑀 ∈ DivRing ↦ {𝑠 ∈ (SubRingβ€˜π‘€) ∣ (𝑀 β†Ύs 𝑠) ∈ DivRing})
21mptrcl 7000 . . 3 (𝑆 ∈ (SubDRingβ€˜π‘…) β†’ 𝑅 ∈ DivRing)
3 fveq2 6884 . . . . . . 7 (𝑀 = 𝑅 β†’ (SubRingβ€˜π‘€) = (SubRingβ€˜π‘…))
4 oveq1 7411 . . . . . . . 8 (𝑀 = 𝑅 β†’ (𝑀 β†Ύs 𝑠) = (𝑅 β†Ύs 𝑠))
54eleq1d 2812 . . . . . . 7 (𝑀 = 𝑅 β†’ ((𝑀 β†Ύs 𝑠) ∈ DivRing ↔ (𝑅 β†Ύs 𝑠) ∈ DivRing))
63, 5rabeqbidv 3443 . . . . . 6 (𝑀 = 𝑅 β†’ {𝑠 ∈ (SubRingβ€˜π‘€) ∣ (𝑀 β†Ύs 𝑠) ∈ DivRing} = {𝑠 ∈ (SubRingβ€˜π‘…) ∣ (𝑅 β†Ύs 𝑠) ∈ DivRing})
7 fvex 6897 . . . . . . 7 (SubRingβ€˜π‘…) ∈ V
87rabex 5325 . . . . . 6 {𝑠 ∈ (SubRingβ€˜π‘…) ∣ (𝑅 β†Ύs 𝑠) ∈ DivRing} ∈ V
96, 1, 8fvmpt 6991 . . . . 5 (𝑅 ∈ DivRing β†’ (SubDRingβ€˜π‘…) = {𝑠 ∈ (SubRingβ€˜π‘…) ∣ (𝑅 β†Ύs 𝑠) ∈ DivRing})
109eleq2d 2813 . . . 4 (𝑅 ∈ DivRing β†’ (𝑆 ∈ (SubDRingβ€˜π‘…) ↔ 𝑆 ∈ {𝑠 ∈ (SubRingβ€˜π‘…) ∣ (𝑅 β†Ύs 𝑠) ∈ DivRing}))
11 oveq2 7412 . . . . . 6 (𝑠 = 𝑆 β†’ (𝑅 β†Ύs 𝑠) = (𝑅 β†Ύs 𝑆))
1211eleq1d 2812 . . . . 5 (𝑠 = 𝑆 β†’ ((𝑅 β†Ύs 𝑠) ∈ DivRing ↔ (𝑅 β†Ύs 𝑆) ∈ DivRing))
1312elrab 3678 . . . 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 819 . 2 (𝑆 ∈ (SubDRingβ€˜π‘…) ↔ (𝑅 ∈ DivRing ∧ (𝑆 ∈ (SubRingβ€˜π‘…) ∧ (𝑅 β†Ύs 𝑆) ∈ DivRing)))
16 3anass 1092 . 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 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  {crab 3426  β€˜cfv 6536  (class class class)co 7404   β†Ύs cress 17180  SubRingcsubrg 20467  DivRingcdr 20585  SubDRingcsdrg 20635
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fv 6544  df-ov 7407  df-sdrg 20636
This theorem is referenced by:  sdrgrcl  20638  sdrgdrng  20639  sdrgsubrg  20640  sdrgid  20641  sdrgss  20642  issdrg2  20644  fldsdrgfld  20647  sdrgint  20653  primefld  20654  primefld0cl  20655  primefld1cl  20656  sdrgdvcl  32900  sdrginvcl  32901  primefldchr  32902  fldgensdrg  32907  fldgenssp  32911  primefldgen1  32914  1fldgenq  32915  irngnzply1lem  33273  irngnzply1  33274  ply1annig1p  33284  minplycl  33286  ply1annprmidl  33287  algextdeglem1  33294  algextdeglem2  33295  algextdeglem3  33296  algextdeglem4  33297  algextdeglem5  33298
  Copyright terms: Public domain W3C validator