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

Theorem isnzr 20531
Description: Property of a nonzero ring. (Contributed by Stefan O'Rear, 24-Feb-2015.)
Hypotheses
Ref Expression
isnzr.o 1 = (1r𝑅)
isnzr.z 0 = (0g𝑅)
Assertion
Ref Expression
isnzr (𝑅 ∈ NzRing ↔ (𝑅 ∈ Ring ∧ 10 ))

Proof of Theorem isnzr
Dummy variable 𝑟 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6907 . . . 4 (𝑟 = 𝑅 → (1r𝑟) = (1r𝑅))
2 isnzr.o . . . 4 1 = (1r𝑅)
31, 2eqtr4di 2793 . . 3 (𝑟 = 𝑅 → (1r𝑟) = 1 )
4 fveq2 6907 . . . 4 (𝑟 = 𝑅 → (0g𝑟) = (0g𝑅))
5 isnzr.z . . . 4 0 = (0g𝑅)
64, 5eqtr4di 2793 . . 3 (𝑟 = 𝑅 → (0g𝑟) = 0 )
73, 6neeq12d 3000 . 2 (𝑟 = 𝑅 → ((1r𝑟) ≠ (0g𝑟) ↔ 10 ))
8 df-nzr 20530 . 2 NzRing = {𝑟 ∈ Ring ∣ (1r𝑟) ≠ (0g𝑟)}
97, 8elrab2 3698 1 (𝑅 ∈ NzRing ↔ (𝑅 ∈ Ring ∧ 10 ))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1537  wcel 2106  wne 2938  cfv 6563  0gc0g 17486  1rcur 20199  Ringcrg 20251  NzRingcnzr 20529
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-nzr 20530
This theorem is referenced by:  nzrnz  20532  nzrringOLD  20534  isnzr2  20535  isnzr2hash  20536  nzrpropd  20537  opprnzrb  20538  ringelnzr  20540  subrgnzr  20611  isdomn3  20732  drngnzr  20765  zringnzr  21489  chrnzr  21563  nrginvrcn  24729  ply1nzb  26177  drngidlhash  33442  qsnzr  33463  mxidlnzr  33475  zrhnm  33930
  Copyright terms: Public domain W3C validator