Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rngo0cl Structured version   Visualization version   GIF version

Theorem rngo0cl 36264
Description: A ring has an additive identity element. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
ring0cl.1 𝐺 = (1st𝑅)
ring0cl.2 𝑋 = ran 𝐺
ring0cl.3 𝑍 = (GId‘𝐺)
Assertion
Ref Expression
rngo0cl (𝑅 ∈ RingOps → 𝑍𝑋)

Proof of Theorem rngo0cl
StepHypRef Expression
1 ring0cl.1 . . 3 𝐺 = (1st𝑅)
21rngogrpo 36255 . 2 (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)
3 ring0cl.2 . . 3 𝑋 = ran 𝐺
4 ring0cl.3 . . 3 𝑍 = (GId‘𝐺)
53, 4grpoidcl 29242 . 2 (𝐺 ∈ GrpOp → 𝑍𝑋)
62, 5syl 17 1 (𝑅 ∈ RingOps → 𝑍𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  ran crn 5632  cfv 6492  1st c1st 7910  GrpOpcgr 29217  GIdcgi 29218  RingOpscrngo 36239
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 2709  ax-sep 5255  ax-nul 5262  ax-pr 5383  ax-un 7663
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 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2888  df-ne 2943  df-ral 3064  df-rex 3073  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3739  df-csb 3855  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-iun 4955  df-br 5105  df-opab 5167  df-mpt 5188  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-iota 6444  df-fun 6494  df-fn 6495  df-f 6496  df-fo 6498  df-fv 6500  df-riota 7306  df-ov 7353  df-1st 7912  df-2nd 7913  df-grpo 29221  df-gid 29222  df-ablo 29273  df-rngo 36240
This theorem is referenced by:  rngolz  36267  rngorz  36268  rngosn6  36271  rngoueqz  36285  rngoidl  36369  0idl  36370  keridl  36377  prnc  36412  isdmn3  36419
  Copyright terms: Public domain W3C validator