Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rngo1cl Structured version   Visualization version   GIF version

Theorem rngo1cl 36802
Description: The unity element of a ring belongs to the base set. (Contributed by FL, 12-Feb-2010.) (New usage is discouraged.)
Hypotheses
Ref Expression
ring1cl.1 𝑋 = ran (1st𝑅)
ring1cl.2 𝐻 = (2nd𝑅)
ring1cl.3 𝑈 = (GId‘𝐻)
Assertion
Ref Expression
rngo1cl (𝑅 ∈ RingOps → 𝑈𝑋)

Proof of Theorem rngo1cl
StepHypRef Expression
1 ring1cl.2 . . . . . 6 𝐻 = (2nd𝑅)
21rngomndo 36798 . . . . 5 (𝑅 ∈ RingOps → 𝐻 ∈ MndOp)
31eleq1i 2824 . . . . . 6 (𝐻 ∈ MndOp ↔ (2nd𝑅) ∈ MndOp)
4 mndoismgmOLD 36733 . . . . . . 7 ((2nd𝑅) ∈ MndOp → (2nd𝑅) ∈ Magma)
5 mndoisexid 36732 . . . . . . 7 ((2nd𝑅) ∈ MndOp → (2nd𝑅) ∈ ExId )
64, 5jca 512 . . . . . 6 ((2nd𝑅) ∈ MndOp → ((2nd𝑅) ∈ Magma ∧ (2nd𝑅) ∈ ExId ))
73, 6sylbi 216 . . . . 5 (𝐻 ∈ MndOp → ((2nd𝑅) ∈ Magma ∧ (2nd𝑅) ∈ ExId ))
82, 7syl 17 . . . 4 (𝑅 ∈ RingOps → ((2nd𝑅) ∈ Magma ∧ (2nd𝑅) ∈ ExId ))
9 elin 3964 . . . 4 ((2nd𝑅) ∈ (Magma ∩ ExId ) ↔ ((2nd𝑅) ∈ Magma ∧ (2nd𝑅) ∈ ExId ))
108, 9sylibr 233 . . 3 (𝑅 ∈ RingOps → (2nd𝑅) ∈ (Magma ∩ ExId ))
11 eqid 2732 . . . 4 ran (2nd𝑅) = ran (2nd𝑅)
12 ring1cl.3 . . . . 5 𝑈 = (GId‘𝐻)
131fveq2i 6894 . . . . 5 (GId‘𝐻) = (GId‘(2nd𝑅))
1412, 13eqtri 2760 . . . 4 𝑈 = (GId‘(2nd𝑅))
1511, 14iorlid 36721 . . 3 ((2nd𝑅) ∈ (Magma ∩ ExId ) → 𝑈 ∈ ran (2nd𝑅))
1610, 15syl 17 . 2 (𝑅 ∈ RingOps → 𝑈 ∈ ran (2nd𝑅))
17 ring1cl.1 . . 3 𝑋 = ran (1st𝑅)
18 eqid 2732 . . . 4 (2nd𝑅) = (2nd𝑅)
19 eqid 2732 . . . 4 (1st𝑅) = (1st𝑅)
2018, 19rngorn1eq 36797 . . 3 (𝑅 ∈ RingOps → ran (1st𝑅) = ran (2nd𝑅))
21 eqtr 2755 . . . 4 ((𝑋 = ran (1st𝑅) ∧ ran (1st𝑅) = ran (2nd𝑅)) → 𝑋 = ran (2nd𝑅))
2221eleq2d 2819 . . 3 ((𝑋 = ran (1st𝑅) ∧ ran (1st𝑅) = ran (2nd𝑅)) → (𝑈𝑋𝑈 ∈ ran (2nd𝑅)))
2317, 20, 22sylancr 587 . 2 (𝑅 ∈ RingOps → (𝑈𝑋𝑈 ∈ ran (2nd𝑅)))
2416, 23mpbird 256 1 (𝑅 ∈ RingOps → 𝑈𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  cin 3947  ran crn 5677  cfv 6543  1st c1st 7972  2nd c2nd 7973  GIdcgi 29738   ExId cexid 36707  Magmacmagm 36711  MndOpcmndo 36729  RingOpscrngo 36757
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fo 6549  df-fv 6551  df-riota 7364  df-ov 7411  df-1st 7974  df-2nd 7975  df-grpo 29741  df-gid 29742  df-ablo 29793  df-ass 36706  df-exid 36708  df-mgmOLD 36712  df-sgrOLD 36724  df-mndo 36730  df-rngo 36758
This theorem is referenced by:  rngoueqz  36803  rngonegmn1l  36804  rngonegmn1r  36805  rngoneglmul  36806  rngonegrmul  36807  isdrngo2  36821  rngohomco  36837  rngoisocnv  36844  idlnegcl  36885  1idl  36889  0rngo  36890  smprngopr  36915  prnc  36930  isfldidl  36931  isdmn3  36937
  Copyright terms: Public domain W3C validator