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 36807
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 36803 . . . . 5 (𝑅 ∈ RingOps → 𝐻 ∈ MndOp)
31eleq1i 2825 . . . . . 6 (𝐻 ∈ MndOp ↔ (2nd𝑅) ∈ MndOp)
4 mndoismgmOLD 36738 . . . . . . 7 ((2nd𝑅) ∈ MndOp → (2nd𝑅) ∈ Magma)
5 mndoisexid 36737 . . . . . . 7 ((2nd𝑅) ∈ MndOp → (2nd𝑅) ∈ ExId )
64, 5jca 513 . . . . . 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 3965 . . . 4 ((2nd𝑅) ∈ (Magma ∩ ExId ) ↔ ((2nd𝑅) ∈ Magma ∧ (2nd𝑅) ∈ ExId ))
108, 9sylibr 233 . . 3 (𝑅 ∈ RingOps → (2nd𝑅) ∈ (Magma ∩ ExId ))
11 eqid 2733 . . . 4 ran (2nd𝑅) = ran (2nd𝑅)
12 ring1cl.3 . . . . 5 𝑈 = (GId‘𝐻)
131fveq2i 6895 . . . . 5 (GId‘𝐻) = (GId‘(2nd𝑅))
1412, 13eqtri 2761 . . . 4 𝑈 = (GId‘(2nd𝑅))
1511, 14iorlid 36726 . . 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 2733 . . . 4 (2nd𝑅) = (2nd𝑅)
19 eqid 2733 . . . 4 (1st𝑅) = (1st𝑅)
2018, 19rngorn1eq 36802 . . 3 (𝑅 ∈ RingOps → ran (1st𝑅) = ran (2nd𝑅))
21 eqtr 2756 . . . 4 ((𝑋 = ran (1st𝑅) ∧ ran (1st𝑅) = ran (2nd𝑅)) → 𝑋 = ran (2nd𝑅))
2221eleq2d 2820 . . 3 ((𝑋 = ran (1st𝑅) ∧ ran (1st𝑅) = ran (2nd𝑅)) → (𝑈𝑋𝑈 ∈ ran (2nd𝑅)))
2317, 20, 22sylancr 588 . 2 (𝑅 ∈ RingOps → (𝑈𝑋𝑈 ∈ ran (2nd𝑅)))
2416, 23mpbird 257 1 (𝑅 ∈ RingOps → 𝑈𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  cin 3948  ran crn 5678  cfv 6544  1st c1st 7973  2nd c2nd 7974  GIdcgi 29743   ExId cexid 36712  Magmacmagm 36716  MndOpcmndo 36734  RingOpscrngo 36762
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  ax-un 7725
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-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  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-iun 5000  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-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fo 6550  df-fv 6552  df-riota 7365  df-ov 7412  df-1st 7975  df-2nd 7976  df-grpo 29746  df-gid 29747  df-ablo 29798  df-ass 36711  df-exid 36713  df-mgmOLD 36717  df-sgrOLD 36729  df-mndo 36735  df-rngo 36763
This theorem is referenced by:  rngoueqz  36808  rngonegmn1l  36809  rngonegmn1r  36810  rngoneglmul  36811  rngonegrmul  36812  isdrngo2  36826  rngohomco  36842  rngoisocnv  36849  idlnegcl  36890  1idl  36894  0rngo  36895  smprngopr  36920  prnc  36935  isfldidl  36936  isdmn3  36942
  Copyright terms: Public domain W3C validator