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Theorem rngo1cl 37947
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 37943 . . . . 5 (𝑅 ∈ RingOps → 𝐻 ∈ MndOp)
31eleq1i 2831 . . . . . 6 (𝐻 ∈ MndOp ↔ (2nd𝑅) ∈ MndOp)
4 mndoismgmOLD 37878 . . . . . . 7 ((2nd𝑅) ∈ MndOp → (2nd𝑅) ∈ Magma)
5 mndoisexid 37877 . . . . . . 7 ((2nd𝑅) ∈ MndOp → (2nd𝑅) ∈ ExId )
64, 5jca 511 . . . . . 6 ((2nd𝑅) ∈ MndOp → ((2nd𝑅) ∈ Magma ∧ (2nd𝑅) ∈ ExId ))
73, 6sylbi 217 . . . . 5 (𝐻 ∈ MndOp → ((2nd𝑅) ∈ Magma ∧ (2nd𝑅) ∈ ExId ))
82, 7syl 17 . . . 4 (𝑅 ∈ RingOps → ((2nd𝑅) ∈ Magma ∧ (2nd𝑅) ∈ ExId ))
9 elin 3966 . . . 4 ((2nd𝑅) ∈ (Magma ∩ ExId ) ↔ ((2nd𝑅) ∈ Magma ∧ (2nd𝑅) ∈ ExId ))
108, 9sylibr 234 . . 3 (𝑅 ∈ RingOps → (2nd𝑅) ∈ (Magma ∩ ExId ))
11 eqid 2736 . . . 4 ran (2nd𝑅) = ran (2nd𝑅)
12 ring1cl.3 . . . . 5 𝑈 = (GId‘𝐻)
131fveq2i 6908 . . . . 5 (GId‘𝐻) = (GId‘(2nd𝑅))
1412, 13eqtri 2764 . . . 4 𝑈 = (GId‘(2nd𝑅))
1511, 14iorlid 37866 . . 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 2736 . . . 4 (2nd𝑅) = (2nd𝑅)
19 eqid 2736 . . . 4 (1st𝑅) = (1st𝑅)
2018, 19rngorn1eq 37942 . . 3 (𝑅 ∈ RingOps → ran (1st𝑅) = ran (2nd𝑅))
21 eqtr 2759 . . . 4 ((𝑋 = ran (1st𝑅) ∧ ran (1st𝑅) = ran (2nd𝑅)) → 𝑋 = ran (2nd𝑅))
2221eleq2d 2826 . . 3 ((𝑋 = ran (1st𝑅) ∧ ran (1st𝑅) = ran (2nd𝑅)) → (𝑈𝑋𝑈 ∈ ran (2nd𝑅)))
2317, 20, 22sylancr 587 . 2 (𝑅 ∈ RingOps → (𝑈𝑋𝑈 ∈ ran (2nd𝑅)))
2416, 23mpbird 257 1 (𝑅 ∈ RingOps → 𝑈𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1539  wcel 2107  cin 3949  ran crn 5685  cfv 6560  1st c1st 8013  2nd c2nd 8014  GIdcgi 30510   ExId cexid 37852  Magmacmagm 37856  MndOpcmndo 37874  RingOpscrngo 37902
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-fo 6566  df-fv 6568  df-riota 7389  df-ov 7435  df-1st 8015  df-2nd 8016  df-grpo 30513  df-gid 30514  df-ablo 30565  df-ass 37851  df-exid 37853  df-mgmOLD 37857  df-sgrOLD 37869  df-mndo 37875  df-rngo 37903
This theorem is referenced by:  rngoueqz  37948  rngonegmn1l  37949  rngonegmn1r  37950  rngoneglmul  37951  rngonegrmul  37952  isdrngo2  37966  rngohomco  37982  rngoisocnv  37989  idlnegcl  38030  1idl  38034  0rngo  38035  smprngopr  38060  prnc  38075  isfldidl  38076  isdmn3  38082
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