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Theorem rngo1cl 36448
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 36444 . . . . 5 (𝑅 ∈ RingOps → 𝐻 ∈ MndOp)
31eleq1i 2825 . . . . . 6 (𝐻 ∈ MndOp ↔ (2nd𝑅) ∈ MndOp)
4 mndoismgmOLD 36379 . . . . . . 7 ((2nd𝑅) ∈ MndOp → (2nd𝑅) ∈ Magma)
5 mndoisexid 36378 . . . . . . 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 3930 . . . 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 6849 . . . . 5 (GId‘𝐻) = (GId‘(2nd𝑅))
1412, 13eqtri 2761 . . . 4 𝑈 = (GId‘(2nd𝑅))
1511, 14iorlid 36367 . . 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 36443 . . 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 3913  ran crn 5638  cfv 6500  1st c1st 7923  2nd c2nd 7924  GIdcgi 29481   ExId cexid 36353  Magmacmagm 36357  MndOpcmndo 36375  RingOpscrngo 36403
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 5260  ax-nul 5267  ax-pr 5388  ax-un 7676
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 2941  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-fo 6506  df-fv 6508  df-riota 7317  df-ov 7364  df-1st 7925  df-2nd 7926  df-grpo 29484  df-gid 29485  df-ablo 29536  df-ass 36352  df-exid 36354  df-mgmOLD 36358  df-sgrOLD 36370  df-mndo 36376  df-rngo 36404
This theorem is referenced by:  rngoueqz  36449  rngonegmn1l  36450  rngonegmn1r  36451  rngoneglmul  36452  rngonegrmul  36453  isdrngo2  36467  rngohomco  36483  rngoisocnv  36490  idlnegcl  36531  1idl  36535  0rngo  36536  smprngopr  36561  prnc  36576  isfldidl  36577  isdmn3  36583
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