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Theorem rngoridm 34062
Description: The unit of a ring is an identity element for the multiplication. (Contributed by FL, 18-Apr-2010.) (New usage is discouraged.)
Hypotheses
Ref Expression
uridm.1 𝐻 = (2nd𝑅)
uridm.2 𝑋 = ran (1st𝑅)
uridm2.2 𝑈 = (GId‘𝐻)
Assertion
Ref Expression
rngoridm ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝐴𝐻𝑈) = 𝐴)

Proof of Theorem rngoridm
StepHypRef Expression
1 uridm.1 . . 3 𝐻 = (2nd𝑅)
2 uridm.2 . . 3 𝑋 = ran (1st𝑅)
3 uridm2.2 . . 3 𝑈 = (GId‘𝐻)
41, 2, 3rngoidmlem 34060 . 2 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → ((𝑈𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑈) = 𝐴))
54simprd 477 1 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝐴𝐻𝑈) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1630  wcel 2144  ran crn 5250  cfv 6031  (class class class)co 6792  1st c1st 7312  2nd c2nd 7313  GIdcgi 27678  RingOpscrngo 34018
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rmo 3068  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fo 6037  df-fv 6039  df-riota 6753  df-ov 6795  df-1st 7314  df-2nd 7315  df-grpo 27681  df-gid 27682  df-ablo 27733  df-ass 33967  df-exid 33969  df-mgmOLD 33973  df-sgrOLD 33985  df-mndo 33991  df-rngo 34019
This theorem is referenced by:  rngoueqz  34064  rngonegmn1r  34066
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