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Theorem rngolidm 34222
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)
uridm.3 𝑈 = (GId‘𝐻)
Assertion
Ref Expression
rngolidm ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝑈𝐻𝐴) = 𝐴)

Proof of Theorem rngolidm
StepHypRef Expression
1 uridm.1 . . 3 𝐻 = (2nd𝑅)
2 uridm.2 . . 3 𝑋 = ran (1st𝑅)
3 uridm.3 . . 3 𝑈 = (GId‘𝐻)
41, 2, 3rngoidmlem 34221 . 2 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → ((𝑈𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑈) = 𝐴))
54simpld 489 1 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝑈𝐻𝐴) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385   = wceq 1653  wcel 2157  ran crn 5314  cfv 6102  (class class class)co 6879  1st c1st 7400  2nd c2nd 7401  GIdcgi 27869  RingOpscrngo 34179
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778  ax-sep 4976  ax-nul 4984  ax-pow 5036  ax-pr 5098  ax-un 7184
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2592  df-eu 2610  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ne 2973  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3388  df-sbc 3635  df-csb 3730  df-dif 3773  df-un 3775  df-in 3777  df-ss 3784  df-nul 4117  df-if 4279  df-sn 4370  df-pr 4372  df-op 4376  df-uni 4630  df-iun 4713  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5221  df-xp 5319  df-rel 5320  df-cnv 5321  df-co 5322  df-dm 5323  df-rn 5324  df-iota 6065  df-fun 6104  df-fn 6105  df-f 6106  df-fo 6108  df-fv 6110  df-riota 6840  df-ov 6882  df-1st 7402  df-2nd 7403  df-grpo 27872  df-gid 27873  df-ablo 27924  df-ass 34128  df-exid 34130  df-mgmOLD 34134  df-sgrOLD 34146  df-mndo 34152  df-rngo 34180
This theorem is referenced by:  rngonegmn1l  34226  zerdivemp1x  34232  isdrngo2  34243  1idl  34311  smprngopr  34337  prnc  34352
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