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Theorem rngoridm 22005
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  =  ( 2nd `  R
)
uridm.2  |-  X  =  ran  ( 1st `  R
)
uridm2.2  |-  U  =  (GId `  H )
Assertion
Ref Expression
rngoridm  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( A H U )  =  A )

Proof of Theorem rngoridm
StepHypRef Expression
1 uridm.1 . . 3  |-  H  =  ( 2nd `  R
)
2 uridm.2 . . 3  |-  X  =  ran  ( 1st `  R
)
3 uridm2.2 . . 3  |-  U  =  (GId `  H )
41, 2, 3rngoidmlem 22003 . 2  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  (
( U H A )  =  A  /\  ( A H U )  =  A ) )
54simprd 450 1  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( A H U )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   ran crn 4871   ` cfv 5446  (class class class)co 6073   1stc1st 6339   2ndc2nd 6340  GIdcgi 21767   RingOpscrngo 21955
This theorem is referenced by:  rngoueqz  22010  rngoridfz  22015  rngonegmn1r  26557
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fo 5452  df-fv 5454  df-ov 6076  df-1st 6341  df-2nd 6342  df-riota 6541  df-grpo 21771  df-gid 21772  df-ablo 21862  df-ass 21893  df-exid 21895  df-mgm 21899  df-sgr 21911  df-mndo 21918  df-rngo 21956
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