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Theorem rngoridm 25554
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 25552 . 2  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  (
( U H A )  =  A  /\  ( A H U )  =  A ) )
54simprd 463 1  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( A H U )  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819   ran crn 5009   ` cfv 5594  (class class class)co 6296   1stc1st 6797   2ndc2nd 6798  GIdcgi 25316   RingOpscrngo 25504
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fo 5600  df-fv 5602  df-riota 6258  df-ov 6299  df-1st 6799  df-2nd 6800  df-grpo 25320  df-gid 25321  df-ablo 25411  df-ass 25442  df-exid 25444  df-mgmOLD 25448  df-sgrOLD 25460  df-mndo 25467  df-rngo 25505
This theorem is referenced by:  rngoueqz  25559  rngoridfz  25564  rngonegmn1r  30558
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