MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rngolidm Structured version   Unicode version

Theorem rngolidm 22012
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
)
uridm.3  |-  U  =  (GId `  H )
Assertion
Ref Expression
rngolidm  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( U H A )  =  A )

Proof of Theorem rngolidm
StepHypRef Expression
1 uridm.1 . . 3  |-  H  =  ( 2nd `  R
)
2 uridm.2 . . 3  |-  X  =  ran  ( 1st `  R
)
3 uridm.3 . . 3  |-  U  =  (GId `  H )
41, 2, 3rngoidmlem 22011 . 2  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  (
( U H A )  =  A  /\  ( A H U )  =  A ) )
54simpld 446 1  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( U H A )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   ran crn 4879   ` cfv 5454  (class class class)co 6081   1stc1st 6347   2ndc2nd 6348  GIdcgi 21775   RingOpscrngo 21963
This theorem is referenced by:  zerdivemp1  22022  rngonegmn1l  26565  zerdivemp1x  26571  isdrngo2  26574  1idl  26636  smprngopr  26662  prnc  26677
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 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fo 5460  df-fv 5462  df-ov 6084  df-1st 6349  df-2nd 6350  df-riota 6549  df-grpo 21779  df-gid 21780  df-ablo 21870  df-ass 21901  df-exid 21903  df-mgm 21907  df-sgr 21919  df-mndo 21926  df-rngo 21964
  Copyright terms: Public domain W3C validator