MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rngo1cl Structured version   Unicode version

Theorem rngo1cl 25107
Description: The unit of a ring belongs to the base set. (Contributed by FL, 12-Feb-2010.) (New usage is discouraged.)
Hypotheses
Ref Expression
ring1cl.1  |-  X  =  ran  ( 1st `  R
)
ring1cl.2  |-  H  =  ( 2nd `  R
)
ring1cl.3  |-  U  =  (GId `  H )
Assertion
Ref Expression
rngo1cl  |-  ( R  e.  RingOps  ->  U  e.  X
)

Proof of Theorem rngo1cl
StepHypRef Expression
1 ring1cl.2 . . . . . 6  |-  H  =  ( 2nd `  R
)
21rngomndo 25099 . . . . 5  |-  ( R  e.  RingOps  ->  H  e. MndOp )
31eleq1i 2544 . . . . . 6  |-  ( H  e. MndOp 
<->  ( 2nd `  R
)  e. MndOp )
4 mndoismgm 25019 . . . . . . 7  |-  ( ( 2nd `  R )  e. MndOp  ->  ( 2nd `  R
)  e.  Magma )
5 mndoisexid 25018 . . . . . . 7  |-  ( ( 2nd `  R )  e. MndOp  ->  ( 2nd `  R
)  e.  ExId  )
64, 5jca 532 . . . . . 6  |-  ( ( 2nd `  R )  e. MndOp  ->  ( ( 2nd `  R )  e.  Magma  /\  ( 2nd `  R
)  e.  ExId  )
)
73, 6sylbi 195 . . . . 5  |-  ( H  e. MndOp  ->  ( ( 2nd `  R )  e.  Magma  /\  ( 2nd `  R
)  e.  ExId  )
)
82, 7syl 16 . . . 4  |-  ( R  e.  RingOps  ->  ( ( 2nd `  R )  e.  Magma  /\  ( 2nd `  R
)  e.  ExId  )
)
9 elin 3687 . . . 4  |-  ( ( 2nd `  R )  e.  ( Magma  i^i  ExId  )  <-> 
( ( 2nd `  R
)  e.  Magma  /\  ( 2nd `  R )  e. 
ExId  ) )
108, 9sylibr 212 . . 3  |-  ( R  e.  RingOps  ->  ( 2nd `  R
)  e.  ( Magma  i^i 
ExId  ) )
11 eqid 2467 . . . 4  |-  ran  ( 2nd `  R )  =  ran  ( 2nd `  R
)
12 ring1cl.3 . . . . 5  |-  U  =  (GId `  H )
131fveq2i 5867 . . . . 5  |-  (GId `  H )  =  (GId
`  ( 2nd `  R
) )
1412, 13eqtri 2496 . . . 4  |-  U  =  (GId `  ( 2nd `  R ) )
1511, 14iorlid 25006 . . 3  |-  ( ( 2nd `  R )  e.  ( Magma  i^i  ExId  )  ->  U  e.  ran  ( 2nd `  R ) )
1610, 15syl 16 . 2  |-  ( R  e.  RingOps  ->  U  e.  ran  ( 2nd `  R ) )
17 ring1cl.1 . . 3  |-  X  =  ran  ( 1st `  R
)
18 eqid 2467 . . . 4  |-  ( 2nd `  R )  =  ( 2nd `  R )
19 eqid 2467 . . . 4  |-  ( 1st `  R )  =  ( 1st `  R )
2018, 19rngorn1eq 25098 . . 3  |-  ( R  e.  RingOps  ->  ran  ( 1st `  R )  =  ran  ( 2nd `  R ) )
21 eqtr 2493 . . . 4  |-  ( ( X  =  ran  ( 1st `  R )  /\  ran  ( 1st `  R
)  =  ran  ( 2nd `  R ) )  ->  X  =  ran  ( 2nd `  R ) )
2221eleq2d 2537 . . 3  |-  ( ( X  =  ran  ( 1st `  R )  /\  ran  ( 1st `  R
)  =  ran  ( 2nd `  R ) )  ->  ( U  e.  X  <->  U  e.  ran  ( 2nd `  R ) ) )
2317, 20, 22sylancr 663 . 2  |-  ( R  e.  RingOps  ->  ( U  e.  X  <->  U  e.  ran  ( 2nd `  R ) ) )
2416, 23mpbird 232 1  |-  ( R  e.  RingOps  ->  U  e.  X
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767    i^i cin 3475   ran crn 5000   ` cfv 5586   1stc1st 6779   2ndc2nd 6780  GIdcgi 24865    ExId cexid 24992   Magmacmagm 24996  MndOpcmndo 25015   RingOpscrngo 25053
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-fo 5592  df-fv 5594  df-riota 6243  df-ov 6285  df-1st 6781  df-2nd 6782  df-grpo 24869  df-gid 24870  df-ablo 24960  df-ass 24991  df-exid 24993  df-mgm 24997  df-sgr 25009  df-mndo 25016  df-rngo 25054
This theorem is referenced by:  rngoueqz  25108  rngonegmn1l  29955  rngonegmn1r  29956  rngoneglmul  29957  rngonegrmul  29958  isdrngo2  29964  rngohomco  29980  rngoisocnv  29987  idlnegcl  30022  1idl  30026  0rngo  30027  smprngopr  30052  prnc  30067  isfldidl  30068  isdmn3  30074
  Copyright terms: Public domain W3C validator