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Theorem rngo1cl 21096
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 21088 . . . . 5  |-  ( R  e.  RingOps  ->  H  e. MndOp )
31eleq1i 2346 . . . . . 6  |-  ( H  e. MndOp 
<->  ( 2nd `  R
)  e. MndOp )
4 mndoismgm 21008 . . . . . . 7  |-  ( ( 2nd `  R )  e. MndOp  ->  ( 2nd `  R
)  e.  Magma )
5 mndoisexid 21007 . . . . . . 7  |-  ( ( 2nd `  R )  e. MndOp  ->  ( 2nd `  R
)  e.  ExId  )
64, 5jca 518 . . . . . 6  |-  ( ( 2nd `  R )  e. MndOp  ->  ( ( 2nd `  R )  e.  Magma  /\  ( 2nd `  R
)  e.  ExId  )
)
73, 6sylbi 187 . . . . 5  |-  ( H  e. MndOp  ->  ( ( 2nd `  R )  e.  Magma  /\  ( 2nd `  R
)  e.  ExId  )
)
82, 7syl 15 . . . 4  |-  ( R  e.  RingOps  ->  ( ( 2nd `  R )  e.  Magma  /\  ( 2nd `  R
)  e.  ExId  )
)
9 elin 3358 . . . 4  |-  ( ( 2nd `  R )  e.  ( Magma  i^i  ExId  )  <-> 
( ( 2nd `  R
)  e.  Magma  /\  ( 2nd `  R )  e. 
ExId  ) )
108, 9sylibr 203 . . 3  |-  ( R  e.  RingOps  ->  ( 2nd `  R
)  e.  ( Magma  i^i 
ExId  ) )
11 eqid 2283 . . . 4  |-  ran  ( 2nd `  R )  =  ran  ( 2nd `  R
)
12 ring1cl.3 . . . . 5  |-  U  =  (GId `  H )
131fveq2i 5528 . . . . 5  |-  (GId `  H )  =  (GId
`  ( 2nd `  R
) )
1412, 13eqtri 2303 . . . 4  |-  U  =  (GId `  ( 2nd `  R ) )
1511, 14iorlid 20995 . . 3  |-  ( ( 2nd `  R )  e.  ( Magma  i^i  ExId  )  ->  U  e.  ran  ( 2nd `  R ) )
1610, 15syl 15 . 2  |-  ( R  e.  RingOps  ->  U  e.  ran  ( 2nd `  R ) )
17 ring1cl.1 . . 3  |-  X  =  ran  ( 1st `  R
)
18 eqid 2283 . . . 4  |-  ( 2nd `  R )  =  ( 2nd `  R )
19 eqid 2283 . . . 4  |-  ( 1st `  R )  =  ( 1st `  R )
2018, 19rngorn1eq 21087 . . 3  |-  ( R  e.  RingOps  ->  ran  ( 1st `  R )  =  ran  ( 2nd `  R ) )
21 eqtr 2300 . . . 4  |-  ( ( X  =  ran  ( 1st `  R )  /\  ran  ( 1st `  R
)  =  ran  ( 2nd `  R ) )  ->  X  =  ran  ( 2nd `  R ) )
2221eleq2d 2350 . . 3  |-  ( ( X  =  ran  ( 1st `  R )  /\  ran  ( 1st `  R
)  =  ran  ( 2nd `  R ) )  ->  ( U  e.  X  <->  U  e.  ran  ( 2nd `  R ) ) )
2317, 20, 22sylancr 644 . 2  |-  ( R  e.  RingOps  ->  ( U  e.  X  <->  U  e.  ran  ( 2nd `  R ) ) )
2416, 23mpbird 223 1  |-  ( R  e.  RingOps  ->  U  e.  X
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    i^i cin 3151   ran crn 4690   ` cfv 5255   1stc1st 6120   2ndc2nd 6121  GIdcgi 20854    ExId cexid 20981   Magmacmagm 20985  MndOpcmndo 21004   RingOpscrngo 21042
This theorem is referenced by:  rngoueqz  21097  rngonegmn1l  26580  rngonegmn1r  26581  rngoneglmul  26582  rngonegrmul  26583  isdrngo2  26589  rngohomco  26605  rngoisocnv  26612  idlnegcl  26647  1idl  26651  0rngo  26652  smprngopr  26677  prnc  26692  isfldidl  26693  isdmn3  26699
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263  df-ov 5861  df-1st 6122  df-2nd 6123  df-riota 6304  df-grpo 20858  df-gid 20859  df-ablo 20949  df-ass 20980  df-exid 20982  df-mgm 20986  df-sgr 20998  df-mndo 21005  df-rngo 21043
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