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Theorem rngo1cl 22019
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 22011 . . . . 5  |-  ( R  e.  RingOps  ->  H  e. MndOp )
31eleq1i 2501 . . . . . 6  |-  ( H  e. MndOp 
<->  ( 2nd `  R
)  e. MndOp )
4 mndoismgm 21931 . . . . . . 7  |-  ( ( 2nd `  R )  e. MndOp  ->  ( 2nd `  R
)  e.  Magma )
5 mndoisexid 21930 . . . . . . 7  |-  ( ( 2nd `  R )  e. MndOp  ->  ( 2nd `  R
)  e.  ExId  )
64, 5jca 520 . . . . . 6  |-  ( ( 2nd `  R )  e. MndOp  ->  ( ( 2nd `  R )  e.  Magma  /\  ( 2nd `  R
)  e.  ExId  )
)
73, 6sylbi 189 . . . . 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 3532 . . . 4  |-  ( ( 2nd `  R )  e.  ( Magma  i^i  ExId  )  <-> 
( ( 2nd `  R
)  e.  Magma  /\  ( 2nd `  R )  e. 
ExId  ) )
108, 9sylibr 205 . . 3  |-  ( R  e.  RingOps  ->  ( 2nd `  R
)  e.  ( Magma  i^i 
ExId  ) )
11 eqid 2438 . . . 4  |-  ran  ( 2nd `  R )  =  ran  ( 2nd `  R
)
12 ring1cl.3 . . . . 5  |-  U  =  (GId `  H )
131fveq2i 5733 . . . . 5  |-  (GId `  H )  =  (GId
`  ( 2nd `  R
) )
1412, 13eqtri 2458 . . . 4  |-  U  =  (GId `  ( 2nd `  R ) )
1511, 14iorlid 21918 . . 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 2438 . . . 4  |-  ( 2nd `  R )  =  ( 2nd `  R )
19 eqid 2438 . . . 4  |-  ( 1st `  R )  =  ( 1st `  R )
2018, 19rngorn1eq 22010 . . 3  |-  ( R  e.  RingOps  ->  ran  ( 1st `  R )  =  ran  ( 2nd `  R ) )
21 eqtr 2455 . . . 4  |-  ( ( X  =  ran  ( 1st `  R )  /\  ran  ( 1st `  R
)  =  ran  ( 2nd `  R ) )  ->  X  =  ran  ( 2nd `  R ) )
2221eleq2d 2505 . . 3  |-  ( ( X  =  ran  ( 1st `  R )  /\  ran  ( 1st `  R
)  =  ran  ( 2nd `  R ) )  ->  ( U  e.  X  <->  U  e.  ran  ( 2nd `  R ) ) )
2317, 20, 22sylancr 646 . 2  |-  ( R  e.  RingOps  ->  ( U  e.  X  <->  U  e.  ran  ( 2nd `  R ) ) )
2416, 23mpbird 225 1  |-  ( R  e.  RingOps  ->  U  e.  X
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726    i^i cin 3321   ran crn 4881   ` cfv 5456   1stc1st 6349   2ndc2nd 6350  GIdcgi 21777    ExId cexid 21904   Magmacmagm 21908  MndOpcmndo 21927   RingOpscrngo 21965
This theorem is referenced by:  rngoueqz  22020  rngonegmn1l  26567  rngonegmn1r  26568  rngoneglmul  26569  rngonegrmul  26570  isdrngo2  26576  rngohomco  26592  rngoisocnv  26599  idlnegcl  26634  1idl  26638  0rngo  26639  smprngopr  26664  prnc  26679  isfldidl  26680  isdmn3  26686
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-fo 5462  df-fv 5464  df-ov 6086  df-1st 6351  df-2nd 6352  df-riota 6551  df-grpo 21781  df-gid 21782  df-ablo 21872  df-ass 21903  df-exid 21905  df-mgm 21909  df-sgr 21921  df-mndo 21928  df-rngo 21966
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