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Theorem rngo1cl 25629
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 25621 . . . . 5  |-  ( R  e.  RingOps  ->  H  e. MndOp )
31eleq1i 2531 . . . . . 6  |-  ( H  e. MndOp 
<->  ( 2nd `  R
)  e. MndOp )
4 mndoismgmOLD 25541 . . . . . . 7  |-  ( ( 2nd `  R )  e. MndOp  ->  ( 2nd `  R
)  e.  Magma )
5 mndoisexid 25540 . . . . . . 7  |-  ( ( 2nd `  R )  e. MndOp  ->  ( 2nd `  R
)  e.  ExId  )
64, 5jca 530 . . . . . 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 3673 . . . 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 2454 . . . 4  |-  ran  ( 2nd `  R )  =  ran  ( 2nd `  R
)
12 ring1cl.3 . . . . 5  |-  U  =  (GId `  H )
131fveq2i 5851 . . . . 5  |-  (GId `  H )  =  (GId
`  ( 2nd `  R
) )
1412, 13eqtri 2483 . . . 4  |-  U  =  (GId `  ( 2nd `  R ) )
1511, 14iorlid 25528 . . 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 2454 . . . 4  |-  ( 2nd `  R )  =  ( 2nd `  R )
19 eqid 2454 . . . 4  |-  ( 1st `  R )  =  ( 1st `  R )
2018, 19rngorn1eq 25620 . . 3  |-  ( R  e.  RingOps  ->  ran  ( 1st `  R )  =  ran  ( 2nd `  R ) )
21 eqtr 2480 . . . 4  |-  ( ( X  =  ran  ( 1st `  R )  /\  ran  ( 1st `  R
)  =  ran  ( 2nd `  R ) )  ->  X  =  ran  ( 2nd `  R ) )
2221eleq2d 2524 . . 3  |-  ( ( X  =  ran  ( 1st `  R )  /\  ran  ( 1st `  R
)  =  ran  ( 2nd `  R ) )  ->  ( U  e.  X  <->  U  e.  ran  ( 2nd `  R ) ) )
2317, 20, 22sylancr 661 . 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 367    = wceq 1398    e. wcel 1823    i^i cin 3460   ran crn 4989   ` cfv 5570   1stc1st 6771   2ndc2nd 6772  GIdcgi 25387    ExId cexid 25514   Magmacmagm 25518  MndOpcmndo 25537   RingOpscrngo 25575
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-fo 5576  df-fv 5578  df-riota 6232  df-ov 6273  df-1st 6773  df-2nd 6774  df-grpo 25391  df-gid 25392  df-ablo 25482  df-ass 25513  df-exid 25515  df-mgmOLD 25519  df-sgrOLD 25531  df-mndo 25538  df-rngo 25576
This theorem is referenced by:  rngoueqz  25630  rngonegmn1l  30592  rngonegmn1r  30593  rngoneglmul  30594  rngonegrmul  30595  isdrngo2  30601  rngohomco  30617  rngoisocnv  30624  idlnegcl  30659  1idl  30663  0rngo  30664  smprngopr  30689  prnc  30704  isfldidl  30705  isdmn3  30711
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