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Theorem rngo1cl 24061
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 24053 . . . . 5  |-  ( R  e.  RingOps  ->  H  e. MndOp )
31eleq1i 2528 . . . . . 6  |-  ( H  e. MndOp 
<->  ( 2nd `  R
)  e. MndOp )
4 mndoismgm 23973 . . . . . . 7  |-  ( ( 2nd `  R )  e. MndOp  ->  ( 2nd `  R
)  e.  Magma )
5 mndoisexid 23972 . . . . . . 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 3640 . . . 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 2451 . . . 4  |-  ran  ( 2nd `  R )  =  ran  ( 2nd `  R
)
12 ring1cl.3 . . . . 5  |-  U  =  (GId `  H )
131fveq2i 5795 . . . . 5  |-  (GId `  H )  =  (GId
`  ( 2nd `  R
) )
1412, 13eqtri 2480 . . . 4  |-  U  =  (GId `  ( 2nd `  R ) )
1511, 14iorlid 23960 . . 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 2451 . . . 4  |-  ( 2nd `  R )  =  ( 2nd `  R )
19 eqid 2451 . . . 4  |-  ( 1st `  R )  =  ( 1st `  R )
2018, 19rngorn1eq 24052 . . 3  |-  ( R  e.  RingOps  ->  ran  ( 1st `  R )  =  ran  ( 2nd `  R ) )
21 eqtr 2477 . . . 4  |-  ( ( X  =  ran  ( 1st `  R )  /\  ran  ( 1st `  R
)  =  ran  ( 2nd `  R ) )  ->  X  =  ran  ( 2nd `  R ) )
2221eleq2d 2521 . . 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 1370    e. wcel 1758    i^i cin 3428   ran crn 4942   ` cfv 5519   1stc1st 6678   2ndc2nd 6679  GIdcgi 23819    ExId cexid 23946   Magmacmagm 23950  MndOpcmndo 23969   RingOpscrngo 24007
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-fo 5525  df-fv 5527  df-riota 6154  df-ov 6196  df-1st 6680  df-2nd 6681  df-grpo 23823  df-gid 23824  df-ablo 23914  df-ass 23945  df-exid 23947  df-mgm 23951  df-sgr 23963  df-mndo 23970  df-rngo 24008
This theorem is referenced by:  rngoueqz  24062  rngonegmn1l  28896  rngonegmn1r  28897  rngoneglmul  28898  rngonegrmul  28899  isdrngo2  28905  rngohomco  28921  rngoisocnv  28928  idlnegcl  28963  1idl  28967  0rngo  28968  smprngopr  28993  prnc  29008  isfldidl  29009  isdmn3  29015
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