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Theorem submcl 14755
Description: Submonoids are closed under the monoid operation. (Contributed by Mario Carneiro, 10-Mar-2015.)
Hypothesis
Ref Expression
submcl.p  |-  .+  =  ( +g  `  M )
Assertion
Ref Expression
submcl  |-  ( ( S  e.  (SubMnd `  M )  /\  X  e.  S  /\  Y  e.  S )  ->  ( X  .+  Y )  e.  S )

Proof of Theorem submcl
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 submrcl 14749 . . . . . . 7  |-  ( S  e.  (SubMnd `  M
)  ->  M  e.  Mnd )
2 eqid 2438 . . . . . . . 8  |-  ( Base `  M )  =  (
Base `  M )
3 eqid 2438 . . . . . . . 8  |-  ( 0g
`  M )  =  ( 0g `  M
)
4 submcl.p . . . . . . . 8  |-  .+  =  ( +g  `  M )
52, 3, 4issubm 14750 . . . . . . 7  |-  ( M  e.  Mnd  ->  ( S  e.  (SubMnd `  M
)  <->  ( S  C_  ( Base `  M )  /\  ( 0g `  M
)  e.  S  /\  A. x  e.  S  A. y  e.  S  (
x  .+  y )  e.  S ) ) )
61, 5syl 16 . . . . . 6  |-  ( S  e.  (SubMnd `  M
)  ->  ( S  e.  (SubMnd `  M )  <->  ( S  C_  ( Base `  M )  /\  ( 0g `  M )  e.  S  /\  A. x  e.  S  A. y  e.  S  ( x  .+  y )  e.  S
) ) )
76ibi 234 . . . . 5  |-  ( S  e.  (SubMnd `  M
)  ->  ( S  C_  ( Base `  M
)  /\  ( 0g `  M )  e.  S  /\  A. x  e.  S  A. y  e.  S  ( x  .+  y )  e.  S ) )
87simp3d 972 . . . 4  |-  ( S  e.  (SubMnd `  M
)  ->  A. x  e.  S  A. y  e.  S  ( x  .+  y )  e.  S
)
9 proplem2 13916 . . . 4  |-  ( ( ( X  e.  S  /\  Y  e.  S
)  /\  A. x  e.  S  A. y  e.  S  ( x  .+  y )  e.  S
)  ->  ( X  .+  Y )  e.  S
)
108, 9sylan2 462 . . 3  |-  ( ( ( X  e.  S  /\  Y  e.  S
)  /\  S  e.  (SubMnd `  M ) )  ->  ( X  .+  Y )  e.  S
)
1110ancoms 441 . 2  |-  ( ( S  e.  (SubMnd `  M )  /\  ( X  e.  S  /\  Y  e.  S )
)  ->  ( X  .+  Y )  e.  S
)
12113impb 1150 1  |-  ( ( S  e.  (SubMnd `  M )  /\  X  e.  S  /\  Y  e.  S )  ->  ( X  .+  Y )  e.  S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   A.wral 2707    C_ wss 3322   ` cfv 5456  (class class class)co 6083   Basecbs 13471   +g cplusg 13531   0gc0g 13725   Mndcmnd 14686  SubMndcsubmnd 14739
This theorem is referenced by:  resmhm  14761  mhmima  14765  gsumwsubmcl  14786  submmulgcl  14926  lsmsubm  15289  gsumzadd  15529  gsumzoppg  15541  symggen  27390
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
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-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  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-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fv 5464  df-ov 6086  df-submnd 14741
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