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Theorem clmgm 20988
Description: Closure of a magma. (Contributed by FL, 14-Sep-2010.) (New usage is discouraged.)
Hypothesis
Ref Expression
clmgm.1  |-  X  =  dom  dom  G
Assertion
Ref Expression
clmgm  |-  ( ( G  e.  Magma  /\  A  e.  X  /\  B  e.  X )  ->  ( A G B )  e.  X )

Proof of Theorem clmgm
StepHypRef Expression
1 clmgm.1 . . . . 5  |-  X  =  dom  dom  G
21ismgm 20987 . . . 4  |-  ( G  e.  Magma  ->  ( G  e.  Magma 
<->  G : ( X  X.  X ) --> X ) )
3 fovrn 5990 . . . . 5  |-  ( ( G : ( X  X.  X ) --> X  /\  A  e.  X  /\  B  e.  X
)  ->  ( A G B )  e.  X
)
433exp 1150 . . . 4  |-  ( G : ( X  X.  X ) --> X  -> 
( A  e.  X  ->  ( B  e.  X  ->  ( A G B )  e.  X ) ) )
52, 4syl6bi 219 . . 3  |-  ( G  e.  Magma  ->  ( G  e.  Magma  ->  ( A  e.  X  ->  ( B  e.  X  ->  ( A G B )  e.  X ) ) ) )
65pm2.43i 43 . 2  |-  ( G  e.  Magma  ->  ( A  e.  X  ->  ( B  e.  X  ->  ( A G B )  e.  X ) ) )
763imp 1145 1  |-  ( ( G  e.  Magma  /\  A  e.  X  /\  B  e.  X )  ->  ( A G B )  e.  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1623    e. wcel 1684    X. cxp 4687   dom cdm 4689   -->wf 5251  (class class class)co 5858   Magmacmagm 20985
This theorem is referenced by:  reacomsmgrp2  25344  reacomsmgrp3  25345  clfsebs  25347  resgcom  25351  fprodadd  25352  isppm  25354  seqzp2  25355  exidcl  26566
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-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-rab 2552  df-v 2790  df-sbc 2992  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-br 4024  df-opab 4078  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-fv 5263  df-ov 5861  df-mgm 20986
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