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Theorem mgmlion 25337
Description: If a magma has a left identity element, it is onto. (Contributed by FL, 25-Sep-2011.)
Hypothesis
Ref Expression
mgmlion.1  |-  X  =  dom  dom  G
Assertion
Ref Expression
mgmlion  |-  ( ( G  e.  Magma  /\  U  e.  X  /\  A. x  e.  X  ( U G x )  =  x )  ->  G : ( X  X.  X ) -onto-> X )
Distinct variable groups:    x, G    x, U    x, X

Proof of Theorem mgmlion
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mgmlion.1 . . . . 5  |-  X  =  dom  dom  G
21ismgm 20987 . . . 4  |-  ( G  e.  Magma  ->  ( G  e.  Magma 
<->  G : ( X  X.  X ) --> X ) )
32ibi 232 . . 3  |-  ( G  e.  Magma  ->  G :
( X  X.  X
) --> X )
433ad2ant1 976 . 2  |-  ( ( G  e.  Magma  /\  U  e.  X  /\  A. x  e.  X  ( U G x )  =  x )  ->  G : ( X  X.  X ) --> X )
5 simpllr 735 . . . . . . 7  |-  ( ( ( ( G  e. 
Magma  /\  U  e.  X
)  /\  x  e.  X )  /\  ( U G x )  =  x )  ->  U  e.  X )
6 simplr 731 . . . . . . . 8  |-  ( ( ( ( G  e. 
Magma  /\  U  e.  X
)  /\  x  e.  X )  /\  ( U G x )  =  x )  ->  x  e.  X )
7 id 19 . . . . . . . . . 10  |-  ( x  =  ( U G x )  ->  x  =  ( U G x ) )
87eqcoms 2286 . . . . . . . . 9  |-  ( ( U G x )  =  x  ->  x  =  ( U G x ) )
98adantl 452 . . . . . . . 8  |-  ( ( ( ( G  e. 
Magma  /\  U  e.  X
)  /\  x  e.  X )  /\  ( U G x )  =  x )  ->  x  =  ( U G x ) )
10 oveq2 5866 . . . . . . . . . 10  |-  ( z  =  x  ->  ( U G z )  =  ( U G x ) )
1110eqeq2d 2294 . . . . . . . . 9  |-  ( z  =  x  ->  (
x  =  ( U G z )  <->  x  =  ( U G x ) ) )
1211rspcev 2884 . . . . . . . 8  |-  ( ( x  e.  X  /\  x  =  ( U G x ) )  ->  E. z  e.  X  x  =  ( U G z ) )
136, 9, 12syl2anc 642 . . . . . . 7  |-  ( ( ( ( G  e. 
Magma  /\  U  e.  X
)  /\  x  e.  X )  /\  ( U G x )  =  x )  ->  E. z  e.  X  x  =  ( U G z ) )
145, 13jca 518 . . . . . 6  |-  ( ( ( ( G  e. 
Magma  /\  U  e.  X
)  /\  x  e.  X )  /\  ( U G x )  =  x )  ->  ( U  e.  X  /\  E. z  e.  X  x  =  ( U G z ) ) )
1514ex 423 . . . . 5  |-  ( ( ( G  e.  Magma  /\  U  e.  X )  /\  x  e.  X
)  ->  ( ( U G x )  =  x  ->  ( U  e.  X  /\  E. z  e.  X  x  =  ( U G z ) ) ) )
16 oveq1 5865 . . . . . . . 8  |-  ( y  =  U  ->  (
y G z )  =  ( U G z ) )
1716eqeq2d 2294 . . . . . . 7  |-  ( y  =  U  ->  (
x  =  ( y G z )  <->  x  =  ( U G z ) ) )
1817rexbidv 2564 . . . . . 6  |-  ( y  =  U  ->  ( E. z  e.  X  x  =  ( y G z )  <->  E. z  e.  X  x  =  ( U G z ) ) )
1918rspcev 2884 . . . . 5  |-  ( ( U  e.  X  /\  E. z  e.  X  x  =  ( U G z ) )  ->  E. y  e.  X  E. z  e.  X  x  =  ( y G z ) )
2015, 19syl6 29 . . . 4  |-  ( ( ( G  e.  Magma  /\  U  e.  X )  /\  x  e.  X
)  ->  ( ( U G x )  =  x  ->  E. y  e.  X  E. z  e.  X  x  =  ( y G z ) ) )
2120ralimdva 2621 . . 3  |-  ( ( G  e.  Magma  /\  U  e.  X )  ->  ( A. x  e.  X  ( U G x )  =  x  ->  A. x  e.  X  E. y  e.  X  E. z  e.  X  x  =  ( y G z ) ) )
22213impia 1148 . 2  |-  ( ( G  e.  Magma  /\  U  e.  X  /\  A. x  e.  X  ( U G x )  =  x )  ->  A. x  e.  X  E. y  e.  X  E. z  e.  X  x  =  ( y G z ) )
23 foov 5994 . 2  |-  ( G : ( X  X.  X ) -onto-> X  <->  ( G : ( X  X.  X ) --> X  /\  A. x  e.  X  E. y  e.  X  E. z  e.  X  x  =  ( y G z ) ) )
244, 22, 23sylanbrc 645 1  |-  ( ( G  e.  Magma  /\  U  e.  X  /\  A. x  e.  X  ( U G x )  =  x )  ->  G : ( X  X.  X ) -onto-> X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544    X. cxp 4687   dom cdm 4689   -->wf 5251   -onto->wfo 5253  (class class class)co 5858   Magmacmagm 20985
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-csb 3082  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-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  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-fo 5261  df-fv 5263  df-ov 5861  df-mgm 20986
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