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Theorem mndfo 14751
Description: The addition operation of a monoid is an onto function (assuming it is a function). (Contributed by Mario Carneiro, 11-Oct-2013.)
Hypotheses
Ref Expression
mndfo.b  |-  B  =  ( Base `  G
)
mndfo.p  |-  .+  =  ( +g  `  G )
Assertion
Ref Expression
mndfo  |-  ( ( G  e.  Mnd  /\  .+  Fn  ( B  X.  B ) )  ->  .+  : ( B  X.  B ) -onto-> B )

Proof of Theorem mndfo
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 449 . . 3  |-  ( ( G  e.  Mnd  /\  .+  Fn  ( B  X.  B ) )  ->  .+  Fn  ( B  X.  B ) )
2 mndfo.b . . . . . . 7  |-  B  =  ( Base `  G
)
3 mndfo.p . . . . . . 7  |-  .+  =  ( +g  `  G )
42, 3mndcl 14726 . . . . . 6  |-  ( ( G  e.  Mnd  /\  x  e.  B  /\  y  e.  B )  ->  ( x  .+  y
)  e.  B )
543expb 1155 . . . . 5  |-  ( ( G  e.  Mnd  /\  ( x  e.  B  /\  y  e.  B
) )  ->  (
x  .+  y )  e.  B )
65ralrimivva 2804 . . . 4  |-  ( G  e.  Mnd  ->  A. x  e.  B  A. y  e.  B  ( x  .+  y )  e.  B
)
76adantr 453 . . 3  |-  ( ( G  e.  Mnd  /\  .+  Fn  ( B  X.  B ) )  ->  A. x  e.  B  A. y  e.  B  ( x  .+  y )  e.  B )
8 ffnov 6203 . . 3  |-  (  .+  : ( B  X.  B ) --> B  <->  (  .+  Fn  ( B  X.  B
)  /\  A. x  e.  B  A. y  e.  B  ( x  .+  y )  e.  B
) )
91, 7, 8sylanbrc 647 . 2  |-  ( ( G  e.  Mnd  /\  .+  Fn  ( B  X.  B ) )  ->  .+  : ( B  X.  B ) --> B )
10 simpr 449 . . . . 5  |-  ( ( G  e.  Mnd  /\  x  e.  B )  ->  x  e.  B )
11 eqid 2442 . . . . . . 7  |-  ( 0g
`  G )  =  ( 0g `  G
)
122, 11mndidcl 14745 . . . . . 6  |-  ( G  e.  Mnd  ->  ( 0g `  G )  e.  B )
1312adantr 453 . . . . 5  |-  ( ( G  e.  Mnd  /\  x  e.  B )  ->  ( 0g `  G
)  e.  B )
142, 3, 11mndrid 14748 . . . . . 6  |-  ( ( G  e.  Mnd  /\  x  e.  B )  ->  ( x  .+  ( 0g `  G ) )  =  x )
1514eqcomd 2447 . . . . 5  |-  ( ( G  e.  Mnd  /\  x  e.  B )  ->  x  =  ( x 
.+  ( 0g `  G ) ) )
16 rspceov 6145 . . . . 5  |-  ( ( x  e.  B  /\  ( 0g `  G )  e.  B  /\  x  =  ( x  .+  ( 0g `  G ) ) )  ->  E. y  e.  B  E. z  e.  B  x  =  ( y  .+  z
) )
1710, 13, 15, 16syl3anc 1185 . . . 4  |-  ( ( G  e.  Mnd  /\  x  e.  B )  ->  E. y  e.  B  E. z  e.  B  x  =  ( y  .+  z ) )
1817ralrimiva 2795 . . 3  |-  ( G  e.  Mnd  ->  A. x  e.  B  E. y  e.  B  E. z  e.  B  x  =  ( y  .+  z
) )
1918adantr 453 . 2  |-  ( ( G  e.  Mnd  /\  .+  Fn  ( B  X.  B ) )  ->  A. x  e.  B  E. y  e.  B  E. z  e.  B  x  =  ( y  .+  z ) )
20 foov 6249 . 2  |-  (  .+  : ( B  X.  B ) -onto-> B  <->  (  .+  : ( B  X.  B ) --> B  /\  A. x  e.  B  E. y  e.  B  E. z  e.  B  x  =  ( y  .+  z ) ) )
219, 19, 20sylanbrc 647 1  |-  ( ( G  e.  Mnd  /\  .+  Fn  ( B  X.  B ) )  ->  .+  : ( B  X.  B ) -onto-> B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1727   A.wral 2711   E.wrex 2712    X. cxp 4905    Fn wfn 5478   -->wf 5479   -onto->wfo 5481   ` cfv 5483  (class class class)co 6110   Basecbs 13500   +g cplusg 13560   0gc0g 13754   Mndcmnd 14715
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432
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 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-reu 2718  df-rmo 2719  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-id 4527  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-fo 5489  df-fv 5491  df-ov 6113  df-riota 6578  df-0g 13758  df-mnd 14721
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