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Theorem mndfo 14683
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 448 . . 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 14658 . . . . . 6  |-  ( ( G  e.  Mnd  /\  x  e.  B  /\  y  e.  B )  ->  ( x  .+  y
)  e.  B )
543expb 1154 . . . . 5  |-  ( ( G  e.  Mnd  /\  ( x  e.  B  /\  y  e.  B
) )  ->  (
x  .+  y )  e.  B )
65ralrimivva 2766 . . . 4  |-  ( G  e.  Mnd  ->  A. x  e.  B  A. y  e.  B  ( x  .+  y )  e.  B
)
76adantr 452 . . 3  |-  ( ( G  e.  Mnd  /\  .+  Fn  ( B  X.  B ) )  ->  A. x  e.  B  A. y  e.  B  ( x  .+  y )  e.  B )
8 ffnov 6141 . . 3  |-  (  .+  : ( B  X.  B ) --> B  <->  (  .+  Fn  ( B  X.  B
)  /\  A. x  e.  B  A. y  e.  B  ( x  .+  y )  e.  B
) )
91, 7, 8sylanbrc 646 . 2  |-  ( ( G  e.  Mnd  /\  .+  Fn  ( B  X.  B ) )  ->  .+  : ( B  X.  B ) --> B )
10 simpr 448 . . . . 5  |-  ( ( G  e.  Mnd  /\  x  e.  B )  ->  x  e.  B )
11 eqid 2412 . . . . . . 7  |-  ( 0g
`  G )  =  ( 0g `  G
)
122, 11mndidcl 14677 . . . . . 6  |-  ( G  e.  Mnd  ->  ( 0g `  G )  e.  B )
1312adantr 452 . . . . 5  |-  ( ( G  e.  Mnd  /\  x  e.  B )  ->  ( 0g `  G
)  e.  B )
142, 3, 11mndrid 14680 . . . . . 6  |-  ( ( G  e.  Mnd  /\  x  e.  B )  ->  ( x  .+  ( 0g `  G ) )  =  x )
1514eqcomd 2417 . . . . 5  |-  ( ( G  e.  Mnd  /\  x  e.  B )  ->  x  =  ( x 
.+  ( 0g `  G ) ) )
16 rspceov 6083 . . . . 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 1184 . . . 4  |-  ( ( G  e.  Mnd  /\  x  e.  B )  ->  E. y  e.  B  E. z  e.  B  x  =  ( y  .+  z ) )
1817ralrimiva 2757 . . 3  |-  ( G  e.  Mnd  ->  A. x  e.  B  E. y  e.  B  E. z  e.  B  x  =  ( y  .+  z
) )
1918adantr 452 . 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 6187 . 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 646 1  |-  ( ( G  e.  Mnd  /\  .+  Fn  ( B  X.  B ) )  ->  .+  : ( B  X.  B ) -onto-> B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2674   E.wrex 2675    X. cxp 4843    Fn wfn 5416   -->wf 5417   -onto->wfo 5419   ` cfv 5421  (class class class)co 6048   Basecbs 13432   +g cplusg 13492   0gc0g 13686   Mndcmnd 14647
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-fo 5427  df-fv 5429  df-ov 6051  df-riota 6516  df-0g 13690  df-mnd 14653
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