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Theorem mndfo 14397
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 447 . . 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 14372 . . . . . 6  |-  ( ( G  e.  Mnd  /\  x  e.  B  /\  y  e.  B )  ->  ( x  .+  y
)  e.  B )
543expb 1152 . . . . 5  |-  ( ( G  e.  Mnd  /\  ( x  e.  B  /\  y  e.  B
) )  ->  (
x  .+  y )  e.  B )
65ralrimivva 2635 . . . 4  |-  ( G  e.  Mnd  ->  A. x  e.  B  A. y  e.  B  ( x  .+  y )  e.  B
)
76adantr 451 . . 3  |-  ( ( G  e.  Mnd  /\  .+  Fn  ( B  X.  B ) )  ->  A. x  e.  B  A. y  e.  B  ( x  .+  y )  e.  B )
8 ffnov 5948 . . 3  |-  (  .+  : ( B  X.  B ) --> B  <->  (  .+  Fn  ( B  X.  B
)  /\  A. x  e.  B  A. y  e.  B  ( x  .+  y )  e.  B
) )
91, 7, 8sylanbrc 645 . 2  |-  ( ( G  e.  Mnd  /\  .+  Fn  ( B  X.  B ) )  ->  .+  : ( B  X.  B ) --> B )
10 simpr 447 . . . . 5  |-  ( ( G  e.  Mnd  /\  x  e.  B )  ->  x  e.  B )
11 eqid 2283 . . . . . . 7  |-  ( 0g
`  G )  =  ( 0g `  G
)
122, 11mndidcl 14391 . . . . . 6  |-  ( G  e.  Mnd  ->  ( 0g `  G )  e.  B )
1312adantr 451 . . . . 5  |-  ( ( G  e.  Mnd  /\  x  e.  B )  ->  ( 0g `  G
)  e.  B )
142, 3, 11mndrid 14394 . . . . . 6  |-  ( ( G  e.  Mnd  /\  x  e.  B )  ->  ( x  .+  ( 0g `  G ) )  =  x )
1514eqcomd 2288 . . . . 5  |-  ( ( G  e.  Mnd  /\  x  e.  B )  ->  x  =  ( x 
.+  ( 0g `  G ) ) )
16 rspceov 5893 . . . . 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 1182 . . . 4  |-  ( ( G  e.  Mnd  /\  x  e.  B )  ->  E. y  e.  B  E. z  e.  B  x  =  ( y  .+  z ) )
1817ralrimiva 2626 . . 3  |-  ( G  e.  Mnd  ->  A. x  e.  B  E. y  e.  B  E. z  e.  B  x  =  ( y  .+  z
) )
1918adantr 451 . 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 5994 . 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 645 1  |-  ( ( G  e.  Mnd  /\  .+  Fn  ( B  X.  B ) )  ->  .+  : ( B  X.  B ) -onto-> B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544    X. cxp 4687    Fn wfn 5250   -->wf 5251   -onto->wfo 5253   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208   0gc0g 13400   Mndcmnd 14361
This theorem is referenced by:  symgfo  25368
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-pow 4188  ax-pr 4214
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-reu 2550  df-rmo 2551  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-riota 6304  df-0g 13404  df-mnd 14367
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