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Theorem mndfo 14496
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 14471 . . . . . 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 2711 . . . 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 6035 . . 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 2358 . . . . . . 7  |-  ( 0g
`  G )  =  ( 0g `  G
)
122, 11mndidcl 14490 . . . . . 6  |-  ( G  e.  Mnd  ->  ( 0g `  G )  e.  B )
1312adantr 451 . . . . 5  |-  ( ( G  e.  Mnd  /\  x  e.  B )  ->  ( 0g `  G
)  e.  B )
142, 3, 11mndrid 14493 . . . . . 6  |-  ( ( G  e.  Mnd  /\  x  e.  B )  ->  ( x  .+  ( 0g `  G ) )  =  x )
1514eqcomd 2363 . . . . 5  |-  ( ( G  e.  Mnd  /\  x  e.  B )  ->  x  =  ( x 
.+  ( 0g `  G ) ) )
16 rspceov 5980 . . . . 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 2702 . . 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 6081 . 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 1642    e. wcel 1710   A.wral 2619   E.wrex 2620    X. cxp 4769    Fn wfn 5332   -->wf 5333   -onto->wfo 5335   ` cfv 5337  (class class class)co 5945   Basecbs 13245   +g cplusg 13305   0gc0g 13499   Mndcmnd 14460
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-fo 5343  df-fv 5345  df-ov 5948  df-riota 6391  df-0g 13503  df-mnd 14466
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