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Theorem mndfo 14708
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 14683 . . . . . 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 2790 . . . 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 6165 . . 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 2435 . . . . . . 7  |-  ( 0g
`  G )  =  ( 0g `  G
)
122, 11mndidcl 14702 . . . . . 6  |-  ( G  e.  Mnd  ->  ( 0g `  G )  e.  B )
1312adantr 452 . . . . 5  |-  ( ( G  e.  Mnd  /\  x  e.  B )  ->  ( 0g `  G
)  e.  B )
142, 3, 11mndrid 14705 . . . . . 6  |-  ( ( G  e.  Mnd  /\  x  e.  B )  ->  ( x  .+  ( 0g `  G ) )  =  x )
1514eqcomd 2440 . . . . 5  |-  ( ( G  e.  Mnd  /\  x  e.  B )  ->  x  =  ( x 
.+  ( 0g `  G ) ) )
16 rspceov 6107 . . . . 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 2781 . . 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 6211 . 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 1652    e. wcel 1725   A.wral 2697   E.wrex 2698    X. cxp 4867    Fn wfn 5440   -->wf 5441   -onto->wfo 5443   ` cfv 5445  (class class class)co 6072   Basecbs 13457   +g cplusg 13517   0gc0g 13711   Mndcmnd 14672
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-fo 5451  df-fv 5453  df-ov 6075  df-riota 6540  df-0g 13715  df-mnd 14678
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