MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mndfo Structured version   Unicode version

Theorem mndfo 15818
Description: The addition operation of a monoid is an onto function (assuming it is a function). (Contributed by Mario Carneiro, 11-Oct-2013.) (Proof shortened by AV, 23-Jan-2020.)
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
StepHypRef Expression
1 mndfo.b . . . 4  |-  B  =  ( Base `  G
)
2 eqid 2467 . . . 4  |-  ( +f `  G )  =  ( +f `  G )
31, 2mndpfo 15817 . . 3  |-  ( G  e.  Mnd  ->  ( +f `  G
) : ( B  X.  B ) -onto-> B )
43adantr 465 . 2  |-  ( ( G  e.  Mnd  /\  .+  Fn  ( B  X.  B ) )  -> 
( +f `  G ) : ( B  X.  B )
-onto-> B )
5 mndfo.p . . . . . 6  |-  .+  =  ( +g  `  G )
61, 5, 2plusfeq 15753 . . . . 5  |-  (  .+  Fn  ( B  X.  B
)  ->  ( +f `  G )  =  .+  )
76eqcomd 2475 . . . 4  |-  (  .+  Fn  ( B  X.  B
)  ->  .+  =  ( +f `  G
) )
87adantl 466 . . 3  |-  ( ( G  e.  Mnd  /\  .+  Fn  ( B  X.  B ) )  ->  .+  =  ( +f `  G )
)
9 foeq1 5797 . . 3  |-  (  .+  =  ( +f `  G )  ->  (  .+  : ( B  X.  B ) -onto-> B  <->  ( +f `  G ) : ( B  X.  B ) -onto-> B ) )
108, 9syl 16 . 2  |-  ( ( G  e.  Mnd  /\  .+  Fn  ( B  X.  B ) )  -> 
(  .+  : ( B  X.  B ) -onto-> B  <-> 
( +f `  G ) : ( B  X.  B )
-onto-> B ) )
114, 10mpbird 232 1  |-  ( ( G  e.  Mnd  /\  .+  Fn  ( B  X.  B ) )  ->  .+  : ( B  X.  B ) -onto-> B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767    X. cxp 5003    Fn wfn 5589   -onto->wfo 5592   ` cfv 5594   Basecbs 14507   +g cplusg 14572   +fcplusf 15743   Mndcmnd 15793
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fo 5600  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-1st 6795  df-2nd 6796  df-0g 14714  df-plusf 15745  df-mgm 15746  df-sgrp 15785  df-mnd 15795
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator