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Theorem ofaddmndmap 33172
Description: The function operation applied to the addition for functions (with the same domain) into a monoid is a function (with the same domain) into the monoid. (Contributed by AV, 6-Apr-2019.)
Hypotheses
Ref Expression
ofaddmndmap.r  |-  R  =  ( Base `  M
)
ofaddmndmap.p  |-  .+  =  ( +g  `  M )
Assertion
Ref Expression
ofaddmndmap  |-  ( ( M  e.  Mnd  /\  V  e.  Y  /\  ( A  e.  ( R  ^m  V )  /\  B  e.  ( R  ^m  V ) ) )  ->  ( A  oF  .+  B )  e.  ( R  ^m  V
) )

Proof of Theorem ofaddmndmap
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl1 997 . . . 4  |-  ( ( ( M  e.  Mnd  /\  V  e.  Y  /\  ( A  e.  ( R  ^m  V )  /\  B  e.  ( R  ^m  V ) ) )  /\  ( x  e.  R  /\  y  e.  R ) )  ->  M  e.  Mnd )
2 simprl 754 . . . 4  |-  ( ( ( M  e.  Mnd  /\  V  e.  Y  /\  ( A  e.  ( R  ^m  V )  /\  B  e.  ( R  ^m  V ) ) )  /\  ( x  e.  R  /\  y  e.  R ) )  ->  x  e.  R )
3 simprr 755 . . . 4  |-  ( ( ( M  e.  Mnd  /\  V  e.  Y  /\  ( A  e.  ( R  ^m  V )  /\  B  e.  ( R  ^m  V ) ) )  /\  ( x  e.  R  /\  y  e.  R ) )  -> 
y  e.  R )
4 ofaddmndmap.r . . . . 5  |-  R  =  ( Base `  M
)
5 ofaddmndmap.p . . . . 5  |-  .+  =  ( +g  `  M )
64, 5mndcl 16069 . . . 4  |-  ( ( M  e.  Mnd  /\  x  e.  R  /\  y  e.  R )  ->  ( x  .+  y
)  e.  R )
71, 2, 3, 6syl3anc 1226 . . 3  |-  ( ( ( M  e.  Mnd  /\  V  e.  Y  /\  ( A  e.  ( R  ^m  V )  /\  B  e.  ( R  ^m  V ) ) )  /\  ( x  e.  R  /\  y  e.  R ) )  -> 
( x  .+  y
)  e.  R )
8 elmapi 7381 . . . . 5  |-  ( A  e.  ( R  ^m  V )  ->  A : V --> R )
98adantr 463 . . . 4  |-  ( ( A  e.  ( R  ^m  V )  /\  B  e.  ( R  ^m  V ) )  ->  A : V --> R )
1093ad2ant3 1017 . . 3  |-  ( ( M  e.  Mnd  /\  V  e.  Y  /\  ( A  e.  ( R  ^m  V )  /\  B  e.  ( R  ^m  V ) ) )  ->  A : V --> R )
11 elmapi 7381 . . . . 5  |-  ( B  e.  ( R  ^m  V )  ->  B : V --> R )
1211adantl 464 . . . 4  |-  ( ( A  e.  ( R  ^m  V )  /\  B  e.  ( R  ^m  V ) )  ->  B : V --> R )
13123ad2ant3 1017 . . 3  |-  ( ( M  e.  Mnd  /\  V  e.  Y  /\  ( A  e.  ( R  ^m  V )  /\  B  e.  ( R  ^m  V ) ) )  ->  B : V --> R )
14 simp2 995 . . 3  |-  ( ( M  e.  Mnd  /\  V  e.  Y  /\  ( A  e.  ( R  ^m  V )  /\  B  e.  ( R  ^m  V ) ) )  ->  V  e.  Y
)
15 inidm 3638 . . 3  |-  ( V  i^i  V )  =  V
167, 10, 13, 14, 14, 15off 6475 . 2  |-  ( ( M  e.  Mnd  /\  V  e.  Y  /\  ( A  e.  ( R  ^m  V )  /\  B  e.  ( R  ^m  V ) ) )  ->  ( A  oF  .+  B ) : V --> R )
17 fvex 5801 . . . 4  |-  ( Base `  M )  e.  _V
184, 17eqeltri 2480 . . 3  |-  R  e. 
_V
19 elmapg 7373 . . 3  |-  ( ( R  e.  _V  /\  V  e.  Y )  ->  ( ( A  oF  .+  B )  e.  ( R  ^m  V
)  <->  ( A  oF  .+  B ) : V --> R ) )
2018, 14, 19sylancr 661 . 2  |-  ( ( M  e.  Mnd  /\  V  e.  Y  /\  ( A  e.  ( R  ^m  V )  /\  B  e.  ( R  ^m  V ) ) )  ->  ( ( A  oF  .+  B
)  e.  ( R  ^m  V )  <->  ( A  oF  .+  B ) : V --> R ) )
2116, 20mpbird 232 1  |-  ( ( M  e.  Mnd  /\  V  e.  Y  /\  ( A  e.  ( R  ^m  V )  /\  B  e.  ( R  ^m  V ) ) )  ->  ( A  oF  .+  B )  e.  ( R  ^m  V
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1399    e. wcel 1836   _Vcvv 3051   -->wf 5509   ` cfv 5513  (class class class)co 6218    oFcof 6459    ^m cmap 7360   Basecbs 14657   +g cplusg 14725   Mndcmnd 16059
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-8 1838  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2020  ax-ext 2374  ax-rep 4495  ax-sep 4505  ax-nul 4513  ax-pow 4560  ax-pr 4618  ax-un 6513
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2236  df-mo 2237  df-clab 2382  df-cleq 2388  df-clel 2391  df-nfc 2546  df-ne 2593  df-ral 2751  df-rex 2752  df-reu 2753  df-rab 2755  df-v 3053  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3729  df-if 3875  df-pw 3946  df-sn 3962  df-pr 3964  df-op 3968  df-uni 4181  df-iun 4262  df-br 4385  df-opab 4443  df-mpt 4444  df-id 4726  df-xp 4936  df-rel 4937  df-cnv 4938  df-co 4939  df-dm 4940  df-rn 4941  df-res 4942  df-ima 4943  df-iota 5477  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-ov 6221  df-oprab 6222  df-mpt2 6223  df-of 6461  df-1st 6721  df-2nd 6722  df-map 7362  df-mgm 16012  df-sgrp 16051  df-mnd 16061
This theorem is referenced by:  lincsumcl  33271
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