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Theorem ofaddmndmap 30883
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 991 . . . 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 755 . . . 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 756 . . . 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 15540 . . . 4  |-  ( ( M  e.  Mnd  /\  x  e.  R  /\  y  e.  R )  ->  ( x  .+  y
)  e.  R )
71, 2, 3, 6syl3anc 1219 . . 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 7345 . . . . 5  |-  ( A  e.  ( R  ^m  V )  ->  A : V --> R )
98adantr 465 . . . 4  |-  ( ( A  e.  ( R  ^m  V )  /\  B  e.  ( R  ^m  V ) )  ->  A : V --> R )
1093ad2ant3 1011 . . 3  |-  ( ( M  e.  Mnd  /\  V  e.  Y  /\  ( A  e.  ( R  ^m  V )  /\  B  e.  ( R  ^m  V ) ) )  ->  A : V --> R )
11 elmapi 7345 . . . . 5  |-  ( B  e.  ( R  ^m  V )  ->  B : V --> R )
1211adantl 466 . . . 4  |-  ( ( A  e.  ( R  ^m  V )  /\  B  e.  ( R  ^m  V ) )  ->  B : V --> R )
13123ad2ant3 1011 . . 3  |-  ( ( M  e.  Mnd  /\  V  e.  Y  /\  ( A  e.  ( R  ^m  V )  /\  B  e.  ( R  ^m  V ) ) )  ->  B : V --> R )
14 simp2 989 . . 3  |-  ( ( M  e.  Mnd  /\  V  e.  Y  /\  ( A  e.  ( R  ^m  V )  /\  B  e.  ( R  ^m  V ) ) )  ->  V  e.  Y
)
15 inidm 3668 . . 3  |-  ( V  i^i  V )  =  V
167, 10, 13, 14, 14, 15off 6445 . 2  |-  ( ( M  e.  Mnd  /\  V  e.  Y  /\  ( A  e.  ( R  ^m  V )  /\  B  e.  ( R  ^m  V ) ) )  ->  ( A  oF  .+  B ) : V --> R )
17 fvex 5810 . . . 4  |-  ( Base `  M )  e.  _V
184, 17eqeltri 2538 . . 3  |-  R  e. 
_V
19 elmapg 7338 . . 3  |-  ( ( R  e.  _V  /\  V  e.  Y )  ->  ( ( A  oF  .+  B )  e.  ( R  ^m  V
)  <->  ( A  oF  .+  B ) : V --> R ) )
2018, 14, 19sylancr 663 . 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 369    /\ w3a 965    = wceq 1370    e. wcel 1758   _Vcvv 3078   -->wf 5523   ` cfv 5527  (class class class)co 6201    oFcof 6429    ^m cmap 7325   Basecbs 14293   +g cplusg 14358   Mndcmnd 15529
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-of 6431  df-1st 6688  df-2nd 6689  df-map 7327  df-mnd 15535
This theorem is referenced by:  lincsumcl  31098
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