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Theorem mhmlin 14750
Description: A monoid homomorphism commutes with composition. (Contributed by Mario Carneiro, 7-Mar-2015.)
Hypotheses
Ref Expression
mhmlin.b  |-  B  =  ( Base `  S
)
mhmlin.p  |-  .+  =  ( +g  `  S )
mhmlin.q  |-  .+^  =  ( +g  `  T )
Assertion
Ref Expression
mhmlin  |-  ( ( F  e.  ( S MndHom  T )  /\  X  e.  B  /\  Y  e.  B )  ->  ( F `  ( X  .+  Y ) )  =  ( ( F `  X )  .+^  ( F `
 Y ) ) )

Proof of Theorem mhmlin
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mhmlin.b . . . . . 6  |-  B  =  ( Base `  S
)
2 eqid 2438 . . . . . 6  |-  ( Base `  T )  =  (
Base `  T )
3 mhmlin.p . . . . . 6  |-  .+  =  ( +g  `  S )
4 mhmlin.q . . . . . 6  |-  .+^  =  ( +g  `  T )
5 eqid 2438 . . . . . 6  |-  ( 0g
`  S )  =  ( 0g `  S
)
6 eqid 2438 . . . . . 6  |-  ( 0g
`  T )  =  ( 0g `  T
)
71, 2, 3, 4, 5, 6ismhm 14745 . . . . 5  |-  ( F  e.  ( S MndHom  T
)  <->  ( ( S  e.  Mnd  /\  T  e.  Mnd )  /\  ( F : B --> ( Base `  T )  /\  A. x  e.  B  A. y  e.  B  ( F `  ( x  .+  y ) )  =  ( ( F `  x )  .+^  ( F `
 y ) )  /\  ( F `  ( 0g `  S ) )  =  ( 0g
`  T ) ) ) )
87simprbi 452 . . . 4  |-  ( F  e.  ( S MndHom  T
)  ->  ( F : B --> ( Base `  T
)  /\  A. x  e.  B  A. y  e.  B  ( F `  ( x  .+  y
) )  =  ( ( F `  x
)  .+^  ( F `  y ) )  /\  ( F `  ( 0g
`  S ) )  =  ( 0g `  T ) ) )
98simp2d 971 . . 3  |-  ( F  e.  ( S MndHom  T
)  ->  A. x  e.  B  A. y  e.  B  ( F `  ( x  .+  y
) )  =  ( ( F `  x
)  .+^  ( F `  y ) ) )
10 oveq1 6091 . . . . . 6  |-  ( x  =  X  ->  (
x  .+  y )  =  ( X  .+  y ) )
1110fveq2d 5735 . . . . 5  |-  ( x  =  X  ->  ( F `  ( x  .+  y ) )  =  ( F `  ( X  .+  y ) ) )
12 fveq2 5731 . . . . . 6  |-  ( x  =  X  ->  ( F `  x )  =  ( F `  X ) )
1312oveq1d 6099 . . . . 5  |-  ( x  =  X  ->  (
( F `  x
)  .+^  ( F `  y ) )  =  ( ( F `  X )  .+^  ( F `
 y ) ) )
1411, 13eqeq12d 2452 . . . 4  |-  ( x  =  X  ->  (
( F `  (
x  .+  y )
)  =  ( ( F `  x ) 
.+^  ( F `  y ) )  <->  ( F `  ( X  .+  y
) )  =  ( ( F `  X
)  .+^  ( F `  y ) ) ) )
15 oveq2 6092 . . . . . 6  |-  ( y  =  Y  ->  ( X  .+  y )  =  ( X  .+  Y
) )
1615fveq2d 5735 . . . . 5  |-  ( y  =  Y  ->  ( F `  ( X  .+  y ) )  =  ( F `  ( X  .+  Y ) ) )
17 fveq2 5731 . . . . . 6  |-  ( y  =  Y  ->  ( F `  y )  =  ( F `  Y ) )
1817oveq2d 6100 . . . . 5  |-  ( y  =  Y  ->  (
( F `  X
)  .+^  ( F `  y ) )  =  ( ( F `  X )  .+^  ( F `
 Y ) ) )
1916, 18eqeq12d 2452 . . . 4  |-  ( y  =  Y  ->  (
( F `  ( X  .+  y ) )  =  ( ( F `
 X )  .+^  ( F `  y ) )  <->  ( F `  ( X  .+  Y ) )  =  ( ( F `  X ) 
.+^  ( F `  Y ) ) ) )
2014, 19rspc2v 3060 . . 3  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( A. x  e.  B  A. y  e.  B  ( F `  ( x  .+  y ) )  =  ( ( F `  x ) 
.+^  ( F `  y ) )  -> 
( F `  ( X  .+  Y ) )  =  ( ( F `
 X )  .+^  ( F `  Y ) ) ) )
219, 20syl5com 29 . 2  |-  ( F  e.  ( S MndHom  T
)  ->  ( ( X  e.  B  /\  Y  e.  B )  ->  ( F `  ( X  .+  Y ) )  =  ( ( F `
 X )  .+^  ( F `  Y ) ) ) )
22213impib 1152 1  |-  ( ( F  e.  ( S MndHom  T )  /\  X  e.  B  /\  Y  e.  B )  ->  ( F `  ( X  .+  Y ) )  =  ( ( F `  X )  .+^  ( F `
 Y ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   A.wral 2707   -->wf 5453   ` cfv 5457  (class class class)co 6084   Basecbs 13474   +g cplusg 13534   0gc0g 13728   Mndcmnd 14689   MndHom cmhm 14741
This theorem is referenced by:  resmhm  14764  resmhm2  14765  resmhm2b  14766  mhmco  14767  mhmima  14768  mhmeql  14769  pwsco2mhm  14775  gsumwmhm  14795  mhmmulg  14927  ghmmhmb  15022  cntzmhm  15142  gsumzmhm  15538  rhmmul  15833  evlslem1  19941  mpfind  19970  dchrzrhmul  21035  dchrmulcl  21038  dchrn0  21039  dchrinvcl  21042  dchrsum2  21057  sum2dchr  21063  mhmhmeotmd  24318  mhmvlin  27443
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-map 7023  df-mhm 14743
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