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Theorem mhmlin 14700
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 2404 . . . . . 6  |-  ( Base `  T )  =  (
Base `  T )
3 mhmlin.p . . . . . 6  |-  .+  =  ( +g  `  S )
4 mhmlin.q . . . . . 6  |-  .+^  =  ( +g  `  T )
5 eqid 2404 . . . . . 6  |-  ( 0g
`  S )  =  ( 0g `  S
)
6 eqid 2404 . . . . . 6  |-  ( 0g
`  T )  =  ( 0g `  T
)
71, 2, 3, 4, 5, 6ismhm 14695 . . . . 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 451 . . . 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 970 . . 3  |-  ( F  e.  ( S MndHom  T
)  ->  A. x  e.  B  A. y  e.  B  ( F `  ( x  .+  y
) )  =  ( ( F `  x
)  .+^  ( F `  y ) ) )
10 oveq1 6047 . . . . . 6  |-  ( x  =  X  ->  (
x  .+  y )  =  ( X  .+  y ) )
1110fveq2d 5691 . . . . 5  |-  ( x  =  X  ->  ( F `  ( x  .+  y ) )  =  ( F `  ( X  .+  y ) ) )
12 fveq2 5687 . . . . . 6  |-  ( x  =  X  ->  ( F `  x )  =  ( F `  X ) )
1312oveq1d 6055 . . . . 5  |-  ( x  =  X  ->  (
( F `  x
)  .+^  ( F `  y ) )  =  ( ( F `  X )  .+^  ( F `
 y ) ) )
1411, 13eqeq12d 2418 . . . 4  |-  ( x  =  X  ->  (
( F `  (
x  .+  y )
)  =  ( ( F `  x ) 
.+^  ( F `  y ) )  <->  ( F `  ( X  .+  y
) )  =  ( ( F `  X
)  .+^  ( F `  y ) ) ) )
15 oveq2 6048 . . . . . 6  |-  ( y  =  Y  ->  ( X  .+  y )  =  ( X  .+  Y
) )
1615fveq2d 5691 . . . . 5  |-  ( y  =  Y  ->  ( F `  ( X  .+  y ) )  =  ( F `  ( X  .+  Y ) ) )
17 fveq2 5687 . . . . . 6  |-  ( y  =  Y  ->  ( F `  y )  =  ( F `  Y ) )
1817oveq2d 6056 . . . . 5  |-  ( y  =  Y  ->  (
( F `  X
)  .+^  ( F `  y ) )  =  ( ( F `  X )  .+^  ( F `
 Y ) ) )
1916, 18eqeq12d 2418 . . . 4  |-  ( y  =  Y  ->  (
( F `  ( X  .+  y ) )  =  ( ( F `
 X )  .+^  ( F `  y ) )  <->  ( F `  ( X  .+  Y ) )  =  ( ( F `  X ) 
.+^  ( F `  Y ) ) ) )
2014, 19rspc2v 3018 . . 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 28 . 2  |-  ( F  e.  ( S MndHom  T
)  ->  ( ( X  e.  B  /\  Y  e.  B )  ->  ( F `  ( X  .+  Y ) )  =  ( ( F `
 X )  .+^  ( F `  Y ) ) ) )
22213impib 1151 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 359    /\ w3a 936    = wceq 1649    e. wcel 1721   A.wral 2666   -->wf 5409   ` cfv 5413  (class class class)co 6040   Basecbs 13424   +g cplusg 13484   0gc0g 13678   Mndcmnd 14639   MndHom cmhm 14691
This theorem is referenced by:  resmhm  14714  resmhm2  14715  resmhm2b  14716  mhmco  14717  mhmima  14718  mhmeql  14719  pwsco2mhm  14725  gsumwmhm  14745  mhmmulg  14877  ghmmhmb  14972  cntzmhm  15092  gsumzmhm  15488  rhmmul  15783  evlslem1  19889  mpfind  19918  dchrzrhmul  20983  dchrmulcl  20986  dchrn0  20987  dchrinvcl  20990  dchrsum2  21005  sum2dchr  21011  mhmhmeotmd  24266  mhmvlin  27320
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-map 6979  df-mhm 14693
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