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Theorem mhmlin 14515
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 2358 . . . . . 6  |-  ( Base `  T )  =  (
Base `  T )
3 mhmlin.p . . . . . 6  |-  .+  =  ( +g  `  S )
4 mhmlin.q . . . . . 6  |-  .+^  =  ( +g  `  T )
5 eqid 2358 . . . . . 6  |-  ( 0g
`  S )  =  ( 0g `  S
)
6 eqid 2358 . . . . . 6  |-  ( 0g
`  T )  =  ( 0g `  T
)
71, 2, 3, 4, 5, 6ismhm 14510 . . . . 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 450 . . . 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 968 . . 3  |-  ( F  e.  ( S MndHom  T
)  ->  A. x  e.  B  A. y  e.  B  ( F `  ( x  .+  y
) )  =  ( ( F `  x
)  .+^  ( F `  y ) ) )
10 oveq1 5949 . . . . . 6  |-  ( x  =  X  ->  (
x  .+  y )  =  ( X  .+  y ) )
1110fveq2d 5609 . . . . 5  |-  ( x  =  X  ->  ( F `  ( x  .+  y ) )  =  ( F `  ( X  .+  y ) ) )
12 fveq2 5605 . . . . . 6  |-  ( x  =  X  ->  ( F `  x )  =  ( F `  X ) )
1312oveq1d 5957 . . . . 5  |-  ( x  =  X  ->  (
( F `  x
)  .+^  ( F `  y ) )  =  ( ( F `  X )  .+^  ( F `
 y ) ) )
1411, 13eqeq12d 2372 . . . 4  |-  ( x  =  X  ->  (
( F `  (
x  .+  y )
)  =  ( ( F `  x ) 
.+^  ( F `  y ) )  <->  ( F `  ( X  .+  y
) )  =  ( ( F `  X
)  .+^  ( F `  y ) ) ) )
15 oveq2 5950 . . . . . 6  |-  ( y  =  Y  ->  ( X  .+  y )  =  ( X  .+  Y
) )
1615fveq2d 5609 . . . . 5  |-  ( y  =  Y  ->  ( F `  ( X  .+  y ) )  =  ( F `  ( X  .+  Y ) ) )
17 fveq2 5605 . . . . . 6  |-  ( y  =  Y  ->  ( F `  y )  =  ( F `  Y ) )
1817oveq2d 5958 . . . . 5  |-  ( y  =  Y  ->  (
( F `  X
)  .+^  ( F `  y ) )  =  ( ( F `  X )  .+^  ( F `
 Y ) ) )
1916, 18eqeq12d 2372 . . . 4  |-  ( y  =  Y  ->  (
( F `  ( X  .+  y ) )  =  ( ( F `
 X )  .+^  ( F `  y ) )  <->  ( F `  ( X  .+  Y ) )  =  ( ( F `  X ) 
.+^  ( F `  Y ) ) ) )
2014, 19rspc2v 2966 . . 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 26 . 2  |-  ( F  e.  ( S MndHom  T
)  ->  ( ( X  e.  B  /\  Y  e.  B )  ->  ( F `  ( X  .+  Y ) )  =  ( ( F `
 X )  .+^  ( F `  Y ) ) ) )
22213impib 1149 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 358    /\ w3a 934    = wceq 1642    e. wcel 1710   A.wral 2619   -->wf 5330   ` cfv 5334  (class class class)co 5942   Basecbs 13239   +g cplusg 13299   0gc0g 13493   Mndcmnd 14454   MndHom cmhm 14506
This theorem is referenced by:  resmhm  14529  resmhm2  14530  resmhm2b  14531  mhmco  14532  mhmima  14533  mhmeql  14534  pwsco2mhm  14540  gsumwmhm  14560  mhmmulg  14692  ghmmhmb  14787  cntzmhm  14907  gsumzmhm  15303  rhmmul  15598  evlslem1  19497  mpfind  19526  dchrzrhmul  20591  dchrmulcl  20594  dchrn0  20595  dchrinvcl  20598  dchrsum2  20613  sum2dchr  20619  mhmhmeotmd  23469  mhmvlin  26775
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-br 4103  df-opab 4157  df-id 4388  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-map 6859  df-mhm 14508
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