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Theorem mhmlin 15585
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 2452 . . . . . 6  |-  ( Base `  T )  =  (
Base `  T )
3 mhmlin.p . . . . . 6  |-  .+  =  ( +g  `  S )
4 mhmlin.q . . . . . 6  |-  .+^  =  ( +g  `  T )
5 eqid 2452 . . . . . 6  |-  ( 0g
`  S )  =  ( 0g `  S
)
6 eqid 2452 . . . . . 6  |-  ( 0g
`  T )  =  ( 0g `  T
)
71, 2, 3, 4, 5, 6ismhm 15580 . . . . 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 464 . . . 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 1001 . . 3  |-  ( F  e.  ( S MndHom  T
)  ->  A. x  e.  B  A. y  e.  B  ( F `  ( x  .+  y
) )  =  ( ( F `  x
)  .+^  ( F `  y ) ) )
10 oveq1 6202 . . . . . 6  |-  ( x  =  X  ->  (
x  .+  y )  =  ( X  .+  y ) )
1110fveq2d 5798 . . . . 5  |-  ( x  =  X  ->  ( F `  ( x  .+  y ) )  =  ( F `  ( X  .+  y ) ) )
12 fveq2 5794 . . . . . 6  |-  ( x  =  X  ->  ( F `  x )  =  ( F `  X ) )
1312oveq1d 6210 . . . . 5  |-  ( x  =  X  ->  (
( F `  x
)  .+^  ( F `  y ) )  =  ( ( F `  X )  .+^  ( F `
 y ) ) )
1411, 13eqeq12d 2474 . . . 4  |-  ( x  =  X  ->  (
( F `  (
x  .+  y )
)  =  ( ( F `  x ) 
.+^  ( F `  y ) )  <->  ( F `  ( X  .+  y
) )  =  ( ( F `  X
)  .+^  ( F `  y ) ) ) )
15 oveq2 6203 . . . . . 6  |-  ( y  =  Y  ->  ( X  .+  y )  =  ( X  .+  Y
) )
1615fveq2d 5798 . . . . 5  |-  ( y  =  Y  ->  ( F `  ( X  .+  y ) )  =  ( F `  ( X  .+  Y ) ) )
17 fveq2 5794 . . . . . 6  |-  ( y  =  Y  ->  ( F `  y )  =  ( F `  Y ) )
1817oveq2d 6211 . . . . 5  |-  ( y  =  Y  ->  (
( F `  X
)  .+^  ( F `  y ) )  =  ( ( F `  X )  .+^  ( F `
 Y ) ) )
1916, 18eqeq12d 2474 . . . 4  |-  ( y  =  Y  ->  (
( F `  ( X  .+  y ) )  =  ( ( F `
 X )  .+^  ( F `  y ) )  <->  ( F `  ( X  .+  Y ) )  =  ( ( F `  X ) 
.+^  ( F `  Y ) ) ) )
2014, 19rspc2v 3180 . . 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 30 . 2  |-  ( F  e.  ( S MndHom  T
)  ->  ( ( X  e.  B  /\  Y  e.  B )  ->  ( F `  ( X  .+  Y ) )  =  ( ( F `
 X )  .+^  ( F `  Y ) ) ) )
22213impib 1186 1  |-  ( ( F  e.  ( S MndHom  T )  /\  X  e.  B  /\  Y  e.  B )  ->  ( F `  ( X  .+  Y ) )  =  ( ( F `  X )  .+^  ( F `
 Y ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   A.wral 2796   -->wf 5517   ` cfv 5521  (class class class)co 6195   Basecbs 14287   +g cplusg 14352   0gc0g 14492   Mndcmnd 15523   MndHom cmhm 15576
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 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477
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 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-sbc 3289  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-br 4396  df-opab 4454  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-fv 5529  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-map 7321  df-mhm 15578
This theorem is referenced by:  mhmf1o  15587  resmhm  15601  resmhm2  15602  resmhm2b  15603  mhmco  15604  mhmima  15605  mhmeql  15606  pwsco2mhm  15613  gsumwmhm  15637  mhmmulg  15773  ghmmhmb  15872  cntzmhm  15970  gsumzmhm  16547  gsumzmhmOLD  16548  rhmmul  16935  evlslem1  17720  mpfind  17741  mhmvlin  18417  mdetunilem7  18551  dchrzrhmul  22713  dchrmulcl  22716  dchrn0  22717  dchrinvcl  22720  dchrsum2  22735  sum2dchr  22741  mhmhmeotmd  26497
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