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Theorem mhmlin 16175
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 2454 . . . . . 6  |-  ( Base `  T )  =  (
Base `  T )
3 mhmlin.p . . . . . 6  |-  .+  =  ( +g  `  S )
4 mhmlin.q . . . . . 6  |-  .+^  =  ( +g  `  T )
5 eqid 2454 . . . . . 6  |-  ( 0g
`  S )  =  ( 0g `  S
)
6 eqid 2454 . . . . . 6  |-  ( 0g
`  T )  =  ( 0g `  T
)
71, 2, 3, 4, 5, 6ismhm 16170 . . . . 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 462 . . . 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 1007 . . 3  |-  ( F  e.  ( S MndHom  T
)  ->  A. x  e.  B  A. y  e.  B  ( F `  ( x  .+  y
) )  =  ( ( F `  x
)  .+^  ( F `  y ) ) )
10 oveq1 6277 . . . . . 6  |-  ( x  =  X  ->  (
x  .+  y )  =  ( X  .+  y ) )
1110fveq2d 5852 . . . . 5  |-  ( x  =  X  ->  ( F `  ( x  .+  y ) )  =  ( F `  ( X  .+  y ) ) )
12 fveq2 5848 . . . . . 6  |-  ( x  =  X  ->  ( F `  x )  =  ( F `  X ) )
1312oveq1d 6285 . . . . 5  |-  ( x  =  X  ->  (
( F `  x
)  .+^  ( F `  y ) )  =  ( ( F `  X )  .+^  ( F `
 y ) ) )
1411, 13eqeq12d 2476 . . . 4  |-  ( x  =  X  ->  (
( F `  (
x  .+  y )
)  =  ( ( F `  x ) 
.+^  ( F `  y ) )  <->  ( F `  ( X  .+  y
) )  =  ( ( F `  X
)  .+^  ( F `  y ) ) ) )
15 oveq2 6278 . . . . . 6  |-  ( y  =  Y  ->  ( X  .+  y )  =  ( X  .+  Y
) )
1615fveq2d 5852 . . . . 5  |-  ( y  =  Y  ->  ( F `  ( X  .+  y ) )  =  ( F `  ( X  .+  Y ) ) )
17 fveq2 5848 . . . . . 6  |-  ( y  =  Y  ->  ( F `  y )  =  ( F `  Y ) )
1817oveq2d 6286 . . . . 5  |-  ( y  =  Y  ->  (
( F `  X
)  .+^  ( F `  y ) )  =  ( ( F `  X )  .+^  ( F `
 Y ) ) )
1916, 18eqeq12d 2476 . . . 4  |-  ( y  =  Y  ->  (
( F `  ( X  .+  y ) )  =  ( ( F `
 X )  .+^  ( F `  y ) )  <->  ( F `  ( X  .+  Y ) )  =  ( ( F `  X ) 
.+^  ( F `  Y ) ) ) )
2014, 19rspc2v 3216 . . 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 1192 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823   A.wral 2804   -->wf 5566   ` cfv 5570  (class class class)co 6270   Basecbs 14719   +g cplusg 14787   0gc0g 14932   Mndcmnd 16121   MndHom cmhm 16166
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-map 7414  df-mhm 16168
This theorem is referenced by:  mhmf1o  16178  resmhm  16192  resmhm2  16193  resmhm2b  16194  mhmco  16195  mhmima  16196  mhmeql  16197  pwsco2mhm  16204  gsumwmhm  16215  mhmmulg  16376  ghmmhmb  16480  cntzmhm  16578  gsumzmhm  17158  gsumzmhmOLD  17159  rhmmul  17574  evlslem1  18382  mpfind  18403  mhmvlin  19069  mdetunilem7  19290  dchrzrhmul  23722  dchrmulcl  23725  dchrn0  23726  dchrinvcl  23729  dchrsum2  23744  sum2dchr  23750  mhmhmeotmd  28147
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