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Theorem mgmhmlin 32865
Description: A magma homomorphism preserves the binary operation. (Contributed by AV, 25-Feb-2020.)
Hypotheses
Ref Expression
mgmhmlin.b  |-  B  =  ( Base `  S
)
mgmhmlin.p  |-  .+  =  ( +g  `  S )
mgmhmlin.q  |-  .+^  =  ( +g  `  T )
Assertion
Ref Expression
mgmhmlin  |-  ( ( F  e.  ( S MgmHom  T )  /\  X  e.  B  /\  Y  e.  B )  ->  ( F `  ( X  .+  Y ) )  =  ( ( F `  X )  .+^  ( F `
 Y ) ) )

Proof of Theorem mgmhmlin
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mgmhmlin.b . . . 4  |-  B  =  ( Base `  S
)
2 eqid 2454 . . . 4  |-  ( Base `  T )  =  (
Base `  T )
3 mgmhmlin.p . . . 4  |-  .+  =  ( +g  `  S )
4 mgmhmlin.q . . . 4  |-  .+^  =  ( +g  `  T )
51, 2, 3, 4ismgmhm 32862 . . 3  |-  ( F  e.  ( S MgmHom  T
)  <->  ( ( S  e. Mgm  /\  T  e. Mgm )  /\  ( F : B
--> ( Base `  T
)  /\  A. x  e.  B  A. y  e.  B  ( F `  ( x  .+  y
) )  =  ( ( F `  x
)  .+^  ( F `  y ) ) ) ) )
6 oveq1 6277 . . . . . . . 8  |-  ( x  =  X  ->  (
x  .+  y )  =  ( X  .+  y ) )
76fveq2d 5852 . . . . . . 7  |-  ( x  =  X  ->  ( F `  ( x  .+  y ) )  =  ( F `  ( X  .+  y ) ) )
8 fveq2 5848 . . . . . . . 8  |-  ( x  =  X  ->  ( F `  x )  =  ( F `  X ) )
98oveq1d 6285 . . . . . . 7  |-  ( x  =  X  ->  (
( F `  x
)  .+^  ( F `  y ) )  =  ( ( F `  X )  .+^  ( F `
 y ) ) )
107, 9eqeq12d 2476 . . . . . 6  |-  ( x  =  X  ->  (
( F `  (
x  .+  y )
)  =  ( ( F `  x ) 
.+^  ( F `  y ) )  <->  ( F `  ( X  .+  y
) )  =  ( ( F `  X
)  .+^  ( F `  y ) ) ) )
11 oveq2 6278 . . . . . . . 8  |-  ( y  =  Y  ->  ( X  .+  y )  =  ( X  .+  Y
) )
1211fveq2d 5852 . . . . . . 7  |-  ( y  =  Y  ->  ( F `  ( X  .+  y ) )  =  ( F `  ( X  .+  Y ) ) )
13 fveq2 5848 . . . . . . . 8  |-  ( y  =  Y  ->  ( F `  y )  =  ( F `  Y ) )
1413oveq2d 6286 . . . . . . 7  |-  ( y  =  Y  ->  (
( F `  X
)  .+^  ( F `  y ) )  =  ( ( F `  X )  .+^  ( F `
 Y ) ) )
1512, 14eqeq12d 2476 . . . . . 6  |-  ( y  =  Y  ->  (
( F `  ( X  .+  y ) )  =  ( ( F `
 X )  .+^  ( F `  y ) )  <->  ( F `  ( X  .+  Y ) )  =  ( ( F `  X ) 
.+^  ( F `  Y ) ) ) )
1610, 15rspc2v 3216 . . . . 5  |-  ( ( 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 ) ) ) )
1716com12 31 . . . 4  |-  ( A. x  e.  B  A. y  e.  B  ( F `  ( x  .+  y ) )  =  ( ( F `  x )  .+^  ( F `
 y ) )  ->  ( ( X  e.  B  /\  Y  e.  B )  ->  ( F `  ( X  .+  Y ) )  =  ( ( F `  X )  .+^  ( F `
 Y ) ) ) )
1817ad2antll 726 . . 3  |-  ( ( ( S  e. Mgm  /\  T  e. Mgm )  /\  ( F : B --> ( Base `  T )  /\  A. x  e.  B  A. y  e.  B  ( F `  ( x  .+  y ) )  =  ( ( F `  x )  .+^  ( F `
 y ) ) ) )  ->  (
( X  e.  B  /\  Y  e.  B
)  ->  ( F `  ( X  .+  Y
) )  =  ( ( F `  X
)  .+^  ( F `  Y ) ) ) )
195, 18sylbi 195 . 2  |-  ( F  e.  ( S MgmHom  T
)  ->  ( ( X  e.  B  /\  Y  e.  B )  ->  ( F `  ( X  .+  Y ) )  =  ( ( F `
 X )  .+^  ( F `  Y ) ) ) )
20193impib 1192 1  |-  ( ( F  e.  ( S MgmHom  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  Mgmcmgm 16072   MgmHom cmgmhm 32856
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-mgmhm 32858
This theorem is referenced by:  mgmhmf1o  32866  resmgmhm  32877  resmgmhm2  32878  resmgmhm2b  32879  mgmhmco  32880  mgmhmima  32881  mgmhmeql  32882
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