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Theorem plusffval 15751
Description: The group addition operation as a function. (Contributed by Mario Carneiro, 14-Aug-2015.)
Hypotheses
Ref Expression
plusffval.1  |-  B  =  ( Base `  G
)
plusffval.2  |-  .+  =  ( +g  `  G )
plusffval.3  |-  .+^  =  ( +f `  G
)
Assertion
Ref Expression
plusffval  |-  .+^  =  ( x  e.  B , 
y  e.  B  |->  ( x  .+  y ) )
Distinct variable groups:    x, y, B    x, G, y    x,  .+ , y
Allowed substitution hints:    .+^ ( x, y)

Proof of Theorem plusffval
Dummy variable  g is distinct from all other variables.
StepHypRef Expression
1 plusffval.3 . 2  |-  .+^  =  ( +f `  G
)
2 fveq2 5872 . . . . . 6  |-  ( g  =  G  ->  ( Base `  g )  =  ( Base `  G
) )
3 plusffval.1 . . . . . 6  |-  B  =  ( Base `  G
)
42, 3syl6eqr 2526 . . . . 5  |-  ( g  =  G  ->  ( Base `  g )  =  B )
5 fveq2 5872 . . . . . . 7  |-  ( g  =  G  ->  ( +g  `  g )  =  ( +g  `  G
) )
6 plusffval.2 . . . . . . 7  |-  .+  =  ( +g  `  G )
75, 6syl6eqr 2526 . . . . . 6  |-  ( g  =  G  ->  ( +g  `  g )  = 
.+  )
87oveqd 6312 . . . . 5  |-  ( g  =  G  ->  (
x ( +g  `  g
) y )  =  ( x  .+  y
) )
94, 4, 8mpt2eq123dv 6354 . . . 4  |-  ( g  =  G  ->  (
x  e.  ( Base `  g ) ,  y  e.  ( Base `  g
)  |->  ( x ( +g  `  g ) y ) )  =  ( x  e.  B ,  y  e.  B  |->  ( x  .+  y
) ) )
10 df-plusf 15745 . . . 4  |-  +f 
=  ( g  e. 
_V  |->  ( x  e.  ( Base `  g
) ,  y  e.  ( Base `  g
)  |->  ( x ( +g  `  g ) y ) ) )
11 df-ov 6298 . . . . . . . 8  |-  ( x 
.+  y )  =  (  .+  `  <. x ,  y >. )
12 fvrn0 5894 . . . . . . . 8  |-  (  .+  ` 
<. x ,  y >.
)  e.  ( ran  .+  u.  { (/) } )
1311, 12eqeltri 2551 . . . . . . 7  |-  ( x 
.+  y )  e.  ( ran  .+  u.  {
(/) } )
1413rgen2w 2829 . . . . . 6  |-  A. x  e.  B  A. y  e.  B  ( x  .+  y )  e.  ( ran  .+  u.  { (/) } )
15 eqid 2467 . . . . . . 7  |-  ( x  e.  B ,  y  e.  B  |->  ( x 
.+  y ) )  =  ( x  e.  B ,  y  e.  B  |->  ( x  .+  y ) )
1615fmpt2 6862 . . . . . 6  |-  ( A. x  e.  B  A. y  e.  B  (
x  .+  y )  e.  ( ran  .+  u.  {
(/) } )  <->  ( x  e.  B ,  y  e.  B  |->  ( x  .+  y ) ) : ( B  X.  B
) --> ( ran  .+  u.  { (/) } ) )
1714, 16mpbi 208 . . . . 5  |-  ( x  e.  B ,  y  e.  B  |->  ( x 
.+  y ) ) : ( B  X.  B ) --> ( ran  .+  u.  { (/) } )
18 fvex 5882 . . . . . . 7  |-  ( Base `  G )  e.  _V
193, 18eqeltri 2551 . . . . . 6  |-  B  e. 
_V
2019, 19xpex 6599 . . . . 5  |-  ( B  X.  B )  e. 
_V
21 fvex 5882 . . . . . . . 8  |-  ( +g  `  G )  e.  _V
226, 21eqeltri 2551 . . . . . . 7  |-  .+  e.  _V
2322rnex 6729 . . . . . 6  |-  ran  .+  e.  _V
24 p0ex 4640 . . . . . 6  |-  { (/) }  e.  _V
2523, 24unex 6593 . . . . 5  |-  ( ran  .+  u.  { (/) } )  e.  _V
26 fex2 6750 . . . . 5  |-  ( ( ( x  e.  B ,  y  e.  B  |->  ( x  .+  y
) ) : ( B  X.  B ) --> ( ran  .+  u.  {
(/) } )  /\  ( B  X.  B )  e. 
_V  /\  ( ran  .+  u.  { (/) } )  e.  _V )  -> 
( x  e.  B ,  y  e.  B  |->  ( x  .+  y
) )  e.  _V )
2717, 20, 25, 26mp3an 1324 . . . 4  |-  ( x  e.  B ,  y  e.  B  |->  ( x 
.+  y ) )  e.  _V
289, 10, 27fvmpt 5957 . . 3  |-  ( G  e.  _V  ->  ( +f `  G
)  =  ( x  e.  B ,  y  e.  B  |->  ( x 
.+  y ) ) )
29 fvprc 5866 . . . . 5  |-  ( -.  G  e.  _V  ->  ( +f `  G
)  =  (/) )
30 mpt20 6362 . . . . 5  |-  ( x  e.  (/) ,  y  e.  (/)  |->  ( x  .+  y ) )  =  (/)
3129, 30syl6eqr 2526 . . . 4  |-  ( -.  G  e.  _V  ->  ( +f `  G
)  =  ( x  e.  (/) ,  y  e.  (/)  |->  ( x  .+  y ) ) )
32 fvprc 5866 . . . . . 6  |-  ( -.  G  e.  _V  ->  (
Base `  G )  =  (/) )
333, 32syl5eq 2520 . . . . 5  |-  ( -.  G  e.  _V  ->  B  =  (/) )
34 mpt2eq12 6352 . . . . 5  |-  ( ( B  =  (/)  /\  B  =  (/) )  ->  (
x  e.  B , 
y  e.  B  |->  ( x  .+  y ) )  =  ( x  e.  (/) ,  y  e.  (/)  |->  ( x  .+  y ) ) )
3533, 33, 34syl2anc 661 . . . 4  |-  ( -.  G  e.  _V  ->  ( x  e.  B , 
y  e.  B  |->  ( x  .+  y ) )  =  ( x  e.  (/) ,  y  e.  (/)  |->  ( x  .+  y ) ) )
3631, 35eqtr4d 2511 . . 3  |-  ( -.  G  e.  _V  ->  ( +f `  G
)  =  ( x  e.  B ,  y  e.  B  |->  ( x 
.+  y ) ) )
3728, 36pm2.61i 164 . 2  |-  ( +f `  G )  =  ( x  e.  B ,  y  e.  B  |->  ( x  .+  y ) )
381, 37eqtri 2496 1  |-  .+^  =  ( x  e.  B , 
y  e.  B  |->  ( x  .+  y ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1379    e. wcel 1767   A.wral 2817   _Vcvv 3118    u. cun 3479   (/)c0 3790   {csn 4033   <.cop 4039    X. cxp 5003   ran crn 5006   -->wf 5590   ` cfv 5594  (class class class)co 6295    |-> cmpt2 6297   Basecbs 14507   +g cplusg 14572   +fcplusf 15743
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-1st 6795  df-2nd 6796  df-plusf 15745
This theorem is referenced by:  plusfval  15752  plusfeq  15753  plusffn  15754  mgmplusf  15755  rlmscaf  17725  istgp2  20458  oppgtmd  20464  submtmd  20471  prdstmdd  20490  ressplusf  27462  pl1cn  27762
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