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Theorem plusffval 14622
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 5661 . . . . . 6  |-  ( g  =  G  ->  ( Base `  g )  =  ( Base `  G
) )
3 plusffval.1 . . . . . 6  |-  B  =  ( Base `  G
)
42, 3syl6eqr 2430 . . . . 5  |-  ( g  =  G  ->  ( Base `  g )  =  B )
5 fveq2 5661 . . . . . . 7  |-  ( g  =  G  ->  ( +g  `  g )  =  ( +g  `  G
) )
6 plusffval.2 . . . . . . 7  |-  .+  =  ( +g  `  G )
75, 6syl6eqr 2430 . . . . . 6  |-  ( g  =  G  ->  ( +g  `  g )  = 
.+  )
87oveqd 6030 . . . . 5  |-  ( g  =  G  ->  (
x ( +g  `  g
) y )  =  ( x  .+  y
) )
94, 4, 8mpt2eq123dv 6068 . . . 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 14611 . . . 4  |-  + f  =  ( g  e. 
_V  |->  ( x  e.  ( Base `  g
) ,  y  e.  ( Base `  g
)  |->  ( x ( +g  `  g ) y ) ) )
11 df-ov 6016 . . . . . . . 8  |-  ( x 
.+  y )  =  (  .+  `  <. x ,  y >. )
12 fvrn0 5686 . . . . . . . 8  |-  (  .+  ` 
<. x ,  y >.
)  e.  ( ran  .+  u.  { (/) } )
1311, 12eqeltri 2450 . . . . . . 7  |-  ( x 
.+  y )  e.  ( ran  .+  u.  {
(/) } )
1413rgen2w 2710 . . . . . 6  |-  A. x  e.  B  A. y  e.  B  ( x  .+  y )  e.  ( ran  .+  u.  { (/) } )
15 eqid 2380 . . . . . . 7  |-  ( x  e.  B ,  y  e.  B  |->  ( x 
.+  y ) )  =  ( x  e.  B ,  y  e.  B  |->  ( x  .+  y ) )
1615fmpt2 6350 . . . . . 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 200 . . . . 5  |-  ( x  e.  B ,  y  e.  B  |->  ( x 
.+  y ) ) : ( B  X.  B ) --> ( ran  .+  u.  { (/) } )
18 fvex 5675 . . . . . . 7  |-  ( Base `  G )  e.  _V
193, 18eqeltri 2450 . . . . . 6  |-  B  e. 
_V
2019, 19xpex 4923 . . . . 5  |-  ( B  X.  B )  e. 
_V
21 fvex 5675 . . . . . . . 8  |-  ( +g  `  G )  e.  _V
226, 21eqeltri 2450 . . . . . . 7  |-  .+  e.  _V
2322rnex 5066 . . . . . 6  |-  ran  .+  e.  _V
24 p0ex 4320 . . . . . 6  |-  { (/) }  e.  _V
2523, 24unex 4640 . . . . 5  |-  ( ran  .+  u.  { (/) } )  e.  _V
26 fex2 5536 . . . . 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 1279 . . . 4  |-  ( x  e.  B ,  y  e.  B  |->  ( x 
.+  y ) )  e.  _V
289, 10, 27fvmpt 5738 . . 3  |-  ( G  e.  _V  ->  ( + f `  G )  =  ( x  e.  B ,  y  e.  B  |->  ( x  .+  y ) ) )
29 fvprc 5655 . . . . 5  |-  ( -.  G  e.  _V  ->  ( + f `  G
)  =  (/) )
30 mpt20 6359 . . . . 5  |-  ( x  e.  (/) ,  y  e.  (/)  |->  ( x  .+  y ) )  =  (/)
3129, 30syl6eqr 2430 . . . 4  |-  ( -.  G  e.  _V  ->  ( + f `  G
)  =  ( x  e.  (/) ,  y  e.  (/)  |->  ( x  .+  y ) ) )
32 fvprc 5655 . . . . . 6  |-  ( -.  G  e.  _V  ->  (
Base `  G )  =  (/) )
333, 32syl5eq 2424 . . . . 5  |-  ( -.  G  e.  _V  ->  B  =  (/) )
34 mpt2eq12 6066 . . . . 5  |-  ( ( B  =  (/)  /\  B  =  (/) )  ->  (
x  e.  B , 
y  e.  B  |->  ( x  .+  y ) )  =  ( x  e.  (/) ,  y  e.  (/)  |->  ( x  .+  y ) ) )
3533, 33, 34syl2anc 643 . . . 4  |-  ( -.  G  e.  _V  ->  ( x  e.  B , 
y  e.  B  |->  ( x  .+  y ) )  =  ( x  e.  (/) ,  y  e.  (/)  |->  ( x  .+  y ) ) )
3631, 35eqtr4d 2415 . . 3  |-  ( -.  G  e.  _V  ->  ( + f `  G
)  =  ( x  e.  B ,  y  e.  B  |->  ( x 
.+  y ) ) )
3728, 36pm2.61i 158 . 2  |-  ( + f `  G )  =  ( x  e.  B ,  y  e.  B  |->  ( x  .+  y ) )
381, 37eqtri 2400 1  |-  .+^  =  ( x  e.  B , 
y  e.  B  |->  ( x  .+  y ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1649    e. wcel 1717   A.wral 2642   _Vcvv 2892    u. cun 3254   (/)c0 3564   {csn 3750   <.cop 3753    X. cxp 4809   ran crn 4812   -->wf 5383   ` cfv 5387  (class class class)co 6013    e. cmpt2 6015   Basecbs 13389   +g cplusg 13449   + fcplusf 14607
This theorem is referenced by:  plusfval  14623  plusfeq  14624  plusffn  14625  mndplusf  14626  rlmscaf  16199  istgp2  18035  oppgtmd  18041  submtmd  18048  prdstmdd  18067  ressplusf  24015
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-plusf 14611
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