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Theorem plusffval 14379
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 5525 . . . . . 6  |-  ( g  =  G  ->  ( Base `  g )  =  ( Base `  G
) )
3 plusffval.1 . . . . . 6  |-  B  =  ( Base `  G
)
42, 3syl6eqr 2333 . . . . 5  |-  ( g  =  G  ->  ( Base `  g )  =  B )
5 fveq2 5525 . . . . . . 7  |-  ( g  =  G  ->  ( +g  `  g )  =  ( +g  `  G
) )
6 plusffval.2 . . . . . . 7  |-  .+  =  ( +g  `  G )
75, 6syl6eqr 2333 . . . . . 6  |-  ( g  =  G  ->  ( +g  `  g )  = 
.+  )
87oveqd 5875 . . . . 5  |-  ( g  =  G  ->  (
x ( +g  `  g
) y )  =  ( x  .+  y
) )
94, 4, 8mpt2eq123dv 5910 . . . 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 14368 . . . 4  |-  + f  =  ( g  e. 
_V  |->  ( x  e.  ( Base `  g
) ,  y  e.  ( Base `  g
)  |->  ( x ( +g  `  g ) y ) ) )
11 df-ov 5861 . . . . . . . 8  |-  ( x 
.+  y )  =  (  .+  `  <. x ,  y >. )
12 fvrn0 5550 . . . . . . . 8  |-  (  .+  ` 
<. x ,  y >.
)  e.  ( ran  .+  u.  { (/) } )
1311, 12eqeltri 2353 . . . . . . 7  |-  ( x 
.+  y )  e.  ( ran  .+  u.  {
(/) } )
1413rgen2w 2611 . . . . . 6  |-  A. x  e.  B  A. y  e.  B  ( x  .+  y )  e.  ( ran  .+  u.  { (/) } )
15 eqid 2283 . . . . . . 7  |-  ( x  e.  B ,  y  e.  B  |->  ( x 
.+  y ) )  =  ( x  e.  B ,  y  e.  B  |->  ( x  .+  y ) )
1615fmpt2 6191 . . . . . 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 199 . . . . 5  |-  ( x  e.  B ,  y  e.  B  |->  ( x 
.+  y ) ) : ( B  X.  B ) --> ( ran  .+  u.  { (/) } )
18 fvex 5539 . . . . . . 7  |-  ( Base `  G )  e.  _V
193, 18eqeltri 2353 . . . . . 6  |-  B  e. 
_V
2019, 19xpex 4801 . . . . 5  |-  ( B  X.  B )  e. 
_V
21 fvex 5539 . . . . . . . 8  |-  ( +g  `  G )  e.  _V
226, 21eqeltri 2353 . . . . . . 7  |-  .+  e.  _V
2322rnex 4942 . . . . . 6  |-  ran  .+  e.  _V
24 p0ex 4197 . . . . . 6  |-  { (/) }  e.  _V
2523, 24unex 4518 . . . . 5  |-  ( ran  .+  u.  { (/) } )  e.  _V
26 fex2 5401 . . . . 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 1277 . . . 4  |-  ( x  e.  B ,  y  e.  B  |->  ( x 
.+  y ) )  e.  _V
289, 10, 27fvmpt 5602 . . 3  |-  ( G  e.  _V  ->  ( + f `  G )  =  ( x  e.  B ,  y  e.  B  |->  ( x  .+  y ) ) )
29 fvprc 5519 . . . . 5  |-  ( -.  G  e.  _V  ->  ( + f `  G
)  =  (/) )
30 mpt20 6199 . . . . 5  |-  ( x  e.  (/) ,  y  e.  (/)  |->  ( x  .+  y ) )  =  (/)
3129, 30syl6eqr 2333 . . . 4  |-  ( -.  G  e.  _V  ->  ( + f `  G
)  =  ( x  e.  (/) ,  y  e.  (/)  |->  ( x  .+  y ) ) )
32 fvprc 5519 . . . . . 6  |-  ( -.  G  e.  _V  ->  (
Base `  G )  =  (/) )
333, 32syl5eq 2327 . . . . 5  |-  ( -.  G  e.  _V  ->  B  =  (/) )
34 mpt2eq12 5908 . . . . 5  |-  ( ( B  =  (/)  /\  B  =  (/) )  ->  (
x  e.  B , 
y  e.  B  |->  ( x  .+  y ) )  =  ( x  e.  (/) ,  y  e.  (/)  |->  ( x  .+  y ) ) )
3533, 33, 34syl2anc 642 . . . 4  |-  ( -.  G  e.  _V  ->  ( x  e.  B , 
y  e.  B  |->  ( x  .+  y ) )  =  ( x  e.  (/) ,  y  e.  (/)  |->  ( x  .+  y ) ) )
3631, 35eqtr4d 2318 . . 3  |-  ( -.  G  e.  _V  ->  ( + f `  G
)  =  ( x  e.  B ,  y  e.  B  |->  ( x 
.+  y ) ) )
3728, 36pm2.61i 156 . 2  |-  ( + f `  G )  =  ( x  e.  B ,  y  e.  B  |->  ( x  .+  y ) )
381, 37eqtri 2303 1  |-  .+^  =  ( x  e.  B , 
y  e.  B  |->  ( x  .+  y ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788    u. cun 3150   (/)c0 3455   {csn 3640   <.cop 3643    X. cxp 4687   ran crn 4690   -->wf 5251   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   Basecbs 13148   +g cplusg 13208   + fcplusf 14364
This theorem is referenced by:  plusfval  14380  plusfeq  14381  plusffn  14382  mndplusf  14383  rlmscaf  15960  istgp2  17774  oppgtmd  17780  submtmd  17787  prdstmdd  17806  ressplusf  23298
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-plusf 14368
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