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Theorem plusffval 16203
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 5851 . . . . . 6  |-  ( g  =  G  ->  ( Base `  g )  =  ( Base `  G
) )
3 plusffval.1 . . . . . 6  |-  B  =  ( Base `  G
)
42, 3syl6eqr 2463 . . . . 5  |-  ( g  =  G  ->  ( Base `  g )  =  B )
5 fveq2 5851 . . . . . . 7  |-  ( g  =  G  ->  ( +g  `  g )  =  ( +g  `  G
) )
6 plusffval.2 . . . . . . 7  |-  .+  =  ( +g  `  G )
75, 6syl6eqr 2463 . . . . . 6  |-  ( g  =  G  ->  ( +g  `  g )  = 
.+  )
87oveqd 6297 . . . . 5  |-  ( g  =  G  ->  (
x ( +g  `  g
) y )  =  ( x  .+  y
) )
94, 4, 8mpt2eq123dv 6342 . . . 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 16197 . . . 4  |-  +f 
=  ( g  e. 
_V  |->  ( x  e.  ( Base `  g
) ,  y  e.  ( Base `  g
)  |->  ( x ( +g  `  g ) y ) ) )
11 df-ov 6283 . . . . . . . 8  |-  ( x 
.+  y )  =  (  .+  `  <. x ,  y >. )
12 fvrn0 5873 . . . . . . . 8  |-  (  .+  ` 
<. x ,  y >.
)  e.  ( ran  .+  u.  { (/) } )
1311, 12eqeltri 2488 . . . . . . 7  |-  ( x 
.+  y )  e.  ( ran  .+  u.  {
(/) } )
1413rgen2w 2768 . . . . . 6  |-  A. x  e.  B  A. y  e.  B  ( x  .+  y )  e.  ( ran  .+  u.  { (/) } )
15 eqid 2404 . . . . . . 7  |-  ( x  e.  B ,  y  e.  B  |->  ( x 
.+  y ) )  =  ( x  e.  B ,  y  e.  B  |->  ( x  .+  y ) )
1615fmpt2 6853 . . . . . 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 210 . . . . 5  |-  ( x  e.  B ,  y  e.  B  |->  ( x 
.+  y ) ) : ( B  X.  B ) --> ( ran  .+  u.  { (/) } )
18 fvex 5861 . . . . . . 7  |-  ( Base `  G )  e.  _V
193, 18eqeltri 2488 . . . . . 6  |-  B  e. 
_V
2019, 19xpex 6588 . . . . 5  |-  ( B  X.  B )  e. 
_V
21 fvex 5861 . . . . . . . 8  |-  ( +g  `  G )  e.  _V
226, 21eqeltri 2488 . . . . . . 7  |-  .+  e.  _V
2322rnex 6720 . . . . . 6  |-  ran  .+  e.  _V
24 p0ex 4583 . . . . . 6  |-  { (/) }  e.  _V
2523, 24unex 6582 . . . . 5  |-  ( ran  .+  u.  { (/) } )  e.  _V
26 fex2 6741 . . . . 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 1328 . . . 4  |-  ( x  e.  B ,  y  e.  B  |->  ( x 
.+  y ) )  e.  _V
289, 10, 27fvmpt 5934 . . 3  |-  ( G  e.  _V  ->  ( +f `  G
)  =  ( x  e.  B ,  y  e.  B  |->  ( x 
.+  y ) ) )
29 fvprc 5845 . . . . 5  |-  ( -.  G  e.  _V  ->  ( +f `  G
)  =  (/) )
30 mpt20 6350 . . . . 5  |-  ( x  e.  (/) ,  y  e.  (/)  |->  ( x  .+  y ) )  =  (/)
3129, 30syl6eqr 2463 . . . 4  |-  ( -.  G  e.  _V  ->  ( +f `  G
)  =  ( x  e.  (/) ,  y  e.  (/)  |->  ( x  .+  y ) ) )
32 fvprc 5845 . . . . . 6  |-  ( -.  G  e.  _V  ->  (
Base `  G )  =  (/) )
333, 32syl5eq 2457 . . . . 5  |-  ( -.  G  e.  _V  ->  B  =  (/) )
34 mpt2eq12 6340 . . . . 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 2448 . . 3  |-  ( -.  G  e.  _V  ->  ( +f `  G
)  =  ( x  e.  B ,  y  e.  B  |->  ( x 
.+  y ) ) )
3728, 36pm2.61i 166 . 2  |-  ( +f `  G )  =  ( x  e.  B ,  y  e.  B  |->  ( x  .+  y ) )
381, 37eqtri 2433 1  |-  .+^  =  ( x  e.  B , 
y  e.  B  |->  ( x  .+  y ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1407    e. wcel 1844   A.wral 2756   _Vcvv 3061    u. cun 3414   (/)c0 3740   {csn 3974   <.cop 3980    X. cxp 4823   ran crn 4826   -->wf 5567   ` cfv 5571  (class class class)co 6280    |-> cmpt2 6282   Basecbs 14843   +g cplusg 14911   +fcplusf 16195
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-id 4740  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-fv 5579  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-1st 6786  df-2nd 6787  df-plusf 16197
This theorem is referenced by:  plusfval  16204  plusfeq  16205  plusffn  16206  mgmplusf  16207  rlmscaf  18176  istgp2  20884  oppgtmd  20890  submtmd  20897  prdstmdd  20916  ressplusf  28103  pl1cn  28403
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