MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  plusffval Structured version   Unicode version

Theorem plusffval 14694
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 5720 . . . . . 6  |-  ( g  =  G  ->  ( Base `  g )  =  ( Base `  G
) )
3 plusffval.1 . . . . . 6  |-  B  =  ( Base `  G
)
42, 3syl6eqr 2485 . . . . 5  |-  ( g  =  G  ->  ( Base `  g )  =  B )
5 fveq2 5720 . . . . . . 7  |-  ( g  =  G  ->  ( +g  `  g )  =  ( +g  `  G
) )
6 plusffval.2 . . . . . . 7  |-  .+  =  ( +g  `  G )
75, 6syl6eqr 2485 . . . . . 6  |-  ( g  =  G  ->  ( +g  `  g )  = 
.+  )
87oveqd 6090 . . . . 5  |-  ( g  =  G  ->  (
x ( +g  `  g
) y )  =  ( x  .+  y
) )
94, 4, 8mpt2eq123dv 6128 . . . 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 14683 . . . 4  |-  + f  =  ( g  e. 
_V  |->  ( x  e.  ( Base `  g
) ,  y  e.  ( Base `  g
)  |->  ( x ( +g  `  g ) y ) ) )
11 df-ov 6076 . . . . . . . 8  |-  ( x 
.+  y )  =  (  .+  `  <. x ,  y >. )
12 fvrn0 5745 . . . . . . . 8  |-  (  .+  ` 
<. x ,  y >.
)  e.  ( ran  .+  u.  { (/) } )
1311, 12eqeltri 2505 . . . . . . 7  |-  ( x 
.+  y )  e.  ( ran  .+  u.  {
(/) } )
1413rgen2w 2766 . . . . . 6  |-  A. x  e.  B  A. y  e.  B  ( x  .+  y )  e.  ( ran  .+  u.  { (/) } )
15 eqid 2435 . . . . . . 7  |-  ( x  e.  B ,  y  e.  B  |->  ( x 
.+  y ) )  =  ( x  e.  B ,  y  e.  B  |->  ( x  .+  y ) )
1615fmpt2 6410 . . . . . 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 5734 . . . . . . 7  |-  ( Base `  G )  e.  _V
193, 18eqeltri 2505 . . . . . 6  |-  B  e. 
_V
2019, 19xpex 4982 . . . . 5  |-  ( B  X.  B )  e. 
_V
21 fvex 5734 . . . . . . . 8  |-  ( +g  `  G )  e.  _V
226, 21eqeltri 2505 . . . . . . 7  |-  .+  e.  _V
2322rnex 5125 . . . . . 6  |-  ran  .+  e.  _V
24 p0ex 4378 . . . . . 6  |-  { (/) }  e.  _V
2523, 24unex 4699 . . . . 5  |-  ( ran  .+  u.  { (/) } )  e.  _V
26 fex2 5595 . . . . 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 5798 . . 3  |-  ( G  e.  _V  ->  ( + f `  G )  =  ( x  e.  B ,  y  e.  B  |->  ( x  .+  y ) ) )
29 fvprc 5714 . . . . 5  |-  ( -.  G  e.  _V  ->  ( + f `  G
)  =  (/) )
30 mpt20 6419 . . . . 5  |-  ( x  e.  (/) ,  y  e.  (/)  |->  ( x  .+  y ) )  =  (/)
3129, 30syl6eqr 2485 . . . 4  |-  ( -.  G  e.  _V  ->  ( + f `  G
)  =  ( x  e.  (/) ,  y  e.  (/)  |->  ( x  .+  y ) ) )
32 fvprc 5714 . . . . . 6  |-  ( -.  G  e.  _V  ->  (
Base `  G )  =  (/) )
333, 32syl5eq 2479 . . . . 5  |-  ( -.  G  e.  _V  ->  B  =  (/) )
34 mpt2eq12 6126 . . . . 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 2470 . . 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 2455 1  |-  .+^  =  ( x  e.  B , 
y  e.  B  |->  ( x  .+  y ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1652    e. wcel 1725   A.wral 2697   _Vcvv 2948    u. cun 3310   (/)c0 3620   {csn 3806   <.cop 3809    X. cxp 4868   ran crn 4871   -->wf 5442   ` cfv 5446  (class class class)co 6073    e. cmpt2 6075   Basecbs 13461   +g cplusg 13521   + fcplusf 14679
This theorem is referenced by:  plusfval  14695  plusfeq  14696  plusffn  14697  mndplusf  14698  rlmscaf  16271  istgp2  18113  oppgtmd  18119  submtmd  18126  prdstmdd  18145  ressplusf  24175
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-plusf 14683
  Copyright terms: Public domain W3C validator