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

Theorem psrbagaddcl 16116
Description: The sum of two finite bags is a finite bag. (Contributed by Mario Carneiro, 9-Jan-2015.)
Hypothesis
Ref Expression
psrbag.d  |-  D  =  { f  e.  ( NN0  ^m  I )  |  ( `' f
" NN )  e. 
Fin }
Assertion
Ref Expression
psrbagaddcl  |-  ( ( I  e.  V  /\  F  e.  D  /\  G  e.  D )  ->  ( F  o F  +  G )  e.  D )
Distinct variable groups:    f, F    f, G    f, I
Allowed substitution hints:    D( f)    V( f)

Proof of Theorem psrbagaddcl
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nn0addcl 9999 . . . 4  |-  ( ( x  e.  NN0  /\  y  e.  NN0 )  -> 
( x  +  y )  e.  NN0 )
21adantl 452 . . 3  |-  ( ( ( I  e.  V  /\  F  e.  D  /\  G  e.  D
)  /\  ( x  e.  NN0  /\  y  e. 
NN0 ) )  -> 
( x  +  y )  e.  NN0 )
3 simp2 956 . . . . 5  |-  ( ( I  e.  V  /\  F  e.  D  /\  G  e.  D )  ->  F  e.  D )
4 psrbag.d . . . . . . 7  |-  D  =  { f  e.  ( NN0  ^m  I )  |  ( `' f
" NN )  e. 
Fin }
54psrbag 16112 . . . . . 6  |-  ( I  e.  V  ->  ( F  e.  D  <->  ( F : I --> NN0  /\  ( `' F " NN )  e.  Fin ) ) )
653ad2ant1 976 . . . . 5  |-  ( ( I  e.  V  /\  F  e.  D  /\  G  e.  D )  ->  ( F  e.  D  <->  ( F : I --> NN0  /\  ( `' F " NN )  e.  Fin ) ) )
73, 6mpbid 201 . . . 4  |-  ( ( I  e.  V  /\  F  e.  D  /\  G  e.  D )  ->  ( F : I --> NN0  /\  ( `' F " NN )  e.  Fin ) )
87simpld 445 . . 3  |-  ( ( I  e.  V  /\  F  e.  D  /\  G  e.  D )  ->  F : I --> NN0 )
9 simp3 957 . . . . 5  |-  ( ( I  e.  V  /\  F  e.  D  /\  G  e.  D )  ->  G  e.  D )
104psrbag 16112 . . . . . 6  |-  ( I  e.  V  ->  ( G  e.  D  <->  ( G : I --> NN0  /\  ( `' G " NN )  e.  Fin ) ) )
11103ad2ant1 976 . . . . 5  |-  ( ( I  e.  V  /\  F  e.  D  /\  G  e.  D )  ->  ( G  e.  D  <->  ( G : I --> NN0  /\  ( `' G " NN )  e.  Fin ) ) )
129, 11mpbid 201 . . . 4  |-  ( ( I  e.  V  /\  F  e.  D  /\  G  e.  D )  ->  ( G : I --> NN0  /\  ( `' G " NN )  e.  Fin ) )
1312simpld 445 . . 3  |-  ( ( I  e.  V  /\  F  e.  D  /\  G  e.  D )  ->  G : I --> NN0 )
14 simp1 955 . . 3  |-  ( ( I  e.  V  /\  F  e.  D  /\  G  e.  D )  ->  I  e.  V )
15 inidm 3378 . . 3  |-  ( I  i^i  I )  =  I
162, 8, 13, 14, 14, 15off 6093 . 2  |-  ( ( I  e.  V  /\  F  e.  D  /\  G  e.  D )  ->  ( F  o F  +  G ) : I --> NN0 )
17 nn0supp 10017 . . . . 5  |-  ( ( F  o F  +  G ) : I --> NN0  ->  ( `' ( F  o F  +  G ) " ( _V  \  { 0 } ) )  =  ( `' ( F  o F  +  G ) " NN ) )
1816, 17syl 15 . . . 4  |-  ( ( I  e.  V  /\  F  e.  D  /\  G  e.  D )  ->  ( `' ( F  o F  +  G
) " ( _V 
\  { 0 } ) )  =  ( `' ( F  o F  +  G ) " NN ) )
19 fvex 5539 . . . . . . . 8  |-  ( F `
 x )  e. 
_V
2019a1i 10 . . . . . . 7  |-  ( ( ( I  e.  V  /\  F  e.  D  /\  G  e.  D
)  /\  x  e.  I )  ->  ( F `  x )  e.  _V )
21 fvex 5539 . . . . . . . 8  |-  ( G `
 x )  e. 
_V
2221a1i 10 . . . . . . 7  |-  ( ( ( I  e.  V  /\  F  e.  D  /\  G  e.  D
)  /\  x  e.  I )  ->  ( G `  x )  e.  _V )
238feqmptd 5575 . . . . . . 7  |-  ( ( I  e.  V  /\  F  e.  D  /\  G  e.  D )  ->  F  =  ( x  e.  I  |->  ( F `
 x ) ) )
2413feqmptd 5575 . . . . . . 7  |-  ( ( I  e.  V  /\  F  e.  D  /\  G  e.  D )  ->  G  =  ( x  e.  I  |->  ( G `
 x ) ) )
2514, 20, 22, 23, 24offval2 6095 . . . . . 6  |-  ( ( I  e.  V  /\  F  e.  D  /\  G  e.  D )  ->  ( F  o F  +  G )  =  ( x  e.  I  |->  ( ( F `  x )  +  ( G `  x ) ) ) )
2625cnveqd 4857 . . . . 5  |-  ( ( I  e.  V  /\  F  e.  D  /\  G  e.  D )  ->  `' ( F  o F  +  G )  =  `' ( x  e.  I  |->  ( ( F `
 x )  +  ( G `  x
) ) ) )
2726imaeq1d 5011 . . . 4  |-  ( ( I  e.  V  /\  F  e.  D  /\  G  e.  D )  ->  ( `' ( F  o F  +  G
) " ( _V 
\  { 0 } ) )  =  ( `' ( x  e.  I  |->  ( ( F `
 x )  +  ( G `  x
) ) ) "
( _V  \  {
0 } ) ) )
2818, 27eqtr3d 2317 . . 3  |-  ( ( I  e.  V  /\  F  e.  D  /\  G  e.  D )  ->  ( `' ( F  o F  +  G
) " NN )  =  ( `' ( x  e.  I  |->  ( ( F `  x
)  +  ( G `
 x ) ) ) " ( _V 
\  { 0 } ) ) )
29 nn0supp 10017 . . . . . . 7  |-  ( F : I --> NN0  ->  ( `' F " ( _V 
\  { 0 } ) )  =  ( `' F " NN ) )
308, 29syl 15 . . . . . 6  |-  ( ( I  e.  V  /\  F  e.  D  /\  G  e.  D )  ->  ( `' F "
( _V  \  {
0 } ) )  =  ( `' F " NN ) )
317simprd 449 . . . . . 6  |-  ( ( I  e.  V  /\  F  e.  D  /\  G  e.  D )  ->  ( `' F " NN )  e.  Fin )
3230, 31eqeltrd 2357 . . . . 5  |-  ( ( I  e.  V  /\  F  e.  D  /\  G  e.  D )  ->  ( `' F "
( _V  \  {
0 } ) )  e.  Fin )
33 nn0supp 10017 . . . . . . 7  |-  ( G : I --> NN0  ->  ( `' G " ( _V 
\  { 0 } ) )  =  ( `' G " NN ) )
3413, 33syl 15 . . . . . 6  |-  ( ( I  e.  V  /\  F  e.  D  /\  G  e.  D )  ->  ( `' G "
( _V  \  {
0 } ) )  =  ( `' G " NN ) )
3512simprd 449 . . . . . 6  |-  ( ( I  e.  V  /\  F  e.  D  /\  G  e.  D )  ->  ( `' G " NN )  e.  Fin )
3634, 35eqeltrd 2357 . . . . 5  |-  ( ( I  e.  V  /\  F  e.  D  /\  G  e.  D )  ->  ( `' G "
( _V  \  {
0 } ) )  e.  Fin )
37 unfi 7124 . . . . 5  |-  ( ( ( `' F "
( _V  \  {
0 } ) )  e.  Fin  /\  ( `' G " ( _V 
\  { 0 } ) )  e.  Fin )  ->  ( ( `' F " ( _V 
\  { 0 } ) )  u.  ( `' G " ( _V 
\  { 0 } ) ) )  e. 
Fin )
3832, 36, 37syl2anc 642 . . . 4  |-  ( ( I  e.  V  /\  F  e.  D  /\  G  e.  D )  ->  ( ( `' F " ( _V  \  {
0 } ) )  u.  ( `' G " ( _V  \  {
0 } ) ) )  e.  Fin )
39 ssun1 3338 . . . . . . . . 9  |-  ( `' F " ( _V 
\  { 0 } ) )  C_  (
( `' F "
( _V  \  {
0 } ) )  u.  ( `' G " ( _V  \  {
0 } ) ) )
4039a1i 10 . . . . . . . 8  |-  ( ( I  e.  V  /\  F  e.  D  /\  G  e.  D )  ->  ( `' F "
( _V  \  {
0 } ) ) 
C_  ( ( `' F " ( _V 
\  { 0 } ) )  u.  ( `' G " ( _V 
\  { 0 } ) ) ) )
418, 40suppssr 5659 . . . . . . 7  |-  ( ( ( I  e.  V  /\  F  e.  D  /\  G  e.  D
)  /\  x  e.  ( I  \  (
( `' F "
( _V  \  {
0 } ) )  u.  ( `' G " ( _V  \  {
0 } ) ) ) ) )  -> 
( F `  x
)  =  0 )
42 ssun2 3339 . . . . . . . . 9  |-  ( `' G " ( _V 
\  { 0 } ) )  C_  (
( `' F "
( _V  \  {
0 } ) )  u.  ( `' G " ( _V  \  {
0 } ) ) )
4342a1i 10 . . . . . . . 8  |-  ( ( I  e.  V  /\  F  e.  D  /\  G  e.  D )  ->  ( `' G "
( _V  \  {
0 } ) ) 
C_  ( ( `' F " ( _V 
\  { 0 } ) )  u.  ( `' G " ( _V 
\  { 0 } ) ) ) )
4413, 43suppssr 5659 . . . . . . 7  |-  ( ( ( I  e.  V  /\  F  e.  D  /\  G  e.  D
)  /\  x  e.  ( I  \  (
( `' F "
( _V  \  {
0 } ) )  u.  ( `' G " ( _V  \  {
0 } ) ) ) ) )  -> 
( G `  x
)  =  0 )
4541, 44oveq12d 5876 . . . . . 6  |-  ( ( ( I  e.  V  /\  F  e.  D  /\  G  e.  D
)  /\  x  e.  ( I  \  (
( `' F "
( _V  \  {
0 } ) )  u.  ( `' G " ( _V  \  {
0 } ) ) ) ) )  -> 
( ( F `  x )  +  ( G `  x ) )  =  ( 0  +  0 ) )
46 00id 8987 . . . . . 6  |-  ( 0  +  0 )  =  0
4745, 46syl6eq 2331 . . . . 5  |-  ( ( ( I  e.  V  /\  F  e.  D  /\  G  e.  D
)  /\  x  e.  ( I  \  (
( `' F "
( _V  \  {
0 } ) )  u.  ( `' G " ( _V  \  {
0 } ) ) ) ) )  -> 
( ( F `  x )  +  ( G `  x ) )  =  0 )
4847suppss2 6073 . . . 4  |-  ( ( I  e.  V  /\  F  e.  D  /\  G  e.  D )  ->  ( `' ( x  e.  I  |->  ( ( F `  x )  +  ( G `  x ) ) )
" ( _V  \  { 0 } ) )  C_  ( ( `' F " ( _V 
\  { 0 } ) )  u.  ( `' G " ( _V 
\  { 0 } ) ) ) )
49 ssfi 7083 . . . 4  |-  ( ( ( ( `' F " ( _V  \  {
0 } ) )  u.  ( `' G " ( _V  \  {
0 } ) ) )  e.  Fin  /\  ( `' ( x  e.  I  |->  ( ( F `
 x )  +  ( G `  x
) ) ) "
( _V  \  {
0 } ) ) 
C_  ( ( `' F " ( _V 
\  { 0 } ) )  u.  ( `' G " ( _V 
\  { 0 } ) ) ) )  ->  ( `' ( x  e.  I  |->  ( ( F `  x
)  +  ( G `
 x ) ) ) " ( _V 
\  { 0 } ) )  e.  Fin )
5038, 48, 49syl2anc 642 . . 3  |-  ( ( I  e.  V  /\  F  e.  D  /\  G  e.  D )  ->  ( `' ( x  e.  I  |->  ( ( F `  x )  +  ( G `  x ) ) )
" ( _V  \  { 0 } ) )  e.  Fin )
5128, 50eqeltrd 2357 . 2  |-  ( ( I  e.  V  /\  F  e.  D  /\  G  e.  D )  ->  ( `' ( F  o F  +  G
) " NN )  e.  Fin )
524psrbag 16112 . . 3  |-  ( I  e.  V  ->  (
( F  o F  +  G )  e.  D  <->  ( ( F  o F  +  G
) : I --> NN0  /\  ( `' ( F  o F  +  G ) " NN )  e.  Fin ) ) )
53523ad2ant1 976 . 2  |-  ( ( I  e.  V  /\  F  e.  D  /\  G  e.  D )  ->  ( ( F  o F  +  G )  e.  D  <->  ( ( F  o F  +  G
) : I --> NN0  /\  ( `' ( F  o F  +  G ) " NN )  e.  Fin ) ) )
5416, 51, 53mpbir2and 888 1  |-  ( ( I  e.  V  /\  F  e.  D  /\  G  e.  D )  ->  ( F  o F  +  G )  e.  D )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   {crab 2547   _Vcvv 2788    \ cdif 3149    u. cun 3150    C_ wss 3152   {csn 3640    e. cmpt 4077   `'ccnv 4688   "cima 4692   -->wf 5251   ` cfv 5255  (class class class)co 5858    o Fcof 6076    ^m cmap 6772   Fincfn 6863   0cc0 8737    + caddc 8740   NNcn 9746   NN0cn0 9965
This theorem is referenced by:  mplmon2mul  16242  evlslem1  19399  tdeglem3  19445
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-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  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-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  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-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  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-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-riota 6304  df-recs 6388  df-rdg 6423  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-n0 9966
  Copyright terms: Public domain W3C validator