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Theorem psrbagaddcl 16427
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 10247 . . . 4  |-  ( ( x  e.  NN0  /\  y  e.  NN0 )  -> 
( x  +  y )  e.  NN0 )
21adantl 453 . . 3  |-  ( ( ( I  e.  V  /\  F  e.  D  /\  G  e.  D
)  /\  ( x  e.  NN0  /\  y  e. 
NN0 ) )  -> 
( x  +  y )  e.  NN0 )
3 simp2 958 . . . . 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 16423 . . . . . 6  |-  ( I  e.  V  ->  ( F  e.  D  <->  ( F : I --> NN0  /\  ( `' F " NN )  e.  Fin ) ) )
653ad2ant1 978 . . . . 5  |-  ( ( I  e.  V  /\  F  e.  D  /\  G  e.  D )  ->  ( F  e.  D  <->  ( F : I --> NN0  /\  ( `' F " NN )  e.  Fin ) ) )
73, 6mpbid 202 . . . 4  |-  ( ( I  e.  V  /\  F  e.  D  /\  G  e.  D )  ->  ( F : I --> NN0  /\  ( `' F " NN )  e.  Fin ) )
87simpld 446 . . 3  |-  ( ( I  e.  V  /\  F  e.  D  /\  G  e.  D )  ->  F : I --> NN0 )
9 simp3 959 . . . . 5  |-  ( ( I  e.  V  /\  F  e.  D  /\  G  e.  D )  ->  G  e.  D )
104psrbag 16423 . . . . . 6  |-  ( I  e.  V  ->  ( G  e.  D  <->  ( G : I --> NN0  /\  ( `' G " NN )  e.  Fin ) ) )
11103ad2ant1 978 . . . . 5  |-  ( ( I  e.  V  /\  F  e.  D  /\  G  e.  D )  ->  ( G  e.  D  <->  ( G : I --> NN0  /\  ( `' G " NN )  e.  Fin ) ) )
129, 11mpbid 202 . . . 4  |-  ( ( I  e.  V  /\  F  e.  D  /\  G  e.  D )  ->  ( G : I --> NN0  /\  ( `' G " NN )  e.  Fin ) )
1312simpld 446 . . 3  |-  ( ( I  e.  V  /\  F  e.  D  /\  G  e.  D )  ->  G : I --> NN0 )
14 simp1 957 . . 3  |-  ( ( I  e.  V  /\  F  e.  D  /\  G  e.  D )  ->  I  e.  V )
15 inidm 3542 . . 3  |-  ( I  i^i  I )  =  I
162, 8, 13, 14, 14, 15off 6312 . 2  |-  ( ( I  e.  V  /\  F  e.  D  /\  G  e.  D )  ->  ( F  o F  +  G ) : I --> NN0 )
17 nn0supp 10265 . . . . 5  |-  ( ( F  o F  +  G ) : I --> NN0  ->  ( `' ( F  o F  +  G ) " ( _V  \  { 0 } ) )  =  ( `' ( F  o F  +  G ) " NN ) )
1816, 17syl 16 . . . 4  |-  ( ( I  e.  V  /\  F  e.  D  /\  G  e.  D )  ->  ( `' ( F  o F  +  G
) " ( _V 
\  { 0 } ) )  =  ( `' ( F  o F  +  G ) " NN ) )
19 fvex 5734 . . . . . . . 8  |-  ( F `
 x )  e. 
_V
2019a1i 11 . . . . . . 7  |-  ( ( ( I  e.  V  /\  F  e.  D  /\  G  e.  D
)  /\  x  e.  I )  ->  ( F `  x )  e.  _V )
21 fvex 5734 . . . . . . . 8  |-  ( G `
 x )  e. 
_V
2221a1i 11 . . . . . . 7  |-  ( ( ( I  e.  V  /\  F  e.  D  /\  G  e.  D
)  /\  x  e.  I )  ->  ( G `  x )  e.  _V )
238feqmptd 5771 . . . . . . 7  |-  ( ( I  e.  V  /\  F  e.  D  /\  G  e.  D )  ->  F  =  ( x  e.  I  |->  ( F `
 x ) ) )
2413feqmptd 5771 . . . . . . 7  |-  ( ( I  e.  V  /\  F  e.  D  /\  G  e.  D )  ->  G  =  ( x  e.  I  |->  ( G `
 x ) ) )
2514, 20, 22, 23, 24offval2 6314 . . . . . 6  |-  ( ( I  e.  V  /\  F  e.  D  /\  G  e.  D )  ->  ( F  o F  +  G )  =  ( x  e.  I  |->  ( ( F `  x )  +  ( G `  x ) ) ) )
2625cnveqd 5040 . . . . 5  |-  ( ( I  e.  V  /\  F  e.  D  /\  G  e.  D )  ->  `' ( F  o F  +  G )  =  `' ( x  e.  I  |->  ( ( F `
 x )  +  ( G `  x
) ) ) )
2726imaeq1d 5194 . . . 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 2469 . . 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 10265 . . . . . . 7  |-  ( F : I --> NN0  ->  ( `' F " ( _V 
\  { 0 } ) )  =  ( `' F " NN ) )
308, 29syl 16 . . . . . 6  |-  ( ( I  e.  V  /\  F  e.  D  /\  G  e.  D )  ->  ( `' F "
( _V  \  {
0 } ) )  =  ( `' F " NN ) )
317simprd 450 . . . . . 6  |-  ( ( I  e.  V  /\  F  e.  D  /\  G  e.  D )  ->  ( `' F " NN )  e.  Fin )
3230, 31eqeltrd 2509 . . . . 5  |-  ( ( I  e.  V  /\  F  e.  D  /\  G  e.  D )  ->  ( `' F "
( _V  \  {
0 } ) )  e.  Fin )
33 nn0supp 10265 . . . . . . 7  |-  ( G : I --> NN0  ->  ( `' G " ( _V 
\  { 0 } ) )  =  ( `' G " NN ) )
3413, 33syl 16 . . . . . 6  |-  ( ( I  e.  V  /\  F  e.  D  /\  G  e.  D )  ->  ( `' G "
( _V  \  {
0 } ) )  =  ( `' G " NN ) )
3512simprd 450 . . . . . 6  |-  ( ( I  e.  V  /\  F  e.  D  /\  G  e.  D )  ->  ( `' G " NN )  e.  Fin )
3634, 35eqeltrd 2509 . . . . 5  |-  ( ( I  e.  V  /\  F  e.  D  /\  G  e.  D )  ->  ( `' G "
( _V  \  {
0 } ) )  e.  Fin )
37 unfi 7366 . . . . 5  |-  ( ( ( `' F "
( _V  \  {
0 } ) )  e.  Fin  /\  ( `' G " ( _V 
\  { 0 } ) )  e.  Fin )  ->  ( ( `' F " ( _V 
\  { 0 } ) )  u.  ( `' G " ( _V 
\  { 0 } ) ) )  e. 
Fin )
3832, 36, 37syl2anc 643 . . . 4  |-  ( ( I  e.  V  /\  F  e.  D  /\  G  e.  D )  ->  ( ( `' F " ( _V  \  {
0 } ) )  u.  ( `' G " ( _V  \  {
0 } ) ) )  e.  Fin )
39 ssun1 3502 . . . . . . . . 9  |-  ( `' F " ( _V 
\  { 0 } ) )  C_  (
( `' F "
( _V  \  {
0 } ) )  u.  ( `' G " ( _V  \  {
0 } ) ) )
4039a1i 11 . . . . . . . 8  |-  ( ( I  e.  V  /\  F  e.  D  /\  G  e.  D )  ->  ( `' F "
( _V  \  {
0 } ) ) 
C_  ( ( `' F " ( _V 
\  { 0 } ) )  u.  ( `' G " ( _V 
\  { 0 } ) ) ) )
418, 40suppssr 5856 . . . . . . 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 3503 . . . . . . . . 9  |-  ( `' G " ( _V 
\  { 0 } ) )  C_  (
( `' F "
( _V  \  {
0 } ) )  u.  ( `' G " ( _V  \  {
0 } ) ) )
4342a1i 11 . . . . . . . 8  |-  ( ( I  e.  V  /\  F  e.  D  /\  G  e.  D )  ->  ( `' G "
( _V  \  {
0 } ) ) 
C_  ( ( `' F " ( _V 
\  { 0 } ) )  u.  ( `' G " ( _V 
\  { 0 } ) ) ) )
4413, 43suppssr 5856 . . . . . . 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 6091 . . . . . 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 9233 . . . . . 6  |-  ( 0  +  0 )  =  0
4745, 46syl6eq 2483 . . . . 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 6292 . . . 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 7321 . . . 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 643 . . 3  |-  ( ( I  e.  V  /\  F  e.  D  /\  G  e.  D )  ->  ( `' ( x  e.  I  |->  ( ( F `  x )  +  ( G `  x ) ) )
" ( _V  \  { 0 } ) )  e.  Fin )
5128, 50eqeltrd 2509 . 2  |-  ( ( I  e.  V  /\  F  e.  D  /\  G  e.  D )  ->  ( `' ( F  o F  +  G
) " NN )  e.  Fin )
524psrbag 16423 . . 3  |-  ( I  e.  V  ->  (
( F  o F  +  G )  e.  D  <->  ( ( F  o F  +  G
) : I --> NN0  /\  ( `' ( F  o F  +  G ) " NN )  e.  Fin ) ) )
53523ad2ant1 978 . 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 889 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 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   {crab 2701   _Vcvv 2948    \ cdif 3309    u. cun 3310    C_ wss 3312   {csn 3806    e. cmpt 4258   `'ccnv 4869   "cima 4873   -->wf 5442   ` cfv 5446  (class class class)co 6073    o Fcof 6295    ^m cmap 7010   Fincfn 7101   0cc0 8982    + caddc 8985   NNcn 9992   NN0cn0 10213
This theorem is referenced by:  mplmon2mul  16553  evlslem1  19928  tdeglem3  19974
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-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  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-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  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-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  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-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-riota 6541  df-recs 6625  df-rdg 6660  df-oadd 6720  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-n0 10214
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