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Theorem gsum2d2lem 15434
Description: Lemma for gsum2d2 15435: show the function is finitely supported. (Contributed by Mario Carneiro, 28-Dec-2014.)
Hypotheses
Ref Expression
gsum2d2.b  |-  B  =  ( Base `  G
)
gsum2d2.z  |-  .0.  =  ( 0g `  G )
gsum2d2.g  |-  ( ph  ->  G  e. CMnd )
gsum2d2.a  |-  ( ph  ->  A  e.  V )
gsum2d2.r  |-  ( (
ph  /\  j  e.  A )  ->  C  e.  W )
gsum2d2.f  |-  ( (
ph  /\  ( j  e.  A  /\  k  e.  C ) )  ->  X  e.  B )
gsum2d2.u  |-  ( ph  ->  U  e.  Fin )
gsum2d2.n  |-  ( (
ph  /\  ( (
j  e.  A  /\  k  e.  C )  /\  -.  j U k ) )  ->  X  =  .0.  )
Assertion
Ref Expression
gsum2d2lem  |-  ( ph  ->  ( `' ( j  e.  A ,  k  e.  C  |->  X )
" ( _V  \  {  .0.  } ) )  e.  Fin )
Distinct variable groups:    j, k, B    ph, j, k    A, j, k    j, G, k    U, j, k    C, k   
j, V    .0. , j,
k
Allowed substitution hints:    C( j)    V( k)    W( j, k)    X( j, k)

Proof of Theorem gsum2d2lem
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 gsum2d2.u . 2  |-  ( ph  ->  U  e.  Fin )
2 gsum2d2.f . . . . 5  |-  ( (
ph  /\  ( j  e.  A  /\  k  e.  C ) )  ->  X  e.  B )
32ralrimivva 2720 . . . 4  |-  ( ph  ->  A. j  e.  A  A. k  e.  C  X  e.  B )
4 eqid 2366 . . . . 5  |-  ( j  e.  A ,  k  e.  C  |->  X )  =  ( j  e.  A ,  k  e.  C  |->  X )
54fmpt2x 6317 . . . 4  |-  ( A. j  e.  A  A. k  e.  C  X  e.  B  <->  ( j  e.  A ,  k  e.  C  |->  X ) :
U_ j  e.  A  ( { j }  X.  C ) --> B )
63, 5sylib 188 . . 3  |-  ( ph  ->  ( j  e.  A ,  k  e.  C  |->  X ) : U_ j  e.  A  ( { j }  X.  C ) --> B )
7 relxp 4897 . . . . . . 7  |-  Rel  ( { j }  X.  C )
87rgenw 2695 . . . . . 6  |-  A. j  e.  A  Rel  ( { j }  X.  C
)
9 reliun 4909 . . . . . 6  |-  ( Rel  U_ j  e.  A  ( { j }  X.  C )  <->  A. j  e.  A  Rel  ( { j }  X.  C
) )
108, 9mpbir 200 . . . . 5  |-  Rel  U_ j  e.  A  ( {
j }  X.  C
)
11 eldifi 3385 . . . . . 6  |-  ( z  e.  ( U_ j  e.  A  ( {
j }  X.  C
)  \  U )  ->  z  e.  U_ j  e.  A  ( {
j }  X.  C
) )
1211adantl 452 . . . . 5  |-  ( (
ph  /\  z  e.  ( U_ j  e.  A  ( { j }  X.  C )  \  U
) )  ->  z  e.  U_ j  e.  A  ( { j }  X.  C ) )
13 elrel 4892 . . . . 5  |-  ( ( Rel  U_ j  e.  A  ( { j }  X.  C )  /\  z  e.  U_ j  e.  A  ( { j }  X.  C ) )  ->  E. j E. k  z  =  <. j ,  k
>. )
1410, 12, 13sylancr 644 . . . 4  |-  ( (
ph  /\  z  e.  ( U_ j  e.  A  ( { j }  X.  C )  \  U
) )  ->  E. j E. k  z  =  <. j ,  k >.
)
15 nfv 1624 . . . . . 6  |-  F/ j
ph
16 nfiu1 4035 . . . . . . . 8  |-  F/_ j U_ j  e.  A  ( { j }  X.  C )
17 nfcv 2502 . . . . . . . 8  |-  F/_ j U
1816, 17nfdif 3384 . . . . . . 7  |-  F/_ j
( U_ j  e.  A  ( { j }  X.  C )  \  U
)
1918nfel2 2514 . . . . . 6  |-  F/ j  z  e.  ( U_ j  e.  A  ( { j }  X.  C )  \  U
)
2015, 19nfan 1834 . . . . 5  |-  F/ j ( ph  /\  z  e.  ( U_ j  e.  A  ( { j }  X.  C ) 
\  U ) )
21 nfmpt21 6040 . . . . . . 7  |-  F/_ j
( j  e.  A ,  k  e.  C  |->  X )
22 nfcv 2502 . . . . . . 7  |-  F/_ j
z
2321, 22nffv 5639 . . . . . 6  |-  F/_ j
( ( j  e.  A ,  k  e.  C  |->  X ) `  z )
2423nfeq1 2511 . . . . 5  |-  F/ j ( ( j  e.  A ,  k  e.  C  |->  X ) `  z )  =  .0.
25 nfv 1624 . . . . . 6  |-  F/ k ( ph  /\  z  e.  ( U_ j  e.  A  ( { j }  X.  C ) 
\  U ) )
26 nfmpt22 6041 . . . . . . . 8  |-  F/_ k
( j  e.  A ,  k  e.  C  |->  X )
27 nfcv 2502 . . . . . . . 8  |-  F/_ k
z
2826, 27nffv 5639 . . . . . . 7  |-  F/_ k
( ( j  e.  A ,  k  e.  C  |->  X ) `  z )
2928nfeq1 2511 . . . . . 6  |-  F/ k ( ( j  e.  A ,  k  e.  C  |->  X ) `  z )  =  .0.
30 simprr 733 . . . . . . . . 9  |-  ( (
ph  /\  ( z  e.  ( U_ j  e.  A  ( { j }  X.  C ) 
\  U )  /\  z  =  <. j ,  k >. ) )  -> 
z  =  <. j ,  k >. )
3130fveq2d 5636 . . . . . . . 8  |-  ( (
ph  /\  ( z  e.  ( U_ j  e.  A  ( { j }  X.  C ) 
\  U )  /\  z  =  <. j ,  k >. ) )  -> 
( ( j  e.  A ,  k  e.  C  |->  X ) `  z )  =  ( ( j  e.  A ,  k  e.  C  |->  X ) `  <. j ,  k >. )
)
32 df-ov 5984 . . . . . . . . 9  |-  ( j ( j  e.  A ,  k  e.  C  |->  X ) k )  =  ( ( j  e.  A ,  k  e.  C  |->  X ) `
 <. j ,  k
>. )
33 simprl 732 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( z  e.  ( U_ j  e.  A  ( { j }  X.  C ) 
\  U )  /\  z  =  <. j ,  k >. ) )  -> 
z  e.  ( U_ j  e.  A  ( { j }  X.  C )  \  U
) )
3430, 33eqeltrrd 2441 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( z  e.  ( U_ j  e.  A  ( { j }  X.  C ) 
\  U )  /\  z  =  <. j ,  k >. ) )  ->  <. j ,  k >.  e.  ( U_ j  e.  A  ( { j }  X.  C ) 
\  U ) )
35 eldifi 3385 . . . . . . . . . . . . 13  |-  ( <.
j ,  k >.  e.  ( U_ j  e.  A  ( { j }  X.  C ) 
\  U )  ->  <. j ,  k >.  e.  U_ j  e.  A  ( { j }  X.  C ) )
3634, 35syl 15 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( z  e.  ( U_ j  e.  A  ( { j }  X.  C ) 
\  U )  /\  z  =  <. j ,  k >. ) )  ->  <. j ,  k >.  e.  U_ j  e.  A  ( { j }  X.  C ) )
37 opeliunxp 4843 . . . . . . . . . . . 12  |-  ( <.
j ,  k >.  e.  U_ j  e.  A  ( { j }  X.  C )  <->  ( j  e.  A  /\  k  e.  C ) )
3836, 37sylib 188 . . . . . . . . . . 11  |-  ( (
ph  /\  ( z  e.  ( U_ j  e.  A  ( { j }  X.  C ) 
\  U )  /\  z  =  <. j ,  k >. ) )  -> 
( j  e.  A  /\  k  e.  C
) )
3938simpld 445 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  ( U_ j  e.  A  ( { j }  X.  C ) 
\  U )  /\  z  =  <. j ,  k >. ) )  -> 
j  e.  A )
4038simprd 449 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  ( U_ j  e.  A  ( { j }  X.  C ) 
\  U )  /\  z  =  <. j ,  k >. ) )  -> 
k  e.  C )
4138, 2syldan 456 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  ( U_ j  e.  A  ( { j }  X.  C ) 
\  U )  /\  z  =  <. j ,  k >. ) )  ->  X  e.  B )
424ovmpt4g 6096 . . . . . . . . . 10  |-  ( ( j  e.  A  /\  k  e.  C  /\  X  e.  B )  ->  ( j ( j  e.  A ,  k  e.  C  |->  X ) k )  =  X )
4339, 40, 41, 42syl3anc 1183 . . . . . . . . 9  |-  ( (
ph  /\  ( z  e.  ( U_ j  e.  A  ( { j }  X.  C ) 
\  U )  /\  z  =  <. j ,  k >. ) )  -> 
( j ( j  e.  A ,  k  e.  C  |->  X ) k )  =  X )
4432, 43syl5eqr 2412 . . . . . . . 8  |-  ( (
ph  /\  ( z  e.  ( U_ j  e.  A  ( { j }  X.  C ) 
\  U )  /\  z  =  <. j ,  k >. ) )  -> 
( ( j  e.  A ,  k  e.  C  |->  X ) `  <. j ,  k >.
)  =  X )
45 eldifn 3386 . . . . . . . . . . . 12  |-  ( z  e.  ( U_ j  e.  A  ( {
j }  X.  C
)  \  U )  ->  -.  z  e.  U
)
4645ad2antrl 708 . . . . . . . . . . 11  |-  ( (
ph  /\  ( z  e.  ( U_ j  e.  A  ( { j }  X.  C ) 
\  U )  /\  z  =  <. j ,  k >. ) )  ->  -.  z  e.  U
)
4730eleq1d 2432 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( z  e.  ( U_ j  e.  A  ( { j }  X.  C ) 
\  U )  /\  z  =  <. j ,  k >. ) )  -> 
( z  e.  U  <->  <.
j ,  k >.  e.  U ) )
48 df-br 4126 . . . . . . . . . . . 12  |-  ( j U k  <->  <. j ,  k >.  e.  U
)
4947, 48syl6bbr 254 . . . . . . . . . . 11  |-  ( (
ph  /\  ( z  e.  ( U_ j  e.  A  ( { j }  X.  C ) 
\  U )  /\  z  =  <. j ,  k >. ) )  -> 
( z  e.  U  <->  j U k ) )
5046, 49mtbid 291 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  ( U_ j  e.  A  ( { j }  X.  C ) 
\  U )  /\  z  =  <. j ,  k >. ) )  ->  -.  j U k )
5138, 50jca 518 . . . . . . . . 9  |-  ( (
ph  /\  ( z  e.  ( U_ j  e.  A  ( { j }  X.  C ) 
\  U )  /\  z  =  <. j ,  k >. ) )  -> 
( ( j  e.  A  /\  k  e.  C )  /\  -.  j U k ) )
52 gsum2d2.n . . . . . . . . 9  |-  ( (
ph  /\  ( (
j  e.  A  /\  k  e.  C )  /\  -.  j U k ) )  ->  X  =  .0.  )
5351, 52syldan 456 . . . . . . . 8  |-  ( (
ph  /\  ( z  e.  ( U_ j  e.  A  ( { j }  X.  C ) 
\  U )  /\  z  =  <. j ,  k >. ) )  ->  X  =  .0.  )
5431, 44, 533eqtrd 2402 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  ( U_ j  e.  A  ( { j }  X.  C ) 
\  U )  /\  z  =  <. j ,  k >. ) )  -> 
( ( j  e.  A ,  k  e.  C  |->  X ) `  z )  =  .0.  )
5554expr 598 . . . . . 6  |-  ( (
ph  /\  z  e.  ( U_ j  e.  A  ( { j }  X.  C )  \  U
) )  ->  (
z  =  <. j ,  k >.  ->  (
( j  e.  A ,  k  e.  C  |->  X ) `  z
)  =  .0.  )
)
5625, 29, 55exlimd 1812 . . . . 5  |-  ( (
ph  /\  z  e.  ( U_ j  e.  A  ( { j }  X.  C )  \  U
) )  ->  ( E. k  z  =  <. j ,  k >.  ->  ( ( j  e.  A ,  k  e.  C  |->  X ) `  z )  =  .0.  ) )
5720, 24, 56exlimd 1812 . . . 4  |-  ( (
ph  /\  z  e.  ( U_ j  e.  A  ( { j }  X.  C )  \  U
) )  ->  ( E. j E. k  z  =  <. j ,  k
>.  ->  ( ( j  e.  A ,  k  e.  C  |->  X ) `
 z )  =  .0.  ) )
5814, 57mpd 14 . . 3  |-  ( (
ph  /\  z  e.  ( U_ j  e.  A  ( { j }  X.  C )  \  U
) )  ->  (
( j  e.  A ,  k  e.  C  |->  X ) `  z
)  =  .0.  )
596, 58suppss 5765 . 2  |-  ( ph  ->  ( `' ( j  e.  A ,  k  e.  C  |->  X )
" ( _V  \  {  .0.  } ) ) 
C_  U )
60 ssfi 7226 . 2  |-  ( ( U  e.  Fin  /\  ( `' ( j  e.  A ,  k  e.  C  |->  X ) "
( _V  \  {  .0.  } ) )  C_  U )  ->  ( `' ( j  e.  A ,  k  e.  C  |->  X ) "
( _V  \  {  .0.  } ) )  e. 
Fin )
611, 59, 60syl2anc 642 1  |-  ( ph  ->  ( `' ( j  e.  A ,  k  e.  C  |->  X )
" ( _V  \  {  .0.  } ) )  e.  Fin )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358   E.wex 1546    = wceq 1647    e. wcel 1715   A.wral 2628   _Vcvv 2873    \ cdif 3235    C_ wss 3238   {csn 3729   <.cop 3732   U_ciun 4007   class class class wbr 4125    X. cxp 4790   `'ccnv 4791   "cima 4795   Rel wrel 4797   -->wf 5354   ` cfv 5358  (class class class)co 5981    e. cmpt2 5983   Fincfn 7006   Basecbs 13356   0gc0g 13610  CMndccmn 15299
This theorem is referenced by:  gsum2d2  15435  gsumcom2  15436
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-er 6802  df-en 7007  df-fin 7010
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