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Theorem gsum2d2lem 15224
Description: Lemma for gsum2d2 15225: 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 2635 . . . 4  |-  ( ph  ->  A. j  e.  A  A. k  e.  C  X  e.  B )
4 eqid 2283 . . . . 5  |-  ( j  e.  A ,  k  e.  C  |->  X )  =  ( j  e.  A ,  k  e.  C  |->  X )
54fmpt2x 6190 . . . 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 4794 . . . . . . 7  |-  Rel  ( { j }  X.  C )
87rgenw 2610 . . . . . 6  |-  A. j  e.  A  Rel  ( { j }  X.  C
)
9 reliun 4806 . . . . . 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 3298 . . . . . 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 4789 . . . . 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 1605 . . . . . 6  |-  F/ j
ph
16 nfiu1 3933 . . . . . . . 8  |-  F/_ j U_ j  e.  A  ( { j }  X.  C )
17 nfcv 2419 . . . . . . . 8  |-  F/_ j U
1816, 17nfdif 3297 . . . . . . 7  |-  F/_ j
( U_ j  e.  A  ( { j }  X.  C )  \  U
)
1918nfel2 2431 . . . . . 6  |-  F/ j  z  e.  ( U_ j  e.  A  ( { j }  X.  C )  \  U
)
2015, 19nfan 1771 . . . . 5  |-  F/ j ( ph  /\  z  e.  ( U_ j  e.  A  ( { j }  X.  C ) 
\  U ) )
21 nfmpt21 5914 . . . . . . 7  |-  F/_ j
( j  e.  A ,  k  e.  C  |->  X )
22 nfcv 2419 . . . . . . 7  |-  F/_ j
z
2321, 22nffv 5532 . . . . . 6  |-  F/_ j
( ( j  e.  A ,  k  e.  C  |->  X ) `  z )
2423nfeq1 2428 . . . . 5  |-  F/ j ( ( j  e.  A ,  k  e.  C  |->  X ) `  z )  =  .0.
25 nfv 1605 . . . . . 6  |-  F/ k ( ph  /\  z  e.  ( U_ j  e.  A  ( { j }  X.  C ) 
\  U ) )
26 nfmpt22 5915 . . . . . . . 8  |-  F/_ k
( j  e.  A ,  k  e.  C  |->  X )
27 nfcv 2419 . . . . . . . 8  |-  F/_ k
z
2826, 27nffv 5532 . . . . . . 7  |-  F/_ k
( ( j  e.  A ,  k  e.  C  |->  X ) `  z )
2928nfeq1 2428 . . . . . 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 5529 . . . . . . . 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 5861 . . . . . . . . 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 2358 . . . . . . . . . . . . 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 3298 . . . . . . . . . . . . 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 4740 . . . . . . . . . . . 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 5970 . . . . . . . . . 10  |-  ( ( j  e.  A  /\  k  e.  C  /\  X  e.  B )  ->  ( j ( j  e.  A ,  k  e.  C  |->  X ) k )  =  X )
4339, 40, 41, 42syl3anc 1182 . . . . . . . . 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 2329 . . . . . . . 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 3299 . . . . . . . . . . . 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 2349 . . . . . . . . . . . 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 4024 . . . . . . . . . . . 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 2319 . . . . . . 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 1803 . . . . 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 1803 . . . 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 5658 . 2  |-  ( ph  ->  ( `' ( j  e.  A ,  k  e.  C  |->  X )
" ( _V  \  {  .0.  } ) ) 
C_  U )
60 ssfi 7083 . 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 1528    = wceq 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788    \ cdif 3149    C_ wss 3152   {csn 3640   <.cop 3643   U_ciun 3905   class class class wbr 4023    X. cxp 4687   `'ccnv 4688   "cima 4692   Rel wrel 4694   -->wf 5251   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   Fincfn 6863   Basecbs 13148   0gc0g 13400  CMndccmn 15089
This theorem is referenced by:  gsum2d2  15225  gsumcom2  15226
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-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
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-ral 2548  df-rex 2549  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-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-1st 6122  df-2nd 6123  df-er 6660  df-en 6864  df-fin 6867
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