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Theorem gsum2d2lem 15502
Description: Lemma for gsum2d2 15503: 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 2758 . . . 4  |-  ( ph  ->  A. j  e.  A  A. k  e.  C  X  e.  B )
4 eqid 2404 . . . . 5  |-  ( j  e.  A ,  k  e.  C  |->  X )  =  ( j  e.  A ,  k  e.  C  |->  X )
54fmpt2x 6376 . . . 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 189 . . 3  |-  ( ph  ->  ( j  e.  A ,  k  e.  C  |->  X ) : U_ j  e.  A  ( { j }  X.  C ) --> B )
7 relxp 4942 . . . . . . 7  |-  Rel  ( { j }  X.  C )
87rgenw 2733 . . . . . 6  |-  A. j  e.  A  Rel  ( { j }  X.  C
)
9 reliun 4954 . . . . . 6  |-  ( Rel  U_ j  e.  A  ( { j }  X.  C )  <->  A. j  e.  A  Rel  ( { j }  X.  C
) )
108, 9mpbir 201 . . . . 5  |-  Rel  U_ j  e.  A  ( {
j }  X.  C
)
11 eldifi 3429 . . . . . 6  |-  ( z  e.  ( U_ j  e.  A  ( {
j }  X.  C
)  \  U )  ->  z  e.  U_ j  e.  A  ( {
j }  X.  C
) )
1211adantl 453 . . . . 5  |-  ( (
ph  /\  z  e.  ( U_ j  e.  A  ( { j }  X.  C )  \  U
) )  ->  z  e.  U_ j  e.  A  ( { j }  X.  C ) )
13 elrel 4937 . . . . 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 645 . . . 4  |-  ( (
ph  /\  z  e.  ( U_ j  e.  A  ( { j }  X.  C )  \  U
) )  ->  E. j E. k  z  =  <. j ,  k >.
)
15 nfv 1626 . . . . . 6  |-  F/ j
ph
16 nfiu1 4081 . . . . . . . 8  |-  F/_ j U_ j  e.  A  ( { j }  X.  C )
17 nfcv 2540 . . . . . . . 8  |-  F/_ j U
1816, 17nfdif 3428 . . . . . . 7  |-  F/_ j
( U_ j  e.  A  ( { j }  X.  C )  \  U
)
1918nfcri 2534 . . . . . 6  |-  F/ j  z  e.  ( U_ j  e.  A  ( { j }  X.  C )  \  U
)
2015, 19nfan 1842 . . . . 5  |-  F/ j ( ph  /\  z  e.  ( U_ j  e.  A  ( { j }  X.  C ) 
\  U ) )
21 nfmpt21 6099 . . . . . . 7  |-  F/_ j
( j  e.  A ,  k  e.  C  |->  X )
22 nfcv 2540 . . . . . . 7  |-  F/_ j
z
2321, 22nffv 5694 . . . . . 6  |-  F/_ j
( ( j  e.  A ,  k  e.  C  |->  X ) `  z )
2423nfeq1 2549 . . . . 5  |-  F/ j ( ( j  e.  A ,  k  e.  C  |->  X ) `  z )  =  .0.
25 nfv 1626 . . . . . 6  |-  F/ k ( ph  /\  z  e.  ( U_ j  e.  A  ( { j }  X.  C ) 
\  U ) )
26 nfmpt22 6100 . . . . . . . 8  |-  F/_ k
( j  e.  A ,  k  e.  C  |->  X )
27 nfcv 2540 . . . . . . . 8  |-  F/_ k
z
2826, 27nffv 5694 . . . . . . 7  |-  F/_ k
( ( j  e.  A ,  k  e.  C  |->  X ) `  z )
2928nfeq1 2549 . . . . . 6  |-  F/ k ( ( j  e.  A ,  k  e.  C  |->  X ) `  z )  =  .0.
30 simprr 734 . . . . . . . . 9  |-  ( (
ph  /\  ( z  e.  ( U_ j  e.  A  ( { j }  X.  C ) 
\  U )  /\  z  =  <. j ,  k >. ) )  -> 
z  =  <. j ,  k >. )
3130fveq2d 5691 . . . . . . . 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 6043 . . . . . . . . 9  |-  ( j ( j  e.  A ,  k  e.  C  |->  X ) k )  =  ( ( j  e.  A ,  k  e.  C  |->  X ) `
 <. j ,  k
>. )
33 simprl 733 . . . . . . . . . . . . . 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 2479 . . . . . . . . . . . . 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 ) )
3534eldifad 3292 . . . . . . . . . . . 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 ) )
36 opeliunxp 4888 . . . . . . . . . . . 12  |-  ( <.
j ,  k >.  e.  U_ j  e.  A  ( { j }  X.  C )  <->  ( j  e.  A  /\  k  e.  C ) )
3735, 36sylib 189 . . . . . . . . . . 11  |-  ( (
ph  /\  ( z  e.  ( U_ j  e.  A  ( { j }  X.  C ) 
\  U )  /\  z  =  <. j ,  k >. ) )  -> 
( j  e.  A  /\  k  e.  C
) )
3837simpld 446 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  ( U_ j  e.  A  ( { j }  X.  C ) 
\  U )  /\  z  =  <. j ,  k >. ) )  -> 
j  e.  A )
3937simprd 450 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  ( U_ j  e.  A  ( { j }  X.  C ) 
\  U )  /\  z  =  <. j ,  k >. ) )  -> 
k  e.  C )
4037, 2syldan 457 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  ( U_ j  e.  A  ( { j }  X.  C ) 
\  U )  /\  z  =  <. j ,  k >. ) )  ->  X  e.  B )
414ovmpt4g 6155 . . . . . . . . . 10  |-  ( ( j  e.  A  /\  k  e.  C  /\  X  e.  B )  ->  ( j ( j  e.  A ,  k  e.  C  |->  X ) k )  =  X )
4238, 39, 40, 41syl3anc 1184 . . . . . . . . 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 )
4332, 42syl5eqr 2450 . . . . . . . 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 )
44 eldifn 3430 . . . . . . . . . . . 12  |-  ( z  e.  ( U_ j  e.  A  ( {
j }  X.  C
)  \  U )  ->  -.  z  e.  U
)
4544ad2antrl 709 . . . . . . . . . . 11  |-  ( (
ph  /\  ( z  e.  ( U_ j  e.  A  ( { j }  X.  C ) 
\  U )  /\  z  =  <. j ,  k >. ) )  ->  -.  z  e.  U
)
4630eleq1d 2470 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( z  e.  ( U_ j  e.  A  ( { j }  X.  C ) 
\  U )  /\  z  =  <. j ,  k >. ) )  -> 
( z  e.  U  <->  <.
j ,  k >.  e.  U ) )
47 df-br 4173 . . . . . . . . . . . 12  |-  ( j U k  <->  <. j ,  k >.  e.  U
)
4846, 47syl6bbr 255 . . . . . . . . . . 11  |-  ( (
ph  /\  ( z  e.  ( U_ j  e.  A  ( { j }  X.  C ) 
\  U )  /\  z  =  <. j ,  k >. ) )  -> 
( z  e.  U  <->  j U k ) )
4945, 48mtbid 292 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  ( U_ j  e.  A  ( { j }  X.  C ) 
\  U )  /\  z  =  <. j ,  k >. ) )  ->  -.  j U k )
5037, 49jca 519 . . . . . . . . 9  |-  ( (
ph  /\  ( z  e.  ( U_ j  e.  A  ( { j }  X.  C ) 
\  U )  /\  z  =  <. j ,  k >. ) )  -> 
( ( j  e.  A  /\  k  e.  C )  /\  -.  j U k ) )
51 gsum2d2.n . . . . . . . . 9  |-  ( (
ph  /\  ( (
j  e.  A  /\  k  e.  C )  /\  -.  j U k ) )  ->  X  =  .0.  )
5250, 51syldan 457 . . . . . . . 8  |-  ( (
ph  /\  ( z  e.  ( U_ j  e.  A  ( { j }  X.  C ) 
\  U )  /\  z  =  <. j ,  k >. ) )  ->  X  =  .0.  )
5331, 43, 523eqtrd 2440 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  ( U_ j  e.  A  ( { j }  X.  C ) 
\  U )  /\  z  =  <. j ,  k >. ) )  -> 
( ( j  e.  A ,  k  e.  C  |->  X ) `  z )  =  .0.  )
5453expr 599 . . . . . 6  |-  ( (
ph  /\  z  e.  ( U_ j  e.  A  ( { j }  X.  C )  \  U
) )  ->  (
z  =  <. j ,  k >.  ->  (
( j  e.  A ,  k  e.  C  |->  X ) `  z
)  =  .0.  )
)
5525, 29, 54exlimd 1820 . . . . 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.  ) )
5620, 24, 55exlimd 1820 . . . 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.  ) )
5714, 56mpd 15 . . 3  |-  ( (
ph  /\  z  e.  ( U_ j  e.  A  ( { j }  X.  C )  \  U
) )  ->  (
( j  e.  A ,  k  e.  C  |->  X ) `  z
)  =  .0.  )
586, 57suppss 5822 . 2  |-  ( ph  ->  ( `' ( j  e.  A ,  k  e.  C  |->  X )
" ( _V  \  {  .0.  } ) ) 
C_  U )
59 ssfi 7288 . 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 )
601, 58, 59syl2anc 643 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 359   E.wex 1547    = wceq 1649    e. wcel 1721   A.wral 2666   _Vcvv 2916    \ cdif 3277    C_ wss 3280   {csn 3774   <.cop 3777   U_ciun 4053   class class class wbr 4172    X. cxp 4835   `'ccnv 4836   "cima 4840   Rel wrel 4842   -->wf 5409   ` cfv 5413  (class class class)co 6040    e. cmpt2 6042   Fincfn 7068   Basecbs 13424   0gc0g 13678  CMndccmn 15367
This theorem is referenced by:  gsum2d2  15503  gsumcom2  15504
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-er 6864  df-en 7069  df-fin 7072
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