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Theorem gsum2d2lem 15552
Description: Lemma for gsum2d2 15553: 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 2800 . . . 4  |-  ( ph  ->  A. j  e.  A  A. k  e.  C  X  e.  B )
4 eqid 2438 . . . . 5  |-  ( j  e.  A ,  k  e.  C  |->  X )  =  ( j  e.  A ,  k  e.  C  |->  X )
54fmpt2x 6420 . . . 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 190 . . 3  |-  ( ph  ->  ( j  e.  A ,  k  e.  C  |->  X ) : U_ j  e.  A  ( { j }  X.  C ) --> B )
7 relxp 4986 . . . . . . 7  |-  Rel  ( { j }  X.  C )
87rgenw 2775 . . . . . 6  |-  A. j  e.  A  Rel  ( { j }  X.  C
)
9 reliun 4998 . . . . . 6  |-  ( Rel  U_ j  e.  A  ( { j }  X.  C )  <->  A. j  e.  A  Rel  ( { j }  X.  C
) )
108, 9mpbir 202 . . . . 5  |-  Rel  U_ j  e.  A  ( {
j }  X.  C
)
11 eldifi 3471 . . . . . 6  |-  ( z  e.  ( U_ j  e.  A  ( {
j }  X.  C
)  \  U )  ->  z  e.  U_ j  e.  A  ( {
j }  X.  C
) )
1211adantl 454 . . . . 5  |-  ( (
ph  /\  z  e.  ( U_ j  e.  A  ( { j }  X.  C )  \  U
) )  ->  z  e.  U_ j  e.  A  ( { j }  X.  C ) )
13 elrel 4981 . . . . 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 646 . . . 4  |-  ( (
ph  /\  z  e.  ( U_ j  e.  A  ( { j }  X.  C )  \  U
) )  ->  E. j E. k  z  =  <. j ,  k >.
)
15 nfv 1630 . . . . . 6  |-  F/ j
ph
16 nfiu1 4123 . . . . . . . 8  |-  F/_ j U_ j  e.  A  ( { j }  X.  C )
17 nfcv 2574 . . . . . . . 8  |-  F/_ j U
1816, 17nfdif 3470 . . . . . . 7  |-  F/_ j
( U_ j  e.  A  ( { j }  X.  C )  \  U
)
1918nfcri 2568 . . . . . 6  |-  F/ j  z  e.  ( U_ j  e.  A  ( { j }  X.  C )  \  U
)
2015, 19nfan 1847 . . . . 5  |-  F/ j ( ph  /\  z  e.  ( U_ j  e.  A  ( { j }  X.  C ) 
\  U ) )
21 nfmpt21 6143 . . . . . . 7  |-  F/_ j
( j  e.  A ,  k  e.  C  |->  X )
22 nfcv 2574 . . . . . . 7  |-  F/_ j
z
2321, 22nffv 5738 . . . . . 6  |-  F/_ j
( ( j  e.  A ,  k  e.  C  |->  X ) `  z )
2423nfeq1 2583 . . . . 5  |-  F/ j ( ( j  e.  A ,  k  e.  C  |->  X ) `  z )  =  .0.
25 nfv 1630 . . . . . 6  |-  F/ k ( ph  /\  z  e.  ( U_ j  e.  A  ( { j }  X.  C ) 
\  U ) )
26 nfmpt22 6144 . . . . . . . 8  |-  F/_ k
( j  e.  A ,  k  e.  C  |->  X )
27 nfcv 2574 . . . . . . . 8  |-  F/_ k
z
2826, 27nffv 5738 . . . . . . 7  |-  F/_ k
( ( j  e.  A ,  k  e.  C  |->  X ) `  z )
2928nfeq1 2583 . . . . . 6  |-  F/ k ( ( j  e.  A ,  k  e.  C  |->  X ) `  z )  =  .0.
30 simprr 735 . . . . . . . . 9  |-  ( (
ph  /\  ( z  e.  ( U_ j  e.  A  ( { j }  X.  C ) 
\  U )  /\  z  =  <. j ,  k >. ) )  -> 
z  =  <. j ,  k >. )
3130fveq2d 5735 . . . . . . . 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 6087 . . . . . . . . 9  |-  ( j ( j  e.  A ,  k  e.  C  |->  X ) k )  =  ( ( j  e.  A ,  k  e.  C  |->  X ) `
 <. j ,  k
>. )
33 simprl 734 . . . . . . . . . . . . . 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 2513 . . . . . . . . . . . . 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 3334 . . . . . . . . . . . 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 4932 . . . . . . . . . . . 12  |-  ( <.
j ,  k >.  e.  U_ j  e.  A  ( { j }  X.  C )  <->  ( j  e.  A  /\  k  e.  C ) )
3735, 36sylib 190 . . . . . . . . . . 11  |-  ( (
ph  /\  ( z  e.  ( U_ j  e.  A  ( { j }  X.  C ) 
\  U )  /\  z  =  <. j ,  k >. ) )  -> 
( j  e.  A  /\  k  e.  C
) )
3837simpld 447 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  ( U_ j  e.  A  ( { j }  X.  C ) 
\  U )  /\  z  =  <. j ,  k >. ) )  -> 
j  e.  A )
3937simprd 451 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  ( U_ j  e.  A  ( { j }  X.  C ) 
\  U )  /\  z  =  <. j ,  k >. ) )  -> 
k  e.  C )
4037, 2syldan 458 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  ( U_ j  e.  A  ( { j }  X.  C ) 
\  U )  /\  z  =  <. j ,  k >. ) )  ->  X  e.  B )
414ovmpt4g 6199 . . . . . . . . . 10  |-  ( ( j  e.  A  /\  k  e.  C  /\  X  e.  B )  ->  ( j ( j  e.  A ,  k  e.  C  |->  X ) k )  =  X )
4238, 39, 40, 41syl3anc 1185 . . . . . . . . 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 2484 . . . . . . . 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 3472 . . . . . . . . . . . 12  |-  ( z  e.  ( U_ j  e.  A  ( {
j }  X.  C
)  \  U )  ->  -.  z  e.  U
)
4544ad2antrl 710 . . . . . . . . . . 11  |-  ( (
ph  /\  ( z  e.  ( U_ j  e.  A  ( { j }  X.  C ) 
\  U )  /\  z  =  <. j ,  k >. ) )  ->  -.  z  e.  U
)
4630eleq1d 2504 . . . . . . . . . . . 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 4216 . . . . . . . . . . . 12  |-  ( j U k  <->  <. j ,  k >.  e.  U
)
4846, 47syl6bbr 256 . . . . . . . . . . 11  |-  ( (
ph  /\  ( z  e.  ( U_ j  e.  A  ( { j }  X.  C ) 
\  U )  /\  z  =  <. j ,  k >. ) )  -> 
( z  e.  U  <->  j U k ) )
4945, 48mtbid 293 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  ( U_ j  e.  A  ( { j }  X.  C ) 
\  U )  /\  z  =  <. j ,  k >. ) )  ->  -.  j U k )
5037, 49jca 520 . . . . . . . . 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 458 . . . . . . . 8  |-  ( (
ph  /\  ( z  e.  ( U_ j  e.  A  ( { j }  X.  C ) 
\  U )  /\  z  =  <. j ,  k >. ) )  ->  X  =  .0.  )
5331, 43, 523eqtrd 2474 . . . . . . 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 600 . . . . . 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 1825 . . . . 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 1825 . . . 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 5866 . 2  |-  ( ph  ->  ( `' ( j  e.  A ,  k  e.  C  |->  X )
" ( _V  \  {  .0.  } ) ) 
C_  U )
59 ssfi 7332 . 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 644 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 360   E.wex 1551    = wceq 1653    e. wcel 1726   A.wral 2707   _Vcvv 2958    \ cdif 3319    C_ wss 3322   {csn 3816   <.cop 3819   U_ciun 4095   class class class wbr 4215    X. cxp 4879   `'ccnv 4880   "cima 4884   Rel wrel 4886   -->wf 5453   ` cfv 5457  (class class class)co 6084    e. cmpt2 6086   Fincfn 7112   Basecbs 13474   0gc0g 13728  CMndccmn 15417
This theorem is referenced by:  gsum2d2  15553  gsumcom2  15554
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-er 6908  df-en 7113  df-fin 7116
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