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Theorem gsum2d2lem 17197
Description: Lemma for gsum2d2 17198: show the function is finitely supported. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by AV, 9-Jun-2019.)
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 ) finSupp  .0.  )
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 eqid 2454 . . . 4  |-  ( j  e.  A ,  k  e.  C  |->  X )  =  ( j  e.  A ,  k  e.  C  |->  X )
21mpt2fun 6377 . . 3  |-  Fun  (
j  e.  A , 
k  e.  C  |->  X )
32a1i 11 . 2  |-  ( ph  ->  Fun  ( j  e.  A ,  k  e.  C  |->  X ) )
4 gsum2d2.u . . 3  |-  ( ph  ->  U  e.  Fin )
5 gsum2d2.f . . . . . 6  |-  ( (
ph  /\  ( j  e.  A  /\  k  e.  C ) )  ->  X  e.  B )
65ralrimivva 2875 . . . . 5  |-  ( ph  ->  A. j  e.  A  A. k  e.  C  X  e.  B )
71fmpt2x 6839 . . . . 5  |-  ( 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 )
86, 7sylib 196 . . . 4  |-  ( ph  ->  ( j  e.  A ,  k  e.  C  |->  X ) : U_ j  e.  A  ( { j }  X.  C ) --> B )
9 relxp 5098 . . . . . . . 8  |-  Rel  ( { j }  X.  C )
109rgenw 2815 . . . . . . 7  |-  A. j  e.  A  Rel  ( { j }  X.  C
)
11 reliun 5111 . . . . . . 7  |-  ( Rel  U_ j  e.  A  ( { j }  X.  C )  <->  A. j  e.  A  Rel  ( { j }  X.  C
) )
1210, 11mpbir 209 . . . . . 6  |-  Rel  U_ j  e.  A  ( {
j }  X.  C
)
13 eldifi 3612 . . . . . . 7  |-  ( z  e.  ( U_ j  e.  A  ( {
j }  X.  C
)  \  U )  ->  z  e.  U_ j  e.  A  ( {
j }  X.  C
) )
1413adantl 464 . . . . . 6  |-  ( (
ph  /\  z  e.  ( U_ j  e.  A  ( { j }  X.  C )  \  U
) )  ->  z  e.  U_ j  e.  A  ( { j }  X.  C ) )
15 elrel 5093 . . . . . 6  |-  ( ( Rel  U_ j  e.  A  ( { j }  X.  C )  /\  z  e.  U_ j  e.  A  ( { j }  X.  C ) )  ->  E. j E. k  z  =  <. j ,  k
>. )
1612, 14, 15sylancr 661 . . . . 5  |-  ( (
ph  /\  z  e.  ( U_ j  e.  A  ( { j }  X.  C )  \  U
) )  ->  E. j E. k  z  =  <. j ,  k >.
)
17 nfv 1712 . . . . . . 7  |-  F/ j
ph
18 nfiu1 4345 . . . . . . . . 9  |-  F/_ j U_ j  e.  A  ( { j }  X.  C )
19 nfcv 2616 . . . . . . . . 9  |-  F/_ j U
2018, 19nfdif 3611 . . . . . . . 8  |-  F/_ j
( U_ j  e.  A  ( { j }  X.  C )  \  U
)
2120nfcri 2609 . . . . . . 7  |-  F/ j  z  e.  ( U_ j  e.  A  ( { j }  X.  C )  \  U
)
2217, 21nfan 1933 . . . . . 6  |-  F/ j ( ph  /\  z  e.  ( U_ j  e.  A  ( { j }  X.  C ) 
\  U ) )
23 nfmpt21 6337 . . . . . . . 8  |-  F/_ j
( j  e.  A ,  k  e.  C  |->  X )
24 nfcv 2616 . . . . . . . 8  |-  F/_ j
z
2523, 24nffv 5855 . . . . . . 7  |-  F/_ j
( ( j  e.  A ,  k  e.  C  |->  X ) `  z )
2625nfeq1 2631 . . . . . 6  |-  F/ j ( ( j  e.  A ,  k  e.  C  |->  X ) `  z )  =  .0.
27 nfv 1712 . . . . . . 7  |-  F/ k ( ph  /\  z  e.  ( U_ j  e.  A  ( { j }  X.  C ) 
\  U ) )
28 nfmpt22 6338 . . . . . . . . 9  |-  F/_ k
( j  e.  A ,  k  e.  C  |->  X )
29 nfcv 2616 . . . . . . . . 9  |-  F/_ k
z
3028, 29nffv 5855 . . . . . . . 8  |-  F/_ k
( ( j  e.  A ,  k  e.  C  |->  X ) `  z )
3130nfeq1 2631 . . . . . . 7  |-  F/ k ( ( j  e.  A ,  k  e.  C  |->  X ) `  z )  =  .0.
32 simprr 755 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  ( U_ j  e.  A  ( { j }  X.  C ) 
\  U )  /\  z  =  <. j ,  k >. ) )  -> 
z  =  <. j ,  k >. )
3332fveq2d 5852 . . . . . . . . 9  |-  ( (
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 >. )
)
34 df-ov 6273 . . . . . . . . . 10  |-  ( j ( j  e.  A ,  k  e.  C  |->  X ) k )  =  ( ( j  e.  A ,  k  e.  C  |->  X ) `
 <. j ,  k
>. )
35 simprl 754 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( z  e.  ( U_ j  e.  A  ( { j }  X.  C ) 
\  U )  /\  z  =  <. j ,  k >. ) )  -> 
z  e.  ( U_ j  e.  A  ( { j }  X.  C )  \  U
) )
3632, 35eqeltrrd 2543 . . . . . . . . . . . . . 14  |-  ( (
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 ) )
3736eldifad 3473 . . . . . . . . . . . . 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 ) )
38 opeliunxp 5040 . . . . . . . . . . . . 13  |-  ( <.
j ,  k >.  e.  U_ j  e.  A  ( { j }  X.  C )  <->  ( j  e.  A  /\  k  e.  C ) )
3937, 38sylib 196 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( z  e.  ( U_ j  e.  A  ( { j }  X.  C ) 
\  U )  /\  z  =  <. j ,  k >. ) )  -> 
( j  e.  A  /\  k  e.  C
) )
4039simpld 457 . . . . . . . . . . 11  |-  ( (
ph  /\  ( z  e.  ( U_ j  e.  A  ( { j }  X.  C ) 
\  U )  /\  z  =  <. j ,  k >. ) )  -> 
j  e.  A )
4139simprd 461 . . . . . . . . . . 11  |-  ( (
ph  /\  ( z  e.  ( U_ j  e.  A  ( { j }  X.  C ) 
\  U )  /\  z  =  <. j ,  k >. ) )  -> 
k  e.  C )
4239, 5syldan 468 . . . . . . . . . . 11  |-  ( (
ph  /\  ( z  e.  ( U_ j  e.  A  ( { j }  X.  C ) 
\  U )  /\  z  =  <. j ,  k >. ) )  ->  X  e.  B )
431ovmpt4g 6398 . . . . . . . . . . 11  |-  ( ( j  e.  A  /\  k  e.  C  /\  X  e.  B )  ->  ( j ( j  e.  A ,  k  e.  C  |->  X ) k )  =  X )
4440, 41, 42, 43syl3anc 1226 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  ( U_ j  e.  A  ( { j }  X.  C ) 
\  U )  /\  z  =  <. j ,  k >. ) )  -> 
( j ( j  e.  A ,  k  e.  C  |->  X ) k )  =  X )
4534, 44syl5eqr 2509 . . . . . . . . 9  |-  ( (
ph  /\  ( z  e.  ( U_ j  e.  A  ( { j }  X.  C ) 
\  U )  /\  z  =  <. j ,  k >. ) )  -> 
( ( j  e.  A ,  k  e.  C  |->  X ) `  <. j ,  k >.
)  =  X )
46 eldifn 3613 . . . . . . . . . . . . 13  |-  ( z  e.  ( U_ j  e.  A  ( {
j }  X.  C
)  \  U )  ->  -.  z  e.  U
)
4746ad2antrl 725 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( z  e.  ( U_ j  e.  A  ( { j }  X.  C ) 
\  U )  /\  z  =  <. j ,  k >. ) )  ->  -.  z  e.  U
)
4832eleq1d 2523 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( z  e.  ( U_ j  e.  A  ( { j }  X.  C ) 
\  U )  /\  z  =  <. j ,  k >. ) )  -> 
( z  e.  U  <->  <.
j ,  k >.  e.  U ) )
49 df-br 4440 . . . . . . . . . . . . 13  |-  ( j U k  <->  <. j ,  k >.  e.  U
)
5048, 49syl6bbr 263 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( z  e.  ( U_ j  e.  A  ( { j }  X.  C ) 
\  U )  /\  z  =  <. j ,  k >. ) )  -> 
( z  e.  U  <->  j U k ) )
5147, 50mtbid 298 . . . . . . . . . . 11  |-  ( (
ph  /\  ( z  e.  ( U_ j  e.  A  ( { j }  X.  C ) 
\  U )  /\  z  =  <. j ,  k >. ) )  ->  -.  j U k )
5239, 51jca 530 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  ( U_ j  e.  A  ( { j }  X.  C ) 
\  U )  /\  z  =  <. j ,  k >. ) )  -> 
( ( j  e.  A  /\  k  e.  C )  /\  -.  j U k ) )
53 gsum2d2.n . . . . . . . . . 10  |-  ( (
ph  /\  ( (
j  e.  A  /\  k  e.  C )  /\  -.  j U k ) )  ->  X  =  .0.  )
5452, 53syldan 468 . . . . . . . . 9  |-  ( (
ph  /\  ( z  e.  ( U_ j  e.  A  ( { j }  X.  C ) 
\  U )  /\  z  =  <. j ,  k >. ) )  ->  X  =  .0.  )
5533, 45, 543eqtrd 2499 . . . . . . . 8  |-  ( (
ph  /\  ( z  e.  ( U_ j  e.  A  ( { j }  X.  C ) 
\  U )  /\  z  =  <. j ,  k >. ) )  -> 
( ( j  e.  A ,  k  e.  C  |->  X ) `  z )  =  .0.  )
5655expr 613 . . . . . . 7  |-  ( (
ph  /\  z  e.  ( U_ j  e.  A  ( { j }  X.  C )  \  U
) )  ->  (
z  =  <. j ,  k >.  ->  (
( j  e.  A ,  k  e.  C  |->  X ) `  z
)  =  .0.  )
)
5727, 31, 56exlimd 1919 . . . . . 6  |-  ( (
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.  ) )
5822, 26, 57exlimd 1919 . . . . 5  |-  ( (
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.  ) )
5916, 58mpd 15 . . . 4  |-  ( (
ph  /\  z  e.  ( U_ j  e.  A  ( { j }  X.  C )  \  U
) )  ->  (
( j  e.  A ,  k  e.  C  |->  X ) `  z
)  =  .0.  )
608, 59suppss 6922 . . 3  |-  ( ph  ->  ( ( j  e.  A ,  k  e.  C  |->  X ) supp  .0.  )  C_  U )
61 ssfi 7733 . . 3  |-  ( ( U  e.  Fin  /\  ( ( j  e.  A ,  k  e.  C  |->  X ) supp  .0.  )  C_  U )  -> 
( ( j  e.  A ,  k  e.  C  |->  X ) supp  .0.  )  e.  Fin )
624, 60, 61syl2anc 659 . 2  |-  ( ph  ->  ( ( j  e.  A ,  k  e.  C  |->  X ) supp  .0.  )  e.  Fin )
63 gsum2d2.a . . . 4  |-  ( ph  ->  A  e.  V )
64 gsum2d2.r . . . . 5  |-  ( (
ph  /\  j  e.  A )  ->  C  e.  W )
6564ralrimiva 2868 . . . 4  |-  ( ph  ->  A. j  e.  A  C  e.  W )
661mpt2exxg 6847 . . . 4  |-  ( ( A  e.  V  /\  A. j  e.  A  C  e.  W )  ->  (
j  e.  A , 
k  e.  C  |->  X )  e.  _V )
6763, 65, 66syl2anc 659 . . 3  |-  ( ph  ->  ( j  e.  A ,  k  e.  C  |->  X )  e.  _V )
68 gsum2d2.z . . . . 5  |-  .0.  =  ( 0g `  G )
69 fvex 5858 . . . . 5  |-  ( 0g
`  G )  e. 
_V
7068, 69eqeltri 2538 . . . 4  |-  .0.  e.  _V
7170a1i 11 . . 3  |-  ( ph  ->  .0.  e.  _V )
72 isfsupp 7825 . . 3  |-  ( ( ( j  e.  A ,  k  e.  C  |->  X )  e.  _V  /\  .0.  e.  _V )  ->  ( ( j  e.  A ,  k  e.  C  |->  X ) finSupp  .0.  <->  ( Fun  ( j  e.  A ,  k  e.  C  |->  X )  /\  (
( j  e.  A ,  k  e.  C  |->  X ) supp  .0.  )  e.  Fin ) ) )
7367, 71, 72syl2anc 659 . 2  |-  ( ph  ->  ( ( j  e.  A ,  k  e.  C  |->  X ) finSupp  .0.  <->  ( Fun  ( j  e.  A ,  k  e.  C  |->  X )  /\  (
( j  e.  A ,  k  e.  C  |->  X ) supp  .0.  )  e.  Fin ) ) )
743, 62, 73mpbir2and 920 1  |-  ( ph  ->  ( j  e.  A ,  k  e.  C  |->  X ) finSupp  .0.  )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398   E.wex 1617    e. wcel 1823   A.wral 2804   _Vcvv 3106    \ cdif 3458    C_ wss 3461   {csn 4016   <.cop 4022   U_ciun 4315   class class class wbr 4439    X. cxp 4986   Rel wrel 4993   Fun wfun 5564   -->wf 5566   ` cfv 5570  (class class class)co 6270    |-> cmpt2 6272   supp csupp 6891   Fincfn 7509   finSupp cfsupp 7821   Basecbs 14716   0gc0g 14929  CMndccmn 16997
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-supp 6892  df-er 7303  df-en 7510  df-fin 7513  df-fsupp 7822
This theorem is referenced by:  gsum2d2  17198  gsumcom2  17199
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