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Theorem gsum2dlem1 16568
Description: Lemma 1 for gsum2d 16570. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by AV, 8-Jun-2019.)
Hypotheses
Ref Expression
gsum2d.b  |-  B  =  ( Base `  G
)
gsum2d.z  |-  .0.  =  ( 0g `  G )
gsum2d.g  |-  ( ph  ->  G  e. CMnd )
gsum2d.a  |-  ( ph  ->  A  e.  V )
gsum2d.r  |-  ( ph  ->  Rel  A )
gsum2d.d  |-  ( ph  ->  D  e.  W )
gsum2d.s  |-  ( ph  ->  dom  A  C_  D
)
gsum2d.f  |-  ( ph  ->  F : A --> B )
gsum2d.w  |-  ( ph  ->  F finSupp  .0.  )
Assertion
Ref Expression
gsum2dlem1  |-  ( ph  ->  ( G  gsumg  ( k  e.  ( A " { j } )  |->  ( j F k ) ) )  e.  B )
Distinct variable groups:    j, k, A    j, F, k    j, G, k    ph, j, k    B, j, k    D, j, k    .0. , j, k
Allowed substitution hints:    V( j, k)    W( j, k)

Proof of Theorem gsum2dlem1
StepHypRef Expression
1 gsum2d.b . 2  |-  B  =  ( Base `  G
)
2 gsum2d.z . 2  |-  .0.  =  ( 0g `  G )
3 gsum2d.g . 2  |-  ( ph  ->  G  e. CMnd )
4 gsum2d.a . . 3  |-  ( ph  ->  A  e.  V )
5 imaexg 6617 . . 3  |-  ( A  e.  V  ->  ( A " { j } )  e.  _V )
64, 5syl 16 . 2  |-  ( ph  ->  ( A " {
j } )  e. 
_V )
7 vex 3073 . . . . 5  |-  j  e. 
_V
8 vex 3073 . . . . 5  |-  k  e. 
_V
97, 8elimasn 5294 . . . 4  |-  ( k  e.  ( A " { j } )  <->  <. j ,  k >.  e.  A )
10 df-ov 6195 . . . . 5  |-  ( j F k )  =  ( F `  <. j ,  k >. )
11 gsum2d.f . . . . . 6  |-  ( ph  ->  F : A --> B )
1211ffvelrnda 5944 . . . . 5  |-  ( (
ph  /\  <. j ,  k >.  e.  A
)  ->  ( F `  <. j ,  k
>. )  e.  B
)
1310, 12syl5eqel 2543 . . . 4  |-  ( (
ph  /\  <. j ,  k >.  e.  A
)  ->  ( j F k )  e.  B )
149, 13sylan2b 475 . . 3  |-  ( (
ph  /\  k  e.  ( A " { j } ) )  -> 
( j F k )  e.  B )
15 eqid 2451 . . 3  |-  ( k  e.  ( A " { j } ) 
|->  ( j F k ) )  =  ( k  e.  ( A
" { j } )  |->  ( j F k ) )
1614, 15fmptd 5968 . 2  |-  ( ph  ->  ( k  e.  ( A " { j } )  |->  ( j F k ) ) : ( A " { j } ) --> B )
17 gsum2d.w . . . . 5  |-  ( ph  ->  F finSupp  .0.  )
1817fsuppimpd 7730 . . . 4  |-  ( ph  ->  ( F supp  .0.  )  e.  Fin )
19 rnfi 7699 . . . 4  |-  ( ( F supp  .0.  )  e.  Fin  ->  ran  ( F supp  .0.  )  e.  Fin )
2018, 19syl 16 . . 3  |-  ( ph  ->  ran  ( F supp  .0.  )  e.  Fin )
219biimpi 194 . . . . . . 7  |-  ( k  e.  ( A " { j } )  ->  <. j ,  k
>.  e.  A )
227, 8opelrn 5171 . . . . . . . 8  |-  ( <.
j ,  k >.  e.  ( F supp  .0.  )  ->  k  e.  ran  ( F supp  .0.  ) )
2322con3i 135 . . . . . . 7  |-  ( -.  k  e.  ran  ( F supp  .0.  )  ->  -.  <.
j ,  k >.  e.  ( F supp  .0.  )
)
2421, 23anim12i 566 . . . . . 6  |-  ( ( k  e.  ( A
" { j } )  /\  -.  k  e.  ran  ( F supp  .0.  ) )  ->  ( <. j ,  k >.  e.  A  /\  -.  <. j ,  k >.  e.  ( F supp  .0.  ) )
)
25 eldif 3438 . . . . . 6  |-  ( k  e.  ( ( A
" { j } )  \  ran  ( F supp  .0.  ) )  <->  ( k  e.  ( A " {
j } )  /\  -.  k  e.  ran  ( F supp  .0.  ) ) )
26 eldif 3438 . . . . . 6  |-  ( <.
j ,  k >.  e.  ( A  \  ( F supp  .0.  ) )  <->  ( <. j ,  k >.  e.  A  /\  -.  <. j ,  k
>.  e.  ( F supp  .0.  ) ) )
2724, 25, 263imtr4i 266 . . . . 5  |-  ( k  e.  ( ( A
" { j } )  \  ran  ( F supp  .0.  ) )  ->  <. j ,  k >.  e.  ( A  \  ( F supp  .0.  ) ) )
28 ssid 3475 . . . . . . . 8  |-  ( F supp 
.0.  )  C_  ( F supp  .0.  )
2928a1i 11 . . . . . . 7  |-  ( ph  ->  ( F supp  .0.  )  C_  ( F supp  .0.  )
)
30 fvex 5801 . . . . . . . . 9  |-  ( 0g
`  G )  e. 
_V
312, 30eqeltri 2535 . . . . . . . 8  |-  .0.  e.  _V
3231a1i 11 . . . . . . 7  |-  ( ph  ->  .0.  e.  _V )
3311, 29, 4, 32suppssr 6822 . . . . . 6  |-  ( (
ph  /\  <. j ,  k >.  e.  ( A  \  ( F supp  .0.  ) ) )  -> 
( F `  <. j ,  k >. )  =  .0.  )
3410, 33syl5eq 2504 . . . . 5  |-  ( (
ph  /\  <. j ,  k >.  e.  ( A  \  ( F supp  .0.  ) ) )  -> 
( j F k )  =  .0.  )
3527, 34sylan2 474 . . . 4  |-  ( (
ph  /\  k  e.  ( ( A " { j } ) 
\  ran  ( F supp  .0.  ) ) )  -> 
( j F k )  =  .0.  )
3635, 6suppss2 6825 . . 3  |-  ( ph  ->  ( ( k  e.  ( A " {
j } )  |->  ( j F k ) ) supp  .0.  )  C_  ran  ( F supp  .0.  )
)
37 ssfi 7636 . . 3  |-  ( ( ran  ( F supp  .0.  )  e.  Fin  /\  (
( k  e.  ( A " { j } )  |->  ( j F k ) ) supp 
.0.  )  C_  ran  ( F supp  .0.  ) )  ->  ( ( k  e.  ( A " { j } ) 
|->  ( j F k ) ) supp  .0.  )  e.  Fin )
3820, 36, 37syl2anc 661 . 2  |-  ( ph  ->  ( ( k  e.  ( A " {
j } )  |->  ( j F k ) ) supp  .0.  )  e.  Fin )
391, 2, 3, 6, 16, 38gsumcl2 16502 1  |-  ( ph  ->  ( G  gsumg  ( k  e.  ( A " { j } )  |->  ( j F k ) ) )  e.  B )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   _Vcvv 3070    \ cdif 3425    C_ wss 3428   {csn 3977   <.cop 3983   class class class wbr 4392    |-> cmpt 4450   dom cdm 4940   ran crn 4941   "cima 4943   Rel wrel 4945   -->wf 5514   ` cfv 5518  (class class class)co 6192   supp csupp 6792   Fincfn 7412   finSupp cfsupp 7723   Basecbs 14278   0gc0g 14482    gsumg cgsu 14483  CMndccmn 16383
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-cnex 9441  ax-resscn 9442  ax-1cn 9443  ax-icn 9444  ax-addcl 9445  ax-addrcl 9446  ax-mulcl 9447  ax-mulrcl 9448  ax-mulcom 9449  ax-addass 9450  ax-mulass 9451  ax-distr 9452  ax-i2m1 9453  ax-1ne0 9454  ax-1rid 9455  ax-rnegex 9456  ax-rrecex 9457  ax-cnre 9458  ax-pre-lttri 9459  ax-pre-lttrn 9460  ax-pre-ltadd 9461  ax-pre-mulgt0 9462
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-int 4229  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-eprel 4732  df-id 4736  df-po 4741  df-so 4742  df-fr 4779  df-se 4780  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-isom 5527  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-om 6579  df-1st 6679  df-2nd 6680  df-supp 6793  df-recs 6934  df-rdg 6968  df-1o 7022  df-oadd 7026  df-er 7203  df-en 7413  df-dom 7414  df-sdom 7415  df-fin 7416  df-fsupp 7724  df-oi 7827  df-card 8212  df-pnf 9523  df-mnf 9524  df-xr 9525  df-ltxr 9526  df-le 9527  df-sub 9700  df-neg 9701  df-nn 10426  df-n0 10683  df-z 10750  df-uz 10965  df-fz 11541  df-fzo 11652  df-seq 11910  df-hash 12207  df-0g 14484  df-gsum 14485  df-mnd 15519  df-cntz 15939  df-cmn 16385
This theorem is referenced by:  gsum2dlem2  16569  gsum2d  16570
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