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Theorem gsumzadd 16808
Description: The sum of two group sums. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 5-Jun-2019.)
Hypotheses
Ref Expression
gsumzadd.b  |-  B  =  ( Base `  G
)
gsumzadd.0  |-  .0.  =  ( 0g `  G )
gsumzadd.p  |-  .+  =  ( +g  `  G )
gsumzadd.z  |-  Z  =  (Cntz `  G )
gsumzadd.g  |-  ( ph  ->  G  e.  Mnd )
gsumzadd.a  |-  ( ph  ->  A  e.  V )
gsumzadd.fn  |-  ( ph  ->  F finSupp  .0.  )
gsumzadd.hn  |-  ( ph  ->  H finSupp  .0.  )
gsumzadd.s  |-  ( ph  ->  S  e.  (SubMnd `  G ) )
gsumzadd.c  |-  ( ph  ->  S  C_  ( Z `  S ) )
gsumzadd.f  |-  ( ph  ->  F : A --> S )
gsumzadd.h  |-  ( ph  ->  H : A --> S )
Assertion
Ref Expression
gsumzadd  |-  ( ph  ->  ( G  gsumg  ( F  oF  .+  H ) )  =  ( ( G 
gsumg  F )  .+  ( G  gsumg  H ) ) )

Proof of Theorem gsumzadd
Dummy variables  k  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 gsumzadd.b . 2  |-  B  =  ( Base `  G
)
2 gsumzadd.0 . 2  |-  .0.  =  ( 0g `  G )
3 gsumzadd.p . 2  |-  .+  =  ( +g  `  G )
4 gsumzadd.z . 2  |-  Z  =  (Cntz `  G )
5 gsumzadd.g . 2  |-  ( ph  ->  G  e.  Mnd )
6 gsumzadd.a . 2  |-  ( ph  ->  A  e.  V )
7 gsumzadd.fn . 2  |-  ( ph  ->  F finSupp  .0.  )
8 gsumzadd.hn . 2  |-  ( ph  ->  H finSupp  .0.  )
9 eqid 2467 . 2  |-  ( ( F  u.  H ) supp 
.0.  )  =  ( ( F  u.  H
) supp  .0.  )
10 gsumzadd.f . . 3  |-  ( ph  ->  F : A --> S )
11 gsumzadd.s . . . 4  |-  ( ph  ->  S  e.  (SubMnd `  G ) )
121submss 15853 . . . 4  |-  ( S  e.  (SubMnd `  G
)  ->  S  C_  B
)
1311, 12syl 16 . . 3  |-  ( ph  ->  S  C_  B )
14 fss 5745 . . 3  |-  ( ( F : A --> S  /\  S  C_  B )  ->  F : A --> B )
1510, 13, 14syl2anc 661 . 2  |-  ( ph  ->  F : A --> B )
16 gsumzadd.h . . 3  |-  ( ph  ->  H : A --> S )
17 fss 5745 . . 3  |-  ( ( H : A --> S  /\  S  C_  B )  ->  H : A --> B )
1816, 13, 17syl2anc 661 . 2  |-  ( ph  ->  H : A --> B )
19 gsumzadd.c . . 3  |-  ( ph  ->  S  C_  ( Z `  S ) )
20 frn 5743 . . . 4  |-  ( F : A --> S  ->  ran  F  C_  S )
2110, 20syl 16 . . 3  |-  ( ph  ->  ran  F  C_  S
)
224cntzidss 16247 . . 3  |-  ( ( S  C_  ( Z `  S )  /\  ran  F 
C_  S )  ->  ran  F  C_  ( Z `  ran  F ) )
2319, 21, 22syl2anc 661 . 2  |-  ( ph  ->  ran  F  C_  ( Z `  ran  F ) )
24 frn 5743 . . . 4  |-  ( H : A --> S  ->  ran  H  C_  S )
2516, 24syl 16 . . 3  |-  ( ph  ->  ran  H  C_  S
)
264cntzidss 16247 . . 3  |-  ( ( S  C_  ( Z `  S )  /\  ran  H 
C_  S )  ->  ran  H  C_  ( Z `  ran  H ) )
2719, 25, 26syl2anc 661 . 2  |-  ( ph  ->  ran  H  C_  ( Z `  ran  H ) )
283submcl 15856 . . . . . . 7  |-  ( ( S  e.  (SubMnd `  G )  /\  x  e.  S  /\  y  e.  S )  ->  (
x  .+  y )  e.  S )
29283expb 1197 . . . . . 6  |-  ( ( S  e.  (SubMnd `  G )  /\  (
x  e.  S  /\  y  e.  S )
)  ->  ( x  .+  y )  e.  S
)
3011, 29sylan 471 . . . . 5  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  e.  S )
31 inidm 3712 . . . . 5  |-  ( A  i^i  A )  =  A
3230, 10, 16, 6, 6, 31off 6549 . . . 4  |-  ( ph  ->  ( F  oF  .+  H ) : A --> S )
33 frn 5743 . . . 4  |-  ( ( F  oF  .+  H ) : A --> S  ->  ran  ( F  oF  .+  H ) 
C_  S )
3432, 33syl 16 . . 3  |-  ( ph  ->  ran  ( F  oF  .+  H )  C_  S )
354cntzidss 16247 . . 3  |-  ( ( S  C_  ( Z `  S )  /\  ran  ( F  oF  .+  H )  C_  S
)  ->  ran  ( F  oF  .+  H
)  C_  ( Z `  ran  ( F  oF  .+  H ) ) )
3619, 34, 35syl2anc 661 . 2  |-  ( ph  ->  ran  ( F  oF  .+  H )  C_  ( Z `  ran  ( F  oF  .+  H
) ) )
3719adantr 465 . . . 4  |-  ( (
ph  /\  ( x  C_  A  /\  k  e.  ( A  \  x
) ) )  ->  S  C_  ( Z `  S ) )
3813adantr 465 . . . . 5  |-  ( (
ph  /\  ( x  C_  A  /\  k  e.  ( A  \  x
) ) )  ->  S  C_  B )
395adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( x  C_  A  /\  k  e.  ( A  \  x
) ) )  ->  G  e.  Mnd )
40 vex 3121 . . . . . . . 8  |-  x  e. 
_V
4140a1i 11 . . . . . . 7  |-  ( (
ph  /\  ( x  C_  A  /\  k  e.  ( A  \  x
) ) )  ->  x  e.  _V )
4211adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( x  C_  A  /\  k  e.  ( A  \  x
) ) )  ->  S  e.  (SubMnd `  G
) )
43 simpl 457 . . . . . . . 8  |-  ( ( x  C_  A  /\  k  e.  ( A  \  x ) )  ->  x  C_  A )
44 fssres 5757 . . . . . . . 8  |-  ( ( H : A --> S  /\  x  C_  A )  -> 
( H  |`  x
) : x --> S )
4516, 43, 44syl2an 477 . . . . . . 7  |-  ( (
ph  /\  ( x  C_  A  /\  k  e.  ( A  \  x
) ) )  -> 
( H  |`  x
) : x --> S )
4627adantr 465 . . . . . . . 8  |-  ( (
ph  /\  ( x  C_  A  /\  k  e.  ( A  \  x
) ) )  ->  ran  H  C_  ( Z `  ran  H ) )
47 resss 5303 . . . . . . . . 9  |-  ( H  |`  x )  C_  H
48 rnss 5237 . . . . . . . . 9  |-  ( ( H  |`  x )  C_  H  ->  ran  ( H  |`  x )  C_  ran  H )
4947, 48ax-mp 5 . . . . . . . 8  |-  ran  ( H  |`  x )  C_  ran  H
504cntzidss 16247 . . . . . . . 8  |-  ( ( ran  H  C_  ( Z `  ran  H )  /\  ran  ( H  |`  x )  C_  ran  H )  ->  ran  ( H  |`  x )  C_  ( Z `  ran  ( H  |`  x ) ) )
5146, 49, 50sylancl 662 . . . . . . 7  |-  ( (
ph  /\  ( x  C_  A  /\  k  e.  ( A  \  x
) ) )  ->  ran  ( H  |`  x
)  C_  ( Z `  ran  ( H  |`  x ) ) )
52 ffun 5739 . . . . . . . . . . 11  |-  ( H : A --> S  ->  Fun  H )
5316, 52syl 16 . . . . . . . . . 10  |-  ( ph  ->  Fun  H )
54 funres 5633 . . . . . . . . . 10  |-  ( Fun 
H  ->  Fun  ( H  |`  x ) )
5553, 54syl 16 . . . . . . . . 9  |-  ( ph  ->  Fun  ( H  |`  x ) )
5655adantr 465 . . . . . . . 8  |-  ( (
ph  /\  ( x  C_  A  /\  k  e.  ( A  \  x
) ) )  ->  Fun  ( H  |`  x
) )
57 fsuppimp 7847 . . . . . . . . . . . 12  |-  ( H finSupp  .0.  ->  ( Fun  H  /\  ( H supp  .0.  )  e.  Fin ) )
5857simprd 463 . . . . . . . . . . 11  |-  ( H finSupp  .0.  ->  ( H supp  .0.  )  e.  Fin )
598, 58syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( H supp  .0.  )  e.  Fin )
6059adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  ( x  C_  A  /\  k  e.  ( A  \  x
) ) )  -> 
( H supp  .0.  )  e.  Fin )
61 fex 6144 . . . . . . . . . . . 12  |-  ( ( H : A --> S  /\  A  e.  V )  ->  H  e.  _V )
6216, 6, 61syl2anc 661 . . . . . . . . . . 11  |-  ( ph  ->  H  e.  _V )
63 fvex 5882 . . . . . . . . . . . 12  |-  ( 0g
`  G )  e. 
_V
642, 63eqeltri 2551 . . . . . . . . . . 11  |-  .0.  e.  _V
65 ressuppss 6931 . . . . . . . . . . 11  |-  ( ( H  e.  _V  /\  .0.  e.  _V )  -> 
( ( H  |`  x ) supp  .0.  )  C_  ( H supp  .0.  )
)
6662, 64, 65sylancl 662 . . . . . . . . . 10  |-  ( ph  ->  ( ( H  |`  x ) supp  .0.  )  C_  ( H supp  .0.  )
)
6766adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  ( x  C_  A  /\  k  e.  ( A  \  x
) ) )  -> 
( ( H  |`  x ) supp  .0.  )  C_  ( H supp  .0.  )
)
68 ssfi 7752 . . . . . . . . 9  |-  ( ( ( H supp  .0.  )  e.  Fin  /\  ( ( H  |`  x ) supp  .0.  )  C_  ( H supp 
.0.  ) )  -> 
( ( H  |`  x ) supp  .0.  )  e.  Fin )
6960, 67, 68syl2anc 661 . . . . . . . 8  |-  ( (
ph  /\  ( x  C_  A  /\  k  e.  ( A  \  x
) ) )  -> 
( ( H  |`  x ) supp  .0.  )  e.  Fin )
70 resfunexg 6137 . . . . . . . . . . 11  |-  ( ( Fun  H  /\  x  e.  _V )  ->  ( H  |`  x )  e. 
_V )
7153, 40, 70sylancl 662 . . . . . . . . . 10  |-  ( ph  ->  ( H  |`  x
)  e.  _V )
72 isfsupp 7845 . . . . . . . . . 10  |-  ( ( ( H  |`  x
)  e.  _V  /\  .0.  e.  _V )  -> 
( ( H  |`  x ) finSupp  .0.  <->  ( Fun  ( H  |`  x )  /\  ( ( H  |`  x ) supp  .0.  )  e.  Fin ) ) )
7371, 64, 72sylancl 662 . . . . . . . . 9  |-  ( ph  ->  ( ( H  |`  x ) finSupp  .0.  <->  ( Fun  ( H  |`  x )  /\  ( ( H  |`  x ) supp  .0.  )  e.  Fin ) ) )
7473adantr 465 . . . . . . . 8  |-  ( (
ph  /\  ( x  C_  A  /\  k  e.  ( A  \  x
) ) )  -> 
( ( H  |`  x ) finSupp  .0.  <->  ( Fun  ( H  |`  x )  /\  ( ( H  |`  x ) supp  .0.  )  e.  Fin ) ) )
7556, 69, 74mpbir2and 920 . . . . . . 7  |-  ( (
ph  /\  ( x  C_  A  /\  k  e.  ( A  \  x
) ) )  -> 
( H  |`  x
) finSupp  .0.  )
762, 4, 39, 41, 42, 45, 51, 75gsumzsubmcl 16801 . . . . . 6  |-  ( (
ph  /\  ( x  C_  A  /\  k  e.  ( A  \  x
) ) )  -> 
( G  gsumg  ( H  |`  x
) )  e.  S
)
7776snssd 4178 . . . . 5  |-  ( (
ph  /\  ( x  C_  A  /\  k  e.  ( A  \  x
) ) )  ->  { ( G  gsumg  ( H  |`  x ) ) } 
C_  S )
781, 4cntz2ss 16242 . . . . 5  |-  ( ( S  C_  B  /\  { ( G  gsumg  ( H  |`  x
) ) }  C_  S )  ->  ( Z `  S )  C_  ( Z `  {
( G  gsumg  ( H  |`  x
) ) } ) )
7938, 77, 78syl2anc 661 . . . 4  |-  ( (
ph  /\  ( x  C_  A  /\  k  e.  ( A  \  x
) ) )  -> 
( Z `  S
)  C_  ( Z `  { ( G  gsumg  ( H  |`  x ) ) } ) )
8037, 79sstrd 3519 . . 3  |-  ( (
ph  /\  ( x  C_  A  /\  k  e.  ( A  \  x
) ) )  ->  S  C_  ( Z `  { ( G  gsumg  ( H  |`  x ) ) } ) )
81 eldifi 3631 . . . . 5  |-  ( k  e.  ( A  \  x )  ->  k  e.  A )
8281adantl 466 . . . 4  |-  ( ( x  C_  A  /\  k  e.  ( A  \  x ) )  -> 
k  e.  A )
83 ffvelrn 6030 . . . 4  |-  ( ( F : A --> S  /\  k  e.  A )  ->  ( F `  k
)  e.  S )
8410, 82, 83syl2an 477 . . 3  |-  ( (
ph  /\  ( x  C_  A  /\  k  e.  ( A  \  x
) ) )  -> 
( F `  k
)  e.  S )
8580, 84sseldd 3510 . 2  |-  ( (
ph  /\  ( x  C_  A  /\  k  e.  ( A  \  x
) ) )  -> 
( F `  k
)  e.  ( Z `
 { ( G 
gsumg  ( H  |`  x ) ) } ) )
861, 2, 3, 4, 5, 6, 7, 8, 9, 15, 18, 23, 27, 36, 85gsumzaddlem 16807 1  |-  ( ph  ->  ( G  gsumg  ( F  oF  .+  H ) )  =  ( ( G 
gsumg  F )  .+  ( G  gsumg  H ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3118    \ cdif 3478    u. cun 3479    C_ wss 3481   {csn 4033   class class class wbr 4453   ran crn 5006    |` cres 5007   Fun wfun 5588   -->wf 5590   ` cfv 5594  (class class class)co 6295    oFcof 6533   supp csupp 6913   Fincfn 7528   finSupp cfsupp 7841   Basecbs 14507   +g cplusg 14572   0gc0g 14712    gsumg cgsu 14713   Mndcmnd 15793  SubMndcsubmnd 15838  Cntzccntz 16225
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-of 6535  df-om 6696  df-1st 6795  df-2nd 6796  df-supp 6914  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-fsupp 7842  df-oi 7947  df-card 8332  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-n0 10808  df-z 10877  df-uz 11095  df-fz 11685  df-fzo 11805  df-seq 12088  df-hash 12386  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-ress 14514  df-plusg 14585  df-0g 14714  df-gsum 14715  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-submnd 15840  df-cntz 16227
This theorem is referenced by:  gsumadd  16811  gsumzsplit  16817
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