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Theorem gsumzadd 17490
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 2429 . 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 16548 . . . 4  |-  ( S  e.  (SubMnd `  G
)  ->  S  C_  B
)
1311, 12syl 17 . . 3  |-  ( ph  ->  S  C_  B )
1410, 13fssd 5755 . 2  |-  ( ph  ->  F : A --> B )
15 gsumzadd.h . . 3  |-  ( ph  ->  H : A --> S )
1615, 13fssd 5755 . 2  |-  ( ph  ->  H : A --> B )
17 gsumzadd.c . . 3  |-  ( ph  ->  S  C_  ( Z `  S ) )
18 frn 5752 . . . 4  |-  ( F : A --> S  ->  ran  F  C_  S )
1910, 18syl 17 . . 3  |-  ( ph  ->  ran  F  C_  S
)
204cntzidss 16942 . . 3  |-  ( ( S  C_  ( Z `  S )  /\  ran  F 
C_  S )  ->  ran  F  C_  ( Z `  ran  F ) )
2117, 19, 20syl2anc 665 . 2  |-  ( ph  ->  ran  F  C_  ( Z `  ran  F ) )
22 frn 5752 . . . 4  |-  ( H : A --> S  ->  ran  H  C_  S )
2315, 22syl 17 . . 3  |-  ( ph  ->  ran  H  C_  S
)
244cntzidss 16942 . . 3  |-  ( ( S  C_  ( Z `  S )  /\  ran  H 
C_  S )  ->  ran  H  C_  ( Z `  ran  H ) )
2517, 23, 24syl2anc 665 . 2  |-  ( ph  ->  ran  H  C_  ( Z `  ran  H ) )
263submcl 16551 . . . . . . 7  |-  ( ( S  e.  (SubMnd `  G )  /\  x  e.  S  /\  y  e.  S )  ->  (
x  .+  y )  e.  S )
27263expb 1206 . . . . . 6  |-  ( ( S  e.  (SubMnd `  G )  /\  (
x  e.  S  /\  y  e.  S )
)  ->  ( x  .+  y )  e.  S
)
2811, 27sylan 473 . . . . 5  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  e.  S )
29 inidm 3677 . . . . 5  |-  ( A  i^i  A )  =  A
3028, 10, 15, 6, 6, 29off 6560 . . . 4  |-  ( ph  ->  ( F  oF  .+  H ) : A --> S )
31 frn 5752 . . . 4  |-  ( ( F  oF  .+  H ) : A --> S  ->  ran  ( F  oF  .+  H ) 
C_  S )
3230, 31syl 17 . . 3  |-  ( ph  ->  ran  ( F  oF  .+  H )  C_  S )
334cntzidss 16942 . . 3  |-  ( ( S  C_  ( Z `  S )  /\  ran  ( F  oF  .+  H )  C_  S
)  ->  ran  ( F  oF  .+  H
)  C_  ( Z `  ran  ( F  oF  .+  H ) ) )
3417, 32, 33syl2anc 665 . 2  |-  ( ph  ->  ran  ( F  oF  .+  H )  C_  ( Z `  ran  ( F  oF  .+  H
) ) )
3517adantr 466 . . . 4  |-  ( (
ph  /\  ( x  C_  A  /\  k  e.  ( A  \  x
) ) )  ->  S  C_  ( Z `  S ) )
3613adantr 466 . . . . 5  |-  ( (
ph  /\  ( x  C_  A  /\  k  e.  ( A  \  x
) ) )  ->  S  C_  B )
375adantr 466 . . . . . . 7  |-  ( (
ph  /\  ( x  C_  A  /\  k  e.  ( A  \  x
) ) )  ->  G  e.  Mnd )
38 vex 3090 . . . . . . . 8  |-  x  e. 
_V
3938a1i 11 . . . . . . 7  |-  ( (
ph  /\  ( x  C_  A  /\  k  e.  ( A  \  x
) ) )  ->  x  e.  _V )
4011adantr 466 . . . . . . 7  |-  ( (
ph  /\  ( x  C_  A  /\  k  e.  ( A  \  x
) ) )  ->  S  e.  (SubMnd `  G
) )
41 simpl 458 . . . . . . . 8  |-  ( ( x  C_  A  /\  k  e.  ( A  \  x ) )  ->  x  C_  A )
42 fssres 5766 . . . . . . . 8  |-  ( ( H : A --> S  /\  x  C_  A )  -> 
( H  |`  x
) : x --> S )
4315, 41, 42syl2an 479 . . . . . . 7  |-  ( (
ph  /\  ( x  C_  A  /\  k  e.  ( A  \  x
) ) )  -> 
( H  |`  x
) : x --> S )
4425adantr 466 . . . . . . . 8  |-  ( (
ph  /\  ( x  C_  A  /\  k  e.  ( A  \  x
) ) )  ->  ran  H  C_  ( Z `  ran  H ) )
45 resss 5148 . . . . . . . . 9  |-  ( H  |`  x )  C_  H
46 rnss 5083 . . . . . . . . 9  |-  ( ( H  |`  x )  C_  H  ->  ran  ( H  |`  x )  C_  ran  H )
4745, 46ax-mp 5 . . . . . . . 8  |-  ran  ( H  |`  x )  C_  ran  H
484cntzidss 16942 . . . . . . . 8  |-  ( ( ran  H  C_  ( Z `  ran  H )  /\  ran  ( H  |`  x )  C_  ran  H )  ->  ran  ( H  |`  x )  C_  ( Z `  ran  ( H  |`  x ) ) )
4944, 47, 48sylancl 666 . . . . . . 7  |-  ( (
ph  /\  ( x  C_  A  /\  k  e.  ( A  \  x
) ) )  ->  ran  ( H  |`  x
)  C_  ( Z `  ran  ( H  |`  x ) ) )
50 ffun 5748 . . . . . . . . . . 11  |-  ( H : A --> S  ->  Fun  H )
5115, 50syl 17 . . . . . . . . . 10  |-  ( ph  ->  Fun  H )
52 funres 5640 . . . . . . . . . 10  |-  ( Fun 
H  ->  Fun  ( H  |`  x ) )
5351, 52syl 17 . . . . . . . . 9  |-  ( ph  ->  Fun  ( H  |`  x ) )
5453adantr 466 . . . . . . . 8  |-  ( (
ph  /\  ( x  C_  A  /\  k  e.  ( A  \  x
) ) )  ->  Fun  ( H  |`  x
) )
558fsuppimpd 7896 . . . . . . . . . 10  |-  ( ph  ->  ( H supp  .0.  )  e.  Fin )
5655adantr 466 . . . . . . . . 9  |-  ( (
ph  /\  ( x  C_  A  /\  k  e.  ( A  \  x
) ) )  -> 
( H supp  .0.  )  e.  Fin )
57 fex 6153 . . . . . . . . . . . 12  |-  ( ( H : A --> S  /\  A  e.  V )  ->  H  e.  _V )
5815, 6, 57syl2anc 665 . . . . . . . . . . 11  |-  ( ph  ->  H  e.  _V )
59 fvex 5891 . . . . . . . . . . . 12  |-  ( 0g
`  G )  e. 
_V
602, 59eqeltri 2513 . . . . . . . . . . 11  |-  .0.  e.  _V
61 ressuppss 6945 . . . . . . . . . . 11  |-  ( ( H  e.  _V  /\  .0.  e.  _V )  -> 
( ( H  |`  x ) supp  .0.  )  C_  ( H supp  .0.  )
)
6258, 60, 61sylancl 666 . . . . . . . . . 10  |-  ( ph  ->  ( ( H  |`  x ) supp  .0.  )  C_  ( H supp  .0.  )
)
6362adantr 466 . . . . . . . . 9  |-  ( (
ph  /\  ( x  C_  A  /\  k  e.  ( A  \  x
) ) )  -> 
( ( H  |`  x ) supp  .0.  )  C_  ( H supp  .0.  )
)
64 ssfi 7798 . . . . . . . . 9  |-  ( ( ( H supp  .0.  )  e.  Fin  /\  ( ( H  |`  x ) supp  .0.  )  C_  ( H supp 
.0.  ) )  -> 
( ( H  |`  x ) supp  .0.  )  e.  Fin )
6556, 63, 64syl2anc 665 . . . . . . . 8  |-  ( (
ph  /\  ( x  C_  A  /\  k  e.  ( A  \  x
) ) )  -> 
( ( H  |`  x ) supp  .0.  )  e.  Fin )
66 resfunexg 6145 . . . . . . . . . . 11  |-  ( ( Fun  H  /\  x  e.  _V )  ->  ( H  |`  x )  e. 
_V )
6751, 38, 66sylancl 666 . . . . . . . . . 10  |-  ( ph  ->  ( H  |`  x
)  e.  _V )
68 isfsupp 7893 . . . . . . . . . 10  |-  ( ( ( H  |`  x
)  e.  _V  /\  .0.  e.  _V )  -> 
( ( H  |`  x ) finSupp  .0.  <->  ( Fun  ( H  |`  x )  /\  ( ( H  |`  x ) supp  .0.  )  e.  Fin ) ) )
6967, 60, 68sylancl 666 . . . . . . . . 9  |-  ( ph  ->  ( ( H  |`  x ) finSupp  .0.  <->  ( Fun  ( H  |`  x )  /\  ( ( H  |`  x ) supp  .0.  )  e.  Fin ) ) )
7069adantr 466 . . . . . . . 8  |-  ( (
ph  /\  ( x  C_  A  /\  k  e.  ( A  \  x
) ) )  -> 
( ( H  |`  x ) finSupp  .0.  <->  ( Fun  ( H  |`  x )  /\  ( ( H  |`  x ) supp  .0.  )  e.  Fin ) ) )
7154, 65, 70mpbir2and 930 . . . . . . 7  |-  ( (
ph  /\  ( x  C_  A  /\  k  e.  ( A  \  x
) ) )  -> 
( H  |`  x
) finSupp  .0.  )
722, 4, 37, 39, 40, 43, 49, 71gsumzsubmcl 17486 . . . . . 6  |-  ( (
ph  /\  ( x  C_  A  /\  k  e.  ( A  \  x
) ) )  -> 
( G  gsumg  ( H  |`  x
) )  e.  S
)
7372snssd 4148 . . . . 5  |-  ( (
ph  /\  ( x  C_  A  /\  k  e.  ( A  \  x
) ) )  ->  { ( G  gsumg  ( H  |`  x ) ) } 
C_  S )
741, 4cntz2ss 16937 . . . . 5  |-  ( ( S  C_  B  /\  { ( G  gsumg  ( H  |`  x
) ) }  C_  S )  ->  ( Z `  S )  C_  ( Z `  {
( G  gsumg  ( H  |`  x
) ) } ) )
7536, 73, 74syl2anc 665 . . . 4  |-  ( (
ph  /\  ( x  C_  A  /\  k  e.  ( A  \  x
) ) )  -> 
( Z `  S
)  C_  ( Z `  { ( G  gsumg  ( H  |`  x ) ) } ) )
7635, 75sstrd 3480 . . 3  |-  ( (
ph  /\  ( x  C_  A  /\  k  e.  ( A  \  x
) ) )  ->  S  C_  ( Z `  { ( G  gsumg  ( H  |`  x ) ) } ) )
77 eldifi 3593 . . . . 5  |-  ( k  e.  ( A  \  x )  ->  k  e.  A )
7877adantl 467 . . . 4  |-  ( ( x  C_  A  /\  k  e.  ( A  \  x ) )  -> 
k  e.  A )
79 ffvelrn 6035 . . . 4  |-  ( ( F : A --> S  /\  k  e.  A )  ->  ( F `  k
)  e.  S )
8010, 78, 79syl2an 479 . . 3  |-  ( (
ph  /\  ( x  C_  A  /\  k  e.  ( A  \  x
) ) )  -> 
( F `  k
)  e.  S )
8176, 80sseldd 3471 . 2  |-  ( (
ph  /\  ( x  C_  A  /\  k  e.  ( A  \  x
) ) )  -> 
( F `  k
)  e.  ( Z `
 { ( G 
gsumg  ( H  |`  x ) ) } ) )
821, 2, 3, 4, 5, 6, 7, 8, 9, 14, 16, 21, 25, 34, 81gsumzaddlem 17489 1  |-  ( ph  ->  ( G  gsumg  ( F  oF  .+  H ) )  =  ( ( G 
gsumg  F )  .+  ( G  gsumg  H ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1870   _Vcvv 3087    \ cdif 3439    u. cun 3440    C_ wss 3442   {csn 4002   class class class wbr 4426   ran crn 4855    |` cres 4856   Fun wfun 5595   -->wf 5597   ` cfv 5601  (class class class)co 6305    oFcof 6543   supp csupp 6925   Fincfn 7577   finSupp cfsupp 7889   Basecbs 15084   +g cplusg 15152   0gc0g 15297    gsumg cgsu 15298   Mndcmnd 16486  SubMndcsubmnd 16532  Cntzccntz 16920
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-se 4814  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-of 6545  df-om 6707  df-1st 6807  df-2nd 6808  df-supp 6926  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-oadd 7194  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-fsupp 7890  df-oi 8025  df-card 8372  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-nn 10610  df-2 10668  df-n0 10870  df-z 10938  df-uz 11160  df-fz 11783  df-fzo 11914  df-seq 12211  df-hash 12513  df-ndx 15087  df-slot 15088  df-base 15089  df-sets 15090  df-ress 15091  df-plusg 15165  df-0g 15299  df-gsum 15300  df-mgm 16439  df-sgrp 16478  df-mnd 16488  df-submnd 16534  df-cntz 16922
This theorem is referenced by:  gsumadd  17491  gsumzsplit  17495
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