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Theorem gsumzadd 16531
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 2454 . 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 15598 . . . 4  |-  ( S  e.  (SubMnd `  G
)  ->  S  C_  B
)
1311, 12syl 16 . . 3  |-  ( ph  ->  S  C_  B )
14 fss 5676 . . 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 5676 . . 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 5674 . . . 4  |-  ( F : A --> S  ->  ran  F  C_  S )
2110, 20syl 16 . . 3  |-  ( ph  ->  ran  F  C_  S
)
224cntzidss 15975 . . 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 5674 . . . 4  |-  ( H : A --> S  ->  ran  H  C_  S )
2516, 24syl 16 . . 3  |-  ( ph  ->  ran  H  C_  S
)
264cntzidss 15975 . . 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 15601 . . . . . . 7  |-  ( ( S  e.  (SubMnd `  G )  /\  x  e.  S  /\  y  e.  S )  ->  (
x  .+  y )  e.  S )
29283expb 1189 . . . . . 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 3668 . . . . 5  |-  ( A  i^i  A )  =  A
3230, 10, 16, 6, 6, 31off 6445 . . . 4  |-  ( ph  ->  ( F  oF  .+  H ) : A --> S )
33 frn 5674 . . . 4  |-  ( ( F  oF  .+  H ) : A --> S  ->  ran  ( F  oF  .+  H ) 
C_  S )
3432, 33syl 16 . . 3  |-  ( ph  ->  ran  ( F  oF  .+  H )  C_  S )
354cntzidss 15975 . . 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 3081 . . . . . . . 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 5687 . . . . . . . 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 5243 . . . . . . . . 9  |-  ( H  |`  x )  C_  H
48 rnss 5177 . . . . . . . . 9  |-  ( ( H  |`  x )  C_  H  ->  ran  ( H  |`  x )  C_  ran  H )
4947, 48ax-mp 5 . . . . . . . 8  |-  ran  ( H  |`  x )  C_  ran  H
504cntzidss 15975 . . . . . . . 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 5670 . . . . . . . . . . 11  |-  ( H : A --> S  ->  Fun  H )
5316, 52syl 16 . . . . . . . . . 10  |-  ( ph  ->  Fun  H )
54 funres 5566 . . . . . . . . . 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 7738 . . . . . . . . . . . 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 6060 . . . . . . . . . . . 12  |-  ( ( H : A --> S  /\  A  e.  V )  ->  H  e.  _V )
6216, 6, 61syl2anc 661 . . . . . . . . . . 11  |-  ( ph  ->  H  e.  _V )
63 fvex 5810 . . . . . . . . . . . 12  |-  ( 0g
`  G )  e. 
_V
642, 63eqeltri 2538 . . . . . . . . . . 11  |-  .0.  e.  _V
65 ressuppss 6819 . . . . . . . . . . 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 7645 . . . . . . . . 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 6053 . . . . . . . . . . 11  |-  ( ( Fun  H  /\  x  e.  _V )  ->  ( H  |`  x )  e. 
_V )
7153, 40, 70sylancl 662 . . . . . . . . . 10  |-  ( ph  ->  ( H  |`  x
)  e.  _V )
72 isfsupp 7736 . . . . . . . . . 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 913 . . . . . . 7  |-  ( (
ph  /\  ( x  C_  A  /\  k  e.  ( A  \  x
) ) )  -> 
( H  |`  x
) finSupp  .0.  )
762, 4, 39, 41, 42, 45, 51, 75gsumzsubmcl 16524 . . . . . 6  |-  ( (
ph  /\  ( x  C_  A  /\  k  e.  ( A  \  x
) ) )  -> 
( G  gsumg  ( H  |`  x
) )  e.  S
)
7776snssd 4127 . . . . 5  |-  ( (
ph  /\  ( x  C_  A  /\  k  e.  ( A  \  x
) ) )  ->  { ( G  gsumg  ( H  |`  x ) ) } 
C_  S )
781, 4cntz2ss 15970 . . . . 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 3475 . . 3  |-  ( (
ph  /\  ( x  C_  A  /\  k  e.  ( A  \  x
) ) )  ->  S  C_  ( Z `  { ( G  gsumg  ( H  |`  x ) ) } ) )
81 eldifi 3587 . . . . 5  |-  ( k  e.  ( A  \  x )  ->  k  e.  A )
8281adantl 466 . . . 4  |-  ( ( x  C_  A  /\  k  e.  ( A  \  x ) )  -> 
k  e.  A )
83 ffvelrn 5951 . . . 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 3466 . 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 16530 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 1370    e. wcel 1758   _Vcvv 3078    \ cdif 3434    u. cun 3435    C_ wss 3437   {csn 3986   class class class wbr 4401   ran crn 4950    |` cres 4951   Fun wfun 5521   -->wf 5523   ` cfv 5527  (class class class)co 6201    oFcof 6429   supp csupp 6801   Fincfn 7421   finSupp cfsupp 7732   Basecbs 14293   +g cplusg 14358   0gc0g 14498    gsumg cgsu 14499   Mndcmnd 15529  SubMndcsubmnd 15583  Cntzccntz 15953
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 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-cnex 9450  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470  ax-pre-mulgt0 9471
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 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-int 4238  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-se 4789  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-isom 5536  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-of 6431  df-om 6588  df-1st 6688  df-2nd 6689  df-supp 6802  df-recs 6943  df-rdg 6977  df-1o 7031  df-oadd 7035  df-er 7212  df-en 7422  df-dom 7423  df-sdom 7424  df-fin 7425  df-fsupp 7733  df-oi 7836  df-card 8221  df-pnf 9532  df-mnf 9533  df-xr 9534  df-ltxr 9535  df-le 9536  df-sub 9709  df-neg 9710  df-nn 10435  df-2 10492  df-n0 10692  df-z 10759  df-uz 10974  df-fz 11556  df-fzo 11667  df-seq 11925  df-hash 12222  df-ndx 14296  df-slot 14297  df-base 14298  df-sets 14299  df-ress 14300  df-plusg 14371  df-0g 14500  df-gsum 14501  df-mnd 15535  df-submnd 15585  df-cntz 15955
This theorem is referenced by:  gsumadd  16534  gsumzsplit  16540
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