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Theorem gsumzadd 15520
Description: The sum of two group sums. (Contributed by Mario Carneiro, 25-Apr-2016.)
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 "
( _V  \  {  .0.  } ) )  e. 
Fin )
gsumzadd.hn  |-  ( ph  ->  ( `' H "
( _V  \  {  .0.  } ) )  e. 
Fin )
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  o F 
.+  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 "
( _V  \  {  .0.  } ) )  e. 
Fin )
8 gsumzadd.hn . 2  |-  ( ph  ->  ( `' H "
( _V  \  {  .0.  } ) )  e. 
Fin )
9 eqid 2436 . 2  |-  ( `' ( F  u.  H
) " ( _V 
\  {  .0.  }
) )  =  ( `' ( F  u.  H ) " ( _V  \  {  .0.  }
) )
10 gsumzadd.f . . 3  |-  ( ph  ->  F : A --> S )
11 gsumzadd.s . . . 4  |-  ( ph  ->  S  e.  (SubMnd `  G ) )
121submss 14743 . . . 4  |-  ( S  e.  (SubMnd `  G
)  ->  S  C_  B
)
1311, 12syl 16 . . 3  |-  ( ph  ->  S  C_  B )
14 fss 5592 . . 3  |-  ( ( F : A --> S  /\  S  C_  B )  ->  F : A --> B )
1510, 13, 14syl2anc 643 . 2  |-  ( ph  ->  F : A --> B )
16 gsumzadd.h . . 3  |-  ( ph  ->  H : A --> S )
17 fss 5592 . . 3  |-  ( ( H : A --> S  /\  S  C_  B )  ->  H : A --> B )
1816, 13, 17syl2anc 643 . 2  |-  ( ph  ->  H : A --> B )
19 gsumzadd.c . . 3  |-  ( ph  ->  S  C_  ( Z `  S ) )
20 frn 5590 . . . 4  |-  ( F : A --> S  ->  ran  F  C_  S )
2110, 20syl 16 . . 3  |-  ( ph  ->  ran  F  C_  S
)
224cntzidss 15129 . . 3  |-  ( ( S  C_  ( Z `  S )  /\  ran  F 
C_  S )  ->  ran  F  C_  ( Z `  ran  F ) )
2319, 21, 22syl2anc 643 . 2  |-  ( ph  ->  ran  F  C_  ( Z `  ran  F ) )
24 frn 5590 . . . 4  |-  ( H : A --> S  ->  ran  H  C_  S )
2516, 24syl 16 . . 3  |-  ( ph  ->  ran  H  C_  S
)
264cntzidss 15129 . . 3  |-  ( ( S  C_  ( Z `  S )  /\  ran  H 
C_  S )  ->  ran  H  C_  ( Z `  ran  H ) )
2719, 25, 26syl2anc 643 . 2  |-  ( ph  ->  ran  H  C_  ( Z `  ran  H ) )
283submcl 14746 . . . . . . 7  |-  ( ( S  e.  (SubMnd `  G )  /\  x  e.  S  /\  y  e.  S )  ->  (
x  .+  y )  e.  S )
29283expb 1154 . . . . . 6  |-  ( ( S  e.  (SubMnd `  G )  /\  (
x  e.  S  /\  y  e.  S )
)  ->  ( x  .+  y )  e.  S
)
3011, 29sylan 458 . . . . 5  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  e.  S )
31 inidm 3543 . . . . 5  |-  ( A  i^i  A )  =  A
3230, 10, 16, 6, 6, 31off 6313 . . . 4  |-  ( ph  ->  ( F  o F 
.+  H ) : A --> S )
33 frn 5590 . . . 4  |-  ( ( F  o F  .+  H ) : A --> S  ->  ran  ( F  o F  .+  H ) 
C_  S )
3432, 33syl 16 . . 3  |-  ( ph  ->  ran  ( F  o F  .+  H )  C_  S )
354cntzidss 15129 . . 3  |-  ( ( S  C_  ( Z `  S )  /\  ran  ( F  o F  .+  H )  C_  S
)  ->  ran  ( F  o F  .+  H
)  C_  ( Z `  ran  ( F  o F  .+  H ) ) )
3619, 34, 35syl2anc 643 . 2  |-  ( ph  ->  ran  ( F  o F  .+  H )  C_  ( Z `  ran  ( F  o F  .+  H
) ) )
3719adantr 452 . . . 4  |-  ( (
ph  /\  ( x  C_  A  /\  k  e.  ( A  \  x
) ) )  ->  S  C_  ( Z `  S ) )
3813adantr 452 . . . . 5  |-  ( (
ph  /\  ( x  C_  A  /\  k  e.  ( A  \  x
) ) )  ->  S  C_  B )
395adantr 452 . . . . . . 7  |-  ( (
ph  /\  ( x  C_  A  /\  k  e.  ( A  \  x
) ) )  ->  G  e.  Mnd )
40 vex 2952 . . . . . . . 8  |-  x  e. 
_V
4140a1i 11 . . . . . . 7  |-  ( (
ph  /\  ( x  C_  A  /\  k  e.  ( A  \  x
) ) )  ->  x  e.  _V )
4211adantr 452 . . . . . . 7  |-  ( (
ph  /\  ( x  C_  A  /\  k  e.  ( A  \  x
) ) )  ->  S  e.  (SubMnd `  G
) )
43 simpl 444 . . . . . . . 8  |-  ( ( x  C_  A  /\  k  e.  ( A  \  x ) )  ->  x  C_  A )
44 fssres 5603 . . . . . . . 8  |-  ( ( H : A --> S  /\  x  C_  A )  -> 
( H  |`  x
) : x --> S )
4516, 43, 44syl2an 464 . . . . . . 7  |-  ( (
ph  /\  ( x  C_  A  /\  k  e.  ( A  \  x
) ) )  -> 
( H  |`  x
) : x --> S )
4627adantr 452 . . . . . . . 8  |-  ( (
ph  /\  ( x  C_  A  /\  k  e.  ( A  \  x
) ) )  ->  ran  H  C_  ( Z `  ran  H ) )
47 resss 5163 . . . . . . . . 9  |-  ( H  |`  x )  C_  H
48 rnss 5091 . . . . . . . . 9  |-  ( ( H  |`  x )  C_  H  ->  ran  ( H  |`  x )  C_  ran  H )
4947, 48ax-mp 8 . . . . . . . 8  |-  ran  ( H  |`  x )  C_  ran  H
504cntzidss 15129 . . . . . . . 8  |-  ( ( ran  H  C_  ( Z `  ran  H )  /\  ran  ( H  |`  x )  C_  ran  H )  ->  ran  ( H  |`  x )  C_  ( Z `  ran  ( H  |`  x ) ) )
5146, 49, 50sylancl 644 . . . . . . 7  |-  ( (
ph  /\  ( x  C_  A  /\  k  e.  ( A  \  x
) ) )  ->  ran  ( H  |`  x
)  C_  ( Z `  ran  ( H  |`  x ) ) )
528adantr 452 . . . . . . . 8  |-  ( (
ph  /\  ( x  C_  A  /\  k  e.  ( A  \  x
) ) )  -> 
( `' H "
( _V  \  {  .0.  } ) )  e. 
Fin )
53 cnvss 5038 . . . . . . . . 9  |-  ( ( H  |`  x )  C_  H  ->  `' ( H  |`  x )  C_  `' H )
54 imass1 5232 . . . . . . . . 9  |-  ( `' ( H  |`  x
)  C_  `' H  ->  ( `' ( H  |`  x ) " ( _V  \  {  .0.  }
) )  C_  ( `' H " ( _V 
\  {  .0.  }
) ) )
5547, 53, 54mp2b 10 . . . . . . . 8  |-  ( `' ( H  |`  x
) " ( _V 
\  {  .0.  }
) )  C_  ( `' H " ( _V 
\  {  .0.  }
) )
56 ssfi 7322 . . . . . . . 8  |-  ( ( ( `' H "
( _V  \  {  .0.  } ) )  e. 
Fin  /\  ( `' ( H  |`  x )
" ( _V  \  {  .0.  } ) ) 
C_  ( `' H " ( _V  \  {  .0.  } ) ) )  ->  ( `' ( H  |`  x ) " ( _V  \  {  .0.  } ) )  e.  Fin )
5752, 55, 56sylancl 644 . . . . . . 7  |-  ( (
ph  /\  ( x  C_  A  /\  k  e.  ( A  \  x
) ) )  -> 
( `' ( H  |`  x ) " ( _V  \  {  .0.  }
) )  e.  Fin )
582, 4, 39, 41, 42, 45, 51, 57gsumzsubmcl 15516 . . . . . 6  |-  ( (
ph  /\  ( x  C_  A  /\  k  e.  ( A  \  x
) ) )  -> 
( G  gsumg  ( H  |`  x
) )  e.  S
)
5958snssd 3936 . . . . 5  |-  ( (
ph  /\  ( x  C_  A  /\  k  e.  ( A  \  x
) ) )  ->  { ( G  gsumg  ( H  |`  x ) ) } 
C_  S )
601, 4cntz2ss 15124 . . . . 5  |-  ( ( S  C_  B  /\  { ( G  gsumg  ( H  |`  x
) ) }  C_  S )  ->  ( Z `  S )  C_  ( Z `  {
( G  gsumg  ( H  |`  x
) ) } ) )
6138, 59, 60syl2anc 643 . . . 4  |-  ( (
ph  /\  ( x  C_  A  /\  k  e.  ( A  \  x
) ) )  -> 
( Z `  S
)  C_  ( Z `  { ( G  gsumg  ( H  |`  x ) ) } ) )
6237, 61sstrd 3351 . . 3  |-  ( (
ph  /\  ( x  C_  A  /\  k  e.  ( A  \  x
) ) )  ->  S  C_  ( Z `  { ( G  gsumg  ( H  |`  x ) ) } ) )
63 eldifi 3462 . . . . 5  |-  ( k  e.  ( A  \  x )  ->  k  e.  A )
6463adantl 453 . . . 4  |-  ( ( x  C_  A  /\  k  e.  ( A  \  x ) )  -> 
k  e.  A )
65 ffvelrn 5861 . . . 4  |-  ( ( F : A --> S  /\  k  e.  A )  ->  ( F `  k
)  e.  S )
6610, 64, 65syl2an 464 . . 3  |-  ( (
ph  /\  ( x  C_  A  /\  k  e.  ( A  \  x
) ) )  -> 
( F `  k
)  e.  S )
6762, 66sseldd 3342 . 2  |-  ( (
ph  /\  ( x  C_  A  /\  k  e.  ( A  \  x
) ) )  -> 
( F `  k
)  e.  ( Z `
 { ( G 
gsumg  ( H  |`  x ) ) } ) )
681, 2, 3, 4, 5, 6, 7, 8, 9, 15, 18, 23, 27, 36, 67gsumzaddlem 15519 1  |-  ( ph  ->  ( G  gsumg  ( F  o F 
.+  H ) )  =  ( ( G 
gsumg  F )  .+  ( G  gsumg  H ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2949    \ cdif 3310    u. cun 3311    C_ wss 3313   {csn 3807   `'ccnv 4870   ran crn 4872    |` cres 4873   "cima 4874   -->wf 5443   ` cfv 5447  (class class class)co 6074    o Fcof 6296   Fincfn 7102   Basecbs 13462   +g cplusg 13522   0gc0g 13716    gsumg cgsu 13717   Mndcmnd 14677  SubMndcsubmnd 14730  Cntzccntz 15107
This theorem is referenced by:  gsumadd  15521  gsumzsplit  15522
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4313  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396  ax-un 4694  ax-cnex 9039  ax-resscn 9040  ax-1cn 9041  ax-icn 9042  ax-addcl 9043  ax-addrcl 9044  ax-mulcl 9045  ax-mulrcl 9046  ax-mulcom 9047  ax-addass 9048  ax-mulass 9049  ax-distr 9050  ax-i2m1 9051  ax-1ne0 9052  ax-1rid 9053  ax-rnegex 9054  ax-rrecex 9055  ax-cnre 9056  ax-pre-lttri 9057  ax-pre-lttrn 9058  ax-pre-ltadd 9059  ax-pre-mulgt0 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2703  df-rex 2704  df-reu 2705  df-rmo 2706  df-rab 2707  df-v 2951  df-sbc 3155  df-csb 3245  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-pss 3329  df-nul 3622  df-if 3733  df-pw 3794  df-sn 3813  df-pr 3814  df-tp 3815  df-op 3816  df-uni 4009  df-int 4044  df-iun 4088  df-br 4206  df-opab 4260  df-mpt 4261  df-tr 4296  df-eprel 4487  df-id 4491  df-po 4496  df-so 4497  df-fr 4534  df-se 4535  df-we 4536  df-ord 4577  df-on 4578  df-lim 4579  df-suc 4580  df-om 4839  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-res 4883  df-ima 4884  df-iota 5411  df-fun 5449  df-fn 5450  df-f 5451  df-f1 5452  df-fo 5453  df-f1o 5454  df-fv 5455  df-isom 5456  df-ov 6077  df-oprab 6078  df-mpt2 6079  df-of 6298  df-1st 6342  df-2nd 6343  df-riota 6542  df-recs 6626  df-rdg 6661  df-1o 6717  df-oadd 6721  df-er 6898  df-en 7103  df-dom 7104  df-sdom 7105  df-fin 7106  df-oi 7472  df-card 7819  df-pnf 9115  df-mnf 9116  df-xr 9117  df-ltxr 9118  df-le 9119  df-sub 9286  df-neg 9287  df-nn 9994  df-2 10051  df-n0 10215  df-z 10276  df-uz 10482  df-fz 11037  df-fzo 11129  df-seq 11317  df-hash 11612  df-ndx 13465  df-slot 13466  df-base 13467  df-sets 13468  df-ress 13469  df-plusg 13535  df-0g 13720  df-gsum 13721  df-mnd 14683  df-submnd 14732  df-cntz 15109
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