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Theorem gsumzadd 15220
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 2296 . 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 14443 . . . 4  |-  ( S  e.  (SubMnd `  G
)  ->  S  C_  B
)
1311, 12syl 15 . . 3  |-  ( ph  ->  S  C_  B )
14 fss 5413 . . 3  |-  ( ( F : A --> S  /\  S  C_  B )  ->  F : A --> B )
1510, 13, 14syl2anc 642 . 2  |-  ( ph  ->  F : A --> B )
16 gsumzadd.h . . 3  |-  ( ph  ->  H : A --> S )
17 fss 5413 . . 3  |-  ( ( H : A --> S  /\  S  C_  B )  ->  H : A --> B )
1816, 13, 17syl2anc 642 . 2  |-  ( ph  ->  H : A --> B )
19 gsumzadd.c . . 3  |-  ( ph  ->  S  C_  ( Z `  S ) )
20 frn 5411 . . . 4  |-  ( F : A --> S  ->  ran  F  C_  S )
2110, 20syl 15 . . 3  |-  ( ph  ->  ran  F  C_  S
)
224cntzidss 14829 . . 3  |-  ( ( S  C_  ( Z `  S )  /\  ran  F 
C_  S )  ->  ran  F  C_  ( Z `  ran  F ) )
2319, 21, 22syl2anc 642 . 2  |-  ( ph  ->  ran  F  C_  ( Z `  ran  F ) )
24 frn 5411 . . . 4  |-  ( H : A --> S  ->  ran  H  C_  S )
2516, 24syl 15 . . 3  |-  ( ph  ->  ran  H  C_  S
)
264cntzidss 14829 . . 3  |-  ( ( S  C_  ( Z `  S )  /\  ran  H 
C_  S )  ->  ran  H  C_  ( Z `  ran  H ) )
2719, 25, 26syl2anc 642 . 2  |-  ( ph  ->  ran  H  C_  ( Z `  ran  H ) )
283submcl 14446 . . . . . . 7  |-  ( ( S  e.  (SubMnd `  G )  /\  x  e.  S  /\  y  e.  S )  ->  (
x  .+  y )  e.  S )
29283expb 1152 . . . . . 6  |-  ( ( S  e.  (SubMnd `  G )  /\  (
x  e.  S  /\  y  e.  S )
)  ->  ( x  .+  y )  e.  S
)
3011, 29sylan 457 . . . . 5  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  e.  S )
31 inidm 3391 . . . . 5  |-  ( A  i^i  A )  =  A
3230, 10, 16, 6, 6, 31off 6109 . . . 4  |-  ( ph  ->  ( F  o F 
.+  H ) : A --> S )
33 frn 5411 . . . 4  |-  ( ( F  o F  .+  H ) : A --> S  ->  ran  ( F  o F  .+  H ) 
C_  S )
3432, 33syl 15 . . 3  |-  ( ph  ->  ran  ( F  o F  .+  H )  C_  S )
354cntzidss 14829 . . 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 642 . 2  |-  ( ph  ->  ran  ( F  o F  .+  H )  C_  ( Z `  ran  ( F  o F  .+  H
) ) )
3719adantr 451 . . . 4  |-  ( (
ph  /\  ( x  C_  A  /\  k  e.  ( A  \  x
) ) )  ->  S  C_  ( Z `  S ) )
3813adantr 451 . . . . 5  |-  ( (
ph  /\  ( x  C_  A  /\  k  e.  ( A  \  x
) ) )  ->  S  C_  B )
395adantr 451 . . . . . . 7  |-  ( (
ph  /\  ( x  C_  A  /\  k  e.  ( A  \  x
) ) )  ->  G  e.  Mnd )
40 vex 2804 . . . . . . . 8  |-  x  e. 
_V
4140a1i 10 . . . . . . 7  |-  ( (
ph  /\  ( x  C_  A  /\  k  e.  ( A  \  x
) ) )  ->  x  e.  _V )
4211adantr 451 . . . . . . 7  |-  ( (
ph  /\  ( x  C_  A  /\  k  e.  ( A  \  x
) ) )  ->  S  e.  (SubMnd `  G
) )
43 simpl 443 . . . . . . . 8  |-  ( ( x  C_  A  /\  k  e.  ( A  \  x ) )  ->  x  C_  A )
44 fssres 5424 . . . . . . . 8  |-  ( ( H : A --> S  /\  x  C_  A )  -> 
( H  |`  x
) : x --> S )
4516, 43, 44syl2an 463 . . . . . . 7  |-  ( (
ph  /\  ( x  C_  A  /\  k  e.  ( A  \  x
) ) )  -> 
( H  |`  x
) : x --> S )
4627adantr 451 . . . . . . . 8  |-  ( (
ph  /\  ( x  C_  A  /\  k  e.  ( A  \  x
) ) )  ->  ran  H  C_  ( Z `  ran  H ) )
47 resss 4995 . . . . . . . . 9  |-  ( H  |`  x )  C_  H
48 rnss 4923 . . . . . . . . 9  |-  ( ( H  |`  x )  C_  H  ->  ran  ( H  |`  x )  C_  ran  H )
4947, 48ax-mp 8 . . . . . . . 8  |-  ran  ( H  |`  x )  C_  ran  H
504cntzidss 14829 . . . . . . . 8  |-  ( ( ran  H  C_  ( Z `  ran  H )  /\  ran  ( H  |`  x )  C_  ran  H )  ->  ran  ( H  |`  x )  C_  ( Z `  ran  ( H  |`  x ) ) )
5146, 49, 50sylancl 643 . . . . . . 7  |-  ( (
ph  /\  ( x  C_  A  /\  k  e.  ( A  \  x
) ) )  ->  ran  ( H  |`  x
)  C_  ( Z `  ran  ( H  |`  x ) ) )
528adantr 451 . . . . . . . 8  |-  ( (
ph  /\  ( x  C_  A  /\  k  e.  ( A  \  x
) ) )  -> 
( `' H "
( _V  \  {  .0.  } ) )  e. 
Fin )
53 cnvss 4870 . . . . . . . . 9  |-  ( ( H  |`  x )  C_  H  ->  `' ( H  |`  x )  C_  `' H )
54 imass1 5064 . . . . . . . . 9  |-  ( `' ( H  |`  x
)  C_  `' H  ->  ( `' ( H  |`  x ) " ( _V  \  {  .0.  }
) )  C_  ( `' H " ( _V 
\  {  .0.  }
) ) )
5547, 53, 54mp2b 9 . . . . . . . 8  |-  ( `' ( H  |`  x
) " ( _V 
\  {  .0.  }
) )  C_  ( `' H " ( _V 
\  {  .0.  }
) )
56 ssfi 7099 . . . . . . . 8  |-  ( ( ( `' H "
( _V  \  {  .0.  } ) )  e. 
Fin  /\  ( `' ( H  |`  x )
" ( _V  \  {  .0.  } ) ) 
C_  ( `' H " ( _V  \  {  .0.  } ) ) )  ->  ( `' ( H  |`  x ) " ( _V  \  {  .0.  } ) )  e.  Fin )
5752, 55, 56sylancl 643 . . . . . . 7  |-  ( (
ph  /\  ( x  C_  A  /\  k  e.  ( A  \  x
) ) )  -> 
( `' ( H  |`  x ) " ( _V  \  {  .0.  }
) )  e.  Fin )
582, 4, 39, 41, 42, 45, 51, 57gsumzsubmcl 15216 . . . . . 6  |-  ( (
ph  /\  ( x  C_  A  /\  k  e.  ( A  \  x
) ) )  -> 
( G  gsumg  ( H  |`  x
) )  e.  S
)
5958snssd 3776 . . . . 5  |-  ( (
ph  /\  ( x  C_  A  /\  k  e.  ( A  \  x
) ) )  ->  { ( G  gsumg  ( H  |`  x ) ) } 
C_  S )
601, 4cntz2ss 14824 . . . . 5  |-  ( ( S  C_  B  /\  { ( G  gsumg  ( H  |`  x
) ) }  C_  S )  ->  ( Z `  S )  C_  ( Z `  {
( G  gsumg  ( H  |`  x
) ) } ) )
6138, 59, 60syl2anc 642 . . . 4  |-  ( (
ph  /\  ( x  C_  A  /\  k  e.  ( A  \  x
) ) )  -> 
( Z `  S
)  C_  ( Z `  { ( G  gsumg  ( H  |`  x ) ) } ) )
6237, 61sstrd 3202 . . 3  |-  ( (
ph  /\  ( x  C_  A  /\  k  e.  ( A  \  x
) ) )  ->  S  C_  ( Z `  { ( G  gsumg  ( H  |`  x ) ) } ) )
63 eldifi 3311 . . . . 5  |-  ( k  e.  ( A  \  x )  ->  k  e.  A )
6463adantl 452 . . . 4  |-  ( ( x  C_  A  /\  k  e.  ( A  \  x ) )  -> 
k  e.  A )
65 ffvelrn 5679 . . . 4  |-  ( ( F : A --> S  /\  k  e.  A )  ->  ( F `  k
)  e.  S )
6610, 64, 65syl2an 463 . . 3  |-  ( (
ph  /\  ( x  C_  A  /\  k  e.  ( A  \  x
) ) )  -> 
( F `  k
)  e.  S )
6762, 66sseldd 3194 . 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 15219 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 358    = wceq 1632    e. wcel 1696   _Vcvv 2801    \ cdif 3162    u. cun 3163    C_ wss 3165   {csn 3653   `'ccnv 4704   ran crn 4706    |` cres 4707   "cima 4708   -->wf 5267   ` cfv 5271  (class class class)co 5874    o Fcof 6092   Fincfn 6879   Basecbs 13164   +g cplusg 13224   0gc0g 13416    gsumg cgsu 13417   Mndcmnd 14377  SubMndcsubmnd 14430  Cntzccntz 14807
This theorem is referenced by:  gsumadd  15221  gsumzsplit  15222
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-oi 7241  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-fzo 10887  df-seq 11063  df-hash 11354  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-0g 13420  df-gsum 13421  df-mnd 14383  df-submnd 14432  df-cntz 14809
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