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Theorem gsumzaddOLD 16536
Description: The sum of two group sums. (Contributed by Mario Carneiro, 25-Apr-2016.) Obsolete version of gsumzadd 16534 as of 5-Jun-2019. (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
gsumzaddOLD.b  |-  B  =  ( Base `  G
)
gsumzaddOLD.0  |-  .0.  =  ( 0g `  G )
gsumzaddOLD.p  |-  .+  =  ( +g  `  G )
gsumzaddOLD.z  |-  Z  =  (Cntz `  G )
gsumzaddOLD.g  |-  ( ph  ->  G  e.  Mnd )
gsumzaddOLD.a  |-  ( ph  ->  A  e.  V )
gsumzaddOLD.fn  |-  ( ph  ->  ( `' F "
( _V  \  {  .0.  } ) )  e. 
Fin )
gsumzaddOLD.hn  |-  ( ph  ->  ( `' H "
( _V  \  {  .0.  } ) )  e. 
Fin )
gsumzaddOLD.s  |-  ( ph  ->  S  e.  (SubMnd `  G ) )
gsumzaddOLD.c  |-  ( ph  ->  S  C_  ( Z `  S ) )
gsumzaddOLD.f  |-  ( ph  ->  F : A --> S )
gsumzaddOLD.h  |-  ( ph  ->  H : A --> S )
Assertion
Ref Expression
gsumzaddOLD  |-  ( ph  ->  ( G  gsumg  ( F  oF  .+  H ) )  =  ( ( G 
gsumg  F )  .+  ( G  gsumg  H ) ) )

Proof of Theorem gsumzaddOLD
Dummy variables  k  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 gsumzaddOLD.b . 2  |-  B  =  ( Base `  G
)
2 gsumzaddOLD.0 . 2  |-  .0.  =  ( 0g `  G )
3 gsumzaddOLD.p . 2  |-  .+  =  ( +g  `  G )
4 gsumzaddOLD.z . 2  |-  Z  =  (Cntz `  G )
5 gsumzaddOLD.g . 2  |-  ( ph  ->  G  e.  Mnd )
6 gsumzaddOLD.a . 2  |-  ( ph  ->  A  e.  V )
7 gsumzaddOLD.fn . 2  |-  ( ph  ->  ( `' F "
( _V  \  {  .0.  } ) )  e. 
Fin )
8 gsumzaddOLD.hn . 2  |-  ( ph  ->  ( `' H "
( _V  \  {  .0.  } ) )  e. 
Fin )
9 eqid 2454 . 2  |-  ( `' ( F  u.  H
) " ( _V 
\  {  .0.  }
) )  =  ( `' ( F  u.  H ) " ( _V  \  {  .0.  }
) )
10 gsumzaddOLD.f . . 3  |-  ( ph  ->  F : A --> S )
11 gsumzaddOLD.s . . . 4  |-  ( ph  ->  S  e.  (SubMnd `  G ) )
121submss 15601 . . . 4  |-  ( S  e.  (SubMnd `  G
)  ->  S  C_  B
)
1311, 12syl 16 . . 3  |-  ( ph  ->  S  C_  B )
14 fss 5678 . . 3  |-  ( ( F : A --> S  /\  S  C_  B )  ->  F : A --> B )
1510, 13, 14syl2anc 661 . 2  |-  ( ph  ->  F : A --> B )
16 gsumzaddOLD.h . . 3  |-  ( ph  ->  H : A --> S )
17 fss 5678 . . 3  |-  ( ( H : A --> S  /\  S  C_  B )  ->  H : A --> B )
1816, 13, 17syl2anc 661 . 2  |-  ( ph  ->  H : A --> B )
19 gsumzaddOLD.c . . 3  |-  ( ph  ->  S  C_  ( Z `  S ) )
20 frn 5676 . . . 4  |-  ( F : A --> S  ->  ran  F  C_  S )
2110, 20syl 16 . . 3  |-  ( ph  ->  ran  F  C_  S
)
224cntzidss 15978 . . 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 5676 . . . 4  |-  ( H : A --> S  ->  ran  H  C_  S )
2516, 24syl 16 . . 3  |-  ( ph  ->  ran  H  C_  S
)
264cntzidss 15978 . . 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 15604 . . . . . . 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 3670 . . . . 5  |-  ( A  i^i  A )  =  A
3230, 10, 16, 6, 6, 31off 6447 . . . 4  |-  ( ph  ->  ( F  oF  .+  H ) : A --> S )
33 frn 5676 . . . 4  |-  ( ( F  oF  .+  H ) : A --> S  ->  ran  ( F  oF  .+  H ) 
C_  S )
3432, 33syl 16 . . 3  |-  ( ph  ->  ran  ( F  oF  .+  H )  C_  S )
354cntzidss 15978 . . 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 5689 . . . . . . . 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 5245 . . . . . . . . 9  |-  ( H  |`  x )  C_  H
48 rnss 5179 . . . . . . . . 9  |-  ( ( H  |`  x )  C_  H  ->  ran  ( H  |`  x )  C_  ran  H )
4947, 48ax-mp 5 . . . . . . . 8  |-  ran  ( H  |`  x )  C_  ran  H
504cntzidss 15978 . . . . . . . 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 ) ) )
528adantr 465 . . . . . . . 8  |-  ( (
ph  /\  ( x  C_  A  /\  k  e.  ( A  \  x
) ) )  -> 
( `' H "
( _V  \  {  .0.  } ) )  e. 
Fin )
53 cnvss 5123 . . . . . . . . 9  |-  ( ( H  |`  x )  C_  H  ->  `' ( H  |`  x )  C_  `' H )
54 imass1 5314 . . . . . . . . 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 7647 . . . . . . . 8  |-  ( ( ( `' H "
( _V  \  {  .0.  } ) )  e. 
Fin  /\  ( `' ( H  |`  x )
" ( _V  \  {  .0.  } ) ) 
C_  ( `' H " ( _V  \  {  .0.  } ) ) )  ->  ( `' ( H  |`  x ) " ( _V  \  {  .0.  } ) )  e.  Fin )
5752, 55, 56sylancl 662 . . . . . . 7  |-  ( (
ph  /\  ( x  C_  A  /\  k  e.  ( A  \  x
) ) )  -> 
( `' ( H  |`  x ) " ( _V  \  {  .0.  }
) )  e.  Fin )
582, 4, 39, 41, 42, 45, 51, 57gsumzsubmclOLD 16528 . . . . . 6  |-  ( (
ph  /\  ( x  C_  A  /\  k  e.  ( A  \  x
) ) )  -> 
( G  gsumg  ( H  |`  x
) )  e.  S
)
5958snssd 4129 . . . . 5  |-  ( (
ph  /\  ( x  C_  A  /\  k  e.  ( A  \  x
) ) )  ->  { ( G  gsumg  ( H  |`  x ) ) } 
C_  S )
601, 4cntz2ss 15973 . . . . 5  |-  ( ( S  C_  B  /\  { ( G  gsumg  ( H  |`  x
) ) }  C_  S )  ->  ( Z `  S )  C_  ( Z `  {
( G  gsumg  ( H  |`  x
) ) } ) )
6138, 59, 60syl2anc 661 . . . 4  |-  ( (
ph  /\  ( x  C_  A  /\  k  e.  ( A  \  x
) ) )  -> 
( Z `  S
)  C_  ( Z `  { ( G  gsumg  ( H  |`  x ) ) } ) )
6237, 61sstrd 3477 . . 3  |-  ( (
ph  /\  ( x  C_  A  /\  k  e.  ( A  \  x
) ) )  ->  S  C_  ( Z `  { ( G  gsumg  ( H  |`  x ) ) } ) )
63 eldifi 3589 . . . . 5  |-  ( k  e.  ( A  \  x )  ->  k  e.  A )
6463adantl 466 . . . 4  |-  ( ( x  C_  A  /\  k  e.  ( A  \  x ) )  -> 
k  e.  A )
65 ffvelrn 5953 . . . 4  |-  ( ( F : A --> S  /\  k  e.  A )  ->  ( F `  k
)  e.  S )
6610, 64, 65syl2an 477 . . 3  |-  ( (
ph  /\  ( x  C_  A  /\  k  e.  ( A  \  x
) ) )  -> 
( F `  k
)  e.  S )
6762, 66sseldd 3468 . 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, 67gsumzaddlemOLD 16535 1  |-  ( ph  ->  ( G  gsumg  ( F  oF  .+  H ) )  =  ( ( G 
gsumg  F )  .+  ( G  gsumg  H ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   _Vcvv 3078    \ cdif 3436    u. cun 3437    C_ wss 3439   {csn 3988   `'ccnv 4950   ran crn 4952    |` cres 4953   "cima 4954   -->wf 5525   ` cfv 5529  (class class class)co 6203    oFcof 6431   Fincfn 7423   Basecbs 14296   +g cplusg 14361   0gc0g 14501    gsumg cgsu 14502   Mndcmnd 15532  SubMndcsubmnd 15586  Cntzccntz 15956
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 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9453  ax-resscn 9454  ax-1cn 9455  ax-icn 9456  ax-addcl 9457  ax-addrcl 9458  ax-mulcl 9459  ax-mulrcl 9460  ax-mulcom 9461  ax-addass 9462  ax-mulass 9463  ax-distr 9464  ax-i2m1 9465  ax-1ne0 9466  ax-1rid 9467  ax-rnegex 9468  ax-rrecex 9469  ax-cnre 9470  ax-pre-lttri 9471  ax-pre-lttrn 9472  ax-pre-ltadd 9473  ax-pre-mulgt0 9474
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 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-se 4791  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-isom 5538  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-of 6433  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-1o 7033  df-oadd 7037  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-oi 7839  df-card 8224  df-pnf 9535  df-mnf 9536  df-xr 9537  df-ltxr 9538  df-le 9539  df-sub 9712  df-neg 9713  df-nn 10438  df-2 10495  df-n0 10695  df-z 10762  df-uz 10977  df-fz 11559  df-fzo 11670  df-seq 11928  df-hash 12225  df-ndx 14299  df-slot 14300  df-base 14301  df-sets 14302  df-ress 14303  df-plusg 14374  df-0g 14503  df-gsum 14504  df-mnd 15538  df-submnd 15588  df-cntz 15958
This theorem is referenced by:  gsumaddOLD  16538  gsumzsplitOLD  16544
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