MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  gsummhm2 Unicode version

Theorem gsummhm2 15228
Description: Apply a group homomorphism to a group sum, mapping version with implicit substitution. (Contributed by Mario Carneiro, 5-May-2015.)
Hypotheses
Ref Expression
gsummhm2.b  |-  B  =  ( Base `  G
)
gsummhm2.z  |-  .0.  =  ( 0g `  G )
gsummhm2.g  |-  ( ph  ->  G  e. CMnd )
gsummhm2.h  |-  ( ph  ->  H  e.  Mnd )
gsummhm2.a  |-  ( ph  ->  A  e.  V )
gsummhm2.k  |-  ( ph  ->  ( x  e.  B  |->  C )  e.  ( G MndHom  H ) )
gsummhm2.f  |-  ( (
ph  /\  k  e.  A )  ->  X  e.  B )
gsummhm2.w  |-  ( ph  ->  ( `' ( k  e.  A  |->  X )
" ( _V  \  {  .0.  } ) )  e.  Fin )
gsummhm2.1  |-  ( x  =  X  ->  C  =  D )
gsummhm2.2  |-  ( x  =  ( G  gsumg  ( k  e.  A  |->  X ) )  ->  C  =  E )
Assertion
Ref Expression
gsummhm2  |-  ( ph  ->  ( H  gsumg  ( k  e.  A  |->  D ) )  =  E )
Distinct variable groups:    x, k, A    B, k, x    C, k    x, D    x, E    ph, k    x, G    x, H    x, X
Allowed substitution hints:    ph( x)    C( x)    D( k)    E( k)    G( k)    H( k)    V( x, k)    X( k)    .0. ( x, k)

Proof of Theorem gsummhm2
StepHypRef Expression
1 gsummhm2.b . . 3  |-  B  =  ( Base `  G
)
2 gsummhm2.z . . 3  |-  .0.  =  ( 0g `  G )
3 gsummhm2.g . . 3  |-  ( ph  ->  G  e. CMnd )
4 gsummhm2.h . . 3  |-  ( ph  ->  H  e.  Mnd )
5 gsummhm2.a . . 3  |-  ( ph  ->  A  e.  V )
6 gsummhm2.k . . 3  |-  ( ph  ->  ( x  e.  B  |->  C )  e.  ( G MndHom  H ) )
7 gsummhm2.f . . . 4  |-  ( (
ph  /\  k  e.  A )  ->  X  e.  B )
8 eqid 2296 . . . 4  |-  ( k  e.  A  |->  X )  =  ( k  e.  A  |->  X )
97, 8fmptd 5700 . . 3  |-  ( ph  ->  ( k  e.  A  |->  X ) : A --> B )
10 gsummhm2.w . . 3  |-  ( ph  ->  ( `' ( k  e.  A  |->  X )
" ( _V  \  {  .0.  } ) )  e.  Fin )
111, 2, 3, 4, 5, 6, 9, 10gsummhm 15227 . 2  |-  ( ph  ->  ( H  gsumg  ( ( x  e.  B  |->  C )  o.  ( k  e.  A  |->  X ) ) )  =  ( ( x  e.  B  |->  C ) `
 ( G  gsumg  ( k  e.  A  |->  X ) ) ) )
12 eqidd 2297 . . . 4  |-  ( ph  ->  ( k  e.  A  |->  X )  =  ( k  e.  A  |->  X ) )
13 eqidd 2297 . . . 4  |-  ( ph  ->  ( x  e.  B  |->  C )  =  ( x  e.  B  |->  C ) )
14 gsummhm2.1 . . . 4  |-  ( x  =  X  ->  C  =  D )
157, 12, 13, 14fmptco 5707 . . 3  |-  ( ph  ->  ( ( x  e.  B  |->  C )  o.  ( k  e.  A  |->  X ) )  =  ( k  e.  A  |->  D ) )
1615oveq2d 5890 . 2  |-  ( ph  ->  ( H  gsumg  ( ( x  e.  B  |->  C )  o.  ( k  e.  A  |->  X ) ) )  =  ( H  gsumg  ( k  e.  A  |->  D ) ) )
171, 2, 3, 5, 9, 10gsumcl 15214 . . 3  |-  ( ph  ->  ( G  gsumg  ( k  e.  A  |->  X ) )  e.  B )
18 eqid 2296 . . . . . . 7  |-  ( Base `  H )  =  (
Base `  H )
191, 18mhmf 14436 . . . . . 6  |-  ( ( x  e.  B  |->  C )  e.  ( G MndHom  H )  ->  (
x  e.  B  |->  C ) : B --> ( Base `  H ) )
206, 19syl 15 . . . . 5  |-  ( ph  ->  ( x  e.  B  |->  C ) : B --> ( Base `  H )
)
21 eqid 2296 . . . . . 6  |-  ( x  e.  B  |->  C )  =  ( x  e.  B  |->  C )
2221fmpt 5697 . . . . 5  |-  ( A. x  e.  B  C  e.  ( Base `  H
)  <->  ( x  e.  B  |->  C ) : B --> ( Base `  H
) )
2320, 22sylibr 203 . . . 4  |-  ( ph  ->  A. x  e.  B  C  e.  ( Base `  H ) )
24 gsummhm2.2 . . . . . 6  |-  ( x  =  ( G  gsumg  ( k  e.  A  |->  X ) )  ->  C  =  E )
2524eleq1d 2362 . . . . 5  |-  ( x  =  ( G  gsumg  ( k  e.  A  |->  X ) )  ->  ( C  e.  ( Base `  H
)  <->  E  e.  ( Base `  H ) ) )
2625rspcv 2893 . . . 4  |-  ( ( G  gsumg  ( k  e.  A  |->  X ) )  e.  B  ->  ( A. x  e.  B  C  e.  ( Base `  H
)  ->  E  e.  ( Base `  H )
) )
2717, 23, 26sylc 56 . . 3  |-  ( ph  ->  E  e.  ( Base `  H ) )
2824, 21fvmptg 5616 . . 3  |-  ( ( ( G  gsumg  ( k  e.  A  |->  X ) )  e.  B  /\  E  e.  ( Base `  H
) )  ->  (
( x  e.  B  |->  C ) `  ( G  gsumg  ( k  e.  A  |->  X ) ) )  =  E )
2917, 27, 28syl2anc 642 . 2  |-  ( ph  ->  ( ( x  e.  B  |->  C ) `  ( G  gsumg  ( k  e.  A  |->  X ) ) )  =  E )
3011, 16, 293eqtr3d 2336 1  |-  ( ph  ->  ( H  gsumg  ( k  e.  A  |->  D ) )  =  E )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   _Vcvv 2801    \ cdif 3162   {csn 3653    e. cmpt 4093   `'ccnv 4704   "cima 4708    o. ccom 4709   -->wf 5267   ` cfv 5271  (class class class)co 5874   Fincfn 6879   Basecbs 13164   0gc0g 13416    gsumg cgsu 13417   Mndcmnd 14377   MndHom cmhm 14429  CMndccmn 15105
This theorem is referenced by:  gsummulglem  15229  prdsgsum  15245  gsummulc1  15406  gsummulc2  15407  lgseisenlem4  20607  gsumvsmul  26867
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-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  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-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-fzo 10887  df-seq 11063  df-hash 11354  df-0g 13420  df-gsum 13421  df-mnd 14383  df-mhm 14431  df-cntz 14809  df-cmn 15107
  Copyright terms: Public domain W3C validator