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Theorem gsummhm2 15464
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 2389 . . . 4  |-  ( k  e.  A  |->  X )  =  ( k  e.  A  |->  X )
97, 8fmptd 5834 . . 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 15463 . 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 2390 . . . 4  |-  ( ph  ->  ( k  e.  A  |->  X )  =  ( k  e.  A  |->  X ) )
13 eqidd 2390 . . . 4  |-  ( ph  ->  ( x  e.  B  |->  C )  =  ( x  e.  B  |->  C ) )
14 gsummhm2.1 . . . 4  |-  ( x  =  X  ->  C  =  D )
157, 12, 13, 14fmptco 5842 . . 3  |-  ( ph  ->  ( ( x  e.  B  |->  C )  o.  ( k  e.  A  |->  X ) )  =  ( k  e.  A  |->  D ) )
1615oveq2d 6038 . 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 15450 . . 3  |-  ( ph  ->  ( G  gsumg  ( k  e.  A  |->  X ) )  e.  B )
18 eqid 2389 . . . . . . 7  |-  ( Base `  H )  =  (
Base `  H )
191, 18mhmf 14672 . . . . . 6  |-  ( ( x  e.  B  |->  C )  e.  ( G MndHom  H )  ->  (
x  e.  B  |->  C ) : B --> ( Base `  H ) )
206, 19syl 16 . . . . 5  |-  ( ph  ->  ( x  e.  B  |->  C ) : B --> ( Base `  H )
)
21 eqid 2389 . . . . . 6  |-  ( x  e.  B  |->  C )  =  ( x  e.  B  |->  C )
2221fmpt 5831 . . . . 5  |-  ( A. x  e.  B  C  e.  ( Base `  H
)  <->  ( x  e.  B  |->  C ) : B --> ( Base `  H
) )
2320, 22sylibr 204 . . . 4  |-  ( ph  ->  A. x  e.  B  C  e.  ( Base `  H ) )
24 gsummhm2.2 . . . . . 6  |-  ( x  =  ( G  gsumg  ( k  e.  A  |->  X ) )  ->  C  =  E )
2524eleq1d 2455 . . . . 5  |-  ( x  =  ( G  gsumg  ( k  e.  A  |->  X ) )  ->  ( C  e.  ( Base `  H
)  <->  E  e.  ( Base `  H ) ) )
2625rspcv 2993 . . . 4  |-  ( ( G  gsumg  ( k  e.  A  |->  X ) )  e.  B  ->  ( A. x  e.  B  C  e.  ( Base `  H
)  ->  E  e.  ( Base `  H )
) )
2717, 23, 26sylc 58 . . 3  |-  ( ph  ->  E  e.  ( Base `  H ) )
2824, 21fvmptg 5745 . . 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 643 . 2  |-  ( ph  ->  ( ( x  e.  B  |->  C ) `  ( G  gsumg  ( k  e.  A  |->  X ) ) )  =  E )
3011, 16, 293eqtr3d 2429 1  |-  ( ph  ->  ( H  gsumg  ( k  e.  A  |->  D ) )  =  E )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2651   _Vcvv 2901    \ cdif 3262   {csn 3759    e. cmpt 4209   `'ccnv 4819   "cima 4823    o. ccom 4824   -->wf 5392   ` cfv 5396  (class class class)co 6022   Fincfn 7047   Basecbs 13398   0gc0g 13652    gsumg cgsu 13653   Mndcmnd 14613   MndHom cmhm 14665  CMndccmn 15341
This theorem is referenced by:  gsummulglem  15465  prdsgsum  15481  gsummulc1  15642  gsummulc2  15643  lgseisenlem4  21005  gsumvsmul  26438
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-cnex 8981  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-int 3995  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-se 4485  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-isom 5405  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-riota 6487  df-recs 6571  df-rdg 6606  df-1o 6662  df-oadd 6666  df-er 6843  df-map 6958  df-en 7048  df-dom 7049  df-sdom 7050  df-fin 7051  df-oi 7414  df-card 7761  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-nn 9935  df-n0 10156  df-z 10217  df-uz 10423  df-fz 10978  df-fzo 11068  df-seq 11253  df-hash 11548  df-0g 13656  df-gsum 13657  df-mnd 14619  df-mhm 14667  df-cntz 15045  df-cmn 15343
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