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Theorem gsummhm2 16834
Description: Apply a group homomorphism to a group sum, mapping version with implicit substitution. (Contributed by Mario Carneiro, 5-May-2015.) (Revised by AV, 6-Jun-2019.)
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 ) finSupp  .0.  )
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 2467 . . . 4  |-  ( k  e.  A  |->  X )  =  ( k  e.  A  |->  X )
97, 8fmptd 6056 . . 3  |-  ( ph  ->  ( k  e.  A  |->  X ) : A --> B )
10 gsummhm2.w . . 3  |-  ( ph  ->  ( k  e.  A  |->  X ) finSupp  .0.  )
111, 2, 3, 4, 5, 6, 9, 10gsummhm 16832 . 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 2468 . . . 4  |-  ( ph  ->  ( k  e.  A  |->  X )  =  ( k  e.  A  |->  X ) )
13 eqidd 2468 . . . 4  |-  ( ph  ->  ( x  e.  B  |->  C )  =  ( x  e.  B  |->  C ) )
14 gsummhm2.1 . . . 4  |-  ( x  =  X  ->  C  =  D )
157, 12, 13, 14fmptco 6065 . . 3  |-  ( ph  ->  ( ( x  e.  B  |->  C )  o.  ( k  e.  A  |->  X ) )  =  ( k  e.  A  |->  D ) )
1615oveq2d 6311 . 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 16796 . . 3  |-  ( ph  ->  ( G  gsumg  ( k  e.  A  |->  X ) )  e.  B )
18 eqid 2467 . . . . . . 7  |-  ( Base `  H )  =  (
Base `  H )
191, 18mhmf 15844 . . . . . 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 2467 . . . . . 6  |-  ( x  e.  B  |->  C )  =  ( x  e.  B  |->  C )
2221fmpt 6053 . . . . 5  |-  ( A. x  e.  B  C  e.  ( Base `  H
)  <->  ( x  e.  B  |->  C ) : B --> ( Base `  H
) )
2320, 22sylibr 212 . . . 4  |-  ( ph  ->  A. x  e.  B  C  e.  ( Base `  H ) )
24 gsummhm2.2 . . . . . 6  |-  ( x  =  ( G  gsumg  ( k  e.  A  |->  X ) )  ->  C  =  E )
2524eleq1d 2536 . . . . 5  |-  ( x  =  ( G  gsumg  ( k  e.  A  |->  X ) )  ->  ( C  e.  ( Base `  H
)  <->  E  e.  ( Base `  H ) ) )
2625rspcv 3215 . . . 4  |-  ( ( G  gsumg  ( k  e.  A  |->  X ) )  e.  B  ->  ( A. x  e.  B  C  e.  ( Base `  H
)  ->  E  e.  ( Base `  H )
) )
2717, 23, 26sylc 60 . . 3  |-  ( ph  ->  E  e.  ( Base `  H ) )
2824, 21fvmptg 5955 . . 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 661 . 2  |-  ( ph  ->  ( ( x  e.  B  |->  C ) `  ( G  gsumg  ( k  e.  A  |->  X ) ) )  =  E )
3011, 16, 293eqtr3d 2516 1  |-  ( ph  ->  ( H  gsumg  ( k  e.  A  |->  D ) )  =  E )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2817   class class class wbr 4453    |-> cmpt 4511    o. ccom 5009   -->wf 5590   ` cfv 5594  (class class class)co 6295   finSupp cfsupp 7841   Basecbs 14507   0gc0g 14712    gsumg cgsu 14713   Mndcmnd 15793   MndHom cmhm 15837  CMndccmn 16671
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-supp 6914  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-map 7434  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-fsupp 7842  df-oi 7947  df-card 8332  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-n0 10808  df-z 10877  df-uz 11095  df-fz 11685  df-fzo 11805  df-seq 12088  df-hash 12386  df-0g 14714  df-gsum 14715  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-mhm 15839  df-cntz 16227  df-cmn 16673
This theorem is referenced by:  gsummulglem  16837  prdsgsum  16880  srgsummulcr  17060  sgsummulcl  17061  gsummulc1  17124  gsummulc2  17125  gsumvsmul  17445  lgseisenlem4  23493
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