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Theorem gsummhm2 16446
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 2443 . . . 4  |-  ( k  e.  A  |->  X )  =  ( k  e.  A  |->  X )
97, 8fmptd 5879 . . 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 16444 . 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 2444 . . . 4  |-  ( ph  ->  ( k  e.  A  |->  X )  =  ( k  e.  A  |->  X ) )
13 eqidd 2444 . . . 4  |-  ( ph  ->  ( x  e.  B  |->  C )  =  ( x  e.  B  |->  C ) )
14 gsummhm2.1 . . . 4  |-  ( x  =  X  ->  C  =  D )
157, 12, 13, 14fmptco 5888 . . 3  |-  ( ph  ->  ( ( x  e.  B  |->  C )  o.  ( k  e.  A  |->  X ) )  =  ( k  e.  A  |->  D ) )
1615oveq2d 6119 . 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 16409 . . 3  |-  ( ph  ->  ( G  gsumg  ( k  e.  A  |->  X ) )  e.  B )
18 eqid 2443 . . . . . . 7  |-  ( Base `  H )  =  (
Base `  H )
191, 18mhmf 15481 . . . . . 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 2443 . . . . . 6  |-  ( x  e.  B  |->  C )  =  ( x  e.  B  |->  C )
2221fmpt 5876 . . . . 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 2509 . . . . 5  |-  ( x  =  ( G  gsumg  ( k  e.  A  |->  X ) )  ->  ( C  e.  ( Base `  H
)  <->  E  e.  ( Base `  H ) ) )
2625rspcv 3081 . . . 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 5784 . . 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 2483 1  |-  ( ph  ->  ( H  gsumg  ( k  e.  A  |->  D ) )  =  E )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2727   class class class wbr 4304    e. cmpt 4362    o. ccom 4856   -->wf 5426   ` cfv 5430  (class class class)co 6103   finSupp cfsupp 7632   Basecbs 14186   0gc0g 14390    gsumg cgsu 14391   Mndcmnd 15421   MndHom cmhm 15474  CMndccmn 16289
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-cnex 9350  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-mulcom 9358  ax-addass 9359  ax-mulass 9360  ax-distr 9361  ax-i2m1 9362  ax-1ne0 9363  ax-1rid 9364  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367  ax-pre-lttri 9368  ax-pre-lttrn 9369  ax-pre-ltadd 9370  ax-pre-mulgt0 9371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rmo 2735  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-int 4141  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-se 4692  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-isom 5439  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-om 6489  df-1st 6589  df-2nd 6590  df-supp 6703  df-recs 6844  df-rdg 6878  df-1o 6932  df-oadd 6936  df-er 7113  df-map 7228  df-en 7323  df-dom 7324  df-sdom 7325  df-fin 7326  df-fsupp 7633  df-oi 7736  df-card 8121  df-pnf 9432  df-mnf 9433  df-xr 9434  df-ltxr 9435  df-le 9436  df-sub 9609  df-neg 9610  df-nn 10335  df-n0 10592  df-z 10659  df-uz 10874  df-fz 11450  df-fzo 11561  df-seq 11819  df-hash 12116  df-0g 14392  df-gsum 14393  df-mnd 15427  df-mhm 15476  df-cntz 15847  df-cmn 16291
This theorem is referenced by:  gsummulglem  16449  prdsgsum  16483  srgsummulcr  16648  sgsummulcl  16649  gsummulc1  16707  gsummulc2  16708  gsumvsmul  17021  lgseisenlem4  22703
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