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Theorem gsummptmhm 16747
Description: Apply a group homomorphism to a group sum expressed with a mapping. (Contributed by Thierry Arnoux, 7-Sep-2018.) (Revised by AV, 8-Sep-2019.)
Hypotheses
Ref Expression
gsummptmhm.b  |-  B  =  ( Base `  G
)
gsummptmhm.z  |-  .0.  =  ( 0g `  G )
gsummptmhm.g  |-  ( ph  ->  G  e. CMnd )
gsummptmhm.h  |-  ( ph  ->  H  e.  Mnd )
gsummptmhm.a  |-  ( ph  ->  A  e.  V )
gsummptmhm.k  |-  ( ph  ->  K  e.  ( G MndHom  H ) )
gsummptmhm.c  |-  ( (
ph  /\  x  e.  A )  ->  C  e.  B )
gsummptmhm.w  |-  ( ph  ->  ( x  e.  A  |->  C ) finSupp  .0.  )
Assertion
Ref Expression
gsummptmhm  |-  ( ph  ->  ( H  gsumg  ( x  e.  A  |->  ( K `  C
) ) )  =  ( K `  ( G  gsumg  ( x  e.  A  |->  C ) ) ) )
Distinct variable groups:    x, A    x, B    x, K    ph, x
Allowed substitution hints:    C( x)    G( x)    H( x)    V( x)    .0. (
x)

Proof of Theorem gsummptmhm
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 gsummptmhm.c . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  C  e.  B )
2 eqidd 2461 . . . 4  |-  ( ph  ->  ( x  e.  A  |->  C )  =  ( x  e.  A  |->  C ) )
3 gsummptmhm.k . . . . . 6  |-  ( ph  ->  K  e.  ( G MndHom  H ) )
4 gsummptmhm.b . . . . . . 7  |-  B  =  ( Base `  G
)
5 eqid 2460 . . . . . . 7  |-  ( Base `  H )  =  (
Base `  H )
64, 5mhmf 15775 . . . . . 6  |-  ( K  e.  ( G MndHom  H
)  ->  K : B
--> ( Base `  H
) )
7 ffn 5722 . . . . . 6  |-  ( K : B --> ( Base `  H )  ->  K  Fn  B )
83, 6, 73syl 20 . . . . 5  |-  ( ph  ->  K  Fn  B )
9 dffn5 5904 . . . . 5  |-  ( K  Fn  B  <->  K  =  ( y  e.  B  |->  ( K `  y
) ) )
108, 9sylib 196 . . . 4  |-  ( ph  ->  K  =  ( y  e.  B  |->  ( K `
 y ) ) )
11 fveq2 5857 . . . 4  |-  ( y  =  C  ->  ( K `  y )  =  ( K `  C ) )
121, 2, 10, 11fmptco 6045 . . 3  |-  ( ph  ->  ( K  o.  (
x  e.  A  |->  C ) )  =  ( x  e.  A  |->  ( K `  C ) ) )
1312oveq2d 6291 . 2  |-  ( ph  ->  ( H  gsumg  ( K  o.  (
x  e.  A  |->  C ) ) )  =  ( H  gsumg  ( x  e.  A  |->  ( K `  C
) ) ) )
14 gsummptmhm.z . . 3  |-  .0.  =  ( 0g `  G )
15 gsummptmhm.g . . 3  |-  ( ph  ->  G  e. CMnd )
16 gsummptmhm.h . . 3  |-  ( ph  ->  H  e.  Mnd )
17 gsummptmhm.a . . 3  |-  ( ph  ->  A  e.  V )
18 eqid 2460 . . . 4  |-  ( x  e.  A  |->  C )  =  ( x  e.  A  |->  C )
191, 18fmptd 6036 . . 3  |-  ( ph  ->  ( x  e.  A  |->  C ) : A --> B )
20 gsummptmhm.w . . 3  |-  ( ph  ->  ( x  e.  A  |->  C ) finSupp  .0.  )
214, 14, 15, 16, 17, 3, 19, 20gsummhm 16743 . 2  |-  ( ph  ->  ( H  gsumg  ( K  o.  (
x  e.  A  |->  C ) ) )  =  ( K `  ( G  gsumg  ( x  e.  A  |->  C ) ) ) )
2213, 21eqtr3d 2503 1  |-  ( ph  ->  ( H  gsumg  ( x  e.  A  |->  ( K `  C
) ) )  =  ( K `  ( G  gsumg  ( x  e.  A  |->  C ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1374    e. wcel 1762   class class class wbr 4440    |-> cmpt 4498    o. ccom 4996    Fn wfn 5574   -->wf 5575   ` cfv 5579  (class class class)co 6275   finSupp cfsupp 7818   Basecbs 14479   0gc0g 14684    gsumg cgsu 14685   Mndcmnd 15715   MndHom cmhm 15768  CMndccmn 16587
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-se 4832  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-isom 5588  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-supp 6892  df-recs 7032  df-rdg 7066  df-1o 7120  df-oadd 7124  df-er 7301  df-map 7412  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-fsupp 7819  df-oi 7924  df-card 8309  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-n0 10785  df-z 10854  df-uz 11072  df-fz 11662  df-fzo 11782  df-seq 12064  df-hash 12361  df-0g 14686  df-gsum 14687  df-mnd 15721  df-mhm 15770  df-cntz 16143  df-cmn 16589
This theorem is referenced by:  evls1gsumadd  18125  evls1gsummul  18126  evl1gsummul  18160  mat2pmatmul  18992  pm2mp  19086  cayhamlem4  19149
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