Theorem gsumfzmhm2 13847

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.) (Revised by Jim Kingdon, 9-Sep-2025.)

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 )
gsumfzmhm2.m	|- ( ph -> M e. ZZ )
gsumfzmhm2.n	|- ( ph -> N e. ZZ )
gsummhm2.k	|- ( ph -> ( x e. B |-> C ) e. ( G MndHom H ) )
gsumfzmhm2.f	|- ( ( ph /\ k e. ( M ... N ) ) -> X e. B )
gsummhm2.1	|- ( x = X -> C = D )
gsumfzmhm2.2	|- ( x = ( G gsumg ( k e. ( M ... N ) |-> X ) ) -> C = E )

Assertion

Ref	Expression
gsumfzmhm2	|- ( ph -> ( H gsumg ( k e. ( M ... N ) |-> D ) ) = E )

Distinct variable groups:   x, k, N   k, M, x   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)   X( k)   .0. ( x, k)

Proof of Theorem gsumfzmhm2

Step	Hyp	Ref	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		gsumfzmhm2.m	. . 3 |- ( ph -> M e. ZZ )
6		gsumfzmhm2.n	. . 3 |- ( ph -> N e. ZZ )
7		gsummhm2.k	. . 3 |- ( ph -> ( x e. B |-> C ) e. ( G MndHom H ) )
8		gsumfzmhm2.f	. . . 4 |- ( ( ph /\ k e. ( M ... N ) ) -> X e. B )
9	8	fmpttd 5763	. . 3 |- ( ph -> ( k e. ( M ... N ) |-> X ) : ( M ... N ) --> B )
10	1, 2, 3, 4, 5, 6, 7, 9	gsumfzmhm 13846	. 2 |- ( ph -> ( H gsumg ( ( x e. B |-> C ) o. ( k e. ( M ... N ) |-> X ) ) ) = ( ( x e. B |-> C ) ` ( G gsumg ( k e. ( M ... N ) |-> X ) ) ) )
11		eqidd 2210	. . . 4 |- ( ph -> ( k e. ( M ... N ) |-> X ) = ( k e. ( M ... N ) |-> X ) )
12		eqidd 2210	. . . 4 |- ( ph -> ( x e. B |-> C ) = ( x e. B |-> C ) )
13		gsummhm2.1	. . . 4 |- ( x = X -> C = D )
14	8, 11, 12, 13	fmptco 5774	. . 3 |- ( ph -> ( ( x e. B |-> C ) o. ( k e. ( M ... N ) |-> X ) ) = ( k e. ( M ... N ) |-> D ) )
15	14	oveq2d 5990	. 2 |- ( ph -> ( H gsumg ( ( x e. B |-> C ) o. ( k e. ( M ... N ) |-> X ) ) ) = ( H gsumg ( k e. ( M ... N ) |-> D ) ) )
16		eqid 2209	. . 3 |- ( x e. B |-> C ) = ( x e. B |-> C )
17		gsumfzmhm2.2	. . 3 |- ( x = ( G gsumg ( k e. ( M ... N ) |-> X ) ) -> C = E )
18	3	cmnmndd 13811	. . . 4 |- ( ph -> G e. Mnd )
19	1, 2, 18, 5, 6, 9	gsumfzcl 13498	. . 3 |- ( ph -> ( G gsumg ( k e. ( M ... N ) |-> X ) ) e. B )
20	17	eleq1d 2278	. . . 4 |- ( x = ( G gsumg ( k e. ( M ... N ) |-> X ) ) -> ( C e. ( Base ` H ) <-> E e. ( Base ` H ) ) )
21		eqid 2209	. . . . . . 7 |- ( Base ` H ) = ( Base ` H )
22	1, 21	mhmf 13464	. . . . . 6 |- ( ( x e. B |-> C ) e. ( G MndHom H ) -> ( x e. B |-> C ) : B --> ( Base ` H ) )
23	7, 22	syl 14	. . . . 5 |- ( ph -> ( x e. B |-> C ) : B --> ( Base ` H ) )
24	16	fmpt 5758	. . . . 5 |- ( A. x e. B C e. ( Base ` H ) <-> ( x e. B |-> C ) : B --> ( Base ` H ) )
25	23, 24	sylibr 134	. . . 4 |- ( ph -> A. x e. B C e. ( Base ` H ) )
26	20, 25, 19	rspcdva 2892	. . 3 |- ( ph -> E e. ( Base ` H ) )
27	16, 17, 19, 26	fvmptd3 5701	. 2 |- ( ph -> ( ( x e. B |-> C ) ` ( G gsumg ( k e. ( M ... N ) |-> X ) ) ) = E )
28	10, 15, 27	3eqtr3d 2250	1 |- ( ph -> ( H gsumg ( k e. ( M ... N ) |-> D ) ) = E )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1375   e. wcel 2180   A.wral 2488   |-> cmpt 4124   o. ccom 4700   -->wf 5290   ` cfv 5294  (class class class)co 5974   ZZcz 9414   ...cfz 10172   Basecbs 12998   0gc0g 13255   gsum_g cgsu 13256   Mndcmnd 13415   MndHom cmhm 13456  CMndccmn 13787

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-13 2182  ax-14 2183  ax-ext 2191  ax-coll 4178  ax-sep 4181  ax-nul 4189  ax-pow 4237  ax-pr 4272  ax-un 4501  ax-setind 4606  ax-iinf 4657  ax-cnex 8058  ax-resscn 8059  ax-1cn 8060  ax-1re 8061  ax-icn 8062  ax-addcl 8063  ax-addrcl 8064  ax-mulcl 8065  ax-addcom 8067  ax-addass 8069  ax-distr 8071  ax-i2m1 8072  ax-0lt1 8073  ax-0id 8075  ax-rnegex 8076  ax-cnre 8078  ax-pre-ltirr 8079  ax-pre-ltwlin 8080  ax-pre-lttrn 8081  ax-pre-apti 8082  ax-pre-ltadd 8083

This theorem depends on definitions:  df-bi 117  df-dc 839  df-3or 984  df-3an 985  df-tru 1378  df-fal 1381  df-nf 1487  df-sb 1789  df-eu 2060  df-mo 2061  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ne 2381  df-nel 2476  df-ral 2493  df-rex 2494  df-reu 2495  df-rmo 2496  df-rab 2497  df-v 2781  df-sbc 3009  df-csb 3105  df-dif 3179  df-un 3181  df-in 3183  df-ss 3190  df-nul 3472  df-if 3583  df-pw 3631  df-sn 3652  df-pr 3653  df-op 3655  df-uni 3868  df-int 3903  df-iun 3946  df-br 4063  df-opab 4125  df-mpt 4126  df-tr 4162  df-id 4361  df-iord 4434  df-on 4436  df-ilim 4437  df-suc 4439  df-iom 4660  df-xp 4702  df-rel 4703  df-cnv 4704  df-co 4705  df-dm 4706  df-rn 4707  df-res 4708  df-ima 4709  df-iota 5254  df-fun 5296  df-fn 5297  df-f 5298  df-f1 5299  df-fo 5300  df-f1o 5301  df-fv 5302  df-riota 5927  df-ov 5977  df-oprab 5978  df-mpo 5979  df-1st 6256  df-2nd 6257  df-recs 6421  df-frec 6507  df-1o 6532  df-er 6650  df-map 6767  df-en 6858  df-fin 6860  df-pnf 8151  df-mnf 8152  df-xr 8153  df-ltxr 8154  df-le 8155  df-sub 8287  df-neg 8288  df-inn 9079  df-2 9137  df-n0 9338  df-z 9415  df-uz 9691  df-fz 10173  df-fzo 10307  df-seqfrec 10637  df-ndx 13001  df-slot 13002  df-base 13004  df-plusg 13089  df-0g 13257  df-igsum 13258  df-mgm 13355  df-sgrp 13401  df-mnd 13416  df-mhm 13458  df-cmn 13789

This theorem is referenced by:  lgseisenlem4  15717