Theorem gsumfzmhm2 13474

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 5717	. . 3 |- ( ph -> ( k e. ( M ... N ) |-> X ) : ( M ... N ) --> B )
10	1, 2, 3, 4, 5, 6, 7, 9	gsumfzmhm 13473	. 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 2197	. . . 4 |- ( ph -> ( k e. ( M ... N ) |-> X ) = ( k e. ( M ... N ) |-> X ) )
12		eqidd 2197	. . . 4 |- ( ph -> ( x e. B |-> C ) = ( x e. B |-> C ) )
13		gsummhm2.1	. . . 4 |- ( x = X -> C = D )
14	8, 11, 12, 13	fmptco 5728	. . 3 |- ( ph -> ( ( x e. B |-> C ) o. ( k e. ( M ... N ) |-> X ) ) = ( k e. ( M ... N ) |-> D ) )
15	14	oveq2d 5938	. 2 |- ( ph -> ( H gsumg ( ( x e. B |-> C ) o. ( k e. ( M ... N ) |-> X ) ) ) = ( H gsumg ( k e. ( M ... N ) |-> D ) ) )
16		eqid 2196	. . 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 13438	. . . 4 |- ( ph -> G e. Mnd )
19	1, 2, 18, 5, 6, 9	gsumfzcl 13131	. . 3 |- ( ph -> ( G gsumg ( k e. ( M ... N ) |-> X ) ) e. B )
20	17	eleq1d 2265	. . . 4 |- ( x = ( G gsumg ( k e. ( M ... N ) |-> X ) ) -> ( C e. ( Base ` H ) <-> E e. ( Base ` H ) ) )
21		eqid 2196	. . . . . . 7 |- ( Base ` H ) = ( Base ` H )
22	1, 21	mhmf 13097	. . . . . 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 5712	. . . . 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 2873	. . 3 |- ( ph -> E e. ( Base ` H ) )
27	16, 17, 19, 26	fvmptd3 5655	. 2 |- ( ph -> ( ( x e. B |-> C ) ` ( G gsumg ( k e. ( M ... N ) |-> X ) ) ) = E )
28	10, 15, 27	3eqtr3d 2237	1 |- ( ph -> ( H gsumg ( k e. ( M ... N ) |-> D ) ) = E )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2167   A.wral 2475   |-> cmpt 4094   o. ccom 4667   -->wf 5254   ` cfv 5258  (class class class)co 5922   ZZcz 9326   ...cfz 10083   Basecbs 12678   0gc0g 12927   gsum_g cgsu 12928   Mndcmnd 13057   MndHom cmhm 13089  CMndccmn 13414

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-0id 7987  ax-rnegex 7988  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-apti 7994  ax-pre-ltadd 7995

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-if 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-iord 4401  df-on 4403  df-ilim 4404  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-frec 6449  df-1o 6474  df-er 6592  df-map 6709  df-en 6800  df-fin 6802  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-inn 8991  df-2 9049  df-n0 9250  df-z 9327  df-uz 9602  df-fz 10084  df-fzo 10218  df-seqfrec 10540  df-ndx 12681  df-slot 12682  df-base 12684  df-plusg 12768  df-0g 12929  df-igsum 12930  df-mgm 12999  df-sgrp 13045  df-mnd 13058  df-mhm 13091  df-cmn 13416

This theorem is referenced by:  lgseisenlem4  15314